Nonlinear Pushover Analysis of Reinforced Concrete Structures

FINAL REPORT Colorado Advanced Software Institute

NONLINEAR PUSHOVER ANALYSIS OF REINFORCED CONCRETE STRUCTURES

Principal Investigator:

Enrico Spacone, Ph.D. Assistant Professor Department of Civil, Env. and Arch. Engineering University of Colorado, Boulder

Graduate Student

Russel Martino, MS student Department of Civil, Env. and Arch. Engineering University of Colorado, Boulder

Collaborating Company

Greg Kingsley, Ph.D., P.E. Principal KL&A of Colorado Golden, Colorado

COLLABORATING COMPANY RELEASE PAGE

Project Title:

NONLINEAR PUSHOVER ANALYSIS OF REINFORCED CONCRETE STRUCTURES

Principal Investigator:

Enrico Spacone, Ph.D.

University:

University of Colorado, Boulder

Collaborating Company:

KL&A of Colorado

Collaborating Company Representative:

Greg Kingsley, Ph.D., P.E.

As authorized representative of the collaborating company, I have reviewed this report and approve it for release to the Colorado Advanced Software Institute.

__________________________________ Signature

_________________ Date

TABLE OF CONTENTS CHAPTERS I

INTRODUCTION ...............................................................................................

1

I-A

Background .................................................................................................

1

I-B

Objectives....................................................................................................

2

THE NON – LINEAR STATIC PUSHOVER ANALYSIS PROCEDURE....

4

II-A Definition of the Non – Linear Static Procedure.........................................

4

II-B

Performing the Non – Linear Static Procedure............................................

6

II-B-1

Vertical Distribution of Lateral Loads ..........................................

6

II-B-2

Building Performance Level .........................................................

8

II-B-3

Calculation of the Seismic Hazard ................................................

9

II-B-4

Calculation fo the Target Displacement........................................

15

Reasons for Performing the Non – Linear Static Procedure........................

18

III LIMITATIONS OF THE NON – LINEAR STATIC PROCEDURE .............

20

III-A Design of Three Reinforced Concrete Moment Resisting Frames ..............

20

III-A-1 Formulation of Gravity Loads Used in Design .............................

22

III-A-2 Formulation of Wind Loads Used in Design ................................

24

III-A-3 Formulation of Earthquake Loads Used in Design .......................

24

III-A-4 Total Design Loads and Section Determination…………............

28

III-B Performing the Pushover Analysis On the Moment Frames .......................

34

III-B-1 Period Determination ....................................................................

34

III-B-2 Vertical Distribution of Lateral Loads ..........................................

36

III-B-3 Element Reduction for Analysis of Frames……………...............

41

III-B-4 Determination of Seismic Hazard for Analysis………….............

42

III-B-5 Calculation of Target Displacement for Analysis……… .............

44

III-C Complete Non – Linear Dynamic Analyses for Frames……………..........

46

III-D Comparisons of Full Dynamic Results with Pushover Results…… ...........

49

II

II-C

i

III-E Dependence of Target Displacement on Choice of Vy……………… .........

53

III-F Conclusions - Limitations and Accuracy of the Pushover Analysis… ........

54

IV FORMULATION OF ELEMENT SHEAR RESPONSE

……………………

55

IV-A Review of Timoshenko Beam Theory……………………………… .........

56

IV-B Non – Linear Force – Based Timoshenko Beam Element………… ...........

58

IV-C Section V – γ Constitutive Law…………………………………… ............

63

IV-C-1 Shape of Shear Hysteretic Law…………………………..............

63

IV-C-2 Theoretical Values of Shear Hysteretic Law…………… .............

65

IV-C-3 Values for Actual Sections – Shear Hysteretic Law………..........

74

IV-D Observations on Element Shear Response Formulation …………… ........

76

NUMERICAL VERIFICATION OR PROPOSED SHEAR MODEL………

78

V-A Column Dimensions and Testing Conditions……………………… ..........

78

V-B Calculated Shear Strength………………………………………… ............

80

V-C Numerical vs. Experimental Column Response……………………...........

89

V-D Conclusions………………………………………………………… ..........

94

VI SHEAR WALL EXAMPLE.…..…………………………………………..........

95

VI-A Wall Configuration…………………………………………………...........

95

VI-B Performing the Pushover Analysis on the Shear Wall……………….. .......

97

VI-C Complete Non-Linear Dynamic Analysis of the Shear Wall………….......

98

VI-D Comparisons of Pushover and Dynamic Analysis……………………. ......

9

VI-E Verification of Flexure Shear Interaction at Element Level…….….… ......

100

VI-F Conclusions…………………………………………………………… ......

101

VII CONCLUSIONS AND FUTURE WORK…………..........................................

102

V

VIII BIBLIOGRAPHY………………………………………………………..……… 104 APPENDICES I

MODIFICATIONS TO PROGRAM FEAP TO PERFORM NON-LINEAR PUSHOVER ANALYSIS ..........................................................

106

AI-A FEAP Pushover Routines ........ ……………………………………………

106

ii

II

AI-A-1 ‘PUSH’ Mesh Command ...........…………………………………

106

AI-A-2 ‘VvsD’ Macro Command...........…………………………………

110

AI-B FEAP Shear Element Modifications ........…………………………………

113

ADDITIONAL CASI REQUIREMENTS ......…………………………………

115

AII-A

Evaluation .............................……………………………………………

115

AII-B

Technology Transfer.............……………………………………………

115

AII-C

Networking ...........................……………………………………………

116

AII-D

Publications ..........................……………………………………………

116

AII-E

Funding.................................……………………………………………

116

iii

ABSTRACT: This report summarizes the results of a research conducted at the University of Colorado, Boulder, aimed at developing a PC-based software tool for performing nonlinear pushover analysis of reinforced concrete buildings. The program links two libraries to an existing finite element program, FEAP, developed at the University of California, Berkeley. The two libraries are a) a frame element library (which includes beam, beam-column and shear wall elements); and b) a library of uniaxial material laws. The project first modified the existing program to perform nonlinear pushover analyses on a routine basis. Current seismic code suggested procedures for nonlinear pushover analyses were then reviewed. The applicability of nonlinear pushover analyses to the seismic design of reinforced concrete frames was evaluated by studying the response of frames of different heights. The responses of static and dynamic nonlinear analyses on the same buildings were compared. A new shear element was then introduced and a typical shear wall of a utility core was analyzed with a push-over analysis. Details on the features added to program FEAP and on the new commands are documented in the appendices.

iv

CHAPTER I INTRODUCTION

I-A Background

As the United States, Japan, and Europe move towards the implementation of Performance Based Engineering philosophies in seismic design of civil structures, new seismic design provisions will require structural engineers to perform nonlinear analyses of the structures they are designing. These analyses can take the form of a full, nonlinear dynamic analysis, or of a static nonlinear Pushover Analysis. Because of the computational time required to perform a full, nonlinear dynamic analysis, the Pushover Analysis, if deemed applicable to the structure at hand, is a very attractive method for use in a design office setting. For this reason, there is a need for easy to use and accurate, nonlinear Pushover Analysis tools which can easily be applied in a design office. Even though recent years have seen a great amount of research in the development of such nonlinear models and techniques, there is still a great deal of knowledge missing for reinforced concrete structures. In particular, the following modeling issues still need to be thoroughly addressed: bond slip, structural walls and shear deformations, joint response, and non – structural members. In the United States, the reference document for performing the Nonlinear Static Procedure, or Pushover Analysis, is currently the Federal Emergency Management Agency Document 273 (FEMA 273) [6]. According to this procedure, a vertical distribution of static, monotonically increasing, lateral loads is applied to a mathematical model of the structure. The loads are increased until the peak response of the structure is obtained on a base shear vs. roof displacement plot. From this plot, and other parameters representing the expected, or design, earthquake, the maximum deformations the structure is expected to undergo during the design seismic event can be estimated. Because the mathematical model must capture the inherent material nonlinearities of the structure, and because the load applied to the structure is increased monotonically, detailed member information can be obtained. This procedure is more involved than applying the approximate static lateral load all at once, as is done in current seismic design codes, in that the loads are applied in increments. This allows the deformations of structural members (for example, the plastic-hinge sequence) to be monitored throughout the nonlinear pushover analysis.

1

The Nonlinear Static Procedure must still be used with caution. The Pushover Analysis is meant to represent a static approximation of the response a structure will undergo when subjected to dynamic earthquake loads. The key word in this definition is approximation. There is a great saving in time when performing the Pushover Analysis as compared with the full nonlinear dynamic analysis. But there are bound to be drawbacks to the method. In particular, the maximum displacement achieved will be directly related to the shape of the lateral load distribution applied to the structure. If the shape of the lateral load differs from the shape the structure attains when loaded dynamically, the calculated maximum displacement could grossly overestimate what the dynamic analysis would predict. While there are currently some programs available to perform the Pushover Analysis on Reinforced Concrete structures, the procedure needs to be refined and more experience is needed to fully access its applicability. One of the several issues still open is modeling the shear deformations in reinfoced concrete columns and structural walls. Shear deformations in Reinforced Concrete members are difficult to model because of the complex mechanisms that govern them.

I-A Objectives

The main objective of this project was to develop an easy to use and accurate nonlinear Pushover Analysis Software tool for civil structures following the procedures outlined in FEMA 273[6]. Even though the procedure is general, the focus of this study is reinforced concrete frames. The objective is to develop an accurate though easy to understand tool that can be routinely used in a design office by a structural engineer that is familiar with both the Pushover Analysis procedure and with basic nonlinear structural analysis techniques. The following are the main tasks of the projects: a) A critical study of the Pushover Analysis procedure as defined by FEMA 273[6]. Comparisons between Nonlinear Pushover Analyses and Nonlinear Dynamic Analyses are key to understanding the limitations of the proposed Pushover Analyses. b) Development of Software Tool for Nonlinear Pushover Analyses. This is achieved by modifying the existing Finite Element Analysis Program (FEAP) developed by Professor Robert Taylor at the University of California at Berkeley [16]. Special steps need to be implemented to perform Nonlinear Pushover Analyses following FEMA 273[6]

2

c) Development of a family of models for Nonlinear Pushover Analysis of Reinforced Concrete structures. Some of these models already exist and need to be linked to program FEAP (in particular, fiber beam column elements with interaction between axial and normal forces). Other models, in particular elements for reinforced concrete members with shear deformations, need to be developed. d) Verification of the new tool via comparisons between experimental and analytical results. e) Application of the new tool to studies of Reinforced Concrete structural systems. With all of the foregoing arguments in mind, the organization of this report is as follows. Chapter II, The Non – Linear Static Pushover Procedure, describes the steps followed in performing the Non – Linear Static Procedure as given by FEMA 273[6]. Chapter III, Applications of the Non – Linear Static Procedure (Pushover Analysis), discusses the applicability and shortcomings of the procedure. Chapter IV, Formulation of Element Shear Response, describes the shear deformation formulation for a force – based beam element. Chapter V, Numerical Verification of Proposed Shear Model, determines the applicability and shortcomings of the shear formulation developed in chapter IV by comparing numerical results with test data obtained from Reinforced Concrete columns failing in shear tested at the University of California at San Diego. Chapter VI, Conclusions, summarizes the results and points to areas for future work. Appendix I, Modifications to FEAP to Perform Pushover Analysis, describes the changes made to the Finite Element Analysis Program to include shear deformations and to run the Pushover Analysis. Appendix I presents the Modifications to Program FEAP to perform Non-Linear Pushover Analyses.. Finally, Appendix II includes Additional CASI Requirements for the Poject Report.

3

CHAPTER II THE NON – LINEAR STATIC PUSHOVER ANALYSIS PROCEDURE

II-A) Definition of the Non – Linear Static Procedure (Pushover Analysis) - FEMA 273 [6]

The Non – Linear Static Procedure or Pushover Analysis is defined in the Federal Emergency Management Agency document 273 (FEMA 273) [6] as a non – linear static approximation of the response a structure will undergo when subjected to dynamic earthquake loading. The static approximation consists of applying a vertical distribution of lateral loads to a model which captures the material non – linearities of an existing or previously designed structure, and monotonically increasing those loads until the peak response of the structure is obtained on a base shear vs. roof displacement plot as shown in figure II-1. Lateral Loads

Roof Disp

Base Shear Structural Response Base Shear

Roof Displacement

Structural Model Figure II-1: Static Approximation Used In the Pushover Analysis The desired condition of the structure after a range of ground shakings, or Building Performance Level, is then decided upon by the owner, architect, and structural engineer.

The Building

Performance Level is a function of the post event conditions of the structural and non – structural components of the structure. Some common Building Performance Levels are shown in figure II-2.

Owner, Architect, Engineer

Operational

Immediate Occupancy

Life Safety

Figure II-2: Building Performance Level

4

Collapse Prevention

Based on the desired Building Performance Level, the Response Spectrum for the design earthquake may be determined. The Response Spectrum gives the maximum acceleration, or Spectral Response Acceleration, a structure is likely to experience under the design ground shaking given the structure’s fundamental period of vibration, T. This relation is shown qualitatively in figure II-3.

Spectral Response Accel, Sa

Response Spectrum

Figure II-3: Response Spectrum

From the Response Spectrum and Base Shear vs. Roof Displacement plot, the Target Displacement, δt, may be determined.

The Target Displacement represents the maximum

displacement the structure will undergo during the design event. One can then find the maximum expected deformations within each element of the structure at the Target Displacement and redesign them accordingly. The Target Displacement is shown qualitatively in figure II-4.

Base Shear

Structural Response

Target Displacement, δt Roof Displacement

Figure II-4: Target Displacement

5

II-B) Performing the Non – Linear Static Procedure (Pushover Analysis)

The steps in performing the Non – Linear Static Procedure or Pushover Analysis are: 1) Determine the gravity loading and the vertical distribution of the lateral loads. 2) Determine the desired Building Performance Level. 3) Calculate the Seismic Hazard. 4) Compute the maximum expected displacement or Target Displacement, δt. Each of these steps are described in the sections following. II-B-1) Determine the Vertical Distribution of the Lateral Loads

In addition to the gravity loads, the first thing that can be determined is the vertical distribution of the lateral loads. The gravity loads to be used in the Pushover Analysis are calculated by equation II.1, while the vertical distribution of lateral loads is given by the FEMA 273 [6] Cvx loading profile reproduced as equation II.2.

QG = 1.1(QD + QL + QS )

(II.1)

Where, QG is equal to the total gravity force, QD is equal to the total dead load effect, QL is equal to the effective live load effect, defined as 25% of the unreduced live load, and QS is equal to 70% of the full design snow load except where the design snow load is less than thirty pounds per square foot in which case it is equal to 0.0.

C vx =

wx h x

k

n

∑w h i =1

(II.2) k

i i

The Cvx coefficient represents the lateral load multiplication factor to be applied at floor level x, wx represents the fraction of the total structural weight allocated to floor level x, hx is the height of floor level x above the base, and the summation in the denominator is the sum of these values over the total number of floors in the structure, n. These values are shown schematically in figure II-5.

6

Cvn

wn

Cv4

w4

Cv3

w3

Cv2

w2

Cv1

w1

hn h4 h1

h2

h3

Figure II-5: Values for Determining the Vertical Distribution of the Lateral Loads

The parameter k varies with the structural fundamental period, T. k is 1.0 for T less than or equal to 0.5 seconds and 2.0 for T greater than or equal to 2.5 seconds. In between these values, k varies linearly as shown in figure II-6. The effect that the parameter k has on the Cvx loading profile is also shown in figure II-6. For shorter, stiffer structures, the fundamental period will be small and the variation of the lateral loading over the height of the building will approach the linear distribution shown in figure II-6 for a k value equal to 1.0. For taller, more flexible structures, the fundamental period will be greater and the variation of the lateral loading over the height of the structure will approach the non – linear distribution shown in figure II-6 for k equal to 2.0. The implication of this is that for stiffer structures the higher mode response of the structure will be less significant and the lateral loading can enforce purely first mode response. As the structure becomes more flexible however, the higher mode effects become much more important and the k value attempts to account for this by adjusting the lateral load distribution. Determination of k

Effect of k on C Floor 5 Floor 4

2.0

k=1

Floor 3

k=2

Floor 2 1.0 Floor 1 0.5

1.5

2.5

0.2 Cvx 0.1 0.3 0.4 Fundamental Period, T (sec) Figure II-6: Variation of k with Fundamental Period T, and Effect of k on Lateral Load

7

0.5

II-B-2) Building Performance Level Determination

The next thing that may be determined is the Building Performance Level. The Building Performance Level is the desired condition of the building after the design earthquake decided upon by the owner, architect, and structural engineer, and is a combination of the Structural Performance Level and the Non – Structural Performance Level. The Structural Performance Level is defined as the post – event conditions of the structural building components. This is divided into three levels and two ranges.

The levels are, S – 1: Immediate Occupancy, S – 3: Life Safety, and S – 5: Collapse

Prevention. The ranges are S – 2: which is a range between S – 1 and S – 3, and S – 4: which is a range between S –3 and S – 5. The ranges are included to describe any building performance level which may be decided upon by the owner, architect, and structural engineer. The Non – Structural Performance Level is defined as the post – event conditions of the non - structural components. This is divided into five levels. They are N – A: Operational, N – B: Immediate Occupancy, N – C: Life Safety, N – D: Hazards Reduced, and N – E: Non – Structural Damage Not Limited. By combining the number from the Structural Performance Level with the second letter from the Non – Structural Performance Level, one can attain the total Building Performance Level. The combinations to achieve the most common Building Performance Levels, 1 – A: Operational, 1 – B: Immediate Occupancy, 3 – C: Life Safety, and 5 – E: Collapse Prevention, are shown in figure II-7. Non - Structural Level

Structural Level Immediate Occupancy

S-1

Operational

Immediate Occupancy

N-A

N-B

1-A

1-B

Life Safety N-C

Hazards Reduced N-D

Damage Not Limited N-E

Range Between S-1 & S-3 S - 2 Life Safety

S-3

3-C

Range Between S-3 & S-5 S - 4 Collapse Prevention

S-5

5-E Building Performance Level

Figure II-7: Determination of Building Performance Level

8

The owner, architect, and structural engineer can now decide what Building Performance Level they want their building to achieve after a range of ground shakings which are expected to occur at a given design location. Referring to figure II-8, A would correspond to a Building Performance level of Operational after a 50% probability of exceedance in 50 year seismic event, F would correspond to a Building Performance Level of Immediate Occupancy after a 20% probability of exceedance in 50 year seismic event and so on. The values K and P shown in bold in figure II-8 correspond to the performance one achieves when designing by the Uniform Building Code (UBC) [17]. This corresponds to Life Safety after a 10% probability of exceedance in 50 year event and Collapse Prevention after a 2% probability of exceedance in 50 year event, respectively. One can easily see that the new design approach allows the designer to advance the state of the art from the UBC code by giving many more design options and allowing the owner, architect, and engineer to predict the post event conditions of the structure for a wide range of ground motions.

Building Performance Level Seismic Event

1-A

1-B

3-C

5-E

50% / 50 years

A

B

C

D

20% / 50 years

E

F

G

H

10% / 50 years

I

J

K

L

2% / 50 years

M

N

O

P

Figure II-8: Building Performance Level for Given Seismic Event

II-B-3) Calculation of the Seismic Hazard

An important parameter that must be determined for the Pushover Analysis is the Seismic Hazard of a given location. The Seismic Hazard is a function of: 1) The Building Performance Level 2) The Mapped Acceleration Parameters (found from contour maps included with FEMA 273) 3) The Site Class Coefficients (which account for soil type) 4) The effective structural damping

9

5) The Fundamental Structural Period The Building Performance Level enters into the Seismic Hazard through the return period of the earthquake under consideration. The return period for the design earthquake, PR, is defined as:

PR =

1 1− e

0.02 ln(1− PE 50 )

(II.3)

Where PE50 is the probability of exceedance in 50 years under consideration. Referring to figure II-8, if the owner, architect, and structural engineer determine that condition A, K, and P must be met, corresponding to Operational after a 50% probability of exceedance in 50 years event, Life Safety after a 10% probability of exceedance in 50 years event, and Collapse Prevention after a 2% probability of exceedance in 50 years event respectively, then the Return Period would be calculated three separate times with PE50 equal to 0.5, 0.1, and 0.02 respectively. Since the Seismic Hazard is a function of this Return Period, as will be shown subsequently, the Pushover Analysis would need to be run separately for each % exceedance considered and the end results compared with the acceptance criteria given in FEMA 273 [6] for the Building Performance Level at each % exceedance. Once the Return Period for the % exceedance under consideration has been determined, the mapped acceleration parameters are used to determine the modified mapped short period response acceleration parameter, SS, and the modified mapped acceleration parameter at one second period, S1. These parameters are found from: If SS2/50 is less than 1.5g and PE50 is between 2% in 50 years and 10% in 50 years then

ln(S i ) = ln(S i10 / 50 ) + [ln(S i 2 / 50 ) − ln(S i10 / 50 )]* [0.606 ln( PR ) − 3.73]

(II.4)

When SS2/50 is greater than or equal to 1.5g or SS2/50 is less than 1.5g and PE50 is greater than 10% probability of exceedance in 50 years then

 P  S i = S i10 / 50  R   475 

n

(II.5)

The subscript i in the above equations is equal to S if the modified mapped short period response acceleration parameter is being determined and it is equal to 1 if the modified mapped response acceleration parameter at a one second period is being determined. The parameter Si2/50 in equation II.4 is the mapped short period acceleration parameter (i =S) or the mapped acceleration parameter at a one

10

second period (i=1) for a 2% probability of exceedance in 50 years event. The parameter Si10/50 in equations II.4 and II.5 is the mapped short period acceleration parameter (i =S) or the mapped acceleration parameter at a one second period (i=1) for a 10% probability of exceedance in 50 years event.

These parameters are found from contour maps which map the short period response

acceleration and the response acceleration at a one second period at probabilities of exceedance of 2% in 50 years and 10% in 50 years for the for the entire United States and are included with FEMA 273. The value n in equation II.5 is a parameter which depends on the mapped parameter SS2/50 and PE50 and is tabulated in FEMA 273. These tables are reproduced in Table II-1.

Table II-1: Values for exponent n for use in equation II.5 Value of n for use with SS

Value of n for use with S1

2% = 1.5g

PE50 > 10% & SS2/50 < 1.5g

PE50 > 10% & SS2/50 >= 1.5g

2% = 1.5g

PE50 > 10% & SS2/50 < 1.5g

PE50 > 10% & SS2/50 >= 1.5g

California

0.29

0.44

0.44

0.29

0.44

0.44

Pacific Northwest

0.56

0.54

0.89

0.67

0.59

0.96

Mountain

0.50

0.54

0.54

0.60

0.59

0.59

Central US

0.98

0.77

0.89

1.09

0.80

0.89

Eastern US

0.93

0.77

1.25

1.05

0.80

1.25

Region

Now that the modified mapped short period response acceleration parameter and the modified mapped response acceleration parameter at a one second period have been determined, these parameters must be further adjusted to account for the soil type at the site. The final design short period spectral response acceleration parameter, SXS, and the final design spectral response acceleration parameter at a one second period, SX1, shall be determined from:

11

S XS = Fa S S

( II.6)

S X 1 = Fv S1

(II.7)

Fa is a function of the soil class at the site and the modified mapped short period response acceleration parameter, SS, and Fv is a function of the soil class at the site and modified mapped response acceleration parameter at a one second period, S1. Values of Fa and Fv are tabulated in FEMA 273. These tables are reproduced in tables II-2 and II-3 respectively. Linear interpolation shall be used for values of SS or S1 between tabulated values and the * represents a condition in which site – specific geotechnical investigation and dynamic site response analyses should be performed.

Table II-2: Values for Site Class coefficient, Fa, for use in equation II.6 Site Class

S <= 0.25

S = 0.50

S = 0.75

S = 1.00

S >= 1.25

A

0.8

0.8

0.8

0.8

0.8

B

1.0

1.0

1.0

1.0

1.0

C

1.2

1.2

1.1

1.0

1.0

D

1.6

1.4

1.2

1.1

1.0

E

2.5

1.7

1.2

0.9

*

F

*

*

*

*

*

Table II-3: Values for Site Class coefficient, Fv, for use in equation II.7 Site Class

S <= 0.1

S = 0.2

S = 0.3

S = 0.4

S >= 0.5

A

0.8

0.8

0.8

0.8

0.8

B

1.0

1.0

1.0

1.0

1.0

C

1.7

1.6

1.5

1.4

1.3

D

2.4

2.0

1.8

1.6

1.5

E

3.5

3.2

2.8

2.4

*

F

*

*

*

*

*

Definitions and classifications of soil type are included in FEMA 273 and are as follows: Class A: Hard rock with measured shear wave velocity, vs > 5,000 ft/s Class B: Rock with 2,500 ft/s < vs < 5,000 ft/s, where vs is the measured shear wave velocity.

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Class C: Very dense soil and soft rock with shear wave velocity, 1,200 ft/s < vs < 2,500 or with either standard blow count, N > 50 or undrained shear strength, su > 2,000 psf. Class D: Stiff soil with shear wave velocity, 600 ft/s < vs <1,200 ft/s or with either standard blow count, 15 < N < 50 or undrained shear strength, 1,000 psf < su < 2,000 psf. Class E: Any profile with more than 10 ft of soft clay defined as soil with plasticity index, PI > 20, or water content, w > 40%, and undrained shear strength, su < 500 psf or a soil profile with shear wave velocity, vs < 600 ft/s. If insufficient data are available to classify a soil profile as type A through D, a type E profile should be assumed. Class F: Soils requiring site – specific evaluations are those soils that are vulnerable to potential failure or collapse under seismic loading, such as liquefiable soils, quick and highly – sensitive clays and collapsible weakly – cemented soils, peats and/or highly organic clays with a thickness greater than 10 ft, very high plasticity clays that have a plasticity index, PI, greater than 75 and with a thickness greater than 25 ft, and soft or medium clays which have a thickness greater than 120 ft. In the above classifications, the shear wave velocity, vs, the Standard Penetration Test blow count, N, and the undrained shear strength, su, are average values over a 100 ft depth of soil. Based on the design spectral response acceleration parameters, SXS and SX1, the General Response Spectrum can be formulated.

The General Response Spectrum graphically relates the

Spectral Response Acceleration, Sa, as a function of Structural Fundamental Period, T. The relation is defined as:

S a = ( S XS / BS ) * (0.4 + 3T / T0 ) S a = S XS / BS

for 0 < T ≤ 0.2T0

for 0.2T0 < T ≤ T0

S a = S X 1 /( B1T )

for T > T0

(II.8) (II.9) ( II.10)

The values BS and B1 in equations II.8 to II.10 are parameters which account for the effective damping coefficient of the structure and are tabulated in FEMA 273. These values are reproduced in table II-4 and linear interpolation shall be used for intermediate values of the effective damping coefficient, β.

13

Table II-4: Damping Coefficients BS and B1 to be used in equations II.8 to II.10 Effective Damping, β (% of critical)

BS

B1

<2

0.8

0.8

5

1.0

1.0

10

1.3

1.2

20

1.8

1.5

30

2.3

1.7

40

2.7

1.9

> 50

3.0

2.0

The value T0 in equations II.8 to II.10 is the characteristic period of the response spectrum, defined as the period associated with the transition from the constant acceleration segment of the spectrum to the constant velocity segment of the spectrum. It is calculated from:

T0 = ( S X 1 BS ) /( S XS B1 )

(II.11)

With the application of equations II.3 through II.11 the General Response Spectrum can be formulated for the design event being considered.

The General Response Spectrum is shown

qualitatively in figure II-9.

Sa = (SXS/BS)(0.4+3T/T0) Sa = SXS/BS Spectral Response Acceleration, Sa

Sa = SX1/(B1T)

SX1/B1 0.4SXS/BS

0.2T0

T0

1.0

Fundamental Structural Period, T Figure II-9: General Response Spectrum

14

The General Response Spectrum is a function of the many site and design event specific parameters which are related by a complicated system of equations. However, once it has been developed, since it is a function only of site location parameters and the design event under consideration, it becomes a very useful tool as it describes the maximum acceleration a structure, with a given fundamental period, must endure during the design event.

II-B-4) Calculation of the Target Displacement

The Target Displacement, i.e. the maximum displacement the structure is expected to undergo during the design event, can now be obtained.

The target displacement is calculated from the

following equation:

δ t = C 0 C1C 2 C 3 S a

Te2 g 4π 2

(II.12)

Where the value C0 is a modification factor that relates spectral displacement and likely building roof displacement. Values for C0 are tabulated in FEMA 273 as a function of the total number of stories of the structure and are included in table II-5. Table II-5: Values for modification factor C0 for use in equation II.12 Number of Stories

Modification Factor C

1

1.0

2

1.2

3

1.3

5

1.4

10 +

1.5

1

1. Linear Interpolation should be used for intermediate values

C1 is a modification factor which relates expected maximum inelastic displacements to displacements calculated for linear elastic response. Values for C1 are obtained from:

C1 = 1.0

for Te ≥ T0

C1 = [1.0 + ( R − 1)T0 / Te ]/ R

15

for Te < T0

(II.13) (II.14)

Te is the effective fundamental period of the structure and is defined as given in equation II.17. To is the characteristic period of the response spectrum, defined as the period associated with the transition from the constant acceleration segment of the spectrum to the constant velocity segment of the spectrum and is calculated as shown in equation II.11. R is the ratio of elastic strength demand to calculated yield strength coefficient. Values for R are obtained from:

R=

Sa 1 V y / W C0

(II.15)

Sa is the Response Spectrum Acceleration, in g’s, ( where g must be in consistent units, usually in/s2) at the effective fundamental period and damping ratio of the building in the direction under consideration as described in section II-B-3 and obtained from equations II.8 through II.10. Vy is the yield strength calculated using the results of the Pushover Analysis, where the non – linear force – displacement curve of the building is characterized by a bilinear relation as shown in figure II-10. W is the total dead load and anticipated live load, as calculated by equation II.1. C0 is as defined above and values are tabulated in table II-5. C2 is a modification factor that represenst the effect of hysteresis shape on the maximum displacement response of the structure. Values for C2 are tabulated in FEMA 273 and are a function of Building Performance Level, framing type, and the fundamental period of the structure. They are included in table II-6. Table II-6: Values for modification factor C2 used in equation II.12 T = 0.1 second Building Performance Level

T >T second

Framing Type 11

Framing Type 22

Framing Type 11

Framing Type 22

Immediate Occupancy

1.0

1.0

1.0

1.0

Life Safety

1.3

1.0

1.1

1.0

1.5

1.0

1.2

1.0

Collapse Prevention 1.

Structures in which more than 30% of the story shear at any level is resisted by components or elements whose strength and stiffness may deteriorate during the design earthquake. Such elements and components include: ordinary moment – resisting frames, concentrically braced frames, frames with partially restrained connections, tension – only braced frames, unreinforced masonry walls, shear – critical walls and piers, or any combination of the above.

2.

All frames not assigned toFraming Type 1.

16

C3 is a modification factor to represent increased displacements due to dynamic P – ∆ effects. For buildings with positive post – yield stiffness, C3 shall be set equal to 1.0. For buildings with negative post – yield stiffness, C3 shall be calculated from:

α ( R − 1) 3 / 2 C 3 = 1.0 + Te

(II.16)

Values for R and Te are obtained from equations II.15 and II.17 respectively, and α is the ratio of post – yield stiffness to effective elastic stiffness, where the non – linear force – displacement relation is characterized by a bilinear relation as shown in figure II-10. The effective fundamental period of the structure in the direction under consideration, Te may be calculated from:

Ki Ke

Te = T

(II.17)

Where T is the elastic fundamental period of the structure (in seconds) in the direction under consideration calculated by elastic dynamic analysis. Ki is the elastic lateral stiffness of the building in the direction under consideration and is found from the initial stiffness of the non – linear base shear vs. roof displacement curve as shown in figure II-10. Ke is the effective lateral stiffness of the building in the direction under consideration and is defined as the slope of the line which connects the point of intersection of the post – yield stiffness line with the horizontal line at the yield base shear value to zero, while intersecting the original base shear vs. roof displacement curve at 60% of the yield base shear value. Ki and Ke are shown in figure II-10. Ki Base Shear

αK Vy Non – Linear Structural Response

0.6 Vy

Ke

Roof Displacement

δ

Figure II-10: Bilinear Relation of Base Shear vs. Roof Displacement Plot

17

II-C) Reasons for Performing the Non – Linear Static Procedure (Pushover Analysis)

The procedure to perform the Pushover Analysis was thoroughly outlined in the previous section. It is easily seen that it is by no means an easy procedure. This brings up the question of why should one perform the pushover analysis, especially when it was defined as a static approximation to an actual dynamic analysis? Also, since the analysis is applied to previously designed or existing structures, why must one perform a more detailed analysis than just designing by an appropriate code such as the UBC? There are two reasons why the Pushover Analysis may be preferred to a full dynamic analysis. The first reason is computational time. To run a full dynamic, non – linear analysis on even a simple structure takes a long time. If the Pushover Analysis is deemed applicable (see chapter III for applicability conditions) to the structure at hand, accurate results can be obtained in fractions of the time it would take to get any useful results from the fully dynamic analysis. Since one of the main goals of this research was to develop a computational tool which could be easily applied in a design office, time is a very important parameter. This makes the Pushover Analysis much more applicable in a design office. The second reason has to do with earthquake unpredictability. When performing a dynamic analysis, it is best to use a series of earthquakes. This further increases the computational time. If we were to redesign a structure based on a maximum displacement achieved from a full dynamic analysis based on one particular earthquake, it is easy to imagine that there could be an earthquake which had the same probability of exceedance percentage but had a different frequency content. Based on the fundamental period of the structure, this would increase or decrease the maximum response. So, one would not know if the design was the maximum that could be expected until a great number of earthquake ground motion records were tested. The Pushover Analysis naturally accounts for all earthquakes with the same probability of exceedance by predicting the maximum displacement that can be expected in the form of the Target Displacement. Now, computational time has been further reduced, since only one analysis must be run for each exceedance probability that the designer is interested in, strengthening the idea that the Pushover Analysis is much more practical in a design office.

18

There are also two reasons why the Pushover Analysis may be preferred to designing according to an existing code, such as the UBC. The first is that it advances the state of the art from code design. The Pushover allows the designer to determine the building’s performance under a range of ground shakings while the current code design just determines that the building won’t fall down or threaten life under the worst possible shaking. This allows owners to choose in advance what the condition of their building will be after a given event which in turn limits their costs in purchasing earthquake insurance. Also, by knowing the resulting condition of the building after any ground motion, including small ground motions which may be just large enough to cause some non – structural damage, the designers can modify their design to protect expensive architectural fixtures or to limit the inconvenience that can be caused to building occupants when mechanical or plumbing components are damaged.

This increases the overall effectiveness of the structure furthering its

applicability in a design office. The second reason is that since the model directly incorporates the actual material nonlinearities of each member, and the structure is monotonically forced into the inelastic response range, the designer is able to get detailed member information at displacements up to and including the maximum displacement. From this information, sections of members which will be most damaged by the ground shaking can be located and these sections can be redesigned to develop the strength or ductility that will be required of them. In comparison, when designing by an appropriate code, the maximum loads are applied directly to the structure and only the maximum response is determined. The relation at specific loading values before the maximum is lost and the interrelation among contributing elements is not available. So, the designer has no idea of what the effect of increasing the strength or ductility at one section will have upon the other. This requires that both sections obtain their maximum strength or ductility, while the Pushover Analysis allows the designer to modify one section which in turn could have a beneficial result on the other section lowering the maximum response it would have to endure. So, the Pushover Analysis increases the effectiveness and efficiency of the design.

19

CHAPTER III APPLICATIONS OF THE NON – LINEAR STATIC PROCEDURE (PUSHOVER ANALYSIS)

The Pushover Analysis was defined in chapter II as a non – linear static approximation of the response a structure will undergo when subjected to dynamic earthquake loading.

Because we are

approximating the complex dynamic loading characteristic of ground motion with a much simpler monotonically increasing static load, there are bound to be limitations to the procedure. The objective of this chapter is to quantify these limitations.

This will be accomplished by performing the Pushover

Analysis and a full non – linear dynamic analysis on reinforced concrete moment resisting frames of six, twelve, and twenty stories. The resulting Target Displacement obtained from the Pushover Analysis may then be compared with the maximum displacement at the roof of each structure obtained from the dynamic analysis. The Pushover Analysis will follow the steps outlined in chapter II, while the steps necessary to perform the dynamic analysis will be described as they are evaluated.

III-A) Design of Three Reinforced Concrete Moment Resisting Frames

The design of each frame will be carried out according to the 1997 Uniform Building Code (UBC) [17] and the American Concrete Institute (ACI) structural concrete building code requirements 318-95 [1]. The frames are located in the Los Angeles, California area which falls under UBC earthquake zone 4. The frames to be designed are each one of four moment resisting frames in the structure and have common bay widths, story heights, and floor plans. The typical floor plan and section are shown in figure III-1, while the frame dimensions are given in figure III-2 . Common floor area loads will be used for each frame as given by the UBC code and are representative of a typical office building. These loads are also shown in figure III-1. The design procedure described here will show the formulation of the gravity loads, wind loads, and earthquake loads used in design.

20

Figure III-1: Typical Floor Plan, Floor Section, and Loads Common to All Frames

21

Figure III-2: Design Frame Dimensions and Member Sizes III-A-1) Formulation of Gravity Loads Used in the Design of Frames. The floor loads typical to each frame are shown in figure III-1. They consist of dead loads which are the partition load, ceiling load, slab weight, and transverse beam weight, and a floor or roof live load. In addition to these loads, the self weight of the girders and columns must be added. However, because the girders and columns must be designed, their weight is not known at the start of the design process. Through an iterative procedure, the required sections for each member can be found and their weight included in the gravity loads. The concrete sections for use in the formulation of gravity loads are shown in figure III-2, while the gravity loads are determined in table III-1.

22

Table III-1: Formulation of Gravity Loads Used in Design

Typical Floor Loads Self - Weight Dead Loads Description

t (in)

h (ft)

L (ft)

Conc wght, 3 wc (k/ft )

P1 (kips)

Slab Transverse Beam

6 9

12 0.67

18 18

0.15 0.15

8.1 1.35

W (ft) 12 12 12 12

L(ft) 18 18 18 18

P2 (kips) P (kips) 16.2 1.35

16.2 1.35

Superimposed Loads Description Ceiling (DL) Partition Walls (DL) Floor Load (LL) Roof Load (LL)

Description 26 x 18 (Girder) 28 x 20 (Girder) 32 x 22 (Girder) C - 1 (Column) C - 2 (Column) C - 3 (Column) C - 4 (Column)

Applied Surface Load (psf) 10 20 50 30

Design Section Loads Conc wght, 1 3 H (in) W (in) L (ft) wc (k/ft ) 26 28 32 20 24 28 32

18 20 22 20 24 28 32

12 12 12 14 14 14 14

0.15 0.15 0.15 0.15 0.15 0.15 0.15

P1 (kips) 1.1 2.2 5.4 3.2

P2 (kips) P (kips) 2.2 2.2 4.3 4.3 10.8 10.8 6.5 6.5

P1 (kips)

P2 (kips)

2.9 3.5 4.4 5.8 8.4 11.4 14.9

5.9 7.0 8.8 5.8 8.4 11.4 14.9

P (kips)

1 All Values Are Illustrated Below

Note : The Force at a Given Location Is the Sum of the Forces Corresponding to the Point Loads at That Location Due to Column Size, Beam Size, and Typical Floor Loads. The Change In Dead Load at the Roof Is Due to 1/2 Column Length There.

23

5.9 7.0 8.8 0.0 0.0 0.0 0.0

III-A-2) Formulation of Wind Loads for Use in Design

The calculation of the wind loading to be applied to each frame will be carried out based on the UBC wind loading profile. The wind pressure associated with each floor level is given by:

P = Ce C q q s I w

(III.1)

P is equal to the design wind pressure and is based on the basic wind speed at the design location and the exposure condition. For the Los Angeles area, the basic wind speed is 70 mph as given on the UBC wind map, and the exposure for a structure which has surrounding buildings is exposure C. From these two conditions, the remaining parameters can be determined. Ce is equal to the combined height, exposure and gust factor, and is a function of the exposure of the building and height of each floor level. Values for this coefficient are tabulated in the UBC. Cq is equal to the pressure coefficient for the structure or portion of the structure under consideration and is tabulated in the UBC. qs is equal to the wind stagnation pressure at a standard height of 33’ at the design location as tabulated in the UBC. Iw is equal to the importance factor of the structure also laid out in the UBC. The total wind force acting at each floor level on a frame is the design wind pressure multiplied by the floor height and the tributary width of the frame. Calculations for each frame are detailed in table III-2. III-A-3) Formulation of Earthquake Loads for Use in Design

The calculation of earthquake loads to be applied to each frame will be carried out based on the UBC earthquake loading profile. The force caused by an earthquake to be applied at each floor level is a function of the Design Base Shear, V, which is given by:

V=

Cv I W RT

(III.2)

However, the Design Base Shear need not exceed:

V=

2.5C a I W R

Further, the Design Base Shear must not be less than the least of the following:

24

(III.3)

Table III-2: UBC Wind Load Calculations Used in Design Data Common To All Frames Wind Importance Factor, Iw = Wind Stagnation Pressure @33', qs = Pressure Coefficient, Cq = Six Story Frame Heigth 2 - Heighth of Heigth 1 UBC tbl 16-G UBC tbl 16-G Floor, H (ft) (ft) (ft) 0 15 14 25 30 28 40 60 42 40 60 56 60 80 70 80 100 84

1.0 12.6 1.4

psf

Ce associated w/ H1 1.06 1.19 1.31 1.31 1.43 1.53

Ce associated w/ H2 1.06 1.23 1.43 1.43 1.53 1.61

Ce associated w/ Floor 1.06 1.214 1.322 1.406 1.48 1.546

Tributary Wind Force, P*WH Width, W (kips) (ft) 18 4.71 18 5.40 18 5.88 18 6.25 18 6.58 18 6.87

Ce associated w/ H1 1.06 1.19 1.31 1.31 1.43 1.53 1.53 1.61 1.67 1.67 1.67 1.79

Ce associated w/ H2 1.06 1.23 1.43 1.43 1.53 1.61 1.61 1.67 1.79 1.79 1.79 1.87

Ce associated w/ Floor 1.06 1.214 1.322 1.406 1.48 1.546 1.602 1.646 1.688 1.73 1.772 1.806

Tributary Wind Force, Width , W P*WH (ft) (kips) 18 4.71 18 5.40 18 5.88 18 6.25 18 6.58 18 6.87 18 7.12 18 7.32 18 7.50 18 7.69 18 7.88 18 8.03

Ce associated w/ H1 1.06 1.19 1.31 1.31 1.43 1.53 1.53 1.61 1.67 1.67 1.67 1.79 1.79 1.79 1.87 1.87 1.87 1.87 1.87 1.87

Ce associated w/ H2 1.06 1.23 1.43 1.43 1.53 1.61 1.61 1.67 1.79 1.79 1.79 1.87 1.87 1.87 2.05 2.05 2.05 2.05 2.05 2.05

Ce associated w/ Floor 1.06 1.214 1.322 1.406 1.48 1.546 1.602 1.646 1.688 1.73 1.772 1.806 1.834 1.862 1.888 1.9132 1.9384 1.9636 1.9888 2.014

Tributary Wind Force, Width, W P*WH (ft) (kips) 18 4.71 18 5.40 18 5.88 18 6.25 18 6.58 18 6.87 18 7.12 18 7.32 18 7.50 18 7.69 18 7.88 18 8.03 18 8.15 18 8.28 18 8.39 18 8.50 18 8.62 18 8.73 18 8.84 18 8.95

Twelve Story Frame Heigth 1 Heigth 2 - Heighth of UBC tbl 16-G UBC tbl 16-G Floor, H (ft) (ft) (ft) 0 15 14 25 30 28 40 60 42 40 60 56 60 80 70 80 100 84 80 100 98 100 120 112 120 160 126 120 160 140 120 160 154 160 200 168 Twenty Story Frame Heigth 1 Heigth 2 - Heighth of UBC tbl 16-G UBC tbl 16-G Floor, H (ft) (ft) (ft) 0 15 14 25 30 28 40 60 42 40 60 56 60 80 70 80 100 84 80 100 98 100 120 112 120 160 126 120 160 140 120 160 154 160 200 168 160 200 182 160 200 196 200 300 210 200 300 224 200 300 238 200 300 252 200 300 266 200 300 280

25

V = 0.11C a IW V=

0.8ZN v I W R

(III.4)

(III.5)

In the above equations: Z is equal to the seismic zone factor. For the Los Angeles area, the seismic zone is Zone 4 and the seismic zone factor is equal to 0.4. Cv and Ca are seismic coefficients and are functions of the soil type and the seismic zone factor. A stiff soil profile will be the basis for design corresponding to soil type SD. I is equal to the seismic importance factor for the structure under consideration. W is equal to the total seismic dead load equal to the total dead load of each structure. Nv is equal to the near source factor of the structure to known faults. Under consideration will be a site with a known fault greater than 15 km away and Nv equals 1.0. R is equal to the overstrength factor based on the lateral force resisting system of the structure. Under consideration will be a system of reinforced concrete Ordinary Moment Resisting Frames since no special designs will be considered (i.e. plastic hinges etc.). T is the elastic fundamental period of the structure (seconds) in the direction under consideration. For UBC calculations, this period is defined as:

T = Ct (hn ) 3 / 4

( III.6)

Ct is a numerical coefficient equal to 0.03 for reinforced concrete moment resisting frames. hn is equal to the height in feet from the base to the roof of the structure. Once the base shear, V, for each frame has been determined, the lateral force to be applied at the top of the structure, Ft, can be calculated. This is given in the UBC as:

Ft = 0.07TV

(III.7)

Where T and V are as described above. From the base shear, V, and the fundamental period, T, as given by the UBC, the lateral force distribution along the height of the structure, Fx, can be determined. This is given as:

26

Fx =

(V − Ft ) wx hx

(III.8)

n

∑w h i =1

i i

Fx is equal to the lateral force to be applied at floor level x, wx is the fraction of the total structure weight, W, allocated to floor level x, hx is the height of floor level x above the base of the structure, and the summation in the denominator is the sum of these values over the total number of floors, n.

The

calculations of the above equations leading to the lateral force distribution over the total height of each structure is illustrated in table III-3.

Table III-3: UBC Earthquake Loads Used in Design Six Story Frame Fundamental Period, T (s) = 0.832 UBC 30-8 Design Base Shear, V = 162 UBC 30-4 - 30-7 Concentrated Force at Top, Ft = 9.4 UBC 30-14 Floor Floor Weight, Wi/Wtotal Height, Floor # Wi (kips) wi (kips) hi (ft) 1 198 0.168 14 2 198 0.168 28 3 198 0.168 42 4 198 0.168 56 5 198 0.168 70 Roof 189 0.160 84 Wtotal = Σwi*hi = 1180

wi * hi 2.35 4.70 7.05 9.40 11.75 13.48 48.74

Quake Force, Fx (kips) 7.4 14.7 22.1 29.4 36.8 51.6

Twelve Story Frame Fundamental Period, T (s) = 1.400 UBC 30-8 Design Base Shear, V = 226 UBC 30-4 - 30-7 Concentrated Force at Top, Ft = 22.1 UBC 30-14 Floor Floor Weight, Wi/Wtotal Height, Floor # Wi (kips) wi (kips) hi (ft) 1 213 0.086 14 2 213 0.086 28 3 213 0.086 42 4 213 0.086 56 5 213 0.086 70 6 213 0.086 84 7 205 0.083 98 8 205 0.083 112 9 198 0.080 126 10 198 0.080 140 11 198 0.080 154 Roof 189 0.077 168 Wtotal = Σwi*hi = 2471

Data Common to All Frames Seismic Zone Factor, Z = 0.4 Near Source Factor, Nv = 1.0 Near Source Factor, Na = 1.0 Seismic Coefficient , Cv = 0.4 Seismic Coefficient , Ca = 0.4 Importance Factor, I = 1.0 Overstrength Factor, R = 3.5

Quake Force, Fx (kips) wi * hi 1.21 2.8 2.41 5.5 3.62 8.3 4.82 11.0 6.03 13.8 7.23 16.5 8.13 18.6 9.30 21.2 10.10 23.1 11.23 25.6 12.35 28.2 12.88 51.5 89.31

Twenty Story Frame Fundamental Period, T (s) = 2.053 UBC 30-8 Design Base Shear, V = 402 UBC 30-4 - 30-7 Concentrated Force at Top, Ft = 57.8 UBC 30-14 Floor Floor Weight, Wi/Wtotal Height, Floor # Wi (kips) wi (kips) hi (ft) 1 243 0.055 14 2 243 0.055 28 3 243 0.055 42 4 233 0.053 56 5 233 0.053 70 6 233 0.053 84 7 233 0.053 98 8 223 0.051 112 9 223 0.051 126 10 223 0.051 140 11 223 0.051 154 12 223 0.051 168 13 223 0.051 182 14 205 0.047 196 15 205 0.047 210 16 205 0.047 224 17 198 0.045 238 18 198 0.045 252 19 198 0.045 266 Roof 189 0.043 280 Wtotal = Σwi*hi = 4400

27

wi * hi 0.77 1.55 2.32 2.96 3.70 4.44 5.18 5.69 6.40 7.11 7.82 8.53 9.25 9.14 9.79 10.44 10.72 11.35 11.98 12.05 141.19

Quake Force, Fx (kips) 1.9 3.8 5.7 7.2 9.0 10.8 12.6 13.9 15.6 17.3 19.1 20.8 22.6 22.3 23.9 25.5 26.1 27.7 29.2 87.2

III-A-4) Total Design Loads and Section Determination

The total design loads are factored combinations of the gravity loads (dead and live), wind loads, and earthquake loads determined previously. The total unfactored design loads are shown in figure III-3, figure III-4, and figure III-5 for the six, twelve, and twenty story frames respectively.

Figure III-3: Gravity, Wind and Earthquake Loads for the Six Story Frame.

From these unfactored loads, we can apply the ACI load combinations and find the factored loads which must be designed for. The ACI load combinations which must be checked are:

1.4 DL + 1.7 LL

(III.9a)

1.05 DL + 1.28 LL ± 1.28WL

(III.9b)

0.9 DL ± 1.3WL

(III.9c)

1.05 DL + 1.28 LL + 1.4 EL

(III.9d)

0.9 DL + 1.43EL

(III.9e)

28

Figure III-4: Gravity, Wind, and Earthquake Loads for the Twelve Story Frame

29

Figure III-5: Gravity, Wind, and Earthquake Loads for the Twenty Story Frame

30

In equations III.9, DL is equal to the dead load, LL is equal to the live load, WL is equal to the wind load, and EL is equal to the earthquake load. Once the loads have been factored, the members can be designed. The resulting section locations are shown in figure III-6, while the resulting sections are shown in figures III-7 and III-8.

Figure III-6: Design Section Locations for Six, Twelve and Twenty Story Frames

31

Figure III-7: Final Design Sections – Beams

32

Figure III-8: Final Design Sections – Columns

33

III-B) Performing the Pushover Analysis on the Moment Resisting Frames

Once the frames have been designed, the Pushover Analysis can be performed on each frame to determine its maximum expected response. As described in chapter II, the first thing that needs to be calculated is the vertical distribution of the lateral loads. However, since this distribution is a function of the fundamental period of the structure through the exponent k, the period must first be determined. The Finite Element Analysis Program (FEAP) [16] will be used to perform the Pushover as well as the dynamic analyses, so this will also be used to find the fundamental period of the structures. Once this period has been determined, the lateral load distribution can be calculated, followed by the Seismic Hazard and the Target Displacement. The Building Performance Level will not be included here since only the Target Displacement is to be compared with the maximum dynamic displacement found at the roof of each structure. So, an earthquake that has a 10% probability of exceedance in 50 years will be used as the Building Performance Level requirement.

III-B-1) Period Determination

The fundamental period of a structure is a function of its mass and stiffness, so the masses at each floor level must be calculated. The stiffness of each structure is a function of the member sizes, which were designed in the previous section, and FEAP calculates this internally. The mass of the structure is a function of the total gravity load to be used in the Pushover Analysis. The total gravity load to be used in the Pushover Analysis was given in equation II.1 and is reproduced here as:

QG = 1.1(QD + QL + QS )

(III.10)

Recall that the live load to be used is 25% of the unreduced live load for the structure and the dead load is the total dead load effect. Since the structures are located in Los Angeles, the design snow load is less than 30 psf and therefore equal to zero. So the total gravity load to be used in the Pushover Analysis is equal to the dead load shown in figures III-3 to III-5 plus 25% of the live load shown in these figures all times the factor 1.1. Note that the dead load and live load used here are the unfactored design loads.

For the

determination of the total mass on the structure, FEAP requires an input mass per unit volume for each element. Since the total gravity load of equation III.10 must be included in the mass, it must be converted

34

into a mass per unit volume for the input. This will be accomplished by including the total load at a floor level, minus the column weight, as the input mass per unit volume for the beams. So, the mass per unit volume input for each beam is the total load at each floor level, minus the column weight, divided by the volume of the member. The volume of each member is its length times its width times its height. Since this division gives a load per unit volume, this value must also be divided by the gravitational constant. The input mass per unit volume for the columns will just be the unit weight of concrete. The masses per each member as a function of the total Pushover Analysis gravity loads are formulated in table III-4. Table III-4: Calculation of Mass Per Unit Volume for Each Member

Sect # Beam 1 - 1a Beam 2 - 2a Beam 3 - 3a

Sect # All Sections

P (kips) 35.9 37.1 39.1

Beam Distributed Masses ΣP = 6*P Width, Heigth, Length, (kips) W (in) H (in) L (in) 215.4 18 26 864 222.6 20 28 864 234.6 22 32 864

Column Distributed Masses 3 3 Conc Unit Weight (k/ft ) Conc Unit Weight (k/in ) 0.15 8.68056E-05

Distributed Mass input 1.378634002E-06 1.190655912E-06 9.981699435E-07

Distributed Mass input 2.246520589E-07

Beam mass per unit volume = (P1 + P2 + P3 + 4P) / (Beam L * W * H * g) Where P1 = P3 = 1/2 P, and P2 = P because column mass is calculated seperately Column mass per unit volume = (concrete unit weight) / g

Once the applicable masses have been included in the input, FEAP can determine the fundamental period of each frame. The fundamental, second, and third periods of each frame are shown for comparison in table III-5. Table III-5: Fundamental, Second, and Third Periods for Each Frame

# Stories in Frame Fundamental Period, T (sec) Second Period, T2 (sec) 6 1.61 0.517 12 2.66 0.954 20 3.59 1.39

35

Third Period, T3 (sec) 0.294 0.550 0.797

The second and third periods for each frame are shown because it is interesting to notice that the second period of the twenty story frame is close to the first period of the six story frame. So, it can be expected that the twenty story’s second mode will contribute in its dynamic response as much as the six story’s first mode will to its dynamic response. A similar statement can be made about the second mode of the twelve story building since its second period is about 60% of the six story’s first period. This may be important when comparing the Pushover Analysis with the dynamic analysis because the Cvx load distribution is enforcing some variation of the first mode response in each of the structures. So, if a first mode response is accurate for the six story frame, it is easy to see that for the other frames it will be less accurate as higher modes contribute more to the dynamic response.

III-B-2) Calculation of the Vertical Distribution of Lateral Loads

Now that the fundamental period for each frame structure has been determined, the vertical distribution of the lateral loads for the Pushover Analysis can be calculated. The vertical distribution of the lateral loads is given by the FEMA 273 Cvx loading profile as given by equation II.2. This equation will be reproduced here for convenience and is:

C vx =

w x hx

k

(III.11)

n

∑w h i =1

i

k x

The Cvx loading coefficient is the value to be applied at each floor level of the frame. wx is the fraction of the total weight, W, allocated to the floor at level x, hx is the height of the floor at level x above the base, and the summation in the denominator is the sum of these values over the total number of stories, six, twelve, or twenty for this example. All of these parameters are illustrated in figure II-5. The exponent k is a factor which varies with fundamental period as shown in figure II-6. These Cvx loading coefficients are input into FEAP and the program multiplies them by the monotonically increasing load to get the lateral loads at each increasing value. The formulation of the Cvx loading coefficients for each frame is detailed in table III-6. Once all of these values have been determined they can be combined with the Pushover Analysis gravity loads determined in equation III.10 to get the total frame loading for analysis as shown in figures III-9, III-10, and III-11 for the six, twelve, and twenty story frames, respectively.

36

Table III-6: Determination of the Vertical Distribution of Lateral Loads for Each Frame Six Story Frame FEAP Fundamental Period, T = 1.61 Period Dependent Factor, k = 1.55 Floor # 1 2 3 4 5 Roof

Weight, QGi (kips) 236 236 236 236 236 209

QGi / QGtotal, wi 0.170 0.170 0.170 0.170 0.170 0.150

Height of floor, hi (ft) 14 28 42 56 70 84

sec

k

wi * hi 10.2 30.0 56.3 88.0 124.5 146.1

Lateral Load Factor, Cvx 0.0225 0.0659 0.1237 0.1934 0.2735 0.3210

Σwi * hi = 455 Twelve Story Frame FEAP Fundamental Period, T = 2.66 sec Period Dependent Factor, k = 2.00

QGtotal =

Floor # 1 2 3 4 5 6 7 8 9 10 11 Roof

k

1388

Weight, QGi (kips) 252 252 252 252 252 252 243 243 236 236 236 209

QGi / QGtotal, wi 0.0864 0.0864 0.0864 0.0864 0.0864 0.0864 0.0836 0.0836 0.0810 0.0810 0.0810 0.0716

Height of floor, hi (ft) 14 28 42 56 70 84 98 112 126 140 154 168

k

wi * hi 16.9 67.7 152.4 270.9 423.4 609.6 802.4 1048.1 1285.6 1587.1 1920.4 2020.0

Lateral Load Factor, Cvx 0.0017 0.0066 0.0149 0.0266 0.0415 0.0597 0.0786 0.1027 0.1260 0.1555 0.1882 0.1979

Σwi * hi = 10205 Twenty Story Frame FEAP Fundamental Period, T = 3.59 sec Period Dependent Factor, k = 2.00

QGtotal =

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Roof

Weight, QGi (kips) 285 285 285 274 274 274 274 264 264 264 264 264 264 243 243 243 236 236 236 209

QGtotal =

5179

Floor #

k

2913

QGi / QGtotal, wi 0.0551 0.0551 0.0551 0.0528 0.0528 0.0528 0.0528 0.0509 0.0509 0.0509 0.0509 0.0509 0.0509 0.0470 0.0470 0.0470 0.0455 0.0455 0.0455 0.0403

Height of floor, hi (ft) 14 28 42 56 70 84 98 112 126 140 154 168 182 196 210 224 238 252 266 280

wi * hi 10.8 43.2 97.2 165.7 258.9 372.9 507.5 638.7 808.3 997.9 1207.5 1437.0 1686.5 1805.4 2072.5 2358.0 2578.9 2891.2 3221.3 3156.1

Σwi * hi =

26315

k

37

k

Lateral Load Factor, Cvx 0.0004 0.0016 0.0037 0.0063 0.0098 0.0142 0.0193 0.0243 0.0307 0.0379 0.0459 0.0546 0.0641 0.0686 0.0788 0.0896 0.0980 0.1099 0.1224 0.1199

Figure III-9: Gravity Loads and Cvx Loading Coefficients for Six Story Frame

38

Figure III-10: Gravity Loads and Cvx Loading Coefficients for Twelve Story Frame

39

Figure III-11: Gravity Loads and Cvx Loading Coefficients for Twenty Story Frame

40

III-B-3) Element Reduction for Analysis of Frames

Because of the length of time required to run a complete non – linear dynamic analysis of these structures, an effort was made to limit the numbers of elements and nodes in each frame. Since a comparison is to be made between dynamic and Pushover analyses, this also applies to the Pushover. The elements and nodes finally adopted for the FEAP input files are shown in figure III-12, while the transformation of the midspan point loads shown in figures III-9 to III-11 are detailed in table III-7.

21

42

63

20

41

62

19

40

61

18

39

60

17

38

59

16

37

58

15

36

57

14

35

56

13

26

39

13

34

55

12

25

38

12

33

54

11

24

37

11

32

53

10

23

36

10

31

52

9

22

35

9

30

51

8

21

34

8

29

50

7

14

21

7

20

33

7

28

49

6

13

20

6

19

32

6

27

48

5

12

19

5

18

31

5

26

47

4

11

18

4

17

30

4

25

46

3

10

17

3

16

29

3

24

45

2

9

16

2

15

28

2

23

44

8

15

1

14

27

1

22

43

1

Figure III-12: Element and Node Configuration Used in FEAP Input

41

Table III-7: Transformation of Midspan Point Loads to Fixed End Forces and Moments

floor #'s 6 5 to 1

floor #'s 12 11 to 9 8 to 7 6 to 1

floor #'s 20 19 to 17 16 to 14 13 to 8 7 to 4 3 to 1

Six Story Frame P1 = P3 P2 P1+P = P2+2*P M = 96*P (kips) P (kips) (kips) P3+P (kips) (kips) (kip-in) +/-20.4 -32.9 -36.1 -53.3 -101.9 3158.4 -25 -35.9 -42.2 -60.9 -114 3446.4 Twelve Story Frame P1 = P3 P2 P1+P = P2+2*P M = 96*P (kips) P (kips) (kips) P3+P (kips) (kips) (kip-in) +/-20.4 -32.9 -36.1 -53.3 -101.9 3158.4 -25 -35.9 -42.3 -60.9 -114.1 3446.4 -25.7 -37.1 -43.6 -62.8 -117.8 3561.6 -28.5 -37.1 -46.3 -65.6 -120.5 3561.6 Twenty Story Frame P1 = P3 P2 P1+P = P2+2*P M = 96*P (kips) P (kips) (kips) P3+P (kips) (kips) (kip-in) +/-20.4 -32.9 -36.1 -53.3 -101.9 3158.4 -25 -35.9 -42.2 -60.9 -114 3446.4 -25.7 -37.1 -43.6 -62.8 -117.8 3561.6 -29.5 -39.1 -48.3 -68.6 -126.5 3753.6 -32.8 -39.1 -51.7 -71.9 -129.9 3753.6 -36.7 -39.1 -55.6 -75.8 -133.8 3753.6

III-B-4) Determination of Seismic Hazard for Analysis

The calculation of the seismic hazard was detailed in section II-B-3 and is a function of the probability of exccedance in 50 years under consideration, the mapped short period acceleration parameter, the mapped acceleration parameter at a one second period, the soil site conditions, the effective structural damping, and the fundamental period of the structure. The current analysis considers an earthquake with a 10% probability of exceedance in 50 years, a Los Angeles location, stiff soil conditions and 5% effective structural damping. The applicable equations are equations II.3 through II.11 to calculate the modified

42

mapped short period and one second period acceleration parameters, SS and S1, at the probability of exceedance under consideration, the Return Period, PR, at the probability of exceedance under consideration, the design acceleration parameters, SXS and SX1, based on the soil type parameters, Fa and Fv, and the Spectral Response Acceleration, Sa, based on structural damping coefficients, BS and B1, the period at which the constant acceleration region of the spectrum intersects with the constant velocity region of the spectrum, T0, and the fundamental structural period, T. These calculations are carried out, and values for the parameters are listed in table III-8, while the General Response Spectrum is shown in figure III-13. Shown on the General Response Spectrum curve is where the six, twelve, and twenty story frames lie. Table III-8: Formulation of the General Response Spectrum for Analysis Mapped 2% / 50yrs S1 (g's) SS (g's) 2 0.75

Mapped 10% / 50yrs SS (g's) S1 (g's) 1.25 0.5

Hazard Level, HL (% / 50yrs) 10

Return Period @ HL, PR (yrs) 475

Accelerations @ HL SS (g's) S1 (g's) 1.25 0.50

Soil Site Class = D Fa Fv 1.0 1.5

Design Accelerations Sxs (g's) Sx1 (g's) 1.25 0.75

Damping Coeff BS B1 1.0 1.0

Intersect Period1 T0 (sec) 0.6

1 Period at Which the Constant Velocity and Acceleration Regions of the Design Spectrum Intersect

1.5

Spectral Response Acceleration, Sa (g's)

1.25

0.2 T0 = 0.12 s 1

6 Story Frame, T = 1.61 s 0.75

12 Story Frame, T = 2.66 s 20 Story Frame, T = 3.59 s 0.5

T0 = 0.6 s

0.25

0 0

0.5

1

1.5

2

2.5

3

3.5

Fundamental Structural Period, T (sec)

Figure III-13: General Response Spectrum for

Analysis

43

4

III-B-5) Calculation of Target Displacement for Analysis Finally the Target Displacement can be calculated for each of the analysis frames. Recall from equation II.12 that the Target Displacement, δt, is equal to:

Te2 g δ t = C 0 C1C 2 C 3 S a 4π 2

(III.12)

All parameters in equation III.12 were defined in chapter II-B-4 and these parameters plus the values to determine these parameters are shown in table III-9. The Effective Stiffness, the Elastic Stiffness, and the post – yield stiffness, which enter into the parameters in equation III.12, must be calculated from the Base Shear vs. Roof Displacement curve as defined in equation II.17 and illustrated for each of the analysis frames in figures III-14 to III-16. Table III-9: Values Used in Obtaining Target Displacements for Analysis

# stories 6 12 20

R 3.27 3.22 2.94

T0 (sec) 0.6 0.6 0.6

# stories 6 12 20

C0 1.42 1.50 1.50

C1 1.0 1.0 1.0

Internal Values α Ki (k/in) Ke (k/in) αKe (k/in) 14.1 12.6 1.35 0.107 8.68 8.26 0.748 0.091 8.11 7.40 0.198 0.027 Equation Values C2 C3 Sa (g's) Te(sec) 1.0 1.0 0.469 1.70 1.0 1.0 0.282 2.73 1.0 1.0 0.209 3.76

Vy (kips) 140 170 245 δt (in) 18.8 30.8 43.3

160

Vy = 140 kips 140

Ki = 14.1 kips/in

αKe = 1.35 kips/in

120

Base Shear (kips)

Original Non - Linear Curve 100

80

Ke = 12.6 kips/in 60

40

δ t = 18.8 in

0.6 Vy = 84 kips 20

0 0

5

10

15

20

25

Roof Displacement (in)

Figure III-14: Base Shear vs. Roof Displacement

With Target Displacement for Six Story Frame

44

200

Vy = 170 kips 180 160

Ki = 8.68 kips/in

α Ke = 0.748 kips/in

140

Ke = 8.26 kips/in

120 100

Original Curve 80 60

0.6 Vy = 102 kips 40

δ t = 30.8 in 20 0 0

5

10

15

20

25

30

35

40

45

Roof Displacement (in)

Figure III-15: Base Shear vs. Roof Displacement With Target Displacement for Twelve Story Frame

300

α Ke = 0.198 kips/in

Vy = 245 kips 250

Original Curve Base Shear (kips)

200

Ke = 7.40 kips/in 150

Ki = 8.11 kips/in 100

0.6 Vy = 147 kips 50

δ t = 43.3 in 0

0

20

40

60

80

100

120

Roof Displacement (in)

Figure III-16: Base Shear vs. Roof Displacement With Target Displacement for Twenty Story Frame

45

III-C) Complete Non – Linear Dynamic Analyses for Each Frame

The full non – linear dynamic analysis will now be run on each frame. The dynamic loading will be a ground motion record as provided by the SAC joint venture [13]. SAC provides a number of ground motions on their web page which [14] are scaled so that the mean response spectrum matches that given by FEMA 273. To be consistent with the Pushover Analysis, the ground motion must be that for a Los Angeles site, on stiff soil, with a 5% structural damping ratio, and a 10% probability of exceedance in 50 years. This happens to be the 1940 El Centro ground motion record as shown in figure III-17.

300

200

Acceleration (in/s2)

100

0 0

2

4

6

8

10

12

14

16

18

20

-100

-200

-300 Time (s)

Figure III-17: 1940 El Centro Ground Motion Used in Dynamic Analysis

FEAP allows dynamic analysis for frames through the frame analysis additions added by Spacone et al [15]. The ground motion is input as a file and the accelerations are converted to forces at each time step by the effective force form of the equation of motion which is:

[M ]{v' '}+ [C ]{v'}+ [K ]{v}= −[M ]{vg ' '}= −{peff }

(III.13)

Where v’’, v’, and v are the relative acceleration, velocity, and displacement vectors of each element respectively, and vg’’ is the acceleration vector due to the ground motion. M is equal to the mass matrix of each element as formulated in table III-4, K is equal to the stiffness matrix of each element, and C is equal

46

to the element damping matrix formulated in FEAP through Rayleigh Damping coefficients. The Rayleigh damping coefficients are given by Chopra [4] as:

α =ζ

2ω iω j

ωi + ω j

,

β =ζ

2 ωi + ω j

(III.14 a)

C = αM + βK

(III.14 b)

Where ζ is equal to the structural damping ratio, 5% in this case, and ωi and ωj are equal to the first and second frequencies (rad/s) of the structure respectively for FEAP input. These values are formulated in table III-10. Table III-10: Rayleigh Damping Coefficients Used in Dynamic Analysis Six Story Frame α β ω1 ω2 ζ1 = ζ2 2.5111956 8.0407019 0.05 0.1913568 0.009477 Twelve Story Frame α β ω1 ω2 ζ1 = ζ2 1.4880482 4.206531 0.05 0.1099207 0.0175606 Twenty Story Frame α β ω1 ω2 ζ1 = ζ2 1.1086985 2.8633657 0.05 0.0799234 0.0251758

The resulting dynamic displacements at the roof of each frame are shown in figures III-18 to III20. Included in these figures are the maximum and minimum dynamic displacements obtained at the roof of each structure. 20 Max Disp = 15.0 in 15

Roof Displacement (in)

10

5

0 0

5

10

15

20

-5

-10

-15 Min Disp = -15.0 in -20 Time (s)

Figure III-18: Dynamic Roof Displacement for Six Story Frame

47

25

20 Max Disp = 14.2 in 15

Roof Displacement (in)

10

5

0 0

5

10

15

20

25

-5

-10

-15 Min Disp = -12.7 in -20 Time (s)

Figure III-19: Dynamic Roof Displacement for Twelve Story Frame

20

15

Max Disp = 10.7 in

Roof Displacement (in)

10

5

0 0

5

10

15

20

-5

-10

-15 Min Disp = -12.5 in -20 Time (s)

Figure III-20: Dynamic Roof Displacement for Twenty Story Frame

48

25

III-D) Comparisons of Full Dynamic Results With Pushover Analysis Results The Target Displacements obtained from the Pushover Analysis and the maximum and minimum displacements obtained at the roof of each structure from the complete dynamic analysis are tabulated in table III-11. Table III-11: Target Displacement and Maximum and Minimum Dynamic Displacements for Frames

# Stories 6 12 20

Maximum Minimum Target Dynamic Roof Dynamic Roof Displacement Displacement Displacement (in) (in) (in) 18.8 15.0 -15.0 30.8 14.2 -12.7 43.3 10.7 -12.5

In comparing these results, one can see that for the six story frame we get results which compare very well. One thing to note is that even though the Target Displacement is greater than the dynamic displacement, the results for the six story frame are good because this Target Displacement is a maximum displacement expected for any earthquake with a 10% probability of exceedance in 50 years. This means that a different earthquake with a different frequency content could give a higher maximum displacement under dynamic loading. To clarify these results, the load distribution for the Pushover Analysis will be compared to the dynamic displaced shape at several times. These times are described in figure III-21(a) when the maximum dynamic roof displacement occurs before the minimum or III-21 (b) when the minimum dynamic roof displacement occurs before the maximum.

Figure III-21: Times at Which Dynamic Displaced Shapes Will be Plotted

49

With the definition of these times, the Pushover static load distribution and the dynamic displaced

7

7

6

6 Floor Level

Floor Number

shape at the defined times can be plotted for the six story frame (figure III-22).

5 4 3 2 1 0.00

min

t3

t5 t6

max

t4

5

t1

4 3

t2

2

t7

1 0 0.10

0.20

0.30

0.40

-20

Cvx Load Factor

-10

0

10

20

Displacement (in)

Figure III-22: Pushover Load Distribution and Dynamic Displaced Shapes – Six Story Frame

It can be seen in the above figure, that even though the displaced shape changes with time, there is a definite participation of the second mode as is apparent by the non – linearity of the displaced shapes. Furthermore, it can be seen that at the maximum and minimum displaced shapes the middle portion of the building has a velocity which is opposite to the top portion. This is apparent by the displaced shapes at times t5, t6, t2, and t3 in which the middle portion of the building has a greater displacement than the roof. So, at the maximum and minimum displacements, when the velocity of the roof is equal to zero, the middle portion of the building must be traveling in the opposite direction. This limits the maximum and minimum displacements at the roof that will be obtained from the dynamic analysis because the middle portion of the building cancels out some of the roof displacement. So, while the Pushover Analysis loading is enforcing a first mode response in the structure, the El Centro earthquake excites the first and second modes of this six story frame, however an earthquake with a different frequency content could excite only the first mode and the maximum and minimum displacements would approach the Target Displacement. In comparing the results for the twelve story frame, a much larger discrepancy between the Pushover Analysis’s prediction and the actual dynamic displacement is recognized. Again, plotting the Pushover load distribution and the dynamic displaced shape at several times (figure III-23), it is seen that again the first mode response is being enforced in the Pushover’s static load, while the dynamic loading is exciting the second, third, and even higher modes.

50

This accounts for the large discrepancy.

14

11

12

9

10

Floor Level

Floor Number

13

7 5 3

min

t3

t7

t6

max

t4

t1

8

t5

6

t2

4 2

1 0.00

0

0.05

0.10

0.15

0.20

0.25

-20

-10

0

10

20

Displacement (in)

Cvx Load Factor

Figure III-23: Pushover Load Distribution and Dynamic Displaced Shapes – Twelve Story Frame

Finally, comparing the results for the twenty story frame, again a very large discrepancy is apparent between the Pushover Analysis’s predicted result and the result obtained from the dynamic analysis. In viewing the comparison of the Pushover’s static load with the dynamic displaced shape at various times (figure III-24), it is seen that once again the Pushover is enforcing a first mode response in the structure while the ground motion is exciting the higher modes.

25

11 6 1 0.00

min

20

16

Floor Level

Floor Number

21

t5 t6

t1

t3

t2

max

15 t4 10

t7

5 0 0.05

0.10

0.15

-20

Cvx Load Factor

-10

0

10

20

Displacement (in)

Figure III-24: Pushover Load Distribution and Dynamic Displaced Shapes– Twenty Story Frame

51

The conclusions that can be drawn from these comparisons is that for stiffer, lower period structures, such as the six story frame, the pushover analysis is accurate and includes expected displacements that could be caused by a multitude of earthquakes. However, as the fundamental period of the structure increases, there is less likely to be an earthquake that excites primarily the first mode response, and the higher modes are more likely to be excited by every earthquake. This was illustrated in the twelve and twenty story frame results. To further illustrate this statement, figure III-25 shows the frequency content of the 1940 El Centro ground motion used in the analysis plotted as a function of period. Also shown in the figure are the first three periods of each structure repeated here for convenience.

Absolute Value of Fourier Transform, |C(w)|

Figure III-25: Frequency Content of 1940 El Centro Ground Motion as a function of Period 250

200

150

Frame # Stories

T1 (s)

T2 (s)

T3 (s)

6

1.61

0.517

0.294

12

2.66

0.954

0.550

20

3.59

1.39

0.797

100

50

0 0

1

2

3

4

5

6

7

8

Period, T (s)

While different earthquakes will vary in frequency content plots, it can easily be seen that as the fundamental period, and therefore the higher periods, increases, an earthquake will be less likely to have a frequency content distribution which excites primarily the first mode of the structure. As the higher periods increase, they are entered into the range which will be excited by the earthquake, therefore contributing to its overall dynamic response. So, the Pushover Analysis is accurate for stiff structures whose higher modes will have less effect on the overall dynamic response.

52

III-E) Dependence of Target Displacement on Choice of Vy

Since the accuracy of the Pushover Analysis is being determined, something must be said about the subjective choice in the structural yield level on the base shear vs. roof displacement plot, Vy. Because the choice is subjective, there may be a significant change in the maximum expected displacement, or Target Displacement, given different choices in the base shear vs. roof displacement plot yield level. To quantify this, the target displacement is calculated for changes in the choice of Vy in figure III-26 for the six story frame.

160

160 Vy = 140 kips

140 120

Base Shear (kips)

Base Shear (kips)

140 120 100

Vy = 120 kips

100

80 60 40 20

δ t = 18.8 in

0

80 60 40 20

δ t = 18.4 in

0 0

5

10 15 20 Roof Displacement (in)

25

0

5

Figure III-26 a

Base Shear (kips)

140 120

Base Shear (kips)

160

140 Vy = 100 kips

δ t = 18.0 in

20

Vy = 152 kips

100

80 60 40

25

Figure III-26 b

160 120 100

10 15 20 Roof Displacement (in)

0

80 60 40 20

δ t = 19.1 in

0 0

5

10 15 20 Roof Displacement (in)

25

0

Figure III-26 c

5

10 15 20 Roof Displacement (in)

25

Figure III-26 d

Figure III-26: Target Displacements for Changes in the Choice of Vy

In figure III-26a, the original choice of Vy is shown with its corresponding Target Displacement. This is clearly the best choice in Vy as the approximate bilinear post yield line is approximately tangent to

53

the actual base shear vs. roof displacement curve. However, the graphs III-26 b-d show other choices that could be possible. Figure III-26b shows an intermediate value of the Vy, and the corresponding change in Target Displacement is approximately 2%. Figure III-26c shows the Vy occurring at a very low value which is clearly not the yield value of the structure. The resulting change in the Target Displacement is approximately 4%. Figure III-26d shows the yield value equal to the ultimate value attained by the structure, meaning the structure has zero post yield stiffness. This is also clearly not the yield value, however the resulting change in the Target Displacement is only about 1.6%. So, the figure shows that for even bad choices of Vy, such as those shown in figures III-26 c and d especially, the resulting change in the Target Displacement is less than 5%. So, the Target Displacement is not greatly affected by changes of the choice in the yield level on the base shear vs. roof displacement plot, even when that choice is not very reasonable.

III-F) Conclusions on the Limitations and Accuracy of the Pushover Analysis

From the comparison of the three reinforced concrete frames analyzed with both the Pushover Analysis and the complete dynamic analysis, it is seen that the Pushover Analysis is accurate for shorter, stiffer structures whose higher modes do not contribute significantly to the overall dynamic response. Further, it is seen that the Pushover Analysis incorporates the maximum expected displacement that may occur from a range of ground motions with a given percent exceedance in 50 years by assuming the worst case scenerio, which is purely first mode response. However, as the higher mode effects become more significant, the Pushover Analysis overestimates the maximum displacement expected during the design event. It has also been shown that, even though the Target Displacement is a function of the subjective value of the structural yield level on the base shear vs. roof displacement plot, the results are not greatly effected by even illogical choices in the value of Vy.

54

CHAPTER IV FORMULATION OF ELEMENT SHEAR RESPONSE

Chapter III describes the application of the NSP analysis to moment resisting frames whose response is governed primarily by flexural deformation. However, seldom, if ever in reinforced concrete structures, is the lateral force resisting system composed entirely of moment resisting frames. More commonly it is composed of structural walls, or a combination of structural walls and moment resisting frames. Since the primary goal in this research is to supply a Pushover Analysis tool which may easily be applied in a design office, the program developed here must have the ability to analyze structural walls. Some common configurations of structural walls used to resist lateral forces are illustrated in figure IV-1. They are coupled walls, walls with openings, walls with varying dimensions or thicknesses, elevator and stair cores, or any combination of these wall systems.

Figure IV-1: Common Configurations of Structural Walls

Even though the limitations of the Pushover Analysis were outlined in Chapter III, it lends itself well to structural walls because they are very stiff and have very short periods, thus their response is typically determined by their fundamental period of vibration.

This benefit is offset by the fact that

structural walls, often called shear walls, have a low span to depth ratio, thus shear deformations must be

55

accounted for. To capture this behavior, the existing force – based fiber beam element presented in Spacone et al [15], must be modified to include shear deformations. This chapter will extend the original formulation to include the shear response of Reinforced Concrete elements. First, the Timoshenko beam theory is reviewed, then the changes in the total element flexibility are illustrated, and the section flexibility is modified to include shear deformations by developing a cyclic shear hysteretic law to relate shear deformations to shear force at each section in the element. Also included are comments on possible enhancements to the proposed formulation. IV-A) Review of Timoshenko Beam Theory In Euler – Bernoulli beam theory, shear deformations are neglected, and plane sections remain plane and normal to the longitudinal axis. In the Timoshenko beam theory, plane sections still remain plane but are no longer normal to the longitudinal axis.

The difference between the normal to the

longitudinal axis and the plane section rotation is the shear deformation. These relations are shown in figure IV-2.

Figure IV-2: Bernoulli and Timoshenko Beam Deformations

It can be seen in figure IV-2 that in the Euler - Bernoulli beam the deformation at a section, dvo/dx, is just the rotation due to bending only, since the plane section remains normal to the longitudinal axis. However, in the Timoshenko beam the section deformation is the sum of two contributions: one is

56

due to bending, dvb/dx, and the other is the shear deformation, dvs/dx. By considering an infinitetesimal length of the beam, as shown in figure IV-3, it is seen that the shear deformation in Timoshenko beam theory, dvs/dx, is the same as the shear strain related to pure shear, γ.

γ= κ=

Figure IV-3: Infinitesimal Length of Beam Showing Bending and Shear Deformations

For linear elastic materials, Hooke’s law for shear applies and:

τ = Gγ

(IV.1)

Where τ is equal to the shear stress applied to the element and G is the shear modulus of elasticity for the material. In the Timoshenko beam theory, the shear stress is assumed constant over the cross section. The shear force, V, is related to the shear stress through:

V = τAs

(IV.2)

where As is equal to the shear area of the section. Combining these two equations:

V = GAsγ

(IV.3)

While this equation only applies to linearly elastic materials, it will be the basis for the formulation of the non-linear shear force - shear strain relation. In this study, it is assumed that V and γ are interrelated though the shear area of the section multiplied by a value which accounts for the non-linear response of Reinforced Concrete to shear force. The existing force based elements available in FEAP account for the deformation resulting from bending alone, however they do not include the shear deformations. The shear deformations will be added according to the following formulation.

57

IV-B) Nonlinear Force – Based Timoshenko Beam Element To include shear deformations in the element, the element flexibility must be established. To express clearly the modifications to the existing Bernoulli formulation, both the Timoshenko and Bernoulli formulations will be carried out. The beam elements added by Spacone et al [15] have five degrees of freedom, without rigid body modes, in three dimensions at the element level. They are two bending moments, Q1 – Q4, or rotations, q1 – q4, at each end and one axial force, Q5, or displacement, q5 (figure IV-4).

Y

y

Q4, q4 Q3, q3

x

Q5, q5 X

Z z

Q2, q2

Q1, q1

Figure IV-4: Three Dimensional Flexibility Based Element

The capital X, Y, and Z in figure IV-4 are the global degrees of freedom while the lowercase x, y, and z are the local degrees of freedom for the element. The element is divided into transverse sections along its length at each one of the classic Guass or Guass – Lobatto integration points. Each of these transverse sections is further divided into longitudinal fibers. This configuration is shown in figure IV-5.

Figure IV-5: Element Divided into Transverse Sections and Longitudinal Fibers

58

To obtain the flexural and axial response along the element, the fibers are analyzed and the stresses, σ, and moduli, E, are summed to compute the section response. The section responses (mainly forces and flexibility) are then summed according to weight factors depending on the location of the section and the integration scheme used to compute the total axial and flexural response along the element. The fiber section model yields interaction between flexural and axial responses. The shear response for each element is calculated at the section level. This response is then summed over all the sections, again according to the weight factors depending on the location of the section and the integration scheme used, to get the total element response due to shear. In the proposed approach shear response is independent of the axial and flexural responses at the section level. However, since force equilibrium is satisfied pointwise along the element, interaction among the axial – flexural and shear responses is enforced at the element level. This will be clarified in the following formulation. There are three major steps in the force – based formulation. In the first step, equilibrium, the force fields are expressed as functions of the nodal forces:

D( x) = b( x)Q

(IV.4)

Where D(x) are the section forces, b(x) contain the element force shape functions, and Q contain the nodal forces as illustrated in figure IV-4. The second step is to write the section constitutive law in which the section deformations, d(x), are related to the section forces, D(x), through the section flexibility matrix, f(x):

d ( x) = f ( x ) D ( x )

(IV.5)

The third step is to satisfy compatibility (in an average sense). Starting from a compatible state of deformations, the element relation between forces, Q, and corresponding deformations, q, is obtained with the application of the principle of virtual forces: L

δQ T q = ∫ δD T ( x)d ( x)dx 0

Putting together IV.4, IV.5 and IV.6, along with noting the arbitrariness of δQ, one obtains:

59

(IV.6)

L

q = ∫ b T ( x) f ( x)b( x)dx Q

(IV.7)

q = FQ

(IV.8a)

F = ∫ b T ( x) f ( x)b( x)dx

(IV.8b)

0

or:

where: L

0

is the element flexibility. The above equations apply to the force – based formulation of both Bernoulli and Timoshenko beams. However there are differences in the individual components which enter into the equations. These differences will be illustrated below. As shown in figure IV-4, the three dimensional element has five degrees of freedom, two moments at each end and an axial load, irrespective of whether shear deformations are included or not. However, on the section level, the number of non – zero deformations depend on whether or not shear deformations are included. The section degrees of freedom are illustrated in figure IV-6.

,γ

,φ

,φ

,ε

,φ

,ε

,φ ,γ

Figure IV-6: Three Dimensional Section Degrees of Freedom – Bernoulli and Timoshenko Beams

60

As can be seen from figure IV-6, the Bernoulli beam section has three non – zero deformations while the Timoshenko beam section has five deformations. Because the shear forces are required in the section formulation, the equilibrium statement must include these forces and therefore the shape functions for the Timoshenko beam must include the relation between shear force applied at a section to the nodal forces of the element. Adding these to the shape functions included in the Bernoulli beam formulation, which assumes constant axial force and linear moment along the element, gives the constant shear force distributions of the Timoshenko beam.

The constitutive relation must also be modified because the

sections now include shear forces and deformations. The section flexibility, which is obtained by inverting the section stiffness, contains the shear flexibility of the section decoupled from the axial and bending terms. Weighted integration of f(x), however, couples shear, axial and bending responses on the element level. These relations are detailed thoroughly in figure IV-7. Referring to figure IV-7, it is interesting to note that while the constitutive equation for the Bernoulli beam exhibits a full 3x3 section flexibility matrix, the same relation for the Timoshenko beam is a full 3x3 matrix for the flexural and axial responses but is merely diagonal for the shear response resulting in a 5x5 section flexibility matrix. This is due to the fact that, as mentioned earlier, the fiber responses are added over the section to obtain the response at each section. So, at the section level, the axial and flexural responses are naturally coupled. The shear response is not calculated at the fiber level, but is determined for the entire section as an average response. However, the flexibility equation for both beam types is a full 5x5 matrix. This is due to the fact that the element response is found by summing the responses of each section weighted through the shape functions, b(x), that impose linear moments and constant shear in equilibrium with the end moments. So, the formulation used here for the Timoshenko beam includes coupling of axial and flexural responses on the section level, but shear responses are independent on the section level. However, shear and axial – flexural responses are coupled at the element level. In other words, because the shear forces are related to the end moments through equilibrium, the shear and bending forces are coupled through equilibrium.

61

Bernoulli Beam

Timoshenko Beam

Element Level – 5 dof’s Section Level – 3 dof’s

Element Level – 5 dof’s Section Level – 5 dof’s  M z ( x)  φ z ( x)  M ( x)  φ ( x)   y   y  D( x) =  N ( x)  d ( x) =  ε ( x)   V ( x)  γ ( x)   y   y   V z ( x)  γ z ( x) 

M y    D( x) =  M z  N   

φ z ( x)    = ( ) d x φ y ( x)  ε ( x)   

D( x) = b( x)Q

1) Equilibrium

x x  L −1 L  0 b( x ) =  0  0  0 

0

0

x x −1 L L 0 0

x  L −1   0  b( x ) =  0  1  L   0 

 0  0  1 

0

x x −1 L L 0 0

0 0 1 L 0

a b b d  f ( x) =  c e  0 0 0 0

c e  f 

f(x) is section flexibility from fiber section. Axial and Bending Deformations Coupled

0

0

0

1 L

1 L

 0  0  1 0   0 

d ( x) = f ( x ) D ( x)

2) Constitutive Law

a b f ( x) = b d  c e

x L

c e f 0 0

0 0 0 g 0

0 0  0  0 h

f(x) is section flexibility from fiber section and Nonlinear V-γ relation. Axial and Bending Deformations Coupled Shear Deformations Uncoupled

q = FQ

3) Compatibility

F = [5 x 5 ]

F = [5 x 5 ]

F is full 5x5 Element Flexibility Matrix With Coupling Among Axial and Flexural Deformations

F is full 5x5 Element Flexibility Matrix With Coupling Among Axial, Flexural and Shear Deformations

Note: a through h are values yet undefined. They are included to represent non – zero flexibility determined values

Figure IV-7: Comparison of Components in Bernoulli and Timoshenko Element Formulations

62

IV-C) Section V – γ Constitutive Law With the element force based formulation carried out in the previous section, all the necessary components of the element formulation are known except for the V – γ constitutive law. In the present formulation, a generalized, nonlinear V – γ law is assumed, in the form:

V = g (γ )

(IV.9)

where g indicates the nonlinear function. If the material was still in the linearly elastic range, then:

g (γ ) = GAs γ

(IV.10)

IV-C-1) Shape of Shear Hysteretic Law The shear hysteretic law implemented here will be a modified version of the bilinear law proposed by Filippou et al [7]. This law, shown in figure IV-8, is clearly not readily applicable to this formulation because the law relates section moment to shear rotation and the flexibility formulation developed previously requires a law which relates shear force to shear strain.

So, this law will be modified to be

applicable to the procedure developed previously and only the shape will be retained.

M, Moment + My M max M* + Mcr

- Mcr

θ max

θ,Shear Rotation

- My

Figure IV-8: Filippou et al’s Moment – Shear Rotation Hysteretic Law

Referring to figure IV-8, Mcr is the cracking moment of the section, My is the yield moment of the section, Mmax is the maximum moment attained on the previous loading cycle, θmax is the maximum

63

previous rotation attained, and M* is a target moment reduced due to section damage. Damage occurs after the section has yielded. The degraded stiffness is due to the closing of cracks which were opened when the original loading occurred. The curve, upon reloading, aims for the intersection of this value with the cracking moment. After reaching this intersection, there is an increase in stiffness as the flexural cracks have now closed. The value of M* is given by Filippou et al [7] as:

M * = M max exp(

− c1θ max ) θy

(IV.11)

Where θy is the yield rotation of the section and c1 is a factor given by Filippou [7] as 0.2 to 0.35. As mentioned above, this law is applied here as a shear force – shear strain relation. Because only the shape of the curve is being retained, the values can be interchanged to give the relation shown in figure IV-9.

V, Shear Force + Vn

GAs

V max V*

+ Vcr γ max

- Vcr

γ ,Shear Strain

- Vn

Figure IV-9: Filippou et al’s Law Converted to a Shear Force – Shear Strain Relation

Because the shape of the curve is being retained, it can be assumed that V* is related to the maximum previous shear force in the same way that M* was related to the maximum previous moment in equation IV.11 or:

V * = Vmax exp(

− c1γ max ) γy

64

(IV.12)

In figure IV-9, Vy has been replaced by Vn. This is because, for reinforced concrete members, the yield strength of the section is not clearly defined, however the nominal strength of the member is easily determined. Assuming a small strain hardening ratio in the steel, which is the only cause for an increase in shear force past the yield level, Vn is approximately equal to Vy. Also shown in figure IV-9 is the elastic shear stiffness of the member as given by equation IV.3. These values will be defined subsequently. IV-C-2) Theoretical Values of Shear Hysteretic Law Now that the shape of the shear force – shear strain hysteretic law has been determined, values for each of the variables must be defined. The shear strength of a reinforced concrete member will be based on Priestley et al’s equation as given by Xiao et al [18]. The shear str ength of a member is given as:

Vn = Vc + V p + Vs

(IV.13)

In equation IV.13, Vn is the nominal shear strength of the section, Vc is the concrete shear contribution, consisting primarily of aggregate interlock and dowel action of the flexural reinforcement, Vp is the increase in shear capacity through arching action provided by an axial load , and Vs is the shear carried by transverse stirrups through a truss mechanism. It is suggested that Priestley’s equation be multiplied by 0.85 for new design (Xiao et al [18]). The concrete shear contribution is given by:

Vc = k f ' c Ae ( psi )

(IV.14)

where f’c is the compressive strength of the concrete, Ae is the effective area of the section approximated as 80% of the gross concrete area, and the factor k varies with the rotational ductility as shown in figure IV10.

3.5 k, psi units 1.2 1 2 3 4 Rotational Ductility, m Figure IV-10: Variation of Factor k with Rotational Ductility in Equation IV.14 The increase in shear resistance due to an applied axial load is defined as:

65

Vp =

D−a Pu 2 D( M / VD)

(IV.15)

Where Pu is the factored axial load, D is the depth of the member, a is the depth of the flexural compression block, and M and V are the moment and shear demands on the section, respectively. The denominator of equation IV.15 can be approximated by assuming equal end moments in the Timoshenko equilibrium equation in figure IV-7. With this assumption V is:

V=

2M L

(IV.16)

and the denominator reduces to the length of the member, L. This same result can also be achieved by recognizing that M/VD is equal to the aspect ratio of the member which equals L/2D. The shear carried by transverse stirrups through the truss mechanism is:

Vs =

Av f yh D' cot θ s

(IV.17)

Where Av is the cross sectional area of the transverse reinforcement in the section, fyh is the yield stress of the transverse stirrups, s is the spacing of the stirrups, and θ is 30o in Priestley’s approach. The value D’ is the distance between the centers of the peripheral hoop or spiral as shown in figure IV-11.

Figure IV-11: Definition of D’ for Equation IV.17

With the substitution of equations IV.15 through IV.17 into equation IV.14, the nominal shear strength of the section to be used in the shear hysteretic law is found. Next, an expression is needed for the cracking shear force, Vcr. The steel does not contribute to the shear resistance until the section is cracked,

66

so the cracking shear force is a function of only the concrete shear contribution and the increase in shear resistance due to axial load. An assumption will be made that the cracking shear force occurs at the largest rotational ductility value of k, i.e. k is equal to 3.5 from figure IV-10. For this reason, the effect of axial load on the cracking shear force will be neglected and Vcr is:

Vcr = 3.5 f ' c Ae

(IV.18)

With the determination of the nominal shear strength and the cracking shear force, the only value left to determine is the shear force value, V*, at which the stiffness changes upon unloading and reloading. The final hysteretic law which was implemented into FEAP does not include V* directly but handles this value through the definition of pinching parameters. The law will now be described in detail followed by the formulation of the pinching parameters and an explanation of their relation to V*. Referring to figure IV-12, consider a load cycle that begins loading in the positive shear force direction.

The same arguments may be made for loading which begins in the negative shear force

direction. The initial loading curve is bi-linear and the slope is equal to the shear modulus, G, multiplied by the shear area, As, up to the nominal shear value, Vn (curve A-B). As will be taken as 80% of the gross concrete area of the section, and G will be as defined later. Once the shear force exceeds the nominal shear value, there is a decrease in stiffness as the only increase in shear force comes from strain hardening in the transverse steel (curve B-C). (The ACI code [1] recommends that strain hardening be neglected) In case of unloading, the law always unloads parallel to the initial stiffness (curve C-D). Once loading in the positive direction has exceeded the nominal shear value and has completely unloaded (curve A-B-C-D), the reloading in the negative direction exhibits a decrease in shear stiffness as the crack which opened in the positive direction must close (branch D-E). Point E must occur at the negative value of the cracking shear because the value at which the cracks close would be equal and opposite to the force at which they opened. So, D-E aims for the cracking shear value in the negative direction upon first reloading after the nominal shear value has been exceeded in the positive direction. It is seen that the slope of D-E depends on the ratio of cracking shear force to nominal shear force. If the cracking shear force was equal to the nominal shear force the stiffness would be much greater as D-E

67

would aim for the nominal shear value in the negative direction. Therefore, the slope of D-E is equal to the cracking shear force divided by the shear strain value at D minus the shear strain value at E. Since E occurs on a line whose slope is parallel to the initial stiffness, the shear strain value at E is the negative cracking shear force value divide by the initial stiffness. Defining a coordinate system y and x, with axis x parallel to the shear strain axis, and axis y parallel to the shear force axis, it is seen that the pinching in the y direction is equal to Vcr / Vn. From point E, there is a significant increase in stiffness as all the shear cracks have now closed, and the curve rejoins the original elastic stiffness curve (curve E-F). At point F, when the nominal shear resistance in the negative direction is reached, there is again a significant reduction in stiffness as the only strength increase is due to strain hardening (curve F-G). The law then unloads elastically again (curve G-H). From point H, there is a further reduction in stiffness as now cracks have been formed in the positive and negative directions. The slope of the line H-I depends on both of the pinching parameters in the x and y directions. Point I is located at the point at which the line connecting point H to V* intersects the cracking shear value in the positive direction ( see figure IV-9). These values are formulated in the discussion referring to figure IV-13. At point I, the cracks formed from negative and positive loading have closed, and the curve exhibits an increase in stiffness (line I-J), as it aims for the last cycle’s maximum shear force – shear strain excursion, point J. At point J, the curve rejoins the strain hardening curve, line J-K. If reloading occurs during an unloading cycle before unloading has completed, the reloading occurs parallel to the initial stiffness (line L-M-N).

68

V, Shear Force

L,N

B

+ Vn

C,J

K y

M + Vcr H

A E

G

F

I

x D

- Vcr

γ ,Shear Strain

- Vn

Figure IV-12: Points Defining Shear Hysteretic Law

To determine the relation between the pinching parameters in the x and y directions and V*, two extreme cases of the slope of line H-I ( line H-I is defined in figure IV-12 ) can be defined. The first is the condition where V* is equal to a value which causes the line γn – V*, where γn is equal to the shear strain at the value of negative unloading ( equal to point H in figure IV-12), to intersect the unloading curve at the positive cracking shear value, figure IV-13a. The shear strain at this intersection is labeled γ2. The second case is that in which V* is equal to the maximum previous shear force, Vmax, figure IV13b. The value γ1 is defined as the shear strain value of the point at which the γn-V* curve in figure IV-13b intersects the initial loading curve at a height equal to the positive cracking shear force. Other values used in figures IV-13 a-c are the shear strain at positive unloading, γp (point D in figure IV-12 ), the pinching in the y direction, py, the pinching in the x direction, px, the initial shear stiffness, GAs, the maximum previous shear strain, γmax, and the unknown shear strain, γ*, at the point where the reloading curve changes slope ( point I in figure IV-13).

69

γ γ γ

γ

γ

γ

γ

γ

(γ −γ )

γ γ γ∗ γ γ γ

γ

Figure IV-13: Determining the Relationship of the Pinching Parameters to V*

As shown in figure IV-13, for small strain hardening ratios, the shear yield level is approximately equal to the maximum shear force attained. Because of the requirement that the first reloading curve in the negative direction, after loading in the positive direction has exceeded the yield shear value, must intersect the elastic curve at the negative cracking shear force value, the pinching in the y direction is equal to the cracking shear force divided by the yield shear force. With the assumption that the maximum shear force and yield shear force are approximately equal for small strain hardening values, the pinching in the y direction becomes:

py =

Vcr V ≈ cr → Vcr = pyVmax V y Vmax

(IV.19)

This substitution is shown in figure IV-13. From figure IV-13a, it is seen that the intermediate shear strain value, γ2, can be expressed as a function of the previous maximum shear strain, γmax, and the previous maximum shear force, Vmax as:

70

pyVmax GAs

(IV.20a)

Vmax GAs

(IV.20b)

Vmax ( py − 1) GAs

(IV.20c)

γ2 =γ p +

γ p = γ max − γ 2 = γ max +

Considering similar triangles m-n-o and m-q-r in figure IV-13b gives for the intermediate shear strain, γ1:

Vmax pyVmax = γ max − γ n γ 1 − γ n

(IV.21a)

γ 1 = py (γ max − γ n ) + γ n

(IV.21b)

Finally, by considering figure IV-13c, an expression for the pinching parameter in the x direction, px, as a function of the unknown shear strain, γ*, can be found:

γ * = γ 1 + (γ 2 − γ 1 )px px =

γ * −γ 1 γ 2 −γ1

(IV.22a) (IV.22b)

In the numerical implementation of the V – γ law, px is an input parameter from which γ* is determined. So, px must be determined as a function of known variables. The first thing that is noticed, referring to figure IV-13c, is that the difference between the maximum shear strain, γmax, and the intermediate shear strain, γ2, can be expressed as a function of the maximum shear force, Vmax :

γ max − γ 2 =

Vmax V − py max GAs GAs

(IV.23)

The same type of relation can also be written for the difference between the yield shear strain, γy, and the intermediate shear strain value, γ1, which with using the approximation that the yield shear, Vy, is approximately equal to the maximum shear value, Vmax, for small strain hardening becomes:

γ y −γ1 =

Vy GAs

− py

Vmax V V → γ y − γ 1 = max − py max GAs GAs GAs

71

(IV.24)

By substituting equation IV.23 into equation IV.24 and rearranging the terms gives:

γ max − γ y = γ 2 − γ 1

(IV.25)

Equation IV.25 is the denominator in equation IV.22 b. A similar simplification for the numerator is needed. By considering similar triangles present in figure IV-13c, a relation for the unknown shear strain value can be written as:

pyVmax V* = γ max − γ n γ * −γ n

γ* =

(IV.26a)

pyVmax (γ max − γ n ) + γ n V*

(IV.26b)

Now, subtracting γ1, as given by equation IV.21b, from each side gives:

V  γ * −γ 1 = py (γ max − γ n ) max − 1 V* 

(IV.27)

This gives the numerator in equation IV.22b. However the value of shear strain at which the negative unloading curve intersects the zero shear force axis, γn, is unknown. To eliminate this value from the equation, an assumption will be made that the member is symmetric and that the negative unloading shear strain value is equal and opposite to the positive unloading shear strain value, or:

γ p = −γ n

(IV.28)

Substituting equations IV.28 and IV.20b into equation IV.27, along with noting that because the assumption is being made that the yield shear force is approximately equal to the maximum shear force, Vmax / GAs can be replaced by the yield shear strain, γy. The resulting relation is:

V  γ * −γ 1 = py (2γ max − γ y ) max − 1 V* 

(IV.29)

Substituting equation IV.12 for V* into equation IV.29, then substituting equations IV.29 and IV.25 into equation IV.22 gives the relation for the pinching parameter in the x direction, px:

 c γ py (2γ max − γ y ) exp 1 max  γ   y  px = γ max − γ y

72

   − 1    

(IV.30)

The final theoretical values to be implemented into the hysteretic law are values which account for damage in the section due to cyclic loading. These values, as of yet, have not been mentioned, but will be explained in detail here. As a reinforced concrete section is cyclically loaded, it loses the ability to achieve the nominal strength of the section as damage occurs in the concrete and in the bond with the steel. Cracks which have previously opened, lose some of their strength upon reclosing as the cracks will never fully attain their original uncracked configuration. In a cyclic loading lateral force vs. tip displacement curve, this damage has the effect of lowering the shear force achieved under a given shear strain for increased number of cycles of loading, and lowering the ability of the concrete to dissipate energy in the shear hysteresis loops for increased cycles. There are two effects of damage, then, the first effect is to lower the maximum resisting force of the section in the current load cycle, and the other is to narrow the hysteresis loop for the given cycle as the section loses the ability to dissipate energy. The shear law then, should account for these decreases in resistance due to damage. In the hysteretic law implemented into FEAP, both of these damage characteristics have the effect of increasing the previous maximum shear strain used in calculating values mentioned previously. These effects are shown in figure IV-14. Figure IV-14: Illustration of Damage In Shear Cyclic Loading

γ

γ

γ

γ

γ

γ

It is seen in figure IV-14 that the effect of damage is to decrease the area under the hysteresis curve, and therefore the energy dissipated, from that under curve q-r-s-t in the undamaged graph, to that under curve q-r-s in the damaged graph. The maximum shear force attained in this loading cycle is decreased from Vmax in the undamaged graph to V in the damaged graph. These effects combine to increase the maximum shear strain, γmax, attained in the cycle.

73

IV-C-3) Input Values for Actual Sections - Shear Hysteretic Law Now that the theoretical values needed to generate the shear hysteretic law have been defined, values for actual reinforced concrete sections must be determined. The first value to be determined in generating the hysteretic curve is the yield shear force, Vy. It has already been mentioned that here an assumption is being made that the yield shear force is approximately equal to the nominal shear of the section, Vn. The nominal shear of the section is determined from equations IV.13 through IV.17. The next value to be determined is the cracking shear force, Vcr. The cracking shear force is given by equation IV.18. From this value and the nominal shear of the section, the pinching parameter in the y direction, py, is determined from a modified version of equation IV.19 as:

py =

Vcr Vn

(IV.31a)

which upon substitution of the previously derived equations becomes:

py =

3.5 f ' c Ae Vc + V p + V s

(IV.31b)

The shear stiffness for the elastic section of the curve must be determined next. The elastic shear stiffness is the shear modulus of concrete multiplied by the shear area of the section. The shear modulus of concrete is given by:

G=

Ec 2(1 + υ )

(IV.32a)

Where E is the elastic modulus of concrete and υ is Poisson’s ratio. The elastic modulus of concrete can be obtained from ACI [1] as:

E c = 57000 f ' c ( psi )

(IV.32b)

Poisson’s ratio for concrete is hard to define, but Nilson [10] def ines a value of 0.2 as appropriate. The next thing that can be determined is the pinching parameter in the x direction, px. This is determined by equation IV.30. The shear ductility of the section, µshr, is equal to the maximum shear strain, γmax, divided by the yield shear strain, γy, or:

74

µ shr =

γ max γy

(IV.33a)

So, the ability of a section to develop shear ductility will be a function of the ratio of the total shear strain developed to the yield strain of the section. Remembering that concrete is a brittle material, it can be assumed that all the ductility in shear of the section comes from the shear reinforcement. So, the shear force ductility of a section can be approximated as:

µ shr =

Vc + V p + Vs Vc + V p

(IV.33b)

Which is to say that if there is no shear reinforcement, the ductility of the section is equal to unity. This is in agreement with concrete being a brittle material. However, for this ductility equation to be applicable, the section must be designed to yield the shear reinforcement before failing the concrete in shear, otherwise the increase in shear reinforcement would not increase the ductility of the section because the concrete would fail first still causing a brittle failure. From equation IV.33a, γmax is:

γ max = µ shrγ y

(IV.33c)

Substituting equations IV.33 into equation IV.30 yields a useful value for the pinching parameter in the x direction composed of all known section values. The last values which must be determined are parameters which account for damage. As was mentioned above, damage has the effect of lowering the shear force achieved under a given shear strain for increased number of cycles of loading, and lowering the ability of the concrete to dissipate energy. Let d1 be a parameter which accounts for the loss of section strength and d2 be a parameter which accounts for the loss of the ability of the section to dissipate energy. With reference to figure IV-14, it can be seen that if all the concrete was damaged, both the ability of the section to resist shear force and dissipate energy would be reliant on the shear reinforcement. The damage factor d1 is multiplied by the ratio of the difference between the maximum shear strain and the yield shear strain to the yield shear strain of the section. Referring to figure IV-14, the damage factor d2 is multiplied by the ratio of the total energy dissipated in the cycle, curve q-r-s in the damaged curve, to the elastic energy dissipated, curve m-n-o in the undamaged curve. So the new maximum shear strain, γmax, given by the hysteretic law is:

75

 γ max − γ y E dissipated γ max = γ p max 1 + d1 + d2  γy E elastic 

   

(IV.34)

Where γpmax is the maximum shear strain in the previous cycle, Edissipated is the area under curve q-r-s in the damaged curve of figure IV-14, and Eelastic is the area under curve m-n-o in the undamaged curve of figure IV-14. It is seen that the increase in the maximum shear strain due to damage is a function of the ability of the section to develop ductility and to dissipate energy in the inelastic range. As was mentioned above, if all the concrete was damaged, these abilities would be purely up to the shear reinforcement. From this statement the damage factors can be approximated as the ratio of the steel shear resistance to that of the concrete, or:

d1 = d 2 =

Vs Vc + V p

(IV.35)

Qualitatively, if a large part of the shear resistance came from the steel, the damage factors would be large, increasing the maximum shear strain developed in the section, which corresponds to the high ductility of steel. On the other hand, if a large part of the shear resistance was provided by the concrete, the damage factors would be small, relating the maximum shear strain to concrete’s brittle behavior. IV-D) Observations on Element Shear Response Formulation The proposed V – γ law is an easy to implement procedure to introduce nonlinear shear deformations in a Reinforced Concrete beam element. One of the main limitations on the shear element formulation developed here is the uncoupling of the axial – flexural responses from shear response on the section level. As was mentioned early on in the formulation, the responses are coupled at the element level but at the section level the shear response is independent of axial – flexural response. This is definitely not the case in real structures as shear diagonal tension cracks are related to flexural cracks. The flexural cracks form on the tension side of the member, perpendicular to the member axis, where only flexural stresses are present. These cracks propagate into the member and rotate in line with the principle tensile stresses, which are due to both flexure and shear. Clearly, the location, orientation, and extent of the shear diagonal tension cracks depends on the flexural stresses as well as the shear stresses. To account for this, a coupled constitutive law should be implemented. Extension of the fiber section model to include shear

76

deformations is being studied. Such laws, however, may become quite complex, and, most importantly, very time consuming. The main advantages of the approach suggested in this work are its simplicity, ease of implementation, and low computational cost that make it applicable to the study of entire Reinforced Concrete Structures. The piecewise linearity is another limitation of the law implemented. Reinforced concrete is a highly non – linear material even for low loads. The law implemented here assumes a bi – linear relation for the material. This bi – linear relation will not capture the reduction in strength as the member begins to fail in shear as will be shown in the next chapter when the numerical results are compared with test results. While this may have some effect on the accuracy of the law when analyzing cyclic loading, the effect it will have on the Pushover Analysis is negligible because the loads are monotonically increased and there is no deterioration in strength of the member as is exhibited with cyclic loading. The initial stiffness of the member and the maximum shear strength it obtains will determine its Pushover response. In short, the hysteretic law implemented here is sufficient for use with the Pushover Analysis (as will be shown in subsequent chapters), while modifications should be made to better describe the shear cyclic response necessary for dynamic analysis.

77

CHAPTER V NUMERICAL VERIFICATION OF PROPOSED SHEAR MODEL

The shear element formulation derived in chapter IV, along with the shear hysteretic law implemented into FEAP, is verified here by modeling scaled Reinforced Concrete bridge piers tested at the University of California at San Diego, UCSD (Xiao et al [18] ).

The ability of the proposed model to

capture shear dominated response is of interest here, so the accuracy of the shear formulation will be determined with respect to test data for columns failing in shear. Six large scale, short columns were tested at UCSD, three as built columns, and three columns retrofitted with elliptical steel jackets. The two as built columns that failed in shear, labeled test column R-3 and test column R-5, will be considered here. V-A) Column Dimensions and Testing Conditions The test columns were designed and built based on scale models of prototype columns, designed according to mid 1960’s construction practice. Test dimensions for the two as built test columns R-3 and R-5 are shown in figure V-1. The columns were designed to achieve complete fixity at each end, and while this part of the testing apparatus is not shown, the resulting support conditions are shown in the figure. Each of the columns used Grade 60 longitudinal steel, Grade 40 transverse stirrups, and 5 ksi concrete, nominally. The axial load on each column was 114 kips and the aspect ratios ( M/VD in equations IV.15 and IV.16) for column R-3 and R-5 were 2 and 1.5, respectively. These design details are listed in table V1. The nominal strengths for the longitudinal steel, transverse stirrups, and concrete are replaced by the actual tested values in the table. A lateral, cyclic, symmetric load was applied to the top of each column in the push and pull directions. The loading consisted of force control up to the first yield of the longitudinal steel, followed by displacement control in increments of increasing ductility as shown in figure V-2.

78

Figure V-1: Dimensions of Test Columns R-3 and R-5

Table V-1: Experimental Design Properties Column

Concrete Strength (ksi)

Longitudinal Steel Yield (ksi)

Stirrup Yield (ksi)

Aspect Ratio (M/VD)

Axial Load (kips)

R-3

4.95

68.1

47.0

2

114

R-5

4.75

68.1

47.0

1.5

114

µ=3 µ=1

µ = 1.5

Figure V-2: Experimental Loading Procedure

79

µ=2

V-B) Design Shear Strength The design shear resistance of each column will be computed here based on Priestley et al’s equations (equation IV.13 – IV.17). These values are already included in Xiao et al [18], however, because the shear law developed is for use in a design office, the procedure to calculate these values is as important as the values themselves. The experimental report gives no procedure for calculating the values needed in Priestley et al’s equation, so one method will be detailed here and the resulting shear strength found for each column will be compared to the values obtained experimentally. The concrete shear contribution, equation IV.14, and increased shear resistance due to axial load, equation IV.15, for use in Priestley’s equation, equation IV.13, depend on the flexural response of the section in that the concrete shear contribution depends on the rotational ductility of the section and the increased shear resistance due to axial load depends on the depth of the flexural compression block at its ultimate flexural strength. The determination of the rotational ductility of each section requires a Moment – Curvature relation, which, because of the complex longitudinal steel configuration in each column, will be obtained through the utilization of a fiber discretization of the section. Because the rotational ductility requires the definition of the curvature at yield and the curvature at the maximum flexural strength ( i.e. rotational ductility equals curvature at maximum flexural strength divided by curvature at yield), a direct byproduct of this formulation will be the depth of the flexural compression block at the sections ultimate flexural strength. The fiber discretization divides the section into concrete fibers and steel fibers as shown in figure V-3. As can be seen, the section is divided into twenty concrete fibers and eight steel fibers. The location of the centroid of the concrete or steel fiber is measured from the top of the section as denoted by yci or ysi for the concrete or steel fiber, respectively, with i ranging from 1 to 20 for concrete and 1 to 8 for steel. The areas of each concrete fiber are equal to the total area of the section divided by the number of concrete fibers, denoted by Ac in figure V-3, and the steel areas are equal to five #6 bars for As1, and two #6 bars for As2. Also shown in figure V-3 is the qualitative strain distribution assumed in the section. In the figure, εtop is equal to the strain at the top of the section, εci is equal to the strain in the concrete fiber i,

εsi is equal to the strain in the steel fiber i, c equals the location of the neutral axis, and κ is the curvature of the section.

80

ε

ε κ ε

Figure V-3: Section Fiber Discretization and Strain Distribution

By specifying the strain in the top of the section, and considering tensile strains negative, the strain at any yci or ysi location can be found by similar triangles.

ε ci = ε si =

ε top (c − y ci ) c

ε top (c − y si ) c

(V.1a)

(V.1b)

The curvature of the section can be related to the top strain and location of the neutral axis by:

κ=

ε top c

(V.2)

Where all values in equations V.1 and V.2 are as defined above. The only parameter left unavailable is the location of the neutral axis, c, given a strain at the top of the section. This value can be found through equilibrium. First, however, the stress at a given concrete or steel section must be defined. The stress at a concrete section, given its strain value, will be determined through the Saenz concrete law with a linear tensile portion included. The law defines a stress – strain relation as:

81

   σ = E0 ε *   ε  1 + A  εc 

   1  2 3  ε  ε    + B  + C      εc   εc  

(V.3a)

 ε  A = C + E 0  c  − 2  f 'c 

(V.3b)

B = 1 − 2C

(V.3c)

 f 'c   −1   ε cf ε  f cf C = E0 c  − 2 f 'c   ε cf   εc − 1    ε    c

(V.3d)

In equations V.3, E0 is equal to the initial elastic stiffness of concrete which is given by ACI [1] and was detailed in equation IV.32b, f’c is the compressive strength of concrete, fcf is equal to 85% of f’c, εcf is equal to 0.0035, εc is equal to 0.0025, and ε is the strain in the concrete section under consideration. The tensile portion of the law is linear elastic with stiffness equal to the initial stiffness of the concrete up to the concrete rupture stress, fr.

The rupture stress is given by ACI [1] as:

f r = 7.5 f 'c ( psi )

(V.4)

The Saenz concrete law shown in figure V-4. For the steel stress – strain relation, an elastic perfectly

Normalized Stress f / f'c

plastic law is used with initial stiffness equal to 29000 ksi.

1.2 1 0.8 0.6 0.4 0.2 0 -0.2 -0.002

0

0.002

0.004

0.006

Strain (in/in) Figure V-4: Normalized Saenz Concrete Curve

82

Now that the concrete stress in a fiber can be found from its strain based on the above curve, and the steel stress can be found from its elastic perfectly plastic relation, the force in each fiber can be found by multiplying its stress by its area. The section resisting moment can be found with respect to its centroid by multiplying the force in each fiber by that fiber’s centroidal distance to the section centroid and summing these moments over all the concrete and steel fibers or:

M sec =

numfib

∑σ i =1

i

D  Ai  − y i  2 

(V.5)

Where D is the section depth, equal to 24” in this case. To find the section neutral axis location, c, the resulting forces must add up to the applied axial load or 114 kips in this case. With this requirement, c relating to the implied εtop can be found giving the moment – curvature relation at this strain. By applying this procedure to many different implied εtop values, the complete moment curvature relation for the section can be found. The moment value obtained when εtop is equal to εcu, the crushing strain of the concrete (εcu is equal to 0.005 from the experimental data ), is the maximum resisting moment of the section. From the neutral axis depth at this maximum moment value, the depth, a, of the flexural compression block can be found from:

a = β 1c

(V.6)

Where β1 varies with concrete compressive strength, f’c , (Nilson [11]). Applying this procedure to both test columns R-3 and R-5, and varying the top strain of the section, yields the complete Moment – Curvature responses for both sections. These responses are shown for column R-3 and

R-5

in

figures

V-5

83

and

V-6

respectively.

8000 Area B 7000

Moment (kip - in)

6000 Area A 5000 Moment - Curvature 4000

Equivalent Elastic Perfectly Plastic

3000 2000

κ yield = 0.000245 (rad / in)

1000

κ at max M = 0.000575 (rad / in)

0 0

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

0.0007

0.0008

0.0009

Curvature (rad/in)

Figure V-5: Moment – Curvature Response for Column R-3

8000 Area B 7000

Moment (kip - in)

6000 Area A 5000 Moment - Curvature 4000

Equivalent Elastic Perfectly Plastic

3000 2000

κ yield = 0.000245 (rad / in)

1000

κ at max M = 0.000559 (rad / in)

0 0

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

0.0007

0.0008

Curvature (rad/in)

Figure V-6: Moment – Curvature Response for Column R-5

84

0.0009

When looking at figures V-5 and V-6, one instantly notices that they are almost identical. This is because the two sections are nearly identical, the only difference being a small change in concrete compressive strength. However, column R-5 has an aspect ratio of 1.5 compared to 2 for column R-3. This is used in the calculation of the increase in shear strength due to an applied axial load and will have a bearing on the shear strength of the section. Also detailed in the figures are the equivalent elastic perfectly plastic responses for each section used to calculate the rotational ductility. The elastic perfectly plastic response is based on equal energy absorption. The areas between the actual Moment – Curvature response and the equivalent response, labeled Area A and Area B, must be equal. With this restriction, the rotational ductility of the section can be found by dividing the curvature associated with the maximum moment of the section by that associated with the yield moment of the equivalent system. As mentioned previously, the depth of the concrete compression block at the ultimate strength of the section, corresponding to the point of maximum curvature (corresponding to εcu = 0.005 from experimental data), is a natural by – product of this formulation. These values are shown in the following tables. Now that the rotational ductility and depth of the concrete compression block have been found for each section, the shear strength of the section can be calculated based on Priestley et al’s equation. Priestley et al’s equation (equation IV.13) is repeated here for convenience:

Vn = Vc + V p + Vs

(V.7)

Where Vn is equal to the nominal shear strength of the section, Vc is the concrete shear contribution, consisting primarily of aggregate interlock and dowel action of the flexural reinforcement, Vp is the increase in shear capacity through arching action provided by an axial load , and Vs is the shear carried by transverse stirrups through a truss mechanism. The concrete shear contribution is given by equation IV.14, repeated here as:

Vc = k f ' c Ae

(V.8)

Where f’ c is the compressive strength of the concrete, Ae is the effective area of the section approximated as 80% of the gross concrete area, and the factor k varies with rotational ductility as shown in figure IV-10. The rotational ductility, µ, k factor, concrete compressive strength, f’c , and effective area for each section

85

are shown in table V-2. With these values known, the concrete contribution to shear resistance, also shown in table V-2, can be calculated for each column.

Table V-2: Calculated Concrete Shear Contribution

µ

Column

f’c (psi)

R-3

4950

2.34

R-5

4750

2.28

Ae (in2)

Vc (kips)

3.11

307.2

67.2

3.18

307.2

67.3

k

The increase in shear resistance due to an applied axial load is defined in equation IV.15 as:

Vp =

D−a Pu 2 D( M / VD)

(V.9)

Where Pu is the applied axial load, D is the depth of the member, and a is the depth of the flexural compression block. The value of D for each column is shown in figure V-1. The axial load for each column is 114 kips and the aspect ration, M/VD, of columns R-3 and R-5 are 2 and 1.5 respectively. The depth of the flexural compression block is determined from the Moment – Curvature relation as described above. These values are shown in table V-3.

Table V-3: Shear Strength Due to Applied Axial Load

Column

D (in)

R-3

24

4.94

R-5

24

5.14

a (in)

P (kips)

Vp (kips)

2.0

114

22.6

1.5

114

29.9

M / VD

Lastly, the shear contribution of the transverse steel can be determined. This was given in equation IV.17 as:

Vs =

Av f yh D' cot θ s

86

(V.10)

The value Av is equal to the area of the transverse reinforcement, fyh is the yielding stress of the transverse reinforcement, D’ is the distance between the centers of the peripheral hoop as shown in figure V-1, s is the spacing of the transverse reinforcement, and θ is equal to 30o for Priestley’s equations. These values are shown in table V-4.

Table V-4: Transverse Steel Shear Contribution Column

Av (in2)

R-3 R-5

fyh (ksi)

D’ (in)

s (in)

Vs (kips)

0.098

47.0

22.0

5.0

35.2

0.098

47.0

22.0

5.0

35.2

Now that all the individual values have been determined, they can be added to obtain the nominal shear strength for each column. These values can then be compared to the experimental values. The calculated shear strengths and the comparisons with the experimental values are shown in table V-5. In the figure, Vcexp is the experimental concrete shear contribution, Vsexp is the experimental steel contribution, and Vtexp is the total experimental shear strength of each column.

Table V-5: Numerical and Experimental Shear Strengths Col

Vs (kips)

Vc + Vp (kips)

Vn (kips)

Vsexp (kips)

Vcexp (kips)

Vtexp (kips)

Vtexp / Vn

R-3

35.2

89.8

125

34.9

106.1

141

1.13

R-5

35.2

97.2

132.4

40.7

127.3

168

1.27

Table V-5 clearly shows that the calculation procedure is conservative and underestimates the nominal shear strength of the section with an average Vtexp / Vn of 1.2. One reason for the discrepancies is that the stress – strain relation did not take into consideration the confining effect of the transverse reinforcement. For a more accurate outcome, a stress – strain law for confined concrete could be used. Another reason for the discrepancies could be the determination of the rotational ductility since the equal area method is approximate. However, in comparison with other procedures as given by Xiao et al, the calculated values

87

are very close to the experimental values. Other procedures are given by various codes and the average Vtexp / Vn varies from 1.4 to 2.03 (The values calculated above were based on the Moment – Curvature relation which utilized a concrete crushing strain of 0.005 in/in based on the experimental data). In new design, an assumed εcu of 0.003, and the 0.85 factor mentioned in chapter IV, would further increase the conservatism of the formulation. V-C) Numerical vs. Experimental Column Response Now that the shear strength of each column has been calculated, the numerical response history based on the implemented hysteretic law can be obtained. Some additional parameters must first be calculated from the shear strength equation components. The first thing that needs to be calculated is the cracking shear force, Vcr. This was given by equation IV.18 as:

Vcr = 3.5 f 'c Ae

(V.11)

With the calculation of the cracking shear force, the remaining values for the hysteretic law can be determined. These are the pinching parameter in the y direction, py, the initial shear stiffness, GAe , the pinching parameter in the x direction, px, which is a function of the shear ductility, µshr ,and the damage factors, d1 and d2. These parameters were developed in chapter IV and are detailed in equations IV.30 to IV.35. The resulting values for columns R-3 and R-5 are shown in table V-6.

Table V-6: Hysteretic Input Values – Column R-3 and R-5 R-3

R-5

f’c (ksi)

4.95

4.75

Vn (kips)

125

132

Vcr (kips)

75.6

74.1

GAe (kips)

513318

502848

py

0.605

0.560

µshr

1.39

1.36

px

0.885

0.836

d1

0.392

0.362

d2

0.392

0.362

88

With the parameters for the shear hysteretic law developed, the numerical analysis can be performed on both columns for comparison with the experimental data. The loading procedure used in the numerical analysis will be a little different than that used in the experimental procedure because of the limitations of the FEAP proportional loading command. The program does not allow a change from displacement control to force control in the same analysis. Further, because the degradation of shear resistance depends on the damage parameters and the pinching parameters, and these depend on the maximum previous shear strain, there is no degradation for repeated loading cycles with the same ductility value. So, only one cycle will be performed at each ductility level. The loading history used in the numerical analysis is shown in figure V-7.

∆

µ = 1 µ = 1.5

µ=2

Figure V-7: Loading Procedure for Numerical Analysis

The results of the Numerical Analysis for columns R-3 and R-5 are shown in figure V-8 and V-9 respectively.

It can be seen from these figures that the numerical response from the hysteretic law

implemented matches the experimental results well in capturing the initial stiffness and the maximum shear resistance of the columns, however there are large discrepancies between the experimental and numerical results once the peak lateral force has been exceeded. Namely, the hysteretic law implemented is not capable of capturing the softening effect (from the opening of large shear diagonal cracks) apparent in the experimental data. Also, the combination of increased pinching and reduced shear resistance once the peak lateral force has been exceeded is missed by the numerical analysis. While the overall shape of the response is satisfactory, some important response aspects are neglected. The question, however, arises as to whether the law can be used accurately for the Pushover Analysis.

89

200 Experimental

150

Numerical

Lateral Force (kips)

100 50 0 -50 -100 -150 -2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Top Displacement (in)

Figure V-8: Numerical and Experimental Results – Column R-3

200 150

Experimental Numerical

Lateral Force (kips)

100 50 0 -50 -100 -150 -200 -1.5

-1

-0.5

0

0.5

1

1.5

Top Displacement (in)

Figure V-9: Numerical and Experimental Results – Column R-5

90

To determine how well the law implemented applies to members subject to monotonically increased loads such as those defined by the Pushover Analysis, this type of loading will be applied to both column R-3 and R-5 for comparison to the experimental result envelopes. The resulting plots are shown in figure V-10 and V-11 for columns R-3 and R-5 respectively. 200 Experimental

150

Numerical

Lateral Force (kips)

100 50 0 -50 -100 -150 -2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Top Displacement (in)

Figure V-10: Monotonically Increasing Loads and Experimental Results – Column R-3 200 150

Experimental Numerical

Lateral Force (kips)

100 50 0 -50 -100 -150 -200 -1.5

-1

-0.5

0

0.5

1

1.5

2

Top Displacement (in)

Figure V-11: Monotonically Increasing Loads and Experimental Results – Column R-5

91

From figures V-10 and V-11, it can be seen that while the hysteretic law implemented misses some important information in comparison with the experimental results for cyclic loads, it is accurate enough for the Pushover Analysis as it captures the initial stiffness of the structure and the maximum resisting shear that was calculated above with the underestimation shown in table V-5. However, one thing to note is that the displacement controlled loading shown in the figures could have increased forever without the section ever losing its shear resistance. This is a definite draw back of the hysteretic law, however since the shear ductility was calculated above, the engineer has a relation to obtain the maximum displacement the structure would be able to undergo. Mentioned in chapter IV was the fact that the shear deformations are uncoupled with the axial – flexural deformations on the section level, but are coupled on the element level. Since the moment and shear responses are linked through equilibrium on the element level, the maximum strength available in the section will depend on the minimum of the individual components, i.e. axial – flexural or shear. This is illustrated by plotting the Moment – Curvature response and the Shear Force – Shear Strain response for column R-3 at different sections. For the numerical analysis, both columns were divided into five Guass – Lobatto integration points as shown in figure V-12. Using the cyclic load analysis results for column R-3, it is seen in figure V-14 that the Moment – Curvature response of the sections one and two vary slightly corresponding to the linear moment distribution along the element. From figure V-13 it is seen that the Shear Force – Shear Strain response of sections one and two are equal corresponding to constant shear along the element. Further it is seen that the shear response of the element controls the total response as a clear yielding point is recognized while the Moment – Curvature response remains basically elastic. This corresponds to the column failing in shear which agrees with the experimental data. So, while the axial – flexural and shear responses are uncoupled at the section level, coupling (through equilibrium) at the element level ensures that the element response will be controlled by the minimum of the individual components. This corresponds to reality in that the maximum strength of a member will depend on its weakest component.

92

Figure V-12: Section Locations of Columns for Numerical Analysis

Section 1 Shear Force, V

150

150 Section 2 Shear Force, V

100

100

50 0 -50

-100 -150 -0.02

50 0 -50

-100

-0.01 0 0.01 Section 1 Shear Strain, γ

-150 -0.02

0.02

-0.01 0 0.01 Section 2 Shear Strain, γ

0.02

Figure V-13: Shear Force vs. Shear Strain Column R-3, Sections 1 and 2

5000 4000 3000 2000 1000 0 -1000 -2000 -3000 -4000 -5000 -0.0002

Section 2 Moment, M

Section 1 Moment, M

8000 6000 4000 2000 0 -2000 -4000 -6000 -8000 -0.0004

-0.0002 0 0.0002 Section 1 Curvature, κ

0.0004

-0.0001 0 0.0001 Section 2 Curvature, κ

Figure V-14: Moment vs. Curvature Column R-3, Sections 1 and 2

93

0.0002

V-D) Conclusions The accuracy of the shear law, presented in chapter IV, was determined through the comparison of numerical results and experimental results of columns failing in shear tested at the University of California at San Diego. The use of Priestley et al’s [18] equ ations for the determination of the initial parameters required for the hysteretic law required the quantification of the section rotational ductility and compression block depth at the ultimate strength of each column section. One method to accomplish this was detailed in that the complete Moment – Curvature response of each column was obtained. Once these necessary values were obtained, the numerical analysis could be performed.

It was shown that the

numerical analysis captures the initial stiffness and maximum lateral force resistance of each section, but under cyclic loading the deterioration of the lateral force resistance and the increase in section pinching is missed. It was further shown, that while this law is not appropriate for cyclic loads for the reasons mentioned above, for monotonically increasing loads, such as those enforced by the Pushover Analysis, it is accurate. Also, though the shear response of the element is uncoupled from the axial – flexural response on the section level, the coupling of these responses on the element level causes element response to be controlled by the weakest of the components.

94

CHAPTER VI SHEAR WALL EXAMPLE

In the previous chapters, the Pushover Analysis Procedure was explained, Applications of the Pushover Analysis were illustrated, a relation for shear deformations in Reinforced Concrete was developed, and the accuracy of this relation was determined. In this chapter, all of the previous information will be combined to perform the Pushover Analysis on an actual shear wall structure under construction. A complete non linear dynamic analysis will be performed on the structure and the results of the two analyses will be compared. Lastly, the nominal shear strength of the wall will be reduced to illustrate the coupling of flexural and shear response at the element level to satisfy equilibrium.

VI-A) Wall Configuration The wall under consideration is part of the lateral force resisting system in a nine story reinforced concrete building. The wall to be analyzed is shown in figure VI-1. Also shown in figure VI-1 are the fundamental, second, and third periods for the wall. The first thing that one notices is that the fundamental period of the structure is much longer than the second and third periods, and it is smaller than the fundamental period of the six story frame analyzed in chapter III. With this and arguments developed in previous chapters in mind, close agreement between the Pushover and Dynamic Analyses is expected. Another issue to point out is the calculation of the mass used in the determination of these periods. In chapter III, when the periods for the three moment resisting frames were calculated, the total dead load and the applicable live load made up the mass on the frame and also made up the gravity load applied to the frame. However, in the case of a lateral force resisting system composed of structural walls, columns often assist in carrying the gravity loads, while the structural walls must carry the entire lateral force. So, the entire mass of the building is divided among the structural walls, while the gravity loads on the structure are divided among the structural walls and columns according to tributary width. In this way, the mass allocated to a particular wall can be different from the vertical loads it must support divided by the gravitational constant.

95

76' - 6"

67' - 4"

58' - 2"

49' - 0"

39' - 10"

30' - 8"

20' - 8"

10' - 8"

0' - 0"

-9' - 0"

Fundamental Period (s)

Second Period (s)

Third Period (s)

0.973

0.187

0.140

Figure VI-1: Dimensions of Structural Wall With Periods Shown

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VI-B) Performing the Pushover Analysis on the Shear Wall The structural wall analyzed here will be subject to the same Pushover Analysis and non linear dynamic analysis conditions as the frames analyzed in chapter III. That is, the wall will be analyzed as if it were located in the Los Angeles, California area, on stiff soil, with five percent structural damping, and a 10% probability of exceedence in 50 year event will be considered. With these conditions, and the fundamental period shown in figure VI-1, the vertical distribution of lateral loads for use in the Pushover analysis can be obtained as detailed in equation III.11 and in table III-6. The gravity loads applicable to the structure can be obtained from equation III.10 where the typical dead load is 125 psf for all floors and the typical live load is 50 psf for the first three floors, and 40 psf for floors four through to the roof. The self weight of the wall must also be included in the dead load. The loads to be applied during the Pushover Analysis are shown in figure VI-2.

Figure VI-2: Gravity Loads and Cvx Loading Coefficients – Shear Wall

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With the above loads determined, the Pushover Analysis, as explained in chapter II and detailed in chapter III, can be performed on the shear wall. The results of this analysis are shown in figure VI-3. It can be seen from the figure that the Target Displacement is 14 in. for a Base Shear Yield Value, Vy, of 325 kips. Also seen in the figure are Ki = 178 kips /in, and Ke = 142 kips /in.

400 Ki = 178 kip / in 350 αKe = 2.7 kip / in

300 Base Shear (kips)

Original Curve 250 Ke = 142 kip / in 200 Vy = 325 k

150

0.6 Vy = 195 k

100

δt = 14 in 50 0 0

2

4

6

8

10

12

14

16

18

Roof Displacement (in)

Figure VI-3: Base Shear vs. Roof Displacement with Target Displacement for Shear Wall

VI-C) Complete Non – Linear Dynamic Analysis for the Shear Wall The full non – linear dynamic analysis will now be run on the shear wall. The dynamic loading will be the same El Centro ground motion that was used in chapter III and illustrated in figure III-17. To be consistent with the Pushover Analysis, 5% structural damping will be considered for the shear wall. The Rayleigh damping parameters will be determined from the first and second periods of the shear wall shown in figure VI-1 using the relations of equations III.14.

The time history of the roof displacement for the

structural wall under consideration is shown if figure VI-4. It is seen from looking at the maximum and minimum displacements at the roof of the structure, that this result matches well with that predicted from the Pushover Analysis. This agreement was foreseen from the fact that the fundamental period of this structure was much greater than the second and third periods. It was also noted that this structure’s fundamental period was shorter than that corresponding to the six story frame analyzed in chapter III. This enforces once again the accuracy of the Pushover Analysis for shorter period structures.

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Another interesting thing to note about figure VI-4 is that the structure’s roof displacement plot shows permanent deformation. After the minimum displacement has been reached, the structure no longer oscillates about the zero displacement axis, but about its permanent deformation. The structure under consideration was designed for a Denver, CO site which falls under UBC earthquake zone 1. Obviously it was not designed for these high seismic loads and that is apparent in its poor structural response.

10 Max Disp = 5.88 in

Roof Displacement (in)

5

0 0

2

4

6

8

10

12

14

16

18

-5

-10

-15 Min Disp = -14.3 in -20 Time (s)

Figure VI-4: Dynamic Roof Displacement for Shear Wall

VI-D) Comparisons of Pushover and Dynamic Analyses Just as the reasons for the well corresponding results for the six story frame analysis, or the poorly matching results for the twelve and twenty story frame analyses performed in chapter III were quantified, the results obtained for the structural wall will now be explained. It has already been mentioned that the Target displacement determined from the Pushover Analysis was expected to match the maximum or minimum dynamic roof displacement well because of the short fundamental period of the structure and the large difference between the first and second periods of the structure. This can further be understood by looking at the frequency content of the ground motion used as shown in figure III-25. It is seen that the fundamental period of this wall lies directly in the range of maximum amplification from the ground motion.

Also, the second and third periods being as short as they are fall in a range of reduced

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APPENDIX I MODIFICATIONS TO PROGRAM FEAP TO PERFORM NON-LINEAR PUSHOVER ANALYSIS

In the preceding chapters the procedure to perform the Pushover Analysis was explained and the formulations to include shear deformations in this analysis were illustrated.

In this chapter, the

modifications made to the Finite Element Analysis Program (FEAP), developed by Professor Taylor at the University of California at Berkeley [16], to accomplish these tasks will be explained.

To perform the

Pushover Analysis as described in chapter II, FEAP must be able to monotonically increase a specified vertical distribution of lateral loads, plot structural base shear vs. roof displacement, and calculate the target displacement based on the seismic event. In chapter IV, the shear element formulation, including the shear hysteretic law which was added to FEAP, was thoroughly detailed.

In chapter V, a procedure for

determining the values needed for the V – γ constitutive relation was outlined. In this chapter the input commands to bring the law and section values together will be discussed.

AI-A) FEAP Pushover Routines

The Pushover Analysis is performed in FEAP through two user commands. These are the mesh command ‘PUSH’ and the macro execution command ‘VVSD’. The ‘PUSH’ mesh command reads all of the input variables necessary to perform the analysis. The ‘VVSD’ macro command reads the element data from the element files specified with ‘PUSH’ and created with the existing frame analysis command ‘SENS’. The commands ‘PUSH’ and ‘VVSD’ are explained in detail below.

AI-A-1) ‘PUSH’ Mesh Command

The new mesh command, ‘PUSH’, will be described in detail with reference to the six story moment resisting frame analyzed in chapter III. The element and node numbers for this frame shown in figure III-12 are reproduced here as figure AI-1. The Pushover Analysis depends on the base shear vs. roof displacement plot for the structure. The roof displacement for this six story frame will be measured at the left hand side of the structure, node 7 in figure AI-1. Since each element output file created by the

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command ‘SENS’ includes relative displacements, the roof displacement is the sum of the columns, elements 1 through 6, on the left side of the structure. The support conditions are not shown in the figure, but they are complete fixity at nodes 1, 8, and 15. The base shear for the structure is the sum of the shear forces in the elements which are restrained, elements 1, 7, and 13 in figure AI-1. In the following description, those elements used to calculate roof displacement, i.e. elements 1 through 6 in this example, will be referred to as displacement elements. The elements used to calculate base shear, i.e. elements 1, 7,

7

14

21

6

13

20

5

12

19

4

11

18

3

10

17

2

9

16

1

8

15

and 13, will be referred to as base shear elements. Figure AI-1: Element and Node Numbers – Six Story Frame

FEAP commands located in the input file are always four letter command names. In this case that is ‘PUSH’. This is followed by the command parameters. For this mesh command, five lines of input commands are required.

The first line has three parameters. They are the number of displacement

elements to read, the displacement element generation, and the displacement element increment. The second line consists of the displacement element numbers to read. This line could have one parameter or many parameters depending on whether generation is used or not. If generation is used, i.e. the second parameter in the first line of commands is equal to 1, then only the first displacement element to be read is listed. If generation is not used, i.e. the second parameter in the first line of commands is equal to 0, all displacement elements to be read are listed.

The third and fourth lines are the base shear element

counterparts of the first and second line for the displacement elements. Line three has the number of base shear elements to read, the base shear element generation, and the base shear element generation increment

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as its parameters, and line four has the element numbers to read for the base shear calculation if generation is not used, and has the first base shear element to read if generation is used. Command line five has the seven independent values necessary to compute the Target Displacement. These are the base shear yield value, Vy, the spectral response acceleration, Sa, the modification factor that relates spectral displacement and likely building roof displacement, C0, the ratio of elastic strength demand to calculated yield strength coefficient, R, the characteristic period of the response spectrum, T0, the modification factor that represents the effect of hysteresis shape on the maximum displacement response of the structure, C2, and the gravitational constant, g. These values were thoroughly detailed in chapter two. The remaining values necessary to compute the Target Displacement are functions of these values and are calculated internally. The gravitational constant must be input because units may change for each problem being analyzed. The command lines are shown in figure AI-2.

push # of displacement elements

displacement element generation

displacement element generation increment

displacement element numbers # of base shear elements

base shear element generation

base shear element generation increment

base shear element numbers Vy

Sa

C0

R

T0

C2

g

Figure AI-2: Input Parameters for Mesh Command ‘PUSH’

To clarify the displacement element and base shear element input commands the parameters for these input commands will be listed for the six story frame example. These are shown in figure AI-3 for both the case of generation and no generation. The values for the fifth command line for the mesh command ‘PUSH’, i.e. the independent parameters for the calculation of the Target Displacement, are listed in table III-9 and only their variable values are listed here.

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push 6 0 0 123456 3 0 0 1 7 13 Vy Sa C0 R T0 C2 g

push 6 1 1 1 3 1 6 1 Vy Sa C0 R T0 C2 g

Generation Not Used

With Generation

Figure AI-3: ‘PUSH’ Input for Six Story Frame Example

It is seen from figure AI-3 that for the case when generation is not used, the second and third parameters in the first command row and the third command row are zero. The first zero in each of these rows specifies that generation will not be used and therefore, the third parameter in each of these rows, which is the generation increment, is also zero. When specifying no generation, all elements that are to be read must be entered. In the case with generation, the second parameter in the first and third rows are set to one. Because of this, the generation increment must be specified. In the displacement element case, the increment is equal to one and the first element to be read is equal to one. The elements to be read subsequently are generated according to the increment and the total number of elements specified to be read. In the base shear element case, the generation increment is equal to six and the first element is equal to one. With these parameters, along with the total number of base shear elements being equal to six, the base shear elements are generated as elements 1, 7, and 13. Because the element responses are read from the element files created by mesh command ‘SENS’, the base shear and displacement elements to be read must be specified in this file. Also, the ‘PUSH’ command must be located somewhere after the ‘SENS’ command. If either of these two conditions are not met, the FEAP program stops with an error code “ERROR: elements in ‘PUSH’ for displacement must be included in ‘SENS’ ” if a displacement element file can not be found or “ERROR: elements in ‘PUSH’ for shear must be included in ‘SENS’ ” if a base shear element file can not be found. The command line number five for ‘PUSH’ must include exactly seven parameters with no zeros. If this condition is not met, the FEAP program stops with an error code “ERROR: Not enough values in line five of mesh command ‘PUSH’ (zero values not allowed)”.

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AI-A-2) ‘VVSD’ Macro Command

Once the parameters have been input using the ‘PUSH’ command, and the Pushover Analysis has been run on the structure, the Target Displacement is determined using the ‘VVSD’ macro command. This command reads the data from the files specified in ‘PUSH’ and created by ‘SENS’. The data written to the element files by the mesh command ‘SENS’ include time, forces and displacements in the global x, y, and z directions, and rotations and moments in the global x, y, and z directions. So, the relative displacement or shear force for each element can be read and these are added to get the total response. In this manner, the total base shear vs. roof displacement plot for the structure can be determined. From this base shear vs. roof displacement plot, the initial stiffness, Ki, and effective stiffness, Ke, along with the post yield stiffness,

αKe, of the structure can be determined. These values are illustrated in figure II-10, and the procedure used to calculate them from the ‘VVSD’ macro command will be described below. The initial stiffness, Ki, of the structure can be found from the stiffness of the base shear vs. roof displacement curve at the start of the analysis. By dividing the base shear value at a particular instant by the corresponding roof displacement at that instant, the initial stiffness of the structure is found. Because the initial loading may be caused by gravity loads and the initial load increment may be extremely small, the first ten values of base shear and roof displacement are bypassed and the average of the next 15 values of base shear divided by roof displacement are taken as the initial stiffness. If the total number of analysis pseudo time steps is less than the required 24 steps to determine the initial stiffness by the previously mentioned method, the last value of base shear divided by the last value of roof displacement is taken as the initial stiffness. It should be noted that if the total number of analysis time steps is less than 24, there may be errors in the computed initial stiffness.

The initial stiffness as calculated above can be written

symbolically as:

if n ≥ 24 then K i =

1 24 Vbi ∑ 15 i =10 ∆ri

(AI.1a)

Vb(n) ∆r (n)

(AI.1b)

if n < 24 then K i =

where Vb is the base shear value, ∆r is the roof displacement, and n is the number of analysis steps.

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The effective stiffness, Ke, of the structure is calculated based on the base shear yield value, Vy, which was input in the mesh command ‘PUSH’. As shown in figure AI-4, the slope of the line which intersects the original base shear vs. roof displacement plot at 0.6Vy is the effective stiffness. So, the effective stiffness of the structure is calculated by locating the displacement value corresponding to the base shear equal to 0.6Vy , ∆r6, and dividing 0.6Vy by this value or:

Ke =

0.6V y

(AI.2)

∆r6

Referring to figure AI-4, the post yield stiffness, αKe, is determined by dividing the maximum base shear achieved, Vbmax, minus the base shear yield value, Vy, by the maximum roof displacement achieved, ∆rmax, minus the roof displacement at yield, ∆ry, or:

αK e =

Vbmax − V y

(AI.3)

∆rmax − ∆ry Ki

Base Shear

αK

Vbmax Vy

Non – Linear Structural Response

0.6 Vy

Ke ∆r

∆r max

∆r y Roof Displacement

Figure AI-4: Values to Define Ke and αKe

Because the calculations mentioned above require the input of the base shear yield value, Vy, and this value is not known at the start of the analysis, the ‘VVSD’ macro command can include three factors as input. These factors are multiplied to the base shear yield value input in the mesh command ‘PUSH’, allowing the analysis of four separate choices of Vy at once (including the one input with the ‘PUSH’ mesh command). To clarify this, the ‘VVSD’ macro command is shown in a typical macro solution statement in figure AI-5.

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macro nopri tang mass,lump subspace,print,1 prop,,1,2 dt,,1 loop,time,153 loop,step,10 tang,,1 next,step stre,all time next,time VvsD,,120/140,100/140,152/140 end

Figure AI-5: Typical Macro Solution Routine Using the ‘VVSD’ Macro Command

Shown in figure AI-5 are the factors used with the ‘VVSD’ macro command when the comparison was made in chapter III of different choices in Vy and the effect that had on target displacement, figure III-25. The original ‘PUSH’ input of Vy was equal to 140. Three other values, 120, 100, and 152, were analyzed at the same time. In this way, four different results were obtained with one analysis. The solution routine in figure AI-5 shows some important properties of using the ‘VVSD’ command. First, because the Target Displacement is a function of the initial elastic period of the structure, the period determination must be included in the solution routine ( tang; mass,lump; subspace,print,1). If this is left out of the routine FEAP execution discontinues and an error message is displayed reading “ERROR: in ‘pushover’ (VvsD) fundamental frequency = 0 must have subspace solution scheme in macro”.

The other thing to notice is that the ‘VVSD’ command is included after the time loop is

completed ( loop,time,153; … next,time). This is where it must be located otherwise it will compute at discrete time intervals instead of considering the whole base shear vs. roof displacement plot. The output from the ‘VVSD’ command includes all the data necessary to generate the plots of figures III-14 to III-16. The data is output into files named ‘pushover1’, ’pushover2’, ‘pushover3’, and ‘pushover4’ if all the available choices for Vy are utilized. There are as many files created as choices in Vy input ( with a maximum of four). The data is output in a format which is easily opened with excel. Upon

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opening the output files in excel, the graphs mentioned above can be created by merely selecting columns A through G. Also located in the output files are the calculated values of the base shear yield value, the effective period, the initial, effective, and post yield stiffnesses, and the Target Displacement.

AI-B) FEAP Shear Element Modifications The shear element formulation, in which a V – γ relation was created for implementation at the section level, was developed in chapter IV. Also defined in that chapter were the values needed as input for the hysteretic law. In chapter V, a procedure was illustrated for determining these values for a given Reinforced Concrete Section. Here, the instructions for entering these values into the FEAP input file are given. For each element, a material type is assigned based on the previously existing mesh command, ‘MATE’. From this command the input parameters representing the one dimensional material laws for shear, located in the mesh command ‘ML1D’ are accessed. The necessary input then, are the parameters representing the implemented hysteretic law into the mesh command ‘ML1D’. For a complete description of the previously existing mesh commands, see Spacone et al [15]. The necessary data for this mesh command, in relation to shear deformations, are shown in figure AI-6. These are, the material ID, the type of material relation, the model used in the material relation and the model parameters.

The model

parameters for shear deformations are the nominal shear strength of the section in the positive loading direction, Vn, the positive elastic section stiffness, GAsp, the positive strain hardening ratio, shp, the nominal strength of the section in the negative loading direction, -Vn, the negative elastic section stiffness, GAsn, the negative strain hardening ratio, shn, the pinching parameters in the x and y directions, px and py respectively and the damage factors d1 and d2. All of these values were defined in chapter IV. The material ID used must be greater than or equal to 9. This material ID must be identical to the number used in ‘MATE’ to call the shear material law (see Spacone et al [15]). The type of material relation is ‘hyste’ for hysteretic diagram and the model is ‘ciam’ for Ciampi law (though this really isn’t Ciampi’s law). Model parameters are such as those defined in chapter V for test columns R-3 and R-5.

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Material ID Material Relation Type Vn

GAsp

shp

-Vn

Material Model GAsn

shn

px

py

d1

Figure AI-6: Input Parameters for Shear Hysteretic Law

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d2

APPENDIX II ADDITIONAL CASI REQUIREMENTS

AII-A) Evaluation

The project was carried out under joint supervision by the University PI and the Collaborating Company representative. This allowed control and evaluation of the results and provided constant guidance to the graduate student working on the project. Even though meetings with the Cooperating Company representative were not as frequent as originally scheduled, the Cooperating Company representative was in continuous contact with the student. At the end of the research a formal presentation was made by the graduate student to a panel that included the Cooperating Company representative, the University PI and two University Professors. Dissemination of the project results via regular mail (to be done in the following months) will encourage feedback from other research groups and from the practicing engineers’ community. A copy of the final report will be distributed to major Universities and Structural Engineering Companies throughout the country and comments will be encouraged. Partial results have been presented at a Seminar on Post-Peak Behavior of RC Structures held in Japan in October 1999. A conference paper will be presented at the ASCE Structures Congress to be held in Philadelphia, PA in May 2000.

AII-B) Technology Transfer

As initially planned, the technology transfer between the University and the Collaborating Company has worked both ways. The cooperating company benefited from the experience the University research group in nonlinear analysis of structures, while the University took advantage of the real design problems and needs the Collaborating Company faces in projects throughout the country. This is a critical point for earthquake engineering design in the United States: researchers and code developing bodies are advancing the level of sophistication required for professionals in building design, when in fact there are no computer programs available to the professional for the task. For many years research in nonlinear structural analysis has not looked at the needs and expectations of the structural engineering community. The Collaborating Company representative's experience in testing and behavior of shear walls helped developing of a new shear-wall model. Finally, Russel Martino, the graduate student who worked on the

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project, will join the Collaborating Company immediately following completion of his MS thesis. The experience he will bring to the Collaborating Company in terms of new seismic code knowledge and nonlinear pushover analysis understanding is part of the knowledge and technology transfer between University and Collaborating Company. From the Collaborating Company's (KL&A of Colorado) point of view, guidance of the project while it was ongoing was satisfactory, and critical to the successful completion of the work. The primary technology transfer is envisioned to occur primarily following the employment of Russel Martino, and with the implementation of the software as a working design tool. Russel will present his work to the company, and will work individually with future program users as they perform seismic design on shear wall buildings. Evaluation of the program user interface will occur at that time. Ongoing collaborations with the PI are expected and encouraged

AII-C) Networking

The work developed in the course of this project has been presented and will be presented at: a)

Seminar on Post-Peak Behavior of RC Structures, Lake Yamanaka, Japan, October 1999;

b) ASCE Structures Congress, Philadelphia, PA, May 2000.

AII-D) Publications

R. Martino, E. Spacone and G. Kingsley "Nonlinear Pushover Analyses of RC Frames", ASCE Structures Congress, Philadelphia, PA, May 2000.

AII-E) Funding

1) NSF Proposal: Models for R/C Structures Reinforced with Fiber Reinforced Plastics, $200,000 + 30,000$ matching from CU Boulder (1998-2001): awarded. 2) NSF Proposal for International Cooperation with Slovenia: Testing of R/C Columns Strengthened with Fiber Reinforced Plastic jackets, (2000-2001), $40,000: submitted. 3) NATO Collaborative Grant Proposal: Cooperation with Slovenia and Italy, (2000-2001), 25,000$: submitted

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4) NSF Proposal: Sensitivity Study of RC Structures Subjected to Ground Motions, under preparation. 5) NSF Career Proposals: Models for Performance-Based Design and Retrofit of Reinforced Concrete Structures, $200,000 + (1999-2003): submitted and declined.

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