Questions and Answers of Mechanics of Materials

The state of strain at the point on the member has components of εx = 180(10-6), εy = -120(10-6), and γxy = -100(10-6). Use the strain transformation equations to determine (a) the in-plane
Solve Prob. 9–9 using the stress transformation equation developed in Sec. 9.2. B 80 MPa 30% 30 MPa 40 MPa
Solve Prob. 10€“4 using Mohr€™s circle.Problem: 10-4Solve Prob. 10€“3 for an element oriented Î¸ = 30° clockwise.
Solve Prob. 10€“6 for an element oriented u = 30° clockwise.
The state of strain on the element has components Îµx= -300(10-6), Îµy= 100(10-6), Î³xy= 150(10-6).Determine the equivalent state of strain, which
The state of strain at the point on a boom of a shop crane has components of Îµx= 250(0-6), Îµy= 300(10-6), and Î³xy= -180(10-6). Use the strain transformation
Solve Prob. 10€“3 using Mohr€™s circle.Problem: 10-3The state of strain at the point on the pin leaf has components of Îµx = 200(10-6), Îµy =
The polyvinyl chloride bar is subjected to an axial force of 900 lb. If it has the original dimensions shown, determine the value of Poisson€™s ratio if the angle Î¸ decreases
The polyvinyl chloride bar is subjected to an axial force of 900 lb. If it has the original dimensions shown, determine the change in the angle u after the load is applied. Epvc= 800(103) psi,
A rod has a radius of 10 mm. If it is subjected to an axial load of 15 N such that the axial strain in the rod is εx = 2.75(10-6), determine the modulus of elasticity E and the change in the rod’s
The principal plane stresses and associated strains in a plane at a point are σ1 = 36 ksi, σ2 = 16 ksi, ε1 = 1.02(10-3), ε2 = 0.180(10-3). Determine the modulus of elasticity and Poisson’s
Use Hooke’s law, Eq. 10–18, to develop the strain transformation equations, Eqs. 10–5 and 10–6, from the stress transformation equations, Eqs. 9–1 and 9–2.
For the case of plane stress, show that Hooke€™s law can be written as (Ey + νε) σ, - στ - (ε, + νε ), σ, (1 – v²) (1 – v)
The 45° strain rosette is mounted on a steel shaft. The following readings are obtained from each gage: Îµa= 800(10-6), Îµb= 520(10-6), Îµc= -450(10-6).
The 60° strain rosette is mounted on the surface of the bracket. The following readings are obtained for each gage: Îµa= -780(10-6), Îµb= 400(10-6), and
The 45° strain rosette is mounted on the surface of a pressure vessel. The following readings are obtained for each gage: Îµa= 475(10-6), Îµb= 250(10-6), and
The 45° strain rosette is mounted on the surface of a shell. The following readings are obtained for each gage: Îµa= -200(10-6), Îµb= 300(10-6), and Îµc=
The strain at point A on the pressure-vessel wall has components Îµx= 480(10-6), Îµy= 720(10-6),Î³xy = 650(10-6). Determine (a) the principal strains at A, in
The strain at point A on a beam has components Îµx= 450(10-6), Îµy= 825(10-6), Î³xy= 275(10-6), Îµz= 0. Determine (a) the principal strains at
The strain at point A on the bracket has components Îµx= 300 (10-6), Îµy= 550 (10-6), Î³xy= -650 (10-6), Îµz= 0. Determine (a) the principal
Solve Prob. 10–7 using Mohr’s circle.Data from in Problem 10-7The state of strain at a point on the bracket has components of εx = 150(10-6), εy = 200(10-6), γxy = -700(10-6). Use
Solve Prob. 10€“8 using Mohr€™s circleData from Problem: 10-8The state of strain at the point on the spanner wrench has components of Îµx = 260(10-6), Îµy
Solve Prob. 10€“5 using Mohr€™s circle.Data from Problem: 10-5.The state of strain at the point on the leaf of the caster assembly has components of Îµx = - 400
The state of strain at the point on the spanner wrench has components of Îµx= 260(10-6), Îµy= 320(10-6), and Î³xy= 180(10-6). Use the strain transformation
If the 2-in.-diameter shaft is made from cast iron having tensile and compressive ultimate stress of (Ïƒult)t= 50 ksi and (Ïƒult)c= 75 ksi, respectively, determine if the
If the 2-in.-diameter shaft is made from brittle material having an ultimate stress of Ïƒult= 50 ksi, for both tension and compression, determine if the shaft fails according to the
The 304-stainless-steel cylinder has an inner diameter of 4 in. and a wall thickness of 0.1 in. If it is subjected to an internal pressure of p = 80 psi, axial load of 500 lb, and a torque of 70 lb
The 304-stainless-steel cylinder has an inner diameter of 4 in. and a wall thickness of 0.1 in. If it is subjected to an internal pressure of p = 80 psi, axial load of 500 lb, and a torque of 70 lb
Solve Prob. 10€“75 using the maximum shear stress theory.Problem: 10-75The components of plane stress at a critical point on a thin steel shell are shown. Determine if failure (yielding)
The components of plane stress at a critical point on a thin steel shell are shown. Determine if failure (yielding) has occurred on the basis of the maximum distortion energy theory. The yield stress
If a machine part is made of titanium (Ti-6A1-4V) and a critical point in the material is subjected to plane stress, such that the principal stresses are σ1 and σ2 = 0.5 σ1, determine the
An aluminum alloy is to be used for a solid drive shaft such that it transmits 30 HP at 1200 rev/min. using a factor of safety of 2.5 with respect to yielding, determine the smallest diameter shaft
The plate is made of Tobin bronze, which yields at ÏƒY= 25 ksi. Using the maximum distortion energy theory, determine the maximum tensile stress Ïƒxthat can be applied to the
The plate is made of Tobin bronze, which yields at ÏƒY= 25 ksi. Using the maximum shear stress theory, determine the maximum tensile stress Ïƒxthat can be applied to the plate if
Derive an expression for an equivalent bending moment Me that, if applied alone to a solid bar with a circular cross section, would cause the same energy of distortion as the combination of an
The short concrete cylinder having a diameter of 50 mm is subjected to a torque of 500 N · m and an axial compressive force of 2 kN. Determine if it fails according to the maximum normal
If the material is machine steel having a yield stress of ÏƒY= 700 MPa, determine the factor of safety with respect to yielding if the maximum shear stress theory is considered. 50 MPa 80
Solve Prob. 10–66 using the maximum shear stress theory.Problem: 10-66If a shaft is made of a material for which σY = 75 ksi, determine the maximum torsional shear stress required to cause yielding
If a shaft is made of a material for which σY = 75 ksi, determine the maximum torsional shear stress required to cause yielding using the maximum distortion energy theory.
Derive an expression for an equivalent torque Te that, if applied alone to a solid bar with a circular cross section, would cause the same energy of distortion as the combination of an applied
Solve Prob. 10–63 using the maximum distortion energy theory.Problem: 10-63If a machine part is made of tool L2 steel and a critical point in the material is subjected to in-plane principal
If a machine part is made of tool L2 steel and a critical point in the material is subjected to in-plane principal stresses σ1 and σ2 = -0.5 σ1, determine the magnitude of σ1 in ksi that will
Solve Prob. 10–61 using the maximum distortion energy theoryData from Problem: 10-61.The yield stress for a zirconium-magnesium alloy is σY = 15.3 ksi. If a machine part is made of this material
The yield stress for a zirconium-magnesium alloy is σY = 15.3 ksi. If a machine part is made of this material and a critical point in the material is subjected to in-plane principal stresses σ1 and
A material is subjected to plane stress. Express the maximum shear stress theory of failure in terms of σx, σy, and τxy. Assume that the principal stresses are of different algebraic signs.
A material is subjected to plane stress. Express the distortion energy theory of failure in terms of σx, σy, and τxy.
A soft material is placed within the confines of a rigid cylinder which rests on a rigid support. Assuming that Îµx= 0 and Îµy= 0, determine the factor by which the stiffness
Estimate the increase in volume of the pressure vessel in Prob. 10€“56.Problem: 10-56.The thin-walled cylindrical pressure vessel of inner radius r and thickness t is subjected to an
The thin-walled cylindrical pressure vessel of inner radius r and thickness t is subjected to an internal pressure p. If the material constants are E and Î½, determine the strains in the
A thin-walled spherical pressure vessel having an inner radius r and thickness t is subjected to an internal pressure p. Show that the increase in the volume within the vessel is ∆V =
Determine the increase in the diameter of the pressure vessel in Prob. 10€“53 if the pistons are replaced by walls connected to the ends of the vessel.Problem: 10-53Air is pumped into the
Air is pumped into the steel thin-walled pressure vessel at C. If the ends of the vessel are closed using two pistons connected by a rod AB, determine the increase in the diameter of the pressure
The A-36 steel pipe is subjected to the axial loading of 60 kN. Determine the change in volume of the material after the load is applied. 30 mm 40 mm 60 kIN 60 kN 0.5 m
The shaft has a radius of 15 mm and is made of L2 tool steel. Determine the strains in the x€² and y€² direction if a torque T = 2 kN · m is applied to the shaft. 45° VT
The steel shaft has a radius of 15 mm. Determine the torque T in the shaft if the two strain gages, attached to the surface of the shaft, report strains of Îµx€²= -80(10-6) and
Initially, gaps between the A-36 steel plate and the rigid constraint are as shown. Determine the normal stresses Ïƒxand Ïƒyin the plate if the temperature is increased by
If the material is graphite for which Eg= 800 ksi and vg= 0.23, determine the principal strains. 26 ksi 15 ksi 10 ksi
A single strain gage, placed in the vertical plane on the outer surface and at an angle of 60° to the axis of the pipe, gives a reading at point A of ÎµA= -250(10-6). Determine the
A single strain gage, placed in the vertical plane on the outer surface and at an angle 60° to the axis of the pipe, gives a reading at point A of ÎµA= -250(10-6). Determine the
A material is subjected to principal stresses Ïƒxand Ïƒy. Determine the orientation u of the strain gage so that its reading of normal strain responds only to Ïƒyand not
A uniform edge load of 500 lb/in. and 350 lb/in. is applied to the polystyrene specimen. If the specimen is originally square and has dimensions of a = 2 in., b = 2 in., and a thickness of t = 0.25
The principal strains at a point on the aluminum surface of a tank are ε1 = 630(10-6) and ε2 = 350(10-6). If this is a case of plane stress, determine the associated principal stresses at the point
The cube of aluminum is subjected to the three stresses shown. Determine the principal strains. Take Eal= 10(103) ksi and Î½al= 0.33. 26 ksi 15 ksi 10 ksi
If a load of P = 3 kip is applied to the A-36 structural-steel beam, determine the strain Îµxand Î³xyat point A. 2 in. |2 in. 12 in. B 6 in. 3 ft 4 ft
The strain in the x direction at point A on the A-36 structural-steel beam is measured and found to be Îµx= 200(10-6). Determine the applied load P. What is the shear strain
The principal strains at a point on the aluminum fuselage of a jet aircraft are ε1 = 780(10-6) and ε2 = 400(10-6). Determine the associated principal stresses at the point in the same plane. Eal =
The strain gage is placed on the surface of the steel boiler as shown. If it is 0.5 in. long, determine the pressure in the boiler when the gage elongates 0.2(10-3) in. The boiler has a thickness of
Determine the bulk modulus for each of the following materials: (a) rubber, Er = 0.4 ksi, νr = 0.48, and (b) glass, Eg = 8(103) ksi, νg = 0.24.
The spherical pressure vessel has an inner diameter of 2 m and a thickness of 10 mm. A strain gage having a length of 20 mm is attached to it, and it is observed to increase in length by 0.012 mm
The state of plane strain on the element is Îµx= -300(10-6), Îµy= 0, and Î³xy= 150(10-6). Determine the equivalent state of strain which represents (a) the
The state of strain on an element has components Îµx= -400(10-6), Îµy= 0, Î³xy= 150(10-6). Determine the equivalent state of strain on an element at the same point
Due to the load P, the state of strain at the point on the bracket has components of Îµx= 500(10-6), Îµy= 350(10-6), and Î³xy= -430(10-6). Use the strain
The state of strain at the point on the support has components of Îµx= 350(10-6), Îµy= 400(10-6),Î³xy = -675(10-6). Use the strain-transformation equations to
The state of strain at the point on the member has components of Îµx= 180(10-6), Îµy= -120(10-6), and Î³xy= -100(10-6). Use the strain transformation equations to
The state of strain at a point on the bracket has components of ÃŽµx= 150(10-6), ÃŽµy= 200(10-6), ÃŽ³xy= -700(10-6). Use the strain transformation
The state of strain at the point on the leaf of the caster assembly has components of ÃŽµx= - 400 (10-6), ÃŽµy= 860(10-6), and ÃŽ³xy=
Solve Prob. 10Ã¢‚¬€œ3 for an element oriented ÃŽ¸ = 30Ã‚° clockwise.
The state of strain at the point on the pin leaf has components of ÃŽµx= 200(10-6), ÃŽµy= 180(10-6), and ÃŽ³xy= -300(10-6). Use the strain
The state of strain at the point on the arm has components of ÃŽµx= 200 (10-6), ÃŽµy= -300 (10-6), and ÃŽ³xy= 400(10-6). Use the strain
Prove that the sum of the normal strains in perpendicular directions is constant, i.e., Îµx + Îµy = Îµxâ€² + Îµyâ€².
Determine the principal stresses and the absolute maximum shear stress. 150 MPa 120 MPa
Determine the principal stresses and the absolute maximum shear stress. 2.5 ksi 4 ksi 5 ksi
The solid shaft is subjected to a torque, bending moment, and shear force. Determine the principal stresses at points A and B and the absolute maximum shear stress. 450 mm 300 N-m 25 mm 45 N-m 800 N
The frame is subjected to a horizontal force and couple moment. Determine the principal stresses and the absolute maximum shear stress at point A. The cross-sectional area at this point is shown. 400
The bolt is fixed to its support at C. If a force of 18 lb is applied to the wrench to tighten it, determine the principal stresses and the absolute maximum shear stress in the bolt shank at point A.
The bolt is fixed to its support at C. If a force of 18 lb is applied to the wrench to tighten it, determine the principal stresses and the absolute maximum shear stress developed in the bolt
The steel pipe has an inner diameter of 2.75 in. and an outer diameter of 3 in. If it is fixed at C and subjected to the horizontal 20-lb force acting on the handle of the pipe wrench, determine the
The steel pipe has an inner diameter of 2.75 in. and an outer diameter of 3 in. If it is fixed at C and subjected to the horizontal 20-lb force acting on the handle of the pipe wrench, determine the
Determine the equivalent state of stress if an element is oriented 40° clockwise from the element shown. Use Mohr€™s circle. 10 ksi 6 ksi
The crane is used to support the 350-lb load. Determine the principal stresses acting in the boom at points A and B. The cross section is rectangular and has a width of 6 in. and a thickness of 3 in.
Determine the equivalent state of stress on an element which represents (a) the principal stresses, and (b) the maximum in-plane shear stress and the associated average normal stress. Also, for each
The propeller shaft of the tugboat is subjected to the compressive force and torque shown. If the shaft has an inner diameter of 100 mm and an outer diameter of 150 mm, determine the principal
Determine the principal stresses in the box beam at points A and B. 1200 lb 800 lb 6 in. 6 in.T B 8 in. „B sat25at -3 ft+2.5 ft+2.5 ft '8 in. -5 ft-
Determine (a) the principal stresses and (b) the maximum in-plane shear stress and average normal stress at the point. Specify the orientation of the element in each case. 60 MPa 30 MPa 45 MPa
Determine the stress components acting on the inclined plane AB. Solve the problem using the method of equilibrium described in Sec. 9.1. 14 ksi 20 ksi 50° в B.
Determine the principal stresses and the absolute maximum shear stress. 10 ksi 8 ksi 20 ksi
Determine the principal stresses and the absolute maximum shear stress. 120 psi 70 psi 30 psi
Draw the three Mohr€™s circles that describe the following state of stress. 25 ksi 25 ksi
Draw the three Mohr€™s circles that describe the following state of stress. 300 psi 400 psi
Draw the three Mohr€™s circles that describe each of the following states of stress. 5 ksi 3 ksi 180 MPa 140 MPa (a) (b)

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