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

Step 2 — Geometry of Deformation: The next question is, “How are the angles of
twist in the two shaft segments related?” The compound shaft is attached to rigid walls
at A and C; therefore, the twisting that occurs in shaft segment (1) plus the twisting in
shaft segment (2) cannot result in any net rotation of the compound shaft. In other
words, the sum of these angles of twist must equal zero:
φ + φ = 0
(b)
1 2
Step 3 — Torque–Twist Relationships: The angles of twists in shaft segments (1)
and (2) can be expressed by the angle-of-twist equation [Equation (6.12)]. Angle-oftwist
equations can be written for both segment (1) and segment (2):
φ
1
TL 1 1
= φ2
=
JG
1 1
TL
JG
2 2
2 2
(c)
Step 4 — compatibility Equation: The torque–twist relationships [Equation (c)]
can be substituted into the geometry-of-deformation equation [Equation (b)] to obtain
a new relationship between the unknown torques T 1 and T 2 :
TL
JG
1 1
1 1
TL 2 2
+ = 0
(d)
JG
2 2
Notice that this relationship is based, not on equilibrium, but rather on the relationship
between deformations that occur in the compound shaft. This type of equation is
termed a compatibility equation.
Step 5 — Solve the Equations: Two equations have been developed in terms of the
internal torques T 1 and T 2 :
Σ Mx = − T1 + T2
+ 32 kip⋅ in. = 0
(a)
TL
JG
1 1
1 1
TL 2 2
+ = 0
(d)
JG
2 2
These two equations must be solved simultaneously for us to determine the torques in
each shaft segment. First, the compatibility equation [Equation (d)] can be solved for
the internal torque T 2 :
T
L1
JG
=−T
⎛ ⎝ ⎜ ⎞
JG ⎠
⎟ ⎛ ⎝ ⎜ L
2 1
1 1
2 2
2
⎞
T
L 1 J
1
⎠
⎟ =− ⎛ ⎝ ⎜ ⎞
L ⎠
⎟ ⎛ ⎝ ⎜ J
2
⎞ G2
⎠
⎟ ⎛ ⎝ ⎜
⎞
G ⎠
⎟
Next, substitute this result into the equilibrium equation [Equation (a)]:
⎛
−T T L 1 − 1
⎝
⎜
L
1
2
⎞ J
⎠
⎟ ⎛ ⎝ ⎜ J
Then solve for the internal torque T 1 :
2
1
⎞ G
⎠
⎟ ⎛ ⎝ ⎜ G
2
1
⎞
32 kip in. 0
⎠
⎟ + ⋅ =
2
1
1
T
1
=
32 kip⋅in.
⎡ L1
J
1 + ⎛ ⎝ ⎜ ⎞
⎢
L2
⎠
⎟ ⎛ ⎣ ⎝ ⎜ J
⎞ G2
⎠
⎟ ⎛ ⎝ ⎜
⎞ ⎤
G1
⎠
⎟ ⎥
⎦
2
1
(e)
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