AXIALLY LOADED
MEMBERS
OUTLINE
• Axially Loaded Members
• Changes in lengths of axially loaded
members
• Bars with intermediate axial loads
• Bars consisting of prismatic segments
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In our previous lessons:
• Normal stress and normal strain
• Mechanical properties of materials
• Elasticity and Plasticity
• Hooke’s Law and Poisson’s ratio
• Shear stress, bearing stress and shear strain
• Shear and bearing capacity of a connection
• Allowable stresses and allowable loads
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OUTCOME
At the end of the lecture:
• students should understand what axially loaded
members are and
• their useful properties necessary for analysis and
design
• Should know techniques for determining changes
in lengths of axially loaded members in uniform
and non-uniform conditions
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Axially Loaded Members
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Axially
loaded
members
are
structural
components subjected only to tension or
compression.
Solid bars with straight longitudinal axes are the
most common type, although cables and coil
springs also carry axial loads.
Examples of axially loaded bars are truss
members, connecting rods in engines, spokes in
bicycle wheels, columns in buildings, and struts
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in aircraft
engine mounts.
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Calculation of changes in
lengths of axially loaded
members
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• When a load is applied along the axis of a
spring, as shown in Fig.1, the spring gets
longer or shorter depending upon the
Springs
direction of the load.
• If the load acts away from the spring, the
spring elongates and we say that the
spring is loaded in tension. If the load acts
toward the spring, the spring shortens
and we say it is in compression
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Fig.1 Spring subjected to an
axial load P
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• The elongation of a spring is pictured in Fig. 2,
where the upper part of the figure shows a spring in
its natural length L, and the lower part of the figure
shows the effects of applying a tensile load.
• Under the action of the force P, the spring lengthens
by an amount δ and its final length becomes L + δ.
• If the material of the spring is linearly elastic, the
load and elongation will be proportional:
Fig. 2: Elongation of an
axially loaded spring
where “k and f” are constants of proportionality
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The constant ‘k’ is called the stiffness of the spring and is defined as the
force required to produce a unit elongation, that is, k = P/δ.
Similarly, the constant ‘f’ is known as the flexibility and is defined as the
elongation produced by a load of unit value, that is, f = δ/P.
From the above, it is apparent that the stiffness and flexibility of a spring
are the reciprocal of each other:
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A prismatic bar is a structural
member having a straight longitudinal
axis
and
constant
cross
section
throughout its length.
Prismatic Bars
Fig.3: Typical cross
sections of structural
members
Structural members may have a
variety of cross-sectional shapes, such
Fig. 4: Prismatic bar of circular
cross section
as those shown in Fig. 3. To analyze
this behavior, let us consider the
prismatic bar shown in Fig. 4
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The elongation δ of a prismatic bar subjected to a tensile
load P is shown in Fig. 5. If the load acts through the
centroid of the end cross section, the uniform normal
stress at cross sections away from the ends is given by
the formula
σ = P/A,
where A is the cross-sectional area.
Fig. 5: Elongation of a prismatic bar in
tension
Furthermore, if the bar is made of a homogeneous
material, the axial strain is
ε = δ/L,
where δ is the elongation and L is the length of the bar.
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Assuming the material is linearly elastic,
Combining these basic relationships, we get the following equation for
the elongation of the bar:
The product EA is known as the axial rigidity of the bar.
The stiffness and flexibility of a prismatic bar are defined in the same way as
for a spring.
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Cables are used to transmit large tensile forces, for
Cables
example raising elevators, guying towers, and
supporting suspension bridges.
Unlike springs and prismatic bars, cables cannot resist
compression.
When determining the elongation of a cable from
the effective modulus should be used for E and the
effective area should be used for A.
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1. Draw the free-body diagram for the
Techniques for
calculating
changes in
length
part or the given body as necessary
2. Apply equations of equilibrium
3. Draw displacement diagram
4. Use
formulas
for
calculating
changes in length
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Example 1
The device shown in Fig. 6 consists of a horizontal
beam ABC supported by two vertical bars BD and CE.
Bar CE is pinned at both ends but bar BD is fixed to
the foundation at its lower end. The distance from A to
B is 450 mm and from B to C is 225 mm. Bars BD and
Fig. 6: Horizontal beam ABC
CE have lengths of 480 mm and 600 mm, respectively, supported by two vertical bars
and their cross-sectional areas are
and Assuming that beam ABC is rigid:
(a) Find the maximum allowable load
respectively. The bars are made of steel
if the displacement of point A is
limited
to
1.0
mm
having a modulus of elasticity E 205 GPa.
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Bars with Intermediate
Axial Loads
Fig. 7: Bar with external loads
acting at intermediate points
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When a prismatic bar of linearly elastic material is
loaded only at the ends, we can obtain its change in
length from the equation δ = PL/EA.
Suppose, for instance, that a prismatic bar is loaded
by one or more axial loads acting at intermediate
points along the axis (Fig. 7).
We can determine the change in length of this bar by
adding
algebraically
the
elongations
shortenings of the individual segments.
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and
Fig. 7: Bar with external loads
acting at intermediate points
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The procedure is as follows:
1. Identify the segments of the bar:
Segments
AB,
BC, and
CD as
segments 1, 2, and 3, respectively.
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2. Determine the internal axial
forces
in segments
1, 2, and 3, respectively, from the
free-body diagrams of Figs. 7b, c,
and d. By summing forces in the
vertical direction
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Fig. 7 (b), (c), and (d) free-body diagrams
showing the internal axial forces N1, N2,
and N3
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3. Determine the changes in the
lengths of the segments from
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Fig. 7 (b), (c), and (d) free-body diagrams
showing the internal axial forces N1, N2,
and N3
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4. Add δ1, δ2, and δ3 to obtain δ,
the change in length of the entire
bar:
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Fig. 7 (b), (c), and (d) free-body diagrams
showing the internal axial forces N1, N2,
and N3
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Bars Consisting of Prismatic
Segments
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Fig. 8: Bar consisting of prismatic
segments having different axial
forces, different dimensions,
and different materials
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Fig. 8: Bar consisting of prismatic
segments having different axial
forces, different dimensions,
and different materials
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Thank you
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