CREEP, SHRINKAGE AND TEMPERATURE EFFECTS IN COMPOSITE ... · PDF file29. th conference on our world in concrete & structures: 25 -26 august 2004, singapore creep, shrinkage and...

CREEP SHRINKAGE AND TEMPERATURE EFFECTS IN COMPOSITE STEEL -CONCRETE BRIDGE BEAMS

F Giussani Politecnico di Milano Italy

F Mola Politecnico di Milano Italy

29th Conference on OUR WORLD IN CONCRETE amp STRUCTURES 25 - 26 August 2004 Singapore

Article Online Id 100029033

The online version of this article can be found at

httpcipremiercom100029033

This article is brought to you with the support of

Singapore Concrete Institute

wwwscinstorgsg

All Rights reserved for CI‐Premier PTE LTD

You are not Allowed to re‐distribute or re‐sale the article in any format without written approval of

CI‐Premier PTE LTD

Visit Our Website for more information

wwwcipremiercom

29th Conference on OUR WORLD IN CONCRETE amp STRUCTURES 25 - 26 August 2004 Singapore

CREEP SHRINKAGE AND TEMPERATURE EFFECTS IN COMPOSITE STEEL - CONCRETE BRIDGE BEAMS

F Giussani Politecnico di Milano Italy F Mola Politecnico di Milano Italy

Abstract The basic aspects of the long-term analysis of simply supported steel-concrete bridge beams

subjected to permanent loads shrinkage and temperature are presented Considering the connector deform ability the general solution of the problem has to be performed by means of a structural analysis leading to a system of integro-differential equations which have to be solved by recurring to numerical algorithms A case study allows to point out the main aspects of the problem

keywords steel-concrete beams deformable connectors creep shrinkage temperature

1 Introduction Imposed deformations due to shrinkage and temperature affect the service stage behaviour 0 f

composite steel-concrete bridge beams Their effects have to be added to those related to permanent and variable loads with particular attention to the stress distributions in the transverse sections which can generate concrete cracking In simply supported beams when a rigid connection between slab and beam is present the problem can be solved by recurring to sectional analysis by assuming suitable constitutive laws for steel and concrete A visco-elastic law is required for concrete owing to the presence of sustained variable in time stresses When deformable connectors are dealt with shrinkage and temperature effects vary along the beam axis and they have to be determined by operating structural analysis In this work the problem is investigated in detail In particular referring to temperature distributions suggested in literature simply supported beams are considered by varying the connector stiffness The stresses in concrete slab are evaluated in order to check the cracking limit state pointing out the favourable effect due to concrete creep both in reducing the interaction forces between beam and slab and in decreasing the self-equilibrated stresses generated by the non-linear temperature distribution Some numerical simulations allow to point out the most interesting aspects of the proposed analyses

2 Formulation of the problem The long term analysis of composite steel-concrete

beams will be performed by recurring to the displacement method [1] by assuming the deformation parameters as the unknowns of the problem When rigid connectors are considered the problem can be described by means of two unknowns identifying the plane of sectional deformation in particular the longitudinal strain f1 and the curvature f3 in the z-y plane Otherwise when taking the connector deformability into account a number of longitudinal strains f11 equal to the

x

z y number of structural elements interacting by means of

Fig 1 Sectional deformations deformable connectors has to be introduced together with the curvature f3 For the sake of clearness but without loosing in generality let us consider the composite section of Fig 1 with one steel element interacting with the concrete slab Referring to a right coordinate system with origin in the steel beam centroid according to the plane section hypothesis and neglecting the

273

y z y

uplift between the slab and the beam the strains at a generic fibre of concrete slab and of steel beam can be written as follows Ee (yzt) = e (z t) - y3 (Z t)

(1 )Es (yz t) = S (Z t) - y 3 (z t)

where e and s are respectively the longitudinal strains of the slab and of the beam evaluated in the origin of the coordinate system At steel-concrete interface the following compatibility equation has to be fulfilled

(2)

The connector deformation Ech can be expressed as a function of the shear flow q

Ech(Zt)= OUch (zt) 8q(zt) 1 (3) ()z ()z kch

where kCh is the connection stiffness and Uch is the longitudinal displacement of the connectors at the abscissa z

The shear flow can be evaluated by imposing the equilibrium of the infinitesimal element having height dy and length dz as shown in Fig 2a It can be expressed as follows

OOZb(y)= a(tzyb)=_8q (4)()z ay ay

The shear flow at a generic coordinate (z I) at(a) (b)

time t can be evaluated by integrating eq (4)Fig 2 Longitudinal equilibrium between y=Yinf where q=O according to Fig 2b

and the reference abscissa y= I We obtain

- t)=_rY b( )OOz(yzt)d( (5)q y z Jyo y ()z y

Assuming for steel beam a linear elastic behaviour and for concrete slab a linear viscoelastic constitutive law according to Mc Henry Principle of Superposition [2] the stress-strain relationships at a generic time t can be expressed as

crs(yzt) = bEsd[Es (yzt)- Es (yt)] = Es [Es (yzt) - Es (yt)] (6)

cre(yz t) = b d[Ee (yz t) - Ec (y t) ]R(t t)

where Es and R(tt) are respectively the elastic modulus of the steel beam and the relaxation function of the concrete slab According to [3] the imposed deformations Es (y t) Ec (y t) acting respectively

on the beam and on the slab are assumed constant along x and z axes By introducing the first of eqs (6) and the second of eqs (1) eq (5) becomes

q(y z t) = -Es[As (I) s (z t) - 5sy (I) (z t)] (7)

where

As = r b(y)dy 5sy = r b(y)ydy (8)

Hence the shear flow at the connection system can be obtained by evaluating the integrals of eqs (8) on the whole steel beam According to eq (3) the connector deformation can be expressed as

Ech (z t) = -~[Ass (z t) - Ssy (z t)] (9) kch

Thus the compatibility relationship (2) assumes its final form

1c (z t) = s (z t) - ~[As ~s (z t) - Ssy (z t) ] (10) kCh

According to eqs (6) by neglecting the reinforcing rebars the equilibrium equations can be expressed as follows

[s~)d 1e (z t) + s~d3 (Z t) ] p( t t) + sWs (Z t) + s~ 3 (Z t) = N (z t) + b dNe(t) p(t t)+ Ns (t) (11)

b[S~cdle (z t) + S~~dl3 (z t)] p( t t)+ S~ss (z t) + sWf 3 (Z t) = -M(z t) - dMe(t) p( t t)- Ms (t) (12)

where

274

(13)Ne (t) = t EeodEe (y t)dAe

(14)Me(t)= I EeoydEe(yt)dAeA

p (t t) = R (t t) IEeo (15)

r(c) - E A r(c) - r(c) - E S r(e) - E I res) - E A res) - res) - E S r es) - E I gt11 - cO e gt12 - gt21 - - cO ey gt22 - cO eyy gt11 - S S gt12 - gt21 - - s Sy gt22 - sSyy (16)

with EcO elastic reference modulus of concrete The Volterra integral equations (11) (12) can be inverted by introducing the solving kernel ()(tt)=EcoJ(tt) obtaining

~~)f le (z t) + ~~f3(z t) + n~~df ls (Z t) + ~~df3(Z t)JP( t t) = ndN (z t) + dNs(t)] P(t t) +Ne (t) (17)

~~ef le (z t)+~~edf3(Z t) + n~~sdf ls (Z t) + i~~df3(Z t) J Nt t) = - ndM (z t) + dMs(t)] P (t t) -Me (t) (18)

Thus the problem can be completely described by means of the second order differential compatibility equation (10) and the two integral equilibrium equations (11) (12) or (17) (18) In order to solve the differential equation (10) adequate boundary conditions have to be prescribed By assuming that no axial forces or bending moments are present at the edges of the composite beam at Zl=O z2=L the following conditions have to be fulfilled Ns (2 t) = 0 Nc (z t) = 0 M (z t) = 0 i = 1 2 j toto (19) Introducing the equilibrium eqs (11) (12) in eqs (19) we obtain

Ns (zp t )=~~)f ls (zp t)+i~f3 (Zi t)-Ns(t)=0

Ne (Zit)= n~~)dfle (zit)+~~df3 (z t)-dNe(t)]p(tt)=O (20)

M (Zi t)= 1~~edf le (Zi t) p (t t)+~Wf ls (Zi t)+ n~~~p (t t)+~~] df3 (Zp t)+ 1dMe(t)p (t t)+Ms (t)=0

As p(ttraquoO j tt and at t=to the second of eqs (20) implies

~~)f le (Zi to) + ~~f3 (Zi to) - Nc (to) = 0 (21 )

the second of eqs (20) at every time t it can be written as follows

~~)f lc (Zi t) + ~~f 3 (Zi t) - Ne (t) = 0 (22)

Furthermore the third of eqs (20) can be inverted by means of the solving kernel ()(tt) resulting

n ~~~)df loCi t) + ~Wdf ls (Zi t) P (t t)] + n~~ + ~~p (t t) J df 3 (Zit) = - ndMe(t) + dMs(t)P ( t f)] (23)

According to Fig 1 and referring to the coordinate system with origin in the centroid of the steel beam the following relationships can be written Ssy = 0 Scy = AcYGc Icyy = IcyyO + AcYG (24) with Icyyo moment of inertia of the slab with respect to its centroid Gc so the first of eqs (20) eqs (22) (23) take the simpler form

EsAsflS (~ t) = Ns (t)

EeoAe [fle (Zit)-yGc f3 (z))J =Nc (t) (25)

1[Ecolcyyo + EslsyyP (t t) J df 3 (Zp t) = 1[YGcdNc (t) - dMc(t) - dMs(t) P (t t)] from which the axial strain of the beam is immediately derived

flS (~ t) = Ns (t)EsAs (26)

On the contrary the axial strain of the slab and the curvature are coupled and they have to be determined by solving the second and the third of eqs (25) The initial conditions are given by the elastic solution of eq (10) and of eqs (11) (12) or (17) (18) respectively with p(toto)=1 or ()(toto)=1 They result

fle (z to) = flS (z to )_5-[Asfs (z to) - Ssyf (z to)] (27)kcll

~~)f lc (z to) + i~)f ls (z to) + [ i~) + ~~] f3 (z to) = N (z to) + Ns (to) + Nc (to) (28)

~~cflC (z to)+ ~~~)f ls (z to) +[~~~ + ~~~] f 3 (Zto)= -M(z to) - Ms (to)-Me (to) (29)

to which the boundary conditions have to be provided In the principal coordinate system of the steel beam at t=to eqs (25) become

275

EsAsfI1S (z to) = Ns (to)

EeoAe [ fI 1e (Z to) - YGe fI 3 (Z to)] =Ne (to) (30)

[Eeolcyyo + ESlsyy ] fI 3 (Zp to) Y GeNe (to) -Me (to) - Ms (to)

from which it results

Ns(to)fI1S (ZitO)~

s s

Mc(to)+Ms(to)-YGcNe(to) fI3(zto) (31 )

Eeolcyyo + Eslsyy

Uto) _ Me (to) + Ms (to) - y GJU to) fI 1e (Zi to )-

_ E A EEl YecO c eolcyyo + ssyy

The problem so formulated can be conveniently solved by means of numerical algorithms in particular by adopting the Finite Difference Method in the space domain expressing the time integrals by means of the formula of trapezia as discussed in [1] When neglecting the connector deformability ie fI1c=fI1S=fI1 eq (10) changes into an identity while eqs (17) (18) assume the following matrix form

1[~ +~P(t t)]dfl(z t)= 1[dg(zt)+ d9s (t)]P(tt)+ge (0 (32)

with Beij = c~c) BSbullij = c~s) gT = [N M] 9 = [Ns MsJ 9 = [Ne Me] Eq (32) represents a system

of Volterra integral equations which can be solved by means of numerical methods as in the previous case or by recurring to the Reduced Relaxation Function Method [4] Eq (32) points out that the long term behaviour of simply supported composite beams with rigid connectors can be determined only by recurring to sectional analysis

3 Case study Let us consider the Simply

supported composite bridge of Fig 3 having the transverse section represented in Fig 4 The analysis 318 m

has been performed by considering i

332 m the f1exurally equivalent composite -------------------------4jr-shy

beam reported in Fig 5 The elastic Fig 3 Structural schememodulus of the steel beam is Es=210000 MPa and the cylindrical characteristic compressive strength of concrete is fck=30 MPa The connection system includes 3 studs with diameter ltD ch=20 mm and 30 cm inter-space along the longitudinal axis According to [5] the connector stiffness can be evaluated by means of the following expression

Ppk~=-~-~----~ in [N mm] (33)

dsh (016 -00017fek )

with dSh diameter of the shank and

2 ()040P = 4 3-~ nltDCh f o35 f065 Eco1 (34)p [ ~ 4 ek su E

~n~ s

where nch is the number of connectors within a shear span and fsu is the strength of the steel studs Assuming fsu=500 MPa eq (34) becomes

4Jo40P = 43- 11 31430035500065 33 middot10 =117852 N (35) p [ J3 15900300 1 (21 104

Hence the connector stiffness results

k = 117852 ~=541 MPa (36) ch 20(016-0001730) 300

The structure is subjected to the self weight the permanent load due to the railway ballast and to the variable load related to the transit of the trains Furthermore concrete shrinkage and seasonal temperature variations are considered

276

1260 250 ~~

430 400 430 lt0

670

0 ltT 2 N

100 6 t---------t

Fig 4 Transverse section of the bridge [cm] Fig 5 Transverse section of the flexurally equivalent beam [cm]

Referring to the beam of Fig 5 the permanent load is g=305+37 5=680 kNm In order to determine the variable load due to the railway carriages the Italian Code [6] has been adopted The most unfavourable load condition results the simultaneous presence of two trains in particular the SW2 train on one track and the SWIO train on the other one Taking the transverse distribution of the load

~ between the four beams into account and introducing the TP=30degC dynamic coefficient lt1gt=2 16(-)318-0 2)+0 73=1 13 the variable

~Tltnr=lOdegC load acting on the considered beam is q=8245 kNm Both ~TsSP=l OdegC permanent and variable loads are applied at to=28 days

Regarding the imposed deformations concrete shrinkage has been evaluated according to CEB MC90 [7] assuming the following rheological parameters RH=70 ho=268 mm According to [8] a thermal gradient with a linear distribution has been applied on concrete slab while a constant distribution is

Fig 6 Thermal gradient assumed to act on steel beam as represented in Fig 6 According to [9] assuming the casting of the concrete slab in

spring the seasonal variation can be expressed in the following form

tT=tT sin2n(t-tO ) (37)max 365

Both shrinkage and thermal gradient act starting from to=7 days

~

x 10 SECTIONAL DEFO RMATION) Let us firstly consider the structural response under static 15---__-~--~--__

g

loads and shrinkage The time evolution of the longitudinal Ie I

strains ~1C ~1S at z=Ll2 is represented in Fig 7 They have been evaluated by solving eqs (10) (11) (12) When kch=oo it results ~1C=~1S while when considering

05 the connector deformability the strain discontinuity allowed at the interface between steel and concrete makes

Or-- __==~--------~ the longitudinal strains of the two structural elements different The influence of the connection stiffness results

middot0 5 quite small in the case of shrinkage As a consequence = k =

Ie I s eh t [days] the axial load in concrete slab is expected to be nearly -1---~~---~-~~_~-J

1 10 100 1000 10000 constant along a large part of the beam span The strain distribution in the transverse section is reported

Fig 7 Time evolution of the longitudinal in Fig 8 referring to z=Ll2 The discontinuity at steelshystrains l1c l1s at mid-span concrete interface is clearly visible A significant increase

of the sectional curvature can also be observed together with the downwards shifting of the neutral axis The corresponding stress distribution is shown in Fig 9 where for the sake of clarity the stresses in the slab are multiplied by the factor 10 A significant decrease of the stresses due to permanent load takes place in the slab By considering the tensile stresses related to restrained shrinkage we can conclude that a marked reduction of the compressive stresses in concrete slab develops in time Shrinkage effects can be particularly unfavourable at the end sections where the stresses related to permanent loads vanish In case of rigid connection the tensile stresses at the supports are Oc=3 MPa gt fc1k=2 MPa For kch=541 MPa tensile stress at the lower fibre of the slab reaches Oc=fc1k at z=Ll10 However the compressive stresses generated by permanent loads keep everywhere Ocltfc1k Furthermore an increase of the stresses takes place in the upper fibres of steel beam Considering also shrinkage effects at final time the total stresses in the beam become more than two times then the elastic ones

277

MID SECfIONmiddot 10 STRAliS MID SECfION I STRAliS MID SECfIONmiddot I STRESSES MID SECfIONmiddot I STRESSES o

30 I---~H-----I

270 L_--------L

0

30 f--1--If---I-----I

o 0 0 r--~-------- 30

270 shy-05 05 IP

f--~r+----I

1 00~-------gt1-J 270 -100---------

Fig 8 Strain distributions at t=to t=oo for kch=541 MPa Fig 9 Stress distributions at t=to t=oo for kch=541 MPa

SHEAR FLOWS The longitudinal distribution of the shear flows generated by permanent and variable loads and by shrinkage at initial and final time are reported in Fig 10 When neglecting the connector deformability the shear flow diagrams are affine to the shear distribution which is linear in case of uniformly distributed loads and vanishes when considering concrete I shrinkage Nevertheless some differences can be noticed The shear flow distribution related to permanent load decreases in time owing to the stress redistribution between concrete slab and steel beam Furthermore for kch~oo the shear flow produced by shrinkage tends to become very large at the end sections and to strongly reduce the related developing length Consequently as previously said the Fig 10 Shear flows at t=to and t=oo axial load in the slab tends to become nearly constant along

Fig 11 Axial loads in concrete slab at t=to t=oo

the beam span When introducing the connector

AXIAL LOADS IN CONRETE SLAB deformability the shear flow distributions smoother at the edge zones of the beam

become

The axial load in concrete slab is reported in Fig 11 while the bending moment diagrams in concrete slab and in steel beam are shown respectively in Fig 12a b The axial load produced by permanent loads decreases in time owing to concrete creep Otherwise the axial load generated by shrinkage increases in time exhibiting a constant distribution along the longitudinal axis for rigid connectors On the contrary the connector flexibility makes the axial load produced by permanent loads decrease of about 10 at mid-span furthermore the

159 Lim] 318 zone of variability of the axial load in the slab due to shrinkage increases

BENDIlK MOMENTS INCONRETE SLAB BENDIlK MOMENTS IN STEEL GIRDER

600

400

200

-400

middot600

sh

q

_ kch~S41 MFa k =

- e11

159 Lim] 31 8

2000

-4000

middot6000

E a t

_ k =S41 MFa eh

60 k = - eh

to 70

0 159 Lim] 318 159 318

30

40

50

(a) (b)

Fig 12 Bending moments in concrete slab (a) and in steel beam (b) at t=to and t=oo

The bending moments acting in concrete slab and steel beam are very different as the flexural rigidity

278

of the beam is significantly higher with respect to that of concrete slab The bending moment induced in the slab by permanent loads presents a significant decrease in time while an increase takes place in the beam Taking also the flexural effects produced by shrinkage in the steel beam into account we observe that at final time the bending moment in the beam become about 15 times the initial one

DISPLACEMENTS The transverse displacements at initial and final time are reported in Fig 13 In the elastic domain the deflections obtained by considering or not the connector deformability differ of about 11 and this percentage decreases in time This as the influence of creep deformations decreases when increasing the connector deformability As a consequence the deflections due to permanent load increase in time of about 34 in case of rigid connection and of 27 when the connector deformability is taken into account The transverse displacements due to shrinkage are less influenced by the connector flexibility because the rate of shear flow

Fig 13 Transverse displacements at t=to t=oo along the longitudinal axis which generates the strain discontinuity between the slab and the beam according to

eq (3) is confined to the edges of the composite beam The total deflections significantly increase in time as a consequence both of creep and of shrinkage In particular they increase of 53 when kch=oo and of 43 when kch=541 MPa

Let us now investigate the long term behaviour of the composite beam of Fig 3 when a non-linear

L [m] 318

thermal gradient is active

270 h [em[

240 1----~+-----_

210

180

150

--_ 1 MP I MT

120

90

60

t I30

o+---~------~

OOE+OO 10E-04 20E-04 30E-04 -4

(a)

____ =cO-eraSt ic-aNl~

__ k=541 lItPa elastIC analysis

k=Q) viscoelastic analysis

k- 541 tvtPa viscoelastic analysis

__ _ _

12

(b)

210

180

E 150 c 120

90

Fig 14 Strains (a) and stresses (b) at z=U2 at t=98 days

- bull shy k- 00 Viscoelasl JC ana lysLS

k- 541 MPa viscoelastic analysis

h [em] I270

240 1---~--T-------i

210 I180 180

150 ] 150

120 c 120 --_ k-oo

90 k- 541 MP

60 a6T

30

O +---~--~--~

OOE+OO 10E-04 20Emiddot04 30E-04 -4

(a) 12

(b)

Fig 15 Strains (a) and stresses (b) at z=U10 at t=98 days

The strains and the stresses in the mid-span section and in the section at z=U10 produced by the thermal strain varying in time according to eq (37) are respectively sketched in Fig 14 Fig 15 By comparing Fig 14 and Fig 15 it clearly appears that the connector deformability only affects the structural response at the edge zones of the beam In fact the constancy of the thermal strain along the longitudinal axis generates a nearly constant shear flow distribution along the beam axis The extension of the zones with variable shear flow increases by increasing the connector deformability According to the hypothesis of plane sections steel beam

I and concrete slab keep thei same curvature and the diagram of the total strains maintains a unique neutral axis even if the thermal strain represented by the dotted line

is non-linear As a consequence a self-equilibrated state of stress arises making the neutral axis of the stress distribution evaluated in the elastic domain coincide with the intersection between the straight line of the total deformation and the diagram of the imposed one The stress diagrams show the favourable effect produced by concrete creep to which a reduction of the stresses obtained by means of elastic analyses is consequent both in steel beam and in concrete slab

279