Page 198 - Shigley's Mechanical Engineering Design

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Deflection and Stiffness 173
For section BC, with the variable of integration θ defined as shown in Fig. 4–13c, sum-
ming moments about the break gives the moment equation for section BC.

M BC = Q(R + R sin θ) + PR sin θ (4)
∂M BC /∂Q = R(1 + sin θ) (5)

From Eq. (4–41), inserting Eqs. (4) and (5) and setting Q = 0, we get

π/2 π/2
1 ∂M BC R
(δ D ) BC = M BC Rdθ = (PR sin θ)[R(1 + sin θ)] dx
0 EI ∂Q EI 0
PR 3 π (6)
= 1 +
EI 4
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,

P 2 3 2 1 3 PR 3 π
δ D = (δ D ) AB + (δ D ) BC = (2R l + l R + l ) + 1 +
EI 2 3 EI 4
(7)
P 3 2 2 3
= (1.785R + 2R l + 1.5 Rl + 0.333l )
EI
4
Substituting values, and noting I = πd /64, and E = 207 GPa for steel, we get
1
2
3
Answer δ D = [1.785(0.05 ) + 2(0.05 )0.04
4
9
207(10 )[π(0.002 )/64]
2
3
+ 1.5(0.05)0.04 + 0.333(0.04 )]
−3
= 3.47(10 ) m = 3.47 mm

EXAMPLE 4–13 Deflection in a Variable-Cross-Section Punch-Press Frame
The general result expressed in Eq. (4–39),

π FR 2 π FR πCF R
δ = − +
2AeE 2AE 2AG
is useful in sections that are uniform and in which the centroidal locus is circular. The
bending moment is largest where the material is farthest from the load axis.
Strengthening requires a larger second area moment I. A variable-depth cross section is
attractive, but it makes the integration to a closed form very difficult. However, if you
are seeking results, numerical integration with computer assistance is helpful.
Consider the steel C frame depicted in Fig. 4–14a in which the centroidal radius is
32 in, the cross section at the ends is 2in × 2in, and the depth varies sinusoidally with