Shear Stress in Beams

Consider a segment of the beam shown. The shear load on the vertical surfaces are generated by shear stress that can be calculated by the following process.

To calculate the shear stress t generated from the shear load V consider removing the segment of the beam shown in red.

By symmetry of stress, shear stresses on the cross section
results in equal shear stresses on the plane perpendicular to the cross section
as shown. This shear stress results in a shear load *F*_{s}.

The side view of this segment, showing only axial loads, will look as follows

As can be seen, the difference in the bending moment on the
two sides of the segment results in the normal stresses being different. The
shear load *F*_{s} is the element that brings the segment into
equilibrium. The axial load due to the normal stresses created by the bending
moment can be calculated by integrating the normal stress over the area *A*^{*}
of the cross section.

Therefore, equilibrium in the axial direction for this segment is written as

_{}

The integral in this expression is the first moment of the
area *A*^{*} about the neutral axis. This first moment will be
denoted by *Q* so that

_{}

The shear stress can now be calculated from the shear load by dividing it by the area it is applied on to get

_{}

Taking the limit as _{} gives

_{}

where we note that _{}

**Calculating the first moment of the area: **The first moment of the area can be calculated from the
relation

_{}

where *A*^{*} is the area of the part of the
cross section that is considered, _{} is the vertical
distance from the centroid of the cross section to the centroid of *A*^{*}.

For composite areas, the first moment of area can be
calculated for each part and then added together. The equation for *Q *in
this case is

_{}

**Shear stress along a slanted direction:** The procedure
presented does not require that the segment be cut with a surface parallel to
the neutral surface. Therefore, one can remove a segment with the lower surface
being at an oblique angle to the neutral surface so that the cross section will
look as shown.

The shear calculated in this way is the average shear
stress perpendicular to the line *RS* of width *b*.

ã Mehrdad Negahban and the
University of Nebraska, 1996-2000.

All
rights reserved

Copy
and distribute freely for personal use only

Department
of Engineering Mechanics, University of Nebraska, Lincoln, NE 68588-0526

Last modified at: 10:40 AM, Thursday, March 02, 2000