Beams are meant to resist transverse load. For this reason, both shear force and bending moment are essential output from structural analysis and input for structural design. Based on the position, magnitude and pattern of loading, the developed internal forces vary with the point we inspect.
We first determine the limit of internal force functions based on the key points, which are basically the point where the loading changes. By using these points, we can segmentize our structure and proceed with determination of internal forces.
The function of shear force is derivative of that for bending moment. By using this correlation, we can identify the point with most critical point without going through the span and perform calculation repeatedly.
The following shows the procedure to derive the internal force functions in beams and frames. Watch the video above for full details.
The internal force is positive for:
- axial force if it is tension.
- shear force if it rotates the structure in clockwise direction.
- bending moment if it is sagging.
Overview of beam example
Loadings on beam are:
- concentrated load 'a'
- uniformly distributed load 'b'
- varying distributed load 'c'
Segmentization of beam
Beam is segmentized into four sections, where their shear force and bending moment functions are different.
Section 1 (valid for 0 to 0.25L from support A)
Equilibrium in y direction and moment about the cutting point to determine the shear force and bending moment functions.
Section 2 (valid for 0.25 to 0.5L from support A)
Equilibrium in y direction and moment about the cutting point to determine the shear force and bending moment functions.
Section 3 (valid for 0.5 to 0.75L from support A)
Equilibrium in y direction and moment about the cutting point to determine the shear force and bending moment functions.
Section 4 (valid for 0 to 0.25L from support B)
Equilibrium in y direction and moment about the cutting point to determine the shear force and bending moment functions.
The internal force functions for corresponding segment.
By differentiating the bending moment functions, we can obtain shear force function.
Based on the properties of derivative, the bending moment is either minimum (critical hogging) or maximum (critical sagging) at the point where shear force is zero, provided the shear force function is a non-constant function.
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