Chapter 11_Temperature Effect

Chapter 11_Temperature Effect - CHAPTER 11 TEMPERATURE...

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1 CHAPTER 11 TEMPERATURE EFFECT Structural components are often subjected to temperature changes during flight operation. high-speed aircraft, especially along the wing leading edge reentry vehicles jet engine combustion chamber and exhaust nozzle spacecraft (bright or dark side relative to the Sun) Temperature change can induce stress as well as structural deformation.
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2 Skin temperature of M 2.4 transport
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3 Strain Induced by Temperature change Consider a rectangular block of isotropic material sitting on a table. Initially, the block is stress free. When the temperature of the block is uniformly raised through heating, the block expands. On the other hand, the block contracts when the temperature is uniformly lowered through cooling. From experimental observation, strain induced by temperature change T Δ can be expressed as 0 0 0 0 0 0 0 0 0 = Δ = = Δ = = Δ = zx zz yz yy xy xx T T T ε α where , called the coefficient of thermal expansion (CTE), is an experimentally determined material constant. The right superscript ‘0’ indicates that the strain is thermally induced. Temperature change does not induce shear strain. The above relationship obtained for a block of finite size under uniform temperature change is assumed to hold for a small material element of infinitesimal volume. Now, consider a uniaxial bar of constant cross-sectional area. The bar is unconstrained and initially stress free. The bar is subjected to temperature change of T Δ as well as axial stress. Then, T E xx xx Δ + = σ 1 where xx : total strain xx E 1 : strain induced by stress T Δ : thermally induced strain x y z x
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4 For a structural component with geometric constraints, temperature change induces stress because it cannot freely expand or contract. Let’s look at the following simple example to appreciate this. Example : Temperature of a uniaxial bar constrained by rigid walls is raised by ) 0 ( > Δ T . The bar is allowed to expand freely in the y and z direction, but it cannot expand in the axial direction. Then, 0 1 = Δ + = T E xx xx α σ ε Δ = T E xx compressive stress for ) 0 ( > Δ T z x rigid wall rigid wall
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5 Resultant force and moments with temperature effect For a slender structure undergoing extension and bending, Axial stress σ xx is related to axial strain ε xx as follows: 12 3 () ( ) xx xx ET E C C z C y T εα α =− Δ = + + Δ (1) where 22 00 0 3 , uwv CC C x xx ∂∂ == = (2) Using the above expression, the resultant force and moments can be expressed as follows: 3 ( ) xx xx F d AE T d C C z C y T d A σε Δ = + + Δ ∫∫ 3 ()( ) yx x x x M
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Chapter 11_Temperature Effect - CHAPTER 11 TEMPERATURE...

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