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## Über dieses Buch

Structural optimization is currently attracting considerable attention. Interest in - search in optimal design has grown in connection with the rapid development of aeronautical and space technologies, shipbuilding, and design of precision mach- ery. A special ?eld in these investigations is devoted to structural optimization with incomplete information (incomplete data). The importance of these investigations is explained as follows. The conventional theory of optimal structural design - sumes precise knowledge of material parameters, including damage characteristics and loadings applied to the structure. In practice such precise knowledge is seldom available. Thus, it is important to be able to predict the sensitivity of a designed structure to random ?uctuations in the environment and to variations in the material properties. To design reliable structures it is necessary to apply the so-called gu- anteed approach, based on a “worst case scenario” or a more optimistic probabilistic approach, if we have additional statistical data. Problems of optimal design with incomplete information also have consid- able theoretical importance. The introduction and investigations into new types of mathematical problems are interesting in themselves. Note that some ga- theoretical optimization problems arise for which there are no systematic techniques of investigation. This monograph is devoted to the exposition of new ways of formulating and solving problems of structural optimization with incomplete information. We recall some research results concerning the optimum shape and structural properties of bodies subjected to external loadings.

## Inhaltsverzeichnis

### Chapter 1. Guaranteed Approaches

Consider optimal design of a cantilever beam clamped at the origin of Cartesian coordinate system (oxyz) and loaded by static transverse forces q acting in the plane xy. We suppose that the beam in its natural unloaded state is placed along the x-axis and that it has a rectangular cross-section with height h = h(x) and width b = const. The beam has length l. The function h = h(x), determining the shape of the beam, is the unknown variable and is to be found in the design optimization process.
N. V. Banichuk, Pekka Neittaanmäki

### Chapter 2. Probabilistic Uncertainties

In this book we will consider different type of uncertainties. In particular, we discuss optimal structural design problems with incomplete information taking into account that some problem parameters take random values with a given probability density. For example, for optimal design problems, considered in Part III in the framework of modern fracture mechanics, the role of random parameters is given by the sizes, positioning and orientations of cracks. In this chapter we will consider the simplest optimization problems with random parameters and discuss some possible approaches to these problems.
N. V. Banichuk, Pekka Neittaanmäki

### Chapter 3. Uncertainties and Worst Case Scenarios

In this section we describe details of problem formulation in abstract form. Let the behavioral system of differential equations with boundary conditions, described the equilibrium of an elastic body, be of the operator form $$L(u,h,q,\xi,\omega ) = 0$$ and is written in the domain Ω in n-dimensional space with boundary Γ = Ω, where u, h, q are respectively the state variable, the design variable, the applied force and the functions ξ, ω characterize the material distribution along the body and the distribution of damages.
N. V. Banichuk, Pekka Neittaanmäki

### Chapter 4. Optimal Design of Beams and Plates with Uncertainties

The equilibrium state of a simply supported beam of length l, situated in the xy plane, with its axis lying on the x-axis and loaded by exterior distributed load q(x) parallel to the y axis, is given by the following system of equations and boundary conditions 4.1 $$\frac{\mathit{dM}} {\mathit{dx}} = Q,\ \frac{\mathit{dQ}} {\mathit{dx}} = -q,$$ 4.2$$M(0) = M\left (l\right ) = 0,$$ where M = M(x) and Q = Q(x) denote, respectively, bending moment and shear load acting on the cross section of the beam, perpendicular to the x-axis. The beam has a rectangular cross-sectional area of width b and height h.
N. V. Banichuk, Pekka Neittaanmäki

### Chapter 5. Uncertainties in Fracture Mechanics and Optimal Design Formulations

Most investigations in the theory of optimal design of structures under strength constraints have been performed within a framework of the deterministic approach. That is, it is assumed that there is regular internal structure of material and that complete information is provided with regard to loading processes and boundary conditions. Corresponding optimal design formulations were typical for structures from elastic-plastic materials [Ban83, Pic88, HNT86, HA79, Ban81, Arm83, Aro89, EO83, HN88, HN96, Nei91, Hau81, HC81, Cea81, OR95, Pra72, Roz76, RK88, JM83, MU81].
N. V. Banichuk, Pekka Neittaanmäki

### Chapter 6. Beams and Plates with Brittle-Fracture Constraints

Consider the problem of optimal design of a beam taking into account the possibility of crack appearance at the beam surfaces. We assume that the beam of length L lies along the x-axis (0 ≤ xL) and that it has a rectangular cross-section with height h = h(x) and constant width b.
N. V. Banichuk, Pekka Neittaanmäki

### Chapter 7. Optimization of Axisymmetric Shells Against Brittle Fracture

Consider a shell which has the shape of a surface of revolution, the axis of which coincides with the x-axis (Fig. 7.1).
N. V. Banichuk, Pekka Neittaanmäki

### Chapter 8. Shape and Thickness Distribution of Pressure Vessels

We consider simultaneous combined optimization of the shape and thickness of membrane shells of revolution under the action of internal pressure. We take account of the constraints concerning the strength of the shell and the volume of its cavity [Ban07]. We give general formulations of optimal design of closed shells of revolution (pressure vessels), and investigate the optimal shape of a shell and the corresponding thickness distribution. We present exact solutions for the optimal design of closed shells of revolution under internal pressure. The simultaneous introduction of two control functions, describing the shape of the shell and the distribution of its thickness not only ensures a substantial reduction in the mass of a shell, but also leads to significant mathematical simplifications: this leads to closed form solution of the optimization problems.
N. V. Banichuk, Pekka Neittaanmäki

### Chapter 9. Brittle and Quasi-Brittle Materials

The optimization problems considered in this section consist in finding the shape and the thickness distribution of axisymmetric shells in the framework of membrane theory, loaded by fixed statical forces in such a way that the cost functional reaches a maximum, while satisfying some strength mechanics constraints. In this section the mass effectiveness of the shell is considered as a cost functional, and as constraints we use bounds on the maximum normal stresses; this is typical for shells made from brittle or quasi-brittle materials. Analytical investigations and corresponding optimal solutions are presented [BRS08].
N. V. Banichuk, Pekka Neittaanmäki

Consider an axisymmetric shell with length L, thickness h = h(x) and radius r = r(x) where 0 ≤ xL. The shell is loaded by gravity forces in the axial direction (see Fig. 10.1).
N. V. Banichuk, Pekka Neittaanmäki

### Chapter 11. Damage Characteristics and Longevity Constraints

Some aspects of optimal design of structures under cyclic loading have been discussed in Chapter 5 taking into account crack appearing and growth. The optimization problems contained a constraint, the number of cyclic before fracture; we call this the longevity constraint. In this section and the next we present some results of optimization of beams, plates, shells and beam-like structures [Ban97, Ban98, BN07, BN08a, BN08b].
N. V. Banichuk, Pekka Neittaanmäki

### Chapter 12. Optimization of Shells Under Cyclic Crack Growth

In the previous chapter we considered the problems of optimal design of bodies with surface cracks. In this chapter we present some results of optimization [BIMS05a, BIMS05b, BRS06] of axisymmetric shells containing through the thickness cracks and loaded by various cyclic loads.
N. V. Banichuk, Pekka Neittaanmäki

### Chapter 13. Uncertainties in Material Characteristics

This section deals with problems of optimal design of structures from various materials. The number of materials is supposed to be finite and consequently the admissible design set consists of separate discrete values. Suppose that material i (i = 1, 2, , r) is characterized by the following property vector (see Fig. 13.1): 13.1$${\xi }_{i} = \left \{{\xi }_{i}^{1},{\xi }_{ i}^{2},\ldots,{\xi }_{ i}^{m}\right \},\quad i = 1,2,\ldots,r,$$ where r is the number of given materials (steel, titanium, …) and m is the number of material properties essential for the problem (material density, Young’s modulus, ).
N. V. Banichuk, Pekka Neittaanmäki

### Chapter 14. Some Basic Notions of Probability Theory

A sample space S associated with an experiment is a set of elements such that any outcome of the experiment corresponds to a unique element of the set. An element A is a subset of a sample space S. An element in a sample space is called an event.
N. V. Banichuk, Pekka Neittaanmäki

### Chapter 15. Probabilistic Approaches for Incomplete Information

Most structural optimization problems that have been investigated use approaches designed for complete data. Nevertheless, some parametric probabilistic problems of optimal structural design with limiting state and buckling constraints have been formulated and investigated in [AB73, ABC84, AC79, Bra84, DFH77, ETB84, AGRZ84, EKS86, Mos77, Mar95, BV07]. Nondeterministic approaches have been used for some optimal design problems when the external conditions (external forces) were taken as a random variables. Fewer probabilistic studies have been devoted to the important class of problems of quasi-brittle elastic body optimization on the basis of modern fracture mechanics criteria [ABC84, BRS99, BRS03a, BRS03b, LM96]. In accordance with fracture mechanics representations it is necessary to consider all possibilities for the appearance of cracks, and to take into account that the crack position, size and mode (opening cracks, shear cracks, etc.) are unknown beforehand. Therefore, the formulation and solution of these optimization problems requires the application of methods which can take into account incomplete information. Minimax methods of game theory (guaranteed approach), probabilistic methods, etc., can be used for this purpose.
N. V. Banichuk, Pekka Neittaanmäki

### Chapter 16. Optimization Under Longevity Constraint

This chapter deals with probabilistic approaches to optimal design of structures made from quasibrittle material and loaded by cyclic forces [YCC97, PKK97, BIMS05a, BIMS05b, BIM07]. Special attention is devoted to different problem formulations and analytical solution methods. First we present some basic assumptions and relations. Then we formulate the optimal structural design problem based on a probabilistic approach. We must minimize the cost functional (volume of material) under constraints on the number of loading cycles before global fracture and on the probability of nondestructive behavior of the body. The original constraints are transformed to inequalities imposed on the stress in the uncracked body at the crack location. The resulting problem of optimal shape design consists of cost functional minimization under stress constraints, and can be solved by conventional methods. Several examples of structural design problems for statically determinate and indeterminate beams and frames are presented in the chapter. Then we use another probabilistic approach, based on the application of moment inequalities, for optimal structural design under a longevity constraint (constraint on the number of cycles). Here we require that the mathematical expectation (first moment) of the critical number of cycles must be greater than the given number of cycles, and the dispersion (second movement) of the critical number of cycles must be less than a given value. It is shown that this problem can be transformed to that of the structural volume minimization under a system of stress constraints. The presentation follows research results of [BRS03a].
N. V. Banichuk, Pekka Neittaanmäki

### Chapter 17. Mixed Probabilistic-Guaranteed Optimal Design

This chapter deals with the mixed probabilistic-guaranteed approach to optimal design of quasi-brittle membrane shells. Special attention is devoted to different problem formulations and analytical methods for their solution. Optimal thickness distributions are presented for various axisymmetric membrane shells. The presentation follows research results of [BRS03b].
N. V. Banichuk, Pekka Neittaanmäki

### Backmatter

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