User:EAS4200C.Fall08.Team12.Watlington.VG

Concept of Displacement
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Strain
Strain in an axial member is defined as
 * $$\ \epsilon=\frac{\Delta L}{L}=\frac{\ell -L}{L}$$

where $$\ \Delta L= u_1-u_0$$ is the total change in length of the member. In the event of nonuniform strain, this equation equals the average strain in the member.

Strain at a Point
Given two points with the following displacements $$\ u_0=u(x_0)$$  and   $$\ u_1=u(x_0 + \Delta x) $$,

one can solve for the difference in displacement by

$$\ \Delta u=u_1-u_0 $$.

Therefore, the strain $$\ \epsilon $$ can be determined by $$\ \epsilon=\lim_{x \to 0}\frac{\Delta u}{\Delta x}=\frac{du}{dx} $$.

The displacement function is given by

Rigid Body Motion
Rigid body motion is defined by translation or rotation of a body in which no strain is induced.

During translation:

$$\ u=u_0=constant $$

$$\ v=v_0=constant $$

$$\ w=w_0=constant $$

During rotation:

$$\ u=-\alpha y $$

$$\ v=\alpha x $$

$$\ w=0 $$

Stress
For an axial member, stress $$\ \sigma $$ is defined as $$\ \sigma=\frac{P}{A} $$.

The stress vector t is defined as $$\ t=\lim_{A \to 0}\frac{\Delta F}{\Delta A} $$.

As t is a vector, it is made up of three components which are denoted by $$\ \sigma_xx, \tau_xy, \tau_xz $$ on the $$\ x $$ face, ... on the $$\ y $$ face, and ... on the $$\ z $$ face.