The question remains, how are [[Shear Stress|shear]], [[Moment|moment]], and overall beam deflection related? The answer lies in performing a [[Static Systems|static]] balance over infinitesimal sections of the beam. Shear, $V$, can be plotted on a shear diagram after reaction [[Force|forces]] are solved for the system.
![[beam_shear_internal.png|center|200]]
Moment throughout the beam is simply the product of reaction shear force and [[Distance|distance]] between the coupling forces. Therefore we can derive beam bending moment from integrating shear force with respect to moment and applying the correct [[Boundary Condition|boundary conditions]].
$M=\int Vdx$
The shear force and bending moment diagrams are sufficient for basic beam stress analysis. However, in some instances the deformation of the beam may be of interest. In such a case the slope, $\theta$, of the beam is derived as follows, again note that appropriate boundary conditions must be applied.
$\theta=\dfrac{1}{EI}\int Mdx$
Finally, the beam deflection, $U_y$, can be derived by integrating the beam slope and applying the correct boundary conditions.
$U_y=\int\theta dx$
![[shear_moment_1.png|center|400]]
![[shear_moment_2.png|center|400]]