Applied Structural Mechanics

The classical fundamentals of modern Structural Mechanics have been founded by two scientists. In his work “Discorsi ”, Galileo GALILEI (1564 – 1642) carried out the first systematic investigations into the fracture process of brittle solids. Besides that, he also described the influence of the shape of a solid (hollow solids, bones, blades of grass) on its stiffness, and thus successfully treated the problem of the Theory of Solids with Uniform Strength. One century later, Robert HOOKE (1635–1703) stated the fundamental law of the linear theory of elasticity by claiming that strain (alteration of length) and stress (load) are proportional (“ut tensio sic vis ”). On the basis of this material law for the Theory of Elasticity, Edme MARIOTTE (1620 – 1684), Gottfried Wilhelm von LEIBNIZ (1646 – 1716), Jakob BERNOULLI (1654 – 1705), Leonard EULER (1707 – 1783), Charles Augustin COULOMB (1736 – 1806) and others treated special problems of bending of beams.

The use of the index notation is advantageous because it normally makes it possible to write in a very compact form mathematical formulas or systems of equations for physical or geometric quantities, which would otherwise contain a large number of terms.

The essential objective of structural analysis is the calculation of stresses and deformations of bodies. As shown in Fig. 3.1 we make a cut through the body, which is in equilibrium under external loads in the form of volume forces f _i, surface tractions p _i and concentrated forces F _k. A resulting force Δ F is transmitted at every element Δ A of the cut.

Description of the deformation of a body with LAGRANGE’s notation:

In the following we are going to deal with bodies for which there exist reversibly unique relations between the components of the strain tensor and the stress tensor, and we furthermore assume that these relations are time independent. The behaviour of the bodies is denoted as elastic, i.e. there are no permanent strains ε _p1 after removing the load of the body (Fig. 5.1). The bodies considered shall furthermore, as it is usual in the classical elasticity theory, be made of a linearly elastic material such that their constitutive law expresses linear relationship between the components of the stress tensor and the strain tensor (range 0 -A - in Fig. 5.1). Such bodies are usually called HOOKEAN bodies.

i.e. altogether 15 equations for 15 unknown field quantities (6 stresses r ^ij, 6 strains γ_ij, 3 displacements v_i).

In the previous chapters we considered elastic structures with small displacements. This simplifying assumption is not always fulfilled; especially in cases of thin —walled structures subjected to larger compressive loads, the deformations may become large compared with the thickness. The equilibrium conditions must then be formulated for the deformed state of the structure and terms of higher order must be taken into account in the strain—deformation relations. This corresponds to the geometrical non—linearity. Here, the material law is considered to be linear. Furthermore, the lemma of mass conservation ( \( \hat \varrho d\hat V = \varrho dv\) ) is assumed to remain valid as well as equality of the volume forces in the deformed and undeformed state \( \left( {\hat f = f} \right)\). The stress — free initial state (LAGRANGE formulation) is taken as a basis. With these assumptions, the equilibrium conditions read as follows [B.1, B.2, B.4]: The strain —displacement relations have been introduced in Chapter 4, and we obtain according to (4.12a)

We assume that there exist one-to-one relationships between the curvilinear coordinates (GAUSSIAN surface parameters) ξ^α (α = 1, 2) and the Cartesian coordinates xⁱ (i = 1, 2, 3) of the points P of a surface (Fig. 11.1).

Assumption: The stresses T ^αß are uniformly distributed over the thickness, i.e. only so-called membrane forces occur, but no bending moments and no shear forces are found.

The previous chapters have introduced fundamentals for determining the structural behaviour required for the dimensioning and design of a structure, i.e. calculation of deformations, stresses, natural vibration frequencies, buckling loads, etc. In view of the development and construction of machines and system components the question arises which measures must be taken in order to reduce costs and to improve quality and reliability; in other words this means that an optimization of the properties is being aimed at. In terms of this demand, the topic Structural Optimization has emerged, over the past years, an extensive field of research that can be described by the following formulation [D.29]:

Structural optimization may be defined as the rational establishment of a structural design that is the best of all possible designs within a prescribed objective and a given set of geometrical and/or behavioral limitations.

Current research in optimal structural design may very roughly be said to follow two main paths. Along the first, the research is primarily devoted to studies of fundamental aspects of structural optimization. Broad conclusions may be drawn on the basis of mathematical properties of governing equations for optimal design. These properties are not only studied analytically in order to derive qualitative results of general validity, but are also often investigated numerically via example problems. Along the other main path of research, the emphasis is laid on the development of effective numerical solution procedures for optimization of complex practical structures [ D.3, D.12, D.21, D.22, D.30 ].

In the following, solution algorithms for optimization problems cast in the standard MP form (15.4) will be considered. One distinguishes between optimization algorithms of zeroth, first and second order depending whether the solution algorithm only requires the function values, or also the first and second derivatives of the functions. It will be assumed in this Chapter that in general the functions f, h_i, g_j in (15.4) are continuous and at least twice continuously differentiable.