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Compressive combine eqs 1

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Unformatted text preview: SOLUTION (conMnued): ²་  SubsMtute force- displacement into compaMbility equaMons Ps L Pc L s = c E s As steel Ps = Pc ✓ E s As E c Ac ◆ = = E c Ac C opper Pc = Ps ✓ E c Ac E s As ◆ ²་  SubsMtute for either Ps or Pc in (verMcal) equilibrium equaMon (and solve for the other force in terms of P) ✓ ◆ ✓ ◆ E s As E c Ac i.e., subsMtute Ps = Pc or, Pc = Ps E c Ac E s As Ps + Pc P = 0 into ✓ ◆ ✓ ◆ E s As E c Ac Ps = P Pc = P E s As + E c Ac E s As + E c Ac Axial force in steel tube Axial force in Copper tube 1/17/13 M. Mello/Georgia Tech Aerospace 14 EXAMPLE 2- 5 (finish) SOLUTION (finish): ²་  Compressive stresses in steel and Copper tubes s = c = = P Es Es As + Ec Ac (axial stress in steel tube) = P Ec (axial stress in Copper tube) E s As + E c A c s As c Ac ²་  Shortening of the assembly = = c = s Pc L Ps L = E c Ac E s As ✓ ◆ E c Ac SubsMtute either Pc = P E s As + E c Ac ✓ ◆ E s As or, Ps = P (derived on previous page) E s As + E c Ac PL yielding: = (shortening of tube assembly) E s As + E c Ac 1/17/13 M. Mello/Georgia Tech Aerospace 15 EXAMPLE 2- 6 ²་  FIND THIS EXAMPLE IN GERE AND GOODNO AND STUDY IT ON YOUR OWN ²་  YOU CAN EASILY DO IT IF YOU HAVE READ THE NOTES TO THIS POINT AND UNDERSTAND THEM WELL ²་  THIS EXAMPLE IS AN ALLOWABLE LOAD CALCULATION ²་  IT’S VERY SIMILAR PRINCIPLES TO PREVIOUS EXAMPLES . . . BASED UPON STATICS, COMPATIBILITY, AND FORCE- DISPLACEMENT RELATIONS ETC.. DETERMINE: (A)  Allowable load P if allowable stresses in wires (σ1 AND σ2) are known. (B)  Allowable load P based upon the given material proper>es, diameters, and allowable stresses for each wire. 1/17/13 M. Mello/Georgia Tech Aerospace 16 THERMAL EFFECTS, MISFITS, AND PRESTRAINS ²་  EXTERNAL LOADS ARE NOT THE ONLY SOURCE OF STRESSES AND STRAIN IN STRUCTURES ²་  OTHER SOURCES INCLUDE: ²་  THERMAL STRAINS ARISING FROM TEMPERATURE CHANGES ²་  MISFITS RESULTING FROM INPERFECTIONSIN CONSTRUCTION ²་  PRESTRAINS PRODUCED BY INITIAL DEFORMATION ²་  INERTIAL LOADS RESULTING FROM ACCLERATED MOTION (EARTHQUAKES) Bridge expansion joint 1/17/13 Deforma>on of the trans- Alaska pipeline following the 7.9 magnitude earthquake of 3 November 2002, Alaska M. Mello/Georgia Tech Aerospace 17 THERMAL EFFECTS Block of material subjected to an increase in temperature thermal strain sign conven>on: ²་  expansion ε > 0 ²་  contrac>on ε < 0 ²་  WATER expands when heated ²་  ICE EXPANDS when cooled (for T < 4°C) thermal strain temperature change ✏ = ↵( T ) (2 15) coefficient of thermal expansion = ✏T L = ↵ ( T )L (2 16) Temperature- displacement rela>on (analogous to force- displacement rela>on) 1/17/13 M. Mello/Georgia Tech Aerospace 18 THERMAL EFFECTS (cont.) ²་  WHAT ABOUT THERMAL STRESSES? ²་  UNIFORM HEATING OF UNCONSTRAINED, HOMOGENEOUS MATERIALS WILL INDUCE THERMAL STRAIN (CONTRACTION/EXPANSION) BUT WILL NOT RESULT IN INTERNAL STRESSES. ²་  STATEMENT GENERALLY APPLIES TO STATICALLY DETERMINATE STRUCTURES STATICALLY DETERMINATE TRUSS WITH A UNIFORM TEMEPRATURE CHANGE IN EACH MEMBER ²་  A STATICALLY INDETERMINATE STRUCTURE MAY OR MAY NOT DEVELOP TEMPERATURE STRESSES. THIS STATICALLY INDETERMINATE TRUSS WON’T EXPERIENCE INTERNAL STRESS DUE TO A UNIFORM TEMERATURE CHANGE. 1/17/13 M. Mello/Georgia Tech Aerospace 19...
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