Numerical Example

Consider the fifty-story frame shown in Fig. 1. The number of degrees of freedom is 600, and the damping ratios for all modes are 0.05. The masses are assumed to be concentrated at the joints, and only horizontal inertia forces are considered. The inertia force is due to the frame self-weight and an additional concentrated mass of 50 ton in an internal joint and 25 ton in an external joint. The width of all elements is 0.5m, the depth of all columns is 1.0m and the depth of all beams is 0.8m. The modulus of elasticity is 3×107 kNm2. The loading is due to the ground acceleration of the El Centro earthquake, shown in Fig. 2. The object is to evaluate the sensitivities of the horizontal displacements at the 1st story and the 50th story with respect to the following four design variables;

X1 – depth of the columns in the 1st story.

X2 – depth of the beams in the 1st story.

X3 – depth of the columns in the 50th story.

X4 – depth of the beams in the 50th story.

Table 1. Eigenvalue sensitivities, fifty-story frame

Sensitivity

Mode

FD(exact)

FD(CA2)

дУ д X

1

0.0480

0.0480

2

0.4723

0.4723

3

1.4404

1.4404

4

2.8998

2.8998

5

4.9780

4.9780

6

7.7050

7.7050

7

11.232

11.232

8

15.617

15.617

д У d X2

1

0.0330

0.0330

2

0.5834

0.5834

3

1.7403

1.7403

4

3.5348

3.5348

5

5.9373

5.9373

6

9.0852

9.0852

7

12.876

12.876

8

17.485

17.485

д У d X3

1

-0.00315

-0.00315

2

-0.02443

-0.02443

3

-0.05199

-0.05198

4

-0.04155

-0.04154

5

0.06455

0.06457

6

0.33255

0.33257

7

0.85281

0.85282

8

1.69010

1.69010

д У d X4

1

-0.00774

-0.00779

2

-0.06148

-0.06149

3

-0.13726

-0.13723

4

-0.12572

-0.12572

5

0.13424

0.13425

6

0.84037

0.84042

7

2.28830

2.28840

8

4.73610

4.73610

Choosing the time-step At = 0.02 sec. and considering the first 8 mode shapes, the results obtained by forward-difference derivatives using exact analysis formulation [FD(exact)] are compared with those

(a) (a)

achieved by the CA approach with only 2 basis vectors [FD(CA2)]. Table 1 shows the eigenvalue sensitivities, Fig. 3 shows the displacements, and Figs. 4, 5 show the displacement sensitivities of the 1st and the 50th stories. It is observed that high accuracy is achieved by the procedure presented.

Solving various frames with different numbers of degrees of freedom, it was found that in all cases only 2 basis vectors provide accurate sensitivities. This result is typical for small perturbations in a single design variable.

7 Conclusions

Calculation of response derivatives with respect to design variables often involves much computational effort, particularly in large structural systems subjected to dynamic loading. Approximation concepts, which are often used to reduce the computational cost involved in repeated analysis, are usually not sufficiently accurate for sensitivity analysis.

In this study efficient sensitivity analysis, using the recently developed combined approximations approach and finite-differences, is presented. Assuming modal analysis, a procedure intended to reduce the number of differential equations that must be solved during the solution process is proposed. Computational 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 show that accurate results can be achieved efficiently. In general, sensitivity analysis by the CA method is used in problems of small perturbations in a single design variable. In such cases a very small number of basis vectors provide accurate results even for structures having large numbers of degrees of freedom.

References

Barthelemy, B., Chon, C. T., Haftka, R. T., 1988. Sensitivity approximation of static structural response. Finite Element in Analysis and Design 4, 249-265.

Bogomolni, M., Kirsch, U., Sheinman, I., In press. Efficient design sensitivities of structures subjected to dynamic loading.

Burton, R. R., 1992. Computing forward-difference derivatives in engineering optimization.

Engineering Optimization 20, 205-224.

Haftka, R. T., Adelman, H. M., 1989. Recent developments in structural sensitivity analysis. Structural Optimization 1, 137-151.

Haftka, R. T., Gurdal, Z., 1993. Elements of Structural Optimization, Third Ed. Kluwer Academic Publishers, Dordrecht.

Haug, E. J., Choi, K. K., Komkov, V., 1986. Design sensitivity analysis of structural system, Academic press, New York.

Kim, N. H. Choi, K. K., 2000. Design sensitivity analysis and optimization of nonlinear transient dynamics, AIAA/USAF/NASA/ISSMO 8th Simposium on Multidisciplinary Analysis and Optimization. 6-8 Sept., Long Beach, CA.

Kirsch, U., 1994. Effective sensitivity analysis for structural optimization. Computer Methods in Applied Mechanics and Engineering 117, 143-156.

Kirsch, U., 2002. Design-Oriented Analysis of Structures, Kluwer Academic Publishers, Dordrecht. Kirsch, U., 2003a. A unified reanalysis approach for structural analysis, design and optimization.

Structural and Multidisciplinary Optimization 25, 67-85.

Kirsch, U., 2003b. Approximate vibration reanalysis of structures. AIAA Journal 41, 504-511.

Kirsch, U., Bogomolni, M., 2004. Procedures for approximate eigenproblem reanalysis of structures.

International Journal for Numerical Methods in Engineering 60, 1969-1986.

Kirsch, U., Bogomolni, M., van Keulen, F., 2005. Efficient finite-difference design-sensitivities. AIAA Journal 43, 399-405.

Kirsch, U., Bogomolni, M., Sheinman, I., submitted. Efficient dynamic reanalysis of structures.

Kirsch, U., Bogomolni, M., Sheinman, I., In press. Nonlinear dynamic reanalysis by combined approximations.

Kirsch, U., Papalambros, P. Y., 2001. Accurate displacement derivatives for structural optimization using approximate reanalysis. Computer Methods in Applied Mechanics and Engineering 190, 3945-3956.

Kramer, G. J., Grierson, D. E., 1989. Computer automated design of structures under dynamic loads. Computers & Structures 32, 313-325,.

Pedersen, P., Cheng, G., Rasmussen, J., 1989. On accuracy problems for semi-analytical sensitivity analysis. Mechanics of Structures and Machines 17, 373-384.

Somerville, P., et al, 1997. Development of ground motion time histories for phase 2 of the FEMA/SAC steel project, Report No. SAC/BD-97/04. van Keulen, F., Haftka, R. T. Kim, N. H., in press. Review of options for design sensitivity analysis. Part 1: linear systems.