School

Texas A&M UniversityDepartment

Aerospace EngineeringCourse Code

AERO 306Professor

John WhitcombStudy Guide

FinalThis

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Chapter 2

Approximate Analysis of Uniaxial

Bars and Beams

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The goal now is to develop a systematic procedure for obtaining approximate solutions. Uniaxial

bars and beams will be used to illustrate the basic concepts. We will begin with the uniaxial bar.

Analysis of a Uniaxial Rod

Solution of differential equations

To refresh your memory, we will begin by solving a uniaxial bar problem using just differential

equations.

The equilibrium equation can be derived by imposing equilibrium on a differential slice of the

bar as follows.

Summing forces gives

( )

0

0

LL

F F dF fdx

dF f

dx

− + + + =

+=

This equation is valid for any uniaxial rod even if the cross sectional area or material properties

are functions of x. Constitutive relations enter the picture when we express F in terms of

displacement. For example, if the material is thermoelastic, then we would substitute

( )

F A AE T

= = −

Warning:

F = internal force, not applied force (Remember Cauchy’s formula)

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Now let’s solve a simple problem.

1

2

12

2

12

Solve differential equation by integrating two times

Assume: EA constant

f=constant

no thermal effects

2

1

2

d du

EA f

dx dx

du

EA fx C

dx

x

EAu f C x C

x

u f C x C

EA

=

=−

= − +

= − + +

= − + +

Now impose the boundary conditions

(0) 0

xL

du

u and EA P

dx =

==

to determine the

integration constants.

( )

( )

22

1

1

2

1

1) (0) 0 (0 0 ) 0 0

1

2)

1 This is the exact solution.

2

xL

u C C

EA

du

EA P EA fL C P

dx EA

C P fL

fx

u P fL x

EA

=

= = + + = = =

= = − + =

= = +

= − + +

Further thought

Assume some variation of temperature along the length and re-solve this problem.

Derivation of Virtual Work

There are various ways to derive the virtual work equation for a uniaxial bar. This document

gives one version here and two more in the appendix to this file. First consider a differential

element of the bar in the equilibrium state.

-F

F + dF

dx

f

u

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