Basic Equations: Differential Form - Mechanics of Structures: Variational and Computational Methods - page 73

1.68 Show that the maximum torsional stress for a bar of rectangular cross-section occurs
at the midpoints of the longer sides of the rectangle.

Hint: See Problem 1.67.

1.69 If the stress function for the torsion of a bar of rectangular cross-section (Fig. P1.66) is

c 1 cos π cos π

π π

y
2 a

z
2 b +

c 2 cos 3

y

cos 3

z
2 b

ψ =

2 a

τ xz .
(b) Are the conditions of equilibrium satisfied?
(c) Find

(a) Find the stresses τ xy and

φ . τ max ,M t , and

Hint: Use Eqs. (1.154), (1.144), (1.148), and (1.149)

1.70 Find relationships between the Prandtl stress function ψ and the warping function

ω.
1.71 Find the torsional characteristics of the thin strip of Fig. P1.71, including, d φ/ dx, J , τ

max .

The results apply to many thin-walled open cross sections.

Hint: Use a Prandtl stress function that is a function of y only. Poisson's equation
becomes

d 2 ψ

dy 2 =−

2 G d

dx

for which a stress funtion can be constructed
Answer:

2

1

3 bt 3 G d

dx

= ψ =

M t dy dz

1
3 bt 3

J =

y =± t / 2 =±

=− ∂ψ
∂

tG d

dx =± tM t

J

τ

max

y

t

b

M t , f

x

y

FIGURE P1.71
Thin strip under torsion.

z