School

Texas A&M UniversityDepartment

Aerospace EngineeringCourse Code

AERO 306Professor

John WhitcombStudy Guide

FinalThis

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Linear Approximation for Two-force Member

This section derives a general linear approximation of the deformation of two-force

members, such as springs and truss members. From earlier classes, you have seen that

even simple truss structures contain a fair number of members. If we do not assume linear

behavior, even simple truss structures would be very difficult to analyze and such

analysis would be well beyond the scope of this class. Fortunately, the linear

approximation is generally quite accurate.

There are various ways to do the derivation. My preferred way is on the next page in

handwritten notes.

Definitions

• Displacements at each = u, v

• Relative x-displacement =

21

u u u=−

• Relative y-displacement =

21

v v v=−

•

21

x x x = −

21

y y y = −

Here are expressions for the length of the member before and after deformation.

( ) ( ) ( ) ( )

1/2 1/2

2 2 2 2

0

L x y L x u y v

= + = + + +

If you multiply out the expression for L, you will find some of the terms combine to

equal Lo. Hence, the expression for L becomes

1/ 2

2 2 2

022L L xu y v u v

= + + + +

Now the challenge is to determine how much the length changes when the displacements

are very small.

Let's get an approximation for L by using a Taylor series expansion

1/2 1/2

22

0 0 0

0

00

0

0, 0

0, 0

( , ) (0,0)

11

22

22

( , ) cos sin

where the angle is determined from the initial (undeformed) state

||

uv

uv

LL

L u v L u v

uv

L x L u y L v

xy

L u v

LL

L u v L u v

−−

==

==

= + +

= + +

= + +

= + +

2 1 2 1

That is, the change in length is simply cos sin

or ( ) cos (v v ) sin

This could also be expressed as the dot product

(relative displacement vector) (un

uv

uu

+

− + −

•it vector parallel to truss member)

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