A cantilever wood beam consists of eight 2 in. thick planks glued together to form a cross section that is 16 in. deep. Each plank has a width of
in. The cantilever beam has a length of
ft and it supports a concentrated load of
lb. (a) Calculate the magnitude of the horizontal shear stress at points A, B, C, and D. (b) From these results, plot a graph showing the distribution of shear stresses from top to bottom of the beam. Answer:
i psi
i psi
i psi
i psi
A cantilever wood beam consists of eight 2 in. thick planks glued together to form a cross section that is 16 in. deep. Each plank has a width of
in. The cantilever beam has a length of
ft and it supports a concentrated load of
lb. (a) Calculate the magnitude of the horizontal shear stress at points A, B, C, and D. (b) From these results, plot a graph showing the distribution of shear stresses from top to bottom of the beam. Answer:
i psi
i psi
i psi
i psi
The Correct Answer and Explanation is:
:
- Width of each plank (in inches)
- Length of the beam (in feet)
- Magnitude of the concentrated load (in pounds)
- Clarification of the locations of points A, B, C, and D (typically they are taken at different heights in the cross-section, such as center of each plank)
However, I can proceed with a general explanation and solution method, assuming:
- 8 planks, each 2 in. thick → total height = 16 in.
- Width of each plank = w in (you need to provide this)
- Load P applied at the free end of the cantilever beam of length L
- Shear stress is highest at the neutral axis and zero at top/bottom surfaces.
General Procedure to Calculate Shear Stress
Shear stress at a horizontal plane is calculated as: τ=VQIb\tau = \frac{VQ}{Ib}
Where:
- VV: Shear force at the section
- QQ: First moment of area above/below the point where stress is calculated
- II: Moment of inertia of the entire cross-section
- bb: Width at the point where the shear stress is being calculated
Step-by-Step Outline
- Cross Section Properties:
- Height h=16 inh = 16 \, \text{in}
- Each plank: 2 in2 \, \text{in} high
- Moment of inertia I=112wh3I = \frac{1}{12} w h^3
- Shear Force (V):
- At the fixed end: V=PV = P
- First Moment of Area (Q):
- For point A (top): zero
- For point B: compute QQ for area above mid-height of plank 2
- For C: include more area, centered around mid-height of plank 4
- For D: near neutral axis, QQ is largest
Distribution of Shear Stress
- Shear stress is parabolic, maximum at mid-height (point D), zero at top and bottom.
- Numerical values depend on the input w, P, L, etc.
Conclusion
To compute exact values in psi at A, B, C, D and plot the graph, I need:
✅ Width of each plank
✅ Beam length
✅ Load magnitude
✅ Exact location of points A–D
