MECHANICS OF MATERIALS
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1 00 The McGraw-Hill Copanies, Inc. All rights reserved. T Edition CHAPTER MECHANICS OF MATERIALS Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf Lecture Notes: J. Walt Oler Texas Tech University Principle Stresses Under a Given Loading
2 00 The McGraw-Hill Copanies, Inc. All rights reserved. 8 - Principle Stresses Under a Given Loading 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 8.5
3 00 The McGraw-Hill Copanies, Inc. All rights reserved. 8-3 Introduction In Chaps. 1 and, you learned how to deterine the noral stress due to centric loads In Chap. 3, you analyzed the distribution of shearing stresses in a circular eber due to a twisting couple In Chap. 4, you deterined the noral stresses caused by bending couples In Chaps. 5 and 6, you evaluated the shearing stresses due to transverse loads In Chap. 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
4 00 The McGraw-Hill Copanies, Inc. All rights reserved. 8-4 Principle Stresses in a Bea Prisatic bea subjected to transverse loading τ x xy My I VQ It 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 Mc I
5 00 The McGraw-Hill Copanies, Inc. All rights reserved. 8-5 Principle Stresses in a Bea
6 00 The McGraw-Hill Copanies, Inc. All rights reserved. 8-6 Principle Stresses in a Bea Cross-section shape results in large values of τ xy near the surface where x is also large. ax ay be greater than
7 00 The McGraw-Hill Copanies, Inc. All rights reserved. 8-7 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
8 00 The McGraw-Hill Copanies, Inc. All rights reserved. 8-8 Saple Proble 8.1 SOLUTION: Deterine shear and bending oent in Section A-A M V A A ( 160kN)( 0.375) 160kN 60kN - Calculate the noral stress at top surface and at flange-web junction. a b M 60kN A S MPa a y c b 10.9 MPa 3 ( 117. MPa)
9 00 The McGraw-Hill Copanies, Inc. All rights reserved. 8-9 Saple Proble 8.1 Evaluate shear stress at flange-web junction. Q τ b ( ) V Q A It 95.5MPa ( )( 3) 160kN ( )( ) Calculate the principal stress at flange-web junction ( 1 ) 1 ax + + b b τb ( 95.5) MPa 3 ( > 150 MPa) Design specification is not satisfied. 3
10 00 The McGraw-Hill Copanies, Inc. All rights reserved 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 all 4 ksi and τ all 14.5 ksi, select the wideflange 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.
11 00 The McGraw-Hill Copanies, Inc. All rights reserved Saple Proble 8. SOLUTION: Deterine reactions at A and D. M M Calculate required section odulus and select appropriate bea section. S in A D 0 0 M ax all R R D 59 kips 41kips select W1 6 bea section A Deterine axiu shear and bending oent fro shear and bending oent diagras. M V ax ax 39.4 kip in 43kips with V 4kip in in 4ksi 1. kips
12 00 The McGraw-Hill Copanies, Inc. All rights reserved. 8-1 Saple Proble 8. Find axiu shearing stress. Assuing unifor shearing stress in web, V 43 kips τ ax ax 5.1 ksi < 14.5ksi 8.40 in A web Find axiu noral stress. M ax 60kip in a ksi S 3 17in τ b b a V A y c web b (.6 ksi) kips 1.45ksii 8.40in 1.3ksi ax 1.3ksi 1.3ksi + + ( 1.45ksi) 1.4 ksi < 4ksi
13 00 The McGraw-Hill Copanies, Inc. All rights reserved 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.
14 00 The McGraw-Hill Copanies, Inc. All rights reserved Design of a Transission Shaft At any section, τ Mc I Tc J where M M y + M z Maxiu shearing stress, τ ax ax c J M + + T ( τ ) Mc I Tc + J for a circular or annular cross - section, I τ Shaft section requireent, J c in M + T τ all ax J
15 00 The McGraw-Hill Copanies, Inc. All rights reserved Saple Proble 8.3 SOLUTION: Deterine the gear torques and corresponding tangential forces. Find reactions at A and B. 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 all 50 MPa, deterine the sallest perissible diaeter for the shaft. Identify critical shaft section fro torque and bending oent diagras. Calculate iniu allowable shaft diaeter.
16 00 The McGraw-Hill Copanies, Inc. All rights reserved Saple Proble 8.3 SOLUTION: Deterine the gear torques and corresponding tangential forces. P 30kW TE 597 N πf π 80Hz F T T E C D T r E E 0kW π ( 80Hz) 10kW π ( 80Hz) ( ) 597 N kN 398N 199 N Find reactions at A and B. A B y y 0.93 kn.80 kn A B z z 6. kn.90 kn F F C D 6.63kN.49 kn
17 00 The McGraw-Hill Copanies, Inc. All rights reserved Saple Proble 8.3 Identify critical shaft section fro torque and bending oent diagras. M + T ax ( ) N + 597
18 00 The McGraw-Hill Copanies, Inc. All rights reserved Saple Proble 8.3 Calculate iniu allowable shaft diaeter. J c For a solid circular shaft, J c π c 3 M + T τ all N 50MPa c d 3 c
19 00 The McGraw-Hill Copanies, Inc. All rights reserved 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.
20 00 The McGraw-Hill Copanies, Inc. All rights reserved. 8-0 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.
21 00 The McGraw-Hill Copanies, Inc. All rights reserved. 8-1 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 conditions of applicability of superposition principle and Saint-Venant s principle are et.
22 00 The McGraw-Hill Copanies, Inc. All rights reserved. 8 - 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.
23 00 The McGraw-Hill Copanies, Inc. All rights reserved. 8-3 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
24 00 The McGraw-Hill Copanies, Inc. All rights reserved. 8-4 Saple Proble 8.5 Evaluate noral stress at H. y + 50kN P A + M I z z a + ( 8.5kN )( 0.05) M I ( 3kN )( 0.00 ) 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 [( )( 0.045) ]( ) 3 ( )( 3) 75kN ( )( ) 4
25 00 The McGraw-Hill Copanies, Inc. All rights reserved. 8-5 Saple Proble 8.5 Calculate principal stresses and axiu shearing stress. Deterine principal planes. τ tan θ θ ax ax in p OC OC p R τ θ p R R CY CD ax ax in MPa MPa MPa MPa 70.4 MPa 7.4 MPa θ p 7.96
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