Experimental and analytical studies of prestressed concrete girders with corrugated steel webs

ORIGINAL ARTICLE

Experimental and analytical studies of prestressed concretegirders with corrugated steel webs

Xiao-Gang Liu • Jian-Sheng Fan • Jian-Guo Nie •

Yu Bai • You-xu Han • Wei-hong Wu

Received: 4 August 2013 / Accepted: 14 May 2014

� RILEM 2014

Abstract In this study, an innovative prestressed

concrete girder with corrugated steel webs and non-

continuous perfobond-rib (PBL) shear connectors is

developed and studied. Experimental research indi-

cates that this composite girder has a reliable shear

connection and several advantages over composite

girder systems with continuous PBL connectors.

These advantages include better cracking resistance

capacity and greater deformability. Using a com-

mercial finite element (FE) package (MSC Software

Corporation. Marc 2010 user’s guide), a detailed

beam-shell FE model is established to simulate the

nonlinear performance of the prestressed composite

girder during the loading process. Comparison of the

FE analysis with the experimental research indicates

that the FE model accurately (with good conver-

gence and satisfactory efficiency) simulates complex

nonlinear mechanical behaviors, including the

stress–strain distributions of the concrete flanges

and the corrugated steel webs, the stress of the

unbonded prestressing tendons and the slippage

behavior of the non-continuous PBL connectors.

From the experimental research and FE analysis, the

mechanisms of the prestressed composite girder

concerning the stress–strain distributions of the

concrete flange and the corrugated steel webs, shear

force distribution mechanisms between the concrete

flange and the corrugated steel webs and the stress

increment law of the unbonded prestressing tendons

are explored in detail, and a shear deformation

prediction approach for prestressed composite gird-

ers is proposed. Finally, a strength and deformation

prediction approach based on elementary beam

theory is proposed for the prestressed composite

girder. This approach accurately considers prestress-

ing tendons that contribute to girder flexural

moment. Furthermore, it predicts stiffness and

deformation of the girder at yielding load, which

can account for deficiencies in the design procedure.

Experimental validation indicates that the proposed

approach provides satisfactory prediction results

regarding strength and deformation in prestressed

composite girders with appropriate reinforcement in

tensioned bottom concrete flanges.

Keywords Prestressed � Composite girder �Corrugated steel web � Nonlinear FE analysis �Strength � Deformation

X.-G. Liu � J.-S. Fan (&) � J.-G. Nie

Key Laboratory of Civil Engineering Safety and

Durability of China Education Ministry, Department of

Civil Engineering, Tsinghua University, Beijing 100084,

China

e-mail: [email protected]

Y. Bai

Department of Civil Engineering, Monash University,

Clayton, VIC 3800, Australia

Y. Han � W. Wu

Gansu Province Transportation Planning Survey and

Design Institute Co., Ltd., Lanzhou 730030, China

Materials and Structures

DOI 10.1617/s11527-014-0334-3

1 Introduction

The prestressed concrete girder with corrugated steel

webs is an innovative structure that was derived from the

traditional prestressed concrete girder and prestressed

composite girder with flat steel webs. Such girders were

first employed in bridge construction by Campenon

Bernard BTP in France [1]. Relative to traditional

girders, this girder has a larger spanning capacity and

requires lighter temporary supports due to its reduced

dead load. Moreover, the reduced impact of concrete

shrinkage and creep and the reduced prestressing loss

due to the web accordion effect, the higher shear

buckling resistance due to corrugation, and the absence

of durability problems due to probable concrete web

cracking make this girder more reliable than traditional

girders. Given these advantages, bridges with such

girders can be built to span large rivers and valleys. A

number of such composite bridges, including the

Cognac, Charolles, Shinkai and Ginzan-Miyuki bridges,

have been constructed [2–4]. Similar composite bridges

are also under construction or have been constructed for

highways in China. Several researchers have investi-

gated the mechanical behavior of prestressed concrete

girders with corrugated steel webs, and some prelimin-

ary conclusions have been recognized. Experimental

studies have confirmed that the flexural moment is

almost entirely attracted by the prestressing tendons and

rebar in the bottom flange in tension and by the concrete

in the top flange in compression. In contrast, most of the

shear stress is attracted by the corrugated steel webs [5–

8]. Moreover, the shear capacity and deformation of the

corrugated steel webs have been proposed and refined

[9–11]. Based on the above conclusions, some compu-

tational programs established by elementary beam

theory have been programmed to simulate the load–

displacement curve of the composite girder. However,

these simulations could not provide detailed stress–

strain distribution for research into the mechanism of the

prestressed composite girder [7].

Until now, some critical issues regarding the mech-

anisms of prestressed composite girders have remained

unclear. The stress increment law of the unbonded

prestressing tendons in such girders remains unclear. In

addition, a simplified model and prediction approach to

tendon stress increment is still lacking. Furthermore,

there is insufficient understanding of the influence of the

diaphragm, arranged to restrain girder torsion, on the

shear stress distribution in the corrugated steel webs in

the longitudinal direction and the shear force distribu-

tion mechanism between the concrete flange and the

corrugated steel webs. Although it is reported that most

of the shear force is attracted by the corrugated webs,

further investigation is needed for girders with a low

span–depth ratio. Moreover, further research is needed

into the approach for deformation prediction in such

girders, which considers the contribution of the shear

deformation of the corrugated steel webs. Nie proposed

an effective stiffness method that can account for the

shear deformation of corrugated webs in the prediction

of girder deformation. However, it is based on the

assumption of linear-elastic material, making it inap-

plicable for predicting girder deformation after concrete

cracking [12]. Li proposed a deformation prediction

approach based on elementary beam theory and a linear-

elastic material assumption, but the shear force distri-

bution between the concrete flange and the corrugated

webs is not specified [13].

The PBL shear connector is widely used in pre-

stressed composite girders. The mechanical behavior

and simplified shear force–slippage relationship of this

connector have been previously studied [14–16]. How-

ever, the continuous PBL connector can possibly impair

the integrity of the concrete flange, causing problems

during construction of the reinforcement assemblage.

In response to the problems above, a new prestressed

composite girder with non-continuous PBL connectors,

as shown in Fig. 1, is designed and tested in this study.

Moreover, using the MSC commercial finite element

(FE) package Marc (2010), a detailed beam–shell FE

model is also established, which can accurately simulate

complex nonlinear mechanical behaviors, including the

stress–strain distribution of the concrete flanges and the

corrugated steel webs as well as the stress of the

unbonded prestressing tendons and the slippage behav-

ior of the non-continuous PBL connectors, to simulate

the nonlinear performance of the prestressed composite

girder during the loading process. On the basis of

experimental research and FE analysis, the mechanism

of the prestressed composite girder and the stress–strain

distribution of the concrete flange is explored in detail.

As well, the shear stress distribution in the corrugated

steel webs and the behavior of the shear force distribu-

tion between the concrete flange and the corrugated steel

webs are analyzed. A shear deformation prediction

approach for the prestressed composite girder is

proposed, the stress increment law of the unbonded

prestressing tendons is discussed, and a simplified stress


increment model of the tendons is proposed. Finally, a

strength and deformation prediction approach based on

elementary beam theory is proposed for such girders.

2 Experimental specimens and measurements

Detailed FE analysis of prestressed composite girders

has indicated that parameters including the rebar

yield strength fyr and the steel flange yield strength fys

have a determinant influence on the stress increment

of unbonded prestressing tendons, and parameters

including the reinforcement ratio q of the concrete

flange in tension and the area of the steel flange in

tension also have a significant influence [17]. There-

fore, three specimens were fabricated with different

parameters of reinforcement yield strength fyr, steel

flange yield strength fys, and reinforcement ratio q of

concrete flange in tension, to validate the mechanical

behavior of the non-continuous PBL shear connector.

These specimens were also fabricated to explore the

shear force distribution mechanism between the

concrete flange and the corrugated webs, as well as

the stress increment law of the unbonded prestressing

tendons.

2.1 Specimen parameters

The geometric dimensions and loading arrangement of

the specimens are shown in Fig. 2. Concrete diaphragms

with thicknesses of 250 and 150 mm were arranged at the

support and loading points respectively to constrain the

torsion of the girder and to satisfy the local pressure

requirement. The basic geometric dimensions and cor-

rugations of the webs were the same for the three

specimens. The parameter variations also included

reinforcement strength, steel flange strength, and rein-

forcement ratio of the bottom flange in tension, as shown

in Table 1. The measured thickness of the steel plates and

the measured strength of the rebar, the steel plate, and the

concrete are shown in Tables 2 and 3. The reinforcement

details are shown in Fig. 3a, b, and all the longitudinal

reinforcement has a diameter of 8 mm. The arrangement

of the prestressing tendons is shown in Fig. 3c. U15.2

tendons with the ultimate strength of 1860 MPa were

used with an initial prestressing stress of approximately

1,000 MPa. Constructional details for the corrugated

webs and the PBL connectors are shown in Fig. 4.

2.2 Specimen measurements

The arrangement of the strain gauges and displacement

gauges is shown in Fig. 5. Some strain rosettes and

crossing displacement gauges were arranged along the

girder in the longitudinal direction to measure the shear

strain and deformation of the corrugated webs. Dis-

placement gauges were also arranged under the bottom

flange to measure the deflection of the girder. Two

displacement gauges were placed between the concrete

flange and the steel flange to measure relative slippage.

Furthermore, strain gauges were arranged to measure

the strain development of the reinforcement, the con-

crete flange, and the steel flange at the midspan cross-

section 1-1. In addition, force gauges were arranged

below the anchorage device to measure the internal

force of the tendons.

3 Finite element model of the prestressed

composite girder

A reliable FE model was necessary to explore

mechanical behavior and perform detailed parametric

analysis. A detailed beam-shell FE model was

established with reference to the literature [18],

where the beam-shell element has been proven

reliable, accurate and efficient.

3.1 Element selection and assemblage

A detailed FE model was established with the MSC

commercial FE package MSC Marc 2010 [19]. In

top concrete flange

bottom concrete flangeunbonded prestressing tendons

corrugatedsteel web

concrete diaphram

PBL connector

corrugated web

steel flange

Fig. 1 Prestressed concrete

girder with corrugated steel

webs


the FE model, the steel plate members are simulated

by three-dimensional thick shell elements, the

concrete flanges and the concrete diaphragms are

simulated by layered shell elements divided into

several concrete layers and reinforcement layers

along the thickness direction, and the unbonded

prestressing tendons with pretension are simulated

by three-dimensional truss elements with an initial

axial stress. Slippage between the concrete flange

and the steel flange induced by the deformation of

the PBL shear connector is simulated by nonlinear

springs in the longitudinal direction of the girder.

Rigid bars with length equal to half the thickness of

the concrete flange are used to link the middle plane

of the layered shell element representing the

concrete flange and the shell elements representing

the steel flange, defining the position of the slippage

surface between the concrete flange and the steel

flange. One node of the bar is coupled with the

corresponding node of the layered shell element

representing the concrete flange. The other node of

the bar coincides with the corresponding node of the

shell element representing the steel flange. In

addition, the nonlinear spring simulating slippage

is used to link the two coincident nodes in girder

longitudinal direction, while the other translational

and rotational displacements of the coincident nodes

are constrained to be the same. Unbonded prestress-

ing tendons are embedded in the bottom concrete

flange, with a rubber bushing to isolate the tendons

and the concrete so that free slippage of the tendons

is allowed. Thus, at the anchorage point of the

tendons, the node of the truss element representing

the tendon is coupled with the corresponding node

of the layered shell element representing the

concrete flange, and all displacements except the

longitudinal one of the other nodes of the truss

elements representing the tendons are coupled with

the corresponding nodes of the layered shell element

representing the concrete flange to simulate the free

slippage of the unbonded tendons in the concrete

flange. The assembly of the FE model is illustrated

in Fig. 6.

100 2000 2000 2000 1006200

8026

650

396 150 250

P/2 P/2

corrugated webs

Fig. 2 Geometric

dimensions and boundary

conditions of specimen

Table 1 Steel plate and

reinforcement of different

specimens

No. Steel

web

Steel

flange

Reinforcement Reinforcement

ratio (%)

CBCW1 Q345B Q235B HPB235 1.26

CBCW2 Q345B Q235B HPB235 2.51

CBCW3 Q345B Q345B HRB335 1.26

Table 2 Material properties of steel plate and reinforcement

Strength Thickness/

diameter (mm)

fy (MPa) fu (MPa)

Steel plate

Q235B t = 2.8 319.7 449.7

Q345B t = 2.8 403.5 516.1

Reinforcement

HPB235 /8 276.7 438.6

HRB335 U8 381.5 601.1

Table 3 Material properties of concrete

No. Position Nominal

strength

fcu (MPa)

CBCW1 Top flange C50 46.9

Bottom flange C50 50.1






(a) reinforcement of CBCW1&CBCW3

800

500

80050

266

8039

6

7x110=770 15

20 4x115=460 20

4040

ϕ15.2-1860

φ8@65

15

φ8@65

800

5026

680

396

7x110=770 1515

φ8@65

φ8@65

50020 9x51=460 20

500125 250 125

5026

680

396

(b) reinforcement of CBCW2 (c) prestressing tendons arrangement

Fig. 3 Reinforcement and arrangement of prestressing tendons of the specimens

60

40

40

37°

36

R30

6048

6048

216

3

3060

260

12 36

60 60 48

601

continuous PBL connector

10

48

40 60 40 60

1

3

12

1010

30

1010

10

33

Non-continuous PBL connector

(a) elevation view of the web (b) sectional view of 1-1 (c) corrugations of the web

t=3mm

Fig. 4 Construction details of corrugated steel webs and PBL connectors

(a) arrangement of the strain rosettes

2100 1000

250 100

396

300 300 300 300 300 300

100

396

500500600600900

200

200

750 400 400

4-14-24-34-44-5

4-8

100

1

1

1

1

4-9

A B C D EF

G

H I J

(b) arrangement of the displacement gauges

Fig. 5 Arrangement of

strain rosettes and

displacement gauges of the

specimens


3.2 Constitutive laws of materials

The materials of the prestressed composite girder are

concrete, steel plates, rebars, prestressing tendons and

PBL connectors. As the composite girder obviously fails

by flexural moment, the nonlinear uniaxial compressive

stress–strain relationship proposed by Rusch [20] was

adopted and idealized, as illustrated in Fig. 7a, where:

fe = 1/3fc and the corresponding strain ee is idealized as

the linear-elastic limit point, and the elasticity module is

idealized as Ec = fe/ee. The cracking behavior of the

concrete is described by the smeared crack model. The

relationship between the tensile stress and the crack width

is idealized to be linear, and the fracture energy Gf, which

is defined as the area under the stress–crack width curve

in Fig. 7a, can be calculated by applying the recommen-

dations of Comite Euro-International du Beton—Fede-

ration Internationale de la Precontrainte (CEB-FIP) [21].

The softening module Ets after cracking was idealized as

linear-elastic. This value was determined by the area gf

under the softening stress–strain curve and the cracking

stress ft, where gf = Gf/hc can be determined by the

fracture energy Gf and the crack bandwidth hc of the shell

element representing concrete flange [22] and ft = 1.4(fc/

10)2/3 [21]. The Poisson ratio can be selected as 0.17, and

the shear factor accounting for the shear force transferred

across the concrete crack can be selected as 0.5 for Mode-

I flexural fracture [18]. A trilinear model was adopted to

describe the uniaxial nonlinear stress–strain relationship

of the rebar and the steel plates, as illustrated in Fig. 7b,

where the elasticity module Es = 206 GPa and the initial

hardening strain eh = 2 %. The uniaxial nonlinear

stress–strain relationship of the prestressing tendons, as

described in Eq. (1), is illustrated in Fig. 7c, where

Ep = 195 GPa, f0.2 = 0.85fb and fb is the ultimate

strength of the tendons. The nonlinear spring representing

the shear deformation of the PBL connectors is described

by a shear-slip curve, as illustrated in Fig. 7d. As the

slippage between the concrete flange and the steel flange

is very slight during the loading procedure, an idealized

bilinear shear–slip relationship, based on research by Ahn

[15], is adopted for the nonlinear spring, where:

s0 = 1 mm, su = 12 mm, and Vu can be calculated by

Eq. (2). In Eq. (2), hsc and tsc are the height and thickness

of the PBL respectively, d is the diameter of the hole in

the PBL, Atr and fyr are the area and the strength of the

rebar crossing the hole in the PBL, and fc is the cylinder

strength of concrete.

e ¼ rE0

þ 0:002r

f0:2

� �13:5

ð1Þ

Vu ¼ 3:14hsctscfc þ 1:21Atrfyr þ 3:79pd

2

� �2 ffiffiffiffifc

p

ð2Þ

4 Results and discussions

4.1 General behavior and numerical model

validation

The three specimens were tested to failure by mono-

tonic loading. During the experimental procedure, all

Z

XY

truss elementof tendons

Layered shell elementof concrete flange

slipping constraintin X direction

nonlinear spring rigid bar

layered shell elementof concrete flange

shell element ofsteel flange

Fig. 6 FE model of unbonded prestressed concrete girder with corrugated webs


the specimens were subjected to a load of 120–130 kN

when the bottom flange cracked, as illustrated in

Fig. 8a. With increasing load, the reinforcement in the

bottom flange and the steel flange in tension gradually

approached the yield stress, and the load–deflection

curve presented a point with the maximum curvature

around the yielding load, as illustrated in Fig. 8a. After

the yielding load was reached, the stiffness of the

composite girder decreased sharply and the deflection

of the girder increased rapidly when the load increment

was almost entirely supported by the prestressing

tendons in tension and the top concrete flange in

compression. All specimens failed by crushing of the

top concrete flange under the compression induced by

the flexural moment, as shown in Fig. 9. The concrete

crack in the bottom flange distributed uniformly in the

σ

fy

-εh -εy 1Es

εy εh

Es

1

1

ε

0.01Es

0.01Es

1 -fy

Es=2×105N/mm2

υs=0.3

εh=2.0%

(a) concrete

σ(σnn)

Et=Ec ft Ets

11gf

−ε0=-0.002 −εeεt ε1

Ecεnn

cr

- fe= - fc/3

σnn

ftGf

ω0 ω- fc

σ

ε1

Ep

fe

fb

s0

0.85Vu

su

Vu

(b) rebar and steel plates

(c) prestressing tendons

(d) PBL connectorss

Fig. 7 Constitutive laws of materials

0 50 100 150 2000

80

160

240

320

load

/kN

mid-span deflection/mm0 50 100 150 200

0

80

160

240

320

0 50 100 150 2000

80

160

240

320

TESTFEA

CBCW1 CBCW2 CBCW3

load

/kN

load

/kN

mid-span deflection/mm mid-span deflection/mm

TESTFEA

TESTFEA

a. cracking load

b. yielding load

a. cracking load

b. yielding load

a. cracking load

b. yielding load

(a) load-midspan deflection curve

0 30 60 90 120

150

180

210

240

tend

onin

t ern

alfo

rce/

k N

0 30 60 90 120

150

180

210

240

0 30 60 90 120

150

180

210

240

CBCW1 CBCW2 CBCW3

tend

onin

tern

alfo

rce/

kN

tend

onin

tern

alfo

r ce/

kN

TESTFEA

TESTFEA

TESTFEA

mid-span deflection/mm mid-span deflection/mm mid-span deflection/mm

c. nominal yeilding point c. nominal yeilding point c. nominal yeilding point

(b) tendon internal force-midspan deflection curve

Fig. 8 Load–midspan deflection curve and tendon internal force–midspan deflection curve


segment, which was only subjected to flexural

moment, between the two loading points, as shown in

Fig. 9.

As the experimental measurements of forces and

displacements are relatively accurate and reliable,

the girder load–midspan deflection curve and the

tendon internal force–midspan deflection curve of

the experimental result and the FEA result were

compared to validate the numerical model, as shown

in Fig. 8a, b. It can be seen that both curves of the

FEA results fits perfectly with the experimental

results, indicating that the proposed numerical

model accurately simulate the mechanical behavior

of the prestressed composite girder and is able to

track the mechanical behavior of the prestressing

tendons during the loading process. The tendon

force of the FEA fits the tested force perfectly when

the deflection does not exceed 30 mm. As the

constitutive law of tendons was idealized in the

numerical model, some differences between the

experimental and FEA tendon forces were observed

around the nominal yielding points of the tendons,

which is assumed in the constitutive laws of the

tendons in the numerical model. Nevertheless, the

results from numerical simulation of the tendons

force were sufficiently accurate, and the numerical

model was reliable for performing detailed research

regarding the mechanical behavior of prestressed

composite girders.

4.2 Strength and deformation

The yielding load Py, the ultimate load Pu, the

yielding deflection Dy, the ultimate deflection Du,

Pu/Py and the ductility coefficient Du/Dy are listed in

Table 4. As the deformation capacity of the pre-

stressed composite girder is excellent, the ultimate

deflection/span ratio will usually exceed 1/20, which

cannot satisfy service demands. Thus, the ultimate

state is defined at a deflection/span ratio of 1/50.

Further, the corresponding load and deflection are

defined as the ultimate load Pu and the ultimate

deflection Du respectively. The yielding load Py and

the yielding deflection Dy can be determined by

Fig. 10. The Pu/Py of all specimens exceeds 1.3, and

the ductility coefficient of all specimens approaches

or exceeds 5, exhibiting excellent ductility and

deformation capacity.

(a) CBCW1

(b) CBCW2

(c) CBCW3

Fig. 9 Failure modes and concrete crack distribution of

specimens

Table 4 Strength and ductility of specimens

No. Yielding point Ultimate point Pu/Py Du/Dy

Py

(kN)

Dy

(mm)

Pu

(kN)

Du

(mm)

CBCW1 177 22.5 245 120 1.38 5.33

CBCW2 230 24.9 305 120 1.33 4.82

CBCW3 203 24.8 283 120 1.39 4.84

CB D

E

A

Pu

Py

O Δ y Δ u mm

kN

Fig. 10 Determination of ultimate and yielding points


4.3 Concrete cracking and shear connector

slippage

The distribution of concrete cracking in the bottom

flange is shown in Fig. 9. Particularly, some cracking

begins at the top surface of the bottom concrete

flange, which cannot be explained by the plane-

section deformation property. This phenomenon is

caused by the anti-arching effect of the prestressing

tendons on the bottom flange, which can induce

significant tensile strain on the top surface of the

bottom concrete flange because the out-of-plane

stiffness of the thin bottom concrete flange is

relatively weak. The relationship of cracking width

to load is shown in Fig. 11, where P/Py is the

indicator of load and Py is the yielding load. The

cracking load Pcr for the specimens is approximately

0.6 Py. If the serviceability limit for concrete

cracking width is defined as 0.2 mm, the correspond-

ing service load for the specimens can approach or

exceed 0.9Py.

Traditional continuous PBL connectors have been

confirmed to have good mechanical performance,

ensuring collaborative work between concrete flange

and corrugated steel webs. The same performance

can be obtained using non-continuous PBL connec-

tors. The load–slippage relationship is provided in

Fig. 12. As can be seen, the slippage between the

concrete flange and the steel flange is very slight

during the whole loading process. Traditional con-

tinuous PBL connectors are subjected to both tensile

force due to girder flexure and interfacial shear force.

In contrast, non-continuous connectors are subjected

only to pure interfacial shear force, which may

relieve the slippage.

4.4 Shear stress of corrugated steel webs

Based on the measurements of the strain rosettes A–G in

Fig. 5, the corresponding relationship of load–shear

stress can be obtained, as shown in Fig. 13. The curves

of rosettes C, F, and G almost completely coincide,

indicating that the shear stress distributes uniformly

along the girder depth direction. Comparison of the A–E

rosette curves indicates that the shear stress varies along

the girder longitudinal direction and the rosettes near the

diaphragms display lower shear stress due to the

constraint of the diaphragm. As all the specimens fail

by flexure and the shear stress is relatively insignificant,

some of the rosettes produce a curve that deviates from

the basic law due to measurement deviation.

Detailed numerical analysis was conducted to

provide additional detail of shear stress distribution,

as shown in Fig. 14a. The shear stress distribution

law was similar for the cracking load, yielding load

and ultimate load.

The shear stress variation in the longitudinal direc-

tion is presented in Fig. 14b, where the distances 0 and

2,000 mm represent the diaphragms at the support and

the loading point respectively. As can be seen, the shear

stress decreases gradually within a distance of 500 mm,

which is approximately twice the web height, from the

concrete diaphragms. The shear stress near the edge of

the diaphragm is approximately 70–75 % of the shear

stress occurring outside the regions affected by the

diaphragm constraint. Moreover, the shearing force

attracted by the corrugated webs Vw can also be

calculated from the web shear stress and geometric

parameters, thereby calculating the ratio of Vw to the

girder shear force V. The relationship of load-Vw/V is

presented in Fig. 15 for the regions where the shear

stress is not affected by the concrete diaphragms. As can

0.6 0.8 1.00.0

0.1

0.2

0.3

0.4cr

acki

ngw

idth

/mm

P/Py

CBCW1CBCW2CBCW3

Fig. 11 Relationship of concrete cracking width–loads

0 100 200 300

0.0

0.05

slip

page

/mm

load/kN

CBCW1CBCW2CBCW30.1

0.15

Fig. 12 Relationship of PBL connector slippage–loads


be seen, Vw/V is approximately 76 % for the specimens,

despite the load variation.

To explore the variation law of Vw/V, equilibrium

analysis was performed on the segment of the

composite girder, as shown in Fig. 16a. Further, Eq.

(3) was obtained by force equilibrium. Experimental

research confirmed that the flexural moment was

almost entirely attracted by the flanges while the

corrugated webs only attracted shearing force. Thus

Fig. 16b can be derived from Fig. 18a and the

relationship of the parameters is shown in Eq. (4),

where h0 can be approximated by the distance

between the top and bottom concrete flanges. Next,

an equilibrium analysis was performed on the isolated

concrete flange and on the corrugated webs, as shown

in Fig. 16c, d. Eq. (5) was obtained by the force

equilibrium of Fig. 16c, where tfu and tfb are the

thickness of the top and bottom concrete flanges

respectively. Equation (6) was obtained by the force

equilibrium of Fig. 16d, where hw is the depth of the

corrugated webs. The final contribution of the

concrete flanges and the corrugated steel webs to

the girder shearing force was derived from Eqs. (3)–

(6), as shown in Eq. (7). The equilibrium analysis

above is based on the assumption that the axial force

T acts on the centroid of the concrete flanges.

However, the acting point of the axial force T should

be adjusted according to actual stress–strain

60 120 180 2400

20

40

60

80τ

/ MPa

load/kN

ABCDE

80 160 240 3200

25

50

75

100

0 75 150 225 3000

20

40

60

80

load/kN load/kN

(a) CBCW1 (b) CBCW2 (c) CBCW3

τ / M

Pa

τ/ M

Pa

ABDE

FG

ABCDEFG

Fig. 13 Relationship of load–web shear stress

6053.547.040.534.027.521.014.58.01.50-5.0

4237.332.627.923.218.513.89.14.4-3.0-5.0cracking load

yielding load

7567.059.051.043.035.027.019.011.03.0-5.0

ultimate load0 500 1000 1500 2000

30

45

60

75

90

/MP

a

distance to the support/mm

CBCW1-yieldingCBCW1-ultimateCBCW2-yieldingCBCW2-ultimateCBCW3-yieldingCBCW3-ultimate

(a) shear stress contour of CBCW1 (b) longitudinal shear stress variations

τ

Fig. 14 Distribution of shear stress on corrugated webs

CBCW1

CBCW2

CBCW3

0 100 200 300 4000.625

0.75

0.875

Vw/V

load/kN

Fig. 15 Relationship of load-Vw/V


distribution if T does not act on the centroid.

Numerical research was also performed to validate

the analytical results. A comparison of the numerical

results and analytical results is provided in Table 5,

where n = Vw/V and np and nfe are the analytical and

numerical results respectively. As can be seen, the

analytical results fit the numerical results perfectly.

V ¼ dM

dxð3Þ

M þ dM ¼ ðT þ dTÞh0

M ¼ Th0

V ¼ Vw þ Vfu þ Vfb

8><>: ð4Þ

Vfu þ Vfb ¼dT

dxðtfu þ tfbÞ=2 ð5Þ

Vw ¼dT

dxhw ð6Þ

Vfu þ Vfb

V¼ ðtfu þ tfbÞ=2

h0

Vw

V¼ hw

h0

8>><>>:

ð7Þ

4.5 Stress–strain distribution of flanges

The cross-sectional normal strain of the composite

girder cannot meet the plane section assumption due to

the accordion effect of the corrugated steel webs. The

cross-sectional normal strain distribution of the com-

posite girders is shown in Fig. 17, where height

indicates the distance of the gauges from the bottom

face of the girder. When P/Pu \ 0.6, the gauge strains

almost distribute along the same line and the sectional

curvatures of the top and bottom flanges are nearly

identical. When P/Pu = 0.8, the sectional curvatures of

the flanges differ due to yielding of the reinforcement

and the steel flange in tension. Moreover, the strain of

the steel flange is greater than that of the reinforcement

in tension due to the interfacial slippage between the

concrete flange and the steel flange.

Next, the relationships between the load and the

strain of the reinforcement and the steel flange are

presented, as shown in Fig. 18a. Initially, the strain

increment of the reinforcement is greater than that of

the steel flange due to its larger cross-sectional

eccentricity. However, the strain increment of the

steel flange is significantly greater than that of the

reinforcement after yielding load due to interfacial

slippage. A detailed stress contour of the numerical

analysis is presented to validate the experimental

research, as shown in Fig. 18b. The normal stresses

of the steel flange and the reinforcement are signif-

icantly different. Thus, there was a significant

difference between the strains induced by interfacial

slippage. Moreover, the reinforcement and the steel

flange yield in the midspan cross-section at yielding

load, indicating that the contributions of the rein-

forcement yield strength and the steel flange yield

strength can be accounted for. Further, because the

ratio of girder width to span was not large, a uniform

distribution of the normal strain along the girder

transverse direction was observed in both experimen-

tal research and numerical analysis. This finding

agrees with the results of previous studies regarding

shear lag effect.

V+dVM+dMM

V

qdx

dx

Vfu

qdx

dx

Vw

Vfb

Vw+dVw

Vfu+dVfu

Vfb+dVfb

T+dT

T+dT

T

T

VwVw+dVw

dT

dT

(a) (b) (c)dx

VfbVfb+dVfb

T+dTTdT

Vfu

qdxVfu+dVfu T+dT

TdT

(d)

Fig. 16 Equilibrium

analysis of girder segment

Table 5 Comparison of analytical and numerical results of

Vw/V

h/(tfu ? tfb) 2 3 4 5

np 0.67 0.80 0.86 0.89

nfe 0.68 0.79 0.87 0.92

np/nfe 0.99 1.01 0.99 0.97


4.6 Stress increment of prestressing tendons

The stress increment of the unbonded prestressing

tendons significantly influences the flexural behavior

of the composite girder. However, the strain of the

unbonded prestressing tendons does not coordinate

with the girder cross-sectional strain due to the free

slippage of tendons in the girder longitudinal direc-

tion. Moreover, the mechanism and prediction

approach of the stress increment of the tendons

remain unclear. The internal force of the tendons was

measured during the loading process. The relation-

ship between tendon internal force and midspan

deflection and that between tendon force increment

and load are shown in Fig. 19a, b respectively.

According to Fig. 19a, the tendon internal force is

almost linearly related to the midspan deflection

before the nominal yielding force of the tendons.

Little difference occurs among the specimens. The

tendon internal force of CBCW2 increases a little

more quickly because it has a greater reinforcement

ratio to the bottom concrete flange and the overall

deformation of the girder is a little larger. According

to Fig. 19b, the relationship between tendon force

increment and girder load obviously follows a

trilinear model that is based on the cracking point,

the yielding point, and the ultimate point, which can

be simplified as shown in Fig. 19c. Before the

cracking point, the internal force increment of the

tendon is limited and its contribution to the cracking

load Pcr can be neglected. After the yielding point,

the internal force increment of the tendons increases

rapidly and the actual force of tendons at the ultimate

load Pu will exceed the nominal yield force of the

tendons or approach the ultimate force due to the

deformation capacity of the prestressed composite

girder. Moreover, the tendon force increment of

CBCW1 is little different from that of CBCW2,

indicating that the reinforcement ratio has little

influence on the tendon force increment at the

girder’s yielding load. The increment rate of the

tendon force of CBCW3 is nearly the same as that of

CBCW1. In contrast, the tendon force increment of

CBCW3 is significantly greater than that of CBCW1,

0

100

200

300

400

-1 0 1 2 3 40

100

200

300

400

-1 0 1 2

(c) CBCW3

heig

ht/m

m

strain/103με

heig

ht/m

m

(b) CBCW2

0

100

200

300

400

-1 0 1 2 3 4

p/pu=0.2p/pu=0.4p/pu=0.6p/pu=0.8

heig

ht/m

m

(a) CBCW1

gauge

rebar

rebar steel flange

concrete p/pu=0.2p/pu=0.4p/pu=0.6p/pu=0.8

p/pu=0.2p/pu=0.4p/pu=0.6p/pu=0.8

strain/103μεstrain/103με

Fig. 17 Cross-sectional normal strain distribution of composite girders

0 50 100 150 200 250-500

0

500

1000

1500

2000

2500

stra

in/μ

ε

load/kN

CBCW1-rebarCBCW1-steel flangeCBCW2-rebarCBCW2-steel flangeCBCW3-rebarCBCW3-steel flange

32429326122919816613410371398

2852422001571157229-13-56-98-141reinforcement layer

steel flange

(a) load-strain relationship (b) stress distribution at yielding load

Fig. 18 Strain comparison of reinforcement and steel flange


indicating that the strength of reinforcement does not

significantly influence the increment rate of the

tendon force, but significantly influences the tendon

force increment at the girder’s yielding load.

5 Strength prediction approach

5.1 Cracking moment

Before the cracking load, the contribution of the

tendon internal force increment is very limited.

Moreover, the cross-sectional strain distributes

almost along the same line, with the same curvature

of the top and the bottom flanges. On the basis of

these phenomena, the cracking moment of the

composite girder can be deduced as shown in Eqs.

(8) and (9), where Ap is the area of the tendons, A0 is

the equivalent area of the concrete flanges, Ac and As

are the areas of the concrete flange and the steel

flange, respectively (mm2), ep is the eccentricity of

the tendons, y is the eccentricity of the bottom face of

the bottom flange (mm), rpe is the initial prestressing

stress of the tendons, ft is the cracking strength of the

concrete (MPa), Ig is the gross section inertia before

cracking and the axial stiffness of the corrugated

webs is neglected (mm4), cm is the sectional plastic

impact factor which can be selected as 1.15–1.2, and

a = Es/Ec, where Es and Ec are the elasticity modules

of the steel and concrete respectively.

Mcr ¼ Aprpeep þAprpeIg

A0yþ cmftIg

yð8Þ

A0 ¼ Ac þ aAs ð9Þ

5.2 Yielding moment

Experimental research indicated that the concrete

compressive stress of the top flange was distributed

approximately as an inverted triangle and that the

reinforcement and the steel flange in tension yielded

at the girder’s yielding load. Thus, the yielding

moment of the composite girder can be deduced from

cross-sectional equilibrium analysis, as shown in Eq.

(10), where yp is the distance from the tendons to the

top face of the top flange, yr is the distance from the

reinforcement in tension to the top face of the top

flange, ys is the distance from the bottom steel flange

in tension to the top face of the top flange, tuf is the

thickness of the top flange (mm), Ar is the area of

reinforcement in tension, As is the area of the steel

flange in tension (mm2), fyr and fys are the yielding

strengths of the reinforcement and steel flange

respectively (MPa), and Drpy is the stress increment

of the prestressing tendons at girder yielding load.

The prediction procedure for Drpy is detailed in paper

[17].

My ¼ Ap rpe þ Drpy

� �yp �

1

3tuf

� �

þ Arfyr yr �1

3tuf

� �þ Asfys ys �

1

3tuf

� �ð10Þ

5.3 Ultimate moment

After the yielding moment of the composite girder,

the load increment is almost entirely attracted by the

prestressing tendons in tension and by the top

concrete flange in compression. If the ultimate state

is defined at a deflection-span ratio of 1/50, the stress

0 30 60 90 120

150

175

200

225 CBCW1CBCW2CBCW3

tend

ons

inte

rnal

for

ce/k

N

mid-span deflection/mm

nonminal yielding forceof tendons

cracking point

0 80 160 240 3200

20

40

60

80

tend

ons

forc

e in

crem

ent /

kN

load/kN

yielding point

CBCW1CBCW2CBCW3

Cracking point(Pcr,ΔFcr)

tend

ons

forc

e in

crem

ent

P

yielding point(Py,ΔFy)

Ultimate point(Pu,ΔFu)

load

ΔF

(a) relationship between the

tendons internal force and

mid-span deflection

(b) relationship between the

tendons force increment and

load

(c) simplified model of the

tendons force increment and load

Fig. 19 Internal force increment of unbonded prestressing tendons


of the prestressing tendons will undoubtedly exceed

the nominal strength rpu = f0.2 = 0.85fb, where fb is

the ultimate strength of the tendons. Thus, the

ultimate moment Mu can be conservatively deduced

as Eq. (11):

Mu ¼ Aprpu yp �1

3tuf

� �þ Arfyr yr �

1

3tuf

� �

þ Asfys ys �1

3tuf

� �ð11Þ

5.4 Experimental validation

Experimental results in this study and previous

research [23] are used to validate this approach, as

shown in Table 6, where the experimental strength is

revised to account for the dead load. The cracking

moment and the yielding moment of the approach fit

well with those of the experimental result. However,

the ultimate moment of this approach is slightly

conservative because the ultimate stress rpu of the

tendons is conservatively selected as rpu =

f0.2 = 0.85fb whereas the ultimate stress of the tendons

approached fb in the experimental research.

6 Deformation prediction approach

Shear deformation of the corrugated steel webs

cannot be neglected and the approach to prediction

of the webs can be deduced by Eqs. (12)–(14). The

shear stress analysis of the corrugated webs in Sect.

4.4 indicates that the contributions of the corrugated

webs to the girder shearing force can be calculated by

Eq. (12). Moreover, the effective shear stiffness can

be selected as Eq. (13) proposed by Johnson [9],

where G is the shear stiffness of steel; b, d and h are

illustrated in Fig. 20. Thus, the shear deformation of

the corrugated steel webs can be calculated by Eq.

(14), where Aw is the cross-sectional area of the webs,

k is the coefficient accounting for the non-uniform

distribution of shear stress and can be selected as 1.0

for the corrugated webs.

Vw

Vp

¼ hw

h0

ð12Þ

Ge ¼ G bþ dð Þ= bþ d sec hð Þ ð13Þ

fs ¼XZ

kVVw

GeAw

dx ¼XZ

hw

h0

� kVVp

GeAw

dx ð14Þ

6.1 Cracking deformation

The flexural deformation of the composite girder can

be calculated by Eq. (15), where Ig is the inertia of the

entire cross-section neglecting the axial stiffness of

the corrugated webs and the prestressing tendons.

The reinforcement and the steel flange can be

converted into equivalent concrete with the coeffi-

cient a = Es/Ec. The total deformation of the girder

can by calculated by superposing the shear and

flexural deformations, as shown in Eq. (16).

fm ¼XZ

MMp

EcIg

dx ð15Þ

fcr ¼ fs þ fm

¼XZ

hw

h0

� kVVpcr

GeAw

dxþXZ

MMpcr

EcIg

dx ð16Þ

6.2 Yielding deformation

Section inertia will decrease as the load increases after

concrete cracking. When the concrete of the bottom

concrete flange is completely inoperative, section inertia

can be selected as Icr, where the concrete of the bottom

flange and the tendons are neglected. As the flexural

moment and the concrete cracking in the bottom flange

of various cross-sections along the longitudinal direc-

tion are different, an equivalent section inertia Ie is

required to account for this phenomenon. According to

the recommendations of ACI318 [24], Ie can be

calculated by interpolating between Ig and Icr, as shown

in Eq. (17). Further, the flexural deformation of the

girder can be calculated by Eq. (18), and the total

deformation can be calculated by Eq. (19).

Ie ¼Mcr

Mp

� �3

Ig þ 1� Mcr

Mp

� �3" #

Icr ð17Þ

fm ¼XZ

MMp

EcIe

dx ð18Þ

fy ¼ fs þ fm ¼XZ

hw

h0

� kVVpy

GeAw

dxþXZ

MMpy

EcIe

dx

ð19Þ

6.3 Experimental validation

Experimental results in this study and previous research

[23] are used to validate the deformation approach, as


shown in Table 7. The cracking deformation and the

yielding deformation of the approach fit well with those

of the experimental results. The predicted yielding

deformation of CBCW1 is a slightly lower because the

reinforcement of the bottom concrete flange is insuffi-

cient to support the tension transferred from concrete

cracking, which is somewhat similar to the case of

insufficiently reinforced concrete beams.

7 Conclusions

Experiments were conducted on three prestressed

concrete girders with corrugated steel webs. Detailed

numerical and analytical research was also performed

in this study. Overall, the following conclusions were

reached:

1. The prestressed concrete girders with corrugated

steel webs and non-continuous PBL connectors

demonstrated reliable shear connections and

satisfactory flexural behavior with large

deformability.

2. The numerical model proposed in this paper

accurately simulated the complex nonlinear

mechanical behavior of the girders. These sim-

ulations had good convergence, satisfactory

computational efficiency and reliability for

performing detailed research into the mechanical

behavior of prestressed composite girders.

3. The shear force of the girders is supported by both

the webs and the concrete flanges. The proposed

distribution mechanisms of the shear force between

the webs and the concrete flanges corresponded

with the experimental and numerical results.

4. The proposed strength and deformation predic-

tion approach based on elementary beam theory

was satisfactory for predicting the results of the

prestressed composite girders with appropriate

reinforcement of the bottom concrete flange in

tension. This approach can be applied in the

engineering design of such girders.

Acknowledgments The authors gratefully appreciate the

financial support provided by the National Natural Science

Foundation of China (51138007, 51222810) and the National

Science and Technology Support Program (2011BAJ09B02).

References

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Table 6 Validation of strength prediction approach

No. Cracking moment (kN m) Yielding moment (kN m) Ultimate moment (kN m)

Mcrt Mcrp Mcrp/Mcrt Myt Myp Myp/Myt Mut Mup Mup/Mut

CBCW1 120 136.0 1.13 187 193.1 1.03 255 231.3 0.91

CBCW2 121 137.2 1.13 240 243.5 1.01 315 278.6 0.88

CBCW3 120 134.3 1.12 213 222.0 1.04 293 257.6 0.88

BXL-B 462 523.9 1.15 1103.5 1085.3 0.98 1392 1200.2 0.86

b d b dh r

θ

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Table 7 Validation of deformation prediction approach

No. Cracking deformation

(mm)

Yielding deformation

(mm)

fcrt fcrp fcrp/fcrt fyt fyp fyp/fyt

CBCW1 6.84 6.32 0.92 24.74 19.95 0.81

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CBCW3 6.03 6.33 1.05 27.82 26.49 0.95

BXL-B 10.41 11.17 1.07 70.33 77.17 1.10


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Experimental and analytical studies of prestressed concrete girders with corrugated steel webs

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