CEE 220 - INTRODUCTION TO MECHANICS OF MATERIALS

Tutorial Session 8 - Solution

1. Pressure Vessels and Circular Rings

a) Figs. (a) and (b) are portions of a cylindrical pressure vessel subjected to an internal pressure p.  In (b), let L be the longitudinal length of the body.  Complete Fig. (a) as a free-body diagram for determining the longitudinal stress s_x . Similarly, complete Fig. (b) for determining  s_q.  In each case show the resultant of the pressure and the stress resultant.

Assume t << r, so that r_i ≈ r_o ≈ r.

See class notes.

b) Repeat the preceding question, but consider the free bodies to contain the fluid of the pressure vessel.

c) Consider a spherical pressure vessel subjected to an internal pressure p.  Draw the free-body diagram of a hemispherical portion, and show the pressure and stress resultants.

d) The figures show in projection view a quarter of a thin spherical pressure vessel contained between a vertical plane and a horizontal plane passing through the center. Fig.(a) includes the fluid, and Fig.(b) does not.

Complete the free body diagram for Fig.(a), showing on each section the pressure resultant and the stress resultant.   The centroid of the area of a half circle is at a distance of 4r/3p from the diameter, and the centroid of a half circumference is at a distance of 2r/p from the diameter. 

Complete the free body diagram for Fig.(b).

e) The axial force in a ring subjected to a radial load of intensity q is found by equilibrium of a half ring as N = qr.   Show that equilibrium of any partial arc of the ring yields the same result.

SF_y = 2qrsinq - 2Nsinq = 0

N = qr

2. Combined Loadings

Let  F_x, F_y, F_z, M_x, M_y, and M_z be the internal forces acting on a positive cross section of a bar, shown in the figure.  In the table below, each cell refers to a stress at a point of the cross-section, as caused by an individual force. 

Write in each cell the formula for the indicated stress.  Use symbols  A, I_y, I_z, Q_y, and Q_z for cross-section properties.  Leave the cell empty if the stress is zero.  Define separately A, I_y, I_z, Q_y, and Q_z in terms of the cross-section dimensions.  Pay attention to the sign of each stress, since superposition is to be applied.  For torsional shear stresses, for which we did not obtain a formula for a rectangular cross section, indicate a non zero stress by its symbol with a plus sign if positive, and a minus sign if negative.

A = 4ab,  I_y = 2a(2b)³/12,   I_z = 2b(2a)³/12,  Q_y = 2ab(b/2),  Q_z = 2ba(a/2)