In some cases, the equilibrium equations are insufficient to determine all of the unknowns of a system. This kind of system is called statically indeterminate. In these cases, we use the following deformation equations to further define the system.

For each of the deformation equations, we must consider the distribution of the internal force throughout the beam. In order to do so we calculate the average value, $\bar{Y}$.

$$ \bar{Y}=\frac 1L\int_TY(x)dx $$

Torsional Loads, ****The rotational deformation of a beam in torsion

$$ \Delta\phi=\frac{\bar{T}L}{GJ} $$

Axial Loads, The axial deformation of a beam is dependent on the tension in the beam, and the change in temperature of the beam

$$ \Delta L=\frac{\bar{N}L}{EA}+L\alpha\overline{\Delta T} $$

Lateral Loads, The angular deformation of a beam under loads perpendicular to the beam

$$ \Delta\theta=\frac{\bar{M}L}{EI} $$

Beam Deflection

We can more precisely describe the deflection of a beam by observing that the vertical deflection of the beam, $v(x)$, (with positive $v$ indicating upward deflection) is related to the internal moments.

$$ v''=\theta'=\frac M{EI} $$

$$ \theta (x)=\frac1{EI}\int M(x)dx $$

$$ v(x)=\frac 1{EI}\int\int M(x)dx $$

The arbitrary constants, $C_1$ and $C_2$, that arise in the indefinite integration, must be defined so as to specify the initial conditions of the beam, such zero positional deflection and zero angular deflection at a fixed support, $v(s)=0$, $\theta(s)=0$