HW10+Solutions - HOMEWORK PROBLEMS Week 10: Due Friday 18...

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HOMEWORK PROBLEMS Week 10: Due Friday 18 November 1. A cylindrical pressure vessel of internal diameter 80 inches and wall thickness 1 inch has two flat ends of thickness 1.5 in, each of which is secured to the cylindrical part by 20 bolts, each of 1 inch diameter. The cross-section of the vessel is shown in Figure 1. The pressure of gas inside the vessel is 300 psi. Find (i) the axial and circumferential membrane stresses in the cylinder wall and (ii) the tensile stress in the bolts. 180 80 1.5 1 20 bolts, 1 in. diameter each end all dimensions in inches Figure 1 Solution The tensile stresses in the vessel wall can be obtained using the formulae σ z = pR 2 t ; σ θ = pR t . Here, we obtain σ z = 300 × 40 2 × 1 = 6000 psi ; σ θ = 300 × 40 1 = 12 , 000 psi . 1
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If we cut through the bolts between the ends of the vessel and the cylindrical portion, we expose the free-body diagram of Figure 1.1. The right figure shows an end view, showing how all 20 bolts are sawn through and the left figure shows a side view, in which the forces acting on the plate due to the pressure p and the tensile stresses in the bolts can be seen. p σ bolts σ bolts sawn-through bolts Figure 1.1 The stress σ bolts acts on a total area equal to 20 times that of a single bolt — i.e. A bolts = 20 × π × 1 2 4 = 15 . 71 in 2 and the pressure p acts on a circular area of diameter 80 inches and hence A p = π × 80 2 4 = 5026 in 2 . For axial equilibrium, we have σ bolts A bolts - pA p = 0 or σ bolts = 300 × 5026 15 . 71 = 95 , 987 psi . Notice that equilibrium statements relate to forces, not stresses, and to write an equilibrium equation involving unknown stresses, we first need to convert them to forces by multiplying by the area on which they act. 2
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This note was uploaded on 02/15/2012 for the course MECHENG 211 taught by Professor ? during the Fall '07 term at University of Michigan.

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HW10+Solutions - HOMEWORK PROBLEMS Week 10: Due Friday 18...

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