Shear Force and Bending Moment Diagrams for Beam


What should we do if we want to determine the internal force developed in an overhanging beam i.e. shear force and bending moment?

First, we check the determinacy of our beam. For statically determinate beam, we are good to proceed using only equilibrium equations. If the beam is statically indeterminate, then we must use either flexibility method or stiffness method.

Then, we determine the support reactions. After the reactions are determined, we can segmentize our structure and proceed with determination of internal forces. For each segment, we obtain internal force functions using equilibrium equations again, and noting that those functions are valid within the limit of the segment.

After we have our internal force functions, we can determine the shear force and bending moment at the interested points. The right functions must be used at the right place to avoid any mistake. The value of internal forces at these points are plotted on respective diagram, and by joining these points we produce shear force and bending moment diagrams.

The following shows the procedure to produce shear force and bending moment diagrams for beams. Watch the video above for full details.
Beam overview
4m simply supported beam with additional 1m overhanging span subjected to concentrated load, uniformly distributed load and varying distributed load.
Support reaction
By taking moment about joint A, reaction force By can be determined considering equilibrium of moment about A.
Support reaction
By taking equilibrium in y direction, reaction force Ay can be calculated.
Segmentization of beam
Beam is divided into 4 segments. The internal force functions of a point are the same if it is located on the same segment.
Segment 1 (valid for 0 to 1.5m from support A)
Segment 2 (valid for 1.5m to 2m from support A)
Segment 4 (valid for 0 to 1m from tip of overhanging span)
Segment 3 (valid for 1m to 3m from tip of overhanging span)
The derived internal force functions are presented here.
Shear force and bending moment at interested points are calculated. 
The determined points are plotted on respective diagram: either shear force or bending moment against location, x.
By joining the plotted points, we obtain shear force and bending moment diagrams. 

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