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

5.5 Statically Indeterminate Axially Loaded members
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STATICALLy INdETERMINATE
AxIALLy LOAdEd MEMbERS
In many simple structures and mechanical systems constructed with axially loaded members,
it is possible to determine the reactions at supports and the internal forces in the individual
members by drawing free-body diagrams and solving equilibrium equations. Such
structures and systems are classified as statically determinate.
For other structures and mechanical systems, the equations of equilibrium alone are not
sufficient for the determination of axial forces in the members and reactions at supports. In
other words, there are not enough equilibrium equations to solve for all of the unknowns in
the system. These structures and systems are termed statically indeterminate. Structures of
this type can be analyzed by supplementing the equilibrium equations with additional equations
involving the geometry of the deformations in the members of the structure or system.
The general solution process can be organized into a five-step procedure:
Step 1 — Equilibrium Equations: Equations expressed in terms of the unknown axial
forces are derived for the structure on the basis of equilibrium considerations.
Step 2 — Geometry of Deformation: The geometry of the specific structure is evaluated
to determine how the deformations of the axial members are related.
Step 3 — Force–Deformation Relationships: The relationship between the internal
force in an axial member and its corresponding elongation is expressed by Equation (5.2).
Step 4 — compatibility Equation: The force–deformation 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 axial forces.
Step 5 — Solve the Equations: The equilibrium equations and the compatibility equation
are solved simultaneously to compute the unknown axial forces.
In engineering literature,
force–deformation
relationships are also called
constitutive relationships
since these relationships
idealize the physical properties
of the material—in other words,
the constitution of the material.
The use of this procedure to analyze a statically indeterminate axial structure is illustrated
in the next example.
As discussed in Chapters 1 and 2, it is convenient to use the notion of a rigid element to
develop axial deformation concepts. A rigid element (such as a bar, a beam, or a plate)
represents an object that is infinitely stiff, meaning that it does not deform in any manner.
While it may translate or rotate, a rigid element does not stretch, compress, skew, or bend.
ExAmpLE 5.5
A 1.5 m long rigid beam ABC is supported by three axial members, as shown in the figure
that follows. A concentrated load of 220 kN is applied to the rigid beam directly under B.
The axial members (1) connected at A and at C are identical aluminum alloy [E =
70 GPa] bars, each having a cross-sectional area A 1 = 550 mm 2 and a length L 1 = 2 m.
Member (2) is a steel [E = 200 GPa] bar with a cross-sectional area A 2 = 900 mm 2 and a
length L 2 = 2 m. All members are connected with simple pins.
If all three bars are initially unstressed, determine
(a) the normal stresses in the aluminum and steel bars, and
(b) the deflection of the rigid beam after application of the 220 kN load.
Plan the Solution
A free-body diagram (FBD) of rigid beam ABC will be drawn, and from this sketch, equilibrium
equations will be derived in terms of the unknown member forces F 1 and F 2 .