LECTURE+7+COE-3001-A

Structures statically determinate

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Unformatted text preview: PERIENCE INTERNAL STRESS DUE TO A UNIFORM TEMERATURE CHANGE. 1/17/13 M. Mello/Georgia Tech Aerospace 5 Example 2- 7: LINEAR ELASTIC BAR BETWEEN RIGID SUPPORTS thermal strain mechanical strain GIVEN: ²་  PrismaOc bar of length (L) held between immovable supports ²་  Temperature of the bar raised uniformly by (ΔT) ²་  α = coefficient of thermal expansion (CTE) of bar material DETERMINE: ²་  Thermal stress (σT) developed in the bar ²་  Assume bar is made of linear elasOc material. = + SOLUTION STRATEGY: ①  Apply some basic mechanical intuiOon before analysis ²་  AS TEMP INCREASES, BAR ATTEMPTS TO ELONGATE ²་  MOVEMENT RESTRICTED BY RIGID SUPPORTS ²་  BAR SUBJECTED TO COMPRESSIVE STRESS ②  EQUILIBRIUM EQUATION: ⌃Fvert = 0 RB RA = 0 (2- 17) ③  COMPATIBILITY EQUATION: ab ab ab =0 = T hermal = 1/17/13 T M NOTE: RB, RA, REPRESENT STRESS MAGNITUDES tendency to increase bar length due to temp. increase OPPOSED BY M echanical =0 =0 tendency to contract bar length due to compressive stress of reacOon forces M. Mello/Georgia Tech Aerospace 6 Example 2- 7 (cont.) SOLUTION: ③  COMPATIBILITY EQUATION (restated): =0 ab = T hermal thermal strain mechanical strain ab ab T = T = ↵ ( T )L (temperature- displacement relaOon) ↵ ( T )L RA = M M echanical =0 =0 M = RA L RB L = EA EA (force- displacement relaOon) = + RA L =0 EA OF EA↵( T ) (MAGNITUDE = R REACTION FORCE RA. (RA B) T = E ↵( 1/17/13 T ) (2- 18) NOTE: IT WAS REASONED AT THE OUTSET THAT THE THERMAL STRESS WILL BE COMPRESSIVE FOR ΔT>0. THE ANSWER HERE REPRESENTS THE AXIAL STRESS MAGNITUDE ONLY (COMPRESSIVE STRESS IS NEGATIVE BY CONVENTION) M. Mello/Georgia Tech Aerospace 7 Example 2- 7 (WHAT ABOUT THE BAR DIAMETER?) RA δT D0 δR D0+ΔDT = D0+ΔDR + Thermal strain Mechanical strain ²་  NOTEL: Gere and Goodno make a passing reference to the transverse strains in the constrained bar (but no analysis presented) Note minus sign here ✓ ◆ ²་  Diameter increase: E↵ T D = (↵ T ) D0 + ⌫ D0 D = ✏thermal D0 + ✏mech D0 E D = (↵ T )D0 + ( ⌫ ✏axial )D0 D = (↵ T ) D 0 + ( ⌫ R E )D0 Note minus sign here 1/17/13 M. Mello/Georgia Tech Aerospace D = ↵ T D0 (1 + ⌫ ) (DIAMETER INCREASE OF CONSTRAINED BAR) 8 MATERIAL (ELASTIC) PROPERTIES MATERIAL MODULUS,...
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This note was uploaded on 09/19/2013 for the course CEE 3001 taught by Professor Zhu during the Spring '09 term at Georgia Tech.

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