Solutions for Elasticity
5.1. Introduction
Most bodies have linear elastic behaviour, or so they are assumed. Using modulus of elasticity of the material and limiting strain for structural element, the engineer can easily determine the limit stress that can be developed in that element. With this, an engineer can determine the element size for any design load.
Concrete is a brittle material. If it exhibits strain beyond its elastic limit, the bond that holds its aggregates together will be permanently destroyed and only the reinforcing steel remains as a functioning component in the structural element. As a result, the occupant of the building will be at risk. Therefore, engineers design a concrete structure in such a way that ensures the material will not be loaded until inelastic behaviour is induced. Design standards such as Eurocode 2 specifies the design concrete strength and its safety factor based on this philosophy.
In engineering, stress and displacement in a body are often concerned. For elasticity, these parameters can be determined with boundary conditions is identified. Two approaches are commonly used, depends on the objectives of the analysis. Navier equation is a displacement-based approach that determines the displacement over a body given its boundary conditions. Beltrami-Michell stress compatibility equation, on the other hand, is a stress-based approach that aims to determine stress over a body is given the compatibility and boundary conditions to be fulfilled. To derive both methods, stress-displacement relationship, which will be discussed in Section 5.2 is essential.
Stress-displacement relationship
For isotropic material, the following stress-strain relationship can be derived from Eq. (4.79):
By applying the relationships in Eqs. (3.2), (3.3), (3.4) to corresponding terms in stress-strain relationships for ctx as shown in Eq. (5.1) results in the follows:
Equation below is obtained by expanding equation above:
After rearrangement, the equation above is transformed to the following:
Normal stress component for у and z directions can be expressed in similar fashion:
The following is obtained by applying the relationships in Eq. (3.6) to corresponding terms in stress-strain relationships for rxyas shown in Eq. (5.1):
Similarly, shear stress component for yz and xz. planes can be expressed as follows:
