The shear force diagram for a serious of beam elements sometimes have a different sign than expected, resulting in a saw-tooth shape. Why does this happen.

When we draw a shear force diagram for a beam or frame on paper, we typically construct the diagram from left to right. We start at the left and let the shear force diagram follow the direction of the forces and reactions that act on the structure. Similarly, for columns we may draw the shear force diagram from the bottom to top.

This convention makes good sense, but has limitations when working in a 3D environment:

- When viewing a beam from the opposite (back) side, "left" and "right" switches, meaning we will draw the shear force diagram in the opposite direction, causing it to be inverted.
- When working with a series of beam elements that are not in a straight line (forming a circle, for example), an
*absolute*"left to right" approach does not work and we may choose to use a*relative*"left to right" view instead. For example, we may arbitrary decide to always look toward the centre point of the structure and let that determine the "left to right" orientation.

**Frame Analysis** avoids arbitrary conventions like the above. Instead, the program uses the local x-axis of each individual beam as the direction for drawing its shear force diagram. The orientation of the local x-axis of a beam element is determined by the numbering of While this approach has its limitation too, it is rational and the outcome completely predictable.

**Beam local axes convention**

The local x-axis of a beam element is taken in the direction of the lower node number (A) to the higher node number (B). The element's shear force diagram is drawn in this direction, i.e. "left to right" is taken from the lower to higher node number, regardless the orientation of the element and the point from where one is viewing the element. In the case of a colum (vertical beam element), the local y-axis direction is used, again from the lower to higher node number.

**Example**

Consider the simple example of a simply-supported beam subject to a UDL. The **Frame Analysis** model shown below uses five beam elements, and the numbering of the nodes has a broken sequence moving left to right (nodes 4 and 5 are switched).

**Shear force diagram**

The shear force diagram shows a discontinuity for element 5-4. This is so because all the other beam elements have their x-axis pointing to the right (in the direction from lower node number to higher node number). In the case of beam 5-4, however, the local x-axis points to the left, and the shear force diagram seems inverted.

This behaviour in the program is by design. If you encounter situations like this and you find the presentation bothersome, you can "correct" the shear force diagram by renumbering the nodes in such a way that their numbers consistently increase in the same direction.

*Note: When designing a beam or column for shear force, one uses absolute shear force values. The fact that a calculated shear force value is positive or negative is purely due to the sign convention used in the analysis.*

**Bending moment diagram **

It should be noted that the above "quirk" does not apply to bending moment diagrams. **Frame Analysis** draws the bending moment diagram on the tension side of a beam element. Since the flexural tension side can be identified, the bending moment diagram can be drawn uniquely regardless the orientation of the beam element local axes.

## Comments (2)

Please explain the benefit of having the shear force diagram deliberately drawn in this manner?

As the shear force diagram is the integral of the loading diagram, and the moment diagram is the integral of the shear force diagram, this "apparent discrepancy" is quite apparent.

Frame Analysis depends on the user's input of nodes. The only way to get a smooth shear diagram is to subsequently number the nodes in one direction. Sumo Structural Modeller, on the other hand, takes care of all the node creation for you and does a markedly better job at subsequently numbering.