Axial Load - Mechanics of Materials

Saint-Venant's Principle: the localized effects caused by a load that acts on a member will dissipate out from where the load is applied. The distance for the effects to dissipate is at least the distance of width of the cross section where the load is applied.

Relative displacement of one end to the other. For a member with varying force and cross section.

δ = ∫P(x)dx / A(x)E

• δ is the displacement of one point on the member with respect to the other point
• P(x) is the internal axial force at the section, located a distance x from the end
• L is the original distance between the 2 points
• A(x) is the cross sectional area as a function of x
• E is the modulus of elasticity

For constant load and cross section:

δ = PL / AE


If a member has different axial forces, or abrupt changes in the cross section and/or the modulus of elasticity:

δ = ∑ PL /AE each of the end displacements are added up for the different sections

The force and displacements are positive if it causes tension and elongation.
The force and displacements are negative if it causes compression and contraction.


Statically Indeterminate: when a member is fixed at both ends so that there are more reactions and there are equations of equilibrium. Therefore, can't solve the reactions.

A compatibility condition is therefore applied and states that since both ends are fixed there is going to be no relative displacement between the two fixed points(points 1 and 2) on the member.
δ1/2 = 0


Thermal Stress: a material can have a change in its dimension if there is a temperature change.
Temperature change is linearly related to the material expansion or contraction.

δT = α ΔT L

• δT is the change in length of the member
• α is the coefficient of thermal expansion. Values are found in tables.
• ΔT is the change in temperature
• L is the original length of the member

Thermal displacements are constrained by supports for a statically indeterminate member.



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