A cantilever wood beam (Figure P9.5a) consists of eight 2 in. thick planks glued together to form a cross section that is 16 in. deep, as shown in Figure P9.5b. Each plank has a width ofb= 5.5 in. The cantilever beam has a length of L = 6 ft and it supports a concentrated load of P = 3,800 lb. (a) Calculate 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 2 in. (typ) 16 in. FIGURE P9.5a Cantilever beam. FIGURE P9.5b Cross-sectional dimensions.
The Correct Answer and Explanation is :
To solve this problem, we need to calculate the horizontal shear stress at different points along the height of the cantilever beam cross-section. Let’s break this problem into two parts.
Part (a): Calculate the horizontal shear stress at points A, B, C, and D
First, we can apply the shear stress formula for a rectangular beam’s cross-section: τ=VQIb\tau = \frac{VQ}{Ib}
Where:
- τ\tau is the shear stress at a point,
- VV is the shear force at the point,
- QQ is the first moment of area about the neutral axis,
- II is the second moment of area (the moment of inertia) of the cross-section,
- bb is the width of the beam at the location of the shear stress.
We know:
- P=3,800P = 3,800 lb (the concentrated load),
- The length of the beam is L=6L = 6 ft, but the location of the applied load (cantilever support) is irrelevant for the shear stress calculation, since shear at the fixed end (near the wall) is assumed to be PP.
- Each plank has a width of b=5.5b = 5.5 in.
- The total depth of the beam is 1616 in.
- The beam is made up of 8 planks, so the total height is 16 in, divided into 8 layers, each 2 in thick.
Step 1: Calculate II (Moment of Inertia)
The moment of inertia II of the rectangular section is given by: I=bh312I = \frac{b h^3}{12}
Where b=5.5b = 5.5 in (width) and h=16h = 16 in (height). Since the beam is made of 8 planks glued together, we calculate for the entire cross-section: I=(5.5 in)(16 in)312=11,264 in4I = \frac{(5.5 \text{ in})(16 \text{ in})^3}{12} = 11,264 \text{ in}^4
Step 2: Calculate QQ (First Moment of Area)
The first moment of area QQ depends on the location of the point where you are calculating the shear stress. For each point AA, BB, CC, and DD, you will calculate QQ relative to the neutral axis (which is in the middle of the cross-section).
For example, for point AA (at the top of the beam), the first moment of area QAQ_A is: QA=Atop⋅ytopQ_A = A_{\text{top}} \cdot y_{\text{top}}
Where AtopA_{\text{top}} is the area of the section above point A, and ytopy_{\text{top}} is the distance from the neutral axis to the centroid of the top area.
Step 3: Calculate the Shear Stress
Using the shear force V=P=3,800V = P = 3,800 lb, the shear stress at each point is: τA=VQAIb\tau_A = \frac{V Q_A}{I b}
Repeat this process for points BB, CC, and DD.
Part (b): Shear Stress Distribution Plot
The shear stress distribution varies from top to bottom of the beam, with maximum shear stress occurring at the neutral axis and decreasing toward the top and bottom surfaces. For a rectangular section, the shear stress is highest at the neutral axis and decreases linearly toward the top and bottom surfaces.
By calculating the shear stress at various points along the height of the beam, we can plot a graph showing how the shear stress varies. At the top and bottom, the shear stress is zero, and it peaks at the neutral axis (the middle of the beam’s height).
Conclusion
The horizontal shear stress distribution in the cantilever beam varies across its depth. The calculation of shear stress involves finding the moment of inertia, the first moment of area, and applying the shear stress formula. The shear stress is maximal at the neutral axis and reduces to zero at the beam’s surfaces.
To summarize:
- The shear stress formula τ=VQIb\tau = \frac{VQ}{Ib} allows us to calculate the shear stress at any point in the beam’s height.
- The shear stress distribution is maximum at the neutral axis and decreases toward the top and bottom.
This process involves detailed calculations at each specified point (A, B, C, D), followed by plotting the results for the shear stress distribution.