07-01 Chap Gere - 7 Analysis of Stress and Strain Plane...

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Plane Stress Problem 7.2-1 An element in plane stress is subjected to stresses s x 5 6500 psi, s y 5 1700 psi, and t xy 5 2750 psi, as shown in the figure. Determine the stresses acting on an element oriented at an angle u5 60° from the x axis, where the angle u is positive when counterclockwise. Show these stresses on a sketch of an element oriented at the angle u . Solution 7.2-1 Plane stress (angle u ) 7 Analysis of Stress and Strain 425 y x O t xy = 2750 psi s y = 1700 psi s x = 6500 psi x O 3450 psi u 5 60 8 5280 psi 2920 psi y s x 5 6500 psi s y 5 1700 psi t xy 5 2750 psi u 5 60 8 5 5280 psi 52 3450 psi s y 1 5 s x 1 s y 2 s x 1 5 2920 psi t x 1 y 1 s x 2 s y 2 sin 2 u 1 t xy cos 2 u s x 1 5 s x 1 s y 2 1 s x 2 s y 2 cos 2 u 1 t xy sin 2 u Problem 7.2-2 Solve the preceding problem for s x 5 80 MPa, s y 5 52 MPa, t xy 5 48 MPa, and 25° (see figure). 52 MPa 48 MPa 80 MPa
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Solution 7.2-2 Plane stress (angle u ) 426 CHAPTER 7 Analysis of Stress and Strain x O 20.1 MPa u 5 25 8 111.8 MPa 20.2 MPa y s x 5 80 MPa s y 5 52 MPa t xy 5 48 MPa u 5 25 8 5 111.8 MPa 5 20.1 MPa s y 1 5 s x 1 s y 2 s x 1 5 20.2 MPa t x 1 y 1 52 s x 2 s y 2 sin 2 u 1 t xy cos 2 u s x 1 5 s x 1 s y 2 1 s x 2 s y 2 cos 2 u 1 t xy sin 2 u Problem 7.2-3 Solve Problem 7.2-1 for s x 9,900 psi, s y 3,400 psi, t xy 5 3,600 psi, and u5 50° (see figure). Solution 7.2-3 Plane stress (angle u ) 3,400 psi 3,600 psi 9,900 psi x O 2,580 psi u 5 50 8 2,540 psi 10,760 psi y s x 9900 psi s y 3400 psi t xy 5 3600 psi u 5 50 8 2540 psi 5 2580 psi s y 1 5 s x 1 s y 2 s x 1 10,760 psi t x 1 y 1 s x 2 s y 2 sin 2 u 1 t xy cos 2 u s x 1 5 s x 1 s y 2 1 s x 2 s y 2 cos 2 u 1 t xy sin 2 u Problem 7.2-4 The stresses acting on element A in the web of a train rail are found to be 42 MPa tension in the horizontal direction and 140 MPa compression in the vertical direction (see figure). Also, shear stresses of magnitude 60 MPa act in the directions shown. Determine the stresses acting on an element oriented at a counterclockwise angle of 48° from the horizontal. Show these stresses on a sketch of an element oriented at this angle. A 60 MPa 140 MPa Side View Cross Section 42 MPa A
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Solution 7.2-4 Plane stress (angle u ) SECTION 7.2 Plane Stress 427 x O 84.2 MPa u 5 48 8 118.2 MPa 20.2 MPa y s x 5 42 MPa s y 52 140 MPa t xy 60 MPa u 5 48 8 118.2 MPa 84.2 MPa s y 1 5 s x 1 s y 2 s x 1 5 20.2 MPa t x 1 y 1 s x 2 s y 2 sin 2 u 1 t xy cos 2 u s x 1 5 s x 1 s y 2 1 s x 2 s y 2 cos 2 u 1 t xy sin 2 u Problem 7.2-5 Solve the preceding problem if the normal and shear stresses acting on element A are 7,500 psi, 20,500 psi, and 4,800 psi (in the directions shown in the figure) and the angle is 30° (counterclockwise). Solution 7.2-5 Plane stress (angle u ) A 4,800 psi 20,500 psi 7,500 psi x O 14,520 psi u 5 30 8 3,660 psi 9,340 psi y s x 5 7,500 psi s y 20,500 psi t xy 4,800 psi u 5 30 8 3,660 psi 14, 520 psi s y 1 5 s x 1 s y 2 s x 1 9,340 psi t x 1 y 1 s x 2 s y 2 sin 2 u 1 t xy cos 2 u s x 1 5 s x 1 s y 2 1 s x 2 s y 2 cos 2 u 1 t xy sin 2 u Problem 7.2-6 An element in plane stress from the fuselage of an airplane is subjected to compressive stresses of magnitude 25.5 MPa in the horizontal direction and tensile stresses of magnitude 6.5 MPa in the vertical direction (see figure). Also, shear stresses of magnitude 12.0 MPa act in the directions shown.
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This note was uploaded on 03/19/2008 for the course E M 316 taught by Professor Korkolis during the Spring '08 term at University of Texas at Austin.

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07-01 Chap Gere - 7 Analysis of Stress and Strain Plane...

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