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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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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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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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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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Normal Strain

strainnormal

stress

==

==

L

A P

L

A P

A P

=

==22

L L

A P

==

=

2

2

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Stress-Strain Test

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Stress-Strain Diagram: Ductile Materials

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Stress-Strain Diagram: Brittle Materials

i d

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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.

i d

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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 .

i d

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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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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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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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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