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physics
mechanics
Mechanics of Materials 7th edition James M. Gere, Barry J. Goodno - Solutions
Draw the shear-force and bending-moment diagrams for a cantilever beam AB carrying a uniform load of intensity q over one-half of its length (see figure).
The compound beam ABCDE shown in the figure consists of two beams (AD and DE) joined by a hinged connection at D. The hinge can transmit a shear force but not a bending moment. The loads on the beam consist of a 4-kN force at the end of a bracket attached at point B and a 2-kN force at the midpoint
The beam shown below has a sliding support at A and an elastic support with spring constant k at B.A distributed load q(x) is applied over the entire beam. Find all support reactions, then plot shear (V) and moment (M) diagrams for beam AB; label all critical V & M values and also the distance to
The shear-force diagram for a simple beam is shown in the figure.Determine the loading on the beam and draw the bendingmoment diagram, assuming that no couples act as loads on the beam.
The shear-force diagram for a beam is shown in the figure. Assuming that no couples act as loads on the beam, determine the forces acting on the beam and draw the bendingmoment diagram.
The compound beam below has an internal moment release just to the left of B and a shear release just to the right of C. Reactions have been computed at A, C and D and are shown in the figure.First, confirm the reaction expressions using statics, then plot shear (V) and moment (M) diagrams. Label
The compound beam below has an shear release just to the left of C and a moment release just to the right of C. A plot of the moment diagram is provided below for applied load P at B and triangular distributed loads w(x) on segments BC and CD.First, solve for reactions using statics, then plot
A simple beam AB supports two connected wheel loads P and 2P that are distance d apart (see figure). The wheels may be placed at any distance x from the left-hand support of the beam.(a) Determine the distance x that will produce the maximum shear force in the beam, and also determine the maximum
The inclined beam below represents the loads applied to a ladder: the weight (W) of the house painter and the self weight (w) of the ladder itself. Find support reactions at A and B, then plot axial force (N), shear (V) and moment (M) diagrams. Label all critical N, V & M values and also the
Beam ABC is supported by a tie rod CD as shown (see Prob. 10.4-9).Two configurations are possible: pin support at A and downward triangular load on AB, or pin at B and upward load on AB. Which has the larger maximum moment?First, find all support reactions, then plot axial force (N), shear (V) and
The plane frame below consists of column AB and beam BC which carries a triangular distributed load. Support A is fixed and there is a roller support at C. Column AB has a moment release just below joint B.Find support reactions at A and C, then plot axial force (N), shear (V) and moment (M)
The cantilever beam AB shown in the figure is subjected to a concentrated load P at the midpoint and a counterclockwise couple of moment M1 = PL/4 at the free end.Draw the shear-force and bending-moment diagrams for this beam.
The plane frame shown below is part of an elevated freeway system. Supports at A and D are fixed but there are moment releases at the base of both columns (AB andDE), as well near in column BC and at the end of beam BE.Find all support reactions, then plot axial force (N), shear (V) and moment (M)
The simple beam AB shown in the figure is subjected to a concentrated load P and a clockwise couple M1 = PL/3 acting at the third points.Draw the shear-force and bending-moment diagrams for this beam.
A simple beam AB subjected to couples M1 and 3M1 acting at the third points is shown in the figure.Draw the shear-force and bending-moment diagrams for this beam.
A simply supported beam ABC is loaded by a vertical load P acting at the end of a bracket BDE (see figure).Draw the shear-force and bending-moment diagrams for beam ABC.
A beam ABC is simply supported at A and B and has an overhang BC (see figure). The beam is loaded by two forces P and a clockwise couple of moment Pa that act through the arrangement shown.Draw the shear-force and bending-moment diagrams for beam ABC.
Beam ABCD is simply supported at B and C and has overhangs at each end (see figure). The span length is L and each overhang has length L/3. A uniform load of intensity q acts along the entire length of the beam.Draw the shear-force and bending-moment diagrams for this beam.
Determine the maximum normal strain ɛmax produced in a steel wire of diameter d = 1/16 in. when it is bent around a cylindrical drum of radius R = 24 in.
A copper wire having diameter d = 3 mm is bent into a circle and held with the ends just touching (see figure). If the maximum permissible strain in the copper is É› max = 0.0024 what is the shortest length L of wire that can be used?
A 4.5 in. outside diameter polyethylene pipe designed to carry chemical wastes is placed in a trench and bent around a quarter-circular 90o bend (see figure). The bent section of the pipe is 46ft long. Determine the maximum compressive strain ɛmax in the pipe.
A cantilever beam AB is loaded by a couple Mo at its free end (see figure.) The length of the beam is L = 15 and the longitudinal normal strain at the top surface is 0.001. The distance from the top surface of the beam to the neutral surface is 75 mm.Calculate the radius of curvature (, the
A thin strip of steel of length L = 20in. and thickness t = 0.2 in. is bent by couples M0 (see figure). The deflection at the midpoint of the strip (measured from a line joining its end points) is found to be 0.20 in. Determine the longitudinal normal strain É› at the top surface of the
A bar of rectangular cross section is loaded and supported as shown in the figure. The distance between supports is L = 1.5m and the height of the bar is h = 120mm. The deflection at the midpoint is measured as 3.0 mm. What is the maximum normal strain at the top and bottom of the bar?
The beams shown in the figure are subjected to bending moments M = 2100lb-in. Each beam has a rectangular cross section with height h = 1.5in. and width b = 0.375in. (Perpendicular to the plane of the figure)(a) For the beam with a hole at mid height, determine the maximum stresses for hole
The beams shown in the figure are subjected to bending moments M = 250N ∙ m. Each beam has a rectangular cross section with height h = 44mm and width b = 10mm (perpendicular to the plane of the figure). (a) For the beam with a hole at mid height, determine the maximum stresses for hole diameters
A rectangular beam with semicircular notches, as shown in part (b) of the figure, has dimensions h = 0.88 in. and h1 = 0.80 in. The maximum allowable bending stress in the metal beam is σmax = 60ksi, and the bending moment is M = 600lb-in. Determine the minimum permissible width bmin of the beam.
A rectangular beam with semicircular notches, as shown in part (b) of the figure, has dimension h = 120mm and h1 = 100mm. The maximum allowable bending stress in the plastic beam is σmax = 6MPa, and the bending moment is M = 150 N ∙ m. Determine the minimum permissible width bmin of the beam.
A rectangular beam with notches and a hole (see figure) has dimensions h = 5.5in., h1 = 5in., and width b = 1.6in. The beam is subjected to a bending moment M = 130k-in., and the maximum allowable bending stress in the material (steel) is σmax = 42000 psi.(a) What is the smallest radius
A thin strip of hard copper (E = 16,000 ksi) having length L = 90 in. and thickness t = 3/32 in. is bent into a circle and held with the ends just touching (see figure).(a) Calculate the maximum bending stress ɛmax in the strip.(b) By what percent does the stress increase or decrease
A railroad tie (or sleeper) is subjected to two rail loads, each of magnitude P = 175kN, acting as shown in the figure. The reaction of the ballast is assumed to be uniformly distributed over the length of the tie, which has cross-sectional dimensions b = 300 mm and h = 250mm. Calculate the maximum
A fiberglass pipe is lifted by a sling, as shown in the figure. The outer diameter of the pipe is 6.0 in., its thickness is 0.25 in., and its weight density is 0.053 lb/in.3. The length of the pipe is and the distance between lifting points is s = 11ft.Determine the maximum bending stress in the
A small dam of height h = 2.0 m is constructed of vertical wood beams AB of thickness t = 120mm, as shown in the figure. Consider the beams to be simply supported at the top and bottom.Determine the maximum bending stress in σ mix the beams, assuming that the weight density of water is
Determine the maximum tensile stress σt (due to pure bending about a horizontal axis through C by positive bending moments M) for beams having cross sections as follows (see figure).(a) A semicircle of diameter d(b) An isosceles trapezoid with bases b1 = b and b2 = 4b/3, and altitude
Determine the maximum bending stress σ max (due to pure bending by a moment M) for a beam having a cross section in the form of a circular core (see figure). The circle has diameter and the angle β = 60o.
A simple beam AB of span length L = 24ft is subjected to two wheel loads acting at distance = 5 ft apart (see figure). Each wheel transmits a load P = 3.0 k, and the carriage may occupy any position on the beam.Determine the maximum bending stress σ max due to the wheel loads if the beam
Determine the maximum tensile stress σt and maximum compressive stress σc due to the load P acting on the simple beam AB(see figure).Data are as follows: P = 6.2KN, L = 3.2m, d= 1.25mb = 80 mm, t = 25 mm, h = 120 mm, and h1 = 90 mm.
A cantilever beam AB, loaded by a uniform load and a concentrated load (see figure), is constructed of a channel section. Find the maximum tensile stress σt and maximum compressive stress σc if the cross section has the dimensions indicated and the moment of inertia about the
A cantilever beam AB of isosceles trapezoidal cross section has length L = 0.8 m, dimensions b1 = 80 mm, b2 = 90 mm, and height h = 110 mm (see figure). The beam is made of brass weighing 85 kN/m3(a) Determine the maximum tensile stress st and maximum compressive stress σc due to the
A beam ABC with an overhang from B to C supports a uniform load of 200 lb/ft throughout its length (see figure). The beam is a channel section with dimensions as shown in the figure. The moment of inertia about the z axis (the neutral axis) equals 8.13 in.4Calculate the maximum tensile stress
A steel wire (E = 200 GPa) of diameter d = 1.25 mm is bent around a pulley of radius R0 = 500 mm (see figure).(a) What is the maximum stress σ max in the wire?(b) By what percent does the stress increase or decrease if the radius of the pulley is increased by 25%?
A frame ABC travels horizontally with an acceleration a0 (see figure). Obtain a formula for the maximum stress σmax in the vertical arm AB, which had length L, thickness t, and mass density (.
A beam of T-section is supported and loaded as shown in the figure. The cross section has width n = 2 1/2in, height h = 3in, and thickness t = 3/8in.Determine the maximum tensile and compressive stresses in the beam.
A cantilever beam AB with a rectangular cross section has a longitudinal hole drilled throughout its length (see figure). The beam supports a load P = 600N. The cross section is 25mm wide and 50mm high, and the hole has a diameter of 10mm.Find the bending stresses at the top of the beam, at the top
A small dam of height h = 6ft is constructed of vertical wood beams , as shown in the figure. The wood beams AB, which have thickness t = 2.5in are simply supported by horizontal steel beams at A and B.Construct a graph showing the maximum bending stress σmax in the wood beams versus the
Consider the non-prismatic cantilever beam of circular cross section shown. The beam has an internal cylindrical hole in segment 1; the bar is solid (radius r) in segment 2. The beam is loaded by a downward triangular load with maximum intensity q0 as shown.Find expressions for maximum tensile and
A steel post (E = 30 × 106 psi) having thickness t = 1/8 in. and height L = 72 in. supports a stop sign (see figure: s = 12.5 in.). The height of the post L is measured from the base to the centroid of the sign. The stop sign is subjected to wind pressure p = 20 lb/ft2 normal to its
A thin, high-strength steel rule (E = 30 × 106 psi) having thickness t = 0.175 in. and length L = 48 in. is bent by couples M0 into a circular are subtending a central angle a = 40° (see figure).(a) What is the maximum bending stress σ max in the rule?(b) By what percent
A simply supported wood beam AB with span length L = 4m carries a uniform load of intensity q = 5.8kN/m (see figure).(a) Calculate the maximum bending stress σmax due to the load if the beam has a rectangular cross section with width b = 140mm and height h = 240 mm.(b) Repeat (a) but use
Each girder of the lift bridge (see figure) is 180 ft long and simply supported at the ends. The design load for each girder is a uniform load of intensity 1.6 k/ft. The girders are fabricated by welding tree steel plates so as to form an I-shaped cross section (see figure) having section modulus S
A freight-car axle AB is loaded approximately as shown in the figure, with the forces P representing the scar loads (transmitted to the axle through the axle boxes) and the forces R representing the rail loads (transmitted to the axle through the wheels). The diameter of the axle is d = 80 mm, the
A seesaw weighing 3 lb/ft of length is occupied by two children, each weighing 90 lb (see figure). The center of gravity of each child is 8 ft from the fulcrum. The board is 19 ft long, 8 in. wide, and 1.5 in. thick. What is the maximum bending stress in the board?
During construction of a highway bridge, the main girders are cantilevered outward from one pier toward the next (see figure). Each girder has a cantilever length of 48 m and an I-shaped cross section with dimensions shown in the figure. The load on each girder (during construction) is assumed to
The horizontal beam ABC of an oil-well pump has the cross section shown in the figure. If the vertical pumping force acting at end C is 9 k and if the distance from the line of action of that force to point B is 16 ft, what is the maximum bending stress in the beam due to the pumping
The cross section of a narrow-gage railway bridge is shown in part (a) of the figure. The bridge is constructed with longitudinal steel girders that support the wood cross ties. The girders are restrained against lateral buckling by diagonal bracing, as indicated by the dashed lines.The spacing of
A so-called "trapeze bar" in a hospital room provides a means for patients to exercise while in bed (see figure). The bar is 2.1m long and has a cross section in the shape of a regular octagon. The design load is 1.2kN applied at the midpoint of the bar, and the allowable bending stress is
A two-axle carriage that is part of an overhead traveling crane in a testing laboratory moves slowly across a simple beam AB (see figure). The load transmitted to the beam from the front axle is 2200 lb and from the rear axle is 3800 lb. The weight of the beam itself may be disregarded.(a)
A cantilever beam AB of circular cross section and length L = 450mm supports a load P = 400N acting at the free end (see figure).The beam is made of steel with an allowable bending stress of 60MPa.Determine the required diameter dmin of the beam, considering the effect of the beam's own weight.
A compound beam ABCD (see figure) is supported at points A, B, and D and has a splice at point C.The distance a = 6.25 ft, and the beam is a S 18 × 70 wide-flange shape with an allowable bending stress of 12,800 psi.(a) If the splice is a moment release, find the allowable uniform load
A small balcony constructed of wood is supported by three identical cantilever beams (see figure). Each beam has length L1 = 2.1m, width b, and height h = 4b.3.The dimensions of the balcon floor are L1 x L2, with L2 = 2.5m. The design load is 5.5kPa acting over the entire floor area. (This load
A beam having a cross section in the form of an unsymmetric wide-flange shape (see figure) is subjected to a negative bending moment acting about the z axis.Determine the width of the top flange in order that the stresses at the top and bottom of the beam will be in the ratio 4:3, respectively.
A beam having a cross section in the form of a channel (see figure) is subjected to a bending moment acting about the z axis.Calculate the thickness t of the channel in order that the bending stresses at the top and bottom of the beam will be in the ratio 7:3, respectively.
Determine the ratios of the weights of three beams that have the same length, are made of the same material, are subjected to the same maximum bending moment, and have the same maximum bending stress if their cross sections are (1) a rectangle with height equal to twice the width, (2) a square, and
A horizontal shelf AD of length L = 915 mm, width b = 305 mm, and thickness t = 22 mm is supported by brackets at B and C [see part (a) of the figure]. The brackets are adjustable and may be placed in any desired positions between the ends of the shelf. A uniform load of intensity q, which includes
A steel plate (called a cover plate) having cross-sectional dimensions 6.0in. x 0.5 in. is welded along the full length of the bottom flange of a W 12 x 50 wide-flange beam (see figure, which shows the beam cross section).What is the percent increase in the smaller section modulus (as compared to
A fiberglass bracket ABCD of solid circular cross section has the shape and dimensions shown in the figure. A vertical load P = 40N acts at the free end D.Determine the minimum permissible diameter dmin of the bracket if the allowable bending stress in the material is 30Mpa and b = 37mm.
A steel beam ABC is simply supported at A and B and has an overhang BC of length L = 150 mm (see figure). The beam supports a uniform load of intensity q = 4.0 kN/m over its entire span AB and 1.5q over BC. The cross section of the beam is rectangular with width b and height 2b. The allowable
A retaining wall 5 ft high is constructed of horizontal wood planks 3 in. thick (actual dimension) that are supported by vertical wood piles of 12 in. diameter (actual dimension), as shown in the figure. The lateral earth pressure is p1 = 100 lb/ft2 at the top of the wall and p2 = 400 lb/ft2 at the
A beam of square cross section (a = length of each side) is bent in the plane of a diagonal (see figure). By removing a small amount of material at the top and bottom corners, as shown by the shaded triangles in the figure, we can increase the section modulus and obtain a stronger beam, even though
The cross section of a rectangular beam having width and height is shown in part (a) of the figure. For reasons unknown to the beam designer, it is planned to add structural projections of width b/9 and height to the top and bottom of the beam [see part (b) of the figure].For what values of is the
A cantilever beam of length L = 7.5ft supports a uniform load of intensity q = 225lb/ft and a concentrated load p = 2750lb (see figure).Calculate the required section modulus S if σAllow = 17000 psi. Then select a suitable wide-flange beam (W shape) from Table E-1(a), Appendix E, and
A simple beam of length carries L = 5m a uniform load of intensity q = 5.8 kN/m and a concentrated load 22.5kN (see figure).Assuming σallow, calculate the required section modulus S. Then select an 200 mm wide-flange beam (W shape) from Table E-1(b) Appendix E, and recalculate taking
A simple beam AB is loaded as shown in the figure. Calculate the required section modulus S if σallow = 17,000 psi, L = 28 ft, P = 2200 lb, and q = 425 lb/ft. Then select a suitable I-beam (S shape) from Table E-2(a), Appendix E, and recalculate S taking into account the weight of the
A pontoon bridge (see figure) is constructed of two longitudinal wood beams, known an balks, that span between adjacent pontoons and support the transverse floor beams, which are called chesses.For purposes of design, assume that a uniform floor load of 8.0kPa acts over the chesses. (This load
A floor system in a small building consists of wood planks supported by 2 in (nominal width) joists spaced at distance s, measured from center to center (see figure). The span length l of each joist is 10.5ft, the spacing of the joists is 16in, and the allowable bending stress in the wood is
The wood joists supporting a plank floor (see figure) are 40mm × 180mm in cross section (actual dimensions) and have a span length L = 4.0m. The floor load is 3.6kPa, which includes the weight of the joists and the floor. Calculate the maximum permissible spacing of the joists if the allowable
A beam ABC with an overhang from B to C is constructed of a C 10 × 30 channel section (see figure). The beam supports its own weight (30 lb/ft) plus a triangular load of maximum intensity q0 acting on the overhang. The allowable stresses in tension and compression are 20 ksi and 11
A tapered cantilever beam AB of length L has square cross sections and supports a concentrated load P at the free end [see figure part (a)]. The width and height of the beam vary linearly from hA at the free end to hB at the fixed end.Determine the distance x from the free end A to the cross
A tall signboard is supported by two vertical beams consisting of thin-walled, tapered circular tubes [see figure]. For purposes of this analysis, each beam may be represented as a cantilever AB of length L = 8.0 m subjected to a lateral load P = 2.4 kN at the free end. The tubes have constant
A tapered cantilever beam AB having rectangular cross sections is subjected to a concentrated load P = 50 lb and a couple M0 = 800 lb-in. acting at the free end [see figure part (a)]. The width b of the beam is constant and equal to 1.0 in., but the height varies linearly from hA = 2.0 in. at the
The spokes in a large flywheel are modeled as beams fixed at one end and loaded by a force P and a couple M0 at the other (see figure). The cross sections of the spokes are elliptical with major and minor axes (height and width, respectively) having the lengths shown in the figure part (a). The
Refer to the tapered cantilever beam of solid circular cross section shown in Fig. 5-24 of Example 5-9. (a) Considering only the bending stresses due to the load P, determine the range of values of the ratio dB/dA for which the maximum normal stress occurs at the support. (b) What is the maximum
A cantilever beam AB having rectangular cross sections with constant width and varying height hx is subjected to a uniform load of intensity (see figure).How should the height hx vary as a function of (measured from the free end of the beam) in order to have a fully stressed beam? (Express hx in
A simple beam ABC having rectangular cross sections with constant height and varying width bx supports a concentrated load P acting at the midpoint (see figure).How should the width bx vary as a function of in order to have a fully stressed beam? (Express bx in terms of the width bB at the midpoint
A cantilever beam AB having rectangular cross sections with varying width bx and varying height hx is subjected to a uniform load of intensity (see figure). If the width varies linearly with x according to the equation bx = bBx/L, how should the height hx vary as a function of in order to have a
The shear stresses in a rectangular beam are given by Eq. (5-39)in which B is the shear force, I is the moment of inertia of the cross-sectional area, is the height of the beam, and y1 is the distance from the neutral axis to the point where the shear stress is being determined (Fig. 5-30). By
A simply supported wood beam of rectangular cross section and span length 1.2m carries a concentrated load P at mid span in addition to its own weight (see figure). The cross section has width 140mm and height 240mm. The weight density of the wood is 5.4kN/m3.Calculate the maximum permissible value
A square wood platform, 8ft x 8ft in area, rests on masonry walls (see figure). The deck of the platform is constructed of 2in nominal thickness tongue-and-groove planks (actual thickness 1.5in; see Appendix F) supported on two 8-ft long beams. The beams have 4in x 6in nominal dimensions (actual
A wood beam ABC with simple supports at A and B and an overhang BC has height h = 300 mm (see figure). The length of the main span of the beam is L = 3.6 m and the length of the overhang is L/3 = 1.2 m. The beam supports a concentrated load 3P = 18 kN at the midpoint of the main span and a moment
Calculate the maximum shear stress τmax and the maximum bending stress σmax in a wood beam (see figure) carrying a uniform load of 22.5 kN/m (which includes the weight of the beam) if the length is 1.95 m and the cross section is rectangular with width 150 mm and height 300
Two wood beams, each of rectangular cross section (3.0 in. x 4.0 in., actual dimensions) are glued together to form a solid beam of dimensions 6.0 in. x 4.0 in. (see figure). The beam is simply supported with a span of 8 ft.What is the maximum moment Mmax that may be applied at the left support if
A cantilever beam of length L = 2m supports a load P = 8.0kN (see figure). The beam is made of wood with cross-sectional dimensions 120mm x 200mm.Calculate the shear stresses due to the load at points located 25mm, 75mm, 100mm and from the top surface of the beam. from these results, plot a graph
A steel beam of length L = 16 in. and cross sectional dimensions b = 0.6 in. and h = 2 in. (see figure) supports a uniform load of intensity q = 240 lb/in., which includes the weight of the beam.Calculate the shear stresses in the beam (at the cross section of maximum shear force) at point's
A beam of rectangular cross section (width b and height h) supports a uniformly distributed load along its entire length L. The allowable stresses in bending and shear are σallow and τ allows, respectively. (a) If the beam is simply supported, what is the span length L0 below which the shear
A laminated wood beam on simple supports is built up by gluing together four 2 in. x 4 in. boards (actual dimensions) to form a solid beam 4 in. x 8 in. in cross section, as shown in the figure. The allowable shear stress in the glued joints is 65 psi, and the allowable bending stress in the wood
A laminated plastic beam of square cross section is built up by gluing together three strips, each 10 mm x 30 mm in cross section (see figure). The beam has a total weight of 3.6 N and is simply supported with span length L = 360 mm.Considering the weight of the beam (q) calculate the maximum
A wood beam AB on simple supports with span length equal to 10 ft is subjected to a uniform load of intensity 125 lb/ft acting along the entire length of the beam, a concentrated load of magnitude 7500 lb acting at a point 3 ft from the right-hand support, and a moment at A of 18,500 ft-lb (see
A wood pole of solid circular cross section (d = diameter) is subjected to a horizontal force P = 450lb (see figure). The length of the pole is L = ft, and the allowable stresses in the wood are 1900 psi in bending and 120 psi in shear.Determine the minimum required diameter of the pole based upon
A simple log bridge in a remote area consists of two parallel logs with planks across them (see figure). The logs are Douglas fir with average diameter 300mm. A truck moves slowly across the bridge, which spans 2.5m. Assume that the weight of the truck is equally distributed between the two
A sign for an automobile service station is supported by two aluminum poles of hollow circular cross section, as shown in the figure. The poles are being designed to resist a wind pressure of 75lb/ft2 against the full area of the sign. The dimensions of the poles and sign are h1 = 20ft, h2 = 5ft,
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