Minimum-weight design of built-up wideflange

Transcription

1 Mimum-weight design of built-up wideflange steel sections Yousef A. Al-Salloum Department of Civil Engeerg, Kg Said University, P. O. Box 800, Riyadh 11421, Saudi Arabia Abstract An algorithm for optimum mimum-weight design of built-up wide-flange sections for flexural steel members is presented. The formulation comprises a general solution subject to compact-section and strength constrats. The design is based on the allowable stress procedure of the American Institute of Steel Construction (ASD-AISC) specifications. The optimum solution is of closedform so that the section proportiong is direct and simple. The solution is governed by activeness of moment alone or moment and shear together. The process is illustrated with the help of an example problem which is implemented on a spreadsheet. 1 Introduction Doubly symmetric built-up wide-flange (BWF) steel beams become practical for cases where the required flexural capacity exceeds that of the largest available rolled steel wide-flange shape. They also offer flexibility limitg plate thicknesses and depth of section to desired values. Traditional procedures for the proportiong of BWF beams do not offer a unique solution and usually proportion a section by analysis of several trial configurations selected on the basis of judgement, tuition and experience. In every trial, the section dimensions are modified order to make the section

2 268 Computer Aided Optimum Design of Structures satisfy the code specified constrats and to obta a weight as low as possible. The design so obtaed, the context of optimization, is not the mimum. Holt and Heithecker [1] and Shive [2] studied the mimum cross-sectional area design of welded doubly symmetric I- beams accordg to the requirements of the allowable stress design (ASD) procedure prescribed by an earlier AISC Specification [3]. The beam was assumed to be laterally supported and the flange was specified by its cross-sectional area, rather than by its width and thickness. In stead of employg a formal optimization procedure, the solution was obtaed by a numerical procedure. Farkas [4] has also addressed the topic. He matas that sce the effect of shear can usually be neglected, the mimum cross-sectional area design may be treated analytically with the help of some approximations. From the resultg simple formulas, comparative calculations can then be performed which should prove useful for designers. However, for complicated problems, like combed effect of bendg and shear, he employed numerical mimization methods, rather than the closed-form one. Al-Salloum [5] studied the optimum proportiong of doubly symmetric BWF sections and developed a comprehensive algorithm to design these sections based on the Load Resistance Factor Design procedure of the AISC specifications (LRFD-AISC) [6]. The optimum solution was of closed-form so that the section proportiong was direct and simple. Although there is a trend toward usg the LRFD-AISC Specifications steel design, the ASD-AISC procedure is still used the design practice. There is a need to fill the gap and develop a closed-form solution for optimum proportiong for mimum-weight or mimum cross-sectional area of the doubly symmetric BWF beams without web stiffeners based on the ASD provisions of the AISC Specifications [7]. The solution is developed this study It optimizes the sectional dimensions subject to compact-section and strength constrats. The effects of both bendg moment and shear force are considered the formulation. Specifically speakg, the formulation determes the mimum-weight section dimensions so that: a) the conditions of 'compact section' are satisfied, b)the condition of web slenderness which permits design accordg to the provisions of chapter F of the ASD-AISC Specifications are met, and c) the web slenderness meets a more restrictive condition which puts allowable shear stress at 0.40Fy and cidentally averts use of termediate stiffeners. The formulation assumes that the compression flange is adequately braced so that the allowable bendg stress is taken at 0.66Fy. The formulation presented is of general form so that it can be employed for design of unstiffened BWF beams without any restrictions.

3 Computer Aided Optimum Design of Structures Formulation of Mimum-Weight Design The cross-section of the BWF beam shown Fig. 1 is required to carry specified bendg moment and shear demands of M and V, respectively. The proportiong of the section volves four design variables: the flange width, bf, the flange thickness, tf, the clear distance between flanges or the web height, h, and the web thickness, tw The dimension b Fig, 1 denotes the unstiffened width of the compression flange. Figure 1: The Built-up Wide Flange Section The area of the cross-section is the objective function which is to be mimized, and is given by, and its section modulus, Sx, is approximated as, (1) 2.1 AISC Design Requirements The ASD-AISC Specifications [7] impose various restrictions for takg allowable values of bendg and shear stresses at 0.66Fy and 0.40Fy, respectively. These conditions are stated below. 1) Web slenderness ratio h/tw does not exceed 970/VFy so that BWF beam may be designed accordg to provisions of chapter F. (2)

4 270 Computer Aided Optimum Design of Structures 2) The flanges are connected contuously to the web, and the unsupported length of compression flange, Ly, does not exceed LC as given by smaller of76bf/vfy or 20,000/[(d%)Fy], 3) Web slenderness ratio also satisfies the constrat, gj, gj = h/t^ < a (3a) where a takes the followg values on different counts, 640/VFy for compact web size, 260 to avoid use of termediate stiffeners, and 380/VFy for usg allowable shear stress at its highest permissible level Of0.4Fy. Of all the three, 380/VFy is the most restrictive value and is employed the formulation. 4) The slenderness ratio of unstiffened compression flange element, satisfies the constrat, g%, p (3b) where p takes a value of 65/>/Fy. 5) The stress duced by bendg, /y = M/S^, is required to satisfy the constrat, gg, 3 - /b * Fb <3c) where view of the conditions (2) to (4) above, the allowable stress bendg, Fy, is taken at its highest permissible value of 0.66 Fy. 6) The stress duced by shear, /^ = VA^h, is required to satisfy the constrat, 4,

5 Computer Aided Optimum Design of Structures 271 where view of limitg h/tw to 380/VFy as stated condition (3) above, the allowable shear stress F^ = 0.40 Fy. The optimization problem thus becomes mimization of the objective function of eqn (1) subject to the behavioral constrats gj through g^, ofeqn (3). 3 Lagrangian Solution The Lagrangian function, L, the basic formulation, is a function of dependent variables: the four design variables bp t^ h and t^ and them Lagrange multipliers, A,j through A,^, of the constrat functions gj through g^ of eqn (3), so that, (4) or X g + g +...+g (5) where m is number of constrats considered the problem. The condition that a derivative of the Lagrangian function with respect to an dependent variable vanishes at an extremum leads to the same number of dependent equations. These equations yield the optimum values of the flange width, bf, flange thickness, t^ web height, h, and web thickness, t^, dependg on the activeness of the bendg moment alone or that of both the bendg moment and the shear force. The Lagrange multiplier method is explaed Appendix A. 4 General Solution The problem, the way it is formulated, leads to a solution which yields separate values of optimum proportions dependg upon activeness the solution of, a) the bendg moment alone, or b) both the bendg moment and the shear force. In the former case, the section design is controlled by flexure, and the shear is automatically satisfied. In the latter case, both the flexure and shear control the design.

6 272 Computer Aided Optimum Design of Structures The limitg moment, M^, which determes the type of activeness is given by,,3/2 (6) When the applied moment is larger than M^ the moment alone controls the design, and the optimum solution becomes: 3M %3/2 1/2 (7a) = 3 3M (7b) 3M a (7c) 3M (7d) When the applied moment is less than M^, the moment and the shear control the design, and the optimum solution becomes: 2PM pv (8a) tf = M (8b) h = (8c) tw = (8d)

7 Computer Aided Optimum Design of Structures Example Problem BWF beam sections are to be designed A36 steel for the design demand of moments and shears given Table 1. The design demands and the constrats are chosen to illustrate the optimum design process. The solution is implemented on a spreadsheet and produced Table 2. The put values are the steel yield stress, and design demands of moment and shear on the section. The calculated values of the design variables, the sectional area, section modulus and moment of ertia are available the tables for each case. Reference to the appropriate solution equations is also made. Table 1 Design Demands of the Example Problem Activeness (1) MOA MSA V (kips) M (kip-) (1) MOA : Moment Only Activeness, MSA : Moment and Shear Activeness The design demands are selected to voke MOA and MSA situations. The general solution for this example is presented Table 2. The first part this table represents the MOA case ( M > M^ ) which the moment is active and the shear is active. The solution for this case is determed from eqn (7). At the optimum, the web slenderness ratio and the compression flange slenderness ratio constrats are at their limitg values or active constrats besides the flexural strength constrat. The second part Table 2 represents the MSA case ( M < M^ ) which all constrats are active at the optimum cludg the shear strength constrat. The solution for this case is determed from eqns. (8). It can be seen that although the required shear strength is doubled the second case, the material is redistributed such that the section modulus remas the same, the moment of ertia creases by 13.4% and the crease the sectional area is only 1.7%. These results show that the optimum solution each case distributes the material the section efficiently such that the sectional area is mimized and all the constrats are satisfied. 6 Conclusions An effective algorithm for the mimum-weight design of unstiffened built-up wide-flange sections for steel beams is developed. The design is based on the ASD procedure of AJSC Specifications. The formulation considers the compact-section and strength constrats. The solution is of closed-form and, therefore, permits direct determation of the section proportions.

8 I Spreadsheet Algorithm for Mimum-Weight Design of BWF Section of the Example Problem GENERAL SOLUTION I ksi ksi FV 14.4 ksi 0.40Fy oc = /VFy (3 = lent only activeness (MO A) k- bf tf M > M^ K lues at optimum t ksi /v = 9.3 ksi h/v = 6J.J bf/2tf = h V A S nent and shear activeness (MSA) k- tf M < M^ ### lues at optimum ksi /v = 14.4 ksi h/tw = 63.3 V2tf = Output h V A S

9 Computer Aided Optimum Design of Structures 275 References 1. Holt, B.C. and Heithecker, G.L., Mimum-weight proportions for steel girders, J. Struct. Div. Proc., ASCE, 95, pp , Shive, A.R., Mimum-Weight Design of Steel Girders, Ph.D. Dissertation, Rice University, Houston, Texas, American Institute of Steel Construction, Specification for the Design, Fabrication and Erection of Structural Steel for Buildgs, AISC, Chicago, Illois, Farkas, J., Optimum Design of Metal Structures. Ellis Horwood Limited, Chichester, West Sussex, England, Al-Salloum, Y.A., Optimum [proportions of built-up wide-flange sections", Journal of Constructional Steel Research, 36, No. 3, pp , American Institute of Steel Construction, Load and Resistance Factor Design Specification for Structural Steel Buildgs, 1st Ed., AISC, Chicago, Illois, American Institute of Steel Construction, Specification for the Design, Fabrication and Erection ofstructural Steel for Bui Idgs, 9th Ed., AISC, Chicago, Illois, APPENDIX A Lagrange Multipliers Method In algebraic terms, the mimum-weight design problem is stated as: Mimize W(X) (Al) T which is an objective function of n design variables X = (xj, ^..., x^), subject to constrats gj(x)<0, j=l,...,m (A2)

10 276 Computer Aided Optimum Design of Structures Nonlear programmg problems can be solved by the well-known Lagrangian function, which combes the objective function and the constrat equations as follows: m = W(X) + ^ ^ gj(x) (A3) where Xj is the Lagrange multiplier for the jth constrat. Each Lagrange multiplier is a sensitivity coefficient that measures the change of the objective function with unit change of the correspondg constrat. By addg slack variables to the constrats, they become: gj(x) + Q-^ = 0, j = 1,..., m (A4) where Q: is the slack variable for the jth equality constrat. The Lagrangian function becomes: r *i (A5) At the stationary pot, the followg conditions must be satisfied: dl + >, A,: - = 0 1= 1,..., n ^~! d*i = gj(x) +Q? =0 j = 1,..., m (A6) From the above (n+2m) nonlear equations, the n optimal values of the design variables Xj, ^..., x^, them values of A,j and m values of the slack variables Q.^ can be calculated. At the optimum, if Q: * 0, the correspondg constrat is active [ gj(x) < 0 ] and its associated Lagrange multiplier is zero ( Xj = 0 ). If Q-^=0, the correspondg constrat is active [ g:(x) = 0 ] and its associated Lagrange multiplier must be nonnegative ( )y > 0 ).