# Shear Force and Bending Moment in Beams and Frames

For statically determinate beams and frames support reactions, bending moment, shear force and axial force are calculated and visualized in diagrams.

## Steps to set up a new model

- define coordinates of all relevant points (bearing and load locations).

The mouse can be used: double click to create new points, click drag to move existing points
- define for points with bearings, which direction is supported
- define loads

## General observations

With straight beams, the curves of the 3 internal forces are continuous except at the points where the associated loads/support reactions act.

At such load application points, there are jumps in the size of the applied load.

The course of the normal force and the course of the shear force are constant beyond the jump points due to the introduction of loads.

In these areas, the bending moment curve is linearly variable.

In the case of structures with differently aligned subareas, the shear force curve and the normal force curve have jumps at the vertices of the structure.

However, the progression of moments at the vertices is usually continuous.

Jumps only occur when a moment acts on a vertex.

## Regarding loads

The program only knows point loads.

For a structure under a *constant line load* q_{0} you can get an *approximate* solution:

You need to create several equidistant points in the loaded area of length L.

With n points (n including the start and end point), each inner point receives a point load of q_{0}L/(n-1) and the two edge points each receive half of that.

## To the results

Only if a *horizontal bar* is examined, the respective internal force is plotted in the diagram as an ordinate.

The signs of the internal forces are based on a *local* xyz coordinate system,

whose x-axis points to the right and whose z-axis points downwards. The y-axis points out of the display plane.

Therefore, a bending moment at the right end of a beam segment (positive cut bank) is positive if it acts to the left.

A shear force is positive at the right end of a beam segment if it acts downwards.

For *frame structures* the signs are based on the *local* xyz coordinate system of the respective frame segment.

The local x-axis points from the first to the second point of the segment. The local z-axis then points to the right in the viewing direction of this local x-axis.

A bending moment is therefore positive at the end of a respective segment if it acts to the left.

Longitudinal forces are easiest to discuss in terms of signs:

Positive normal forces represent a tensile load, negative normal forces represent a compressive load.

The program does not know any line loads.

*Constant line loads* can, however, be represented relatively well by a group of individual loads (see above).

The associated moment curve is then a polygon.

At the corners of the polygon the results agree with the exact results,

which are given by a square course in the area of the line load, correspond exactly.

The associated shear force progression is then step-shaped.

It then only agrees with the continuous exact course in the middle of the interval.

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