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Mechanics of Materials 7th edition James M. Gere, Barry J. Goodno - Solutions

The cross section of a sign post of constant thickness is shown in the figure. Derive the formula below for the distance e from the centerline of the wall of the post to the shear center S. Also, compare this formula with that given in Prob. 6.9-11 for the special case of Î² = 0 here and
A cross section in the shape of a circular arc of constant thickness is shown in the figure. Derive the following formula for the distance e from the center of the arc to the shear center S:In which b is in radians. Also, plot a graph showing how the distance e varies as Î² varies from 0
Calculate the distance e from the centerline of the web of a C 310 x 45 channel section to the shear center S (see figure). (For purposes of analysis, consider the flanges to be rectangles with thickness tf equal to the average flange thickness given in Table E-3b in Appendix E.)
The cross section of an unbalanced wide-flange beam is shown in the figure. Derive the following formula for the distance h1 from the centerline of one flange to the shear center S:Also, check the formula for the special cases of a T-beam (b2 = t2 = 0) and a balanced wide-flange beam (t2 = t1 and
The cross section of an unbalanced wide-flange beam is shown in the figure. Derive the following formula for the distance e from the centerline of the web to the shear center S:Also, check the formula for the special cases of a channel section (b1 = 0 and b2 = b) and a doubly symmetric beam (b1 =
The cross section of a channel beam with double flanges and constant thickness throughout the section is shown in the figure.Derive the following formula for the distance e from the centerline of the web of the shear center S:
The cross section of a slit circular tube of constant thickness is shown in the figure.(a) Show that the distance e from the center of the circle to the shear center S is equal to 2r in the figure part a.(b) Find an expression for e if flanges with the same thickness as that of the tube are added,
The cross section of a slit square tube of constant thickness is shown in the figure. Derive the following formula for the distance e from the corner of the cross section to the shear center S:
The cross section of a slit rectangular tube of constant thickness is shown in the figures.(a) Derive the following formula for the distance e from the centerline of the wall of the tube in figure part (a) to theShear centerb) Find an expression for e if flanges with the same thickness as that of
A U-shaped cross section of constant thickness is shown in the figure. Derive the following formula for the distance e from the center of the semicircle to the shear center S:Also, plot a graph showing how the distance e (expressed as the non-dimensional ratio e/r) varies as a function of the ratio
Determine the shape factor f for a cross section in the shape of a double trapezoid having the dimensions shown in the figure. Also, check your result for the special cases of a rhombus (b1 = 0) and a rectangle (b1 = b2).
Solve the preceding problem for a W 410 x 85 wide-flange beam. Assume that σY = 250 MPa.
A hollow box beam with height h = 16 in., width b = 8 in., and constant wall thickness t = 0.75 in. is shown in the figure. The beam is constructed of steel with yield stress ÏƒY = 32 ksi.Determine the yield moment MY, plastic moment MP, and shape factor f.
Solve the preceding problem for a box beam with dimensions h = 0.5 m, b = 0.18 m, and t = 22 mm. The yield stress of the steel is 210 MPa
A hollow box beam with height h = 9.5 in., inside height h1 = 8.0 in., width b = 5.25 in., and inside width b1 = 4.5 in. is shown in the figure. Assuming that the beam is constructed of steel with yield stress ÏƒY = 42 ksi, calculate the yield moment MY, plastic moment MP, and shape
Solve the preceding problem for a box beam with dimensions h = 200 mm, h1 = 160 mm, b = 150 mm, and b1 = 130 mm. Assume that the beam is constructed of steel with yield stress σY = 220 MPa.
The hollow box beam shown in the figure is subjected to a bending moment M of such magnitude that the flanges yield but the webs remain linearly elastic. (a) Calculate the magnitude of the moment M if the dimensions of the cross section are h = 15 in., h1 = 12.75 in., (b) b = 9 in., and b1 = 7.5
Solve the preceding problem for a box beam with dimensions h = 400 mm, h1 = 360 mm, b = 200 mm, and b1 = 160 mm, and with yield stress σY = 220 MPa.
A W 10 x 60 wide-flange beams is subjected to a bending moment M of such magnitude that the flanges yield but the web remains linearly elastic. (a) Calculate the magnitude of the moment M if the yield stress is σY = 36 ksi. (b) What percent of the moment M is produced by the elastic core?
A singly symmetric beam of T-section (see figure) has cross-sectional dimensions b = 140 mm, a = 190.8 mm, tw = 6.99 mm, and tf = 11.2 mm.Calculate the plastic modulus Z and the shape factor f.
A wide-flange beam of unbalanced cross section has the dimensions shown in the figure.Determine the plastic moment MP if ÏƒY = 36 ksi.
(a) Determine the shape factor f for a hollow circular cross section having inner radius r1 and outer radius r2 (see figure).(b) If the section is very thin, what is the shape factor?
Determine the plastic moment MP for a beam having the cross section shown in the figure if ÏƒY = 210 MPa.
A propped cantilever beam of length L = 54 in. with a sliding support supports a uniform load of intensity q (see figure). The beam is made of steel (ÏƒY = 36 ksi) and has a rectangular cross section of width b = 4.5 in. and height h = 6.0 in. What load intensity q will produce a fully
A steel beam of rectangular cross section is 40 mm wide and 80 mm high (see figure). The yield stress of the steel is 210 MPa.(a) What percent of the cross-sectional area is occupied by the elastic core if the beam is subjected to a bending moment of 12.0 kNm acting about the z axis?(b) What is the
Calculate the shape factor f for the wide flange beam shown in the figure ifh = 12.2 in.,b = 8.08 in.,tf = 0.64 in., and tw = 0.37 in.
Solve the preceding problem for a wide-flange beam with h = 404 mm, b = 140 mm, tf = 11.2 mm, and tw = 6.99 mm.
Determine the plastic modulus Z and shape factor f for a W 12 x 14 wide-flange beam.
Solve the preceding problem for a W 250 x 89 wide-flange beam.
Determine the yield moment MY, plastic moment MP, and shape factor f for a W 16 x 100 wide-flange beam if σY = 36 ksi.
An element in plane stress is subjected to stresses (x = 4750 psi, (x = 1200 psi, and Txy= 950 psi, as shown in the figure.Determine the stresses acting on an element oriented at an angle ( = 60° from the x axis, where the angle u is positive when counterclockwise. Show these stresses on a
Solve the preceding problem if the normal and shear stresses acting on the element are (x = 2100 kPa. (y = 300 kPa, and Txy = - 560 kPa, the seam is oriented at an angle of 22.5° to the element (see figure).
A rectangular plate of dimensions 3.0 in. ( 5.0 in. is formed by welding two triangular plates (see figure). The plate is subjected to a tensile stress of 500 psi in the long direction and a compressive stress of 350 psi in the short direction.Determine the normal stress (w acting perpendicular to
Solve the preceding problem for a plate of dimensions 100 mm ( 250 mm subjected to a compressive stress of 2.5 MPa in the long direction and a tensile stress of 12.0 MPa in the short direction (see figure).
At a point on the surface of a machine the material is in biaxial stress with (x = 3600 psi, and (y = - 1600 psi, as shown in the first part of the figure. The second part of the figure shows an inclined plane aa cut through the same point in the material but oriented at an angle (.Determine the
Solve the preceding problem for (x = 32 MPa and (y = - 50 MPa (see figure).
An element in plane stress from the frame of a racing car is oriented at a known angle ( (see figure). On this inclined element, the normal and shear stresses have the magnitudes and directions shown in the figure.Determine the normal and shear stresses acting on an element whose sides are parallel
Solve the preceding problem for the element shown in the figure.
A plate in plane stress is subjected to normal stresses (x and (y and shear stress Txy, as shown in the figure. At counterclockwise angles ( = 35° and ( = 75° from the x axis, the normal stress is 4800 psi tension.If the stress (x equals 2200 psi tension, what are the stresses (y and Txy?
The surface of an airplane wring is subjected to plane stress with normal stresses (x and (y and shear stress Txy, as shown in the figure. At a counter clock wise angle ( = 32o from the x axis, the normal stress is 37 MPa tension, and at an angle ( = 48o, it is 12 MPa compression.If the stress (x
At a point in a structure subjected to plane stress, the stresses are (x = - 4100 psi,(y = 2200 psi, and Txy = 2900 psi (the sign convention for these stresses is shown in Fig. 7-1), A stress element located at the same point in the structure (but oriented at a counterclockwise angle (1 with
Solve the preceding problem for an element in plane stress subjected to stresses (x - 100 MPa, (y = 80 MPa, and Txy = 28 MPa, as shown in the figure.Determine the stresses acting on an element oriented at an angle ( = 30° from the x axis, where the angle ( is positive when counterclockwise.
Solve Problem 7.2-1 for an element in plane stress subjected toStresses (x = - 5700 psi, (y = - 2300 psi, and Txy = 2500 psi, as shown in the figure.Determine the stresses acting on an element oriented at an angle ( = 50° from the x axis, where the angle u is positive when counterclockwise.
The stresses acting on element A in the web of a train rail are found to be 40 MPa tension in the horizontal direction and 160 MPa compression in the vertical direction (see figure). Also, shear stresses of magnitude 54 MPa act in the directions shown.Determine the stresses acting on an element
Solve the preceding problem if the normal and shear stresses acting on element A are 6500 psi, 18,500 psi, and 3800 psi (in the directions shown in the figure).Determine the stresses acting on an element oriented at a counterclockwise angle of 30° from the horizontal. Show these stresses on a
An element in plane stress from the fuselage of an airplane is subjected to compressive stresses of magnitude 27 MPa in the horizontal direction and tensile stresses of magnitude 5.5 MPa in the vertical direction (see figure). Also, shear stresses of magnitude 10.5 MPa act in the directions
The stresses acting on element B in the web of a wide-flange beam are found to be 14,000 psi compression in the horizontal direction and 2600 psi compression in the vertical direction (see figure). Also, shear stresses of magnitude 3800 psi act in the directions shown.Determine the stresses acting
Solve the preceding problem if the normal and shear stresses acting on element B are 46 MPa, 13 MPa, and 21 MPa (in the directions shown in the figure) and the angle is 42.5° (clockwise).
The polyethylene liner of a settling pond is subjected to stresses (x = 350 psi, (y = 112 psi, and Txy= -120 psi, as shown by the plane-stress element in the first part of the figure.Determine the normal and shear stresses acting on a seam oriented at an angle of 30o to the element, as shown in the
An element in plane stress is subjected to stresses (x = 4750 psi, (y = 1200 psi, and Txy = 950 (see the figure for Problem 7.2-1). Determine the principal stresses and show them on a sketch of a properly oriented element?
A propeller shaft subjected to combined torsion and axial thrust is designed to resist a shear stress of 56 MPa and a compressive stress of 85 MPa (see figure).(a) Determine the principal stresses and show them on a sketch of a properly oriented element.(b) Determine the maximum shear stresses and
(x = 2500 psi, (y = 1020 psi, Txy = - 900 psi(a) Determine the principal stresses and show them on a sketch of a properly oriented element.(b) Determine the maximum shear stresses and associated normal stresses and show them on a sketch of a properly oriented element.
(x = 2150 kPa, (y = 375 kPa, Txy = - 460 kPa (a) Determine the principal stresses and show them on a sketch of a properly oriented element. (b) Determine the maximum shear stresses and associated normal stresses and show them on a sketch of a properly oriented element?
(x = 14,500 psi, (y = 1070 psi, Txy = 1900 psi (a) Determine the principal stresses and show them on a sketch of a properly oriented element. (b) Determine the maximum shear stresses and associated normal stresses and show them on a sketch of a properly oriented element.
(x = 16.5 MPa, (y = - 91 MPa, Txy = - 39 MPa (a) Determine the principal stresses and show them on a sketch of a properly oriented element. (b) Determine the maximum shear stresses and associated normal stresses and show them on a sketch of a properly oriented element?
(x = - 3300 psi, (y = - 11,000 psi, Txy = 4500 psi (a) Determine the principal stresses and show them on a sketch of a properly oriented element. (b) Determine the maximum shear stresses and associated normal stresses and show them on a sketch of a properly oriented element?
(x = - 108 MPa, (y = 58 MPa, Txy = - 58 MPa (a) Determine the principal stresses and show them on a sketch of a properly oriented element. (b) Determine the maximum shear stresses and associated normal stresses and show them on a sketch of a properly oriented element?
At a point on the surface of a machine component, the stresses acting on the x face of a stress element are (x = 5900 psi and Txy = 1950 psi (see figure).What is the allowable range of values for the stress (y if the maximum shear stress is limited to T0 = 2500 psi?
At a point on the surface of a machine component the stresses acting on the x face of a stress element are (x = 42 MPa and Txy = 33 MPa (see figure).What is the allowable range of values for the stress (y if the maximum shear stress is limited to T0 = 35 MPa?
An element in plane stress is subjected to stresses (x = 5700 psi and Txy = - 2300 psi (see figure). It is known that one of the principal stresses equals 6700 psi in tension.(a) Determine the stress (y.(b) Determine the other principal stress and the orientation of the principal planes, then show
An element in plane stress is subjected to stresses (x = 100 MPa, (y = 80 MPa, and Txy = 28 MPa (see the figure for Problem 7.2-2). Determine the principal stresses and show them on a sketch of a properly oriented element?
An element in plane stress is subjected to stresses (x = - 50 MPa and Txy = 42 MPa (see figure). It is known that one of the principal stresses equals 33 MPa in tension.(a) Determine the stress (y.(b) Determine the other principal stress and the orientation of the principal planes, then show the
The stresses acting on element A in the web of a train rail are found to be 40 MPa tension in the horizontal direction and 160 MPa compression in the vertical direction (see figure). Also, shear stresses of magnitude 54 MPa act in the directions shown (see the figure for Problem 7.2-4). Determine
The normal and shear stresses acting on element A are 6500 psi, 18,500 psi, and 3800 psi (in the directions shown in the figure) (see the figure for Problem 7.2-5)? Determine the maximum shear stresses and associated normal stresses and show them on a sketch of a properly oriented element.
An element in plane stress from the fuselage of an airplane is subjected to compressive stresses of magnitude 27 MPa in the horizontal direction and tensile stresses of magnitude 5.5 MPa in the vertical direction. Also, shear stresses of magnitude 10.5 MPa act in the directions shown (see the
The stresses acting on element B in the web of a wide-flange beam are found to be 14,000 psi compression in the horizontal direction and 2600 psi compression in the vertical direction. Also, shear stresses of magnitude 3800 psi act in the directions shown (see the figure for Problem
The normal and shear stresses acting on element B are (x = - 46 MPa, (y = - 13 MPa, and Txy = 21 MPa (see figure for Problem 7.2-8). Determine the maximum shear stresses and associated normal stresses and show them on a sketch of a properly oriented element?
A shear wall in a reinforced concrete building is subjected to a vertical uniform load of intensity q and a horizontal force H, as shown in the first part of the figure. (The force H represents the effects of wind and earthquake loads.) As a consequence of these loads, the stresses at point A on
An element in uniaxial stress is subjected to tensile stresses(x = 11,375 psi, as shown in the figure. Using Mohr's circle, determine:(a) The stresses acting on an element oriented at a counterclockwise angle ( = 24° from the x axis.(b) The maximum shear stresses and associated normal stresses.
(x = 27 MPa, (y = 14 MPa, Txy = 6 MPa, ( = 40°Using Mohr's circle, determine the stresses acting on an element oriented at an angle ( from the x axis. Show these stresses on a sketch of an element oriented at the angle (. (The angle ( is positive when counterclockwise and negative when
(x = 3500 psi, (y = 12,200 psi, Txy - 3300 psi,( = - 51° Using Mohr's circle, determine the stresses acting on an element oriented at an angle ( from the x axis. Show these stresses on a sketch of an element oriented at the angle (. (The angle ( is positive when counterclockwise and negative when
(x = - 47 MPa, (y = - 186 MPa, Txy = - 29 MPa, ( = - 33° Using Mohr's circle, determine the stresses acting on an element oriented at an angle ( from the x axis. Show these stresses on a sketch of an element oriented at the angle (. (The angle ( is positive when counterclockwise and negative when
(x = - 1720 psi, (y = - 680 psi, Txy = 320 psi, ( = 14° Using Mohr's circle, determine the stresses acting on an element oriented at an angle ( from the x axis. Show these stresses on a sketch of an element oriented at the angle (. (The angle ( is positive when counterclockwise and negative when
(x = 33 MPa, (y = - 9 MPa, Txy = 29 MPa, ( = 35° Using Mohr's circle, determine the stresses acting on an element oriented at an angle ( from the x axis. Show these stresses on a sketch of an element oriented at the angle (. (The angle ( is positive when counterclockwise and negative when
(x = - 5700 psi, (y = 950 psi, Txy 2100 psi, ( = 65° Using Mohr's circle, determine the stresses acting on an element oriented at an angle ( from the x axis. Show these stresses on a sketch of an element oriented at the angle (. (The angle ( is positive when counterclockwise and negative when
(x = - 29.5 MPa, (y = 29.5 MPa, Txy = 27 MPaUsing Mohr's circle, determine (a) the principal stresses and (b) the maximum shear stresses and associated normal stresses. Show all results on sketches of properly oriented element?
(x = 7300 psi, (y = 0 psi, Txy = 1300 psi Using Mohr's circle, determine (a) the principal stresses and (b) the maximum shear stresses and associated normal stresses. Show all results on sketches of properly oriented elements?
(x = 0 MPa, (y = 23.4 MPa, Txy = - 9.6 MPa Using Mohr's circle, determine (a) the principal stresses and (b) the maximum shear stresses and associated normal stresses. Show all results on sketches of properly oriented elements.
(x = 2050 psi, (y = 6100 psi, Txy = 2750 psi Using Mohr's circle, determine (a) the principal stresses and (b) the maximum shear stresses and associated normal stresses. Show all results on sketches of properly oriented elements?
An element in uniaxial stress is subjected to tensile stresses (x = 49 MPa, as shown in the figure Using Mohr's circle, determine:(a) The stresses acting on an element oriented at an angle ( = - 27° from the x axis (minus means clockwise).(b) The maximum shear stresses and associated normal
(x = 2900 kPa, (y = 9100 kPa, Txy = - 3750 kPa Using Mohr's circle, determine (a) the principal stresses and (b) the maximum shear stresses and associated normal stresses. Show all results on sketches of properly oriented elements?
(x = - 11,500 psi, (y = - 18,250 psi, Txy = - 7200 psi Using Mohr's circle, determine (a) the principal stresses and (b) the maximum shear stresses and associated normal stresses. Show all results on sketches of properly oriented elements?
(x = - 3.3 MPa, (y = 8.9 MPa, Txy = - 14.1 MPa Using Mohr's circle, determine (a) the principal stresses and (b) the maximum shear stresses and associated normal stresses. Show all results on sketches of properly oriented elements?
(x = 800 psi, (y = - 2200 psi, Txy = 2900 psi Using Mohr's circle, determine (a) the principal stresses and (b) the maximum shear stresses and associated normal stresses. Show all results on sketches of properly oriented elements.
An element in uniaxial stress is subjected to compressive stresses of magnitude 6100 psi, as shown in the figure. Using Mohr's circle, determine:(a) The stresses acting on an element oriented at a slope of 1 on 2 (see figure).(b) The maximum shear stresses and associated normal stresses. Show all
An element in biaxial stress is subjected to stresses (x = - 48 MPa and (y = 19 MPa, as shown in the figure. Using Mohr's circle, determine:(a) The stresses acting on an element oriented at a counterclockwise angle ( = 25° from the x axis.(b) The maximum shear stresses and associated normal
An element in biaxial stress is subjected to stresses (x = 6250 psi and (y = - 1750 psi, as shown in the figure. Using Mohr's circle, determine:(a) The stresses acting on an element oriented at a counterclockwise angle ( = 55° from the x axis.(b) The maximum shear stresses and associated normal
An element in biaxial stress is subjected to stresses (x = - 29 MPa and (y = 57 MPa, as shown in the figure. Using Mohr's circle, determine:(a) The stresses acting on an element oriented at a slope of 1 on 2.5 (see figure).(b) The maximum shear stresses and associated normal stresses. Show all
An element in pure shear is subjected to stresses Txy = 2700 psi, as shown in the figure. Using Mohr's circle, determine:(a) The stresses acting on an element oriented at a counterclockwise angle ( = 52° from the x axis.(b) The principal stresses. Show all results on sketches of properly
An element in pure shear is subjected to stresses Txy = - 14.5 MPa, as shown in the figure. Using Mohr's circle, determine:(a) The stresses acting on an element oriented at a counterclockwise angle ( = 22.5° from the x axis(b) The principal stresses. Show all results on sketches of properly
An element in pure shear is subjected to stresses Txy = 3750 psi, as shown in the figure. Using Mohr's circle, determine:(a) The stresses acting on an element oriented at a slope of 3 on 4 (see figure).(b) The principal stresses. Show all results on sketches of properly oriented elements.
A rectangular steel plate with thickness t = 0.25 in. is subjected to uniform normal stresses (x and (y, as shown in the figure. Strain gages A and B, oriented in the x and y directions, respectively, are attached to the plate. The gage readings give normal strains (x = 0.0010 (elongation) and (y =
A square plate of width b and thickness t is loaded by normal forces Px and Py, and by shear forces V, as shown in the figure.These forces produce uniformly distributed stresses acting on the side faces of the place.Calculate the change (V in the volume of the plate and the strain energy U stored
Solve the proceeding problem for an aluminum plate with b = 12 in., t = 1.0 in., E = 10,00 ksi, v = 0.33, Px = 90 k, Py = 20 k, and V = 15 k?
A circle of diameter d = 200 mm is etched on a brass plate (see figure). The plate has dimensions 400 ( 400 ( 20 mm.Forces are applied to the plate, producing uniformly distributed normal stresses (x = 42 MPa and (y = 14 MPa.Calculate the following quantities: (a) the change in length (ac of
Solve the preceding problem if the thickness of the steel plate is t = 10 mm, the gage readings are (x = 480 ( 10-6 (elongation) and (y = 130 ( 10-6 (elongation), the modulus is E = 200 GPa, and Poisson's ratio is v = 0.30.
Assume that the normal strains (x and (y for an element in plane stress (see figure) are measured with strain gages.(a) Obtain a formula for the normal strain (z in the z direction in terms of (x, (y, and Poisson's ratio v.(b) Obtain a formula for the dilatation e in terms of (x, (y, and Poisson's
A magnesium plate in biaxial stress is subjected to tensile stresses (x = 24 MPa and (x = 12 MPa (see figure).The corresponding strains in the plate are (x = 440 ( 10-6 and (y = 80 ( 10-6.Determine Poisson's ratio and the modulus of elasticity E for the material.
Solve the preceding problem for a steel plate with (x = 10,800 psi (tension), (y = - 5400 psi (compression), (x = 420 ( 10-6 (elongation), and (x = - 300 ( 10-6 (shortening)?

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