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physics
thermodynamics

Questions and Answers of Thermodynamics

For the rigid stairway frame shown in Figure P5-3, determine (1) the displacements at node 2, (2) the support reactions, and (3) the local nodal forces acting on each element. Draw the bending moment
Consider the plane structure shown in Figure P5-39. First assume the structure to be a plane frame with rigid joints, and analyze using a frame element. Then assume the structure to be pin-jointed
For the rigid frame shown in Figure P5-4, determine (1) the nodal displacements and rotation at node 4, (2) the reactions, and (3) the forces in each element. Then check equilibrium at node 4.
For the two-story, two-bay rigid frame shown in Figure P5-40, determine (1) the nodal displacement components and (2) the shear force and bending moments in each member. Let E = 200 GPa, I = 2
For the two-story, three-bay rigid frame shown in Figure P5-41, determine (1) the nodal displacements and (2) the member end shear forces and bending moments. (3) Draw the shear force and bending
For the rigid frame shown in Figure P5-42, determine (1) the nodal displacements and rotations and (2) the member shear forces and bending moments. Let E = 200 GPa, I = 0.795 Ã— 10-4m4
For the rigid frame shown in Figure P5-43, determine (1) the nodal displacements and rotations and (2) the shear force and bending moments in each member. Let E = 29 Ã— 106 psi, I = 3100
A structure is fabricated by welding together three lengths of I-shaped members as shown in Figure P5-44. The yield strength of the members is 36 ksi, E = 29 × 106 psi, and Poisson's ratio is 0.3.
For the tapered beam shown in Figure P5-45, determine the maximum deflection using one, two, four, and eight elements. Calculate the moment of inertia at the midlength station for each element. Let E
Derive the stiffness matrix for the nonprismatic torsion bar shown in Figure P5-46. The radius of the shaft is given by r = r0 + (x /L)r0, where r0 is the radius at x = 0.
Derive the total potential energy for the prismatic circular cross-section torsion bar shown in Figure P5-47. Also determine the equivalent nodal torques for the bar subjected to uniform torque per
For the grid shown in Figure P5-48, determine the nodal displacements and the local element forces. Let E = 30 Ã— 106 psi, G = 12 Ã— 106 psi, I = 200 in.4, and J = 100 in.4
For the rigid frames shown in Figures P5-5 through P5-15, determine the displacements and rotations of the nodes, the element forces, and the reactions. The values of E, A, and I to be used are
For the grids shown in Figures P5-50 and P5-51, determine the nodal displacements and the local element forces. Let E = 210 GPa, G = 84 GPa, I = 2 Ã— 10-4m4, J = 1310-4 m4, and A =
Solve the grid structures shown in Figures P5-52 through P5-57 using a computer program. For grids P5-52-P5-54, let E = 30 Ã— 106 psi, G = 12 Ã— 106 psi, I = 200 in.4, and J =
Determine the displacements and reactions for the space frames shown in Figures P5-58 and P5-59. Let Ix = 100 in.4, Iy = 200 in.4, Iz = 1000 in.4, E = 30,000 ksi, G = 12,000 ksi, and A = 100 in.2 for
Design a jib crane as shown in Figure P5-60 that will support a downward load of 6000 lb. Choose a common structural steel shape for all members. Use allowable stresses of 0.66Sy (Sy is the yield
Design the support members AB and CD for the platform lift shown in Figure P5-61. Select a mild steel and choose suitable cross-sectional shapes with no more than a 4 : 1 ratio of moments of inertia
A two-story building frame is to be designed as shown in Figure P5-62. The members are all to be I-beams with rigid connections. We would like the floor joists beams to have a 15-in. depth and the
A pulpwood loader as shown in Figure P5-63 is to be designed to lift 2.5 kip. Select a steel and determine a suitable tubular cross section for the main upright member BF that has attachments for the
A small hydraulic floor crane as shown in Figure P5-65 carries a 5000-lb load. Determine the size of the beam and column needed. Select either a standard box section or a wide-flange section. Assume
Design the gabled frame subjected to the external wind load shown in Figure P5-68 (comparable to an 80 mph wind speed) for an industrial building. Assume this is one of a typical frame spaced every
Design the gabled frame shown for a balanced snow load shown in Figure P5-69 (typical of the Midwest) for an apartment building. Select a wide flange section for the frame. Assume the allowable
Design a gantry crane that must be able to lift 10 tons as it must lift compressors, motors, heat exchangers, and controls. This load should be placed at the center of one of the main 12-foot-long
Design the rigid highway bridge frame structure shown in Figure P5-71 for a moving truck load (shown below) simulating a truck moving across the bridge. Use the load shown and place it along the top
The curved semi-circular frame shown in Figure P5-73 is supported by a pin on the left end and a roller on the right end and is subjected to a load P 51000 lb at its apex. The frame has a radius to
Sketch the variations of the shape functions Nj and Nm, given by Eqs. (6.2.18), over the surface of the triangular element with nodes i, j, and m. Check that Ni 1Nj 1Nm 51 anywhere on the
For the plane strain elements shown in Figure P6-10, the nodal displacements are given asu1 = 0.005 mm............v1 = 0.002 mm............u2 = 0.0 mmv2 = 0.0 mm...............u3 = 0.005
Determine the nodal forces for (1) a linearly varying pressure px on the edge of the triangular element shown in Figure P6-11(a); and (2) the quadratic varying pressure shown in Figure P6-11(b) by
Determine the nodal forces for (1) the quadratic varying pressure loading shown in Figure P6-12(a) and (2) the sinusoidal varying pressure loading shown in Figure P6-12(b) by the work equivalence
Determine the nodal displacements and the element stresses, including principal stresses, for the thin plate of Section 6.5 with a uniform shear load (instead of a tensile load) acting on the right
Determine the nodal displacements and the element stresses, including principal stresses, due to the loads shown for the thin plates in Figure P6-14. Use E = 105 GPa, v = 0.30, and t = 5 mm. Assume
Determine the nodal displacements and the element stresses, including principal stresses, due to the loads shown for the thin plates in Figure P6-15 on the next page. Use E = 30 Ã— 106 psi, v =
Evaluate the body force matrix for the plates shown in Figures P6-14(a) and (c). Assume the weight density to be 154.2 kN/m3.
Why is the triangular stiffness matrix derived in Section 6.2 called a constant-strain triangle?
How do the stresses vary within the constant-strain triangle element?
Can you use the plane stress or plane strain element to model the following? If so, indicate which ones are best modeled using plane stress or best modeled using plane strain elements.a. A flat slab
For a simple three-noded triangular element, show explicitly that differentiation of Eq. (6.2.47) indeed results in Eq. (6.2.48); that is, substitute the expression for [B] and the plane stress
The plane stress element only allows for in-plane displacements, while the frame or beam element resists displacements and rotations. How can we combine the plane stress and beam elements and still
For the plane structures modeled by triangular elements shown in Figure P6-21, show that numbering in the direction that has fewer nodes, as in Figure P6-21(a) (as opposed to numbering in the
Go through the detailed steps to evaluate Eq. (6.3.6).In Eq. (6.3.6)
Show that the sum N1 + N2 + N3 + N4 is equal to 1 anywhere on a rectangular element, where N1 through N4 are defined by Eqs. (6.6.5).In Eqs (6.6.5)
For the rectangular element of Figure 6-20 on page 374 the nodal displacements are given byFor b = 2 in., h = 1 in., E = 30 Ã— 106 psi, and v = 0.3, determine the element strains and stresses at
Evaluate the stiffness matrix for the elements shown in Figure P6-3. The coordinates are in units of inches. Assume plane stress conditions. Let E = 10 Ã— 106 psi, v = 0.25, and thickness t =
For the elements given in Problem 6.3, the nodal displacements are given asu1 = 0.0 in.............v1 = 0.0025 in.............u2 0.0012 in.v2 = 0.0 in.............u3 = 0.0 in...............v3 =
Determine the von Mises stress for Problem 6.4.u1 = 0.0 in.............v1 = 0.0025 in.............u2 0.0012 in.v2 = 0.0 in.............u3 = 0.0 in...............v3 = 0.0025 in.Determine the element
Evaluate the stiffness matrix for the elements shown in Figure P6-6. The coordinates are given in units of millimeters. Assume plane stress conditions. Let E = 105 GPa, v = 0.25, and t = 10 mm.
For the elements given in Problem 6.6, the nodal displacements are given asu1 = 2.0 mm...............v1 = 1.0 mm............u2 = 0.5 mmv2 = 0.0 mm...............u3 = 3.0 mm.............v3 = 1.0
Determine the von Mises stress for Problem 6.7.For the elements given in Problem 6.6, the nodal displacements are given asu1 = 2.0 mm...............v1 = 1.0 mm............u2 = 0.5 mmv2 = 0.0
For the plane strain elements shown in Figure P6-9, the nodal displacements are given asu1 = 0.001 in...............v1 = 0.005 in..............u3 = 0.001 in.v2 = 0.0025 in..............u3 = 0.0
For the finite element mesh shown in Figure P7-1, comment on the appropriateness of the mesh. Indicate the mistakes in the model. Explain and show how to correct them.Figure P7-1
Consider the bar with two elements shown in Figure P7.10. Perform a patch test using these two elements. Let E = 200 GPa, and A = 1 Ã— 10-4 m2. Use the standard bar element stiffness matrix [Eq.
Determine the free-end displacements and the element stresses for the plate discretized into four triangular elements and subjected to the tensile forces shown in Figure P7-12. Compare your results
Determine the stresses in the plate with the hole subjected to the tensile stress shown in Figure P7-13. Graph the stress variation (x versus the distance y from the hole. Let E = 200 GPa, v = 0.25,
Solve the following problem of a steel tensile plate with a concentrated load applied at the top, as shown in Figure P7-14. Determine at what depth the effect of the load dies out. Plot stress (y
For the flat connecting rod shown in Figure P7-15, determine the maximum principal stresses and their location. Let E = 30 Ã— 106 psi, v = 0.25, t = 1 in., and P = 1000 lb.Figure P7-15
Determine the maximum principal stresses and their locations for the member with fillet subjected to tensile surface tractions shown in Figure P7-16. Let E = 200 GPa and v = 0.25. Then let E = 73 GPa
Determine the maximum principal stresses in the member with a re-entrant corner as shown in Figure P7-17. At what location are the principal stresses largest? Let E = 30 Ã— 106 psi and v = 0.25.
For the tooth implant subjected to loads shown in Figure P7-19, determine the maximum principal stresses. Let E = 1.6 Ã— 106 psi and v = 0.3 for the dental restorative implant material
Comment on the mesh sizing in Figure P7-2. Is it reasonable? If not, explain why not.Figure P7-2
Determine the middepth deflection at the free end and the maximum principal stresses and their location for the beam subjected to the shear load variation shown in Figure P7-20. Do this using 64
Determine the stresses in the shear wall shown in Figure P7-21. At what location are the principal stresses largest? Let E = 21 GPa, v = 0.25, twall 5 0.10 m, and tbeam = 0.20 m. Use 0.1 m radii at
Determine the stresses in the plates with the round and square holes subjected to the tensile stresses shown in Figure P7-22. Compare the largest principal stresses for each plate. Let E = 210 GPa, v
For the concrete overpass structure shown in Figure P7-23, determine the maximum principal stresses and their locations. Assume plane strain conditions. Let E = 3.0 Ã— 106 psi and v = 0.30.Figure
For the tensile member shown in Figure P7-25 on the next page with two holes, determine the maximum principal stresses and their locations. Let E = 210 GPa, v = 0.25, and t = 10 mm. Then let E = 70
For the plate shown in Figure P7-26 on the next page, determine the maximum von Mises stresses and their locations. Let E = 210 GPa and v = 0.25?Figure P7-26
Determine the von Mises stresses in the wrench shown in Figure P7-28. Let E = 200 GPa and v = 0.25, and assume uniform thickness t = 10 mm. Assume the bottom inside surface is fixed.Figure P7-28
What happens if the material property v = 0.5 in the plane strain case? Is this possible? Explain.
For the 1 in. thick canopy hook shown in Figure P7-31 on page 432, used to hold down an aircraft canopy, determine the maximum von Mises stress and maximum deflection. The hook is subjected to a
For the Â¼ in. thick L-shaped steel bracket shown in Figure P7-32 on page 432, show that the stress at the 90o re-entrant corner never converges. Try models with increasing numbers of elements to
The machine shown in Figure P7-33 on page 433 is an overload protection device that releases the load when the shear pin S fails. Determine the maximum von Mises stress in the upper part ABE if the
One arm of a crimper tool shown in Figure P7-37 is to be designed of 1080 as-rolled steel. The loads are shown in the figure. Fix the nodes around the two holes. Select a thickness for the arm based
Design the bicycle wrench with the approximate dimensions shown in Figure P7-38. If you need to change dimensions explain why. The wrench should be made of steel or aluminum alloy. Determine the
For the various parts shown in Figure P7-39 on the next page determine the best one to relieve stress. Make the original part have a small radius of 0.1 in. at the inside reentrant corners. Place a
Under what conditions is the structure in Figure P7-4 a plane strain problem? Under what conditions is the structure a plane stress problem?Figure P7-4
When do problems occur using the smoothing (averaging of stress at the nodes from elements connected to the node) method for obtaining stress results?
What thickness do you think is used in computer programs for plane strain problems?
Which one of the CST models shown in Figure P7-7 is expected to give the best results for a cantilever beam subjected to an end shear load? Why?Figure P7-7(a)
The plane stress element only has in-plane displacements, while the frame element resists displacements and rotations. How can we combine the plane stress and beam elements and still insure
In considering the patch test, answer the following questions:(a) Can elements of different mechanical properties be used? Why?(b) Can the patch be arbitrary in shape? Why?(c) Can we mix triangular
Evaluate the shape functions given by Eq. (8.2.6). Sketch the variation of each function over the surface of the triangular element shown in Figure 8-3?Figure 8-3 LST triangle for evaluation of a
Express the strains εx, εy, and γxy for the element of Figure 8-3 by using the results given in Section 8.2. Evaluate these strains at the centroid of the element; then evaluate the stresses at
For the element of Figure 8-3 (shown again as Figure P8-3) subjected to the uniform pressure shown acting over the vertical side, determine the nodal force replacement system using Eq. (6.3.7).
For the element of Figure 8-3 (shown as Figure P8-4) subjected to the linearly varying line load shown acting over the vertical side, determine the nodal force replacement system using Eq. (6.3.7).
For the linear-strain elements shown in Figure P8-5, determine the strains (x, (y, and (xy. Evaluate the stresses (x, (y, and (xy at the centroids. The coordinates of the nodes are shown in units of
For the linear-strain element shown in Figure P8-6, determine the strains (x, (y, and (xy. Evaluate these strains at the centroid of the element; then evaluate the stresses (x, (y, and (xy at the
Evaluate the shape functions for the linear-strain triangle shown in Figure P8-7. Then evaluate the [B] matrix. Units are millimeters.Figure P8-7
For the elements shown in Figure P9-1, evaluate the stiffness matrices using Eq. (9.2.2). The coordinates are shown in the figures. Let E = 15 Ã— 106 psi and v = 0.25 for each element.Figure
How should one model the boundary conditions of nodes acting on the axis of symmetry?
How would you evaluate the circumferential strain, εθ, at r = 0? What is this strain in terms of the a's given in Eq. (9.1.15). Elasticity theory tells us that the radial strain must equal the
What will be the stresses σr and σθ at r = 0? Look at Eq. (9.1.2) after considering Problem 9.11.
The soil mass in Figure P9-13 is loaded by a force transmitted through a circular footing as shown. Determine the stresses in the soil. Compare the values of σr using an axisymmetric model with the
Perform a stress analysis of the pressure vessel shown in Figure P9-14. Let E = 5 × 106 psi and v = 0.15 for the concrete, and let E = 29 × 106 psi and v = 0.25 for the steel liner. The steel liner
Perform a stress analysis of the concrete pressure vessel with the steel liner shown in Figure P9-15. Let E = 30 GPa and v = 0.15 for the concrete, and let E = 205 GPa and v = 0.25 for the steel
For the axisymmetric connecting rod shown in Figure P9-18, determine the stresses σz, σr, σθ, and τrz. Plot stress contours (lines of constant stress) for each of the normal stresses. Let E = 30
For the thick-walled open-ended cylindrical pipe subjected to internal pressure shown in Figure P9-19, use five layers of elements to obtain the circumferential stress, σθ, and the principal
Evaluate the nodal forces used to replace the linearly varying surface traction shown in Figure P9-2. Use Eq. (9.1.34).Figure P9-2
A steel cylindrical pressure vessel with flat plate end caps is shown in Figure P9-20 with vertical axis of symmetry. Addition of thickened sections helps to reduce stress concentrations in the
For the cylindrical vessel with ellipsoidal heads shown in Figure P9-22a under loading p = 500 psi, determine if the vessel is safe against yielding. Use the same material and factor of safety as in
The syringe with plunger is shown in Figure P9-23. The material of the syringe is glass with E = 69 GPa, v = 0.15, and tensile strength of 5 MPa. The bottom hole is assumed to be closed under test

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