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MECH 321 - Solid Mechanics II

Week 2, Lecture 3

Thermal Stress and

Stress Concentrations

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Thermal Stresses

A change in temperature can cause a material to change its dimensions.

In general if the temperature increases, the material expands, if the temperature decreases, the material contacts.

Ordinarily, the change in dimension is linearly related to the change in temperature.

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Thermal Stresses

Because of the linear relationship between the change in temperature and the change in dimensions, we can calculate the change in length of a member using the formula...

TLT Δ=αδ

Where α - Linear coefficient of thermal expansion (determined experimentally)

TΔL

Tδ

- Change in temperature- Original length of the member

- Change in length of the member

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Thermal Stresses

If the temperature or the coefficient of thermal expansion varies alone the length of the member, we need to integrate along the length of the member.

∫ Δ=L

T Tdx0αδ

If there is more than one distinct material making up the length of the member, we can sum the individual materials together.

∑=

Δ=N

iiiT TL

1αδ

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Thermal Stresses

If the ends of the member are not constrained, the member will simply expand.

However, if the ends are constrained, the member must be treated as statically indeterminate.

These members are the ones that develop thermal stresses

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Example 4.12The rigid (does not bend) beam is fixed to the top of the three posts made of A-36 steel and 2014-T6 aluminum. The posts each have a length of 250 mm when no load is applied to the bar, and the temperature is T1 = 20°C. Determine the force supported by each post if the bar is subjected to a uniform distributed load of 150 kN/m and the temperature is raised to T2 = 80°C.

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Stress Concentrations• Stress concentrations occur when cross-sectional area

changes.• Maximum stress is determined using a stress concentration

factor, K, which is a function of geometry.

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• The stress concentration factor K is a ratio of the maximum stress to the average stress acting at the smallest cross section; i.e.

Stress Concentrations

avg

Kσσmax=

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Stress Concentrations

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Stress Concentrations

avg

Kσσmax=

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Example 4.14The steel strap is subjected to an axial load of 80 kN. Find the maximum normal stress developed in the strap and the displacement of one end of the strap with respect to the other end. The steel has a yield stress of σY = 700 MPa, and Est = 200 GPa.

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Next Time

Inelastic (Plastic) Deformation – Axial Loads