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Deformation Due To Axial Loading

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Equation

δ =

PL AE

Nomenclature

symbol	description
δ	elongation of the rod
P	force
L	original length
A	cross-sectional area
E	modulus of elasticity
σ	stress
ε	normal strain

Explanation

Consider a homogeneous rod of length L and uniform cross section of area A subjected to a centric axial load P. If the resulting axial stress:

σ =

P A

does not exceed the proportional limit of the material, Hooke's law may be used:

σ = Eε

rewriting for the strain ε:

ε =

σ E

substituting P/A for σ:

(Eq1)

ε = P AE

As strain ε = δ/L:

(Eq2)      δ = εL

Substituting Eq1 into Eq2:

(Eq3)

δ = PL AE

Conditions for use of Eq3

the rod is homogeneous (constant E)

rod has uniform cross-sectional area A

the rod is loaded at its ends

An easy way to remember the equation, if A and E are switched around is plea, or flea if F is used for force instead of P.

If the rod is loaded at other points, or if it consists of several portions of various cross sections and possible of different materials, it must be divided into component parts that satisfy individually the required conditions for the application of Eq3. Denoting respectively by P_i, L_i, A_i, and E_i the internal force, length, cross-sectional area, and modulus of elasticity corresponding to part i, the deformation of the entire rod is expressed as:

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Related
▪ P - Deformation of a Component of Two Materials
▪ P - Statically Indeterminate Tension in Wires with Deformation