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MAE 314
Solid Mechanics
Chapter 2
Dr. H.Y. S. Huang
Mechanical & Aerospace Engineering Department
North Carolina State University
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Stress-Strain Relation
σ = Eε
P
δ
=E
A
L
PL
⇒ δ=
AE
normalization
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Example: Deformation
The rigid yokes B & C are security fastened to the 2-in square
steel (E = 30,000 ksi) bar AD. Determine
(a) The deformation of bar AB.
(b) The deformation of bar BC.
(c) The deformation of the complete bar.
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In-class activity: deformation
A single axial load of magnitude P = 15 kips is applied at end C
of the steel rod ABC. Knowing that E = 30 × 106 psi, determine
the diameter d of portion BC for which the deflection of point
C will be 0.05 in.
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Example: Deformation
A tie rod and a pipe strut are used to support a 50-kN load. The
cross-sectional areas are 650 mm2 for the rod AB and 925 mm2
for pipe strut BC. Both members are made of structural steel
that has a Young’s modulus of 200 GPa. Determine (a) the
normal stresses in AB and BC. (b) the lengthening or shortening
of AB and BC.
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In-class activity: Deformation
Two tie rods are used to support a load P = 16-kip. Rod AB is
made of an aluminum alloy with a Young’s modulus of 10,600
ksi. A length of 80-in and a cross-sectional area of 0.6 in2. Rod
BC is made of structural steel with a Young’s modulus of 29,000
ksi, a length of 160 in and a cross-sectional area of 1.25 in2.
Determine the elongation of rod AB and BC.
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Statically Indeterminate
a. Equilibrium equation:
How to draw FBD?
How to determine the internal forces?
b. Deformation equation
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Reinforcement
•Most concrete columns are reinforced with steel rods.
•These two materials work together in supporting the applied
loads.
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Example: rigid supports at both ends
A rigid plate C is used to transfer a 20-kip load P to a steel (E =
30,000 ksi) rod A and to an aluminum alloy (E = 10,000 ksi) pipe
B. The supports at the top of the rod and bottom of the pipe are
rigid. The cross-sectional areas of A & B are 0.8-in2 & 3-in2,
respectively. Determine
(a) the normal stresses in A & B.,
(b) the displacement of plate C.
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In-class activity: rigid supports at both ends
A hollow brass (E = 100 GPa) tube A with an outside diameter
of 100-mm & an inside diameter of 50-mm is fastened to a 50mm diameter-steel (E = 200 GPa) rod B. The supports at the
top and bottom of the assembly & the collar C used to apply
the 500 kN load P are rigid. Determine the deflection of the
collar C.
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Example: Reinforcement
The aluminum post is reinforced with a brass core. If this
assembly supports an axial compressive load of P = 9-kip, applied
to the rigid cap, determine the normal stresses in the aluminum
and the brass. Eal = 10x103 ksi, and E br = 15x103 ksi
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In-class activity: Reinforcement
The 7.5 x 7.5x 20-in. oak (E = 1800 ksi) block was reinforced
by bolting two 2 x 7.5 x 20-in. steel (E = 29,000 ksi) plates to
opposite sides of the block. If the stresses in the wood &
the steel are to be limited to 4.6 ksi & 22 ksi, respectively.
Determine the
(a) maximum axial load P that can be applied to the
reinforced block.
(b) The change in length of the block when the load P is
applied.
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Example: indeterminate problem
The rigid bar CDE is horizontal before the load P is applied. Tie rod A is a
hot-rolled steel (E = 210 GPa) bar with a length of 450 mm and a crosssectional area of 300 mm2. Post B is an oak timber (E = 12 GPa) with a
length of 375 mm and a cross-sectional area of 4500 mm2. After the 225kN load P is applied, determine
(a) The normal stresses in bar A and post B.
(b) The vertical displacement of point D.
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In-class activity: indeterminate problem
A pin-connected structure is loaded and supported. Member CD is rigid
and is horizontal before the load P is applied. Member A is an aluminum
alloy bar with a modulus of elasticity of 10,600 ksi and a cross-sectional
area of 2.25 in2. Member B is a stainless steel bar with a modulus of
elasticity of 28,000 ksi and a cross-sectional area of 1.75 in2. After the
load is applied to the structure, determine
(a) The normal stresses in bars A and B.
(b) The vertical displacement of point D.
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Temperature Effects
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Temperature Effects
a. Deformation equation (temp
effects)
b. Deformation equation (force
effects)
δ Temp = α (ΔT )L
PL
δ Force =
AE
c. Equilibrium equation
How to draw FBD?
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Example: Temperature Effects
A rod consisting of 2 cylindrical portions AB & BC is restrained
at both ends. Portion AB is made of brass (Eb = 15 x 106 psi,
αb = 11.6 x 10-6/oF) and portion BC is made of steel (Es = 29 x
106 psi, αs = 6.5 x 10-6/oF). Determine the normal stresses
induced in portion AB and BC by a temperature rise of 90 oF.
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In-class activity: Temperature effects
A rod consisting of 2 cylindrical portions AB & BC is restrained
at both ends. Portion AB is made of steel (Es = 29 x 106 psi, αs
= 6.5 x 10-6/oF) and portion BC is made of brass (Eb = 15 x 106
psi, αb = 11.6 x 10-6/oF). Determine the normal stresses
induced in portion AB by a temperature rise of 90 oF.
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Example: force and temp
Bar B of the pin-connected system is made of an aluminum alloy (Ea = 70
GPa, Aa = 300 mm2, and αa = 22.5(10-6)/°C) and bar A is made of a
hardened carbon steel (Es = 210 GPa, As = 1200 mm2, and αs =
11.9(10-6)/°C). Bar CDE is to be considered rigid. When the system is
unloaded at 40°C, bars A and B are unstressed. After the load P is
applied, the temperature of both bars decreases to 15°C.
Determine
(a) The normal stresses in bars A and B.
(b) The vertical displacement (deflection) of pin E.
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in-class activities: Force + temp
A 40-kip load P will be supported by a structure consisting of a rigid
bar A, two aluminum alloy (Ea = 10,600 ksi and αa = 12.5(10-6)/°F) bars
B, and a stainless steel (Es = 28,000 ksi and αs = 9.6(10-6)/°F) bar C. The
bars are unstressed when the structure is assembled at 72°F. Each bar
has a cross- sectional area of 2.00 in2. Determine the normal stresses
in the bars after the 40-kip load is applied and the temperature is
increased to 240°F.
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Axial loading - elongation
•Axial loading along x direction results in elongation in the xdirection.
•But what happens in y & z directions?
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Axial loading - elongation
•Axial loading along x direction results in elongation in the xdirection.
•But what happens in y & z directions?
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εy
εz
υ=− =−
εx
εx
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Example: Poisson Ratio
A 500-mm-long, 16-mm-diameter rod is observed to increase
in length by 300 μm, and to decrease in diameter by 2.4 μm
when subjected to an axial 12-kN load. Determine the Young’s
modulus and Poisson ratio of the rod.
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In-class activity: Poisson ratio
A 20-mm-diameter rod is subjected to an axial force of P = 6
kN. Knowing that an elongation of 14-mm and a decrease in
diameter of 0.85-mm are observed in a 150-mm length.
Determine the Young’s modulus and the poisson ratio for the
rod.
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Chapter 2 Summary
• Stress-Strain curve
• Hookes’ law; Young’s modulus.
• Force to Deformation, Deformation to Force
• Rigid ends and Reinforcement
• Indeterminate problems
• Temperature effects
• Poisson ratio
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