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Thermal Effects and Indeterminacy: Notation

Thermal effects can cause indeterminacy in structures. When a material experiences a change in temperature, it will expand or contract based on its coefficient of thermal expansion. If the material is restrained from moving freely, thermal strains will induce internal stresses. The superposition method can be used to analyze these statically indeterminate problems. We treat the restraint as an applied force and impose the geometric restriction of the restraint. For example, a restrained bar being heated will push against its supports, creating compressive stresses within the bar as the supports push back and restrain its expansion.

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78 views4 pages

Thermal Effects and Indeterminacy: Notation

Thermal effects can cause indeterminacy in structures. When a material experiences a change in temperature, it will expand or contract based on its coefficient of thermal expansion. If the material is restrained from moving freely, thermal strains will induce internal stresses. The superposition method can be used to analyze these statically indeterminate problems. We treat the restraint as an applied force and impose the geometric restriction of the restraint. For example, a restrained bar being heated will push against its supports, creating compressive stresses within the bar as the supports push back and restrain its expansion.

Uploaded by

JR Zuniga
Copyright
© Attribution Non-Commercial (BY-NC)
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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ARCH 331

Note Set 6.3

F2009abn

Thermal Effects and Indeterminacy


Notation: A E f L P P

= area = modulus of elasticity or Youngs modulus = stress = length = name for axial force vector = name of reaction force = coefficient of thermal expansion for a material

t P restr T

= = = = =

thermal strain (no units) elongation or length change elongation due to axial load restrained length change elongation due or length change due to temperature = change in temperature

Thermal Strains Physical restraints limit deformations to be the same, or sum to zero, or be proportional with respect to the rotation of a rigid body. We know axial stress relates to axial strain:

PL which relates to P AE

Deformations can be caused by the material reacting to a change in energy with temperature. In general (there are some exceptions): Solid materials can contract with a decrease in temperature. Solid materials can expand with an increase in temperature. The change in length per unit temperature change is the coefficient of thermal expansion, . It has units of Thermal Strain:

or

C and the deformation is related by: T = (T )L

T = T

There is no stress associated with the length change with free movement, BUT if there are restraints, thermal deformations or strains can cause internal forces and stresses.

ARCH 331

Note Set 6.3

F2009abn

How A Restrained Bar Feels with Thermal Strain 1. Bar pushes on supports because the material needs to expand with an increase in temperature. 2. Supports push back. 3. Bar is restrained, cant move and the reaction causes internal stress.

Superposition Method If we want to solve a statically indeterminate problem that has extra support forces: We can remove a support or supports that makes the problem look statically determinate Replace it with a reaction and treat it like it is an applied force Impose geometry restrictions that the support imposes

For Example:

T = ( T )L

p =

PL AE

P + T = 0
P = ( T )L /

PL + ( T )L = 0 AE f = P = ( T )E A

AE = (T )AE / L

ARCH 331

Note Set 6.3

F2009abn

Example 1 (pg 228)

ALSO: If the beam is anchored to a concrete slab, and the steel sees a temperature change of 50 F while the concrete only sees a change of 30 F, determine the compressive stress in the beam. c = 5.5 x 10-6 / F s = 6.5 x 10-6 / F Ec = 3 x 106 psi Es = 29 x 106 psi

ARCH 331

Note Set 6.3

F2009abn

Example 2

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