Edward A. Sadek

A finite element model for the analysis of stiffened laminated plates

Abstract A refined higher-order displacement model for the study of the behavior of concentrically and eccentrically stiffened laminated plates based on C0 finite element discretization is presented. The model incorporates non-linear variations of longitudinal displacements through the thickness and hence eliminates the need to use shear correction coefficient(s). Transverse shear deformations are included in the formulation making the model applicable for both moderately thick and thin composite stiffened plates. The plate element used is a nine-noded isoparametric one with seven degrees of freedom at each node. The stiffener element is a three-noded isoparametric beam element with four degrees of freedom at each node. The stiffness of a stiffener is reflected at all nine nodes of the plate element in which it is placed. Accordingly, the stiffeners can be positioned anywhere within the plate element along lines of constant natural coordinates and need not necessarily be placed on nodal lines which gives a great flexibility in the choice of mesh size. The integrals are evaluated by a selectively reduced integration (SRI) technique with three and two Gauss quadrature rules for membrane-flexure and shear parts, respectively. The present formulation is checked for different examples of stiffened plates made of isotropic and fiber-reinforced composites, and results are compared with existing analytical and other finite element solutions.

Optimization of structures having general cross-sectional relationships using an optimality criterion method

Abstract An optimality criterion-based method for locating the optimal designs of structures with members having general cross-sectional relationships is presented. The method exploits the concept of one most active constraint to avoid the need of calculating a large set of Lagrange multipliers for the active constraints and also eliminates the need to decide as to whether or not a particular constraint should be considered active. The method also avoids the scaling procedure normally used in other optimality criterion techniques and can deal with multiple load conditions and design-variables linkage. Several example structures, designed under stress, displacement, minimum and maximum area constraints and design-variables linkage are presented. The results demonstrate the simplicity, efficiency and reliability of the method even when used on small microcomputers. Although this research is based on the concept of one most active constraint, several constraints may be active at the optimal designs.

Some serendipity finite elements for the analysis of laminated plates

Abstract Three rectangular eight-noded serendipity elements are developed and employed in the analysis of flexible laminated composite plate problems including transverse shear deformations. The elements are based on inplane displacement models which incorporate first-order to higher-order terms in thickness coordinate for the consideration of transverse shear deformations. In the first two elements designated REC56-Z0 (seven degrees of freedom per node i.e. 56 degrees of freedom per element) and REC72-Z0 (nine degrees of freedom per node i.e. 72 degrees of freedom per element), the transverse deflection across plate thickness is assumed constant. While in the third element REC88-Z2 (11 degrees of freedom per node i.e. 88 degrees of freedom per element), the transverse deflection varies in the form of a second order power series expansion of thickness coordinate. Stiffness, load and stress matrices are derived. Accuracy of the results are examined by employing the elements in the study of cross-ply square and rectangular simply supported composite plates with different thickness ratio ranging between 2 and 100 under a sinusoidal load distribution. The results are compared with the exact solutions.

Minimum weight design of structures under frequency and frequency response constraints

A technique for the determination of the least weight design which satisfies the desired frequency and frequency response constraints, plus upper and lower bounds on the design variables, is introduced. The design algorithm is based on Zoutendijks method of feasible directions [1]. This method requires the analytical gradients of the objective function and the constraint functions which are active at a given stage in the design process. A considerable improvement in convergence has been achieved by considering each pushoff factor as a linear function of the corresponding active constraint. An initial step length based on a pre-selected decrement of the objective function is used. The results, obtained using an IBM-compatible microcomputer, assert that the method leads to good results even when a very limited number of modes is used in calculating the sensitivities of the frequency response function. This makes the method very efficient with regard to both accuracy, computation time and suitability for use on microcomputers.

Dynamic optimization of framed structures

Abstract An analytical procedure for the determination of the least weight structure which satisfies a specific frequency requirement plus upper and lower bounds on the design variables is presented. The design algorithm is an iterative solution of the Kuhn-Tucker optimality criterion. The procedure is to modify an existing design to first obtain the correct structural frequency and then, while the frequency is held constant, to minimize the weight. This is accomplished using gradient equations derived in matrix notation for direct application to the finite element method of analysis. The most important features of the algorithm are: (a) a small number of design iterations are needed to reach optimal or near-optimal design, (b) structural elements with a wide variety of size stiffness may be used. The procedure has been completely automated in a computer program. Results of two numerical examples show that the method is convergent and that optimized configurations can be determined in as few as 10 redesign cycles.

Application of methods of feasible directions to structural optimization problems

Abstract A general approach to structural optimization which has received much attention in recent years is that of using mathematical programming (numerical search) techniques. These techniques may be separated into direct and indirect methods. Of the direct methods of attack on general nonlinear inequality constrained problems, the largest class is called methods of feasible directions. This paper presents the application fo Zoutendijks method of feasible directions [5] to structural optimization problems. The algorithm requires the analytic gradient of the objective function and the constraint functions which are active at a given stage in the design process. A considerable improvement in convergence has been achieved by considering each pushoff factor as a linear function of the corresponding active constraint. A comparison of the half-step versus full-step search procedure is presented. An initial step length based on a present decrement of objective function is used. A discussion of the linear versus quadratic interpolations of a constraint function in search for a bound point is presented. The algorithm is demonstrated with elastic design of a 25-bar space tower, a 3-bay single storey and a double bay double storey rigid jointed plane frames. Data on the differences in the optimum designs obtained from different starting points is reported.

An optimality criterion method for structural optimization problems

Abstract This work presents the application of an optimality criterion method which exploits the concept of one most critical constraint. The method eliminates the need to calculate a large set of Lagrange multipliers for the active constraints, and also eliminates the need for a decision as to whether or not a particular constraint should be considered active. The method can treat multiple load conditions under both stress and displacement constraints. A study of the effect of using different starting designs and how to deal with passive members has been made. Also some suggestions which have been found to improve the results obtained are presented.

Direct versus indirect methods in structural optimization

Abstract Mathematical programming methods are among the most powerful optimization techniques. These techniques may be separated into direct and indirect methods. Of the direct methods of attack on general nonlinear inequality constrained problems, the largest class is the method of feasible directions. Of the indirect methods, the interior penalty function appears to be the most reliable one while the variable metric method seems to be an extremely powerful algorithm. This paper presents a comparison between the results obtained using Zoutendijks method of feasible directions and the method of interior penalty function coupled with the variable metric method as a minimizing algorithm. A considerable improvement in convergence has been achieved by considering each push-off factor as a linear function of the corresponding active constraint. A comparison of the half-step vs full-step search procedure is presented. Also a comparison between the use of either the normalized or the non-normalized gradients is illustrated. A discussion of the linear vs quadratic interpolations of a constraint function in search for a bound point is presented. An initial step length based on a present decrement of objective function is used. The two algorithms are demonstrated with elastic design of a 25-bar space tower, a 3-bay single-storey frame and a double-bay double-storey rigid jointed plane frame. Data on the differences in the optimal designs obtained from different starting points are reported.

An optimality criterion method for structures having general cross-sectional relationships

Abstract An optimality criterion-based method is presented which is capable of locating the optimal designs of structures with members having general cross-sectional relationships among the area, section modulus and second moment of area. The method exploits the concept of one most critical constraint to avoid the need to calculate a large set of Lagrange multipliers for the active constraints and also eliminates the need for a decision as to whether or not a particular constraint should be considered active. The method also avoids the scaling procedure normally used in other optimality criterion techniques and can deal with multiple load conditions and design-variables linkage. Several example structures, designed under stress, displacement, minimum and maximum area constraints, and design-variables linkage demonstrate the simplicity, efficiency and reliability of the method, even on small microcomputers. Although this research uses the concept of one most critical constraint, several constraints may be active at the optimal designs.

On the dynamics of framed structures

Abstract The paper presents a comparison between the eigen-values of a structure obtained using the distributed mass-stiffness technique proposed by the author[l] and those obtained using a static stiffness matrix coupled with a mass matrix both of them based on the same shape function (consistent mass matrix). The paper also includes some suggestions for improving the results obtained using the latter method.