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physics
mechanics
Mechanics of Materials 7th edition James M. Gere, Barry J. Goodno - Solutions
A heavy object of weight W is dropped onto the midpoint of a simple beam AB from a height h (see figure).Obtain a formula for the maximum bending stress σmax due to the falling weight in terms of h, σst, and δst, where σst is the maximum bending
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 deflection of the beam when the weight W is applied statically, obtain a formula for the maximum
A cantilever beam AB of length L = 6 ft is constructed of a W 8 x 21 wide-flange section (see figure). A weight W = 1500 lb falls through a height h = 0.25 in. onto the end of the beam.Calculate the maximum deflection δmax of the end of the beam and the maximum bending stress
A weight W = 20 kN falls through a height h = 1.0 mm onto the midpoint of a simple beam of length L = 3 m (see figure). The beam is made of wood with square cross section (dimension d on each side) and E = 12 GPa. If the allowable bending stress in the wood is σallow = 10 MPa, what is
A weight W = 4000 lb falls through a height h = 0.5 in. onto the midpoint of a simple beam of length L = 10 ft (see figure).Assuming that the allowable bending stress in the beam is σallow = 18,000 psi and E = 30 x 106 psi, select the lightest wide-flange beam listed in Table E-1a in
An overhanging beam ABC of rectangular cross section has the dimensions shown in the figure. A weight W = 750 N drops onto end C of the beam. If the allowable normal stress in bending is 45 MPa, what is the maximum height h from which the weight may be dropped? (Assume E = 12 GPa.)
A heavy flywheel rotates at an angular speed v (radians per second) around an axle (see figure). The axle is rigidly attached to the end of a simply supported beam of flexural rigidity EI and length L (see figure). The flywheel has mass moment of inertia Im about its axis of rotation. If the
A simple beam AB of length L and height h undergoes a temperature change such that the bottom of the beam is at temperature T2 and the top of the beam is at temperature T1 (see figure).Determine the equation of the deflection curve of the beam, the angle of rotation θA at the left-hand
A cantilever beam AB of length L and height h (see figure) is subjected to a temperature change such that the temperature at the top is T1 and at the bottom is T2.Determine the equation of the deflection curve of the beam, the angle of rotation θB at end B, and the deflection
An overhanging beam ABC of height h has a guided support at A and a roller at B. The beam is heated to a temperature T1 on the top and T2 on the bottom (see figure).Determine the equation of the deflection curve of the beam, the angle of rotation θC at end C, and the deflection
A simple beam AB of length L and height h (see figure) is heated in such a manner that the temperature difference T2 - T1 between the bottom and top of the beam is proportional to the distance from support A; that is, assume the temperature difference varies linearly along the beam:T2 - T1 = T0xin
Beam AB, with elastic support kR at A and pin support at B, of length L and height h (see figure) is heated in such a manner that the temperature difference T2 - T1 between the bottom and top of the beam is proportional to the distance from support A; that is, assume the temperature difference
The deflection curve for a simple beam AB (see figure) is given by the following equation:Describe the load acting on the beam.
The deflection curve for a simple beam AB (see figure) is given by the following equation:(a) Describe the load acting on the beam. (b) Determine the reactions RA and RB at the supports. (c) Determine the maximum bending moment Mmax.
The deflection curve for a cantilever beam AB (see figure) is given by the following equation:Describe the load acting on the beam.
The deflection curve for a cantilever beam AB (see figure) is given by the following equation:(a) Describe the load acting on the beam. (b) Determine the reactions RA and MA at the support.
A wide-flange beam (W 12 x 35) supports a uniform load on a simple span of length L = 14 ft (see figure).Calculate the maximum deflection δmax at the midpoint and the angles of rotation u at the supports if q = 1.8 k/ft and E = 30 x 106 psi. Use the formulas of Example 9-1.
A cantilever beam AB supporting a triangularly distributed load of maximum intensity q0 is shown in the figure.Derive the equation of the deflection curve and then obtain formulas for the deflection δB and angle of rotation θB at the free end.
A cantilever beam AB is acted upon by a uniformly distributed moment (bending moment, not torque) of intensity m per unit distance along the axis of the beam (see figure).Derive the equation of the deflection curve and then obtain formulas for the deflection δB and angle of rotation
The beam shown in the figure has a guided support at A and a spring support at B. The guided support permits vertical movement but no rotation.Derive the equation of the deflection curve and determine the deflection δB at end B due to the uniform load of intensity q.
Derive the equations of the deflection curve for a simple beam AB loaded by a couple M0 acting at distance a from the left-hand support (see figure). Also, determine the deflection δ0 at the point where the load is applied.
Derive the equations of the deflection curve for a cantilever beam AB carrying a uniform load of intensity q over part of the span (see figure). Also, determine the deflection δB at the end of the beam.
Derive the equations of the deflection curve for a cantilever beam AB supporting a distributed load of peak intensity q0 acting over one-half of the length (see figure). Also, obtain formulas for the deflections δB and δC at points B and C, respectively.
Derive the equations of the deflection curve for a simple beam AB with a distributed load of peak intensity q0 acting over the left-hand half of the span (see figure). Also, determine the deflection δC at the midpoint of the beam.
The beam shown in the figure has a guided support at A and a roller support at B. The guided support permits vertical movement but no rotation. Derive the equation of the deflection curve and determine the deflection δA at end A and also δC at point C due to the uniform
A uniformly loaded steel wide-flange beam with simple supports (see figure) has a downward deflection of 10 mm at the midpoint and angles of rotation equal to 0.01 radians at the ends. Calculate the height h of the beam if the maximum bending stress is 90 MPa and the modulus of elasticity is 200
What is the span length L of a uniformly loaded simple beam of wide-flange cross section (see figure) if the maximum bending stress is 12,000 psi, the maximum deflection is 0.1 in., the height of the beam is 12 in., and the modulus of elasticity is 30 x 106 psi? (Use the formulas of Example 9-1.)
Calculate the maximum deflection δmax of a uniformly loaded simple beam (see figure) if the span length L = 2.0 m, the intensity of the uniform load q = 2.0 kN/m, and the maximum bending stress s = 60 MPa. The cross section of the beam is square, and the material is aluminum having
A cantilever beam with a uniform load (see figure) has a height h equal to 1/8 of the length L. The beam is a steel wideflange section with E = 28 x 106 psi and an allowable bending stress of 17,500 psi in both tension and compression.Calculate the ratio d/L of the deflection at the free end to the
A gold-alloy microbeam attached to a silicon wafer behaves like a cantilever beam subjected to a uniform load (see figure). The beam has length L = 27.5 μm and rectangular cross section of width b = 4.0 μm and thickness t = 0.88 μm.The total load on the
Obtain a formula for the ratio dC/dmax of the deflection at the midpoint to the maximum deflection for a simple beam supporting a concentrated load P (see figure).From the formula, plot a graph of δC/δmax versus the ratio a/L that defines the position of the load (0.5
Derive the equation of the deflection curve for a cantilever beam AB supporting a load P at the free end (see figure). Also, determine the deflection δB and angle of rotation θB at the free end.
Derive the equation of the deflection curve for a simple beam AB loaded by a couple M0 at the left-hand support (see figure). Also, determine the maximum deflection δmax.
Derive the equation of the deflection curve for a cantilever beam AB when a couple M0 acts counterclockwise at the free end (see figure). Also, determine the deflection δB and slope θB at the free end. Use the third-order differential equation of the deflection curve (the
Derive the equations of the deflection curve for beam AB, with guided support at A and roller support at B, supporting a distributed load of maximum intensity q0 acting on the right-hand half of the beam (see figure).Also, determine deflection δA, angle of rotation θB, and
A simple beam AB is subjected to a distributed load of intensity q = q0 sin πx/L, where q0 is the maximum intensity of the load (see figure). Derive the equation of the deflection curve, and then determine the deflection δmax at the midpoint of the beam. Use the fourth-order
The simple beam AB shown in the figure has moments 2M0 and M0 acting at the ends. Derive the equation of the deflection curve, and then determine the maximum deflection δmax. Use the third-order differential equation of the deflection curve (the shear-force equation).
A beam with a uniform load has a guided support at one end and spring support at the other. The spring has stiffness k = 48EI/L3. Derive the equation of the deflection curve by starting with the third-order differential equation (the shear-force equation). Also, determine the angle of rotation
The distributed load acting on a cantilever beam AB has an intensity q given by the expression q0 cos πx/2L, where q0 is the maximum intensity of the load (see figure).Derive the equation of the deflection curve, and then determine the deflection dB at the free end. Use the fourth-order
A cantilever beam AB is subjected to a parabolically varying load of intensity q = q0 (L2 x2)/L2, where q0 is the maximum intensity of the load (see figure).Derive the equation of the deflection curve, and then determine the deflection δB and angle of rotation θB at the
A beam on simple supports is subjected to a parabolically distributed load of intensity q = 4q0x(L x)/L2, where q0 is the maximum intensity of the load (see figure).Derive the equation of the deflection curve, and then determine the maximum deflection δmax. Use the fourth order
Derive the equation of the deflection curve for beam AB, with guided support at A and roller at B, carrying a triangularly distributed load of maximum intensity q0 (see figure).Also, determine the maximum deflection δmax of the beam. Use the fourth-order differential equation of the
Derive the equations of the deflection curve for beam ABC, with guided support at A and roller support at B, supporting a uniform load of intensity q acting on the over-hang portion of the beam (see figure).Also, determine deflection δC and angle of rotation θC. Use the
A cantilever beam AB carries three equally spaced concentrated loads, as shown in the figure. Obtain formulas for the angle of rotation θB and deflection δB at the free end of the beam.
A beam ABC having flexural rigidity EI = 75 kNˆ™m2 is loaded by a force P = 800 N at end C and tied down at end A by a wire having axial rigidity EA = 900 kN (see figure).What is the deflection at point C when the load P is applied?
Determine the angle of rotation δB and deflection θB at the free end of a cantilever beam AB supporting a parabolic load defined by the equation q = q0x2/L2 (see figure).
A simple beam AB supports a uniform load of intensity q acting over the middle region of the span (see figure).Determine the angle of rotation θA at the left-hand support and the deflection δmax at the midpoint.
The overhanging beam ABCD supports two concentrated loads P and Q (see figure).(a) For what ratio P/Q will the deflection at point B be zero? (b) For what ratio will the deflection at point D be zero? (c) If Q is replaced by uniform load with intensity q (on the overhang), repeat (a) and (b) but
A thin metal strip of total weight W and length L is placed across the top of a flat table of width L/3 as shown in the figure. What is the clearance d between the strip and the middle of the table? (The strip of metal has flexural rigidity EI.)
An overhanging beam ABC with flexural rigidity EI = 15 k-in.2 is supported by a guided support at A and by a spring of stiffness k at point B (see figure). Span AB has length L = 30 in. and carries a uniform load. The over-hang BC has length b = 15 in. For what stiffness k of the spring will the
A beam ABCD rests on simple supports at B and C (see figure). The beam has a slight initial curvature so that end A is 18 mm above the elevation of the supports and end D is 12 mm above. What moments M1 and M2, acting at points A and D, respectively, will move points A and D downward to the level
The compound beam ABC shown in the figure has a guided 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 deflection d under the load P.
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 δE at the free end E due to the load P acting at that point.
A stekel beam ABC is simply supported at A and held by a high-strength steel wire at B (see figure). 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 flexural rigidity EI = 36 x 106 lb-in.2 What is the deflection δC of point C
A simple beam AB supports five equally spaced loads P (see figure).(a) Determine the deflection δ1 at the midpoint of the beam. (b) If the same total load (5P) is distributed as a uniform load on the beam, what is the deflection δ2 at the midpoint? (c) Calculate the ratio
The compound beam shown in the figure consists of a cantilever beam AB (length L) that is pin-connected to a simple beam BD (length 2L). After the beam is constructed, a clearance c exists between the beam and a support at C, midway between points B and D. Subsequently, a uniform load is placed
Find the horizontal deflection δh and vertical deflection δn at the free end C of the frame ABC shown in the figure. (The flexural rigidity EI is constant throughout the frame.)
The frame ABCD shown in the figure is squeezed by two collinear forces P acting at points A and D. What is the decrease d in the distance between points A and D when the loads P are applied? (The flexural rigidity EI is constant throughout the frame.)
A beam ABCDE has simple supports at B and D and symmetrical overhangs at each end (see figure). The center span has length L and each overhang has length b. A uniform load of intensity q acts on the beam.(a) Determine the ratio b/L so that the deflection δC at the midpoint of the beam
A frame ABC is loaded at point C by a force P acting at an angle α to the horizontal (see figure). Both members of the frame have the same length and the same flexural rigidity.Determine the angle a so that the deflection of point C is in the same direction as the load. (Disregard the
The cantilever beam AB shown in the figure has an extension BCD attached to its free end. A force P acts at the end of the extension.(a) Find the ratio a/L so that the vertical deflection of point B will be zero.(b) Find the ratio a/L so that the angle of rotation at point B will be zero.
Beam ACB hangs from two springs, as shown in the figure. The springs have stiffnesses k1 and k2 and the beam has flexural rigidity EI.(a) What is the downward displacement of point C, which is at the midpoint of the beam, when the moment M0 is applied? Data for the structure are as follows: M0 =
What must be the equation y = f(x) of the axis of the slightly curved beam AB (see figure) before the load is applied in order that the load P, moving along the bar, always stays at the same level?
Determine the angle of rotation θB and deflection δB at the free end of a cantilever beam AB having a uniform load of intensity q acting over the middle third of its length (see figure).
The cantilever beam ACB shown in the figure has flexural rigidity EI = 2.1 x 106 k-in.2 Calculate the downward deflections δC and δB at points C and B, respectively, due to the simultaneous action of the moment of 35 k-in. applied at point C and the concentrated load of
A beam ABCD consisting of a simple span BD and an overhang AB is loaded by a force P acting at the end of the bracket CEF (see figure).(a) Determine the deflection δA at the end of the overhang. (b) Under what conditions is this deflection upward? Under what conditions is it downward?
A horizontal load P acts at end C of the bracket ABC shown in the figure.(a) Determine the deflection δC of point C. (b) Determine the maximum upward deflection δmax of member AB.
A cantilever beam AB is subjected to a uniform load of intensity q acting throughout its length (see figure).Determine the angle of rotation θB and the deflection δB at the free end.
The simple beam AB shown in the figure supports two equal concentrated loads P, one acting downward and the other upward. Determine the angle of rotation θA at the left-hand end, the deflection δ1 under the downward load, and the deflection δ2 at the midpoint
A simple beam AB is subjected to couples M0 and 2M0 as shown in the figure. Determine the angles of rotation θA and θB at the beam and the deflection d at point D where the load M0 is applied.
The load on a cantilever beam AB has a triangular distribution with maximum intensity q0 (see figure).Determine the angle of rotation θB and the deflection δB at the free end.
A cantilever beam AB is subjected to a concentrated load P and a couple M0 acting at the free end (see figure). Obtain formulas for the angle of rotation θB and the deflection δB at end B.
Determine the angle of rotation θB and the deflection δB at the free end of a cantilever beam AB with a uniform load of intensity q acting over the middle third of the length (see figure).
Calculate the deflections δB and δC at points B and C, respectively, of the cantilever beam ACB shown in the figure. Assume M0 = 36 k-in., P = 3.8 k, L = 8 ft, and EI = 2.25 x 109 lb-in2.
A cantilever beam ACB supports two concentrated loads P1 and P2 as shown in the figure.Calculate the deflections δB and δC at points B and C, respectively. Assume P1 = 10 kN, P2 = 5 kN, L = 2.6 m, E = 200 GPa, and I = 20.1 x 106 mm4.
Obtain formulas for the angle of rotation θA at support A and the deflection δmax at the midpoint for a simple beam AB with a uniform load of intensity q (see figure).
A simple beam AB supports two concentrated loads P at the positions shown in the figure. A support C at the midpoint of the beam is positioned at distance d below the beam before the loads are applied. Assuming that d = 10 mm, L = 6 m, E = 200 GPa, and I = 198 x 106 mm4, calculate the magnitude of
A simple beam AB is subjected to a load in the form of a couple M0 acting at end B (see figure). Determine the angles of rotation θA and θB at the supports and the deflection d at the midpoint.
The cantilever beam ACB shown in the figure has moments of inertia I2 and I1 in parts AC and CB, respectively.(a) Using the method of superposition, determine the deflection δB at the free end due to the load P.(b) Determine the ratio r of the deflection δB to the
A tapered cantilever beam AB supports a concentrated load P at the free end (see figure).The cross sections of the beam are rectangular tubes with constant width b and outer tube depth dA at A, and outer tube depth dB = 3dA/2 at support B. The tube thickness is constant, t = dA/20. IA is the moment
Repeat Problem 9.7-10 but now use the tapered propped cantilever tube AB, with guided support at B, shown in the figure which supports a concentrated load P at the guided end. Find the equation of the deflection curve and the deflection dB at the guided end of the beam due to the load P.
A simple beam ACB is constructed with square cross sections and a double taper (see figure).The depth of the beam at the supports is dA and at the midpoint is dC = 2dA. Each half of the beam has length L. Thus, the depth d and moment of inertia I at distance x from the left-hand end are,
The cantilever beam ACB shown in the figure supports a uniform load of intensity q throughout its length. The beam has moments of inertia I2 and I1 in parts AC and CB, respectively.(a) Using the method of superposition, determine the deflection δB at the free end due to the uniform
Beam ACB hangs from two springs, as shown in the figure. The springs have stiffnesses k1 and k2 and the beam has flexural rigidity EI.(a) What is the downward displacement of point C, which is at the midpoint of the beam, when the moment M0 is applied? Data for the structure are as follows: M0 =
A simple beam ABCD has moment of inertia I near the supports and moment of inertia 2I in the middle region, as shown in the figure. A uniform load of intensity q acts over the entire length of the beam. Determine the equations of the deflection curve for the left-hand half of the beam. Also, find
A beam ABC has a rigid segment from A to B and a flexible segment with moment of inertia I from B to C (see figure).A concentrated load P acts at point B. Determine the angle of rotation θA of the rigid segment, the deflection δB at point B, and the maximum deflection
A simple beam ABC has moment of inertia 1.5I from A to B and I from B to C (see figure).A concentrated load P acts at point B. Obtain the equations of the deflection curves for both parts of the beam. From the equations, determine the angles of rotation θA and θC at the
The tapered cantilever beam AB shown in the figure has thin-walled, hollow circular cross sections of constant thickness t.The diameters at the ends A and B are dA and dB = 2dA, respectively. Thus, the diameter d and moment of inertia I at distance x from the free end are, respectively, in which
The tapered cantilever beam AB shown in the figure has a solid circular cross section.The diameters at the ends A and B are dA and dB = 2dA, respectively. Thus, the diameter d and moment of inertia I at distance x from the free end are, respectively, in which IA is the moment of inertia at end A of
A tapered cantilever beam AB supports a concentrated load P at the free end (see figure).The cross sections of the beam are rectangular with constant width b, depth dA at support A, and depth dB = 3dA/2 at the support. Thus, the depth d and moment of inertia I at distance x from the free end are,
A uniformly loaded simple beam AB (see figure) of span length L and rectangular cross section (b = width, h = height) has a maximum bending stress σmax due to the uniform load. Determine the strain energy U stored in the beam.
A simple beam AB of length L supports a concentrated load P at the midpoint (see figure).(a) Evaluate the strain energy of the beam from the bending moment in the beam.(b) Evaluate the strain energy of the beam from the equation of the deflection curve.(c) From the strain energy, determine the
A propped cantilever beam AB of length L, and with guided support at A, supports a uniform load of intensity q (see figure).(a) Evaluate the strain energy of the beam from the bending moment in the beam.(b) Evaluate the strain energy of the beam from the equation of the deflection curve.
A simple beam AB of length L is subjected to loads that produce a symmetric deflection curve with maximum deflection d at the midpoint of the span (see figure). How much strain energy U is stored in the beam if the deflection curve is(a) A parabola,(b) A half wave of a sine curve?
A beam ABC with simple supports at A and B and an overhang BC supports a concentrated load P at the free end C (see figure).(a) Determine the strain energy U stored in the beam due to the load P. (b) From the strain energy, find the deflection δC under the load P. (c) Calculate the
A simple beam ACB supporting a uniform load q over the first half of the beam and a couple of moment M0 at end B is shown in the figure.Determine the strain energy U stored in the beam due to the load q and the couple M0 acting simultaneously.
The frame shown in the figure consists of a beam ACB supported by a struct CD. The beam has length 2L and is continuous through joint C. A concentrated load P acts at the free end B. Determine the vertical deflection δB at point B due to the load P.
A simple beam AB of length L is loaded at the left-hand end by a couple of moment M0 (see figure).
An overhanging beam ABC is subjected to a couple MA at the free end (see figure). The lengths of the overhang and the main span are a and L, respectively.
An overhanging beam ABC rests on a simple support at A and a spring support at B (see figure). A concentrated load P acts at the end of the overhang. Span AB has length L, the overhang has length a, and the spring has stiffness k.
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