Figure 4–23a shows a workpiece clamped to a milling machine table by a bolt tightened to a tension of 2000 lbf. The clamp contact is offset from the centroidal axis of the strut by a distance e = 0.10 in, as shown in part b of the figure. The strut, or block, is steel, 1 in square and 4 in long,

Question

Figure 4–23a shows a workpiece clamped to a milling machine table by a bolt tightened to a tension of 2000 lbf.

The clamp contact is offset from the centroidal axis of the strut by a distance [latex]e = 0.10[/latex] in, as shown in part b of the figure.

The strut, or block, is steel, 1 in square and 4 in long, as shown. Determine the maximum compressive stress in the block.

Accepted Answer

First we find [latex] A = bh = 1\left(1\right) = 1 \ in^2 \ , \ I = {bh^3}/{12} = {1\left(1\right)^3}/{12} = 0.0833 in^4 \, \ k^2={I}/{A} = {0.0833}/{1} = 0.0833 \ in^2 \ , \ and {l}/{k} = {4}/{0.0833}^{{1}/{2}}= 13.9 [/latex] . Equation (4–56) gives the limiting slenderness ratio as

[latex]\left(\frac{l}{k} \right) _2=0.282\left(\frac{AE}{P} \right) ^{{1}/{2}}[/latex] (4–56)

[latex]\left(\frac{l}{k} \right) _2=0.282\left(\frac{AE}{P} \right) ^{{1}/{2}}=0.282\left[\frac{1\left(30\right)\left(10\right)^6 }{1000} \right]^{{1}/{2}} =48.8 [/latex]

Thus the block could be as long as

[latex]l=48.8 \ k=48.8\left(0.0833\right) ^{{1}/{2}}=14.1 \ in [/latex]

before it need be treated by using the secant formula. So Eq. (4–55) applies and the maximum compressive stress is

[latex]\sigma _c= \frac{P}{A} +\frac{Mc}{I} = \frac{P}{A} +\frac{PecA}{IA} = \frac{P}{A} \left(1+\frac{ec}{k^2} \right) [/latex] (4–55)

[latex]\sigma _c= \frac{P}{A} +\frac{Mc}{I} = \frac{P}{A} +\frac{PecA}{IA} = \frac{P}{A} \left(1+\frac{ec}{k^2} \right) =\frac{1000}{1} \left[1+\frac{0.1\left(0.5\right) }{0.833} \right] =1600 \ psi[/latex]

Question 1.4.20: Figure 4–23a shows a workpiece clamped to a milling machine ...

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