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Analytical investigation of access holes in load-bearing diaphragms of steel box girders
Journal of Constructional Steel Research 61 (2005) 113–131 www.elsevier.com/locate/jcsr
Analytical investigation of access holes in load-bearing diaphragms of steel box girders Dong Y. Yoona, Sung C. Leeb, Chai H. Yooa,∗ a Department of Civil Engineering, Auburn University, Auburn, AL 36849-5337, United States b Department of Civil and Environmental Engineering, Dong Guk University, Seoul 100-715,
Republic of Korea Received 22 September 2003; accepted 26 July 2004
Abstract This paper presents a series of parametric studies on the effect of access holes on the strength of the load-bearing diaphragms in steel box girder bridges. These parameters include the number of bearings, initial imperfections, the size and location of access holes, and the size of stiffeners around the openings. The load-bearing diaphragms were modeled using three-dimensional finite element analyses for material and geometric nonlinearities. It was found that the opening can be made large enough for practical purposes in order to provide convenient access to the inside of the box girder with no significant reduction in the ultimate strength of the load-bearing diaphragm. The location of the access hole has no detrimental effect on the diaphragm strength. © 2004 Elsevier Ltd. All rights reserved. Keywords: Access holes; Finite element analysis; Load-bearing diaphragm
1. Introduction Steel box girder bridges incorporate two kinds of diaphragms. One is a load-bearing diaphragm positioned at supports, as shown in Fig. 1, and the other is an intermediate ∗ Corresponding author. Tel.: +1 334 844 6279; fax: +1 334 844 6290.
E-mail address: [email protected] (C.H. Yoo). 0143-974X/$ - see front matter © 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jcsr.2004.07.005
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Nomenclature B Bb Bt D d E E sh Fy Fult q td tw εsh εy
average width of diaphragm [(Bb + Bt )/2] bottom width of diaphragm top width of diaphragm depth of diaphragm distance from bottom flange to center of access hole modulus of elasticity tangent modulus yield stress ultimate tensile stress shorter dimension of plate or distance between bearing stiffeners thickness of diaphragm thickness of web strain at strain hardening yield strain
cross-frame used to reduce distortion and to retain the original shape of the box. This study examines only the load-bearing diaphragms. The primary function of load-bearing diaphragms at the supports of box girders is to transfer vertical and horizontal forces from the webs and the flanges to the bearings. Load-bearing diaphragms are subjected to high concentrated bearing forces and hence are designed to avoid premature buckling failure due to the large compressive loads. Bearing stiffeners are typically placed on both sides of the load-bearing diaphragm to help carry these large concentrated loads. Fig. 2 shows the three major forces acting on a load-bearing diaphragm. Stresses in the load-bearing diaphragm due to each force are affected by the following factors: (1) its shape; (2) the locations and number of bearings; (3) the location, shape, and size of the access hole; and (4) the stiffening around the access hole. Due to its multi-dimensional nature, it is not an easy task to accurately evaluate the resulting stresses in the load-bearing diaphragm. As a consequence of the Silver Bridge failure, reported in NTSB [16], highway bridges in the United States are mandated to be inspected periodically under the 1968 Federal Aid Highway Act. The inspection cycle is determined by the age of the bridge, but every bridge must be inspected at least once every two years. Inspection crews walk or crawl through the inside of the box girders and maintenance repair work is made on an as-needed basis. Therefore, it is necessary for load-bearing diaphragms to have access holes installed. Access holes have various shapes and locations, but all must be as large as practical in order for inspectors and/or equipment to pass conveniently. However, the access-hole dimensions are usually designed to be as small as possible due to the traditional design concern for a possible strength reduction of the load-bearing diaphragms. Frequently,
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Fig. 1. Load-bearing diaphragm.
Fig. 2. Stress in plate diaphragm at bearing, from Heins and Hall [12].
stiffeners are provided around access holes to increase the strength and to alleviate stress concentration. As a result, it is often inconvenient to pass through these small access holes. In this study, the strength reduction effect on the load-bearing diaphragms due to these access holes was investigated. Some of the variables that may affect the strength include the material properties, the location and size of the access holes, the presence of access hole stiffeners, and the number of bearings. Geometric and material incremental nonlinear analyses were performed in order to characterize and quantify the load-bearing behavior of solid-plate diaphragms with various sizes of access holes installed at different locations. The tool used was a commercially available three-dimensional finite element
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Fig. 3. Stiffened diaphragm, from Allen and Bulson [6].
code, ADINA [3], that is capable of incorporating the geometric and material nonlinear characteristics of the load-bearing diaphragms and the other box girder members. 2. Brief reviews The load-bearing diaphragms in most steel box girders are designed to be either very stocky, unstiffened diaphragms or stiffened diaphragms with one or more pairs of bearing stiffeners. Design concepts for load-bearing diaphragms can be divided into the plate and column approach, respectively. The design concepts used in the current specifications for each are briefly reviewed below. 2.1. Plate concept BS 5400 [10]: Part 3, Code of Practice for Design of Steel Bridges, is based on the plate concept. Two different schemes are contained in the code: one is used for simple diaphragms, as illustrated in Fig. 2 where only vertical bearing stiffeners divide the diaphragm into sub-panels; and the other is used for diaphragms of considerably complicated geometries, as shown in Fig. 3 where horizontal stiffeners and stub stiffeners are incorporated. In each case, the design procedure includes an examination of the geometric proportions, analyses of diaphragm stresses, and strength and stability checks. Approximate expressions are used for the elastic critical buckling strength of rectangular and trapezoidal sub-panels loaded along the edges of the diaphragm. The rotational restraints provided by the flanges and webs to the diaphragm are conservatively neglected. Full length bearing stiffeners are assumed to be placed symmetrically on both sides of the diaphragm plate unless a special analysis is made on the effects of any eccentricity with respect to the plate. Article 9.17.2.7(b), BS 5400 [10] stipulates that unstiffened openings in unstiffened load-bearing diaphragms must be circular and of a diameter not exceeding the least value given by Eq. (1): 6td , D/20 or B/20
(1)
where td = diaphragm thickness, D = depth of the diaphragm, and B = width of the diaphragm taken as the average of the widths of the top and bottom flange levels for boxes with sloping webs. In practice, however, all openings satisfying the limitation
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Fig. 4. Bearing stiffener and effective width of web.
imposed by Eq. (1) are not sufficiently large for access holes for the passage of inspectors and/or maintenance equipment. Consequently, any access holes for such purposes must be stiffened. The provision also specifies that holes are not to be placed within the highly stressed area bounded by the bottom flange and one third of the height of the diaphragm. The limiting stresses are to be evaluated using a two-dimensional plate theory. 2.2. Column concept A column concept is adopted to design the bearing stiffener in plate girders in the AISC [4], AASHTO [2], and AASHTO LRFD [1] specifications. In the column concept, the bearing stiffeners plus portions of the web are assumed to act as a column to resist the compressive force transmitted to the bearing. According to Article 10.34.6.1, AASHTO [2], the effective area of the column section is taken as the area of the stiffener, plus a centrally located strip of web extending not more than nine times the thickness of the web on each side of the outer projecting elements of the stiffener group, as shown in Fig. 4. However, neither AASHTO [2] nor AASHTO LRFD [1] give explicit provisions for the design of the load-bearing diaphragms and access holes: the design of bearing stiffeners attached to the diaphragms of box girders is assumed to follow the provisions for plate girders; and the design of access holes is not directly addressed. 3. Finite element analysis For finite element analyses, 3/4-node shell elements are employed to model the steel components and 6/8-node solid elements are used for the concrete decks and bearings. These elements are readily available from the ADINA [3] element library and they are compatible with all elastic buckling analyses and incremental iterative analyses for both material and geometric nonlinearities. In order to determine an appropriate, reliable mesh refinement, a convergence analysis was carried out by varying the number of elements for an isolated bearing diaphragm, as shown in Fig. 5. The diaphragm is simply supported at all the four edges and a uniform load is applied on the top edge. Table 1 shows the results of collapse analyses of the three meshes. The mesh shown in Fig. 5(b) was selected for this study because it yields close to the same numerical values as those obtained from the finer mesh shown in Fig. 5(c) with a considerable reduction in computation time.
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Table 1 Convergence test Mesh
Number of elements
Collapse load (kN)
1 2 3
174 1030 2000
2653.7 2628.9 2600.4
Fig. 5. Convergence test models.
3.1. Finite element model description Five different types of load-bearing diaphragms were examined in this study. These types of diaphragms include those most likely to be used in practical steel box girders. Four straight box girders and one horizontally curved box girder were considered. The details of the models are summarized in Table 2. The first character in the model designation represents whether the box girder is straight or horizontally curved, i.e., S for a straight girder and C for a curved girder. The second character stands for the shape of the box section, i.e., T for a trapezoidal section and R for a rectangular section. The third character indicates the number of spans. The fourth character represents the number of bearings at each load-bearing diaphragm location. As per Article 6.11.1, AASHTO LRFD [1], the inclination of web plates is kept constant at 1 to 4 for all trapezoidal sections. Typical finite element models of straight and horizontally curved box girders are shown in Fig. 6. As the focus of the study is the behavior of the diaphragm, the diaphragm was divided with finer meshes compared to other parts of the model. In order to reduce the computation time and modeling effort, only one third of the span was considered in the models. The models having two equal spans, ST22, ST21, and SR22, are shown in Fig. 6(a). At the cut point, only translation along the vertical axis and rotation with respect to the horizontal axis were permitted, suppressing all other degrees of freedom, to simulate the vertical bending behavior of the girder. The difference in the ultimate strength of the diaphragm based on the full and partial models was noted to be less than 1%. Models ST32 and CT32, having three spans, were analyzed using full models.
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Table 2 Bridge models Model
ST32
ST22
ST21
SR22
CT32
Number of spans Span length
3 58 m-72 m-58 m
Global geometry
Straight
2 45 m-45 m (l = 14 m) Straight
2 40 m-40 m (l = 13 m) Straight
2 50 m-50 m (l = 15 m) Straight
Cross section Number of girders Number of bearings Number of bearing stiffeners at each bearing D (mm) Bt (mm) Bb (mm) Br (mm) Max. t f (mm) tw (mm) Reference
Trapezoid 2 2 1
Trapezoid 1 2 2
Trapezoid 1 1 2
Rectangle 2 2 1
3 48.8 m-64 m48.8 m Horizontally curved Trapezoid 2 2 1
1800 3400 2500 12,880 35.0 14.0 AISI [5]
2100 3450 2400 6200 28.5 12.7
2000 3300 2300 6000 30.0 15.0
2300 2400 2400 12,000 28.5 14.0
Fig. 6. Finite element models.
1980 3048 2060 12,350 38.1 14.3 Hall and Yoo [13]
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For all straight girder models, two types of bearing devices were tested: elastomeric bearings, where relatively small vertical deflections and rotations were permitted, and other bearing devices using only steel sole plates, where vertical deflections were suppressed. It was intended in this study to investigate the potential effect of an uneven vertical displacement at the level of the elastomeric bearing pad on the development of the transverse compressive stress at the bottom part of the diaphragm plate caused by the horizontal force component resulting from the inclined webs. Examination of test runs indicated that there was no appreciable difference (less than 1%) in the magnitude of the induced transverse compressive stresses between the two types. Thus, sole plate bearings were used for all the girder models. In the incremental iterative analyses, a uniformly distributed vertical load was applied at the top of the composite deck. Based on the uniform surface load applied, ADINA [3] automatically determines the load intensity at each mesh point, reflecting the different tributary areas caused by different mesh dimensions. No horizontal load was considered in this study. Vertical loads applied at the cut point in the partial models reflect the accumulation of the uniformly distributed load of the truncated parts of the structure. The bridge models were intentionally designed so that the load-bearing diaphragm would collapse before the flange and web did. The flange and web were designed to remain elastic while the load-bearing diaphragm was proportioned to be stressed beyond the linear elastic limit. As a result, the analysis results can be interpreted to represent the ultimate strength of the load-bearing diaphragm under continuity conditions provided by the flanges and webs. 3.2. Initial geometric imperfections It has been shown that initial geometric imperfections tend to reduce the ultimate strengths of two-dimensional compression members with moderate thicknesses or one-dimensional columns with intermediate slenderness ratios (Lee and Yoo [14]; Galambos [11]). Although Article 9.19, Bridge Welding Code [8] stipulates the maximum dimensional tolerances of plate girder webs and stiffeners, the magnitudes of the allowable initial geometric deformations or out-of-flatnesses for thin-walled elements differ from code to code. Furthermore, there is no specific limit stipulated for the initial geometric imperfections of load-bearing diaphragms of box girders. A strategy of imposing an initial imperfection on the finite element mesh, a technique frequently used in nonlinear analysis, was employed in order to simulate out-ofplane deformations of the load-bearing diaphragms reflecting fabrication tolerances. As suggested by Bathe [7], the first elastic buckling mode shape of the load-bearing diaphragm was adopted as the initial geometric imperfection shape, with the largest amplitude being equal to the largest fabrication tolerance, q/150, permitted by Article 9.19, Bridge Welding Code [8]. The symbol q defines the least lateral dimension of an element. A very small initial imperfection of q/150,000 was used to simulate the initially straight model in the incremental analyses. The imposition of the normalized second and/or higher mode buckling shapes on the perfect geometry does not reduce the ultimate load-carrying capacity significantly, as discussed in Bathe [7]. Hence, it was not implemented for this study. Fig. 7 illustrates the
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Fig. 7. First elastic buckling mode shape of load-bearing diaphragms.
Fig. 8. Stress–strain curve of steel.
first elastic buckling mode shapes of the load-bearing diaphragms. The first elastic buckling mode shape of the entire load-bearing diaphragm includes the twisting mode of the bearing stiffeners, as shown in Fig. 7. The finite element meshes reflecting these initial geometric imperfections were then used with the maximum amplitude of the normalized eigenvector being equal to that permitted by the Bridge Welding Code [8] for the incremental (load) analysis. 3.3. Material properties The stress–strain curve of steel assumed in this study is shown in Fig. 8. For a typical construction steel having a minimum yield stress of 345 MPa (50 ksi) and a modulus of elasticity of 200 GPa (29,000 ksi), the yield strain, ε y , was computed to be 0.001725. The plastic plateau, εsh , was assumed to be 10ε y . According to Bruneau et al. [9], the plastic plateau generally varies from 5 to 15. The tangent modulus, E sh , at the onset of strain hardening was taken to be approximately 1/30th (6667 MPa) of the modulus of elasticity of steel, E. For a minimum yield stress, Fy , of 345 MPa (50 ksi) and an ultimate tensile stress, Fult , of 485 MPa
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Table 3 Strength reduction of load-bearing diaphragms due to initial imperfections Ultimate strength with geometric initial imperfections Ultimate strength without geometric initial imperfections Maximum initial deformations
ST32
ST22
ST21
SR22
CT32
q/1500 q/500 q/150
0.98 0.96 0.92
0.98 0.97 0.94
0.99 0.97 0.94
0.95 0.92 0.89
0.98 0.96 0.92
(70 ksi), approximately 12ε y is required to reach the ultimate stress after the initial strain hardening. Poisson’s ratios for steel and concrete were taken to be 0.3 and 0.16, respectively. The modulus of elasticity of the concrete deck was taken to be 25,000 MPa (3600 ksi), thereby giving a modular ratio of 8. The relatively thin layer of elastomer is also assumed to behave in an elastic manner. Based on the AASHTO LRFD [1], the elastic modulus and Poisson’s ratio were taken to be 240 MPa (35 ksi) and 0.49 because the elastomer is an almost incompressible material. 4. Data analysis 4.1. Effect of initial geometric imperfections The initial geometric imperfections considered were q/1500, q/500, and q/150. In Table 3, the weakening effect of an initial out-of-flatness is summarized for all five models examined. As the magnitude of the initial geometric imperfection increases, the ultimate strength of the load-bearing diaphragms decreases, as expected. In the SR22 model, the maximum strength is reduced by 11.0% when the maximum initial deformation is taken to be q/150 compared to that of an almost perfect girder. As has been shown elsewhere (Choi and Yoo [15]), the strength reductions of the load-bearing diaphragm due to the initial geometric imperfections are not negligible, particularly for plates with an intermediate width-to-thickness ratio. Therefore, in order to accurately determine the ultimate strength of the load-bearing diaphragm, the initial geometric imperfection must be considered. It is unconservative practice to neglect the ultimate strength reduction due to the initial geometric imperfection. 4.2. Induced stresses The principal stresses of model ST32 with two bearings at the ultimate load are illustrated in Fig. 9(a). It can be seen that the stresses resulted from the three major actions (bearing, bending, and shear actions) shown in Fig. 2. The horizontal compressive stresses shown below the access hole and the horizontal tensile stresses near the top of the diaphragm between the bearings are the bending stresses. The magnitude of the maximum compressive bending stress below the access hole accounts for 73% of the maximum vertical compressive stress at failure. The bottom horizontal stiffener shown in Fig. 3, therefore, may be necessary in order to prevent buckling failure due to compressive bending
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Fig. 9. Stresses in two-bearing diaphragm at failure (model: ST32, hole: 800 mm × 1000 mm).
stresses. High compressive bearing stresses are developed near the bearings and emanate radially to the outer panels due to the combination with shear stresses. The diagonal tensile stresses in the outer panels are due to the shearing force. There is no discernible stress concentration around the access hole. Fig. 9(b) shows the yield zones formed immediately above the bearings due to bearing stresses at failure. Despite the different geometry, the principal stress distribution in other models with two bearings exhibited a similar pattern. It appears that bearing failure is the most critical consideration in the design of two-bearing systems. Fig. 10(a) shows the principal stresses in the diaphragm of the one-bearing system (Model ST21). A significantly different response characteristic is evident when compared to that of the two-bearing system. Significant tensile stresses, up to the yield stress, develop around the access hole in the diagonal direction. These tensile stresses develop due to the
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Fig. 10. Stresses in one-bearing diaphragm at failure (model: ST21, hole: 600 mm × 800 mm).
diagonal tension field caused by a cantilever action of the right-hand side of the diaphragm, which acts as a short bracket. In the lower left part of the panel with no access hole, the diagonal tensile stresses are greater than the diagonal compressive stresses, which implies the diagonal tension field also develops after shear buckling. Fig. 10(b) shows the yield zones formed around the access hole due to the tension field action in addition to the bearing stresses. Fig. 11 shows the deformed shape at failure. It appears that the shear strength is one of the critical design parameters in the design of one-bearing systems. Stiffening the access hole may be necessary in the diaphragm of a one-bearing system in order to prevent excessive deformations. Bending stresses are insignificant when compared to the bearing and shearing stresses.
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Fig. 11. Deformed shape of one-bearing diaphragm at failure (model: ST21, hole: 600 mm × 800 mm).
Table 4 Dimensions of access holes Access hole size b (mm)
d (mm)
r (mm)
400
500
200
600 800
800 1000
300 400
1000
1500
500
4.3. Effect of access hole size The effect of the size of the access hole on the ultimate strength of the load-bearing diaphragm was also investigated in this study. Four different sizes, shown in Table 4 were examined. Fig. 12 shows the total reaction versus the out-of-plane displacement for model ST32. Table 5 summarizes the ultimate strengths, showing the reductions for all five models analyzed. As the size of the access holes increased, the ultimate strength of the load-bearing diaphragms decreased, as expected. For model ST32, the difference in the ultimate strength between the model with no access hole and the model with a relatively large access hole with an opening of 800 mm × 1000 mm was 11%. The weakening effect is thus relatively insignificant, even for large access holes. This is because a large portion of the applied loads for box girder bridges is transmitted to the bearing devices through bearing stiffeners and a portion of the diaphragm plate adjacent to the bearing stiffener(s). Therefore, it appears feasible to provide access holes in the loadbearing diaphragm with adequate dimensions to facilitate the passage of the inspectors and portable equipment. The loss of strength due to the access hole can be easily recovered by increasing the diaphragm thickness.
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Table 5 Effect of access hole size on strength of load-bearing diaphragms Ultimate strength with access hole/Ultimate strength without access hole Access hole
ST32
ST22
ST21
SR22
CT32
400 mm × 500 mm 600 mm × 800 mm 800 mm × 1000 mm 1000 mm × 1500 mm
0.97 0.92 0.89 –
0.97 0.94 0.90 –
0.98 0.95 0.92 –
0.98 0.96 0.94 0.92
0.95 0.92 0.90 –
Fig. 12. Changes in strength of load-bearing diaphragms due to access holes (model ST32).
4.4. Effect of access hole stiffeners The purpose of installing the access hole stiffeners is to minimize the undesirable effect of the stress concentration around the access hole and to compensate for the possible strength reduction of load-bearing diaphragms due to the presence of the access hole. The effects of access hole stiffeners on the strength of load-bearing diaphragms were examined using the ST32 bridge model. Fig. 13 shows the total reaction versus the out-of-plane displacement for a variety of access hole stiffeners attached around an opening of 800 mm × 1000 mm. The ultimate strength of the load-bearing diaphragm increased 9.3% when a pair of 130 mm × 26 mm stiffeners were provided. In the case of a relatively small access hole of 400 mm×500 mm, the ultimate strength increased only 4.6% when a pair of 100 mm × 20 mm stiffeners were used, as shown in Fig. 14.
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Fig. 13. Total reaction vs. displacement of load-bearing diaphragms depending on dimensions of hole stiffener (model ST32, hole: 800 mm × 1000 mm).
Fig. 14. Total reaction vs. displacement of load-bearing diaphragms depending on dimensions of hole stiffener (model ST32, hole: 400 mm × 500 mm).
The effect of access hole stiffeners attached around a typical access hole with an opening of 600 mm × 800 mm is summarized in Table 6. It can be seen from Table 6
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Table 6 Effect of access hole stiffener on ultimate strength of load-bearing diaphragms, Model ST32 (hole: 600 mm × 800 mm) Access hole stiffener (width × thickness)
Area of hole stiffener (mm2 )
Ultimate strength (kN)
Strength ratio (with/without hole)
No hole No hole stiffener 50 mm × 8 mm × 2 100 mm × 8 mm × 2 150 mm × 8 mm × 2 100 mm × 12 mm × 2 100 mm × 18 mm × 2
–
21,550 18,876 19,355 19,561 19,729 19,937 20,153
1.00 0.88 0.90 0.91 0.92 0.93 0.94
0 800 1600 2400 2400 3600
Table 7 Ultimate strength of load-bearing diaphragm vs. hole dimension, Model ST21 Hole dimension
Stiffener
Ultimate strength (kN)
Difference (%)
No hole 400 mm × 500 mm 800 mm × 1000 mm
– 120 mm × 10 mm 120 mm × 10 mm
12,877 13,053 11,925
– +1.3 −8.0
that the ultimate strength is increased 6% by attaching a pair of 100 mm × 18 mm stiffeners around the opening compared to that with no stiffeners. However, the difference in the ultimate strength is only 4.1% between the load-bearing diaphragms stiffened by a pair of the smallest stiffeners (50 mm × 8 mm) and by a pair of the largest stiffeners (100 mm × 18 mm). Table 7 shows the variations in the ultimate strengths for model ST21 for two different sizes of openings stiffened by a pair of 120 mm × 10 mm stiffeners. It is of interest to note that the ultimate strength of the load-bearing diaphragm with a smaller opening of 400 mm × 500 mm stiffened by a pair of 120 mm × 10 mm actually increases by 1.3% over that with no opening. This may be attributable to the fact that the restraining effect of the stiffeners against the out-of-plane deformation of the diaphragm is more significant than the potential weakening due to the small opening. Overall, it appears that practical stiffeners attached around moderate size access holes cannot completely compensate for the weakening effect of access holes, as shown in Table 6. 4.5. Effect of access hole location Article 9.17.2.7, BS 5400 [10] suggests that the centerline of the access hole is preferably located within the upper-third of an unstiffened diaphragm, to avoid losing an excessive amount of strength. Article 9.17.2.8, BS 5400 [10] stipulates that the distance of any edge from an adjacent wall of the box should be examined in order to prevent stress concentration and/or excessive deformation in a stiffened diaphragm.
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Table 8 Effect of access hole location on ultimate strength of load-bearing diaphragms, Model ST22 (hole: 600 mm × 800 mm) Position of access hole
D /D
Ultimate strength Pu (kN)
Pu /Pu,4th Level
4th level 3rd level 2nd level 1st level
0.75 0.58 0.42 0.25
19,561 18,876 18,511 18,310
1.00 0.97 0.95 0.94
Fig. 15. Position of access hole.
The relationship between the location of the access hole and the strength of load-bearing diaphragms was examined using the ST22 bridge model. Fig. 15 shows the positions of access holes in the load-bearing diaphragms examined in this study. The symbol © shows the geometric center of the access hole and d is the distance measured from the top of the bottom flange to the center of the access hole. Table 8 shows the ultimate strength for each of the possible locations of the access hole. As the location of the access hole moved downward, the ultimate strength of the load-bearing diaphragm decreased. The difference in the ultimate strength of the diaphragm with an access hole at the third level and the fourth level was only 3.5% and the difference due to locating the access hole at the first level (d /D = 0.25) and at the fourth level was only 6.4%. The effect of the location of the access hole on the ultimate strength of the load-bearing diaphragm does not appear to be too detrimental. This is again attributable to the fact that a large portion of the applied loads for box girder bridges is transmitted to the bearing devices through bearing stiffeners and the portion of the diaphragm plate adjacent to the bearing stiffener(s). It thus seems entirely feasible to provide access holes at a convenient location for the passage of inspectors and equipment. 5. Summary and conclusions In this paper, a series of parametric studies were performed in order to investigate the effect of access holes on the strength of the load-bearing diaphragms in steel box girder bridges. It was found that the weakening effect of an initial geometric imperfection needs to be considered in the evaluation of the ultimate strength of the load-bearing
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diaphragm, particularly for plates with a moderate width-to-thickness ratio. In the twobearing system, very high compressive bending stresses develop below the access hole between the bearings and there is no noticeable stress concentration found around the access hole. In the one-bearing system, partial yielding could take place around the access hole due to the action of the tension field. The addition of access hole stiffeners around the opening in load-bearing diaphragms with a large sized hole cannot fully restore the reduction of the ultimate strength due to the opening. The reduction in strength due to a moderate access hole is not excessive in load-bearing diaphragms. Also, the location of the access hole has no detrimental influence on the diaphragm strength. These phenomena are attributable to the fact that a large portion of the applied loads is transmitted to the bearing devices through bearing stiffeners and the portion of the diaphragm plate adjacent to the bearing stiffener(s). It is recommended that the access hole be provided at a convenient location, with adequate dimensions to facilitate the passage of the inspectors and portable equipment. Any related loss of diaphragm strength can be easily recovered by only a small increment of the diaphragm thickness. In the design of one-bearing systems, the shear strength needs to be carefully examined in order to prevent premature failure. Acknowledgements This research was supported by the Korean Ministry of Construction & Transportation (MOCT) through the Korean Bridge Design & Engineering Research Center at Seoul National University and SISTec (Smart Infra-Structure Technology Center) in KAIST, Korea. The authors wish to express their gratitude for the financial support. References [1] AASHTO. LRFD Bridge design specifications. 3rd ed. Washington, DC: American Association of State Highway and Transportation Officials; 2004. [2] AASHTO. Standard specifications for highway bridges. 17th ed. Washington DC: American Association of State Highway and Transportation Officials; 2002. [3] ADINA. Theory and modeling guide, vol. 1. Report ARD 00-7, ADINA R & D, Inc.; 2001. [4] AISC. Load and resistance factor design for structural steel building. 3rd ed. Chicago (IL): American Institute of Steel Construction; 2001. [5] AISI. Four LRFD design examples of steel highway bridges, prepared by HDR Engineering, Inc. for National Steel Bridge Alliance, American Iron and Steel Institute; 1997. [6] Allen HG, Bulson PS. Background to buckling. UK: McGraw-Hill; 1980. [7] Bathe KJ. Finite element procedures. NJ: Prentice Hall; 1996. [8] Bridge welding code. ANSI/AASHTO/AWS D1.5-96, A joint publication of AASHTO and American Welding Society, Washington DC; 1996. [9] Bruneau M, Uang C, Whittaker A. Ductile design of steel structures. New York (NY): McGraw-Hill; 1998. [10] BSI. Steel, concrete and composite bridges, BS5400, Part 3. Code of practice for design of steel bridges, British Standards Institution, London; 1982. [11] Galambos TV. Guide to stability design criteria for metal structures. 5th ed. New York (NY): John Wiley & Sons; 1998. [12] Heins CP, Hall DH. Designer’s guide to steel box girder bridges. Bethlehem (PA): Bethlehem Steel Corporation; 1981.
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[13] Hall DH, Yoo CH. Recommended specifications for steel curved-girder bridges. National Cooperative Highway Research Program Transportation Research Board National Research Council, NCHRP Project 12-38; 1998. [14] Lee SC, Yoo CH. Strength of plate girder web panels under pure shear. J Struct Engng, ASCE 1998;124(2): 184–94. [15] Choi BH, Yoo CH. Strength of stiffened flanges in horizontally curved box girders. J Engng Mech, ASCE 2004 [in press]. [16] NTSB. Collapse of U.S. 35 Highway Bridge, Point Pleasant, West Virginia, December 15, 1967. Highway Accident Report No. NTSB-HAR-71-1, National Transportation Safety Board, Washington DC; 1970.
Analytical investigation of access holes in load-bearing diaphragms of steel box girders Dong Y. Yoona, Sung C. Leeb, Chai H. Yooa,∗ a Department of Civil Engineering, Auburn University, Auburn, AL 36849-5337, United States b Department of Civil and Environmental Engineering, Dong Guk University, Seoul 100-715,
Republic of Korea Received 22 September 2003; accepted 26 July 2004
Abstract This paper presents a series of parametric studies on the effect of access holes on the strength of the load-bearing diaphragms in steel box girder bridges. These parameters include the number of bearings, initial imperfections, the size and location of access holes, and the size of stiffeners around the openings. The load-bearing diaphragms were modeled using three-dimensional finite element analyses for material and geometric nonlinearities. It was found that the opening can be made large enough for practical purposes in order to provide convenient access to the inside of the box girder with no significant reduction in the ultimate strength of the load-bearing diaphragm. The location of the access hole has no detrimental effect on the diaphragm strength. © 2004 Elsevier Ltd. All rights reserved. Keywords: Access holes; Finite element analysis; Load-bearing diaphragm
1. Introduction Steel box girder bridges incorporate two kinds of diaphragms. One is a load-bearing diaphragm positioned at supports, as shown in Fig. 1, and the other is an intermediate ∗ Corresponding author. Tel.: +1 334 844 6279; fax: +1 334 844 6290.
E-mail address: [email protected] (C.H. Yoo). 0143-974X/$ - see front matter © 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jcsr.2004.07.005
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Nomenclature B Bb Bt D d E E sh Fy Fult q td tw εsh εy
average width of diaphragm [(Bb + Bt )/2] bottom width of diaphragm top width of diaphragm depth of diaphragm distance from bottom flange to center of access hole modulus of elasticity tangent modulus yield stress ultimate tensile stress shorter dimension of plate or distance between bearing stiffeners thickness of diaphragm thickness of web strain at strain hardening yield strain
cross-frame used to reduce distortion and to retain the original shape of the box. This study examines only the load-bearing diaphragms. The primary function of load-bearing diaphragms at the supports of box girders is to transfer vertical and horizontal forces from the webs and the flanges to the bearings. Load-bearing diaphragms are subjected to high concentrated bearing forces and hence are designed to avoid premature buckling failure due to the large compressive loads. Bearing stiffeners are typically placed on both sides of the load-bearing diaphragm to help carry these large concentrated loads. Fig. 2 shows the three major forces acting on a load-bearing diaphragm. Stresses in the load-bearing diaphragm due to each force are affected by the following factors: (1) its shape; (2) the locations and number of bearings; (3) the location, shape, and size of the access hole; and (4) the stiffening around the access hole. Due to its multi-dimensional nature, it is not an easy task to accurately evaluate the resulting stresses in the load-bearing diaphragm. As a consequence of the Silver Bridge failure, reported in NTSB [16], highway bridges in the United States are mandated to be inspected periodically under the 1968 Federal Aid Highway Act. The inspection cycle is determined by the age of the bridge, but every bridge must be inspected at least once every two years. Inspection crews walk or crawl through the inside of the box girders and maintenance repair work is made on an as-needed basis. Therefore, it is necessary for load-bearing diaphragms to have access holes installed. Access holes have various shapes and locations, but all must be as large as practical in order for inspectors and/or equipment to pass conveniently. However, the access-hole dimensions are usually designed to be as small as possible due to the traditional design concern for a possible strength reduction of the load-bearing diaphragms. Frequently,
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Fig. 1. Load-bearing diaphragm.
Fig. 2. Stress in plate diaphragm at bearing, from Heins and Hall [12].
stiffeners are provided around access holes to increase the strength and to alleviate stress concentration. As a result, it is often inconvenient to pass through these small access holes. In this study, the strength reduction effect on the load-bearing diaphragms due to these access holes was investigated. Some of the variables that may affect the strength include the material properties, the location and size of the access holes, the presence of access hole stiffeners, and the number of bearings. Geometric and material incremental nonlinear analyses were performed in order to characterize and quantify the load-bearing behavior of solid-plate diaphragms with various sizes of access holes installed at different locations. The tool used was a commercially available three-dimensional finite element
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Fig. 3. Stiffened diaphragm, from Allen and Bulson [6].
code, ADINA [3], that is capable of incorporating the geometric and material nonlinear characteristics of the load-bearing diaphragms and the other box girder members. 2. Brief reviews The load-bearing diaphragms in most steel box girders are designed to be either very stocky, unstiffened diaphragms or stiffened diaphragms with one or more pairs of bearing stiffeners. Design concepts for load-bearing diaphragms can be divided into the plate and column approach, respectively. The design concepts used in the current specifications for each are briefly reviewed below. 2.1. Plate concept BS 5400 [10]: Part 3, Code of Practice for Design of Steel Bridges, is based on the plate concept. Two different schemes are contained in the code: one is used for simple diaphragms, as illustrated in Fig. 2 where only vertical bearing stiffeners divide the diaphragm into sub-panels; and the other is used for diaphragms of considerably complicated geometries, as shown in Fig. 3 where horizontal stiffeners and stub stiffeners are incorporated. In each case, the design procedure includes an examination of the geometric proportions, analyses of diaphragm stresses, and strength and stability checks. Approximate expressions are used for the elastic critical buckling strength of rectangular and trapezoidal sub-panels loaded along the edges of the diaphragm. The rotational restraints provided by the flanges and webs to the diaphragm are conservatively neglected. Full length bearing stiffeners are assumed to be placed symmetrically on both sides of the diaphragm plate unless a special analysis is made on the effects of any eccentricity with respect to the plate. Article 9.17.2.7(b), BS 5400 [10] stipulates that unstiffened openings in unstiffened load-bearing diaphragms must be circular and of a diameter not exceeding the least value given by Eq. (1): 6td , D/20 or B/20
(1)
where td = diaphragm thickness, D = depth of the diaphragm, and B = width of the diaphragm taken as the average of the widths of the top and bottom flange levels for boxes with sloping webs. In practice, however, all openings satisfying the limitation
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Fig. 4. Bearing stiffener and effective width of web.
imposed by Eq. (1) are not sufficiently large for access holes for the passage of inspectors and/or maintenance equipment. Consequently, any access holes for such purposes must be stiffened. The provision also specifies that holes are not to be placed within the highly stressed area bounded by the bottom flange and one third of the height of the diaphragm. The limiting stresses are to be evaluated using a two-dimensional plate theory. 2.2. Column concept A column concept is adopted to design the bearing stiffener in plate girders in the AISC [4], AASHTO [2], and AASHTO LRFD [1] specifications. In the column concept, the bearing stiffeners plus portions of the web are assumed to act as a column to resist the compressive force transmitted to the bearing. According to Article 10.34.6.1, AASHTO [2], the effective area of the column section is taken as the area of the stiffener, plus a centrally located strip of web extending not more than nine times the thickness of the web on each side of the outer projecting elements of the stiffener group, as shown in Fig. 4. However, neither AASHTO [2] nor AASHTO LRFD [1] give explicit provisions for the design of the load-bearing diaphragms and access holes: the design of bearing stiffeners attached to the diaphragms of box girders is assumed to follow the provisions for plate girders; and the design of access holes is not directly addressed. 3. Finite element analysis For finite element analyses, 3/4-node shell elements are employed to model the steel components and 6/8-node solid elements are used for the concrete decks and bearings. These elements are readily available from the ADINA [3] element library and they are compatible with all elastic buckling analyses and incremental iterative analyses for both material and geometric nonlinearities. In order to determine an appropriate, reliable mesh refinement, a convergence analysis was carried out by varying the number of elements for an isolated bearing diaphragm, as shown in Fig. 5. The diaphragm is simply supported at all the four edges and a uniform load is applied on the top edge. Table 1 shows the results of collapse analyses of the three meshes. The mesh shown in Fig. 5(b) was selected for this study because it yields close to the same numerical values as those obtained from the finer mesh shown in Fig. 5(c) with a considerable reduction in computation time.
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Table 1 Convergence test Mesh
Number of elements
Collapse load (kN)
1 2 3
174 1030 2000
2653.7 2628.9 2600.4
Fig. 5. Convergence test models.
3.1. Finite element model description Five different types of load-bearing diaphragms were examined in this study. These types of diaphragms include those most likely to be used in practical steel box girders. Four straight box girders and one horizontally curved box girder were considered. The details of the models are summarized in Table 2. The first character in the model designation represents whether the box girder is straight or horizontally curved, i.e., S for a straight girder and C for a curved girder. The second character stands for the shape of the box section, i.e., T for a trapezoidal section and R for a rectangular section. The third character indicates the number of spans. The fourth character represents the number of bearings at each load-bearing diaphragm location. As per Article 6.11.1, AASHTO LRFD [1], the inclination of web plates is kept constant at 1 to 4 for all trapezoidal sections. Typical finite element models of straight and horizontally curved box girders are shown in Fig. 6. As the focus of the study is the behavior of the diaphragm, the diaphragm was divided with finer meshes compared to other parts of the model. In order to reduce the computation time and modeling effort, only one third of the span was considered in the models. The models having two equal spans, ST22, ST21, and SR22, are shown in Fig. 6(a). At the cut point, only translation along the vertical axis and rotation with respect to the horizontal axis were permitted, suppressing all other degrees of freedom, to simulate the vertical bending behavior of the girder. The difference in the ultimate strength of the diaphragm based on the full and partial models was noted to be less than 1%. Models ST32 and CT32, having three spans, were analyzed using full models.
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Table 2 Bridge models Model
ST32
ST22
ST21
SR22
CT32
Number of spans Span length
3 58 m-72 m-58 m
Global geometry
Straight
2 45 m-45 m (l = 14 m) Straight
2 40 m-40 m (l = 13 m) Straight
2 50 m-50 m (l = 15 m) Straight
Cross section Number of girders Number of bearings Number of bearing stiffeners at each bearing D (mm) Bt (mm) Bb (mm) Br (mm) Max. t f (mm) tw (mm) Reference
Trapezoid 2 2 1
Trapezoid 1 2 2
Trapezoid 1 1 2
Rectangle 2 2 1
3 48.8 m-64 m48.8 m Horizontally curved Trapezoid 2 2 1
1800 3400 2500 12,880 35.0 14.0 AISI [5]
2100 3450 2400 6200 28.5 12.7
2000 3300 2300 6000 30.0 15.0
2300 2400 2400 12,000 28.5 14.0
Fig. 6. Finite element models.
1980 3048 2060 12,350 38.1 14.3 Hall and Yoo [13]
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For all straight girder models, two types of bearing devices were tested: elastomeric bearings, where relatively small vertical deflections and rotations were permitted, and other bearing devices using only steel sole plates, where vertical deflections were suppressed. It was intended in this study to investigate the potential effect of an uneven vertical displacement at the level of the elastomeric bearing pad on the development of the transverse compressive stress at the bottom part of the diaphragm plate caused by the horizontal force component resulting from the inclined webs. Examination of test runs indicated that there was no appreciable difference (less than 1%) in the magnitude of the induced transverse compressive stresses between the two types. Thus, sole plate bearings were used for all the girder models. In the incremental iterative analyses, a uniformly distributed vertical load was applied at the top of the composite deck. Based on the uniform surface load applied, ADINA [3] automatically determines the load intensity at each mesh point, reflecting the different tributary areas caused by different mesh dimensions. No horizontal load was considered in this study. Vertical loads applied at the cut point in the partial models reflect the accumulation of the uniformly distributed load of the truncated parts of the structure. The bridge models were intentionally designed so that the load-bearing diaphragm would collapse before the flange and web did. The flange and web were designed to remain elastic while the load-bearing diaphragm was proportioned to be stressed beyond the linear elastic limit. As a result, the analysis results can be interpreted to represent the ultimate strength of the load-bearing diaphragm under continuity conditions provided by the flanges and webs. 3.2. Initial geometric imperfections It has been shown that initial geometric imperfections tend to reduce the ultimate strengths of two-dimensional compression members with moderate thicknesses or one-dimensional columns with intermediate slenderness ratios (Lee and Yoo [14]; Galambos [11]). Although Article 9.19, Bridge Welding Code [8] stipulates the maximum dimensional tolerances of plate girder webs and stiffeners, the magnitudes of the allowable initial geometric deformations or out-of-flatnesses for thin-walled elements differ from code to code. Furthermore, there is no specific limit stipulated for the initial geometric imperfections of load-bearing diaphragms of box girders. A strategy of imposing an initial imperfection on the finite element mesh, a technique frequently used in nonlinear analysis, was employed in order to simulate out-ofplane deformations of the load-bearing diaphragms reflecting fabrication tolerances. As suggested by Bathe [7], the first elastic buckling mode shape of the load-bearing diaphragm was adopted as the initial geometric imperfection shape, with the largest amplitude being equal to the largest fabrication tolerance, q/150, permitted by Article 9.19, Bridge Welding Code [8]. The symbol q defines the least lateral dimension of an element. A very small initial imperfection of q/150,000 was used to simulate the initially straight model in the incremental analyses. The imposition of the normalized second and/or higher mode buckling shapes on the perfect geometry does not reduce the ultimate load-carrying capacity significantly, as discussed in Bathe [7]. Hence, it was not implemented for this study. Fig. 7 illustrates the
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Fig. 7. First elastic buckling mode shape of load-bearing diaphragms.
Fig. 8. Stress–strain curve of steel.
first elastic buckling mode shapes of the load-bearing diaphragms. The first elastic buckling mode shape of the entire load-bearing diaphragm includes the twisting mode of the bearing stiffeners, as shown in Fig. 7. The finite element meshes reflecting these initial geometric imperfections were then used with the maximum amplitude of the normalized eigenvector being equal to that permitted by the Bridge Welding Code [8] for the incremental (load) analysis. 3.3. Material properties The stress–strain curve of steel assumed in this study is shown in Fig. 8. For a typical construction steel having a minimum yield stress of 345 MPa (50 ksi) and a modulus of elasticity of 200 GPa (29,000 ksi), the yield strain, ε y , was computed to be 0.001725. The plastic plateau, εsh , was assumed to be 10ε y . According to Bruneau et al. [9], the plastic plateau generally varies from 5 to 15. The tangent modulus, E sh , at the onset of strain hardening was taken to be approximately 1/30th (6667 MPa) of the modulus of elasticity of steel, E. For a minimum yield stress, Fy , of 345 MPa (50 ksi) and an ultimate tensile stress, Fult , of 485 MPa
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Table 3 Strength reduction of load-bearing diaphragms due to initial imperfections Ultimate strength with geometric initial imperfections Ultimate strength without geometric initial imperfections Maximum initial deformations
ST32
ST22
ST21
SR22
CT32
q/1500 q/500 q/150
0.98 0.96 0.92
0.98 0.97 0.94
0.99 0.97 0.94
0.95 0.92 0.89
0.98 0.96 0.92
(70 ksi), approximately 12ε y is required to reach the ultimate stress after the initial strain hardening. Poisson’s ratios for steel and concrete were taken to be 0.3 and 0.16, respectively. The modulus of elasticity of the concrete deck was taken to be 25,000 MPa (3600 ksi), thereby giving a modular ratio of 8. The relatively thin layer of elastomer is also assumed to behave in an elastic manner. Based on the AASHTO LRFD [1], the elastic modulus and Poisson’s ratio were taken to be 240 MPa (35 ksi) and 0.49 because the elastomer is an almost incompressible material. 4. Data analysis 4.1. Effect of initial geometric imperfections The initial geometric imperfections considered were q/1500, q/500, and q/150. In Table 3, the weakening effect of an initial out-of-flatness is summarized for all five models examined. As the magnitude of the initial geometric imperfection increases, the ultimate strength of the load-bearing diaphragms decreases, as expected. In the SR22 model, the maximum strength is reduced by 11.0% when the maximum initial deformation is taken to be q/150 compared to that of an almost perfect girder. As has been shown elsewhere (Choi and Yoo [15]), the strength reductions of the load-bearing diaphragm due to the initial geometric imperfections are not negligible, particularly for plates with an intermediate width-to-thickness ratio. Therefore, in order to accurately determine the ultimate strength of the load-bearing diaphragm, the initial geometric imperfection must be considered. It is unconservative practice to neglect the ultimate strength reduction due to the initial geometric imperfection. 4.2. Induced stresses The principal stresses of model ST32 with two bearings at the ultimate load are illustrated in Fig. 9(a). It can be seen that the stresses resulted from the three major actions (bearing, bending, and shear actions) shown in Fig. 2. The horizontal compressive stresses shown below the access hole and the horizontal tensile stresses near the top of the diaphragm between the bearings are the bending stresses. The magnitude of the maximum compressive bending stress below the access hole accounts for 73% of the maximum vertical compressive stress at failure. The bottom horizontal stiffener shown in Fig. 3, therefore, may be necessary in order to prevent buckling failure due to compressive bending
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Fig. 9. Stresses in two-bearing diaphragm at failure (model: ST32, hole: 800 mm × 1000 mm).
stresses. High compressive bearing stresses are developed near the bearings and emanate radially to the outer panels due to the combination with shear stresses. The diagonal tensile stresses in the outer panels are due to the shearing force. There is no discernible stress concentration around the access hole. Fig. 9(b) shows the yield zones formed immediately above the bearings due to bearing stresses at failure. Despite the different geometry, the principal stress distribution in other models with two bearings exhibited a similar pattern. It appears that bearing failure is the most critical consideration in the design of two-bearing systems. Fig. 10(a) shows the principal stresses in the diaphragm of the one-bearing system (Model ST21). A significantly different response characteristic is evident when compared to that of the two-bearing system. Significant tensile stresses, up to the yield stress, develop around the access hole in the diagonal direction. These tensile stresses develop due to the
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Fig. 10. Stresses in one-bearing diaphragm at failure (model: ST21, hole: 600 mm × 800 mm).
diagonal tension field caused by a cantilever action of the right-hand side of the diaphragm, which acts as a short bracket. In the lower left part of the panel with no access hole, the diagonal tensile stresses are greater than the diagonal compressive stresses, which implies the diagonal tension field also develops after shear buckling. Fig. 10(b) shows the yield zones formed around the access hole due to the tension field action in addition to the bearing stresses. Fig. 11 shows the deformed shape at failure. It appears that the shear strength is one of the critical design parameters in the design of one-bearing systems. Stiffening the access hole may be necessary in the diaphragm of a one-bearing system in order to prevent excessive deformations. Bending stresses are insignificant when compared to the bearing and shearing stresses.
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Fig. 11. Deformed shape of one-bearing diaphragm at failure (model: ST21, hole: 600 mm × 800 mm).
Table 4 Dimensions of access holes Access hole size b (mm)
d (mm)
r (mm)
400
500
200
600 800
800 1000
300 400
1000
1500
500
4.3. Effect of access hole size The effect of the size of the access hole on the ultimate strength of the load-bearing diaphragm was also investigated in this study. Four different sizes, shown in Table 4 were examined. Fig. 12 shows the total reaction versus the out-of-plane displacement for model ST32. Table 5 summarizes the ultimate strengths, showing the reductions for all five models analyzed. As the size of the access holes increased, the ultimate strength of the load-bearing diaphragms decreased, as expected. For model ST32, the difference in the ultimate strength between the model with no access hole and the model with a relatively large access hole with an opening of 800 mm × 1000 mm was 11%. The weakening effect is thus relatively insignificant, even for large access holes. This is because a large portion of the applied loads for box girder bridges is transmitted to the bearing devices through bearing stiffeners and a portion of the diaphragm plate adjacent to the bearing stiffener(s). Therefore, it appears feasible to provide access holes in the loadbearing diaphragm with adequate dimensions to facilitate the passage of the inspectors and portable equipment. The loss of strength due to the access hole can be easily recovered by increasing the diaphragm thickness.
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Table 5 Effect of access hole size on strength of load-bearing diaphragms Ultimate strength with access hole/Ultimate strength without access hole Access hole
ST32
ST22
ST21
SR22
CT32
400 mm × 500 mm 600 mm × 800 mm 800 mm × 1000 mm 1000 mm × 1500 mm
0.97 0.92 0.89 –
0.97 0.94 0.90 –
0.98 0.95 0.92 –
0.98 0.96 0.94 0.92
0.95 0.92 0.90 –
Fig. 12. Changes in strength of load-bearing diaphragms due to access holes (model ST32).
4.4. Effect of access hole stiffeners The purpose of installing the access hole stiffeners is to minimize the undesirable effect of the stress concentration around the access hole and to compensate for the possible strength reduction of load-bearing diaphragms due to the presence of the access hole. The effects of access hole stiffeners on the strength of load-bearing diaphragms were examined using the ST32 bridge model. Fig. 13 shows the total reaction versus the out-of-plane displacement for a variety of access hole stiffeners attached around an opening of 800 mm × 1000 mm. The ultimate strength of the load-bearing diaphragm increased 9.3% when a pair of 130 mm × 26 mm stiffeners were provided. In the case of a relatively small access hole of 400 mm×500 mm, the ultimate strength increased only 4.6% when a pair of 100 mm × 20 mm stiffeners were used, as shown in Fig. 14.
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Fig. 13. Total reaction vs. displacement of load-bearing diaphragms depending on dimensions of hole stiffener (model ST32, hole: 800 mm × 1000 mm).
Fig. 14. Total reaction vs. displacement of load-bearing diaphragms depending on dimensions of hole stiffener (model ST32, hole: 400 mm × 500 mm).
The effect of access hole stiffeners attached around a typical access hole with an opening of 600 mm × 800 mm is summarized in Table 6. It can be seen from Table 6
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Table 6 Effect of access hole stiffener on ultimate strength of load-bearing diaphragms, Model ST32 (hole: 600 mm × 800 mm) Access hole stiffener (width × thickness)
Area of hole stiffener (mm2 )
Ultimate strength (kN)
Strength ratio (with/without hole)
No hole No hole stiffener 50 mm × 8 mm × 2 100 mm × 8 mm × 2 150 mm × 8 mm × 2 100 mm × 12 mm × 2 100 mm × 18 mm × 2
–
21,550 18,876 19,355 19,561 19,729 19,937 20,153
1.00 0.88 0.90 0.91 0.92 0.93 0.94
0 800 1600 2400 2400 3600
Table 7 Ultimate strength of load-bearing diaphragm vs. hole dimension, Model ST21 Hole dimension
Stiffener
Ultimate strength (kN)
Difference (%)
No hole 400 mm × 500 mm 800 mm × 1000 mm
– 120 mm × 10 mm 120 mm × 10 mm
12,877 13,053 11,925
– +1.3 −8.0
that the ultimate strength is increased 6% by attaching a pair of 100 mm × 18 mm stiffeners around the opening compared to that with no stiffeners. However, the difference in the ultimate strength is only 4.1% between the load-bearing diaphragms stiffened by a pair of the smallest stiffeners (50 mm × 8 mm) and by a pair of the largest stiffeners (100 mm × 18 mm). Table 7 shows the variations in the ultimate strengths for model ST21 for two different sizes of openings stiffened by a pair of 120 mm × 10 mm stiffeners. It is of interest to note that the ultimate strength of the load-bearing diaphragm with a smaller opening of 400 mm × 500 mm stiffened by a pair of 120 mm × 10 mm actually increases by 1.3% over that with no opening. This may be attributable to the fact that the restraining effect of the stiffeners against the out-of-plane deformation of the diaphragm is more significant than the potential weakening due to the small opening. Overall, it appears that practical stiffeners attached around moderate size access holes cannot completely compensate for the weakening effect of access holes, as shown in Table 6. 4.5. Effect of access hole location Article 9.17.2.7, BS 5400 [10] suggests that the centerline of the access hole is preferably located within the upper-third of an unstiffened diaphragm, to avoid losing an excessive amount of strength. Article 9.17.2.8, BS 5400 [10] stipulates that the distance of any edge from an adjacent wall of the box should be examined in order to prevent stress concentration and/or excessive deformation in a stiffened diaphragm.
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Table 8 Effect of access hole location on ultimate strength of load-bearing diaphragms, Model ST22 (hole: 600 mm × 800 mm) Position of access hole
D /D
Ultimate strength Pu (kN)
Pu /Pu,4th Level
4th level 3rd level 2nd level 1st level
0.75 0.58 0.42 0.25
19,561 18,876 18,511 18,310
1.00 0.97 0.95 0.94
Fig. 15. Position of access hole.
The relationship between the location of the access hole and the strength of load-bearing diaphragms was examined using the ST22 bridge model. Fig. 15 shows the positions of access holes in the load-bearing diaphragms examined in this study. The symbol © shows the geometric center of the access hole and d is the distance measured from the top of the bottom flange to the center of the access hole. Table 8 shows the ultimate strength for each of the possible locations of the access hole. As the location of the access hole moved downward, the ultimate strength of the load-bearing diaphragm decreased. The difference in the ultimate strength of the diaphragm with an access hole at the third level and the fourth level was only 3.5% and the difference due to locating the access hole at the first level (d /D = 0.25) and at the fourth level was only 6.4%. The effect of the location of the access hole on the ultimate strength of the load-bearing diaphragm does not appear to be too detrimental. This is again attributable to the fact that a large portion of the applied loads for box girder bridges is transmitted to the bearing devices through bearing stiffeners and the portion of the diaphragm plate adjacent to the bearing stiffener(s). It thus seems entirely feasible to provide access holes at a convenient location for the passage of inspectors and equipment. 5. Summary and conclusions In this paper, a series of parametric studies were performed in order to investigate the effect of access holes on the strength of the load-bearing diaphragms in steel box girder bridges. It was found that the weakening effect of an initial geometric imperfection needs to be considered in the evaluation of the ultimate strength of the load-bearing
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diaphragm, particularly for plates with a moderate width-to-thickness ratio. In the twobearing system, very high compressive bending stresses develop below the access hole between the bearings and there is no noticeable stress concentration found around the access hole. In the one-bearing system, partial yielding could take place around the access hole due to the action of the tension field. The addition of access hole stiffeners around the opening in load-bearing diaphragms with a large sized hole cannot fully restore the reduction of the ultimate strength due to the opening. The reduction in strength due to a moderate access hole is not excessive in load-bearing diaphragms. Also, the location of the access hole has no detrimental influence on the diaphragm strength. These phenomena are attributable to the fact that a large portion of the applied loads is transmitted to the bearing devices through bearing stiffeners and the portion of the diaphragm plate adjacent to the bearing stiffener(s). It is recommended that the access hole be provided at a convenient location, with adequate dimensions to facilitate the passage of the inspectors and portable equipment. Any related loss of diaphragm strength can be easily recovered by only a small increment of the diaphragm thickness. In the design of one-bearing systems, the shear strength needs to be carefully examined in order to prevent premature failure. Acknowledgements This research was supported by the Korean Ministry of Construction & Transportation (MOCT) through the Korean Bridge Design & Engineering Research Center at Seoul National University and SISTec (Smart Infra-Structure Technology Center) in KAIST, Korea. The authors wish to express their gratitude for the financial support. References [1] AASHTO. LRFD Bridge design specifications. 3rd ed. Washington, DC: American Association of State Highway and Transportation Officials; 2004. [2] AASHTO. Standard specifications for highway bridges. 17th ed. Washington DC: American Association of State Highway and Transportation Officials; 2002. [3] ADINA. Theory and modeling guide, vol. 1. Report ARD 00-7, ADINA R & D, Inc.; 2001. [4] AISC. Load and resistance factor design for structural steel building. 3rd ed. Chicago (IL): American Institute of Steel Construction; 2001. [5] AISI. Four LRFD design examples of steel highway bridges, prepared by HDR Engineering, Inc. for National Steel Bridge Alliance, American Iron and Steel Institute; 1997. [6] Allen HG, Bulson PS. Background to buckling. UK: McGraw-Hill; 1980. [7] Bathe KJ. Finite element procedures. NJ: Prentice Hall; 1996. [8] Bridge welding code. ANSI/AASHTO/AWS D1.5-96, A joint publication of AASHTO and American Welding Society, Washington DC; 1996. [9] Bruneau M, Uang C, Whittaker A. Ductile design of steel structures. New York (NY): McGraw-Hill; 1998. [10] BSI. Steel, concrete and composite bridges, BS5400, Part 3. Code of practice for design of steel bridges, British Standards Institution, London; 1982. [11] Galambos TV. Guide to stability design criteria for metal structures. 5th ed. New York (NY): John Wiley & Sons; 1998. [12] Heins CP, Hall DH. Designer’s guide to steel box girder bridges. Bethlehem (PA): Bethlehem Steel Corporation; 1981.
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