Grillage analysis of cellular decks with inclined webs

Structures and Buildin Structures Buildings gs Volume 164 Issue SB1 Grillage analysis of cellular decks with inclined webs Mozos and Aparicio

Proceedings of the Institution of Civil Engineers Proceedings Engineers Structures and Buildings 164 February 2011 Issue SB1 Pages 13–18 doi: 10.1680/stbu.8.00056 Paper 800056 Rece Re ceiv ived ed 14 14/0 /04/ 4/20 2009 09 Acce Ac cept pted ed 24 24/1 /11/ 1/20 2009 09 Published online 10/01/2011 Keywords: bridges/buildings, Keywords: bridges/buildings, structure & design/concrete structures ICE Publishing: All rights reserved

Grillage analysis of cellular decks with inclined webs Carlos M. Mozos PhD

Ańgel C. Aparicio PhD

Professor, Department of Civil Engineering and Buildings, University of Castilla-La Mancha, Ciudad Real, Spain

Professor, Department of Construction Engineering, Technical University of Catalonia, Barcelona, Spain

Although grillage analysis is a very popular method for obtaining the response of flat slabs and bridge decks, there are very few guidelines on how to apply the method to decks with inclined webs. This paper presents new and more accurate criteria for obtaining the torsion constant of longitudinal and transverse grillage beams and for splitting a deck into longitudinal beams. These criteria are applied in a numerical study of three different decks with inclined webs. The static response of the numerical models under three load cases is compared with the response of threedimensional finite-element numerical models.

Notation E e1 e2 G J J 12 J eq eq M t S T u v

w x y z z 1 z 2 Æ ª xy Ł12    xy  xy

1.

Young’s modulus slab thickness web thickness shear modulus torsion constant torsion constant of portion of deck between y ¼ y1 and y ¼ y2 torsion constant of equivalent transverse beam with constant cross-section torque distance between transverse beams in the grillage model displacement in x -direction displacement in y -direction displacement in z -direction -direction x-axis coordinate y-axis coordinate z-axis coordinate distance dista nce of a point of the the slab slab to grillag grillagee plane plane in z -direction -direction distance dista nce of a point of the the incline inclined d web web to the grillag grillagee plane in z -direction -direction angle of the inclined web shear strain in xy plane relative rotation of the end of transverse beam about its longitudinal axis Poisson’s ratio shear stress shear stress in xy plane curvature in xy plane

Intr In trod oduc ucti tion on

Matrix analysis of structures is an efficient method for evaluating the static and dynamic behaviour of structures (Ryall (Ryall et al.,

2000). For th 2000). this is re reas ason on,, ex exte tens nsiv ivee th theo eore reti tica call re rese sear arch ch ha hass been focused on developing techniques for modelling continuous three three-dim -dimension ensional al (3D) structu structures res using exclu exclusive sively ly onedimens dim ension ional al (1D (1D)) ele eleme ments nts wit with h tw two o nod nodes es and six deg degree reess of freedo fre edom m per node (Apa Aparic ricio io and Cas Casas, as, 200 2001 1; Hambly, Hambly, 1991 1991;; Hambly and Pennells, 1975; 1975; Jaeger and Bakht, 1982; 1982; Manterola, 1977, 197 7, 200 2000 0; O’Br O’Brie ien n an and d Ke Keoh oh,, 19 1999 99;; Sa Sawk wko, o, 19 1968 68). ). Th This is meth me thod od ha hass be been en us used ed fo forr mo mode dell llin ing g br brid idge ge de deck ckss th thro roug ugh h grillag gril lagee ana analys lysis is and se sever veral al res resear earch ch wo works rks and gui guidel deline iness (Aparicio and Casas, 2001; Hambly, 1991; Hambly and Pennells, 1975; Jaeger and Bakht, 1982; 1982; Manterola, 1977, 2000; 2000; O’Brien and Ke Keoh, oh, 199 1999 9) pro provide vide criteria for appl applying ying grillage anal analysis ysis to different types of bridge decks. In spite of this, only Hambly (1991)) has pro (1991 propos posed ed a gen genera erall pro proced cedure ure for app applyi lying ng gril grillag lagee analysis to decks with inclined webs. Hambly suggests using, for non-rectang non-re ctangular ular cells, torsio torsion n stiff stiffness ness provided provided by the top and bottom slabs, and considering that the centres of the slabs are separated by a distance equal to the average height of the cell. On the other hand, for decks with rectangular cells, cells, Jaeger and Bakht (1982) (1982) and Aparicio Aparicio and Casas (2001) (2001) sugge suggest st distri distributing buting the Saint Vena enant nt tor torsio sion n con consta stant nt of the deck in pro propor portio tion n to the area enclosed by the cell. This pa This pape perr de deal alss wi with th ho how w to sp spli litt a de deck ck in into to lo long ngit itud udin inal al beams and how to calculate the torsion constant of the longitudinall and transverse itudina transverse beam beamss when grillage analysis analysis is appli applied ed to determine the behaviour of a deck with inclined webs. A new criterion, crite rion, based on integ integratio ration n of the torsion stiff stiffness ness provided provided by each portion of the deck, is presented in Section 2. In the numerical study presented in Section 3, three cellular decks with inclin inc lined ed we webs bs and dif differ ferent ent dep depths ths are stu studie died d fro from m a sta static tic point of view view.. The behaviour of each deck is investigated with a plane grillage model obtained with the proposed criteria for splitting the deck and for calculating the torsion constant of the beams. The response is compared with the behaviour of 3D reference models.

13

Structures and Buildin Structures Buildings gs Volume 164 Issue SB1

Grillage analysis of cellular decks with inclined webs Mozos and Aparicio

2.

and the shear stress is

Torsion Torsio n const constant ant of beam beams s in gri grilla llage ge analysis applied to decks with inclined webs

A total torque M acting acting on the deck gener generates ates torques M i and shear forces V i on the grillage longitudinal beams and torques on the transverse beams (Figure 1). Each torque and force produces vertical vertic al displa displaceme cements nts and rotat rotations ions about the longit longitudinal udinal axis on the grillage beams and a shear flow on the slabs and webs of the deck (Calgaro (Calgaro and Virlogeux, 1994; 1994; Kollbrunner et 1970; et al., 1970; Vlassov Vlasso v, 1962 1962). ). Co Cons nseq eque uent ntly ly,, th thee to tors rsio ion n co const nstan antt mu must st be calculated considering that each portion of the deck rotates along the longitudinal axis of the grillage beam to which it has been assigned. assign ed. Thus, the torsio torsion n consta constant nt J 12 of a po porti rtion on of de deck ck ¼ ¼ between y y1 and y y2 (Figure 2) can be ob obta tain ined ed by summin sum ming g the tor torsio sion n stif stiffne fness ss pro provid vided ed by ev every ery dif differ ferent ential ial portion d y. For a portion d y between y ¼ y1 and y y ¼ yc, the torque is

1:

d M t ¼  e1 d yz yz 1 þ  cos Æ

e2 d yz yz 2 cos Æ

4:

 xy ¼ 2 zG  xy

the torsion curvature curvature can be obtai obtained ned by integr integrating ating the diff differerential torque d M t from y1 to y2

 xy ¼ 12

5:

2:

yz 1   cos Æ d M t ¼  e1 d yz

3:

"ð ð

yc y

¼

2

the torque is

y1

e2 yz 2 d yz cos Æ

y2 y

þ

6:

Since the shear strain is

ª xy ¼

( e1 z 21 þ

@ u @ v @ 2 w  xy þ ¼ 2 z ¼ 2 z @ y @ x y @ x x x@ y y

¼

yc

 xy G

7:

M

V i

d y y þ

y2 y

ð

yc

2( e1 z 21  e2 z 22 ) d y G 2( y

( e1 z 21 þ e2 z 22 ) d y y

#

( e1 z 21  e2 z 22 ) d y y

In the grillage model, a transverse beam 1–2 (Figure 3) with a non-variable torsion constant J eq eq is used. This equivalent torsion consta con stant nt is obt obtain ained ed by con consid sideri ering ng equ equali ality ty of the rel relati ative ve rota ro tati tion onss of en ends ds 1 an and d 2 of bo both th va vari riab able le an and d no nonn-va vari riab able le beams

xT x

M i

e2 z 22 )

Thus, the torsion constant of a longitudinal beam that represents the portion of deck between y ¼ y1 and y y ¼ y2 is

J ¼ yc and y ¼ y2,

G 2

y1

12

and for d y between y

 M t

yc y

ð

Ł12 ¼

ð 0

M t M t d x ¼ xT GJ GJ eq eq

where xT is defined in Figure 3. Considering the shear stress at sections A–A and B–B (Figure 3), the torsion constant at these sections is

Torquess and shear forces on grillage beams due to Figure 1. Torque torsion

z y 2

d y z

1 1

z

2 2

z

2 2

y c y 1

d y 1 1

e

e 2 α

Figure 2. Shear flow on deck cross-section

14

c.o.g.

Grillage plane

y



B

A

x T

S T

x c

z z

z

c

1 1

e

y

T T

1 1

z

2 2

z

1 1

c.o.g. x

1

z

S T

2 α

e

z

2 2

2

A–A B

B–B

A

coordinate dinatess for transv transverse erse beam 1 –2, Figure 3. Local system of coor BT12

8:



J ¼ 2



e2 z 2 S T cos Æ 2

e1 z 21 

3.

where the  sign depends on the relative position of the shear stress flow on the inclined web and the centre of gravity (c.o.g.). Thus, the relative rotation of the beam ends becomes

3.1 xc x

Ł12 ¼

ð x0

M t d x G 2[ 2[ e1 z 21  ( e2 = cos Æ) z 22 ] S T xT x

þ

ð xc

¼

9:

G 2[ 2[e1 z 21 þ

M t d x ( e2 = cos Æ) z 22 ] S T

M t xT GJ eq eq

3.3

x0 ¼ 10: 10:

e1 z 21 

e21 z 1 2

# 9= ; 1=2

xT z T

in order to provide a minimum torsion stiffness at the initial end of the beam. Thus, the equivalent torsion constant of the transverse beam is

J eq eq ¼ xT

(ð ð xc x

d x

x0

2[ e1 z 21  ( e2 = cos Æ) z 22 ] S T

xT x

þ

11: 11:

2 xc 2[ e1 z 1 þ

Reference Refer ence numer numerical ical mode models ls

A 3D fini finitete-ele eleme ment nt ref refere erence nce mod model el wa wass devel develope oped d for eac each h . deck. Figur Figuree 5 shows shows the referen reference ce mod model el of the 1 55 m dee deep p deck (Figure 4 (Figure 4(b)). (b)).

where the lower limit x0 is defined by

cos Æ z c  e2

Studie Stu died d dec decks ks

Figure 4 illustrate illustratess the three deck decks, s, taken from the para parametr metric ic study of Walther Walther et (1999). (1999 ). The distance between diaphragms et al. (6.20 m) is de defin fined ed by th thee an anch chor orag ages es of th thee st staays ys,, an and d th thee . thick thi ckne ness ss of th thee di diap aphr hrag agms ms is 0 35 m. Th Thee sla slabs bs,, we webs bs an and d dia iap phragms are concrete with an elast stiic modulus . . E ¼ 36 342 4 MPa and Poisson’s ratio  ¼ 0 2.

3.2

8< "  :

Numeri Num erical cal ana analys lysis is

The sta static tic beh behav aviou iourr of thr three ee dec decks ks wit with h inc inclin lined ed we webs bs wa wass studied using grillage models and compared with the behaviour of 3D finite-element reference models (in all cases, a 31 .0 m long cantile can tileve ver). r). The sof softw tware are use used d in the num numeri erical cal ana analys lysis is wa wass Sofistik version 2.3.

d x ( e2 = cos Æ) z 22 ] S T

)

Grillage Grilla ge nume numerical rical mode models ls

The grillage mesh, with longitudinal beams BL1 and BL2 and transv tra nsvers ersee bea beams ms BT1 BT12 2 and BT22, is sho shown wn in Fig Figure ure 6. The grillage plane coincides with the principal axis of the deck as a whole. who le. Lon Longit gitudi udinal nal and tra transv nsvers ersee cut cutss are pla placed ced mid midwa way y between the grillage members and the deck portions are assigned to the clo closed sed gri grilla llage ge bea beam. m. The cri criter teria ia use used d for cal calcul culati ating ng areas and inertia of the grillage members are as follows. (a) The cross-sectional area assigned to each beam is the area of the portion of the deck that is represented by the beam. (b) In-plane In-plane shear deformation deformation has been included included in the bendin bending g about z -axis -axis deformation. Consequently, the beams are considered as shear-rigid for forces acting along the y- axis. (c) The shear area area about the z -axis -axis was obtained by considering the criteria suggested by Hambly by Hambly (1991). (1991).

15



(d ) The torsion constant of the longitudinal longitudinal and transverse transverse beams was obtained by applying the criterion presented in Section 2. 2·50

2·50

(e) The bending bending inertia about about the y -axis of the longitudinal longitudinal beams is the second moment of area about the y- axis, which is located at the same level as the c.o.g. of the deck. Otherwise, Other wise, the bendi bending ng inertia of the transverse transverse beams is the second moment of area about the c.o.g. of the portion of the deck in each section.

0·22 5 1 · 1

0·21 0·20

0·16

(a) 0·20 5 5 · 1

( f ) The in-plane bending bending inertia suggested suggested by by Hambly Hambly (1991) (1991) for cruciform elements was considered.

0·21 0·16 0·20 (b)

3.4 3. 4

The sta static tic beh behav aviou iourr of the mod models els wa wass in inves vestig tigate ated d for tw two o concentrated loads Q acting in the three different cases (Q1 to Q3) shown in Figure 7 Figure 7..

0·18 0·17

5 2 · 2

Load Lo ad ca case sess

0·17 0·18

3.5 3. 5

(c)

Resu Re sult ltss

The displacements w and v of nodes 1 to 4 (Figure 6) alo along ng the z- and y-ax -axes es we were re obt obtain ained ed by sta static tic lin linear ear ana analys lysis is for each ea ch gr gril illa lage ge mo mode dell an and d fo forr th thee eq equi uiva vale lent nt no node dess of th thee reference models for load cases Q1 to Q3. Figure 8 shows the ratio rat io of the displacem displacement entss of the nodes of the grillage grillage mod model el to th thee re refe fere renc ncee mo mode del. l. No Note te th that at al alll of th thee gr gril illa lage ge mo mode dels ls show homogeneous behaviour and provide an accurate response under und er lon longitu gitudin dinal al ben bendin ding, g, tor torsion sion and tra transv nsvers ersee ben bendin ding. g. The cri criter teria ia pro propos posed ed her heree for spl splitt itting ing the dec deck k and cal calcul culatating the torsio torsion n consta constant nt of the beam beamss thus allows reproduction reproduction of th thee st stat atic ic re resp spon onse se of th thee de deck ckss un unde derr to tors rsio ion n (l (loa oad d ca case se . Q2). Q2 ). In th this is ca case se,, th thee ma maxi ximu mum m err error or,, 4 5%, is sma smalle llerr tha than n thee er th erro rorr in incu curre rred d us usin ing g th thee cr crit iter erio ion n pr prop opos osed ed in Hambly (1991), (1991 ), wh whic ich h is cl clos osee to 14 14%, %, or th thee cr crit iter erio ion n pr prop opose osed d by Aparicio and Casas (2001) (2001) and and Jaeger and Bakht (1982) (1982 ) (30%) for the sam samee dec decks ks (se (seee Mozos (2007 (2007)). )). Load ca case se Q3 sho shows ws an acceptable acceptable respo response nse of the grillage models under transv transverse erse bending.

Figure 4. Deck dimensions (in metres): (a) deck D1 .15; (b) deck D1.55; (c) deck D2 .25

z y

finite-element reference model for deck D1.55 Figure 5. 3D finite-element

z y 3 1 · 0 00

x

0 ·4 6

3·54

5·00

BT12

3·54

BT22

BL1 BL2

BL2

Grillage e mesh and reference nodes 1– 4 (dimensions (dimensions in Figure 6. Grillag metres)

16

B L 1

BT12

BL1

B L 2

0·46

B L 1

B L 2

1 2 B T

2 2 B T

1 2 B T

4 3 2

1 · 5 5

1



Q1

Q2 z

z

y

Q3

z

y x

x

y x

Q Q

Q

Q

Q Q

Figure 7. Load cases; Q ¼ 1000 kN

M E F

1·013

1·08

0·97

1·012

1·06

0·96

M E F

1·011

w / e

g a l l i r g

w

M E F

1·04

w / e

g a l l i r g

1·010

w

1·009

1·008

g a l l i r g

1·02

w

1·00

1

2

3

4

0·95

w / e

0·98

0·94

0·93

1

Node (a)

3

2

4

0·92

1

Node (b) Deck D1·15

Deck D1·55

3

2

4

Node (c) Deck D2·25

Figure 8. Ratio of the displacements of grillage models to finiteelement reference model for decks D1 .15, D1.55 and D2 .25 (nodes 1–4) under (a) load Q1, (b) load Q2 and (c) load Q3

4.

Conc Co nclu lusi sion ons s

The objectiv objectives es of thi thiss stu study dy we were re to pre presen sentt a ne new w and more accura acc urate te cri criter terion ion for obt obtain aining ing the tor torsio sion n con consta stant nt of lon longgitudinal and transverse grillage beams, and to define a criterion for splitting a deck into longitudinal beams. Three cellular decks of different depths were studied from a static point of view using plane grillage models. Such grillage models were obtained by placing longitudinal ‘cuts’ midwa midway y between the grillage members and ass assign igning ing dec deck k port portion ionss to the clo closed sed gril grillag lagee bea beam. m. The torsio tor sion n con consta stant nt of the lon longit gitudi udinal nal and tra transve nsverse rse bea beams ms wa wass obtain obt ained ed usi using ng the cri criter terion ion des descri cribed bed.. The res respon ponse se of the grillage models was compared with the 3D finite-element reference models. The fol follo lowin wing g con conclu clusio sions ns can be dra drawn wn fro from m the num numeri erical cal analysis.

(a) The proposed criterion criterion for obtaining the torsion constant of the beams allows one to obtain a grillage model with a torsion stiffness very close to the cellular deck and to reproduce its static behaviour under torsional load with adequate precision. (b) Splitting Splitting the deck into longitudinal longitudinal beams mainly affects affects the transverse bending response of the grillage model. The results obtained by placing cuts midway between the longitudinal beams yield acceptable accuracy. accuracy.

Acknowledgements The authors thank the Spanish Junta de Comunidades de CastillaLa Mancha and the Spanish Ministry of Education and Science for fina financi ncial al sup support port und under er res resear earch ch pro projec jects ts PB PBI-0 I-05-0 5-031 31 and BIA2006-15471-C02-02 respectivel respectively y.

17



REFERENCES

Manterola J (2000) Puentes. E.T.S. de Ingenieros de Caminos,

Aparicio AC and Casas JR (2001) Apuntes de Puentes. E.T.S. de

Canales y Puertos de Madrid, Madrid. Mozos CM (2007) Theoretical and Experimental Study on the Structural Response of Cable Stayed Bridges to a Stay Failuree. Universidad de Castilla-La Mancha, Ciudad Real (in Failur Spanish). O’Brien E and Keoh D (1999) Bridge Deck Analysis, vol. 1, 1st edn. Taylor & Francis, London. Ryall M, Parke G and Harging J (2000) Manual of Bridge Engineering , 1st edn. Thomas Telford, London. Sawko F (1968) Recent developments in the analysis of steel bridges using electronic computers. Proceedings of Conference on Steel Bridges. British Constructional Steelw Stee lwork ork Association, Association, London London.. pp. 1 –10. Vlassov BA (1962) Pie´ ces ces Longues En Voiles Minces , 10th edn. ´ ditions Editions Nationales Physico-mathe´matiques, ´matiques, E Eyrolles, Paris. Walther R, Houriet B, Isler W, Moıä P and Klein J-F (1999) Cable Stayed Bridges, 2nd edn. Thomas Telford, London.

Ingenieros de Caminos, Canales y Puertos de Barcelona, Barcelona. Calgaro J and Virlogeux M (1994) Projet et Construction des Ponts.. Analyse Structurale des Tabliers de Ponts, vol. 2, 2nd Ponts edn. Presses Pre sses de d e l’ećo ćole le Nationale Nati onale des d es Ponts et Chausse Chaus seés, és , Paris. Hambly EC (1991) Bridge Deck Behaviour , 2nd edn. Taylor & Francis, London. Hambly E and Pennells Pennells E (1975) Grillage analysis applied to cellular bridge decks. The Structural Engineer 53(7) 275.. 53(7): 267– 275 Jaeger L and Bakht B (1982) The grillage analogy in bridge analysis. Canadian Journal of Civil Engineering 9(2) 9(2): 224– 235. Kollbrunner C, Basler K and Eperon PA (1970) Torsion: ` l’e´ ´ tude Application a tude des Structures. Spes S.A., Lausanne. Manterola Manterol a J (1977) Ca´lculo ´lcu lo de d e tableros table ros de d e puente puen te por po r el me´todo ´t odo ´ n y Acero 122(1): 93–149. del emparrillado. Hormigo´

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Grillage analysis of cellular decks with inclined webs

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