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Fourth 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 © 2006 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Principle Stresses Under a Given Loading Introduction Principle Stresses in a Beam Sample Problem 8.1 Sample Problem 8.2 Design of a Transmission Shaft Sample Problem 8.3 Stresses Under Combined Loadings Sample Problem 8.5 © 2006 The McGraw-Hill Companies, Inc. All rights reserved. 8-2 Fourth Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Introduction • In Chapters 1 and 2, you learned how to determine the normal stress due to centric loads. • In Chapter 3, you analyzed the distribution of shearing stresses in a circular member due to a twisting couple. • In Chapter 4, you determined the normal 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 components of stress are transformed by a rotation of the coordinate axes and how to determine the principal planes, principal stresses, and maximum shearing stress at a point. • In Chapter 8, you will learn how to determine the stress in a structural member or machine element due to a combination of loads and how to find the corresponding principal stresses and maximum shearing stress. © 2006 The McGraw-Hill Companies, Inc. All rights reserved. 8-3 Fourth Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Principle Stresses in a Beam • Prismatic beam subjected to transverse loading My Mc σm = I I VQ VQ τ xy = − τm = It It σx = − • Principal stresses determined from methods discussed in Chapter 7 • Determine if the maximum normal stress within the cross-section is larger than Mc σm = I © 2006 The McGraw-Hill Companies, Inc. All rights reserved. 8-4 Fourth Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Principle Stresses in a Beam © 2006 The McGraw-Hill Companies, Inc. All rights reserved. 8-5 Fourth Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Principle Stresses in a Beam • Cross-section shape results in large values of τxy near the surface where σx is also large. • σmax may be greater than σm . © 2006 The McGraw-Hill Companies, Inc. All rights reserved. 8-6 Fourth Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Sample Problem 8.1 SOLUTION: • Determine shear and bending moment in Section A−A’. • Calculate the normal stress at top surface and at flange-web junction. A 160-kN force is applied at the end of a W200 × 52 rolled-steel beam. • Evaluate the shear stress at flangeweb junction. Neglecting the effects of fillets and of stress concentrations, determine whether the normal stresses satisfy a design specification that they be equal to or less than 150 MPa at section A−A’. • Calculate the principal stress at flange-web junction. © 2006 The McGraw-Hill Companies, Inc. All rights reserved. 8-7 Fourth Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Sample Problem 8.1 SOLUTION: • Determine shear and bending moment in Section A−A’. M A = (160 kN )(0.375 m ) = 60 kN - m V A = 160 kN • Calculate the normal stress at top surface and at flange-web junction. MA 60 kN ⋅ m = S 512 × 10−6 m3 = 117.2 MPa σa = 90.4 mm y σb = σ a b = (117.2 MPa ) 103 mm c = 102.9 MPa © 2006 The McGraw-Hill Companies, Inc. All rights reserved. 8-8 Fourth Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Sample Problem 8.1 • Evaluate shear stress at flange-web junction. Q = (204 × 12.6 )96.7 = 248.6 × 103 mm3 = 248.6 × 10− 6 m3 ( ) V AQ (160 kN ) 248.6 × 10− 6 m3 τb = = It 52.7 × 10− 6 m 4 (0.0079 m ) ( ) = 95.5 MPa • Calculate the principal stress at flange-web junction. σ max = 12 σ b + (12 σ b )2 + τ b2 2 102.9 102.9 2 = + + (95.5) 2 2 = 159.9 MPa (> 150 MPa ) Design specification is not satisfied. © 2006 The McGraw-Hill Companies, Inc. All rights reserved. 8-9 Fourth Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Sample Problem 8.2 SOLUTION: • Determine reactions at A and D. • Determine maximum shear and bending moment from shear and bending-moment diagrams. The overhanging beam supports a uniformly distributed load and a concentrated load. Knowing that for the grade of steel to be used σall = 165 MPa and τall = 100 MPa, select the wide-flange beam which should be used. • Calculate required section modulus and select appropriate beam section. • Find maximum normal stress. • Find maximum shearing stress. © 2006 The McGraw-Hill Companies, Inc. All rights reserved. 8 - 10 Fourth Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Sample Problem 8.2 SOLUTION: • Determine reactions at A and D. • Determine maximum shear and bending moment from shear and bending-moment diagrams. • Calculate required section modulus and select appropriate beam section. © 2006 The McGraw-Hill Companies, Inc. All rights reserved. 8 - 11 Fourth Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Sample Problem 8.2 • Find maximum shearing stress. Assuming uniform shearing stress in web, • Find maximum normal stress. © 2006 The McGraw-Hill Companies, Inc. All rights reserved. 8 - 12