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

170
TORSION
The five-step procedure demonstrated in the previous example provides a versatile
method for the analysis of statically indeterminate torsion structures. Additional problemsolving
considerations and suggestions for each step of the process are discussed in the
following table:
Solution method for Statically Indeterminate Torsion Systems
Step 1
Step 2
Step 3
Equilibrium
Equations
Geometry of
Deformation
Torque–Twist
Relationships
Draw one or more free-body diagrams (FBDs) for the structure, focusing on the joints, which
connect the members. Joints are located wherever (a) an external torque is applied, (b) one or
more cross-sectional properties (such as the diameter) change, (c) the material properties (i.e., G)
change, or (d) a member connects to a rigid element (such as a gear, pulley, support, or flange).
Generally, FBDs of reaction joints are not useful.
Write equilibrium equations for the FBDs. Note the number of unknowns involved and
the number of independent equilibrium equations. If the number of unknowns exceeds the
number of equilibrium equations, a deformation equation must be written for each extra
unknown.
Comments:
• Label the joints with capital letters, and label the members with numbers. This simple
scheme can help you clearly recognize effects that occur in members (such as angles
of twist) and effects that pertain to joints (such as rotation angles of rigid
elements).
• As a rule, when cutting an FBD through a torsion member, assume that the internal
torque is positive, as detailed in Section 6.6. The consistent use of positive internal
torques along with positive angles of twist (in Step 3) proves quite effective for many
situations.
This step is unique to statically indeterminate problems. The structure or system should be studied
to assess how the deformations of the torsion members are related to each other. Most of the
statically indeterminate torsion systems can be categorized as either
1. systems with coaxial torsion members or
2. systems with torsion members connected end to end in series.
The relationship between the internal torque and the angle of twist in a torsion member is
expressed by
φ = TL i i
i
JG
i
i
Step 4
Step 5
Compatibility
Equation(s)
Solve the
Equations
As a practical matter, writing down torque–twist relationships for the torsion members is a helpful
routine at this stage of the calculation procedure. These relationships will be used to construct the
compatibility equation(s) in Step 4.
The torque–twist relationships (from Step 3) are incorporated into the geometric relationship of
member angles of twist (from Step 2) to derive a new equation, which is expressed in terms of the
unknown internal torques. Together, the compatibility and equilibrium equations provide sufficient
information to solve for the unknown variables.
The compatibility equation(s) and the equilibrium equation(s) are solved simultaneously. While
conceptually straightforward, this step requires careful attention to calculation details, such as sign
conventions and consistency of units.