CE5510 Advanced Structural Concrete Design - STRUT-AND-TIE METHODS -
Assoc Prof Tan Kiang Hwee Department of Civil Engineering National University of Singapore 2/16/2004
In this lecture
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We will explore !the
concept of strut-and-tie models !their applications to new construction (and strengthening works)
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At the end of the lecture You should be able to !identify
cases where strut-and-tie models are applicable or appropriate !formulate strut-and-tie models in structural concrete members !design the reinforcement according to the strut-and-tie models 2/16/2004
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Contents !B-
and D-regions
!Concept
of Strut-and-Tie Models
• Geometric Layout • Design of Struts • Nodes and Nodal Zones • Design of Ties • Detailing
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! Design
• • • • • •
Examples for New Construction
High Wall Corbel Dapped-Beam Transfer Girder Deep Beam with Opening (Stepped (Non-Prismatic) Beams)
! (Examples
for Strengthening Works)
• Dapped Beams • Beam with Openings or Recesses 2/16/2004
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Main (B-) & Local (D-) regions D-region B-region
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Main (B-) regions
!regions
of relatively uniform stresses !Bernoulli hypothesis of linear strain distribution applies !internal forces or stresses are derived from statics !“Standard” methods of Codes apply
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Local (D-) regions
!significantly
non-linear strain
distribution !near concentrated loads, corners, bends, openings and other discontinuities !internal flow of forces well described by strut-and-tie models !conventionally design by thumb-rule 2/16/2004
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Concept of Strut-and-Tie Models !
Components ! concrete
compression
struts ! steel tension ties ! nodes (nodal zone) where struts and ties meet !
concrete
Dual purpose ! describe
essential aspects of structural behaviour ! provide tools for structural dimensioning
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steel
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Geometric Layout of strut-and-tie models
Load path ?
Boundary forces/stresses
follows the flow of internal forces in the structure 2/16/2004
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!
Major requirements ! S-T
model must be in equilibrium with applied loads (statically admissible field) ! Strength of struts, ties and nodal zones must equal or exceed forces in these members (safe) ! Sufficient to consider only axes of struts and ties in the early design stage; need to consider widths in general ! Struts must not overlap each other ! Ties may cross struts or other ties ! Angle between a strut and a tie joined at a node should not be less than 25 degrees. 2/16/2004
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Basic steps !
! ! !
Compute internal stresses on boundaries, subdivide boundary and compute force resultants on each sub-length; or Compute action effects on boundaries Draw truss to transmit forces Check stresses in individual truss member
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Some rules for estabilshing strut-and tie model
Elastic stress trajectories 2/16/2004
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Minimum steel content
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ΣFiliεmi=minimum
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×
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Agreement with Crack Pattern
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Superposition of models
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Truss 2 can form only if truss 1 does not fail prematurely
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Exercise 1 !
Explore the application of strut-and-tie model in the design of anchorage zone of a post-tensioned beam compression
or tension
Principal compressive 2/16/2004Stress trajectories
Stress contours Tan K H, NUS
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Exercise 2 !
A T-beam is post-tensioned with a cable anchored at the centroid of the section at its end. Given that the area of the flange is one-third of the overall crosssection, explain by sketching in the following figures, how you would obtain the required reinforcement to resist bursting tension in the web due to the prestressing force.
x-section 2/16/2004
strut-&-tie model
reinforcement Tan K H, NUS
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Elements of strut-and-tie model !
Compression struts ! line
along centre-line of strut ! strut with width !
Tension ties ! band
of steel reinforcement ! anchorage (hooks, development length) !
Nodes ! bounded
by compressive forces (CCC) ! anchoring one tension tie (CCT) ! anchoring more than one tie (CTT, TTT) 2/16/2004
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Forces in struts and ties
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In general, φFn ≥ Fu φ : strength reduction factor Fn : nominal strength of the member Fu : force in the member due to factored loads
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Struts ! Types
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of struts
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!
Design of struts Fns = fcuAc fcu : effective compressive strength fcu = ν fc’ ACI Code: φ fcu = φ ν fc’ = φSTM α1 βs fc’ (to ensure same load capacity as FIP Recommendations, consistency between AC1 1999 and 2002 Codes, & consistency between B-and D- regions)
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! Factors
affecting fcu
!Load
duration effects (α1 = 0.85) !Cracking of struts • Bottle-shaped struts • Cracked struts • Transverse tensile strains !Confinement
from surrounding concrete (e.g. pile caps)
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Prismatic strut
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Bottle-shaped strut
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Nodal zones ! Forces
! CCC,
must be in equilibrium
CCT, CTT, TTT joints C C
C T
C C
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CCC 2/16/2004
CCT Tan K H, NUS
Extended Nodal Zones
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Ws = wt cos θ + lb sin θ 2/16/2004
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Ties ! Spread
of ties wt = (Fu/ φ)/(fcu bw)
! Strength
of ties Tn = Asfy
! Anchorage 2/16/2004
of ties Tan K H, NUS
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Reinforcing Requirements ! Minimum
reinforcement
!To
ensure ductility !For crack control !Bottle-shaped
struts: Σ(Asi/bsi)sin γi ≥ 0.003 !Other code requirements 2/16/2004
γi
(A s
) s i b i/
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Summary
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Isolate D-region and compute force resultants on boundaries ! Draw truss to transmit forces !
! use
of elastic analysis, crack patterns ! equilibrium of forces, width of struts, anchorage of ties !
Provide steel reinforcement for ties & check concrete stresses in struts and nodes where necessary
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100 mm
534 kN 534 kN
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305 x 305mm column
Example 1 –
187
267
Column on wall
565
2438 x 305 mm wall
263 187
534
fc’=20 MPa fy=414 MPa 187
187
534
σ=P/A+M/I 4.67 MPa
Based on fce=0.66fc’ 2T13 each face 1.80 MPa
3T13 each face 187
187 2T13 each face
427
655
2/16/2004 518
203
678 586
678 904
MacGregor 226 mm
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213
158
371
457 mm
785 309
Final layout 1510
b=406 mm h=508 mm d=457 mm
89
96
292
745
530
5
474
763 115
457 mm
213 687 687
99 2
222
241 mm
890
785 kN
222
890 kN
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100 mm
w=1732/ (0.61fc’) =200mm Tan K H, NUS
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Example 2 -
Corbel fc’=35 MPa fy=414 MPa Short member cantilevering from a column or wall
305 x 127mm bearing plate
486 x 486mm
MacGregor 2/16/2004
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4T25 3T25
a=241 mm 178x102x9.5 angle
279 3T13 closed stirrups
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2T13
406 356 2T13 C = 1155 kN; a = C/(0.8ν ν2fc’b) = 127mm bef = a+ll/6 = 127+413/6 = 196 mm Asfy ≥ Σ[(C/4)(1-a/bef)] = 203 kN As ≥ 490 mm2
229
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Example 3 - Dapped ends Check strut width; compute steel required in ties
Bearing area = V/(0.85ν ν2fc’) 516
443 419 mm
MacGregor 2/16/2004
686 mm
H=74 V=369
369
443 6 7 5
37
38 5
52 3
369
369
381 mm
381 mm
914 mm
553
516
37
762 mm deep by 381 mm width beam fc’= 20 MPa, fy=414 MPa Tan K H, NUS
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4T13 closed stirrups
4T13 U stirrups 2T13 1T13 U bar 4T20 welded to angle 2T13 U bars 2T20 U bars 4T25 bars
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178 mm
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Example 4 - Transfer girder 11600 kN 3600
700
6850
140.4 kN/m 3600
700 2/16/2004 MacGregor
b=700 mm
fc’=35 MPa fy=410 MPa
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6579 kN
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Combined truss and strut action
6543 kN
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Right end: V=6543 - 11x(140.4x0.6) = 5616 kN ∴req’d Av=(5616/9)x103 ÷ (410x610) =2537 mm2/m ∴use φ 22 U-stirrups @ 300 mm c/c (2540 mm2/m)
Left end:
For K-UU, D= (624+84.2)/sin 280 = 1508 kN ∴req’d width w =D/(bfce) =D/(0.5bfc’) =123 mm For S-UU, w=65 mm 6579w = 94 mm Average kN →assume all struts to be 100 mm and lower tensile tie located at mid-height 2/16/2004 of truss node at UU.
∴V transmitted by stirrups = 3x854 = 2562 kN = 39% of 6579 kN
To ensure ductility, at least 30% of shear to be transmitted by stirrups; the rest by a major diagonal strut. →try φ 22 U-stirrups @ 225 mm c/c (Avfyv=854 kN per 600 mm spacing)
V transmitted by strut H-AA = 6579-2562-6x84.2 =3512 kN 6543 For H-AA, D = 5102 kN; w=416 mm. kN Combined truss For E-AA, D = 1174 kN; w=96 mm.
and strut action
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Example 5 - Deep beam with opening
fyd=434 MPa fcd=17 MPa
Schlaich 2/16/2004
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Right side, complete model
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left side, model 1 2/16/2004
left side, model 2
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Check concrete stresses: Stresses under bearing plates: σp=3000x103/(700x400) = 10.7 MPa < 1.0 fcd=17 MPa σA=1070x103/(500x400) = 5.4 MPa < 0.8 fcd=13.6 MPa σB=1930x103/(500x400) = 9.7 MPa < 0.8 fcd=13.6 MPa Required depth of compression zone: C=T= 1070 kN d ≥ 1070x103/(400x1.0fcd) = 135 mm < 400 mm ∴OK (Nodes taken 200 mm below top surface.) 2/16/2004
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Check anchorage length of reinforcing bars
Other critical anchorages - C, D
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> Anchorage length Tan K H, NUS
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Further reinforcement
mesh on either face of wall
stirrups 2/16/2004
nominal column reinforcement
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References !
!
!
!
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J.G. MacGregor, “REINFORCED CONCRETE: Mechanics and Design”, 3rd Ed., Prentice-Hall, 1997, Ch. 18. A.H. Nilson, D. Darwin and C.W. Dolan, “Design of Concrete Structures”, McGraw-Hill, 2003, pp. K.H. Reineck (Ed), “Examples for the Design of Structural Concrete with Strut-and-Tie Models”, ACI SP-208, 2002, 244 pp. Strut-and-Tie Resource Web Site http://www.cee.uiuc.edu/kuchma/strut_and_ti e/STM/
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Further reading: !
!
!
!
!
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J. Schlaich, et al., “Toward a Consistent Design of Structural Concrete”, J. of Prestressed Concrete Institute, V.32, No. 3, 1987, pp.74-150. P. Marti, “Basic Tools of Reinforced Concrete Beam Design”, ACI Journal, V. 82, No. 1, Jan-Feb 1985, pp. 46-56. Tan, K.H. and Naaman, A.E., "Strut-and-Tie Model for Externally Prestressed Concrete Beams", ACI Structural Journal, Vol. 90, No. 6, USA, November-December 1993, pp. 683-691. Tan, K.H., “Shear Strengthening of Dapped Beams Using FRP Systems", Fifth International Symposium on Fibre Reinforced Plastics for Reinforced Concrete Structures (FRPRCS-5), Cambridge, UK, July 16-18, 2001, Vol. 1, pp. 249-258. Mansur, M.A., Tan, K.H. and Weng, W., “Effects of Creating an Opening in Existing Beams”, ACI Structural Journal, Vol. 96, No. 6, USA, November-December 1999, pp. 899-905.
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