Normal Load (Axial load): Load is perpendicular to the supporting material.
Axial Load In
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Transcript of Axial Load In
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2002 The McGraw-Hill Companies, Inc. All rights reserved.
irdition Beer Johnston DeWolf
MECHANICS OFMATERIALS
Third Edition
Ferdinand P. Beer E. Russell Johnston, Jr.John T. DeWolf
Lecture Notes:J. Walt Oler Texas Tech University
CHAPTER
2002 The McGraw-Hill Companies, Inc. All rights reserved.
2Stress and Strain Axial Loading
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2002 The McGraw-Hill Companies, Inc. All rights reserved.
irdition Beer Johnston DeWolf
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Learning Outcomes
Students will be able to:
a)To understand the strength of the materials as mechanical properties.
b)Determine the elastic deformation of axially loaded member
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2002 The McGraw-Hill Companies, Inc. All rights reserved.
irdition Beer Johnston DeWolf
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Stress & Strain: Axial Loading
Suitability of a structure or machine may depend on the deformations inthe structure as well as the stresses induced under loading. Staticsanalyses alone are not sufficient.
Considering structures as deformable allows determination of member forces and reactions which are statically indeterminate .
Determination of the stress distribution within a member also requiresconsideration of deformations in the member.
Chapter 2 is concerned with deformation of a structural member under axial loading. Later chapters will deal with torsional and pure bendingloads.
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2002 The McGraw-Hill Companies, Inc. All rights reserved.
irdition Beer Johnston DeWolf
9.2 The StressStrain Diagram
Conventional StressStrain DiagramConventional StressStrain DiagramNominal or engineering stress is
obtained by dividing the applied load P
by the specimens original cross-sectional area.
Nominal or engineering strain isobtained by dividing the change in thespecimens gauge length by thespecimens original gauge length.
0 A P
=
0 L
=
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2002 The McGraw-Hill Companies, Inc. All rights reserved.
irdition Beer Johnston DeWolf
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Normal Strain
strainnormal
stress
==
==
L
A P
L
A P
A P
=
==22
L L
A P
==
=
2
2
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2002 The McGraw-Hill Companies, Inc. All rights reserved.
irdition Beer Johnston DeWolf
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Stress-Strain Test
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irdition Beer Johnston DeWolf
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Stress-Strain Diagram: Ductile Materials
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irdition Beer Johnston DeWolf
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Stress-Strain Diagram: Brittle Materials
i d
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irdition Beer Johnston DeWolf
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Hookes Law: Modulus of Elasticity
Below the yield stress
Elasticityof Modulus or ModulusYoungs=
=
E E
Strength is affected by alloying, heattreating, and manufacturing process
but stiffness (Modulus of Elasticity) is not.
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irdition Beer Johnston DeWolf
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Elastic vs. Plastic Behavior
If the strain disappears when thestress is removed, the material issaid to behave elastically .
When the strain does not returnto zero after the stress isremoved, the material is said to
behave plastically .
The largest stress for which thisoccurs is called the elastic limit .
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irdition Beer Johnston DeWolf
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Deformations Under Axial Loading
AE P
E E ===
From Hookes Law:
From the definition of strain:
L
=
Equating and solving for the deformation,
AE PL=
With variations in loading, cross-section or material properties,
=i ii
ii E A L P
ird
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irdition Beer Johnston DeWolf
Example
Determine the deformation of the steel rod shown under thegiven loads.
SOLUTION: Divide the rod into components at
the load application points.
Apply a free-body analysis on each
component to determine theinternal force.
Evaluate the total of the componentdeflections.
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irdition Beer Johnston DeWolf
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Poissons Ratio
For a slender bar subjected to axial loading:
0=== z y x
x E
The elongation in the x-direction isaccompanied by a contraction in the other
directions. Assuming that the material isisotropic (no directional dependence),
0= z y
Poissons ratio is defined as
x
z
x
y
===
strainaxial
strainlateral
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irdition Beer Johnston DeWolf
The Shear StressStrain Diagram
For most engineering materials the elasticbehaviour is linear , so Hookes Law for shear applies.
3 material constants, v , E , and G are actuallyrelated by the equation
G=
G = shear modulus of elasticity or the modulus of rigidity
( )v E
G+
=12