MECHANICS OF MATERIALS
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1 00 The cgraw-hill Copanies, Inc. All rights reserved. Third E CHAPTER 8 Principle ECHANICS OF ATERIALS Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf Lecture Notes: J. Walt Oler Texas Tech University Stresses Under a Given Loading
2 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 The cgraw-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 00 The cgraw-hill Copanies, Inc. All rights reserved. 8-3
4 Principle Stresses in a Bea Prisatic bea subjected to transverse loading x xy y I VQ It c I VQ It Principal stresses deterined fro ethods of Chapter 7 Can the axiu noral stress within the cross-section be larger than c I 00 The cgraw-hill Copanies, Inc. All rights reserved. 8-4
5 Principle Stresses in a Bea 00 The cgraw-hill Copanies, Inc. All rights reserved. 8-5
6 Principle Stresses in a Bea Cross-section shape results in large values of xy near the surface where x is also large. ax ay be greater than 00 The cgraw-hill Copanies, Inc. All rights reserved. 8-6
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 Pa at section A-A. Evaluate the shear stress at flangeweb junction. Calculate the principal stress at flange-web junction 00 The cgraw-hill Copanies, Inc. All rights reserved. 8-7
8 Saple Proble 8.1 SOLUTION: Deterine shear and bending oent in Section A-A V A A 160kN kN 60kN - Calculate the noral stress at top surface and at flange-web junction. σ a b 60kN A S Pa a y c b 10.9 Pa 117. Pa The cgraw-hill Copanies, Inc. All rights reserved. 8-8
9 Saple Proble 8.1 Evaluate shear stress at flange-web junction. Q b Pa V Q A It kN Calculate the principal stress at flange-web junction ax 1 b Pa 150 Pa Design specification is not satisfied. b b The cgraw-hill Copanies, Inc. All rights reserved. 8-9
10 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 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. 00 The cgraw-hill Copanies, Inc. All rights reserved. 8-10
11 Saple Proble 8. SOLUTION: Deterine reactions at A and D. Calculate required section odulus and select appropriate bea section. S in A D 0 0 ax all R R D 59kips 41kips select W1 6 bea section A Deterine axiu shear and bending oent fro shear and bending oent diagras. V ax ax 39.4kip in 43kips with V 4kip in in 4ksi 1. kips 00 The cgraw-hill Copanies, Inc. All rights reserved. 8-11
12 Saple Proble 8. Find axiu shearing stress. Assuing unifor shearing stress in web, ax V ax A web 43 kips 8.40 in 5.1 ksi 14.5ksi Find axiu noral stress. σ a b b S a V A ax y c web b 60kip in in.6 ksi 1. kips 1.45ksii 8.40in ksi 1.3ksi ax 1.3ksi 1.3ksi 1.45ksi 1.4ksi 4ksi 00 The cgraw-hill Copanies, Inc. All rights reserved. 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. 00 The cgraw-hill Copanies, Inc. All rights reserved. 8-13
14 00 The cgraw-hill Copanies, Inc. All rights reserved. hird 8-14 Design of a Transission Shaft At any section, J Tc I c z y where axiu shearing stress, ax ax circular or annular cross - section, for a T J c J I J Tc I c Shaft section requireent, all T c J ax in
15 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 Pa, deterine the sallest perissible diaeter for the shaft. Identify critical shaft section fro torque and bending oent diagras. Calculate iniu allowable shaft diaeter. 00 The cgraw-hill Copanies, Inc. All rights reserved. 8-15
16 Saple Proble 8.3 SOLUTION: Deterine the gear torques and corresponding tangential forces. T F T T E E C D P f T r E E 0kW 80Hz 10kW 80Hz 30kW 80Hz 597 N kN 398N 199 N Find reactions at A and B. A B y y 0.93kN.80kN A B z z 597 N 6.kN.90kN F F C D 6.63kN.49 kn 00 The cgraw-hill Copanies, Inc. All rights reserved. 8-16
17 Saple Proble 8.3 Identify critical shaft section fro torque and bending oent diagras. T ax N The cgraw-hill Copanies, Inc. All rights reserved. 8-17
18 Saple Proble 8.3 Calculate iniu allowable shaft diaeter. J c For a solid J c c 3 T all circular shaft, N 50Pa 6 c d 3 c The cgraw-hill Copanies, Inc. All rights reserved. 8-18
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. 00 The cgraw-hill Copanies, Inc. All rights reserved. 8-19
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. 00 The cgraw-hill Copanies, Inc. All rights reserved. 8-0
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. 00 The cgraw-hill Copanies, Inc. All rights reserved. 8-1
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. 00 The cgraw-hill Copanies, Inc. All rights reserved. 8 -
23 Saple Proble 8.5 SOLUTION: Deterine internal forces in Section EFG. V x x y 30 kn 0 50kN kN kN z P 50kN Note: Section properties, A I I x z V 75kN 30kN kN z The cgraw-hill Copanies, Inc. All rights reserved. 8-3
24 Saple Proble 8.5 Evaluate noral stress at H. y 50kN P A I z z a 8.5kN I 3kN x x Pa 66.0Pa Evaluate shearing stress at H. Q yz V I A y z Q t x 17.5 Pa b kN The cgraw-hill Copanies, Inc. All rights reserved. 8-4
25 Saple Proble 8.5 Calculate principal stresses and axiu shearing stress. Deterine principal planes. tan ax ax in p OC R OC R p R CY CD p ax ax in Pa Pa Pa Pa 70.4 Pa 7.4 Pa p The cgraw-hill Copanies, Inc. All rights reserved. 8-5
MECHANICS OF MATERIALS
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
MECHANICS OF MATERIALS Design of a Transmission Shaft
Design of a Transission Shaft If power is transferred to and fro the shaft by ygears or sprocket wheels, the shaft is subjected to transverse loading as well as shear loading. Noral stresses due to transverse
MECHANICS OF MATERIALS Design of a Transmission Shaft
Design of a Transmission Shaft If power is transferred to and from the shaft by gears or sprocket wheels, the shaft is subjected to transverse loading as well as shear loading. Normal stresses due to transverse
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CHATER MECHANICS OF MATERIAS Ferdinand. Beer E. Russell Johnston, Jr. John T. DeWolf Energy Methods ecture Notes: J. Walt Oler Teas Tech niversity 6 The McGraw-Hill Copanies, Inc. All rights reserved.
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Third E CHAPTER 6 Shearing MECHANCS OF MATERALS Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf Lecture Notes: J. Walt Oler Texas Tech University Stresses in Beams and Thin- Walled Members Shearing
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CHAPTER MECHANCS OF MATERALS Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf Lecture Notes: J. Walt Oler Texas Tech University Shearing Stresses in Beams and Thin- Walled Members 006 The McGraw-Hill
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00 The Graw-Hill Copanies, n. All rights reserved. Third E CHAPTER Pure ECHANCS OF ATERALS Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf Bending Leture Notes: J. Walt Oler Teas Teh Universit
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CHAPTER 6 MECHANCS OF MATERALS Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf David F. Mazurek Lecture Notes: J. Walt Oler Texas Tech University Shearing Stresses in Beams and Thin- Walled Members
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Third E CHAPTER 1 Introduction MECHANICS OF MATERIALS Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf Lecture Notes: J. Walt Oler Texas Tech University Concept of Stress Contents Concept of Stress
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2009 The McGraw-Hill Companies, Inc. All rights reserved. Fifth SI Edition CHAPTER 3 MECHANICS OF MATERIALS Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf David F. Mazurek Torsion Lecture Notes:
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009 The McGraw-Hill Companies, nc. All rights reserved. Fifth S E CHAPTER 6 MECHANCS OF MATERALS Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf David F. Mazurek Lecture Notes: J. Walt Oler Texas
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006 The Graw-Hill Copanies, n. ll rights reserved. Fourth E CHTER ure ECHNCS OF TERLS Ferdinand. Beer E. Russell Johnston, Jr. John T. DeWolf Bending Leture Notes: J. Walt Oler Teas Teh Universit ECHNCS
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Third E CHAPTER 2 Stress MECHANICS OF MATERIALS Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf Lecture Notes: J. Walt Oler Texas Tech University and Strain Axial Loading Contents Stress & Strain:
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2009 The McGraw-Hill Companies, Inc. All rights reserved. Fifth SI Edition CHAPTER 6 MECHANICS OF MATERIALS Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf David F. Mazurek Lecture Notes: J.
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009 The McGraw-Hill Companies, Inc. All rights reserved. Fifth SI Edition CHAPTER 7 MECHANICS OF MATERIALS Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf David F. Mazurek Transformations of
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