Shear Force, Bending Moment Diagrams

Beam design requires knowing the variation of the internal shear force $V$ and bending moment $M$ acting at each point along the axis of the beam.

To do this, method of sections must be used to determine load variation by solving for $V(x)$ and $M(x)$ to plot on Shear and Bending moment diagrams respectively. These equations must be derived for each section of a beam that has a different loading, since they’re discontinuous at points where distributed load changes or where concentrated forces or couple moments are applied.

Procedure

1. Solve for support reactions and couple moments acting on the beam, and resolve into components acting parallel and perpendicular to the beam’s axis
2. Assuming(usually) the left end of the beam is the origin,
   1. Specify x coordinates extending to regions of the beam between concentrated forces and/or couple moments or where distributed loading is continuous
   2. Section the beam at each distance x and draw the FBD of each segment, showing $V$ and $M$ in the positive sense.
   3. Solve for $V$ by summing forces perpendicular to the beam’s axis, and solve for $M$ by summing moments about the sectioned end of the segment
3. Plot the $V$ and $M$ versus x

Overall

• Internal forces allow us to assess material failure
• Computing internal forces is simple when virtually ‘cutting’ at the desired location to expose actions/reactions

physics statics