See exam 1 and exam 2 study guides for previous materials covered in exam 1 and 2. Stress transformation. Positive τ xy : τ xy

Transcription

1 ME33: Mechanic of Material Final Eam Stud Guide 1 See eam 1 and eam tud guide for previou material covered in eam 1 and. Stre tranformation In ummar, the tre tranformation equation are: + ' + co θ + in θ ( ' ' in θ + co θ + ' co θ in θ Important: The angle θ i the angle meaured counterclockwie from to : ' θ ' oitive : oitive : rincipal lane: The maimum and minimum normal tree are called the principal tree. The orientation angle of thi plane, θ p, i: The tree in thi plane are: tan ( θ p + ( 1, ± + p 0 *See net page on how to ue thi eqn Therefore, the tre element in the principal (p 1 -p ae plane i: p θ p θ p1 +90 o 1 p 1 θ p1 Maimum in plane hear tre MES 1

2 ME33: Mechanic of Material Final Eam Stud Guide The orientation angle of thi plane, θ, i: The tree in thi plane are: ( ma + in plane tan ( θ + 1 *See net page on how to ue thi eqn Therefore, the tre element in the principal ( 1 - ae plane i: θ θ o ma in-plane 1 θ 1 Important point: 1. θ p1 i meaured CCW from to p 1. θ 1 i meaured CCW from to 1 3. poitive hear tre and poitive normal tre are: oitive : oitive : 4. When calculating the angle from tan ( θ p (, we can get two angle, but we don t know which i θ p1 which i θ p. To find out, here i what we can do: Calculate ' uing one of the angle. If ' 1, then that angle i θ p1, if ', then that angle i θ p. (See eample 9.3 on page 448 of tetbook. MES

3 ME33: Mechanic of Material Final Eam Stud Guide 3 Mohr Circle Mohr circle i a graphical repreentation of the tre tranformation equation. rocedure to contruct a Mohr Circle, given a tre tate: 1. Draw the and ae a follow: poitive to the right and poitive down: + +. oition the center of the circle at ; 0 3. The radiu of the circle i R + + R + 4. The tre tate in the - plane hown above are repreented b a traight line connecting two point on the Mohr Circle: X (, and (, (, + X (, MES 3

4 ME33: Mechanic of Material Final Eam Stud Guide 4 How to ue the Mohr Circle 1. Stre tranformation from - plane to - plane in a Mohr Circle: ' ' (, X '(, ' ' ' θ '(, ' ' ' θ X (, + θ CCW rotation ( to in tre element θ CCW rotation in Mohr Circle (X to X. lane of principal tre p (, 1 p 1 (,0 θ p1 1( 1,0 + θ p1 X (, 3. lane of maimum in-plane hear tre (, S (, ma ma + θ 1 1 θ 1 S (, 1 ma X (, MES 4

5 ME33: Mechanic of Material Final Eam Stud Guide 5 Failure Theorie Undertand how to ue failure theorie e.g. finding minimum cro ectional area, finding maimum weight that a tructure can upport, etc. uing failure theorie with factor of afet Theorie of failure for brittle material, 1. Maimum Normal Stre Theor Thi failure theor aume that the ultimate tre of the material in tenion and compreion are equal. Failure occur when the maimum normal tre (principal tre in the material reache a value that i equal to the ultimate normal tre. 1 u u. Mohr Failure Criterion Thi failure theor i for brittle material whoe ultimate trength in tenion and compreion are different. In tenion, ma, tenion U, tenion In compreion, ma, compreion U, compreion MES 5

6 ME33: Mechanic of Material Final Eam Stud Guide 6 Theorie of failure for ductile material, 1. Maimum Shear Stre Theor (Treca ield criterion Failure occur when the abolute maimum hear tre in the material i equal to the hear tre that caue the material to ield in uniaial tenion tet. ab ma Cae I: the -D principal tree are both poitive Abolute maimum hear tre larget radiu Cae II: the -D principal tree are both negative + 1 ab ma Cae III: the -D principal tree have oppoite ign Abolute maimum hear tre larget radiu Abolute maimum hear tre larget radiu 1 ab ma Therefore, in term of the principal tree, 1 if 1 and have ame ign if and have oppoite ign 1 ab ma 1 1. Maimum Ditortion Energ Theor Thi failure theor ue the maimum ditortion energ (the energ required to change the hape of the material without changing the volume to characterize failure MES 6

7 ME33: Mechanic of Material Final Eam Stud Guide 7 Beam Deflection Beam deflection: nd order integration method 1. Determine the internal bending moment equation for each continuou egment a. Find the reaction force b. Derive the internal bending moment through equilibrium method Note: It i alo poible to ue the graphical method. However, it i important to remember that the actual equation of M mut be derived.. Moment-Deflection equation for each continuou egmentt u M Integrate to find the deflection equation EI 3. Boundar Condition & Continuit Equation Ue the boundar condition and continuit equation to find the integration contant in the deflection equation found in tep. Boundar Condition: Roller in Fied end Zero deflection u 0 Zero deflection u 0 Zero deflection u 0 No deflection lope retriction No deflection lope retriction Zero deflection lope du 0 d Continuit Equation: For each continuou ection, we have different internal bending moment equation. Conequentl, the deflection equation are different. To enure that the different equation reult in a continuou deflection hape, we will enforce continuit equation. At each dicontinuit point, u1 u du1 du d d 1 Staticall determinate v. taticall indeterminate MES 7

8 ME33: Mechanic of Material Final Eam Stud Guide 8 From Table 1- of tetbook: Dicontinuit function 1. Calculate upport reaction. Ue the dicontinuit function in the table above to epre M( a a function of. Note: Since we are epreing the equation uing dicontinuit function, onl 1 equation i needed for an beam. du 3. Ue the moment-diplacement equation EI M and integrate to get u( d 4. Ue boundar condition to find integration contant (continuit not needed for thi method MES 8

9 ME33: Mechanic of Material Final Eam Stud Guide 9 Buckling Buckling can happen when the internal load > cr. For variou tpe of upport, the critical force equation for buckling become, π EI cr in general e where e K i the effective length of the column and K i lited in the table below, inned-inned end: K 1 Fied-Fied end: K 0.5 inned-fied end: K 0.7 Fied-Free end: K Failure anali: buckling v. ielding (cruhing MES 9

10 ME33: Mechanic of Material Final Eam Stud Guide 10 Energ Method Work-Energ in deformable material: Work done to deform an elatic material W e Total train energ (tored energ in tructure U i Work done to deform an elatic material B a Force 1 We Fd B a Moment 1 We Mθ F Force applied at point d Deformation of point in the direction of the force M Moment applied at point θ Deformation angle/lope at point in the direction of the moment Component of train energ in the tructure: B aial loading N UiN, AE N internal normal force B torion T UiT, JG T internal torque Bending B bending moment M M internal bending moment UiM, d EI B tranvere hear force fv V internal hear force UiV, d GA A Q f form factor da I t A For a rectangular cro ection, f 6 5 Total train energ of the tructure: N T M fv Ui + + d d AE JG + EI GA Note: The contribution of train energ from tranvere hear force i negligible for long lender beam, therefore it can be neglected. Impact loading: ue the equation KE1+ E1+ We KE + E For purel elatic deformation, train energ i conerved. Therefore, train energ can be included in potential energ. MES 10

11 ME33: Mechanic of Material Final Eam Stud Guide 11 imitation of the Work-Energ principle: Onl diplacement in the direction of a ingle applied load can be computed. To get around thi limitation, we will ue a more powerful method: - rinciple of Virtual Work (Ch in tetbook rinciple of Virtual Work For true: To find a diplacement at point ( Δ, replace real loading with a virtual load of magnitude 1 at point, in the direction of the diplacement. FvirtualFreal nn Δ i AE i i AE i where, i tru i F virtual n internal normal force of tru i for the cae of virtual loading F N internal normal force of tru i for the cae of real loading real For beam: To find a diplacement at point ( Δ, replace real loading with a virtual load of magnitude 1 at point, in the direction of the diplacement. MvirtualMreal mm Δ d d EI EI where, M virtual m internal bending moment of tru i for the cae of virtual loading M M internal bending moment of tru i for the cae of real loading real Note: If angle at point i needed, appl a virtual point moment of magnitude 1 at point. Strateg: 1. Replace real loading with a virtual load (or moment of magnitude 1 at the point where diplacement i to be computed (in the ame direction a the diplacement.. Determine internal reultant for both virtual loading cae and real loading cae. 3. Appl the equation, nn Δ i AE i for true mm Δ d EI for beam (tranvere hear negligible MES 11