MECHANICS OF MATERIALS

Transcription

1 Fourth Edton CHTER MECHNICS OF MTERIS Ferdnand. Beer E. Russell Johnston, Jr. John T. DeWolf ecture Notes: J. Walt Oler Texas Tech Unversty Stress and Stran xal oadng

2 Contents Stress & Stran: xal oadng Normal Stran Stress-Stran Test Stress-Stran Dagram: Ductle Materals Stress-Stran Dagram: Brttle Materals Hooke s aw: Modulus of Elastcty Elastc vs. lastc Behavor Fatgue Deformatons Under xal oadng Example.0 Sample roblem. Statc Indetermnacy Example.04 Thermal Stresses osson s Rato Generalzed Hooke s aw Dlataton: Bulk Modulus Shearng Stran Example. Relaton mong E, ν, and G Sample roblem.5 Composte Materals Sant-Venant s rncple Stress Concentraton: Hole Stress Concentraton: Fllet Example. Elastoplastc Materals lastc Deformatons Resdual Stresses Example.4,.5,.6 -

3 Stress & Stran: xal oadng Sutablty of a structure or machne may depend on the deformatons n the structure as well as the stresses nduced under loadng. Statcs analyses alone are not suffcent. Consderng structures as deformable allows determnaton of member forces and reactons whch are statcally ndetermnate. Determnaton of the stress dstrbuton wthn a member also requres consderaton of deformatons n the member. Chapter s concerned wth deformaton of a structural member under axal loadng. ater chapters wll deal wth torsonal and pure bendng loads. -

4 Normal Stran σ stress δ ε normal stran σ δ ε σ δ δ ε - 4

5 Stress-Stran Test - 5

6 Stress-Stran Dagram: Ductle Materals - 6

7 Stress-Stran Dagram: Brttle Materals - 7

8 Hooke s aw: Modulus of Elastcty Below the yeld stress σ Eε E Youngs Modulus or Modulus of Elastcty Strength s affected by alloyng, heat treatng, and manufacturng process but stffness (Modulus of Elastcty) s not. - 8

9 Elastc vs. lastc Behavor If the stran dsappears when the stress s removed, the materal s sad to behave elastcally. The largest stress for whch ths occurs s called the elastc lmt. When the stran does not return to zero after the stress s removed, the materal s sad to behave plastcally. - 9

10 Fatgue Fatgue propertes are shown on S-N dagrams. member may fal due to fatgue at stress levels sgnfcantly below the ultmate strength f subjected to many loadng cycles. When the stress s reduced below the endurance lmt, fatgue falures do not occur for any number of cycles. -

11 Deformatons Under xal oadng From Hooke s aw: σ σ E ε ε E E From the defnton of stran: δ ε Equatng and solvng for the deformaton, δ E Wth varatons n loadng, cross-secton or materal propertes, δ E -

12 Example.0 E 9 6 ps D.07 n. d 0.68 n. SOUTION: Dvde the rod nto components at the load applcaton ponts. pply a free-body analyss on each component to determne the nternal force Determne the deformaton of the steel rod shown under the gven loads. Evaluate the total of the component deflectons. -

13 SOUTION: Dvde the rod nto three components: pply free-body analyss to each component to determne nternal forces, lb lb lb Evaluate total deflecton, δ E E ( 60 ) ( 5 ) ( 0 ) n n. 0.9 n 6 n. 0.n δ 75.9 n. -

14 Sample roblem. SOUTION: The rgd bar BDE s supported by two lnks B and CD. nk B s made of alumnum (E 70 Ga) and has a cross-sectonal area of 500 mm. nk CD s made of steel (E 00 Ga) and has a cross-sectonal area of (600 mm ). For the 0-kN force shown, determne the deflecton a) of B, b) of D, and c) of E. pply a free-body analyss to the bar BDE to fnd the forces exerted by lnks B and DC. Evaluate the deformaton of lnks B and DC or the dsplacements of B and D. Work out the geometry to fnd the deflecton at E gven the deflectons at B and D. - 4

15 Sample roblem. SOUTION: Free body: Bar BDE Dsplacement of B: δ B E ( 60 N)( 0.m) -6 9 ( 500 m )( 70 a) 54 6 m M F F B 0 M CD D 0 B 0 ( 0kN 0.6m) 0 ( 0kN 0.4m) + F + 90kN tenson F CD B 60kN compresson 0.m 0.m Dsplacement of D: δ D E δ B 0.54 mm ( 90 N)( 0.4m) -6 9 ( 600 m )( 00 a) 00 6 m δ D 0.00 mm - 5

16 Sample roblem. Dsplacement of D: BB DD BH HD 0.54 mm 0.00 mm x 7.7 mm ( 00 mm) x x EE DD δ E 0.00 mm δ E HE HD.98 mm ( ) mm 7.7 mm δ E.98 mm - 6

17 Statc Indetermnacy Structures for whch nternal forces and reactons cannot be determned from statcs alone are sad to be statcally ndetermnate. structure wll be statcally ndetermnate whenever t s held by more supports than are requred to mantan ts equlbrum. Redundant reactons are replaced wth unknown loads whch along wth the other loads must produce compatble deformatons. Deformatons due to actual loads and redundant reactons are determned separately and then added or superposed. δ δ + δ R 0-7

18 Example.04 Determne the reactons at and B for the steel bar and loadng shown, assumng a close ft at both supports before the loads are appled. SOUTION: Consder the reacton at B as redundant, release the bar from that support, and solve for the dsplacement at B due to the appled loads. Solve for the dsplacement at B due to the redundant reacton at B. Requre that the dsplacements due to the loads and due to the redundant reacton be compatble,.e., requre that ther sum be zero. Solve for the reacton at due to appled loads and the reacton found at B. - 8

19 Example.04 SOUTION: Solve for the dsplacement at B due to the appled loads wth the redundant constrant released, δ 0 E m.5 E 9 N 0.50 m Solve for the dsplacement at B due to the redundant constrant, δ R 400 R E m B m (.95 ) E 50 R B 6 m N 6 m - 9

20 Example.04 Requre that the dsplacements due to the loads and due to the redundant reacton be compatble, δ δ + δ.5 δ E R B R (.95 ) E N 577 kn R B 0 Fnd the reacton at due to the loads and the reacton at B F 0 R 00 kn 600kN kn R y kn R R B kn 577 kn - 0

21 Thermal Stresses temperature change results n a change n length or thermal stran. There s no stress assocated wth the thermal stran unless the elongaton s restraned by the supports. Treat the addtonal support as redundant and apply the prncple of superposton. δ T α ( T ) α thermal expanson coef. δ δ T + δ 0 δ E The thermal deformaton and the deformaton from the redundant support must be compatble. α( T ) + 0 E Eα σ ( T ) Eα ( T ) -