Section 9.2

The Combined Effects of Axial and Bending Forces

Last Revised:11/04/2014

Figure 9.2.1Basic Internal ForcesClick on image for larger view

The three internal translational forces on a section, as seen in Figure 9.2.1, are axial, shear in the x and shear in the y directions. The three internal rotational forces are torque, bending about the x axis, and bending about the y axis.

The axial force, P, and bending moments about the x axis and y axis, Mx and My, each create normal stress on the cross section. The other three forces (T, Vx, and Vy) create shear stress on the cross section.

The magnitude of the internal forces changes as a function of location along the member. It is also most likely thatthe maximums for each internal force do not occur at the same location along the member. One challenge of the analysis process is to find the location in the member where the maximum total normal stress is found.

The total normal stress at any point (along the member and on the section - for example at dA in Figure 9.2.1), using superposition, is the sum of the normal stresses created by the axial and bending forces at that location on the member.

total=axial+bending-x+bending-y

Using the appropriate equations from mechanics, the total stress at a point can be expressed as:

total= P/A + Mxy/Ix+ Myx/Iy

Where:

P, Mx, and Myaresimultaneously occurring internal forces at the same location on the memberand

x and y are the respective distances from the y and x axes.

Be careful to avoid the mistake of finding the maximum P, maximum Mx, and maximum Myalong the member then combining the effects from these three maximum forces. To do so creates a fictionally high total stress that does not occur anywhere in the memberunless these peak values just happen to occur at the same location on the member. Finding the maximum value oftotalmay require you to look at several locations along the member as it may not be readily apparent where this will occur when the maximum values of P, Mxand Mydo not occur simultaneously at the same location on the member.

The maximum compressive and/or tensile normal stresses will each occur at a point on the perimeter of a section when bending is present. This simplifies your search for the maximum occurring stress on a given cross section.

In mechanics, you are taught that the maximum tensile stress and maximum compressive stress in a member must be kept within acceptable limits in order to be safe. These limits on bending and axial stress are not always the same for each type of stress.

A typical limit state equation takes the form:

axial