Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

Step 2 — Geometry of Deformation: The geometry of the specific system is evaluated
to determine how the deformations of the torsion members are related.
Step 3 — Torque–Twist Relationships: The relationships between the internal torque
in a member and its corresponding angle of twist are expressed by Equation (6.12).
Step 4 — compatibility Equation: The torque–twist relationships are substituted into
the geometry-of-deformation equation to obtain an equation that is based on the structure’s
geometry, but expressed in terms of the unknown internal torques.
Step 5 — Solve the Equations: The equilibrium equations and the compatibility equation
are solved simultaneously to compute the unknown internal torques.
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STATICALLy INdETERMINATE
TORSION MEMbERS
The use of this procedure to analyze a statically indeterminate torsion system is illustrated
in the next example.
ExAmpLE 6.8
A compound shaft consists of two solid shafts that are connected
at flange B and securely attached to rigid walls at A
and C. Shaft (1) is a 3.00 in. diameter solid aluminum [G =
4,000 ksi] shaft that is 60 in. long. Shaft (2) is a 2.00 in. diameter
solid bronze [G = 6,500 ksi] shaft that is 40 in. long.
If a concentrated torque of 32 kip · in. is applied to flange B,
determine
y
A
32 kip.in.
(1)
(2)
B
60 in. 40 in.
C
x
(a) the maximum shear stress magnitudes in shafts (1) and (2).
(b) the rotation angle of flange B relative to support A.
Plan the Solution
The solution begins with a free-body diagram at flange B. The equilibrium equation
obtained from this free-body diagram reveals that the compound shaft is statically
indeterminate. We can obtain the additional information needed to solve the problem
by considering the relationship between the angles of twist in the aluminum and
bronze segments of the shaft.
SOLUTION
Step 1 — Equilibrium Equation: Draw a free-body diagram
of flange B. Assume positive internal torques in shaft
segments (1) and (2). [See the sign convention detailed in
Section 6.6.] From this free-body diagram, the following
moment equilibrium equation can be obtained:
y
A
32 kip.in.
(1)
(2)
T 1
T 2
B
60 in. 40 in.
C
x
Σ Mx = − T1 + T2
+ 32 kip⋅ in. = 0
(a)
There are two unknowns in Equation (a): T 1 and T 2 . Consequently, statics alone does
not provide enough information for this problem to be solved. To obtain another
relationship involving the unknown torques T 1 and T 2 , we will consider the general
relationship between the twist angles in the compound shaft.