Using beam theory, calculate the theoretical value for transverse shear stress at the gage location of 0.70 for a 1000 pound applied load. The equation is . The moment of inertia is 1/3 and the thickness is 0.5 inches. The transverse shear force at the gage location is 500 pounds with a Q value of 0.1275.
The correct answer and explanation is:
To calculate the transverse shear stress using beam theory, we use the standard shear stress formula: τ=VQIT\tau = \frac{VQ}{IT}
Where:
- τ\tau = transverse shear stress (psi)
- VV = transverse shear force (pounds)
- QQ = first moment of area about the neutral axis (in³)
- II = moment of inertia (in⁴)
- TT = thickness of the material at the point of interest (in)
Given:
- V=500V = 500 lb
- Q=0.1275Q = 0.1275 in³
- I=13I = \frac{1}{3} in⁴
- T=0.5T = 0.5 in
Substituting into the formula:
τ=(500)(0.1275)(1/3)(0.5)=63.75(0.1667)≈382.5 psi\tau = \frac{(500)(0.1275)}{(1/3)(0.5)} = \frac{63.75}{(0.1667)} \approx 382.5 \text{ psi}
Final Answer:
τ≈382.5 psi\boxed{\tau \approx 382.5 \text{ psi}}
Explanation (300 words):
The transverse shear stress (τ\tau) in a beam is a measure of how internal shear forces are distributed across the cross-section of the beam. It is especially significant near regions where the cross-section changes or where external shear forces are applied. Beam theory provides a framework for estimating these stresses based on the geometry and loading conditions of the beam.
The formula used, τ=VQIT\tau = \frac{VQ}{IT}, derives from integrating the equilibrium of a differential beam element. Each term in this formula has a distinct physical meaning:
- V (shear force): This is the internal shear force acting perpendicular to the beam’s longitudinal axis at the location of interest.
- Q (first moment of area): It represents the moment of the area above (or below) the point where the shear stress is being calculated, about the beam’s neutral axis.
- I (moment of inertia): This reflects the beam’s resistance to bending, calculated about the neutral axis.
- T (thickness): This is the width of the material at the level where the shear stress is being computed.
In this problem, a 1000-pound external load induces a 500-pound transverse shear force at a gage location of 0.70. Using the given Q=0.1275Q = 0.1275, I=13I = \frac{1}{3}, and T=0.5T = 0.5, we calculated a shear stress of approximately 382.5 psi. This theoretical value helps in structural analysis and ensuring safety by comparing with material limits like shear yield strength.