Chapter 7 Transformations of Stress and Strain INTRODUCTION Transformation of Plane Stress Mohr s Circle for Plane Stress Application of Mohr s Circle to 3D Analsis 90 60 60 0 0 50 90 Introduction 7-1
Stresses in Thin-Walled Pressure Vessels Clindrical Pressure Vessel Spherical Pressure Vessel Transformation of Plane Strain ' ' ' ' Measurements of Strain; Strain Rosette 7- Introduction
TRANSFORMATION OF PLANE STRESS θ + = + cos θ + sin θ + = cos θ sin θ = sin θ + cos θ Transformation of Plane Stress 7-3
Eample The state of stress at a point on the surface of a pressure vessel is represented on the element shown. Represent the state of stress at the point on another element that is orientated 30 clockwise from the position shown. Units: MPa. + = + cos θ + sin θ 5 50 80 80 + = cos θ sin θ 5 50 = sin θ + cos θ 7-4
Eample Determine the stresses on a surface that is rotated (a) 30 clockwise, (b) 15 counterclockwise. Units: MPa 80 + = + cos θ + sin θ 50 + = cos θ sin θ 50 80 = sin θ + cos θ + = + cos θ + sin θ + = cos θ sin θ = sin θ + cos θ 7-5
Eample For the piece of wood, determine the in-plane shear stress parallel to the grain, (b) the normal stress perpendicular to the grain. The grain is rotated 30 from the horizontal. Units: psi 400 = sin θ + cos θ 400 + = cos θ sin θ 7-6
PRINCIPAL STRESSES: MAXIMUM SHEARING STRESS ma min ma min ave ma ave ave ma ave + = + cos θ + sin θ = sin θ + cos θ ma,min + = ± + ma = + tan θ p = θ s = θ ± 45 p ma = ma min Principal Stresses: Maimum Shearing Stress 7-7
Eample Determine the (a) principal stresses, (b) maimum in-plane shear stress and the associated normal stress. Units: MPa 80 ma,min + = ± + 50 50 ma = + 80 ave = + 7-8
Eample Determine the (a) principal stresses, (b) maimum in-plane shear stress and the associated normal stress. Sketch the resulting stresses on the element and the corresponding orientation. Units: MPa 80 tan θ p = 50 + = + cos θ + sin θ 50 80 + = cos θ sin θ = sin θ + cos θ ave = + 7-9
Eample Determine the (a) principal stresses, (b) maimum in-plane shear stress and the associated normal stress. Units: MPa 48 60 ma,min + = ± + 16 16 60 48 ma = + ave = + 7-10
Eample Determine the (a) principal stresses, (b) maimum in-plane shear stress and the associated normal stress. Units: psi 300 ma,min + = ± + 8800 8800 300 ma = + ave = + 7-11
MOHR S CIRCLE FOR PLANE STRESS R + = + ave = ma min ma min ave ma ave ave ma ave 7-1 Mohr s Circle for Plane Stress
Eample Using Mohr s circle, determine the stresses on a surface that is rotated 30 clockwise. Units: MPa 80 ave = + 50 R = + 50 80 7-13
Eample Using Mohr's circle, determine the (a) principal stresses, (b) maimum in-plane shear stress and the associated normal stress. Units: MPa 80 ave = + 50 R = + 50 80 7-14
Eample Using Mohr's circle, determine the (a) principal stresses, (b) maimum in-plane shear stress and the associated normal stress. Units: MPa 48 ave = + 60 16 16 R = + 48 60 7-15
Eample Using Mohr's circle, determine the (a) principal stresses, (b) maimum in-plane shear stress and the associated normal stress. Units: MPa 90 ave = + 60 0 0 R = + 60 90 7-16
3D APPLICATIONS OF MOHR S CIRCLE Eample Using Mohr's circle, determine the maimum shear stress. (Hint: Consider both in-plane and out-of-plane shearing stresses). Units: MPa 34.7 17.5 17.5 95.3 34.7 34.7 30 30 17.5 95.3 100 100 100 100 30 30 3D Applications of Mohr s Circle 7-17
Eample Using Mohr's circle, determine the maimum shear stress. (Hint: Consider both in-plane and out-of-plane shearing stresses). Units: MPa 90 60 60 0 0 90 90 116 60 0 46.4 46.4 116 7-18
Eample Using Mohr's circle, determine the maimum shear stress. (Hint: Consider both in-plane and out-of-plane shearing stresses). Units: MPa 90 60 60 0 0 50 90 90 116 60 0 46.4 46.4 116 7-19
Eample Using Mohr's circle, determine the maimum shear stress. (Hint: Consider both in-plane and out-of-plane shearing stresses). Units: MPa 34.7 17.5 60 17.5 95.3 48 34.7 30 17.5 95.3 100 100 30 7-0
STRESSES IN THIN-WALLED PRESURE VESSELS Clindrical Pressure Vessels Clindrical Pressure Vessel = pr t = 1 pr t 1 1 Stresses in Thin-Walled Pressure Vessels 7-1
Spherical Pressure Vessels Spherical Pressure Vessel = = 1 pr t = ma pr 4t 1 1 7- Stresses in Thin-Walled Pressure Vessels
Eample The viewport is attached to the submersible with 16 bolts and has an internal air pressure of 95 psi. The viewport material used has an allowable maimum tensile and shear stress of 700 and 400 psi respectivel. The inside diameter of the viewport is 18". Determine the force in each bolt and the wall thickness of the viewport. Units: in 7-3
Eample The pressure vessel has an inside diameter of meters and an internal pressure of 3 MPa. If the spherical ends have a wall thickness of 10 mm and the clindrical portion has a wall thickness of 30 mm, determine the maimum normal and shear stress in each section. 7-4
Eample The open water tank has an inside diameter of 50 ft and is filled to a height of 60 ft. Determine the minimum wall thickness due to the water pressure onl if the allowable tensile stress is 4 ksi. 7-5
Eample 18 mm thick plates are welded as shown to form the clindrical pressure tank. Knowing that the allowable normal stress perpendiculer to the weld is 60 MPa, determine the maimum allowable internal pressure and the height of the tank. Units: m. h 8 7-6
Eample The clindrical portion of the compressed air tank is made of 10 mm thick plate welded along a heli forming an angle of 45. Knowing that the allowable stress normal to the weld is 80 MPa, determine the largest gage pressure that can be used in the tank. Units: m 0.5 0.75 7-7
TRANSFORMATION OF PLANE STRAIN ' ' ' ' ε + ε ε ε γ ε = + cos θ + sin θ γ = ( ε ε )sinθ + γ cosθ ε + ε ε ε γ ε = cos θ sin θ PRINCIPAL STRAINS tan θ ε p γ = ε ε ma,min ab, ε + ε ε ε γ = ε = ± + γ ma ε ε γ = + υ εc = ( εma + ε min ) 1 υ 7-8
MOHR S CIRCLE FOR PLANE STRAIN γ ( μ ) ε ave ε = + ε ε( μ ) ε ε γ R = + ε ma,min ab, ε + ε ε ε γ = ε = ± + γma = εma ε min 7-9
Eample Given the strains below, determine the strains if the element is rotated 30 counterclockwise. ε ε γ = 300μ = 00μ =+ 175μ ' ε + ε ε ε γ ε = + cos θ + sin θ ' ε + ε ε ε γ ε = cos θ sin θ ' ' γ = ( ε ε )sinθ + γ cosθ 7-30
Eample Given the strains below, determine (a) the direction and magnitude of the principal strains, (b) the maimum in-plane shearing strain, (c) the maimum strain. Assume plane stress. υ = 1/3 ε = 300μ ε = 00μ γ =+ 175μ tan θ p γ = ε ε ' ' ε + ε ε ε γ ε = + cos θ + sin θ ε + ε ε ε γ ε = cos θ sin θ ' ' ε ε γ R = + υ εc = ( ε1+ ε ) 1 υ 7-31
MEASUREMENTS OF STRAIN; STRAIN ROSETTE Clindrical Pressure Vessel ε = ε cos θ + ε sin θ + γ sinθ cosθ 1 1 1 1 1 ε = ε cos θ + ε sin θ + γ sinθ cosθ ε = ε cos θ + ε sin θ + γ sinθ cosθ 3 3 3 3 3 ε 3 θ 3 ε θ θ 1 ε 1 7-3
Eample Given the following strains, determine (a) the in-plane principal strains, (b) the in-plane maimum shearing strain. ε ε ε 1 3 =+ 600μ =+ 450μ = 175μ ε 3 30 ε ε 1 30 ε = ε cos θ + ε sin θ + γ sinθ cosθ 1 1 1 1 1 ε = ε cos θ + ε sin θ + γ sinθ cosθ ε = ε cos θ + ε sin θ + γ sinθ cosθ 3 3 3 3 3 ε ma,min ε + ε ε ε γ = ± + γ ma ε ε γ = + 7-33
Eample Given the strain measurements below for the 30" diameter, 0.5" thick tank, determine the gage pressure, (b) the principal stresses and the maimum in-plane shearing stress. ε 1 =+ 160μ υ = 0.3 θ = 30 E = 6 9 10 psi ε 1 γ ( μ ) ε( μ ) 7-34
SUMMARY Transformation of Plane Stress Mohr s Circle for Plane Stress Application of Mohr s Circle to 3D Analsis 90 60 60 0 0 50 90 7-35
SUMMARY Stresses in Thin-Walled Pressure Vessels Clindrical Pressure Vessel Transformation of Plane Strain ' Spherical Pressure Vessel ' ' ' Measurements of Strain; Strain Rosette 7-36
INTRODUCTION Principal Stresses under a Given Loading PRINCIPAL STRESSES IN A BEAM 4 48 Wide Flange Stresses Principal Wide Flange Stresses Rectangular Crosssection Stresses 8-1
Eample (a) Knowing that the allowable normal stress is 80 MPa and the allowable shear stress is 50 MPa, determine the height of the rectangular section if the width is 100 mm. Units: kn, m 10 8 8 C D A 3 B 3 3 3 8-
Eample (a) Knowing that the allowable normal stress is 80 MPa and the allowable shear stress is 50 MPa, select the most economical wide-flange shape that should be used to support the loading shown. (b) Determine the principal stresses at the junction between the flange and web on a section just to the right of the 4 kn load. Units: kn, m kn/m A 3. B 0.8 4 8-3
Eample (a) Knowing that the allowable normal stress is 4 ksi and the allowable shear stress is 15 ksi, select the most economical W8 wide-flange shape that should be used to support the loading shown. (b) Determine the principal stresses at the junction between the flange and web. Units: lb, in. 50 lb/in 3000 A 48 8-4
STRESSES UNDER COMBINED LOADINGS Eample Determine the stresses at A and B. Units: k, k-in, in. 9 B A 9 1.5 9 1.5 B.5.5 A Cross-section A B 8-5
Eample Determine the stresses at A. The disk has a diameter of 4" and the solid shaft has a diameter of 1.8". Units: kips, in. 6 6.5 8 A z A 8-6
Eample Determine the stresses at points A and B. The beam is a W60. Units: kips, in. 4 B 0 A 48 W60 4 in 4 in 3 in 3 in in Area, A 5.87 in Depth, d 6.0 in Flange Width, bf 6.0 Flange Thickness, tf 0.365 in Web Thickness, t 0.60 I 41.4 I 13.3 S S 13.4 4.41 w in A B 8-7
Eample Determine the principal stresses s and maimum in-plane shearing stress at A and B. Units: k, k-in. From a previous solution: 5.31 5.31 B A 9 A 9 1.5 B 8.5 5.6 B 8.5 5.6.5.5 A Cross-section ma,min ma ave ma,min ma 8-8 ave
Eample Determine the principal stresses and maimum in-plane shearing stress at A. The disk has a diameter of 4" and the solid shaft has a diameter of 1.8". From a previous solution: A 4.7 5.69 5.69 6 6.5 4.7 8 ma,min A z ma ave 8-9
Eample Determine the principal stresses and maimum in-plane shearing stress at A and B. The beam is a W60. From a previous solution: 17.8 A 17.8 B 4 0 A 48 3.41 B 3.41.75.75 ma,min ma ave ma,min ma ave 8-10
SUMMARY Principal Stresses in a Beam 4 48 Wide Flange Stresses Principal Wide Flange Stresses Rectangular Crosssection Stresses 8-11