Introduction to Mechanics for Solids and Structures (Finite Element Method) Part 2

Constitutive Equations

Hooke’s law for 2D solids has the following matrix form with σ and ε from Eqs. (2.24) and (2.25):

where c is a matrix of material constants, which have to be obtained through experiments. For plane stress, isotropic materials, we have

To obtain the plane stress c matrix above, the conditions ofare imposed on the generalized Hooke’s law for isotropic materials. For plane strain problems,are imposed, or alternatively, replace E and ν in Eq. (2.31), respectively, withwhich    leads    to

Dynamic Equilibrium Equations

The dynamic equilibrium equations for 2D solids can be easily obtained by removing the terms related to the z coordinate from the 3D counterparts of Eqs. (2.15)-(2.17):

These equilibrium equations can be written in a concise matrix form of

where fb is the external force vector given by

For static problems, the dynamic inertia term is removed, and the equilibrium equations can be written as

Equations (2.35) or (2.37) will be much easier to solve and computationally less expensive as compared with equations for the 3D solids.

Equations For truss Members

A typical truss structure is shown in Figure 2.7. Each truss member in a truss structure is a solid whose dimension in one direction is much larger than in the other two directions as shown in Figure 2.8. The force is applied only in the x direction. Therefore a truss member is actually a one-dimensional (1D) solid. The equations for 1D solids can be obtained by further omitting the stress related to the y direction,from    the 2D case.

Figure 2.7. A typical structure made up of truss members. The entrance of the faculty of Engineering, National University of Singapore.

Figure 2.8. Truss member. The cross-sectional dimension of the solid is much smaller than that in the axial (x) directions, and the external forces are applied in the x direction, and hence the axial displacement is a function of x only.

Stress and Strain

Omitting the stress terms in the y direction, the stress in a truss member is only σχχ, which is often simplified as σχ. The corresponding strain in a truss member is εχχ, which is simplified as εχ. The strain-displacement relationship is simply given by

Constitutive Equations

Hooke’s law for 1D solids has the following simple form, with the exclusion of the y dimension and hence the Poisson effect:

This is actually the original Hooke’s law in one dimension. The Young’s module E can be obtained using a simple tensile test.

Dynamic Equilibrium Equations

By eliminating the y dimension term from Eq. (2.33), for example, the dynamic equilibrium equation for 1D solids is

Substituting Eqs. (2.38) and (2.39) into Eq. (2.40), we obtain the governing equation for elastic and homogenous (E is independent of x) trusses as follows:

The static equilibrium equation for trusses is obtained by eliminating the inertia term in Ea. (2.40):

The static equilibrium equation in terms of displacement for elastic and homogenous trusses is obtained by eliminating the inertia term in Eq. (2.41):

For bars of constant cross-sectional area A, the above equation can be written as

whereis    the external force applied in the axial direction of the bar.

Equations For Beams

A beam possesses geometrically similar dimensional characteristics as a truss member, as shown in Figure 2.9. The difference is that the forces applied on beams are transversal, meaning the direction of the force is perpendicular to the axis of the beam. Therefore, a beam experiences bending, which is the deflection in the y direction as a function of x.

Stress and Strain

The stresses on the cross-section of abeam are the normal stress, σχζ, and shear stress, σχζ. There are several theories for analysing beam deflections. These theories can be basically divided into two major categories: a theory for thin beams and a theory for thick beams.

Figure 2.9. Simply supported beam. The cross-sectional dimensions of the solid are much smaller than in the axial (x) directions, and the external forces are applied in the transverse (z) direction, hence the deflection of the beam is a function of x only.

Figure 2.10. Euler-Bernoulli assumption for thin beams. The plane cross-sections that are normal to the undeformed, centroidal axis, remain plane and normal to the deformed axis after bending deformation. We hence have

This topic focuses on the thin beam theory, which is often referred to as the Euler-Bernoulli beam theory. The Euler-Bernoulli beam theory assumes that the plane cross-sections, which are normal to the undeformed, centroidal axis, remain plane after bending and remain normal to the deformed axis, as shown in Figure 2.10. With this assumption, one can first have

which simply means that the shear stress is assumed to be negligible. Secondly, the axial displacement, u, at a distance z from the centroidal axis can be expressed by

where θ is the rotation in the x-z plane. The rotation can be obtained from the deflection of the centroidal axis of the beam, w, in the z direction:

The relationship between the normal strain and the deflection can be given by

where L is the differential operator given by

Constitutive Equations

Similar to the equation for truss members, the original Hooke’s law is applicable for beams: