THE BASICS FOR SOLIDS AND STRUCTURES

The basic concepts and classical theories of solids and structures are defined here in a brief and easy to understand manner. Solids and structures are stressed when they are subjected to loads or forces. The stresses are, in general, not uniform, and lead to strains,which can be observed as either deformation or displacement. Solid mechanics and structural mechanics deal with the relationships between

Stresses and strains,

Displacements and forces,

Stresses (strains) and forces

for given boundary conditions of solids and structures.

These relationships are vitally important in modelling, simulating and designing engineered structural systems.Forces can be static and/or dynamic.

Statics deals with the mechanics of solids andstructures subjected to static loads such as the deadweight on the floor of buildings. Solids and structures will experience vibration under the action of dynamic forces varying withtime, such as excitation forces generated by a running machine on the floor. In this case,the stress, strain and displacement will be functions of time, and the principles and theoriesof dynamics must apply. As statics can be treated as a special case of dynamics, the staticequations can be derived by simply dropping out the dynamic terms in the general, dynamic equations. We will adopt this approach of deriving the dynamic equation first, and obtaining the static equations directly from the dynamic equations derived.

Depending on the property of the material, solids can be elastic, meaning that the deformation in the solids disappears fully if it is unloaded. There are also solids that are considered plastic, meaning that the deformation in the solids cannot be fully recovered when it is unloaded. Elasticity deals with solids and structures of elastic materials, and plasticity deals with those of plastic materials.

The scope of this Course deals mainly with solids and structures of elastic materials. In addition, to the problems of very small deformation, where the deformation and load has a linear relationship. Therefore,our problems will mostly be LINEAR ELASTIC.

The material is ISOTROPIC.

Boundary conditions are another important consideration in mechanics. There are DISPLACEMENT and FORCE boundary conditions for solids and structures. For heat transfer problems there are TEMPERATURE and CONVECTION boundary conditions. Treatment of the boundary conditions is a very important topic, and will be covered in detail later.

Structures are made of structural components that are in turn made of solids. There are generally four most commonly used structural components: truss, beam, plate, and shell,as shown in Figure

In physical structures, the main purpose of using these structural components is to effectively utilize the material and reduce the weight and cost of thestructure. A practical structure can consist of different types of structural components, including solid blocks. Theoretically, the principles and methodology in solid mechanics can be applied to solve a mechanics problem for all structural components, but this is usuallynot a very efficient method. Theories and formulations for taking geometrical advantages ofthe structural components have therefore been developed.

EQUATIONS FOR THREE-DIMENSIONAL SOLIDS

Stress and Strain
Let us consider a continuous three-dimensional (3D) elastic solid with a volume V anda surface S, as shown in Figure

The surface of the solid is further divided into two typesof surfaces: a surface on which the external forces are prescribed is denoted Sf; and surfaceon which the displacements are prescribed is denoted Sd. The solid can also be loaded bybody force fb and surface force fs in any distributed fashion in the volume of the solid.At any point in the solid, the components of stress are indicated on the surface of an'infinitely' small cubic volume, as shown in Figure.

On each surface, there will be the normal stress component, and two components of shearing stress. The sign convention for the subscript is that the first letter represents the surface on which the stress is acting, and the second letter represents the direction of the stress. The directions of the stresses shown in the figure are taken to be the positive directions. By taking moments of forces about the central axes of the cube at the state of equilibrium, it is easy to confirm that

Therefore, there are six stress components in total at a point in solids. These stresses are often called a stress tensor. They are often written in a vector form of

Corresponding to the six stress tensors, there are six strain components at any point in a solid, which can also be written in a similar vector form of

Strain is the change of displacement per unit length, and therefore the components of strain can be obtained from the derivatives of the displacements as follows:

where u, v and w are the displacement components in the x, y and z directions, respectively.
The six strain-displacement relationships in the above Eqn. can be rewritten in the following matrix form:

e= LU

where U is the displacement vector, and has the form of

and L is a matrix of partial differential operators obtained simply by inspection on Eqn.