Transactions on the Built Environment vol 13, 1995 WIT Press, ISSN

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1 Optimum design of welded plate girders subjected to highway bridge loading M. Asghar Bhatti,* A.S. Al-Gahtani^ "Department of Civil and Environmental Engineering, University of 7owa, /owa CzYy, A4 J2242, L%4 ^Department of Civil Engineering, University of Petroleum and Minerals, Dhahran, Saudi Arabia Abstract Optimum design of plate girders subjected to highway bridge loading is presented in this paper. The formulation is capable of handling composite or noncomposite designs, shored or unshored construction, stiffened or unstiffened design, symmetric or unsymmetric cross-section, simple or continuous spans, and prismatic or nonprismatic girders. The bridge loading is applied through influence line functions at selected points along the girder. The design problem is constrained to satisfy the AASHTO specifications. 1 Introduction The use of welded plate girders for short to medium span highway bridges offers greatest flexibility in choosing the most economical section. They offer the following advantages over the standard rolled sections [1]. Welded plate girders can be designed as long and as deep as required by the design and fabrication method. Rolled beams are of constant cross section along their length but in plate girders dimensions can be easily varied if desired. For composite girders, an unsymmetrical section offers considerable savings in weight since typically the top flanges need only be large enough to accommodate the shear connectors. With advancement of welding technology the welded girders can be fabricated on fully automatic welding lines with reasonable cost and uniform quality control. Thus the delivery time for welded girders may be more predictable than that for rolled shapes. Use of optimization techniques is necessary to take advantage of this flexibility in achieving the most economical design. Razani and Goble [2] studied the optimum design of constant-depth plate girders based on 1961 AASHTO specifications. The work was extended by Goble and DeSantis [3] to include the composite sections. Holt and Heithecker [4] presented formulas for optimum proportions for laterally supported symmetrical plate girders. The problem was formulated on the basis of the 1967 American Institute of Steel

2 226 Structural Optimization Construction (AISC) specifications. Schilling [5] presented some numerical ratios for optimum plate girder dimensions based on 1969 AISC specifications. Vachajitpan and Rockey [6] developed a nonlinear programming method to obtain minimum weight design of unstiffened constant depth symmetric plate girders. Burns and Ramamurthy [7] presented a formulation to optimize a uniform plate girder design for composite or noncomposite situation. This study shows the efficiency of using a general formulation which can be used for more than one design. Fleisher [8] presented a mathematical formulation and nomographic design for symmetric plate girders based on the web buckling theory and on the assumption of flange configuration that do not necessitate bending stress reduction in the compressive flange stress under the AISC specifications. Except for few early references, none of the other papers have dealt with optimum design under moving highway loads. In this paper a method for optimum weight/cost design of welded plate girders under AASHTO specifications [9] is presented. The formulation is very general and includes a variety of design options such as: single or continuous span highway bridge girders stiffened or unstiffened girders shored or unshored construction fixed or variable depth along the span symmetrical or unsymmetrical design variable flange thicknesses along span Section 2 briefly reviews analysis and design of composite highway bridge girders. Optimal design formulation is presented in section 3. Section 4 presents few numerical results. 2 Analysis and Design of Highway Bridge Girders From analysis point of view the noncomposite girders and shored composite girders are special cases of the unshored composite case. The unshored composite case involves consideration of three different cross sectional properties under three different loading conditions. Only one of these section properties is needed in the case of noncomposite design and two of them are used in the shored composite design. As shown in Figure 1, a composite welded plate girder consists of four parts; (a) reinforced concrete slab, (b) steel section, (c) shear connectors, and (d) web stiffeners. The purpose of the slab deck is to distribute wheel loads to the girders. The shear connectors are designed to transfer longitudinal shear between the slab and the steel section. They are needed in the positive moment regions only. If the calculated web shear stress exceeds the allowable shear stress of the web, stiffeners may be used to increase the web shear capacity without increasing the web thickness. Analysis of a composite girder A composite beam finite element is used for analysis of the girder. The element consists of unsymmetrical steel section with a concrete slab at the top acting as a cover plate.. The transformed cross-sectional area is calculated by dividing the effective slab area by the modular ratio m = ES/EC, where ES is the modulus

3 Structural Optimization 227 of elasticity of the steel beam and EC is modulus of elasticity of the concrete slab. Based on concrete strength the modular ratio varies between 6 and 11 [9]. The modular ratio is multiplied by a factor KC to approximately account for creep effects. This factor is equal to 3 for sustained dead loads and is equal to 1 for other loads. Corresponding to three different loading conditions, following three different sections properties (moment of inertia and plastic section modulii) must be computed for each element: (a) Steel beam section properties for noncomposite dead loads (b) Composite section properties with creep effects (K^ = 3) for dead loads (c) Composite section properties with K^ = 1 for live loads Each span can be divided into a number of finite elements. For a variable depth girder enough elements must be defined to get a reasonable approximation of the depth using constant depth elements. Since the loads are applied at nodal points, even for a prismatic girder, enough elements must be used to get a fairly smooth influence line. Based on numerical experiments it is suggested to use 10 elements per span. top slab top flange plate -sheer studs web plate- transverse stiff ener bottom flange^ plate I - section Figure 1. Typical Highway Bridge Cross Section Design forces and stresses A composite girder must be analyzed under several different loadings depending upon the construction method. An influence line analysis is required for truck loads. Figures 2 shows typical bending moment diagrams. Bending moments due to four different loadings are shown identified as MS (noncomposite dead loads), Mrj (composite dead loads), M^ (positive live loads), and M~ (negative live loads). The total design forces at any point are

4 228 Structural Optimization the algebraic sum of the forces from different loadings. For additional details see reference [10]. HS20-44 (k-ft.) TOO Bridge length,ft. Bending moments from different loads M ( k-ft.) Bridge length,ft. b. Design bending moments CT0 Figure 2. Typical Design Bending Moments from Different Bridge Loads 3 Optimization Formulation The design of welded plate girders requires determination of six geometrical dimensions; depth and thickness of web, width and thickness of top flange, and width and thickness of bottom flange. Changing the flange thickness along the bridge is more practical and less costly than changing the flange width.

5 Structural Optimization 229 therefore, the flange widths are kept constant along the girder and are not considered as design variables. Thus for each beam element or a group of elements the design variables are defined as: 1) Top flange area (At) 2) bottom flange area (Ay) 3) web depth (d\j 4) web thickness (tw) 5) stiffener spacing (a) in case of stiffened design The volume of the steel beam and the stiffeners is the total weight to be minimized. In most practical bridge designs, the cost is related to the weight of the structure. Cost optimization requires estimation of construction cost that depends on many variables such as, the contractor, construction equipment, method of erection, fabrication, transportation, labor, etc. These variables change continuously with time and differ from one location to another. The volume (or weight) as an objective function is more universal and requires no input or change in input parameters, which allows the designer to try different designs in short time with less difficulty especially at the preliminary design stage. The minimized volume/weight can then be transformed to a cost design if the construction cost is determined. The objective function is minimized subject to the bending stress, shear stress, slenderness ratios, deflection, and depth limitations. It is necessary to set reasonable and practical limits for the design variables to avoid calculations for impractical values. A 5/16 inch plate thickness is the minimum allowed by the AASHTO specifications. The constraints are applied to welded plate beams with welding metal having an equal or greater ultimate strength than the base metal. Flange plates must be joined end to end by full penetration butt welds. The weld metals must conform to AASHTO standard specifications for welding of structural steel for highway bridges. 4 Numerical Results The interior girder of a 100 feet long highway bridge subjected to HS20-44 loading is designed with uniform section over the span. The design information is shown in Figure 3. Different design options under the formulated AASHTO constraints are used. The optimum sections are summarized in Table 1. It can be seen that the unshored composite design reduced the beam cross section area by 20% compared to the noncomposite design. The shored composite design reduced the beam cross section even more by almost 30% of the noncomposite design. More savings could be obtained with shored composite design if narrower top flange is chosen because the flange width to thickness constraint governed the design in this case. The savings will be more with the composite design if the noncomposite design is obtained with the compression flange assumed to be partially supported since the allowable compression stress of the flange will be reduced and a larger flange area would be required for moment resistance. The beam depth also shows wide variation. The composite design as expected offers shallower depth. Shored composite design reduces the depth by more than 25% of the noncomposite design. The unstiffened design in this problem has deeper beam than the stiffened design. This is because the unstiffened design is optimized under both bending and shear stresses while the stiffened design has abundant shear strength and the beam depth is mainly decided by the moment action and is limited by depth to thickness ratio. The unstiffened design requires larger web area for shear resistance and thus thicker and deeper

6 230 Structural Optimization web plates. The stiffened design has smaller web area and larger flange areas than the unstiffened design. 22ft. 8ft. -*- 8ft. Span length,100. ft. Number of lanes, 2 Beam spacing, 8. ft. Steel type,a36 Slab thickness, 7.5 in. Concrete corepressive srength, 4 ksi. Modular ratio, 8 Design live load, HS20-44 Super imposed dead load, 45 Ib/sq. ft. Top flange width, 1 2 i nches Bottom flange width, 20 inches Allowable live load deflection, 1.5 inch Allowable noncomposite dead load deflection, 2. in. Total allowable deflection, 5. in Unsupported flange length, 0.0 in. I. bt j 4 H DESIGN OPTION Unstiffened noncomposite composite unshored composite shored Stiffened noncomposite composite unshored composite shored Figure 3. Data for Numerical Example Table 1 Optimum Design Dimensions of Example 1 dw (in) tw (in) tt (i n) , , tb (in) AS (in2) Stiffeners Spacing ratio (a/dw) 1.5* 1.5* 1.5* Bottom flange width = 20" Top flange width = 12" No. of Iters. The economical design can be selected from these designs based on minimum beam cross section area or minimum depth or other design criteria such as its suitability and applicability to the site conditions and material and labor availability. The most important fact is the availability of these designs in a

7 Structural Optimization 231 very short time. The CPU time for these design ranged from 7 to 12 seconds. The optimization process took about 9 to 12 iterations. Effect of Stiffeners Cost on Beam Design The effect of stiffeners cost on girder design is shown here by designing the girder with different values of stiffener cost coefficient (%) while keeping cost coefficient of plate girder (Ci) constant. Only the noncomposite stiffened design case is used to show this effect. The effect of C2 value on web depth is shown in Figure 4. It can be seen that as the stiffeners cost increases the web depth decrease. This is because the stiffeners are mainly a function of the web depth. Good estimate for stiffeners cost and beam cost will results in an economical beam design accounting for full construction cost. 4 y = beam depth without sitteners cost (.2 C2/C, Figure 4. Change in girder depth with increase in stiffeners cost Effect of Flange Width on the Optimum Beam Cross Section Area Since the beam flange widths are not considered design variables in this paper for the reasons mentioned in reference [10], it is important to study their effect on the optimum girder cross section area. Figure 5 shows changes in the optimum corss-section area with flange width. Concluding Remarks The ability to create and compare different designs for the same structure is the real benefit from a general optimization based formulation. This approach enables the designer to investigate different designs in relatively short time, especially at the preliminary design stage. The efficiency of the formulation presented in this study for optimum design of bridge girders subjected to AASHTO loads is very encouraging. Similar formulations can be applied to other type of bridges, especially long span bridges, where the dead loads are dominant forces in the optimum design. A similar formulation can also be applied to hybrid plate girders, box girders and truss bridges.

8 232 Structural Optimization 1 - Lateral supports spaced at 6 ft. 2- Lateral supports spaced at 12 ft. 3- Lateral supports spaced at 24ft Flange width Jnches Figure 5. Changes in optimum cross-section area with flange width References 1. Douwen, A.A., "Optimum Design of Plate Girders", Proceedings of the International Symposium on Steel Plated Structures, London, Razani, R. and Goble, G.G., "Optimum Design of Constant-Depth Plate Girders", Journal of Structural Division, ASCE, ST2, Vol. 92, April Goble, G.G. and DeSantis, P.V., "Automated Optimum Design of Unstiffened Girder Cross Sections", Journal of the Structural Division, ASCE, ST6, Vol. 92, December Holt, E.C., and Heithecker, G.L., "Minimum Weight Proportions for Steel Girders", Journal of Structural Division, ASCE, ST10, Vol. 95, Schilling, G.C., "Optimum Properties for I-Shaped Beams", Journal of the Structural Division, ASCE, ST12, Vol. 100, December Vachajitapan, P. and Rockey, K.C., "Design Method for Optimum Unstiffened Girders", Journal of the Structural Division, ASCE, ST1, Vol. 104, January Burns, S.A., and Ramamurthy, S., "Mathematical Programming in Structural Design", Journal of the Structural Division, ASCE, ST7, Vol. 109, July Fleisher, W., "Design and Optimization of Plate Girders and Welded Fabricated Beams for Building Construction", Engineering Journal, AISC, No. 1, 1st Quarter, American Association of State Highway and Transportation Officials (AASHTO) Standard Specifications for Highway Bridges. 10. Al-Gahtani, Ahmad Saad, "Optimum Design of Welded I-Beams Subjected to Highway Bridge Loads", Ph.D. thesis, University of Iowa, Iowa City, IA 52242, May 1986.