Shear Force in Inclined Beam

Suppose I have an inclined beam with a UDL applied vertically downwards, will the shear force act normal to the beam, or vertically?

2 Answers

It depends on the type of loading. If it is horizontal UDL: The shear will be normal to the beam, and it will be the same as the shear of a horizontal beam loaded with

$$w_{perp}=w_{inclined}cos\theta$$

• $$\theta$$= angle of the beam wrt horizon.

There will be an axial stress in the beam as well. It will be the same as a column loaded with a load. $$UDL*sin\theta$$ If it is a snow type of loading, refer to the diagram.

source

For a vertical UDL applied to an inclined beam, the UDL can be resolved into two components, the shear component normal to the inclined beam and the axial component parallel and inline with the inclined beam. That is the mathematically correct answer based on engineering Statics. The other answer posted by @kamran gives a nice diagram of some loading conditions and provides the basic equations to separate the shear component from the axial component. Diagram from kamran's post However, in common practice, the angle of inclination in structural beams is typically relatively small, and engineers can conservatively ignore the inclination angle, calculating the shear stress in the beam using the horizontal beam formulas. The axial force needs to be considered for the reaction that occurs at the beam supports, as it will appear as a thrust force, and anchorage needs to include that additional force. You didn't ask about the bending moment and the flexural stress in the inclined beam, but conservatively the same small angle conservative approach is a valid solution.

Note that cosine of 0 degrees is 1.0 and a cosine of 10 degrees is 0.985, so in this example, a small inclination angle of 10 degrees is only off by about less than 2 percent, well within the typical engineering factors of safety and overstrength present in engineered materials (e.g., steel specified as 36,000 psi minimum yield strength will test at a higher value as a result of manufacturing, and reinforcing bars specified as 60,000 psi ultimate strength will test at a higher ultimate breaking strength for the same reasons). A roof beam inclined at a pitch of 4/12 is inclined at an angle of 18.43 degrees, which calculates to a cosine value of 0.9487. So for your inclined beam, is a conservative error of 5% acceptable when designing for shear resistance in a beam? Here is where engineering judgement needs to be applied. Your answers may vary depending if it is a steel beam vs. a reinforced concrete beam, since one material performs well in shear loading, whereas the other performs in a brittle and non-ductile manner.