PROBLEM 7.1 SOLUTION. σ = 5.49 ksi. τ = ksi

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1 PROBLEM 7.1 For the given state of stress, determine the normal and shearing stresses exerted on the oblique face of the shaded triangular element shon. Use a method of analysis based on the equilibrium of that element, as as done in the derivations of Sec. 7.. Stresses Areas Forces F = : σ A 15 A sin 3 cos 3 15 A cos 3 sin A cos 3 cos 3 = σ = 3 sin 3 cos 3 1 cos 3 Σ F = : τ A+ 15 A sin 3 sin 3 15 A cos 3 cos 3 1 A cos 3 sin 3 = τ = 15(cos 3 sin 3 ) + 1 cos 3 sin 3 σ = 5.49 ksi τ = ksi PROPRIETARY MATERIAL. 1 The McGra-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed,

2 PROBLEM 7.4 For the given state of stress, determine the normal and shearing stresses exerted on the oblique face of the shaded triangular element shon. Use a method of analysis based on the equilibrium of that element, as as done in the derivations of Sec. 7.. Stresses Areas Forces Σ F = : σ A+ 18 A cos 15 sin A cos 15 cos 15 7 A sin 15 sin A sin 15 cos 15 = σ = 18 cos 15 sin cos sin sin 15 cos 15 σ = 49. MPa Σ F = : τ A+ 18 A cos 15 cos A cos 15 sin 15 7 A sin 15 cos A sin 15 sin 15 = τ = 18(cos 15 sin 15 ) + (45 + 7) cos 15 sin 15 τ =.41 MPa PROPRIETARY MATERIAL. 1 The McGra-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed,

3 PROBLEM 7.5 For the given state of stress, determine (a) the principal planes, (b) the principal stresses. σ = MPa σ = 4 MPa τ = 35 MPa x y xy (a) τ xy ()(35) tan θ p = 3.5 σ σ = + 4 = x y θ p = 74.5 θ p = 37., 53. (b) σ max, min σx + σy σx σy = ± + τ xy = ± + (35) = 5 ± 3.4 MPa σ = 13. MPa max σ = 8.4 MPa min PROPRIETARY MATERIAL. 1 The McGra-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed,

4 PROBLEM 7.8 For the given state of stress, determine (a) the principal planes, (b) the principal stresses. σ = 8ksi σ = 1ksi τ = 5ksi x y xy (a) τ xy (5) tan θ p = =.5 σ σ 8 1 = x y θ p =.551 θ p = 13.3, 7.7 (b) σ max, min σx + σy σx σ y = ± + τ xy = ± + (5) = ± σ max = ksi σ min = 9.18 ksi PROPRIETARY MATERIAL. 1 The McGra-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed,

5 PROBLEM 7.19 A steel pipe of 1-in. outer diameter is fabricated from 1 -in.-thick plate by 4 elding along a helix that forms an angle of.5 ith a plane perpendicular to the axis of the pipe. Knoing that a 4-kip axial force P and an 8-kip in. torque T, each directed as shon, are applied to the pipe, determine σ and τ in directions, respectively, normal and tangential to the eld. Stresses: 1 d = 1 in., c = d = in., t =.5 in. c = c t = 5.75 in. 1 ( 1 ) ( 1 ) A= π c c = π( 5.75 ) = 9.84 in π π J = c c = ( 5.75 ) = in P σ = A 4 = = ksi 9.84 Tc τ = J (8)() = = 1.53 ksi σ =, σ = ksi, τ = 1.53 ksi x y xy Choose the x and y axes, respectively, tangential and normal to the eld. Then σ = σ and τ = τ θ =.5 y xy σx + σy σx σ y σy = cos θ τxysin θ ( ) [ ( )] = cos sin 45 = 4.7 ksi σ = 4.7 ksi σx σy τxy = sin θ + τxycos θ [ ( )] = sin cos45 =.47 ksi τ =.47 ksi PROPRIETARY MATERIAL. 1 The McGra-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed,

6 PROBLEM 7. To steel plates of uniform cross section 1 8 mm are elded together as shon. Knoing that centric 1-kN forces are applied to the elded plates and that the in-plane shearing stress parallel to the eld is 3 MPa, determine (a) the angle β, (b) the corresponding normal stress perpendicular to the eld. Area of eld: A = = 3 3 (1 1 )(8 1 ) cos β 8 1 cos β m (a) (b) F = : F 1sin β = F = 1sin β kn = 1 1 sin β N s s s sin β Fs τ = 3 1 = = 15 1 sin βcos β A 8 1 / cos β sin βcos β = sin β = = F = : F 1cosβ = F = 1cos14.34 = 9.88 kn A n n n 8 1 = = m cos Fn σ = = = Pa A β = σ = MPa PROPRIETARY MATERIAL. 1 The McGra-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed,

PROBLEM 7.31 SOLUTION. β = τ max = 36.4 MPa. Solve Probs. 7.5 and 7.9, using Mohr s circle.

PROBLEM 7.31 SOLUTION. β = τ max = 36.4 MPa. Solve Probs. 7.5 and 7.9, using Mohr s circle. PROBLEM 7.1 Solve Probs. 7.5 an 7.9, using Mohr s circle. PROBLEM 7.5 through 7.8 For the given state of stress, etermine (a) the principal planes, (b) the principal stresses. PROBLEM 7.9 through 7.1 For

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