Experimental studies on shear lag of box girders

Engineering Structures 24 (2002) 469–477 www.elsevier.com/locate/engstruct

Experimental studies on shear lag of box girders Q.Z. Luo a, Y.M. Wu a, J. Tang a

b,*

, Q.S. Li

b

Department of Civil Engineering and Architecture, University of Foshan, 18 Jiang Wan Rd, Foshan City, Guangdong 528000, People’s Republic of China b Department of Building and Construction, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong Received 9 April 2000; received in revised form 4 June 2001; accepted 25 September 2001

Abstract Two experimental studies on shear lag effect are conducted, one for box girders with varying depth in cross section and the other for box girders under simultaneous axial and lateral loads. Three Perspex glass models built specifically for the research work were used to perform the experiments with the purpose of investigating the shear lag phenomenon of box girders and providing a benchmark for future analytical and numerical studies of shear lag effect. The experiments intend to address two issues, the beam– column action and the effect of varying depth upon the shear lag of box girders. Finite element analysis is also conducted to check the accuracy of the numerical methods in predicting shear lag effect in these two situations. In general, the experimental results match reasonably well with the numerical predictions. However, there are some noticeable differences. In some cases, the difference in the stress predicted by the finite element method and the experimental results can be as high as 25%.  2002 Published by Elsevier Science Ltd. Keywords: Box girder; Experiment; Beam–column; Shear lag

1. Introduction The distribution of bending stresses in thin-walled box girders at any transverse section is non-uniform, and in general at the web–flange junction the stresses reach their maximum, decreasing towards the middle point of the top and bottom slabs and the cantilever flanges. This phenomenon is called the shear lag effect. Many theoretical and experimental research works have been performed by various researchers on shear lag effect (e.g. [1–18]). The problem was first examined by aeronautical engineers and was later studied for bridge structures with reference to the rectangular thin-walled box beam [8,10–12]. Stress larger than the elementary bending uniform stress was developed at the web–flange connection. An “effective width” of the plate—with the uniform stress equal to the maximum longitudinal stress—has been widely used by engineers in conjunction with the elementary beam theory.

* Corresponding author. Tel.: +852-2784-7646; fax: +852-27887612. E-mail address: [email protected] (J. Tang). 0141-0296/02/$ - see front matter  2002 Published by Elsevier Science Ltd. PII: S 0 1 4 1 - 0 2 9 6 ( 0 1 ) 0 0 1 1 3 - 4

Shear lag effect for various geometric and loading conditions were studied by many researchers (e.g. [1– 7]). Both analytical (or numerical) and experimental approaches were used in shear lag analysis (e.g. [13– 18]). However, shear lag effect under some unique situations has not yet been fully studied. Two such situations, box girders with varying depth and box girders under simultaneous axial and lateral loads, were identified in this research. Chang and Yun [16] conducted an experimental study on shear lag using a Perspex glass model of a cantilever box girder with linearly varying depth. Although their work has contributed towards the understanding of the shear lag effect on a cantilever box girder, they did not consider cases other than a cantilever box girder with linearly varying depth. Their work did show that there is a need to study shear lag effect for box girders with varying depth under various support and load conditions. For box girders under simultaneous axial and lateral loads, no experimental study has been reported so far. However, there is definitely a need for such a study as many bridge decks are under simultaneous axial and lateral loads, such as the cable-stayed bridges. Furthermore,

470

Q.Z. Luo et al. / Engineering Structures 24 (2002) 469–477

the combined load condition which is often referred to as beam–column action is not easy to analyse analytically, as the beam action and column action cannot be analysed separately. The interaction of the two actions makes the problem more complicated. Thus, experimental results in this kind of situation are really desired to check any assumption used in the analytical or numerical methods. Chang [14] attempted to use superposition to study the shear lag effect in prestressed box girder bridges. However, no beam–column action is considered and in fact no axial deformation is considered in the study. Also, the results are not verified by experiment or other means. In this paper, three Perspex glass models, a three-span continuous box girder with varying depth and two simply supported box girders with constant cross section were used to provide experimental results. The experimental results obtained from the models were presented and discussed in detail. The present research intends to address two issues, the beam–column action and the effect of varying depth upon the shear lag. It should be mentioned that the availability of the experimental data could provide adequate insight into the physics of the problem and further serve the benchmark to examine the accuracy of the approximate or numerical solutions. Therefore, it is always desirable to obtain the experimental results to such problems. Finite element analysis is also conducted to check the accuracy of the numerical methods in predicting the shear lag effect in these two situations. It is found that overall, the finite element method can predict the shear lag phenomenon reasonably well, however, in some cases the difference in the stress predicted by the finite element method and the experimental results can be as high as 25%.

2. Right above the two webs, on the top slab, two linearly distributed loads were applied symmetrically. The stress distribution at the three cross sections mentioned above was measured. 3. Right above the two webs, on the cantilever beam, two linearly distributed loads were applied symmetrically. The stress distribution at the fixed end cross section was measured.

2.2. Model design Fig. 1 shows the longitudinal layout of the bridge model where, 1. The thickness of the transverse diaphragms is 0.8 cm. 2. The variation of the depth follows the curve: y=4+0.0025x2. 3. The calculation span length is 46+86+46=178 cm. 4. The total length of the top slab is 178.8 cm. Fig. 2 shows the cross section layout of the model. Fig. 3 shows the transverse diaphragms of the model. The thickness of the diaphragms is 0.8 cm. There are five diaphragms in all, two in the middle supports, as shown in Fig. 3(a), and the other two at the two ends and one in the middle span, as shown in Fig. 3(b). 2.3. Location of the measuring points Strain gauges are used in both longitudinal and transverse directions. They are 2x2 mm glue strain gauges, of resistance 120 ⍀. In order to increase the measuring accuracy, reducing the interference of the measuring points by the loading point, bi-directional strain gauges on the upper and lower sides of the same location were installed. At the same time, several measuring points on

2. Three-span continuous box girder 2.1. Experiment content 1. On the top slab of the middle span, a pair of concentrated forces right above the two webs were exerted symmetrically. The stress distributions at three cross sections, two at each side of the support and one at the middle span, were measured.

Fig. 1.

Fig. 2.

Longitudinal layout (unit: cm).

Cross section layout (unit: cm).

Q.Z. Luo et al. / Engineering Structures 24 (2002) 469–477

Fig. 3.

471

Transverse diaphragms (unit: cm).

the other sides of the symmetric line of the cross section were arranged. See Fig. 4. 2.4. Loading procedure 2.4.1. Vertical concentrated load A lever was used; through the steel ball under the distributing beam, force was applied. See Fig. 5(a). 2.4.2. Vertical distributed load Two rubber strips above the two webs were used. Specially made standard weights on the strips were added. See Fig. 5(b). 2.5. Measuring method Static resistance strain measurement, equilibrium box, electric measure reading stability apparatus, full bridge measurement devices, and the temperature compensation method were used to measure the strain values.

Fig. 5.

Loading devices.

2.6. Material test The main properties of the Perspex glass material are Young’s elastic modulus and Poisson’s ratio, as the test uses the same material for the whole model. The stretch test specimen of the material uses bi-directional strain gauges, adhered to both sides of the measuring location, to measure the longitudinal and transverse strain, and determine Young’s elastic modulus and Poisson’s ratio under tension. The flexure test specimen of the material is made of a rectangular plate that is simply supported, with strain gauges at the surfaces of the upper and lower sides of the middle span. The four-point loading method is used to make the middle portion of the beam under pure bending action; see Fig. 6.

Fig. 4.

Location of the measuring points (unit: cm).

Fig. 6. Elastic modulus measurement.

472

Q.Z. Luo et al. / Engineering Structures 24 (2002) 469–477

Young’s modulus and Poisson’s ratio are obtained from the following formulas:

can be approximated by the second or third order parabolic curves. It is expected that these findings can serve as benchmarks for future analytical and numerical analysis of the shear lag effect for continuous box girders with varying depth. The results from the finite element analysis match with experimental results quite well (see Figs. 7 and 8). However, comparing node 1 in Tables 1 and 2, for example, shows the difference can be as high as 20% between the experimental results and those from the finite element analysis. For large span continuous bridges, construction is often carried out using the cantilever construction method. One of the by-products of this experimental study is the results for shear lag effect of a cantilever girder with varying depth, as shown in the figure given in Table 3. Table 3 lists the experimental results as well as those from the finite element analysis. As can be seen, the shear lag effect in this situation also cannot be overlooked. Comparing Tables 1 and 2, it can be seen that the shear lag effect is more severe at the centre span under the concentrated load than that under the distributed load. The stress ratio between point 3 and point 1 is 1.64 for the former and 1.45 for the latter. It is interesting to note that at the supports, this phenomenon is reversed (1.33 and 1.78).

−ey s E⫽ , m⫽ ex ex where s=6Pa/bh2 for the flexure test and s=N/bh for the tension test. The width of the specimen is b, the height is h, and ex, ey are longitudinal and transverse strain, respectively. Through these tests, we obtained m=0.4, and elastic modulus E=2600 MPa. The temperatures are controlled at 18苲20 °C. The test is conducted in short duration, therefore creep effect is avoided. 2.7. Experimental results A concentrated force (P=137.33 N) is applied to the bridge model at the middle span, and the experimental results as well as those from the finite element analysis are listed in Table 1. Under the distributed load (q=5 N/cm); the results are listed in Table 2. It is clear that the shear lag effect is very obvious in all sections and shear lag effect cannot be overlooked in these situations. By examining Figs. 7 and 8, it is seen that the shapes of the stress distribution along the width of the bridge at various locations are similar to those found by other researchers for other geometric bridge models under different loading conditions, i.e. the shape Table 1 Stress distribution under the concentrated load (unit: MPa) Section Nodes

Flange

Top slab

Flange

I–I

1 2 3 4 5 6 7 8 9

II–II

III–III

Experiment

Finite element

Experiment

Finite element

Experiment

Finite element

⫺0.1354 ⫺0.1662 ⫺0.22191 ⫺0.21727 ⫺0.14913 ⫺0.12071 ⫺0.21773 ⫺0.21634 ⫺0.13906

⫺0.16247 ⫺0.17116 ⫺0.22377 ⫺0.22295 ⫺0.15215 ⫺0.13948 ⫺0.22295 ⫺0.22377 ⫺0.16247

0.06964 0.07483 0.09285 0.08914 0.06221 0.05664 0.09378 0.09842 0.06806

0.07134 0.07538 0.09986 0.09883 0.06443 0.05828 0.09883 0.09986 0.07134

0.07799 0.07892 0.09564 0.09749 0.06871 0.06546 0.09378 0.10307 0.07428

0.08009 0.08350 0.10414 0.10303 0.07348 0.06819 0.10303 0.10414 0.08009

Q.Z. Luo et al. / Engineering Structures 24 (2002) 469–477

473

Table 2 Stress distribution under the distributed load (unit: MPa) Section Nodes

Flange

Top slab

Flange

I–I

1 2 3 4 5 6 7 8 9

II–II

III–III

Experiment

Finite element

Experiment

Finite element

Experiment

Finite element

⫺0.14763 ⫺0.16806 ⫺0.21355 ⫺0.21476 ⫺0.16714 ⫺0.14670 ⫺0.21383 ⫺0.21262 ⫺0.1755

⫺0.18065 ⫺0.18583 ⫺0.21724 ⫺0.21516 ⫺0.16924 ⫺0.16104 ⫺0.21516 ⫺0.21724 ⫺0.18065

0.12590 0.14856 0.22470 0.21727 0.12163 0.08914 0.21263 0.22191 0.12888

0.13991 0.15237 0.22784 0.22686 0.12577 0.10770 0.22686 0.22784 0.13991

0.12163 0.14763 0.21077 0.22098 0.12906 0.10214 0.21820 0.22563 0.11885

0.14673 0.15871 0.23130 0.23031 0.13294 0.11553 0.23031 0.23130 0.14673

Fig. 7.

Three-span continuous bridge under concentrated load.

3. Two simply supported beams To study the shear lag effect for box girders under simultaneous axial and lateral loads, two simply supported Perspex glass beams were built for this study. The axial loads were applied as concentrated end loads to the

webs at the height of the centroid of the cross section. Fig. 9 shows the cross sections of the two models, models 1 and 2. Tables 4 and 5 list the results of the experiment as well as the results from the finite element analysis in which the simply supported beams are under axial load

474

Q.Z. Luo et al. / Engineering Structures 24 (2002) 469–477

Fig. 8.

Three-span continuous bridge under distributed load.

Table 3 Stress distribution for a cantilever beam (unit: MPa) Section Nodes

Flange

Top slab

Flange

III–III (fixed end)

1 2 3 4 5 6 7 8 9

Experiment

Finite element

0.51346 0.5571 0.65924 0.68059 0.48282 0.42525 0.71866 0.70009 0.52553

0.53191 0.56066 0.73481 0.73132 0.49521 0.45299 0.73132 0.73481 0.53191

alone. The shear lag effect in this kind of situation can be clearly seen. It is noted that at the section ( x=10 cm), the ratio of the stress at point 3 to that at point 1 is 1.74 for model 1, 2.82 for model 2 and at the section (x=15 cm), the ratio becomes 1.22 for model 1 and 1.66 for model 2. From these results, we can see that:

1. The shear lag effect is related to the width of the box girder, the larger the width, the more severe the shear lag effect. This observation can be obtained by comparing the ratios of the stress at point 3 to that at point 1 between model 1 and model 2. Clearly, the ratio in model 2 (B=40 cm) is larger than that in model 1 (B=30 cm) in every section.

Q.Z. Luo et al. / Engineering Structures 24 (2002) 469–477

475

this situation is very obvious and cannot be overlooked. It should be mentioned that these kind of experimental data have not been reported in the literature before. It is believed that these results could be thus used as a benchmark for future analytical and numerical analysis of shear lag effect under simultaneous axial and lateral loads. As stated before, the shear lag effect under simultaneous axial and lateral loads is more complicated due to the interaction of these two loads. The pattern observed from Tables 6 and 7 is not as clear as in Tables 4 and 5. However, some phenomena are observed here:

Fig. 9.

Cross sections of the two simply supported models.

2. Also the shear lag effect gets more serious for sections that are near the load application point. The further away from the point of load application, the less effect of the shear lag. This can be seen by observing the sections at x=10 and x=15 cm away from the support for the same model. This phenomenon can also be attributed to the well known St Venant’s principle. Tables 6 and 7 present the results of the experiment as well as the results from the finite element analysis in which the simply supported beams are under simultaneous axial and lateral loads. The shear lag effect in

1. The shear lag effect under the concentrated load is more severe than that under the distributed load. The ratio of stress at point 3 to point 1 is 1.27 and 1.39 for model 1 and 2, respectively, under the concentrated load while the values are 1.10 and 1.13, respectively, under the distributed load. 2. The width effect is also observed, as in the situation under axial load alone. See the discussion presented above. Comparison can be made between the results from the finite element analysis and those from the experiment: 1. The finite element analysis consistently underestimates the stress on the top slab in comparison with the experimental results. The difference could be as high as 25%.

Table 4 Simply supported beam under axial loads, model I (unit: MPa) Section, B=30 cm, N=68.7 kg Nodes

Top slab

Web Bottom slab

X=10 cm

1 2 3 4 5 6 7 8 9 10 11

X=15 cm

Experiment

Finite element

Experiment

Finite element

⫺1.073 ⫺1.379 ⫺1.862 ⫺1.904 ⫺1.533 ⫺1.379 ⫺2.422 ⫺2.282 ⫺1.740 ⫺1.260 ⫺1.190

⫺0.937 ⫺1.293 ⫺1.814 ⫺1.872 ⫺1.489 ⫺1.363 ⫺2.430 ⫺2.382 ⫺1.728 ⫺1.324 ⫺1.209

⫺1.540 ⫺1.617 ⫺1.883 ⫺1.918 ⫺1.771 ⫺1.582 ⫺2.149 ⫺2.072 ⫺1.743 ⫺1.652 ⫺1.568

⫺1.373 ⫺1.530 ⫺1.763 ⫺1.817 ⫺1.695 ⫺1.643 ⫺2.025 ⫺2.002 ⫺1.804 ⫺1.632 ⫺1.567

476

Q.Z. Luo et al. / Engineering Structures 24 (2002) 469–477

Table 5 Simply supported beam under axial loads, model II (unit: MPa) Section, B=40 cm, N=91.52 kg Nodes

Top slab

Web Bottom slab

X=10 cm

1 2 3 4 5 6 7 8 9 10 11

X=15 cm

Experiment

Finite element

Experiment

Finite element

⫺0.910 ⫺1.512 ⫺2.562 ⫺2.520 ⫺1.610 ⫺1.302 ⫺3.038 ⫺2.884 ⫺2.030 ⫺1.246 ⫺1.050

⫺0.717 ⫺1.323 ⫺2.275 ⫺2.315 ⫺1.532 ⫺1.297 ⫺3.015 ⫺2.831 ⫺2.031 ⫺1.280 ⫺1.093

⫺1.316 ⫺1.792 ⫺2.184 ⫺2.212 ⫺1.918 ⫺1.652 ⫺2.380 ⫺2.282 ⫺2.030 ⫺1.624 ⫺1.512

⫺1.231 ⫺1.588 ⫺2.076 ⫺2.125 ⫺1.810 ⫺1.677 ⫺2.395 ⫺2.352 ⫺2.034 ⫺1.642 ⫺1.502

Table 6 Simply supported beam under both axial and concentrated lateral loads (unit: MPa) Section Nodes

Top slab

Bottom slab

1 2 3 4 5 6 9 10 11

Model I, B=30, X=40 cm, N=47.48 kg, P=21.19 kg

Model II, B=40, X=40 cm, N=91.52 kg, P=53.94 kg

Experiment

Finite element

Experiment

Finite element

⫺3.640 ⫺3.924 ⫺4.634 ⫺4.666 ⫺4.163 ⫺3.985 4.043 3.002 2.865

⫺3.540 ⫺3.844 ⫺4.431 ⫺4.476 ⫺3.999 ⫺3.889 3.855 3.190 3.000

⫺6.510 ⫺7.449 ⫺9.072 ⫺9.138 ⫺7.770 ⫺6.909 9.770 6.532 5.724

⫺5.788 ⫺6.667 ⫺8.605 ⫺8.705 ⫺7.118 ⫺6.802 9.972 6.511 5.963

2. For the bottom slab, the situation is quite different. In fact, at many points, the finite element analysis overestimates the stresses in comparison with the experimental results. However, the difference is not as large as in the case of the top slab. 3. In general, the results from the finite element analysis match reasonably well with those from the experiment.

4. Conclusions Two experimental studies on the shear lag effect of box girders are conducted, one for box girders with varying depth in cross section and the other for box girders under simultaneous axial and lateral loads. Three Perspex glass models, a three-span continuous box girder bridge model and two simply supported girder bridge

Q.Z. Luo et al. / Engineering Structures 24 (2002) 469–477

477

Table 7 Simply supported beam under both axial and distributed lateral loads (unit: MPa) Section Nodes

Top slab

Bottom slab

Model I, B=30, X=40cm, N=68.70 kg, q=0.666 Model II, B=40, X=40 cm, N=66.93 kg, kg/m q=0.597 kg/m

1 2 3 4 5 6 9 10 11

Experiment

Finite element

Experiment

Finite element

⫺5.502 ⫺5.719 ⫺6.048 ⫺6.146 ⫺5.845 ⫺5.803 4.605 4.316 4.106

⫺5.381 ⫺5.506 ⫺5.723 ⫺5.759 ⫺5.615 ⫺5.559 4.645 4.400 4.298

⫺3.857 ⫺3.976 ⫺4.361 ⫺4.284 ⫺3.949 ⫺3.843 3.101 2.849 2.706

⫺3.623 ⫺3.796 ⫺4.100 ⫺4.142 ⫺3.967 ⫺3.908 3.118 2.818 2.719

models, are made to meet the specific requirements of this research work. The experimental results are tabulated and plotted to demonstrate the shear lag phenomenon of box girders together with the computational results from the finite element analysis. It is shown through the experimental results as well as the finite element analysis that the shear lag effects cannot be overlooked as they are obviously present in all these situations. It is hoped that the experimental results presented here can provide some insight into the physics of the problem and will serve as a benchmark for future analytical and numerical analysis of the shear lag effect under these situations. Furthermore, more experimental researches and analytical or numerical studies may build upon this study towards a better understanding of the shear lag effect of box girders with various geometric cross sections under different loading conditions. Acknowledgements The authors are thankful to the reviewers for their useful comments and suggestions. Financial support for this study by a City University of Hong Kong Strategic Research Grant (No. 7001176) is gratefully acknowledged. References [1] Moffatt PJ, Dowling PJ. British shear lag rules for composite girders. J Struct Div, ASCE 1978;104(ST7):1123–30.

[2] Lee JAN. Effective width of Tee-beams. The Structural Engineer 1962;40(1):21–7. [3] Song Q, Scordelis AC. Shear lag analysis of T-, I-, and box beams. J Struct Engng, ASCE 1990;116(5):1290–305. [4] Song Q, Scordelis AC. Formulas for shear lag effect of T-, I-, and box beams. J Struct Engng, ASCE 1990;116(5):1306–18. [5] Evans HR, Ahmad MKH, Kristek V. Shear lag in composite box girders of complex cross section. J Construct Steel Res 1993;24(3):183–204. [6] Evans HR, Taherian AR. A design aid for shear lag calculations. Proc Instn Civ Engrs 1980;69(Part 2):403–24. [7] Taherian AR, Evans HR. The bar simulation method for the calculation of shear lag in multi-cell and continuous box girder. Proc Instn Civ Engrs 1977;63(Part 2):881–97. [8] Kuzmanovic BO, Graham HJ. Shear lag in box girders. J Struct Div, ASCE 1981;107(9):1701–12. [9] Foutch DA, Chang PC. A shear lag anomaly. J Struct Engng, ASCE 1982;108(7):1653–8. [10] Dezi L, Mentrasti L. Nonuniform bending-stress distribution (shear lag). J Struct Engng, ASCE 1986;111(12):2675–90. [11] Reissner E. On the problem of stress distribution in wide flanged box beam. J Aero Sci 1938;5:295–9. [12] Reissner E. Analysis of shear lag in box beam by principle of minimum potential energy. Q Appl Math 1946;5(3):268–78. [13] Cheung YK. Finite strip method in structural analysis. Oxford (UK): Pergamon International Library, 1976. [14] Chang ST. Prestress influence on shear lag effect in continuous box girder bridge. J Struct Engng, ASCE 1992;118(11):3113–21. [15] Chang ST, Zheng FZ. Negative shear lag in cantilever box girders with constant depth. J Struct Engng, ASCE 1987;113(1):20–35. [16] Chang ST, Yun D. Shear lag effect in box girder with varying depth. J Struct Engng, ASCE 1988;114(10):2280–92. [17] Luo QZ, Li QS. Shear lag of thin-walled curved girder bridges. J Engng Mech, ASCE 2000;126(10):1111–4. [18] Moffatt KR, Dowling PJ. Shear lag in steel box girder bridges. Struct Engng 1975;53(10):439–48.