Questions and Answers of Mechanics Of Materials

A cantilever beam has two triangular loads as shown in the figure.(a) Find an expression for beam deflection σC using superposition.(b) Find the required magnitude of load intensity q2 in terms of
Find required distance d (in terms of L) so that rotation B = 0 is due to M and q loadings applied at the same time. Also, what is the resulting net rotation θA at support A? Moment M is applied at
Find an expression for required moment MA (in terms of q and L) that will result in rotation θB = 0 due to MA and q loadings applied at the same time. Also, what is the resulting net rotation at
Find an expression for the required moment MA (in terms of q and L) that will result in rotation θB = 0 due to MA and q loadings applied at the same time. Also, what is the resulting net rotation at
The wing of a small plane is represented by a simplified prismatic cantilever beam model acted on by the distributed loads shown in the figure. Assume constant EI =1200 kN.m2. Find the tip deflection
The wing of a large commercial jet is represented by a simplified prismatic cantilever beam model with uniform load w and concentrated loads P at the two engine locations (see figure). Find
A framework ABCD is acted on by force P at 2L/3 from. Assume that EI is constant.(a) Find expressions for reactions at supports B and C.(b) Find expressions for angles of rotation at A, B, C, and
Compound beam ABC is loaded by point load P = 1.5 kips at a distance 2a/3 from point A and a triangularly distributed load on segment BC with peak intensity q0 = 0.5 kips/ft. If length a = ft and
Calculate the deflection at point C of a beam subjected to uniformly distributed load w = 275 N/m on span AB and point load P = 10 kN at C. Assume that L = 5 m and EI = 1.50 X 107 N.m2.
A steel beam ABC is simply supported at A and held by a high-strength steel wire at B. A load P = 240 lb acts at the free end C. The wire has axial rigidity EA = 1500 X 103 lb, and the beam has
A compound beam ABCDE (see figure) consists of two parts (ABC and CDE) connected by a hinge (i.e., moment release) at C. The elastic support at B has stiffness k = EI / b3. Determine the deflection
The compound beam ABC shown in the figure has a sliding support at A and a fixed support at C. The beam consists of two members joined by a pin connection (i.e., moment release) at B. Find the
A cantilever beam is subjected to a quadratic distributed load q(x) over the length of the beam (see figure). Find an expression for moment M in terms of the peak distributed load intensity q0 so
A simple beam with an overhang is subjected to a point load P = 6 kN. If the maximum allowable deflection at point C is 0.5 mm, select the lightest W 360 section from Table F-1(b) that can be used
Repeat Problem 9.5-15 for the anti-symmetric loading shown in the figure.Staticsso reactions at A and B are equal and opposite but neither is equal to P (unlike symmetnc load case)Problem 9.5-15Use
A cantilever beam has a length L = 12 ft and a rectangular cross-section (b = 16 in., h = 24 in.). A linearly varying distributed load with peak intensity q0 acts on the beam.(a) Find peak intensity
Use the method of superposition to find the angles of rotation θA and θB at the supports, and the maximum deflection θmax for a simply supported beam subjected to symmetric loads P at a
Derive the equation of the deflection curve for a cantilever beam AB when a couple of M0 acts counterclockwise at the free end (see figure). Also, determine the deflection σB and slope πB at the
A cantilever beam is subjected to load P at mid-span and counterclockwise moment M at B (see figure).(a) Find an expression for moment M in terms of the load P so that the reaction moment MA at A is
Beams AB and CDE are connected using rigid link DB with hinges (or moment releases) at ends D and B (see figure a). Beam AB is fixed at joint A and beam CDE is pin-supported at joint E. Load P =150
A cantilever beam carries a trapezoidal distributed load (see figure). Let wB = 2.5 kN/m, wA =.0 kN/m, and L = 2.5 m. The beam has a modulus E = 45GPa and a rectangular cross-section with
A cantilever beam ACB supports two concentrated loads P1 and P2, as shown in the figure. Determine the deflections dC and dB at points C and B, respectively. P₁ C P2 B
A beam ABC with simple supports at A and B and an overhang BC supports a concentrated load P at the free end C.(a) Determine the strain energy U stored in the beam due to the load P.(b) From the
Beam ABC is loaded by a uniform load q and point load P at joint C. Using the method of superposition, calculate the deflection at joint C. Assume that L = 4 m, a = 2 m, q = 15 kN/m, P = 7.5 kN, E =
Copper beam AB has a circular cross-section with a radius of 0.25 in. and length L = 3 ft. The beam is subjected to a uniformly distributed load w = 3.5 lb/ft. Calculate the required load P at joint
An overhanging beam ABC supports a concentrated load P at the end of the overhang. Span AB has length L, and the overhang has length a. Determine the deflection dC at the end of the overhang.
A simply supported beam (E = 12 GPa) carries a uniformly distributed load q = 125 N/m, and a point load P = 200 N at mid-span. The beam has a rectangular cross-section (b = 75 mm, h = 200 mm) and a
An object of weight W is dropped onto the midpoint of a simple beam AB from a height h (see figure). The beam has a rectangular cross-section of area A. Assuming that h is very large compared to the
A simply supported beam is loaded with a point load, as shown in the figure. The beam is a steel wide flange (W 12 X 35) in strong axis bending. Calculate the maximum deflection of the beam and the
A simply supported beam (E = 1600 ksi) is loaded by a triangular distributed load from A to C (see figure). The load has a peak intensity q0 = 10 lb/ ft, and the deflection is known to be 0.01 in. at
A heavy object of weight W is dropped onto the midpoint of a simple beam AB from a height h. Obtain a formula for the maximum bending stress smax due to the falling weight in terms of h, σst, and
A simple beam AB of length L is loaded at the left-hand end by a couple of moments M0. Determine the angle of rotation uA at support A. Mo L B 30
A crank arm consists of a solid segment of length b1 and diameter d, a segment of length b2, and a segment of length b3, as shown in the figure. Two loads P act as shown: one parallel to -x and
An arm ABC lying in a horizontal plane and supported at A (see figure) is made of two identical solid steel bars AB and BC welded together at a right angle. Each bar is 22 in. long.(a) Knowing that
A horizontal bracket ABC consists of two perpendicular arms AB of a length 0.75 m and BC of a length 0.5 m. The bracket has a solid, circular cross section with a diameter equal to 65 mm. The bracket
An L-shaped bracket lying in a horizontal plane supports a load P = 150 lb (see figure). The bracket has a hollow rectangular cross section with thickness t = 0.125 in. and outer dimensions b = 2.0
A semicircular bar AB lying in a horizontal plane is supported at B (see figure part a). The bar has a centerline radius R and weight q per unit of length (total weight of the bar equals πqR). The
A double-decker bicycle rack made up of square steel tubing is fixed at A (figure a). The weight of a bicycle is represented as a point load applied at B on a plane frame model of the rack (figure
Beam ABCD has sliding support at A, roller supports at C and D, and a pin connection at B (see figure). Assume that the beam has a rectangular cross-section (b = 4 in., h = 12 in.). Uniform load q
Determine the maximum tensile, compressive, and shear stresses at points A and B on the bicycle pedal crank shown in the figure. The pedal and crank are in a horizontal plane and points A and B are
A gondola on a ski lift is supported by two bent arms, as shown in the figure. Each arm is offset by the distance b = 180 mm from the line of action of the weight force W. The allowable stresses in
A W 12 X 14 wide-f lange beam (see TableF-1(a), Appendix F) is simply supported with a span length of 120 in. (see figure). The beam supports two anti-symmetrically placed concentrated loads of
A W 360 X 79 steel beam is fixed at A. The beam has a length of 2.5 m and is subjected to a linearly varying distributed load with maximum intensity q0 = 500 N/m on segment AB and a uniformly
Repeat the preceding problem but now find the stress state on Element A at the base. Let WS = 240 N, WL = 250 N, t = 5 mm, d = 360 mm. See the figure for the locations of element A and all loads. 240
A W 12 X 35 steel beam is fixed at A. The beam has a length L = 6 ft and is subjected to a linearly varying distributed load with peak intensity q0 = 830 lb/ft. Calculate the state of plane stress at
A traffic light and signal pole is subjected to the weight of each traffic signal WS = 45 lb and the weight of the road lamp WL = 55 lb. The pole is fixed at the base. Find the principal normal
A sign is supported by a pipe having an outer diameter 110 mm and inner diameter 90mm. The dimensions of the sign are 2.0 m X 1.0 m, and its lower edge is 3.0 m above the base. Note that the center
A sign is supported by a pole of hollow circular cross section, as shown in the figure. The outer and inner diameters of the pole are 10.5 in. and 8.5 in., respectively. The pole is 42 ft high and
A post having a hollow, circular cross section supports a P = 3.2 kN load acting at the end of an arm that is b = 1.5 m long (see figure). The height of the post is L = 9 m, and its section modulus
A segment of a generator shaft with a hollow circular cross-section is subjected to a torque T = 240 kip-in. (see figure). The outer and inner diameters of the shaft are 8.0 in. and 6.25 in.,
A cantilever beam (width b = 3 in. and depth h = 6 in.) has a length L = 5 ft and is subjected to a point load P and a concentrated moment M = 20 kip-ft at end B. If normal stress σx = 0 at
The hollow drill pipe for an oil well (see figure) is 6.2 in. in outer diameter and 0.75 in. in thickness. Just above the bit, the compressive force in the pipe (due to the weight of the pipe) is 62
Beam ABC with an overhang BC is subjected to a linearly varying distributed load on span AB with peak intensity q0 = 2500 N/m and a point load P = 1250 N applied at C. The beam has a width b =100 mm
A segment of a generator shaft is subjected to a torque T and an axial force P, as shown in the figure. The shaft is hollow (outer diameter d2 = 300 mm and inner diameter d1 =stress 250 mm) and
A cantilever wood beam with a width b =100 mm and depth h = 150 mm has a length L = 2 m and is subjected to point load P at mid-span and uniform load q =15 N/m. (a) If the normal stress σx = 0 at
The torsional pendulum shown in the figure consists of a horizontal circular disk of a mass M = 60 kg suspended by a vertical steel wire (G = 80 GPa) of a length L = 2 m and diameter d = 4 mm.
A cylindrical pressure vessel with flat ends is subjected to a torque T and a bending moment M (see figure). The outer radius is 12.0 in. and the wall thickness is 1.0 in. The loads are T = 800
A pressurized cylindrical tank with flat ends is loaded by torques T and tensile forces P (see figure). The tank has a radius of r = 125 mm and wall thickness t = 6.5 mm. The internal pressure p =
A cylindrical pressure vessel having a radius r =14 in. and wall thickness t = 0.5 in. is subjected to internal pressure p = 375 psi. In addition, a torque T = 90 kip-ft acts at each end of the
A cylindrical tank having a diameter d = 2.5 in. is subjected to internal gas pressure p = 600 psi and an external tensile load T = 1000 lb (see figure). Determine the minimum thickness t of the wall
Solve the preceding problem using transverse load V = 300 N and torque T = 3.5 Nm applied at point B. The bar has length L = 1.5 m and diameter d = 8 mm. Calculate the principal stresses and the
A cantilever beam with a width b =100 mm and depth h = 150 mm has a length L = 2 m and is subjected to a point load P = 500 N at B. Calculate the state of plane stress at point C located 50 mm below
A solid circular bar is fixed at point A. The bar is subjected to transverse load V = 70 lb and torque T = 300 lb-in. at point B. The bar has a length L = 60 in. and a diameter d = 3 in. Calculate
A scuba tank (see figure) is being designed for an internal pressure of 2640 psi with a factor of safety of 2.0 with respect to yield. The yield stress of the steel is 65,000 psi in tension and
A W 12 x 35 steel cantilever beam is subjected to an axial load P = 10 kips and a transverse load V = 15 kips. The beam has length L = 6 ft. (a) Calculate the principal normal stresses and the
A W 310 x 52 steel beam is subjected to a point load P = 45 kN and a transverse load V = 20 kN at B. The beam has a length L = 2 m. (a) Calculate the principal normal stresses and the maximum shear
Repeat Problem 8.3-1 for a fire extinguisher tank with an internal pressure of 1.8 MPa, a diameter of 130 mm, and thickness 1.5 mm.Problem 8.3-1A fire extinguisher tank is designed for an internal
A wood beam with a cross-section 4 x 6 in. is simply supported at A and B. The beam has a length of 9 ft and is subjected to point load P = 5 kips at mid-span. Calculate the state of stress at point
A fire extinguisher tank is designed for an internal pressure of 825 psi. The tank has an outer diameter of 4.5 in. and a thickness of 0.08 in. Calculate the longitudinal stress, the circumferential
Solve Problem 7.7-14 by using Mohr’s circle for plane strain.Solve the preceding problem for the following strains: Yxy &x=-1120 × 10-6, Ey = -430 × 10-6, = 780 x 10-6, and 0 = 45°.
Solve Problem 7.7-13 by using Mohr’s circle for plane strain.Problem 7.7-13An element of material in plane strain is subjected to strains. Ex 480 × 107 X Ey = = 70 × 10-6, and Y.xv = 420 ×
Solve Problem 7.7-12 by using Mohr’s circle for plane strain.Problem 7.7-12Solve the preceding problem for the following strains: Yxy & = 120 × 10-6, y E X =-360 × 10-6. =-450 × 10-6, and
Solve Problem 7.7-11 by using Mohr’s circle for plane strain.Problem 7.7-11The strains for an element of material in plane strain (see figure) are as follows:Determine the principal strains and
A simply supported wood beam is subjected to point load P at mid-span. The normal stress on the element C is known to be σx = 12 MPa. Find the maximum shear stress on the element and show the
A simply supported wood beam is subjected to point load P at mid-span. The stresses on element C are known to be σx = –92 psi and πxy = –7 psi. Find the principal stresses on the element and
Solve Problem 7.7-9 by using Mohr’s circle for plane strain.Problem 7.7-9An element of material subjected to plane strain (see figure) has strains ofCalculate the strains for an element oriented at
A square plate with side dimension of 2 in, is subjected to compressive stress σx and tensile stress σy. The stresses on element A oriented at angle u = 458 are σx = - 75 psi, σy = - 75 psi, and
An element in plane stress on the surface of an automobile drive shaft is subjected to stresses of σx = -45 MPa and πxy = 39 MPa. It is known that one of the principal stresses equals 41 MPa in
The stresses at a point on the down tube of a bicycle frame are σx = 4800 psi and πxy = –1950 psi. It is known that one of the principal stresses equals 6375 psi in tension.(a) Determine the
An element in plane stress is subjected to stresses σx, σy, and θxy (see figure). Using Mohr’s circle, determine the stresses acting on an element oriented at an angle u from the x-axis. Show
The stresses acting on a stress element on the arm of a power excavator (see figure) are σx = 52 MPa and txy = 33 MPa. What is the allowable range of values for the stress σy if the maximum shear
An element in plane stress is subjected to stresses σx, σy, and θxy (see figure). Using Mohr’s circle, determine the stresses acting on an element oriented at an angle u from the x-axis. Show
At a point on the web of a girder on a gantry crane, the stresses acting on the x face of a stress element are σx = 6250 psi and πxy = 1425 psi. What is the allowable range of values for the stress
An element in plane stress is subjected to stresses σx, σy, and πxy.(a) Determine the principal stresses and show them on a sketch of a properly oriented element.(b) Determine the maximum shear
The state of stress on an element along the hydraulic lift cylinder on a truck is σy = -5 MPa. Find the maximum shear stress on the element and show the state of stress on a sketch of a properly
An element in plane stress is subjected to stresses σx, σy, and πxy.(a) Determine the principal stresses and show them on a sketch of a properly oriented element.(b) Determine the maximum shear
An element in plane stress is subjected to stresses σx, σy, and θxy (see figure). Using Mohr’s circle, determine the stresses acting on an element oriented at an angle u from the-axis. Show
An element in plane stress is subjected to stresses σx, σy, and θxy (see figure). Using Mohr’s circle, determine the stresses acting on an element oriented at an angle u from the-axis. Show
An element in plane stress is subjected to stresses σx, σy, and θxy (see figure). Using Mohr’s circle, determine the stresses acting on an element oriented at an angle u from the x-axis. Show
An element in plane stress is subjected to stresses σx, σy, and πxy.(a) Determine the principal stresses and show them on a sketch of a properly oriented element.(b) Determine the maximum shear
An element in plane stress is subjected to stresses σx, σy, and θxy. Using Mohr’s circle, determine the stresses acting on an element oriented at an angle u from the -axiss. Show these stresses
An element in plane stress is subjected to stresses σx, σy, and θxy (see figure). Using Mohr’s circle, determine the stresses acting on an element oriented at an angle u from the x-axis. Show
A rubber sheet in biaxial stress is subjected to tensile stresses σx = 270 Pa and σy = 144 Pa. The corresponding strains in the sheet are εx = 0.0002 and εy = 0.000015. Determine Poisson’s
An element in plane stress is subjected to stresses σx, σy, and θxy (see figure). Using Mohr’s circle, determine the stresses acting on an element oriented at an angle u from the x-axis. Show
Repeat the preceding problem using σy = - 750 psi.
The normal stress on an elastomeric rubber pad in a test machine is σy = -100 psi (see figure). Assume E = 312 psi and shear modulus G =105 psi. (a) Calculate the strains in the pad in the
An element in plane stress is subjected to stresses σx, σy, and θxy (see figure). Using Mohr’s circle, determine the stresses acting on an element oriented at an angle u from the x-axiss. Show
An element in plane stress is subjected to stresses σx, σy, and θxy (see figure). Using Mohr’s circle, determine the stresses acting on an element oriented at an angle u from the x-axis. Show
An element in plane stress is subjected to stresses σx, σy, and θxy (see figure). Using Mohr’s circle, determine the stresses acting on an element oriented at an angle u from thex-axiss. Show
The T-beam shown in the figure is fabricated by welding together two steel plates. If the allowable load for each weld is 1.8 kips/in. in the longitudinal direction, what is the maximum allowable

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