Two-Dimensional relationship of compliance and stiffness matrix
A thin plate is a prismatic member having a small thickness, and it is the case for a typical lamina. If a plate is thin and there are no out-of-plane loads, it can be considered to be under plane stress. If the upper and lower surfaces of the plate are free from external loads, then σ 3 = 0, τ31 = 0,and τ23 = 0. For an orthotropic plane stress problem can then be written as
(1) where Sij are the elements of the compliance matrix. Note the four independent compliance elements in the matrix. Inverting Equation (1) gives the stress – – strain strain relationship as
(2) where Qij are the reduced stiffness coefficients, which are related to the compliance coefficients as
(3)
Relationship of Compliance and Stiffness Matrix to Engineering Elastic Constants of a Lamina
Equation (1) and Equation (2) show the relationship of stress and strain through the compliance [S] and reduced stiffness [Q] matrices. However, stress and strains are generally related through engineering elastic constants. For a unidirectional lamina, these engineering elastics constants are E1 = longitudinal Young’s modulus E2 = transverse Young’s modulus ν12 = major Poisson’s ratio G12 = in-plane shear modulus Experimentally, the four independent engineering e lastic constants are measured as follows and can be related to the four independent elements of the compliance matrix [S] of Equation (1).
Apply a pure tensile load in direction 1 (Figure 1), i.e.,
Figure 1
Then, from Equation (1)
Apply a pure tensile load in direction 2 (Figure 2), i.e. ,
Figure 2
Then, from Equation (1)
Apply a pure shear stress in the plane 1 – 2 (Figure 3), i.e.
Figure 3
Then, from Equation (1)
Thus, we have proved that
Also, the stiffness coefficients Q ij are related to the engineering constants
,
,
,
Problems:
1. Using the following data with respect to unidirectional graphite/epoxy lamina obtain the reduced stiffness and compliance matrices. E1= 181 GPa, E2= 10.3 GPa, ν12 = 0.28, G12 = 7.17 GPa
The compliance matrix elements are
The minor Poisson’s ratio is
The reduced stiffness matrix [Q] elements are
The compliance matrix
Using Equation (1), the strains in the 1 – 2 coordinate system are
Thus, the strains in the local axes are
