Analysis of Stress and Strain

07Ch07.qd 9/7/08 1:18 PM Page 571 7 Analsis of Stress and Strain Plane Stress Problem 7.-1 An element in plane stress is subjected to stresses s 4750 psi, s 100 psi, and t 950 psi, as shown in the figure. Determine the stresses acting on an element oriented at an angle u 60 from the ais, where the angle u is positive when counterclockwise. Show these stresses on a sketch of an element oriented at the angle u. 100 psi 950 psi 4750 psi Solution 7.-1 s 4750 psi s 100 psi t 950 psi u 60 s 1 s + s s 1 910 psi cos(u) + t sin(u) t 11 s s t 11 01 psi s 1 s + s s 1 s 1 3040 psi sin(u) + t cos(u) Problem 7.- Solve the preceding problem for an element in plane stress subjected to stresses s 100 MPa, s 80 MPa, and t 8 MPa, as shown in the figure. Determine the stresses acting on an element oriented at an angle u 30 from the ais, where the angle u is positive when counterclockwise. Show these stresses on a sketch of an element oriented at the angle u. 80 MPa 8 MPa 100 MPa 571

07Ch07.qd 9/7/08 1:18 PM Page 57 57 CHAPTER 7 Analsis of Stress and Strain Solution 7.- s 100 MPa s 80 MPa u 30 s 1 s + s s 1 119. MPa t 8MPa cos (u) + t sin (u) t 11 s s t 11 5.30 MPa s 1 s + s s 1 s 1 60.8 MPa sin(u) + t cos(u) Problem 7.-3 Solve Problem 7.-1 for an element in plane stress subjected to stresses s 5700 psi, s 300 psi, and t 500 psi, as shown in the figure. Determine the stresses acting on an element oriented at an angle u 50 from the ais, where the angle u is positive when counterclockwise. Show these stresses on a sketch of an element oriented at the angle u. 300 psi 500 psi 5700 psi Solution 7.-3 s 5700 psi s 300 psi t 500 psi u 50 s 1 s + s s 1 143 psi cos (u) + t sin (u) t 11 s s t 11 140 psi s 1 s + s s 1 s 1 6757 psi sin (u) + t cos (u) Problem 7.-4 The stresses acting on element A in the web of a train rail are found to be 40 MPa tension in the horizontal direction and 160 MPa compression in the vertical direction (see figure). Also, shear stresses of magnitude 54 MPa act in the directions shown. Determine the stresses acting on an element oriented at a counterclockwise angle of 5 from the horizontal. Show these stresses on a sketch of an element oriented at this angle. A Side View Cross Section 160 MPa 40 MPa A 54 MPa

07Ch07.qd 9/7/08 1:18 PM Page 573 SECTION 7. Plane Stress 573 Solution 7.-4 s 40 MPa s 160 MPa t 54 MPa u 5 s 1 s + s s 1 136.6 MPa cos(u) + t sin(u) t 11 s s t 11 84.0 MPa s 1 s + s s 1 s 1 16.6 MPa sin (u) + t cos (u) Problem 7.-5 Solve the preceding problem if the normal and shear stresses acting on element A are 6500 psi, 18,500 psi, and 3800 psi (in the directions shown in the figure). Determine the stresses acting on an element oriented at a counterclockwise angle of 30 from the horizontal. Show these stresses on a sketch of an element oriented at this angle. A Side View Cross Section 18,500 psi 6500 psi A 3800 psi Solution 7.-5 s 6500 psi s 18500 psi t 3800 psi u 30 s 1 s + s s 1 3041 psi cos (u) + t sin (u) t 11 s s t 11 175 psi s 1 s + s s 1 s 1 8959 psi sin (u) + t cos (u) Problem 7.-6 An element in plane stress from the fuselage of an airplane is subjected to compressive stresses of magnitude 7 MPa in the horizontal direction and tensile stresses of magnitude 5.5 MPa in the vertical direction (see figure). Also, shear stresses of magnitude 10.5 MPa act in the directions shown. Determine the stresses acting on an element oriented at a clockwise angle of 35 from the horizontal. Show these stresses on a sketch of an element oriented at this angle. 5.5 MPa 7 MPa 10.5 MPa

07Ch07.qd 9/7/08 1:18 PM Page 574 574 CHAPTER 7 Analsis of Stress and Strain Solution 7.-6 s 7 MPa s 5.5 MPa u 35 s 1 s + s s 1 6.4 MPa t 10.5 MPa cos (u) + t sin (u) t 11 s s t 11 18.9 MPa s 1 s s s 1 s 1 15.1 MPa sin (u) + t cos (u) Problem 7.-7 The stresses acting on element B in the web of a wide-flange beam are found to be 14,000 psi compression in the horizontal direction and 600 psi compression in the vertical direction (see figure). Also, shear stresses of magnitude 3800 psi act in the directions shown. Determine the stresses acting on an element oriented at a counterclockwise angle of 40 from the horizontal. Show these stresses on a sketch of an element oriented at this angle. B Side View Cross Section 600 psi 14,000 psi B 3800 psi Solution 7.-7 s 14000 psi s 600 psi t 3800 psi u 40 s 1 s + s s 1 1303 psi cos (u) + t sin (u) t 11 s s t 11 4954 psi s 1 s + s s 1 s 1 3568 psi sin (u) + t cos (u) Problem 7.-8 Solve the preceding problem if the normal and shear stresses acting on element B are 46 MPa, 13 MPa, and 1 MPa (in the directions shown in the figure) and the angle is 4.5 (clockwise). 13 MPa 1 MPa B 46 MPa

07Ch07.qd 9/7/08 1:18 PM Page 575 SECTION 7. Plane Stress 575 Solution 7.-8 s 46 MPa s 13 MPa t 1 MPa u 4.5 s 1 s + s s 1 51.9 MPa cos (u) + t sin (u) t 11 s s t 11 14.6 MPa s 1 s + s s 1 s 1 7.1 MPa sin (u) + t cos (u) Problem 7.-9 The polethlene liner of a settling pond is subjected to stresses s 350 psi, s 11 psi, and t 10 psi, as shown b the plane-stress element in the first part of the figure. Determine the normal and shear stresses acting on a seam oriented at an angle of 30 to the element, as shown in the second part of the figure. Show these stresses on a sketch of an element having its sides parallel and perpendicular to the seam. O 11 psi 350 psi 10 psi 30 Seam Solution 7.-9 Plane stress (angle ) s 350 psi s 11 psi t 10 psi u 30 s 1 s + s 187 psi cos u + t sin u The normal stress on the seam equals 187 psi tension. The shear stress on the seam equals 163 psi, acting clockwise against the seam. t 1 1 s s 163 psi sin u + t cos u s 1 s + s s 1 75 psi

07Ch07.qd 9/7/08 1:18 PM Page 576 576 CHAPTER 7 Analsis of Stress and Strain Problem 7.-10 Solve the preceding problem if the normal and shear stresses acting on the element are s 100 kpa, s 300 kpa, and t 560 kpa, and the seam is oriented at an angle of.5 to the element (see figure). 300 kpa.5 O 100 kpa 560 kpa Seam Solution 7.-10 Plane stress (angle ) s 1 s + s 1440 kpa t 1 1 s s + s s 1030 kpa cos u + t sin u sin u + t cos u s 1 s + s s 1 960 kpa s 100 kpa s 300 kpa t 560 kpa u.5 The normal stress on the seam equals 1440 kpa tension. The shear stress on the seam equals 1030 kpa, acting clockwise against the seam. Problem 7.-11 A rectangular plate of dimensions 3.0 in. * 5.0 in. is formed 350 psi b welding two triangular plates (see figure). The plate is subjected to a tensile stress of 500 psi in the long direction and a compressive stress of 350 psi in the short direction. Determine the normal stress s w acting perpendicular to the line of the weld and the shear t w acting parallel to the weld. (Assume that the normal stress s w is positive when it acts in tension against the weld and the shear stress t w is 3 in. 5 in. positive when it acts counterclockwise against the weld.) Weld 500 psi

07Ch07.qd 9/7/08 1:18 PM Page 577 SECTION 7. Plane Stress 577 Solution 7.-11 Biaial stress (welded joint) t 1 1 s s sin u + t cos u 375 psi s 1 s + s s 1 15 psi STRESSES ACTING ON THE WELD s 500 psi s 350 psi t 0 u arctan 3 in. 5 in. s 1 s + s 75 psi arctan 0.6 30.96 cos u + t sin u s w 15 psi t w 375 psi Problem 7.-1 Solve the preceding problem for a plate of dimensions 100 mm * 50 mm subjected to a compressive stress of.5 MPa in the long direction and a tensile stress of 1.0 MPa in the short direction (see figure). Weld 1.0 MPa 100 mm 50 mm.5 MPa Solution 7.-1 Biaial stress (welded joint) t 1 1 s s s 1 s + s s 1 10.0 MPa sin u + t cos u 5.0 MPa STRESSES ACTING ON THE WELD s.5 MPa s 1.0 MPa t 0 u arctan s 1 s + s 100 mm 50 mm 0.5 MPa arctan 0.4 1.80 cos u + t sin u s w 10.0 MPa t w 5.0 MPa

07Ch07.qd 9/7/08 1:18 PM Page 578 578 CHAPTER 7 Analsis of Stress and Strain Problem 7.-13 At a point on the surface of a machine the material is in biaial stress with s 3600 psi, and s 1600 psi, as shown in the first part of the figure. The second part of the figure shows an inclined plane aa cut through the same point in the material but oriented at an angle u. Determine the value of the angle u between zero and 90 such that no normal stress acts on plane aa. Sketch a stress element having plane aa as one of its sides and show all stresses acting on the element. O 1600 psi 3600 psi a a u Solution 7.-13 Biaial stress s 3600 psi s 1600 psi t 0 Find angle u for s 0. s normal stress on plane a-a STRESS ELEMENT s 1 0 u 56.31 s 1 s + s s 1 000 psi t 1 1 s s 400 psi sin u + t cos u s 1 s + s 1000 + 600 cos u(psi) For s 1 0, we obtain cos u + t sin u cos u 1000 600 u 11.6 and u 56.31 Problem 7.-14 Solve the preceding problem for s 3 MPa and s 50 MPa (see figure). 50 MPa a O 3 MPa u a

07Ch07.qd 9/7/08 1:18 PM Page 579 SECTION 7. Plane Stress 579 Solution 7.-14 Biaial stress s 3 MPa s 50 MPa t 0 Find angles u for s 0. s normal stress on plane a-a STRESS ELEMENT s 1 0 u 38.66 s 1 s + s s 1 18 MPa t 1 1 s s 40 MPa sin u + t cos u s 1 s + s cos u + t sin u 9 + 41 cos u ( MPa) For s, we obtain cos u 9 1 0 41 u 77.3 and u 38.66 Problem 7.-15 An element in plane stress from the frame of a racing car is oriented at a known angle u (see figure). On this inclined element, the normal and shear stresses have the magnitudes and directions shown in the figure. Determine the normal and shear stresses acting on an element whose sides are parallel to the aes, that is, determine s, s, and t. Show the results on a sketch of an element oriented at u 0. 3950 psi O 475 psi u = 40 14,900 psi Solution 7.-15 Transform from u 40 to u 0 s 14900 psi s 3950 psi t 475 psi u 40 s 1 s + s s 1 1813 psi cos (u) + t sin (u) t 11 s s t 11 496 psi s 1 s + s s 1 s 1 6037 psi sin (u) + t cos (u)

07Ch07.qd 9/7/08 1:18 PM Page 580 580 CHAPTER 7 Analsis of Stress and Strain Problem 7.-16 figure. Solve the preceding problem for the element shown in the 4.3 MPa 6.5 MPa u = 55 4.0 MPa O Solution 7.-16 Transform from u 55 to u 0 s 4.3 MPa s 6.5 MPa t 4 MPa u 55 s 1 s + s s 1 56.5 MPa cos (u) + t sin (u) t 11 s s t 11 3.6 MPa s 1 s + s s 1 s 1 18.3 MPa sin (u) + t cos (u) Problem 7.-17 A plate in plane stress is subjected to normal stresses s and s and shear stress t, as shown in the figure. At counterclockwise angles u 35 and u 75 from the ais, the normal stress is 4800 psi tension. If the stress s equals 00 psi tension, what are the stresses s and t? s O t s = 00 psi Solution 7.-17 s 00 psi s unknown t unknown For At u 35 and u 75, s 1 4800 psi Find s and t s 1 s + s cos (u) + t sin (u) u 35 s 1 4800 psi 4800 psi 00 psi + s + 00 psi s * cos (70 ) + t sin(70 ) or 0.3899 s + 0.93969 t 333.8 psi (1)

07Ch07.qd 9/7/08 1:18 PM Page 581 SECTION 7. Plane Stress 581 For u 75 : or 0.93301s + 0.50000 t 465.6 psi () s 1 4800 psi Solve Eqs. (1) and (): 4800 psi 00 psi + s + 00 psi s s 3805 psi t 05 psi * cos (150 ) + t sin(150 ) Problem 7.-18 The surface of an airplane wing is subjected to plane stress with normal stresses s and s and shear stress t, as shown in the figure. At a counterclockwise angle u 3 from the ais, the normal stress is 37 MPa tension, and at an angle u 48, it is 1 MPa compression. If the stress s equals 110 MPa tension, what are the stresses s and t? O s t s = 110 MPa Solution 7.-18 s 110 MPa s unknown s unknown or 0.8081s + 0.89879t 4.11041 MPa (1) At u 3, s 1 37 MPa (tension) For u 48 : At u 48, s 1 1 MPa (compression) s 1 1 MPa Find s and t s 1 s + s + s s cos(u) + t sin(u) 1 MPa 110 MPa + s + 110 MPa s * cos (96 ) + t sin (96 ) or 0.556s + 0.9945t 61.5093 MPa () For u 3 s 1 37 MPa Solve Eqs. (1) and (): s 60.7 MPa t 7.9 MPa 37 MPa 110 MPa + s + 110 MPa s * cos (64 ) + t sin (64 )

07Ch07.qd 9/7/08 1:18 PM Page 58 58 CHAPTER 7 Analsis of Stress and Strain Problem 7.-19 At a point in a structure subjected to plane stress, the stresses are s 4100 psi, s 00 psi, and t 900 psi (the sign convention for these stresses is shown in Fig. 7-1). A stress element located at the same point in the structure (but oriented at a counterclockwise angle u 1 with respect to the ais) is subjected to the stresses shown in the figure (s b, t b, and 1800 psi). Assuming that the angle u 1 is between zero and 90, calculate the normal stress s b, the shear stress t b, and the angle u 1 s b O t b 1800 psi u 1 Solution 7.-19 s 4100 psi s 00 psi t 900 psi For s 1 1800 psi s 1 s b t 11 t b Find s b, t b, and u 1 Stress s b s + s 1800 psi s b 3700 psi Angle u u 1 : u 1 s b s 1 s + s cos (u) + t sin (u) 1800 psi 950 psi 3150 psi cos1u 1 + 900 psi sin 1u 1 SOLVE NUMERICALLY: u 1 87.3 u 1 43.7 Shear Stress t b t b s s t b 38 psi sin 1u 1 + t cos 1u 1 Principal Stresses and Maimum Shear Stresses When solving the problems for Section 7.3, consider onl the in-plane stresses (the stresses in the plane). Problem 7.3-1 An element in plane stress is subjected to stresses s 4750 psi, s 100 psi, and t 950 psi (see the figure for Problem 7.-1). Determine the principal stresses and show them on a sketch of a properl oriented element. Solution 7.3-1 s 4750 psi s 100 psi t 950 psi PRINCIPAL STRESSES atana t b s s u p1 u p1 14.08 u p u p1 + 90 u p 104.08 s 1 s + s s s + s s 1 4988 psi s 96 psi cos 1u p1 + t sin 1u p1 cos 1u p + t sin 1u p

07Ch07.qd 9/7/08 1:18 PM Page 583 SECTION 7.3 Principal Stresses and Maimum Shear Stresses 583 Problem 7.3- An element in plane stress is subjected to stresses s 100 MPa, s 80 MPa, and t 8 MPa (see the figure for Problem 7.-). Determine the principal stresses and show them on a sketch of a properl oriented element. Solution 7.3- s 100 MPa s 80 MPa t 8 MPa PRINCIPAL STRESSES atana t b s s u p1 u p1 35. u p u p1 + 90 u p 15.17 s 1 s + s s s + s s 1 10 MPa s 60 MPa cos1u p1 + t sin1u p1 cos1u p + t sin1u p Problem 7.3-3 An element in plane stress is subjected to stresses s 5700 psi, s 300 psi, and t 500 psi (see the figure for Problem 7.-3). Determine the principal stresses and show them on a sketch of a properl oriented element. Solution 7.3-3 s 5700 psi s 300 psi t 500 psi PRINCIPAL STRESSES t atana b s s u p u p 7.89 u p1 u p + 90 u p1 6.1 s 1 s + s s s + s s 1 977 psi s 703 psi cos1u p1 + t sin1u p1 cos1u p + t sin1u p Problem 7.3-4 The stresses acting on element A in the web of a train rail are found to be 40 MPa tension in the horizontal direction and 160 MPa compression in the vertical direction (see figure). Also, shear stresses of magnitude 54 MPa act in the directions shown (see the figure for Problem 7.-4). Determine the principal stresses and show them on a sketch of a properl oriented element.

07Ch07.qd 9/7/08 1:18 PM Page 584 584 CHAPTER 7 Analsis of Stress and Strain Solution 7.3-4 s 40 MPa s 160 MPa t 54 MPa PRINCIPAL STRESSES t atana b s s u p1 u p1 14. u p u p1 + 90 u p 75.8 s 1 s + s s s + s s 1 53.6 MPa s 173.6 MPa cos 1u p1 + t sin 1u p1 cos 1u p + t sin 1u p Problem 7.3-5 The normal and shear stresses acting on element A are 6500 psi, 18,500 psi, and 3800 psi (in the directions shown in the figure) (see the figure for Problem 7.-5). Determine the maimum shear stresses and associated normal stresses and show them on a sketch of a properl oriented element. Solution 7.3-5 s 6500 psi s 18500 psi t 3800 psi PRINCIPAL ANGLES atana b s s u p1 u p1 8.45 u p u p1 + 90 u p 81.55 s 1 s + s t cos 1u p1 + t sin 1u p1 s 1 7065 psi s 19065 psi MAXIMUM SHEAR STRESSES t ma a s s b + t A t ma 13065 psi u s1 u p1 45 u s1 53.4 s aver s + s s aver 6000 psi s s + s cos 1u p + t sin 1u p Problem 7.3-6 An element in plane stress from the fuselage of an airplane is subjected to compressive stresses of magnitude 7 MPa in the horizontal direction and tensile stresses of magnitude 5.5 MPa in the vertical direction. Also, shear stresses of magnitude 10.5 MPa act in the directions shown (see the figure for Problem 7.-6). Determine the maimum shear stresses and associated normal stresses and show them on a sketch of a properl oriented element.

07Ch07.qd 9/7/08 1:18 PM Page 585 SECTION 7.3 Principal Stresses and Maimum Shear Stresses 585 Solution 7.3-6 s 7 MPa s 5.5 MPa t 10.5 MPa PRINCIPAL ANGLES t atana b s s u p u p 16.43 u p1 u p + 90 u p1 106.43 s 1 s + s s s + s cos 1u p1 + t sin 1u p1 cos 1u p + t sin 1u p s 1 8.6 MPa s 30.1 MPa MAXIMUM SHEAR STRESSES t ma a s s b + t A t ma 19.3 MPa u s1 u p1 45 u s1 61.4 s aver s + s s aver 10.8 MPa Problem 7.3-7 The stresses acting on element B in the web of a wide-flange beam are found to be 14,000 psi compression in the horizontal direction and 600 psi compression in the vertical direction. Also, shear stresses of magnitude 3800 psi act in the directions shown (see the figure for Problem 7.-7). Determine the maimum shear stresses and associated normal stresses and show them on a sketch of a properl oriented element. Solution 7.3-7 s 14000 psi s 600 psi t 3800 psi PRINCIPAL ANGLES atana b s s u p u p 16.85 u p1 u p + 90 u p1 106.85 s 1 s + s t cos1u p1 + t sin1u p1 s s + s s 1 1449 psi s 15151 psi MAXIMUM SHEAR STRESSES cos1u p + t sin1u p t ma a s s b + t A t ma 6851 psi u s1 u p1 45 u s1 61.8 s aver s + s s aver 8300 psi Problem 7.3-8 The normal and shear stresses acting on element B are s 46 MPa, s 13 MPa, and t 1 MPa (see figure for Problem 7.-8). Determine the maimum shear stresses and associated normal stresses and show them on a sketch of a properl oriented element.

07Ch07.qd 9/7/08 1:18 PM Page 586 586 CHAPTER 7 Analsis of Stress and Strain Solution 7.3-8 s 46 MPa s 13 MPa t 1 MPa PRINCIPAL ANGLES atana b s s u p u p 5.9 u p1 u p + 90 u p1 64.08 s 1 s + s t cos 1u p1 + t sin 1u p1 s 1.8 MPa s 56. MPa MAXIMUM SHEAR STRESSES t ma a s s b + t A t ma 6.7 MPa u s1 u p1 45 u s1 19.08 s aver s s s aver 9.5 MPa s s + s cos 1u p + t sin 1u p Problem 7.3-9 A shear wall in a reinforced concrete building is subjected to a vertical uniform load of intensit q and a horizontal force H, as shown in the first part of the figure. (The force H represents the effects of wind and earthquake loads.) As a consequence of these loads, the stresses at point A on the surface of the wall have the values shown in the second part of the figure (compressive stress equal to 1100 psi and shear stress equal to 480 psi). A q H 480 psi A 1100 psi (a) Determine the principal stresses and show them on a sketch of a properl oriented element. (b) Determine the maimum shear stresses and associated normal stresses and show them on a sketch of a properl oriented element. Solution 7.3-9 Shear wall s 0 s 1100 psi t 480 psi (a) PRINCIPAL STRESSES Therefore, s 1 180 psi and u p1 0.56 s 180 psi and u p 69.44 f tan u p t s s 0.8773 u p 41.11 and u p 0.56 u p 138.89 and u p 69.44 s 1 s + s cos u + t sin u For u p 41.11 : s 1 180 psi For u p 138.89 : s 1 180 psi

07Ch07.qd 9/7/08 1:18 PM Page 587 SECTION 7.3 Principal Stresses and Maimum Shear Stresses 587 (b) MAXIMUM SHEAR STRESSES t ma a s s b + t A 730 psi u s1 u p1 45 65.56 and t 730 psi u s u p1 + 45 4.44 and t 730 psi f s aver s + s 550 psi Problem 7.3-10 A propeller shaft subjected to combined torsion and aial thrust is designed to resist a shear stress of 56 MPa and a compressive stress of 85 MPa (see figure). (a) Determine the principal stresses and show them on a sketch of a properl oriented element. (b) Determine the maimum shear stresses and associated normal stresses and show them on a sketch of a properl oriented element. 85 MPa 56 MPa Solution 7.3-10 s 85 MPa s 0 MPa T 56 MPa (a) PRINCIPAL STRESSES s 1 7.8 MPa s 11.8 MPa atana b s s u p u p 6.4 u p1 u p + 90 u p1 116.4 s 1 s + s t cos 1u p1 + t sin 1u p1 (b) MAXIMUM SHEAR STRESSES t ma a s s b + t A t ma 70.3 MPa u s1 u p1 45 u s1 71.4 s aver s + s s aver 4.5 MPa s s + s cos 1u p + t sin 1u p

07Ch07.qd 9/7/08 1:18 PM Page 588 588 CHAPTER 7 Analsis of Stress and Strain Problems 7.3-11 s 500 psi, s 100 psi, t 900 psi (a) Determine the principal stresses and show them on a sketch of a properl oriented element. (b) Determine the maimum shear stresses and associated normal stresses and show them on a sketch of a properl oriented element. s t s O Probs. 7.3-11 through 7.3-16 Solution 7.3-11 s 500 psi s 100 psi t 900 psi (a) PRINCIPAL STRESSES tan(u p ) u p1 5.9 u p 90 + u p1 u p 64.71 s 1 s + s s s + s t s s u p1 t atana b s s cos 1u p1 + t sin 1u p1 cos 1u p + t sin 1u p Therefore, For u p1 5.3 : s 1 95 psi For u p 64.7 : (b) MAXIMUM SHEAR STRESSES t ma A t ma 1165 psi s 595 psi a s s b + t u s1 u p1 45 u s1 70.3 and t 1 1165 psi u s u p1 45 u s 19.71 and t 1165 psi s aver s + s s aver 1760 psi Problems 7.3-1 s 150 kpa, s 375 kpa, t 460 kpa (a) Determine the principal stresses and show them on a sketch of a properl oriented element. (b) Determine the maimum shear stresses and associated normal stresses and show them on a sketch of a properl oriented element. Solution 7.3-1 s 150 kpa s 375 kpa t 460 kpa (a) PRINCIPAL STRESSES u p1 13.70 u p 90 + u p1 u p 76.30 tan(u p ) t s s u p1 t atana b s s s 1 s + s s s + s cos 1u p1 + t sin 1u p1 cos 1u p + t sin 1u p

07Ch07.qd 9/7/08 1:18 PM Page 589 SECTION 7.3 Principal Stresses and Maimum Shear Stresses 589 Therefore, For u p1 13.70 s 1 6 kpa For u p 76.3 (b) MAXIMUM SHEAR STRESSES s 63 kpa t ma a s s b + t A t ma 145 psi u s1 u p1 45 u s1 58.7 and t 1 1000 kpa u s u p1 + 45 u s 31.3 and t 1000 kpa s aver s + s s aver 163 kpa Problems 7.3-13 s 14,500 psi, s 1070 psi, t 1900 psi (a) Determine the principal stresses and show them on a sketch of a properl oriented element. (b) Determine the maimum shear stresses and associated normal stresses and show them on a sketch of a properl oriented element. Solution 7.3-13 s 14500 psi s 1070 psi t 1900 psi (a) PRINCIPAL STRESSES tan(u p ) u p1 7.90 u p 90 + u p1 u p 97.90 s 1 s + s s s + s t s s u p1 t atana b s s cos 1u p1 + t sin 1u p1 cos 1u p + t sin 1u p Therefore, For u p1 7.90 s 1 14764 psi For u p 97.9 s 806 psi (b) MAXIMUM SHEAR STRESSES t ma a s s b + t A t ma 6979 psi u s1 u p1 45 u s1 37.1 and t 1 6979 psi u s u p1 + 45 u s 5.9 and t 6979 psi s aver s + s s aver 7785 psi Problems 7.3-14 s 16.5 MPa, s v 91 MPa, t 39 MPa (a) Determine the principal stresses and show them on a sketch of a properl oriented element. (b) Determine the maimum shear stresses and associated normal stresses and show them on a sketch of a properl oriented element.

07Ch07.qd 9/7/08 1:18 PM Page 590 590 CHAPTER 7 Analsis of Stress and Strain Solution 7.3-14 s 16.5 MPa s 91 MPa t 39 MPa (a) PRINCIPAL STRESSES tan(u p ) u p1 17.98 u p 90 + u p1 u p 7.0 s 1 s + s s s + s t s s u p1 cos 1u p1 + t sin 1u p1 cos 1u p + t sin 1u p Therefore, For u p1 17.98 s 1 9. MPa For u p 7.0 s 103.7 MPa t atana b s s (b) MAXIMUM SHEAR STRESSES t ma a s s b + t A t ma 9631.7 psi u s1 u p1 45 u s1 63.0 and t 1 66.4 MPa u s u p1 + 45 u s 7.0 and t 66.4 MPa s aver s + s s aver 37.3 MPa Problems 7.3-15 s 3300 psi, s 11,000 psi, t 4500 psi (a) Determine the principal stresses and show them on a sketch of a properl oriented element. (b) Determine the maimum shear stresses and associated normal stresses and show them on a sketch of a properl oriented element. Solution 7.3-15 s 3300 psi s 11000 psi t 4500 psi Therefore, For u (a) p1 4.7 s 1 18 psi PRINCIPAL STRESSES For u p 114.7 s 1307 psi tan(u p ) u p1 4.73 u p 90 + u p1 u p 114.73 s 1 s + s s s + s t s s u p1 t atana b s s cos 1u p1 + t sin 1u p1 cos 1u p + t sin 1u p (b) MAXIMUM SHEAR STRESSES t ma a s s b + t A t ma 59 psi u s1 u p1 45 u s1 0.3 and t 1 59 psi u s u p1 + 45 u s 69.7 and t 59 psi s aver s + s s aver 7150 psi

07Ch07.qd 9/7/08 1:18 PM Page 591 SECTION 7.3 Principal Stresses and Maimum Shear Stresses 591 Problems 7.3-16 s 108 MPa, s 58 MPa, t 58 MPa (a) Determine the principal stresses and show them on a sketch of a properl oriented element. (b) Determine the maimum shear stresses and associated normal stresses and show them on a sketch of a properl oriented element. Solution 7.3-16 s 108 MPa s 58 MPa t 58 MPa (a) PRINCIPAL STRESSES tan(u p ) u p 17.47 u p1 90 + u p u p1 107.47 s 1 s + s t s s u p t atana b s s cos 1u p1 + t sin 1u p1 (b) MAXIMUM SHEAR STRESSES t ma a s s b + t A t ma 14686.1 psi u s1 u p1 45 u s1 6.47 and t 1 101.3 MPa u s u p1 + 45 u s 15.47 and t 101.3 MPa s aver s + s s aver 5.0 MPa s s + s cos 1u p + t sin 1u p Therefore, For u p1 107.47 s 1 76.3 MPa For u p 17.47 s 16.3 MPa Problem 7.3-17 At a point on the surface of a machine component, the stresses acting on the face of a stress element are s 5900 psi and t 1950 psi (see figure). What is the allowable range of values for the stress s if the maimum shear stress is limited to t 0 500 psi? O s t = 1950 psi s = 5900 psi

07Ch07.qd 9/7/08 1:18 PM Page 59 59 CHAPTER 7 Analsis of Stress and Strain Solution 7.3-17 s 5900 psi s unknown t 1950 psi Find the allowable range of values for s if the maimum allowable shear stresses is t ma 500 psi t ma a s s b + t (1) A Solve for s Therefore, 771 psi s 909 psi From Eq. (1): t ma (s 1 ) a s s 1 b + t A t ma t s s J at ma t b K s a 909 771 b psi.771 ksi 9.09 ksi t ma (σ 1 ).5 ksi σ 1 Problem 7.3-18 At a point on the surface of a machine component the stresses acting on the face of a stress element are s 4 MPa and t 33 MPa (see figure). What is the allowable range of values for the stress s if the maimum shear stress is limited to t 0 35 MPa? s O t = 33 MPa s = 4 MPa Solution 7.3-18 s 4 MPa s unknown t 33 MPa Find the allowable range of values for s if the maimum allowable shear stresses is t ma 35 MPa t ma a s s b + t A (1)

07Ch07.qd 9/7/08 1:18 PM Page 593 SECTION 7.3 Principal Stresses and Maimum Shear Stresses 593 Solve for s From Eq. (1): t ma t s s J at ma t b K Therefore, 18.7 MPa s 65.3 MPa s a 65.3 18.7 b MPa t ma (s 1 ) a s s 1 b + t A 18.7 MPa 65.3 MPa t ma (σ 1 ) 35 MPa σ 1 Problem 7.3-19 An element in plane stress is subjected to stresses s 5700 psi and t 300 psi (see figure). It is known that one of the principal stresses equals 6700 psi in tension. (a) Determine the stress s. (b) Determine the other principal stress and the orientation of the principal planes, then show the principal stresses on a sketch of a properl oriented element. O s 5700 psi 300 psi Solution 7.3-19 s 5700 psi s unknown t 300 psi Solve for s s 1410 psi (a) STRESS s Because s is smaller than a given principal stress, we know that the given stress is the larger principal stress. s 1 6700 psi s 1 s + s + a s s b + t A (b) PRINCIPAL STRESSES t tan (u p ) t atana b s s u s s p1 u p1 3.50 u p 90 + u p1 u p 66.50

07Ch07.qd 9/7/08 1:18 PM Page 594 594 CHAPTER 7 Analsis of Stress and Strain s 1 s + s cos 1u p1 + t sin1u p1 Therefore, For u p1 3.5 : s 1 6700 psi For u p 66.5 : s 410 psi s s + s cos 1u p + t sin 1u p Problem 7.3-0 An element in plane stress is subjected to stresses s 50 MPa and t 4 MPa (see figure). It is known that one of the principal stresses equals 33 MPa in tension. (a) Determine the stress s. (b) Determine the other principal stress and the orientation of the principal planes, then show the principal stresses on a sketch of a properl oriented element. O s 4 MPa 50 MPa Solution 7.3-0 s 50 MPa s unknown t 4 MPa (a) STRESS s Because s is smaller than a given principal stress, we know that the given stress is the larger principal stress. s 1 33 MPa s 1 s + s Solve for s s 11.7 MPa (b) PRINCIPAL STRESSES tan(u p ) + a s s b + t A t s s u p t atana b s s u p 6.85 u p1 90 + u p u p1 63.15 s 1 s + s cos 1u p1 + t sin 1u p1 s s + s cos 1u p + t sin 1u p Therefore, For u p1 63. : s 1 33.0 MPa For u p 6.8 : s 71.3 MPa

07Ch07.qd 9/7/08 1:19 PM Page 595 SECTION 7.4 Mohr s Circle 595 Mohr s Circle The problems for Section 7.4 are to be solved using Mohr s circle. Consider onl the in-plane stresses (the stresses in the plane). Problem 7.4-1 An element in uniaial stress is subjected to tensile stresses s 11,375 psi, as shown in the figure. Using Mohr s circle, determine: (a) The stresses acting on an element oriented at a counterclockwise angle u 4 from the ais. (b) The maimum shear stresses and associated normal stresses. Show all results on sketches of properl oriented elements. O 11,375 psi Solution 7.4-1 s 11375 psi s 0 psi t 0 psi s 1 188 psi (a) ELEMENT AT u 4 (b) MAXIMUM SHEAR STRESSES u 48 R s R 5688 psi Point C: s c R s c 5688 psi Point D: s 1 R + R cos(u) s 1 9493 psi t 11 R sin (u) t 11 47 psi PointD œ : s1 R R cos (u) Point S1: u s1 90 u s1 45 t ma R t ma 5688 psi Point S: u s 90 u s 45 t ma R t ma 5688 psi s aver R s aver 5688 psi Problem 7.4- An element in uniaial stress is subjected to tensile stresses s 49 MPa, as shown in the figure Using Mohr s circle, determine: (a) The stresses acting on an element oriented at an angle u 7 from the ais (minus means clockwise). (b) The maimum shear stresses and associated normal stresses. Show all results on sketches of properl oriented elements. O 49 MPa Solution 7.4- s 49 MPa s 0 MPa t 0 MPa Point D: (a) ELEMENT AT u 7 u 54.0 R s R 4.5 MPa Point C: s c R s c 4.5 MPa s 1 R + R cos ( u ) s 1 38.9 MPa t 11 R sin (u) t 11 19.8 MPa Point D œ s 1 R R cos ( u ) s 1 10.1 MPa

07Ch07.qd 9/7/08 1:19 PM Page 596 596 CHAPTER 7 Analsis of Stress and Strain (b) MAXIMUM SHEAR STRESSES Point S1: u s1 90 u s1 45.0 t ma R t ma 4.5 MPa Point S: u s 90 u s 45.0 t ma R t ma 4.5 MPa s aver R s aver 4.5 MPa Problem 7.4-3 An element in uniaial stress is subjected to compressive stresses of magnitude 6100 psi, as shown in the figure. Using Mohr s circle, determine: (a) The stresses acting on an element oriented at a slope of 1 on (see figure). (b) The maimum shear stresses and associated normal stresses. Show all results on sketches of properl oriented elements. O 6100 psi 1 Solution 7.4-3 s 6100 psi s 0 psi t 0 psi (a) ELEMENT AT A SLOPE OF 1 ON u atana 1 b u 6.565 u 53.130 R s Point C: Point D: s c R s c 3050 psi s 1 R + R cos (u) R 3050 psi s 1 4880 psi t 11 R sin (u) t 11 440 psi Point D œ : s 1 R R cos (u) (b) MAXIMUM SHEAR STRESSES Point S1: t ma R t ma 3050 psi Point S: s 1 10 psi u s1 90 u s1 45 u s 90 u s 45 t ma R t ma 3050 psi s aver R s aver 3050 psi Problem 7.4-4 An element in biaial stress is subjected to stresses s 48 MPa and s 19 MPa, as shown in the figure. Using Mohr s circle, determine: (a) The stresses acting on an element oriented at a counterclockwise angle u 5 from the ais. (b) The maimum shear stresses and associated normal stresses. 19 MPa Show all results on sketches of properl oriented elements. O 48 MPa

07Ch07.qd 9/7/08 1:19 PM Page 597 SECTION 7.4 Mohr s Circle 597 Solution 7.4-4 s 48 MPa s 19 MPa t 0 MPa (a) ELEMENT AT u 5 u 50.0 deg R s + s R 33.5 MPa Point C: s s + R Point D: s 1 s c R cos(u) s 1 36.0 MPa t 11 R sin(u) t 11 5.7 MPa Point D œ : s 1 s c + R cos(u) s 1 7.0 MPa s c 14.5 MPa (b) MAXIMUM SHEAR STRESSES Point S1: u s1 90 u s1 45.0 t ma R t ma 33.5 MPa Point S: u s 90 u s 45.0 t ma R t ma 33.5 MPa s aver s c s aver 14.5 MPa Problem 7.4-5 An element in biaial stress is subjected to stresses s 650 psi and s 1750 psi, as shown in the figure. Using Mohr s circle, determine: (a) The stresses acting on an element oriented at a counterclockwise angle u 55 from the ais. (b) The maimum shear stresses and associated normal stresses. Show all results on sketches of properl oriented elements. O 1750 psi 650 psi Solution 7.4-5 s 650 psi s 1750 psi t 0 psi (a) ELEMENT AT u 60 u 10 R s + s R 4000 psi Point C: s c s R s c 50 psi Point D: s 1 s c + R cos(u) s 1 50 psi t 11 R sin (u) t 11 3464 psi Point D œ : s 1 s c R cos(u) s 1 450 psi (b) MAXIMUM SHEAR STRESSES Point S1: u s1 90 u s1 45 t ma R t ma 4000 psi Point S: u s 90 u s 45 t ma R t ma 4000 psi s aver s c s aver 50 psi

07Ch07.qd 9/7/08 1:19 PM Page 598 598 CHAPTER 7 Analsis of Stress and Strain Problem 7.4-6 An element in biaial stress is subjected to stresses s 9 MPa and s 57 MPa, as shown in the figure. Using Mohr s circle, determine: (a) The stresses acting on an element oriented at a slope of 1 on.5 (see figure). (b) The maimum shear stresses and associated normal stresses. Show all results on sketches of properl oriented elements. 57 MPa 1.5 O 9 MPa Solution 7.4-6 s 9 MPa s 57 MPa t 0 MPa (a) ELEMENT AT A SLOPE OF 1 ON.5 u atana 1 b u 1.801.5 u 43.603 R s + s R 43.0 MPa Point C: s c s + R s c 14.0 MPa Point D: s 1 s c R cos (u) t 11 R sin (u) t 11 9.7 MPa Point D œ : s 1 s c + R cos (u) s 1 45.1 MPa s 1 17.1 MPa (b) MAXIMUM SHEAR STRESSES Point S1: u s1 90 u s1 45.0 t ma R t ma 43.0 MPa Point S: u s 90 u s 45.0 t ma R t ma 43.0 MPa s aver s c s aver 14.0 MPa Problem 7.4-7 An element in pure shear is subjected to stresses t 700 psi, as shown in the figure. Using Mohr s circle, determine: (a) The stresses acting on an element oriented at a counterclockwise angle u 5 from the ais. (b) The principal stresses. Show all results on sketches of properl oriented elements. O 700 psi

07Ch07.qd 9/7/08 1:19 PM Page 599 SECTION 7.4 Mohr s Circle 599 Solution 7.4-7 s 0 psi s 0 psi t 700 psi (a) ELEMENT AT u 5 u 104.0 R t R 700 psi Point D: s 1 R cos (u 90 ) s 1 60 psi t 11 R sin (u 90 ) t 11 653 psi Point D œ : s 1 R cos (u 90 ) s 1 60 psi (b) PRINCIPAL STRESSES Point P1: u p1 90 u p1 45 s 1 R s 1 700 psi Point P: u p 90 u p 45 s R s 700 psi Problem 7.4-8 An element in pure shear is subjected to stresses t 14.5 MPa, as shown in the figure. Using Mohr s circle, determine: (a) The stresses acting on an element oriented at a counterclockwise angle u.5 from the ais (b) The principal stresses. Show all results on sketches of properl oriented elements. O 14.5 MPa Solution 7.4-8 s 0 MPa s 0 MPa t 14.5 MPa (a) ELEMENT AT u.5 u 45.00 R t R 14.50 MPa Point D: s 1 R cos (u 90 ) s 1 10.5 MPa t 11 R sin (u 90 ) t 11 10.5 MPa Point D œ : s 1 R cos (u 90 ) s 1 10.5 MPa (b) PRINCIPAL STRESSES Point P1: u p1 70 s 1 R s 1 14.50 MPa Point P: u p 70 u p 135.0 s R u p1 135.0 s 14.50 MPa

07Ch07.qd 9/7/08 1:19 PM Page 600 600 CHAPTER 7 Analsis of Stress and Strain Problem 7.4-9 An element in pure shear is subjected to stresses t 3750 psi, as shown in the figure. Using Mohr s circle, determine: (a) The stresses acting on an element oriented at a slope of 3 on 4 (see figure). (b) The principal stresses. 4 3 Show all results on sketches of properl oriented elements. O 3750 psi Solution 7.4-9 s 0 psi s 0 psi t 3750 psi (a) ELEMENT AT A SLOPE OF 3 ON 4 u atana 3 b u 36.870 4 u 73.740 R t R 3750 psi Point D: s 1 R cos (u 90 ) s 1 3600 psi t 11 R sin (u 90 ) t 11 1050 psi Point D œ : s 1 R cos (u 90 ) s 1 3600 psi (b) PRINCIPAL STRESSES Poin P1: u p1 90 u p1 45 s 1 R s 1 3750 psi Point P: u p 90 u p 45 s R s 3750 psi Problem 7.4-10 s 7 MPa, s 14 MPa, t 6 MPa, u 40 Using Mohr s circle, determine the stresses acting on an element oriented at an angle u from the ais. Show these stresses on a sketch of an element oriented at the angle u. (Note: The angle u is positive when counterclockwise and negative when clockwise.) s t O s Probs. 7.4-10 through 7.4-15

07Ch07.qd 9/7/08 1:19 PM Page 601 SECTION 7.4 Mohr s Circle 601 Solution 7.4-10 s 7 MPa s 14 MPa t 6 MPa u 40 s aver s + s s aver 0.50 MPa R (s s aver ) + t R 8.8459 MPa t a atana b a 4.71 s s aver b u a b 37.9 Point D: s 1 s aver + R cos (b) s 1 7.5 MPa t 11 R sin (b) t 11 5.36 MPa Point D œ : s 1 s aver R cos (b) s 1 13.46 MPa Problem 7.4-11 s 3500 psi, s 1,00 psi, t 3300 psi, u 51 Using Mohr s circle, determine the stresses acting on an element oriented at an angle u from the ais. Show these stresses on a sketch of an element oriented at the angle u.(note: The angle u is positive when counterclockwise and negative when clockwise.) Solution 7.4-11 s 3500 psi s 100 psi t 3300 psi u 51 s aver s + s s aver 7850 psi R (s s aver ) + t R 5460 psi t a atana b a 37.18 s s aver b 180 + u a b 40.8 Point D: s 1 s aver + R cos (b) s 1 1198 psi t 11 R sin (b) t 11 3569 psi Point D œ : s 1 s aver R cos (b) s 1 3718 psi Problem 7.4-1 s 47 MPa, s 186 MPa, t 9 MPa, u 33 Using Mohr s circle, determine the stresses acting on an element oriented at an angle u from the ais. Show these stresses on a sketch of an element oriented at the angle u.(note: The angle u is positive when counterclockwise and negative when clockwise.) Solution 7.4-1 s 47MPa s 186MPa t 9MPa u 33 s aver s + s s aver 116.50 MPa R (s s aver ) + t R 75.3077 MPa t a atana ` ` b a.65 s s aver b u a b 43.35 Point D: s 1 s aver + R cos (b) s 1 61.7 MPa t 11 R sin (b) t 11 51.7 MPa Point D œ : s 1 s aver R cos (b) s 1 171.3 MPa

07Ch07.qd 9/7/08 1:19 PM Page 60 60 CHAPTER 7 Analsis of Stress and Strain Problem 7.4-13 s 170 psi, s 680 psi, t 30 psi, u 14 Using Mohr s circle, determine the stresses acting on an element oriented at an angle u from the ais. Show these stresses on a sketch of an element oriented at the angle u. (Note: The angle u is positive when counterclockwise and negative when clockwise.) Solution 7.4-13 s 170 psi s 680 psi t 380 psi u 14 s aver s + s s aver 100 psi R (s s aver ) + t R 644.0 psi Point D: s 1 s aver R cos(b) s 1 1481 psi t 11 R sin (b) t 11 580 psi Point D œ : s 1 s aver + R cos (b) s 1 919 psi t a atana b a 36.16 s s aver b u + a b 64.16 Problem 7.4-14 s 33 MPa, s 9 MPa, t 9 MPa, u 35 Using Mohr s circle, determine the stresses acting on an element oriented at an angle u from the ais. Show these stresses on a sketch of an element oriented at the angle u. (Note: The angle u is positive when counterclockwise and negative when clockwise.) Solution 7.4-14 s 33 MPa s 9 MPa t 9 MPa u 35 s aver s + s s aver 1.00 MPa R (s s aver ) + t R 35.8050 MPa t a atana ` ` b a 54.09 s s aver b u a b 15.91 Point D: s 1 s aver + R cos (b) s 1 46.4 MPa t 11 R sin (b) t 11 9.81 MPa Point D œ : s 1 s aver R cos (b) s 1.4 MPa Problem 7.4-15 s 5700 psi, s 950 psi, t 100 psi, u 65 Using Mohr s circle, determine the stresses acting on an element oriented at an angle u from the ais. Show these stresses on a sketch of an element oriented at the angle u. (Note: The angle u is positive when counterclockwise and negative when clockwise.)

07Ch07.qd 9/7/08 1:19 PM Page 603 SECTION 7.4 Mohr s Circle 603 Solution 7.4-15 s 5700 psi s 950 psi t 100 psi u 65 s aver s + s s aver 375 psi R (s s aver ) + t R 3933 psi t a atana b a 3.8 s s aver Point D: s 1 s aver + R cos (b) s 1 1846 psi t 11 R sin (b) t 11 3897 psi Point D œ : s 1 s aver R cos (b) s 1 904 psi b 180 u + a b 8.8 Problems 7.4-16 s 9.5 MPa, s 9.5 MPa, t 7 MPa Using Mohr s circle, determine (a) the principal stresses and (b) the maimum shear stresses and associated normal stresses. Show all results on sketches of properl oriented elements. s t s O Probs. 7.4-16 through 7.4-3 Solution 7.4-16 s 9.5 MPa s 9.5 MPa t 7 MPa s aver s + s s aver 0 MPa R (s s aver ) + t R 39.9906 MPa a atana ` ` b a 4.47 s s aver (a) PRINCIPAL STRESSES u p1 t 180 a u p1 68.8 u p u p1 90 u p 1. Point P1: s 1 R s 1 40.0 MPa Point P: s R s 40.0 MPa (b) MAXIMUM SHEAR STRESSES u s1 90 a u s1 3.8 u s 90 + u s1 u s 113.8 Point S1: s aver 0 MPa t ma R t ma 40.0 MPa

07Ch07.qd 9/7/08 1:19 PM Page 604 604 CHAPTER 7 Analsis of Stress and Strain Problems 7.4-17 s 7300 psi, s 0 psi, t 1300 psi Using Mohr s circle, determine (a) the principal stresses and (b) the maimum shear stresses and associated normal stresses. Show all results on sketches of properl oriented elements. Solution 7.4-17 s 7300 psi s 0 psi t 1300 psi s aver s + s s aver 3650 psi R (s s aver ) + t R 3875 psi t a atana ` ` b a 19.60 s s aver (a) PRINCIPAL STRESSES u p1 a u p1 9.80 u p a + 180 u p 99.8 Point P1: s 1 R + s aver s 1 755 psi Point P: s R + s aver s 5 psi (b) MAXIMUM SHEAR STRESSES u s1 90 + a u s1 35. u s 90 + u s1 u s 54.8 Point S1: s aver 3650 psi t ma R t ma 3875 psi Problems 7.4-18 s 0 MPa, s 3.4 MPa, t 9.6 MPa Using Mohr s circle, determine (a) the principal stresses and (b) the maimum shear stresses and associated normal stresses. Show all results on sketches of properl oriented elements. Solution 7.4-18 s 0 MPa s 3.4 MPa t 9.6 MPa s aver s + s s aver 11.70 MPa R (s s aver ) + t R 15.1344 MPa t a atana ` ` b a 39.37 s s aver (a) PRINCIPAL STRESSES u p1 a u p1 19.68 u p u p1 + 90 u p 70.3 Point P1: s 1 R + s aver (b) MAXIMUM SHEAR STRESSES u s1 s 1 3.43 MPa Point P: s R + s aver s 6.8 MPa 90 a u s1 64.7 u s 90 + u s1 u s 5.3 Point S1: s aver 11.70 MPa t ma R t ma 15.13 MPa

07Ch07.qd 9/7/08 1:19 PM Page 605 SECTION 7.4 Mohr s Circle 605 Problems 7.4-19 s 050 psi, s 6100 psi, t 750 psi Using Mohr s circle, determine (a) the principal stresses and (b) the maimum shear stresses and associated normal stresses. Show all results on sketches of properl oriented elements. Solution 7.4-19 s 050 psi s 6100 psi t 750 psi s aver s + s s aver 4075 psi R (s s aver ) + t R 3415 psi a atana ` ` b a 53.63 s s aver (a) PRINCIPAL STRESSES u p1 t 180 a u p1 63. u p a u p 6.8 Point P1: s 1 R + s aver s 1 7490 psi Point P: s R + s aver s 660 psi (b) MAXIMUM SHEAR STRESSES u s1 90 + a u s1 18. u s 90 + u s1 u s 71.8 Point S1: s aver 4075 psi t ma R t ma 3415 psi Problems 7.4-0 s 900 kpa, s 9100 kpa, t 3750 kpa Using Mohr s circle, determine (a) the principal stresses and (b) the maimum shear stresses and associated normal stresses. Show all results on sketches of properl oriented elements. Solution 7.4-0 s 900 kpa s 9100 kpa t 3750 kpa s aver s + s s aver 6000 kpa R (s s aver ) + t R 4865.4393 kpa t a atana ` ` b a 50.4 s s aver (a) PRINCIPAL STRESSES u p1 a + 180 u p1 115. u p a u p 5. Point P1: s 1 R + s aver s 1 10865 KPa Point P: s R + s aver s 1135 kpa (b) MAXIMUM SHEAR STRESSES u s1 90 + a u s1 70. u s 90 + u s1 u s 160. Point S1: s aver 6000 kpa t ma R t ma 4865 kpa

07Ch07.qd 9/7/08 1:19 PM Page 606 606 CHAPTER 7 Analsis of Stress and Strain Problems 7.4-1 s 11,500 psi, s 18,50 psi, t 700 psi Using Mohr s circle, determine (a) the principal stresses and (b) the maimum shear stresses and associated normal stresses. Show all results on sketches of properl oriented elements. Solution 7.4-1 s 11500 psi t 700 psi s 1850 psi Point P1: s 1 R + s aver s 1 693 psi s aver s + s s aver 14875 psi R (s s aver ) + t R 795 psi a atana ` ` b a 64.89 s s aver (a) PRINCIPAL STRESSES u p1 a u p1 3.4 u p t 180 a u p 57.6 Point P: s R + s aver s 87 psi (b) MAXIMUM SHEAR STRESSES u s1 70 a u s1 10.6 u s 90 + u s1 u s 19.6 Point S1: s aver 14875 psi t ma R t ma 795 psi Problems 7.4- s 3.3 MPa, s 8.9 MPa, t 14.1 MPa Using Mohr s circle, determine (a) the principal stresses and (b) the maimum shear stresses and associated normal stresses. Show all results on sketches of properl oriented elements. Solution 7.4- s 3.3 MPa s 8.9 MPa t 14.1 MPa s aver s + s s aver.8 MPa R (s s aver ) + t R 15.4 MPa t a atana ` ` b a 66.6 s s aver (a) PRINCIPAL STRESSES u p1 a + 180 u p1 13.3 u p a u p 33.3 Point P1: s 1 R + s aver s 1 18. MPa Point P: s R + s aver

07Ch07.qd 9/7/08 1:19 PM Page 607 SECTION 7.4 Mohr s Circle 607 s 1.6 MPa (b) MAXIMUM SHEAR STRESSES u s1 90 + a u s1 78.3 u s 90 + u s1 u s 168.3 Point S1: s aver.8 MPa t ma R t ma 15.4 MPa Problems 7.4-3 s 800 psi, s 00 psi, t 900 psi Using Mohr s circle, determine (a) the principal stresses and (b) the maimum shear stresses and associated normal stresses. Show all results on sketches of properl oriented elements. Solution 7.4-3 s 800 psi s 00 psi t 900 psi s aver s + s s aver 700 psi R (s s aver ) + t R 365 psi t a atana ` ` b a 6.65 s s aver (a) PRINCIPAL STRESSES u p1 a u p1 31.3 u p 180 + a u p 11.3 Point P1: s 1 R + s aver s 1 565 psi Point P: s R + s aver s 3965 psi (b) MAXIMUM SHEAR STRESSES u s1 90 + a u s1 13.7 u s 90 + u s1 u s 76.3 Point S1: s aver 700 psi t ma R t ma 365 psi

07Ch07.qd 9/7/08 1:0 PM Page 608 608 CHAPTER 7 Analsis of Stress and Strain Hooke s Law for Plane Stress When solving the problems for Section 7.5, assume that the material is linearl elastic with modulus of elasticit E and Poisson s ratio n. Problem 7.5-1 A rectangular steel plate with thickness t 0.5 in. is subjected to uniform normal stresses and, as shown in the figure. Strain gages A and B, oriented in the and directions, respectivel, are attached to the plate. The gage readings give normal strains e 0.0010 (elongation) and e 0.0007 (shortening). Knowing that E 30 10 6 psi and 0.3, determine the stresses and and the change t in the thickness of the plate. s B A O Probs. 7.5-1 and 7.5- s Solution 7.5-1 E 30 * 10 6 psi 0.3 SUBSTITUTE NUMERICAL VALUES: Eq. (7-40a): s Eq. (7-40b): s Rectangular plate in biaial stress t 0.5 in. â 0.0010 â 0.0007 E (1 ) (â + â ) 6,040 psi E (1 ) (â + â ) 13,190 psi Eq. (7-39c): â z E (s + s ) 18.5 * 10 6 t â z t 3.1 * 10 6 in. (Decrease in thickness) Problem 7.5- Solve the preceding problem if the thickness of the steel plate is t 10 mm, the gage readings are e 480 10 6 (elongation) and e 130 10 6 (elongation), the modulus is E 00 GPa, and Poisson s ratio is 0.30. Solution 7.5- Rectangular plate in biaial stress t 10 mm â 480 * 10 6 â 130 * 10 6 E 00 GPa 0.3 SUBSTITUTE NUMERICAL VALUES: Eq. (7-40a): s E (1 ) (â + â ) 114.1 MPa Eq. (7-40b): s E (1 ) (â + â ) 60. MPa Eq. (7-39c): â z E (s + s ) 61.4 * 10 6 t â z t 610 * 10 6 mm (Decrease in thickness)

07Ch07.qd 9/7/08 1:0 PM Page 609 SECTION 7.5 Hooke s Law for Plane Stress 609 Problem 7.5-3 Assume that the normal strains P and P for an element in plane stress (see figure) are measured with strain gages. s (a) Obtain a formula for the normal strain P z in the z direction in terms of P, P, and Poisson s ratio. (b) Obtain a formula for the dilatation e in terms of P, P, and Poisson s ratio. O t s z Solution 7.5-3 Given: â, â, (a) NORMAL STRAIN â z Plane stress Eq. (7-34c): â z E (s + s ) E Eq. (7-36a): s (1 ) (â + â ) E Eq. (7-36b): s (1 ) (â + â ) Substitute s and s into the first equation and simplif: â z 1 (â + â ) (b) DILATATION Eq. (7-47): e Substitute and from above and simplif: e s 1 E s (s + s ) 1 1 (â + â ) Problem 7.5-4 A magnesium plate in biaial stress is subjected to tensile stresses s 4 MPa and s 1 MPa (see figure). The corresponding strains in the plate are 440 10 6 and 80 10 6. Determine Poisson s ratio and the modulus of elasticit E for the material. s O s Probs. 7.5-4 through 7.5-7 Solution 7.5-4 Biaial stress s 4 MPa s 1 MPa â 440 * 10 6 â 80 * 10 6 POISSON S RATION AND MODULUS OF ELASTICITY Eq. (7-39a): â 1 E (s s ) Substitute numerical values: E (440 * 10 6 ) 4 MPa (1 MPa) E (80 * 10 6 ) 1 MPa (4 MPa) Solve simultaneousl: 0.35 E 45 GPa Eq. (7-39b): â 1 E (s s )

07Ch07.qd 9/7/08 1:0 PM Page 610 610 CHAPTER 7 Analsis of Stress and Strain Problem 7.5-5 Solve the preceding problem for a steel plate with s 10,800 psi (tension), s 5400 psi (compression), e 40 10 6 (elongation), and e 300 10 6 (shortening). Solution 7.5-5 Biaial stress s 10,800 psi s 5400 psi â 40 * 10 6 â 300 * 10 6 POISSON S RATIO AND MODULUS OF ELASTICITY Eq. (7-39a): â 1 E (s s ) Substitute numerical values: E (40 * 10 6 ) 10,800 psi ( 5400 psi) E ( 300 * 10 6 ) 5400 psi (10,800 psi) Solve simultaneousl: 1/3 E 30 * 10 6 psi Eq. (7-39b): â 1 E (s s ) Problem 7.5-6 A rectangular plate in biaial stress (see figure) is subjected to normal stresses 90 MPa (tension) and 0 MPa (compression). The plate has dimensions 400 * 800 * 0 mm and is made of steel with E 00 GPa and 0.30. (a) Determine the maimum in-plane shear strain g ma in the plate. (b) Determine the change t in the thickness of the plate. (c) Determine the change V in the volume of the plate. Solution 7.5-6 Biaial stress s 90 MPa s 0 MPa E 00 GPa 0.30 Dimensions of Plate: 400 mm * 800 mm * 0 mm Shear Modulus (Eq. 7-38): G E 76.93 GPa (1 + ) (b) CHANGE IN THICKNESS Eq. (7-39c): â z E (s + s ) 105 * 10 6 t â z t 100 * 10 6 mm (Decrease in thickness) (a) MAXIMUM IN-PLANE SHEAR STRAIN Principal stresses: s 1 90 MPa s 0 MPa Eq. (7-6): t ma s 1 s 55.0 MPa Eq. (7-35): g ma t ma G 715 * 10 6 (c) CHANGE IN VOLUME 1 From Eq. (7-47): V V 0 a b(s + s ) E V 0 (400)(800)(0) 6.4 * 10 6 mm 3 1 Also, a b(s + s ) 140 * 10 6 E V (6.4 * 10 6 mm 3 )(140 * 10 6 ) 896 mm 3 (Increase in volume)

07Ch07.qd 9/7/08 1:0 PM Page 611 SECTION 7.5 Hooke s Law for Plane Stress 611 Problem 7.5-7 Solve the preceding problem for an aluminum plate with s 1,000 psi (tension), s 3,000 psi (compression), dimensions 0 30 0.5 in., E 10.5 10 6 psi, and 0.33. Solution 7.5-7 Biaial stress s 1,000 psi s 3,000 psi E 10.5 * 10 6 psi 0.33 Dimensions of Plate: 0 in. * 30 in. * 0.5 in. Shear Modulus (Eq. 7-38): G E (1 + ) 3.9474 * 106 psi (b) CHANGE IN THICKNESS Eq. (7-39c): â z E (s + s ) 8.9 * 10 6 t â z t 141 * 10 6 in. (Decrease in thickness) (a) MAXIMUM IN-PLANE SHEAR STRAIN Principal stresses: s 1 1,000 psi s 3,000 psi (c) CHANGE IN VOLUME 1 From Eq. (7-47): V V 0 a b(s + s ) E Eq. (7-6): t ma s 1 s 7,500 psi Eq. (7-35): g ma t ma G 1,900 * 10 6 V 0 (0)(30)(0.5) 300 in. 3 1 Also, a b(s E + s ) 91.4 * 10 6 V (300 in. 3 )(91.4 * 10 6 ) 0.0874 in. 3 (Increase in volume) Problem 7.5-8 A brass cube 50 mm on each edge is compressed in two perpendicular directions b forces P 175 kn (see figure). Calculate the change V in the volume of the cube and the strain energ U stored in the cube, assuming E 100 GPa and 0.34. P = 175 kn P = 175 kn Solution 7.5-8 Biaial stress-cube s s P b kn) (175 70.0 MPa (50 mm) Side b 50 mm P 175 kn E 100 GPa 0.34 ( Brass) CHANGE IN VOLUME Eq. (7-47): e 1 E V 0 b 3 (50 mm) 3 15 * 10 3 mm 3 V ev 0 56 mm 3 (Decrease in volume) (s + s ) 448 * 10 6

07Ch07.qd 9/7/08 1:0 PM Page 61 61 CHAPTER 7 Analsis of Stress and Strain STRAIN ENERGY Eq. (7-50): u 1 E (s + s s s ) 0.0334 MPa U uv 0 (0.0334 MPa)(15 * 10 3 mm 3 ) 4.04 J Problem 7.5-9 A 4.0-inch cube of concrete (E 3.0 * 10 6 psi, 0.1) is compressed in biaial stress b means of a framework that is loaded as shown in the figure. Assuming that each load F equals 0 k, determine the change V in the volume of the cube and the strain energ U stored in the cube. F F Solution 7.5-9 Biaial stress concrete cube b 4 in. E 3.0 * 10 6 psi 0.1 F 0 kips CHANGE IN VOLUME Eq. (7-47): e 1 E V 0 b 3 (4 in.) 3 64 in. 3 V ev 0 0.0603 in. 3 (Decrease in volume) (s + s ) 0.000949 Joint A: P F1 8.8 kips s s P 1768 psi b STRAIN ENERGY Eq. (7-50): u 1 E (s + s s s ) 0.9377 psi U uv 0 60.0 in.-lb Problem 7.5-10 A square plate of width b and thickness t is loaded b normal forces P and P, and b shear forces V, as shown in the figure. These forces produce uniforml distributed stresses acting on the side faces of the place. Calculate the change V in the volume of the plate and the strain energ U stored in the plate if the dimensions are b 600 mm and t 40 mm, the plate is made of magnesium with E 45 GPa and v 0.35, and the forces are P 480 kn, P 180 kn, and V 10 kn. P t V V P b O P V b V P Probs. 7.5-10 and 7.5-11