Mechanics of Materials, Ch01-Axial Load

Russell C. Hibbeler

Chapter 4: Axial Load

Elastic Deformation of an Axially Loaded Member

� Using Hooke’s law and the definitions of stress and

strain, we are able to develop the elastic deformation of a member subjected to axial loads.of a member subjected to axial loads.

� Suppose an element subjected to loads,

( )( )

and P x dδ

εA x dx E

σσ = = =

( )( )∫=

L

ExA

dxxP

0

δ

= small displacementδ

( )( )

P x dxd

A x Eδ =

© 2008 Pearson Education South Asia Pte Ltd© 2008 Pearson Education South Asia Pte Ltd

Chapter 4: Axial LoadChapter 4: Axial Load

Mechanics of Material 7Mechanics of Material 7thth EditionEdition

= small displacement

L = original length

P(x) = internal axial force

A(x) = cross-sectional area

E = modulus of elasticity

δ

Constant Load and CrossConstant Load and Cross--Sectional AreaSectional Area

� When a constant external force is applied at each end of the member, end of the member,

Sign ConventionSign Convention

� Force and displacement is positive when tension and

AE

PL=δ




elongation and negative will be compression and contraction.

For small elements, small deflection approximation assumes that higher order terms can be ignored:

2 2 2 2, , , ,dx dy dw dz dwdz




Example 4.4A member is made from a material that has a specific weight

and modulus of and elasticity E. If it is formed into a cone,

find how far its end is displaced due to gravity when it is

suspended in the vertical position.

γ

suspended in the vertical position.

Solution:Radius x of the cone as a function of y is determined by proportion,

yL

rx

L

r

y

x oo == ;

The volume of a cone having a base of radius x and height y is




The volume of a cone having a base of radius x and height y is

3

2

2

2

33y

L

ryxV oππ

==

Since , the internal force at the section becomes

Solution:

( ) 3

2

2

3 ;0 y

L

ryPF o

y

γπ==↑+ ∑

VW γ=

3L

( ) 2

2

2

2y

L

rxyA oπ

π ==

The area of the cross section is also a function of position y,

Between the limits of y =0 and L yields

( ) ( )[ ]( )[ ] (Ans)

3222

LdyLrdyyPL

o

L γγπδ === ∫∫




( )( )

( )[ ]( )[ ] (Ans)

6

3

0

22

0E

L

ELr

dyLr

EyA

dyyP

o

o γγπ

γπδ === ∫∫

Principle of Superposition

� Principle of superposition is to simplify stress and

displacement problems by subdividing the loading into components and adding the results.

� A member is statically indeterminate when equations

of equilibrium are not sufficient to determine the reactions on a member.

Statically Indeterminate Axially Loaded Member




0, 0x B AF F F P= + − =∑

Statically Indeterminate Axially Loaded Member

Equilibrium equation:

/ 0A Bδ =

A ACAC

F L

AEδ = B CB

CB

F L

AEδ

−=

Method of solving 1: Compatibility




0A AC B CBF L F L

AE AE− =

CB ACA B

L LF P and F P

L L

= =

Method of solving 2: Superposition

0AC BPL F L

AE AE− =

Displacement without support at B

Reaction at support to compress bar to original length: remove P

ACB p

PL

AEδ δ= =AF P=




ACB

LF P

L

=

0ACA

LP F P

L

+ − =

ACB ABB

PLF L

AE AEδ

−− = = −

Thermal Stress

� Change in temperature cause a material to change

its dimensions.

Since the material is homogeneous and isotropic,� Since the material is homogeneous and isotropic,

TLT ∆−= αδ

linear coefficient of thermal expansionα

0

L

T Tdxδ α= ∆∫Td T dxδ α= ∆




= linear coefficient of thermal expansion, property of the material

= algebraic change in temperature of the member

= original length of the member

= algebraic change in length of the member

αT∆T

Tδ

*Inelastic Axial Deformation and Residual stresses

� Member may be designed to yield and permanently

deform; this is referred to as being elastic perfectly

plastic or elastoplastic.plastic or elastoplastic.

AdAP Y

A

Yp σσ == ∫




� The yield stress and A is the bar’s cross-sectional

area at section a–a.

CB ACA B

L LF P and F P

L L

= =

Since AC yielded re-calculate FA




P applied in reverse direction

Force P removed:

Residual Stress:

Permanent displacement:




Alternative way:

Mechanics of Materials, Ch01-Axial Load

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