# Question 7.5: An I-beam is made by gluing five wood planks together, as sh......

An I-beam is made by gluing five wood planks together, as shown. At a given axial position, the beam is subjected to a shear force V = 6000 lb. (a) What is the average shear stress at the neutral axis z = 0? (b) What are the magnitudes of the average shear stresses acting on each glued joint?

Find: Average shear stresses.
Assume: Hooke’s law applies.

Step-by-Step
The 'Blue Check Mark' means that this solution was answered by an expert.

We obtained a formula for shear stress at a given height, $σ_{xz} = V Q^{\prime}$/I b, where $Q^{\prime}$ and b depend on the z position in question. I in this relationship is always the second moment of area of the entire cross section about the z-axis. We have been given V. So, we must calculate I once and then calculate the appropriate values of $Q^{\prime}$ and b for both parts of this problem.

By inspection of the cross section’s symmetry, we see that the centroid is at the geometric center of the I-beam. For the central vertical segment, therefore d, the distance between the centroid of the segment and the centroid of the entire cross section, is zero. The four remaining segments will each have the same second moment of area about their own horizontal bisectors, and the same areas and distances d. Thus, we can write

\begin{aligned} & I=I_{\text {vertical }}+4 I_{\text {smaller }}=\left(\frac{1}{12} b h^3\right)_{\text {vertical }}+4\left[\frac{1}{12} b h^3+A d^2\right]_{\text {smaller }}{ }^{\prime} \\ & I=\frac{1}{12}(2 \text { in })(8 \text { in })^3+4\left[\frac{1}{12}(4 \text { in })(2 \text { in })^3+(2 \text { in } \times 4 \text { in })(3 \text { in })^2\right]=384 \text { in }^4 \end{aligned}

We can calculate $Q^{\prime}$ at the neutral axis by finding the centroid and area of the shaded area on the left, or by summing the contributions due to the individual planks, as shown at right. The values of z for planks 2 and 3 are the same as their d values used in the I calculation.

\begin{aligned} Q^{\prime} & =\int_{A^{\prime}} z \mathrm{~d} A=\sum z^{\prime} A^{\prime} \\ & =(2)(2 \cdot 4)+3(4 \cdot 2)+3(4 \cdot 2)=64 \mathrm{in}^3 . \end{aligned}

So, the average shear stress at the neutral axis is $\sigma_{x z}=V Q^{\prime} / I b=(6000 \mathrm{lb})\left(64 \mathrm{in}^3\right) / \left(384 \text { in }^4\right)(2 \text { in })=500 \text { psi. }$

As an exercise, verify that each glued joint is subjected to the same average shear stress. We will determine only the average shear stress acting on the lower-right glued joint by using the area A and length of contact b as shown below. The value of $Q^{\prime}$ is (3)(4 · 2) = 24 in³, and the average shear stress is V$Q^{\prime}$ /I b = 188 psi.

Question: 7.9

## An axial load is applied to a solid circular bar that contains an offset in order to fit in a tight space in a machine. Compute the maximum tensile and compressive normal stresses at section a. Given: Magnitude of axial force and bent bar geometry. Find: Maximum normal stresses at specified ...

Although we have learned about bars subject to axi...
Question: 7.8

## Having confirmed the normal stress equation in Example 7.7, derive the shear stress distribution in a rectangular beam (b × h) subject to loading that causes bending moment M(x) and shear force V(x), starting from the plane stress elasticity equations of equilibrium from Chapter 4: ...

The first of the equilibrium equations represents ...
Question: 7.7

## For a thin elastic beam with rectangular cross section (b × h) and loading that causes bending moment M(x), derive Equation 7.22, the expression for the normal stress σxx(x, z) in a beam, starting from the fact that, as we have shown, the normal stress varies linearly through the thickness. Given: ...

We begin with a general statement of the fact that...
Question: 7.6

## The beam shown is subjected to a distributed load. For the cross section at x = 0.6 m, determine the average shear stress (a) at the neutral axis and (b) at z = 0.02 m. Given: Dimensions of and loading on simply supported beam. Find: Shear stress at two locations along height of cross section at x ...

First we need to consult our FBD and find the reac...
Question: 7.4

## A steel T-beam is used in an inverted position to span 400 mm. If, due to the application of the three forces as shown in the figure, the longitudinal strain gage at A (3 mm down from top of beam and at the x location shown) registers a compressive strain of −50 × 10^−5, how large are the applied ...

The gage at A, in the upper portion of the cross s...
Question: 7.3

## Find the centroid and the second moment of area about the horizontal (y) axis of the cross section shown. All dimensions are in millimeters. If a beam is constructed with the cross section shown from steel whose maximum allowable tensile stress is 400 MPa, what is the maximum bending moment that ...

The symmetry of the cross section shown suggests t...
Question: 7.2