A two-cell anisotropic rotor blade with a SIKORSKY DBLN-526 Airfoil cross-section is the first example and it is depicted in Figure 4. Differences in the laminate design are highlighted by different line formats in Figure 4 including the webs, and they are given in Tables 3-5.

The solid-line region has elastic modula values that are the same as those in Table 3 but the wall thickness is 0.03 in, and the laminate stacking sequence is [(40°/ – 40°/(30°/0°)2)].

The stiffness matrix of this cross-section about the origin of the axes in Figure 4 was found to be:










0.9970 x 107 lb – in

0.3120 x 1012 lb – in

-0.1725 x 1012 lb – in


x 107

0.1094 x 109 lb – in2

0.7177 x 107 lb – in2

-0.4463 x 108 lb – in2

0.3120 x 1012

0.7177 x 107

1.2372 x 1012 lb – in2

-0.1792 x 1010 lb – in2





-0.4463 x 108

-0.1792 x 1010

1.3014 x 1012 lb – in2

Table 5. Laminate design: web region in Figure 4.



wall thickness ‘h’

0.3 in

laminate design

[60°], ply thickness = 0.3 in


0.26 x 1012 psi


0.26 x 1012 psi


0.1 x 1011 psi


0.96 x 1011 psi

vlt = vtn


A second application example is the three-cells rotor blade cross-section shown in Figure 5.

The laminate design convention is the same as that for the previous example. The only exception is the second web, which has its ply fibre orientation at -60° instead of 60° as in the first web. Similarly, the stiffness matrix of this cross-section about the origin of the axes in Figure 5 was found to be:

0.4018 x 1011 lb

0.7104 x 106 lb – in

0.2683 x 1011 lb – in

1.2344 x 1011 lb

– in

5 =

0.7104 x 106

0.2818 x 107 lb – in2

0.1652 x 106 lb – in2

-0.4188 x 107 lb

– in2

0.2683 x 1011

0.1652 x 106

0.2769 x 1011 lb – in2

0.6500 x 1011 lb

– in2

1.2344 x 1011

-0.4188 x 107

0.6500 x 1011

9.8795 x 1011 lb

– in2


One must enumerate the advantages of this implementation in light of the fact that it will be utilized in tackling the dynamics of rotating blades for various research efforts in the Applied Dynamics Group at Carleton University. Relative to other specialized cross-sectional finite element packages like VABS and ABAQUS, the thin-walled asymptotic theory offers the following advantages:

• It provides a vast design space for parametric studies and easily interfaces with other disciplines like dynamics and control, which may necessitate keeping the information about the elastic deformation in a maximally compressed, yet correct, form.

• The meshing step and preparation of special input files are not required. The only input in­formation required is the geometry of the airfoil (contour of the cross-section), which is readily available for any airfoil in simple and compact format, as well as the laminate design.

• As the thinness ratio increases, the analytical method becomes superior to VABS from the numerical point of view, since the finite element method results become unstable for high aspect ratio elements.

• It allows all but the essential variables to be eliminated offering a simple output and almost instantaneous execution time compared to more specialized software packages.

E-mail: kirsch@tx. technion. ac. il

1. Introduction

Design sensitivity analysis of structures deals with the calculation of the response derivatives with respect to the design variables. These derivatives, called the sensitivity coefficients, are used in the solution of various problems. In design optimization, the sensitivity coefficients are often required to select a search direction. These coefficients are used also in generating approximations for the response of a modified system. In addition, the sensitivities are required for assessing the effects of uncertainties in the structural properties on the system response. Calculation of the sensitivities involves much computational effort, particularly in large structural systems with many design variables. As a result, there has been much interest in efficient procedures for calculating the sensitivity coefficients. Developments in methods for sensitivity analysis are discussed in many studies (e. g. Haug et. al. 1986; Haftka and Adelman, 1989; Haftka and Gurdal, 1993; van Keulen et. al. in press). Methods of sensitivity analysis for discretized systems can be divided into the following classes:

a. Finite-difference methods, which are easy to implement but might involve numerous repeated analyses and high computational cost, particularly in problems with many design or response variables. In addition, finite-difference approximations might have accuracy problems. The efficiency can be improved by using fast reanalysis techniques.

b. Analytical methods, which provide exact solutions but might not be easy to implement in some problems such as shape optimization.

c. "Semi-analytical" methods, which are based on a compromise between finite-difference methods and analytical methods. These methods use finite-difference evaluation of the right-hand-side vector. They are easy to implement but might provide inaccurate results.

In general, the following factors are considered in choosing a suitable sensitivity analysis method for a specific application: the accuracy of the calculations, the computational effort involved and the ease-of-implementation. The implementation effort is weighted against the performance of the algorithms as reflected in their computational efficiency and accuracy. The quality of the results and efficiency of the calculations are usually two conflicting factors. That is, higher accuracy is often achieved at the expense of more computational effort.

Dynamic sensitivity analysis has been demonstrated by several authors. Using the mode superposition approach and assuming harmonic loading, the response sensitivities were evaluated by direct differentiation of the equations of motion in the generalized coordinates (Kramer and Grierson, 1989). In cases of earthquake loading the ground acceleration is usually given in discrete time steps, thus the loading is not given analytically. In several studies (Kim and Choi, 2000; van Keulen et. al, in press) the unconditionally stable implicit numerical equation was directly derived. It was found that the analysis equations and the sensitivity equations have the same left-hand side expression. Thus, it was possible to use the available factorized coefficient matrix. A numerical procedure was applied for calculation of the sensitivity of the response.

1 This paper is a shortened version of the paper "Efficient design sensitivities of structures subjected to dynamic loading" by the authors, in press in the International Journal of Solid and Structures.


M. Pandey et al. (eds), Advances in Engineering Structures, Mechanics & Construction, 549-561.

© 2006 Springer. Printed in the Netherlands.

Approximation concepts are often used to reduce the computational cost involved in repeated analysis of structures (Kirsch, 2002). However, most approximations that are adequate for structural reanalysis are not sufficiently accurate for sensitivity analysis. In this study, approximate reanalysis is used to improve the efficiency of dynamic sensitivity analysis by finite-differences. Given the results of exact analysis for an initial design, the displacements for various modified designs are evaluated efficiently by the recently developed Combined Approximations (CA) approach (Kirsch, 2002; 2003a). Originally, the approach was developed for linear static problems. Recently, accurate results were reported also for eigenproblem (Kirsch, 2003b; Kirsch and Bogomolni, 2004) and dynamic reanalysis problems (Kirsch et. al. submitted, in press).

Calculation of analytical derivatives using approximate analysis models have been demonstrated previously (Kirsch, 1994; Kirsch and Papalambros, 2001). It was found that accurate results can be achieved but, as noted earlier, analytical derivatives might not be easy to implement. It was demonstrated recently (Kirsch et. al. 2005, Bogomolni et. al. in press) that accurate derivatives can be achieved efficiently by CA and finite-differences for linear static problems and eigenproblems.

The present study deals with the design sensitivity analysis for discrete linear systems subjected to dynamic loading. The problem of dynamic analysis by mode superposition is first introduced, and the response derivatives with respect to design variables are presented as a combination of sensitivities of the eigenvectors and the generalized displacements. A procedure for reducing the number of differential equations that must be solved during the solution process is then proposed. Procedures intended to improve the accuracy of the approximations are developed, and efficient evaluation of the response derivatives by the combined approximations approach is presented. Numerical examples demonstrate the accuracy of the results.