V = Vertical shear force
Q = Static moment of area
I = Moment of inertia of the beam cross section
t = thickness of material
This post will show how the above equation can be used to calculate shear stress in a rectangular. The results will be validated using a simple patran/nastran model.
Assumed a beam with pinned-pinned boundary and a point load applied at the mid point as shown below:
Let,
P=1000lbs
A=25in
B=25in
Hence using any beam software or simple hand calcs, it can be shown that the vertical shear load is 500lbs between x=0in and application point P and 500lbs (but opposite direction) between application point P and x=50in.
Assuming the beam is a rectangular plate with a width of 0.5in and a height of 10in. The moment of inertia can be calculated.
I = (width x height^3)/12 = 41.67in^4
Calculating the Q (Static moment of area) at different height on the beam. the following table can be constructed.
Where y is the area/segment of interest to the neutral axis.
A = area of interest.
VQ/it = shear stress of beam.
The next step is to show how to validate this shear stress in patran/nastran.
First create a beam in Patran using Quads elements as shown below. The thickness of each quad should be 0.5in and the material properties are based on aluminum which has a youngs modulus of 10300000psi and poison ratio of 0.33. Using an applied load of 1000lbs as per the example problem above, the patran model is shown below.
note that there is 50 elements in width and 10 element in height.
next we can verify the shear stress using RESULTS->CREATE->MARKER->TENSOR. Select tensor and select component xy. in target entities select element. In the figure below a few elements near the neutral axis is selected. The shear stress should be around 148.5 psi.
The shear values are consistent with "hand calculations".
P=1000lbs
A=25in
B=25in
Hence using any beam software or simple hand calcs, it can be shown that the vertical shear load is 500lbs between x=0in and application point P and 500lbs (but opposite direction) between application point P and x=50in.
Assuming the beam is a rectangular plate with a width of 0.5in and a height of 10in. The moment of inertia can be calculated.
I = (width x height^3)/12 = 41.67in^4
Calculating the Q (Static moment of area) at different height on the beam. the following table can be constructed.
| y | A | Q | VQ/it |
| 4.5 | 0.25 | 1.125 | 27 |
| 4.25 | 0.75 | 3.1875 | 76.5 |
| 3.75 | 1.25 | 4.6875 | 112.5 |
| 3.25 | 1.75 | 5.6875 | 136.5 |
| 2.75 | 2.25 | 6.1875 | 148.5 |
Where y is the area/segment of interest to the neutral axis.
A = area of interest.
VQ/it = shear stress of beam.
The next step is to show how to validate this shear stress in patran/nastran.
First create a beam in Patran using Quads elements as shown below. The thickness of each quad should be 0.5in and the material properties are based on aluminum which has a youngs modulus of 10300000psi and poison ratio of 0.33. Using an applied load of 1000lbs as per the example problem above, the patran model is shown below.
note that there is 50 elements in width and 10 element in height.
next we can verify the shear stress using RESULTS->CREATE->MARKER->TENSOR. Select tensor and select component xy. in target entities select element. In the figure below a few elements near the neutral axis is selected. The shear stress should be around 148.5 psi.
The shear values are consistent with "hand calculations".
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