Deflection 03

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For section BC, with the variable of integration θ, summing moments about the break gives the moment equation for section BC.

Noting that the break in section CD contains nothing but Q, and after setting Q = 0, we can conclude that there is no actual strain energy contribution in this section.

Combining terms from Eqs. (3) and (6) to get the total vertical deflection at D,

inserting Eqs. (4) and (5) and setting Q = 0, we get

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EXAMPLEÂ 08 Given as shown , find the deflection at point 1

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Angular deflection at wall is zero

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EXAMPLEÂ 09

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The solution for the boundary conditions that y = 0 at x = 0, l is

By substituting x = l/2

Since the shaded area showed failures in actual tests Most designers select point T such that Pcr/A = Sy/2

The parabolic curve is

ec/k2

=

eccentricity ratio.

M i Maximum bending b di momentt also l occurs att midspan id

then a = Sy

Substituting Mmax By imposing the compressive yield strength Syc as the maximum value of σc 47

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Secant Column Formula.

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Magnitude of the maximum compressive stress

where k = (I/A)1/2 and is the radius of gyration, This difference between the two formulas suggests that one way of differentiating between a “secant column” and a strut, or short compression member, is to say that in a strut, the effect of bending deflection must be limited to a certain small percentage of the eccentricity.

If we decide that the limiting percentage is to be 1 percent of e, then, the limiting slenderness ratio turns out to be

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EXAMPLE 10

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If the actual slenderness ratio is greater than (l/k)2 , then use the secant formula

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EXAMPLE 11

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

First we find A = bh = 1(1) = 1 in2, I = bh3/12 = 1(1)3/12 = 0.0833 in4, k2 = I/A = 0.0833/1 = 0.0833 in2, and l/k = 4/(0.0833)1/2 = 13.9. The limiting slenderness ratio is

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before it need be treated by using the secant formula.

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So the maximum compressive stress is

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