MECHANICS OF MATERIALS

Transcription

1 009 The McGraw-Hill Copanies, Inc. All rights reserved. Fifth SI E CHAPTER 8 MECHANICS OF MATERIALS Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf David F. Mazurek Lecture Notes: J. Walt Oler Texas Tech University Principle Stresses Under a Given Loading

2 ifth Contents Introduction Principle Stresses in a Bea Saple Proble 8.1 Saple Proble 8. Design of a Transission Shaft Saple Proble 8.3 Stresses Under Cobined Loadings Saple Proble The McGraw-Hill Copanies, Inc. All rights reserved. 8-

3 ifth Introduction In Chapter 1 and, you learned how to deterine the noral stress due to centric loads. In Chapter 3, you analyzed the distribution of shearing stresses in a circular eber due to a twisting couple. In Chapter 4, you deterined the noral stresses caused by bending couples. In Chapters 5 and 6, you evaluated the shearing stresses due to transverse loads. In Chapter 7, you learned how the coponents of stress are transfored by a rotation of the coordinate axes and how to deterine the principal planes, principal stresses, and axiu shearing stress at a point. In Chapter 8, you will learn how to deterine the stress in a structural eber or achine eleent due to a cobination of loads and how to find the corresponding principal stresses and axiu shearing stress. 009 The McGraw-Hill Copanies, Inc. All rights reserved. 8-3

4 ifth Principle Stresses in a Bea Prisatic bea subjected to transverse loading σ τ x xy My I VQ It σ τ Mc I VQ It Principal stresses deterined fro ethods of Chapter 7 Can the axiu noral stress within the cross-section be larger than σ Mc I 009 The McGraw-Hill Copanies, Inc. All rights reserved. 8-4

5 ifth Principle Stresses in a Bea 009 The McGraw-Hill Copanies, Inc. All rights reserved. 8-5

6 ifth Principle Stresses in a Bea Cross-section shape results in large values of τ xy near the surface where σ x is also large. σ ax ay be greater than σ 009 The McGraw-Hill Copanies, Inc. All rights reserved. 8-6

7 ifth Saple Proble 8.1 SOLUTION: Deterine shear and bending oent in Section A-A Calculate the noral stress at top surface and at flange-web junction. A 160-kN force is applied at the end of a W00x5 rolled-steel bea. Neglecting the effects of fillets and of stress concentrations, deterine whether the noral stresses satisfy a design specification that they be equal to or less than 150 MPa at section A-A. Evaluate the shear stress at flangeweb junction. Calculate the principal stress at flange-web junction 009 The McGraw-Hill Copanies, Inc. All rights reserved. 8-7

8 ifth Saple Proble 8.1 SOLUTION: Deterine shear and bending oent in Section A-A M V A ( 160kN)( 0.375) 160kN A 60kN - Calculate the noral stress at top surface and at flange-web junction. σ σ a b M 60kN A S MPa σ a y c b 10.9 MPa 3 ( 117. MPa) The McGraw-Hill Copanies, Inc. All rights reserved. 8-8

9 ifth Saple Proble 8.1 Evaluate shear stress at flange-web junction. Q τ b Calculate the principal stress at flange-web junction σ ( ) V Q A It 95.5MPa ( )( 6 3) 160kN ( )( ) ( 1σ ) 1 ax σ + + b b τb ( 95.5) MPa ( > 150 MPa) Design specification is not satisfied The McGraw-Hill Copanies, Inc. All rights reserved. 8-9

10 ifth Saple Proble 8. SOLUTION: Deterine reactions at A and D. The overhanging bea supports a uniforly distributed load and a concentrated load. Knowing that for the grade of steel to used σ all 165 MPa and τ all 100 MPa, select the wide-flange bea which should be used. Deterine axiu shear and bending oent fro shear and bending oent diagras. Calculate required section odulus and select appropriate bea section. Find axiu noral stress. Find axiu shearing stress. 009 The McGraw-Hill Copanies, Inc. All rights reserved. 8-10

11 Fifth E Saple Proble 8. SOLUTION: Deterine reactions at A and D. M M A 0 RD 65.5 kn D 0 RA kn Deterine axiu shear and bending oent fro shear and bending oent diagras. M V 33. kn with V 54.9 kn ax ax kn Calculate required section odulus and select appropriate bea section. S in M ax σ all N Pa select W530 9 bea section 009 The McGraw-Hill Copanies, Inc. All rights reserved. 8-11

12 Fifth E Saple Proble 8. Find axiu shearing stress. Assuing unifor shearing stress in web, τ ax Vax N 35.6 MPa < 100 MPa Aweb Find axiu noral stress. M ax 3300 N MPa S y MPa σ b σ a b (156.1 MPa ) 66.5 c N V τb 10.1 MPa Aweb σa σ ax 009 The McGraw-Hill Copanies, Inc. All rights reserved. 147 MPa 147 MPa + + (10.1 MPa ) MPa < 165 MPa 8-1

13 Design of a Transission Shaft If power is transferred to and fro the shaft by gears or sprocket wheels, the shaft is subjected to transverse loading as well as shear loading. Noral stresses due to transverse loads ay be large and should be included in deterination of axiu shearing stress. Shearing stresses due to transverse loads are usually sall and contribution to axiu shear stress ay be neglected. 009 The McGraw-Hill Copanies, Inc. All rights reserved. 8-1

14 Design of a Transission Shaft At any section, σ τ Maxiu shearing stress, τ ax ax Shaft section requireent, J c in c J σ M + T M + T τ + ( τ ) all ax Mc I Tc + J for a circular or annular cross - section, I τ Mc I Tc J where M M y + M z J 009 The McGraw-Hill Copanies, Inc. All rights reserved. 8-

15 Saple Proble 8.3 SOLUTION: Deterine the gear torques and corresponding tangential forces. Find reactions at A and B. Identify critical shaft section fro torque and bending oent diagras. Solid shaft rotates at 480 rp and transits 30 kw fro the otor to gears G and H; 0 kw is taken off at gear G and 10 kw at gear H. Knowing that τ all 50 MPa, deterine the sallest perissible diaeter for the shaft. Calculate iniu allowable shaft diaeter. 009 The McGraw-Hill Copanies, Inc. All rights reserved. 8-3

16 Saple Proble 8.3 SOLUTION: Deterine the gear torques and corresponding tangential forces. T F T T E E C D P 30kW 597 N πf π ( 8Hz) T 597 N E 3.73kN re kW 398N F π ( 8Hz) 10kW 199 N F π ( 8Hz) Find reactions at A and B. A B y y 0.93 kn.80 kn A B z z 6. kn.90 kn C D 6.63kN.49 kn 009 The McGraw-Hill Copanies, Inc. All rights reserved. 8-4

17 Saple Proble 8.3 Identify critical shaft section fro torque and bending oent diagras. ( ) M + M + T y z ax N T e,c 1,319 N. T e,e 1,003 N. 009 The McGraw-Hill Copanies, Inc. All rights reserved. 8-5

18 Saple Proble 8.3 Calculate iniu allowable shaft diaeter. J c M y + M τ all z + T 1357 N MPa 6 3 For a solid circular shaft, J c c π 3 6 c d c The McGraw-Hill Copanies, Inc. All rights reserved. 8-6

19 Stresses Under Cobined Loadings Wish to deterine stresses in slender structural ebers subjected to arbitrary loadings. Pass section through points of interest. Deterine force-couple syste at centroid of section required to aintain equilibriu. Syste of internal forces consist of three force coponents and three couple vectors. Deterine stress distribution by applying the superposition principle. 009 The McGraw-Hill Copanies, Inc. All rights reserved. 8-7

20 Stresses Under Cobined Loadings Axial force and in-plane couple vectors contribute to noral stress distribution in the section. Shear force coponents and twisting couple contribute to shearing stress distribution in the section. 009 The McGraw-Hill Copanies, Inc. All rights reserved. 8-8

21 Stresses Under Cobined Loadings Noral and shearing stresses are used to deterine principal stresses, axiu shearing stress and orientation of principal planes. Analysis is valid only to extent that cons of applicability of superposition principle and Saint-Venant s principle are et. 009 The McGraw-Hill Copanies, Inc. All rights reserved. 8-9

22 Saple Proble 8.5 SOLUTION: Deterine internal forces in Section EFG. Evaluate noral stress at H. Evaluate shearing stress at H. Three forces are applied to a short steel post as shown. Deterine the principle stresses, principal planes and axiu shearing stress at point H. Calculate principal stresses and axiu shearing stress. Deterine principal planes. 009 The McGraw-Hill Copanies, Inc. All rights reserved. 8-10

23 Saple Proble 8.5 SOLUTION: Deterine internal forces in Section EFG. V M M x x y 30 kn 0 ( 50kN)( ) ( 75kN)( 0.00 ) 8.5kN M z P 50kN Note: Section properties, A I I x z ( )( ) V 75kN ( 30kN)( ) 3kN ( )( ) ( )( ) z The McGraw-Hill Copanies, Inc. All rights reserved. 8-11

24 Saple Proble 8.5 Evaluate noral stress at H. σ y + 50kN P A + M I z z a + ( 8.5kN )( 0.05) M I ( 3kN )( 0.00 ) 6 x x ( ) MPa 66.0MPa Evaluate shearing stress at H. Q τ yz V I A y z Q t x 17.5 MPa b 4 6 [( )( 0.045) ]( ) 6 3 ( )( 6 3) 75kN ( )( ) The McGraw-Hill Copanies, Inc. All rights reserved. 8-1

25 Saple Proble 8.5 Calculate principal stresses and axiu shearing stress. Deterine principal planes. τ σ σ ax ax in OC OC R R CY 17.5 tan θ p CD 33.0 θ p R τ σ σ θ p ax ax in MPa MPa MPa 37.4 MPa 70.4 MPa 7.4 MPa θ p The McGraw-Hill Copanies, Inc. All rights reserved. 8-13