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

106
AxIAL dEFORMATION
The five-step procedure demonstrated in the previous example provides a versatile
method for the analysis of statically indeterminate structures. Additional problemsolving
considerations and suggestions for each step of the process are discussed in the
table that follows.
Solution method for Statically Indeterminate Axial Structures
Step 1 Equilibrium Equations Draw one or more free-body diagrams (FBDs) for the structure, focusing on the joints
that connect the members. Joints are located wherever (a) an external force is applied,
(b) the cross-sectional properties (such as area or diameter) change, (c) the properties of
the material (i.e., E) change, or (d) a member connects to a rigid element (such as a rigid
bar, beam, plate, 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.
Step 2
Geometry of
Deformation
Comments:
• Label the joints with capital letters and label the members with numbers. This simple
scheme can help you to clearly recognize effects that occur in members (such as
deformations) and effects that pertain to joints (such as deflections of rigid elements).
• As a rule, assume that the internal force in an axial member is tensile, an assumption
that is consistent with a positive deformation (i.e., an elongation) of the axial
member. This practice will make it easier for us to incorporate the effects of
temperature change into our analyses of the deformations in structures made up of
axial members. Temperature change effects will be discussed in Section 5.6.
This step is unique to statically indeterminate problems. The structure or system should
be scrutinized to assess how the deformations in the members are related to each other.
An equilibrium equation and the corresponding geometry-of-deformation equation
must be consistent. This condition means that, when a tensile force is assumed for a
member in an FBD, a tensile deformation must be indicated for the same member in the
deformation diagram.
Most of the statically indeterminate axial structures fall into one of three general
configurations:
1. Coaxial or parallel axial members.
2. Axial members connected end to end in series.
3. Axial members connected to a rotating rigid element.
Characteristics of these three categories are discussed in more detail shortly.
Step 3
Force–Deformation
Relationships
The relationship between the internal force in, and the deformation of, axial member i is
expressed by
FL i i
δ i =
AE
As a practical matter, writing down force–deformation relationships for the axial
members at this stage of the solution is a helpful routine. These relationships will be
used to construct the compatibility equation(s) in Step 4.
i
i
Step 4 Compatibility Equation The force–deformation relationships (from Step 3) are incorporated into the geometric
relationship of member deformations (from Step 2) to derive a new equation, which is
expressed in terms of the unknown member forces. Together, the compatibility and
equilibrium equations provide sufficient information to solve for the unknown variables.
Step 5 Solve the Equations The compatibility equation 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.