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physics
thermodynamics

A First Course in the Finite Element Method 6th edition Daryl L. Logan - Solutions

For the basement wall shown in Figure P13-40, determine the temperature distribution and the heat transfer through the wall and soil. The wall is constructed of concrete (k = 1.0 Btu/h-ft-oF). The soil has an average thermal conductivity of k = 0.85 Btu/h-ft-8F. The inside air is maintained at 70
Now add a 6 in. thick concrete floor to the model of Figure P13-40 (as shown in Figure P13-41). Determine the temperature distribution and the heat transfer through the concrete and soil. Use the same properties as shown in Problem 13-40.Figure P13-41,Refer to Problem 13-40, For the basement wall
Aluminum fins (k = 170 W/m-K) with triangular profiles shown in Figure P13-42 are used to remove heat from a surface with a temperature of 200 oC. The temperature of the surrounding air is 10 oC. The natural convection coefficient is h = 25 W/m2 -K. Determine the temperature distribution throughout
The Allen wrench shown in Figure P13-44 is unloaded but now exposed to a temperature of 300 K, at its lower end, while the other end has a heat flux of 10 W/m2 acting over the end surface. Determine the temperature distribution throughout the wrench. The thermal conductivity of the material is 43.6
The forklift from Figure P13-45 has its load removed. The fork is made of AISI 4130 steel. The thermal conductivity of the steel is 35 W/m oC. The top surface of the fork is at 50 oC. The other surfaces of the L-shaped appendages located at the upper and lower left sides of the forklift are at room
The radio control car front steering unit in Figure P13-46 (detailed dimensions shown in Figure P11-22) is now relieved of stress, but its base has an applied temperature of 100 oF. The lower surface of the lower right-side flange has an applied temperature of 50 8F. Other surfaces are exposed to
Air is flowing at a rate of 10 lb/h inside a round tube with a diameter of 1.5 in. and length of 10 in., similar to Figure 13-32 on page 645. The initial temperature of the air entering the tube is 508F. The wall of the tube has a uniform constant temperature of 200Â°F. The specific heat of the
The fin shown in Figure P13-5 is insulated on the perimeter. The left end has a constant temperature of 100 oC. A positive heat flux of q* = 500 W/m2 acts on the right end. Let Kxx = 6 W/(m ( oC) and cross-sectional area A = 0.1 m2. Determine the temperatures at L/4, L/2, 3L/4, and L, where L = 0.4
For the composite wall shown in Figure P13-6, determine the interface temperatures. What is the heat flux through the 8-cm portion? Use the finite element method. Use three elements with the nodes shown. 1 cm = 0.01 m.Figure P13-6
For the composite wall idealized by the one-dimensional model shown in Figure P13-7, determine the interface temperatures. For element 1, let Kxx = 5 W (m ( oC) for element 2, Kxx = 10 W (m ( oC); and for element 3, Kxx = 15 W (m ( oC). The left end has a constant temperature of 1008C and the right
A composite wall is shown in Figure P13-8. For element 1, let Kxx = 5 W (m-oC), for element 2 let Kxx = 10 W (m-oC), for element 3 let Kxx = 15 W (m-oC). The left end has a heat source of 600 W applied to it. The right end is held at 10oC. Determine the left end temperature and the interface
A double-pane glass window shown in Figure P13-9, consists of two 4-mm thick layers of glass with k = 0.80 W/m-oC separated by a 10 mm thick stagnant air space with k = 0.025 W/m-oC. Determine(a) The temperature at both surfaces of the inside layer of glass and the temperature at the outside
For the one-dimensional flow through the porous media shown in Figure P14 -1, determine the potentials at one-third and two-thirds of the length. Also determine the velocities in each element. Let A = 0.2 m2.Figure P14-1
For the two-dimensional fluid flow shown in Figure P14 -10, determine the potentials at the center and right edge.Figure P14-10
Using a computer program, determine the potential distribution in the two-dimensional bodies shown in Figures P14 -11 to P14 -16.Figure P14-11Figure P14-12Figure P14-13
For the direct current (DC) electrical networks shown in Figures P14 -17 and P14 -18, determine the currents through each loop and in branches AD and BC in Figure P14 -17 or branches AB and BC in Figure P14 -18. Figure P14-17, Figure P14-18
For the direct current (DC) networks consisting of batteries, resistors (shown by the rectangular shapes), and light emitting diodes (LEDs) (shown by the triangular shapes) in Figures P14 -19 and P14 -20, determine the currents through each loop, in branches AD and BC in Figure P14 -19, and in
For the one-dimensional flow through the porous medium shown in Figure P14 -2 with fluid flux at the right end, determine the potentials at the third points. Also determine the velocities in each element. Let A = 2m2.Figure P14-2
For the infinitely long air-enclosed channel shown in Figure P14-22, determine the voltage variation through the air (( = 1) and the largest electric field magnitude and where it is located.Figure P14-22
A busbar is a rectangular conductor used in distribution of electric power in a distribution box. The ground and busbar are considered perfect insulators. Assume the potential of the busbar is 240 V. For the system shown in Figure P14-23, determine the voltage distribution in the air (( = 1) around
For the one-dimensional fluid flow through the stepped porous medium shown in Figure P14-3, determine the potentials at the junction of each area. Also determine the velocities in each element. Let Kxx = 1 in./s.Figure P14-3
For the one-dimensional fluid-flow problem (Figure P14 -4) with velocity known at the right end, determine the velocities and the volumetric flow rates at nodes 1 and 2. Let Kxx = 2 cm/s.Figure P14-4
Derive the stiffness matrix, Eq. (14.2.15), using the first term on the right side of Eq. (13.4.17)?
For the one-dimensional fluid-flow problem in Figure P14-6, determine the velocities and volumetric flow rates at nodes 2 and 3. Let Kxx = 10-1 in./s?Figure P14-6
For the simple pipe networks shown in Figure P14 -7, determine the pressures at nodes 1, 2, and 3 and the volumetric flow rates through the branches. Assume the pressure at node 4 is zero. In network(a) Let Q = 1 m3 / s. Let the resistances be, R1 = 1, R2 = 2, R3 = 3, R4 = 4, and R5 = 5 all in
For the triangular element subjected to a fluid source shown in Figure P14-8, determine the amount of Q* allocated to each node.Figure P14-8
For the triangular element subjected to the surface fluid source shown in Figure P14 -9, determine the amount of fluid force at each node.Figure P14-9
For the one-dimensional steel bar fixed at the left end, free at the right end, and subjected to a uniform temperature rise T = 100°F as shown in Figure P15 - 1, determine the free-end displacement, the displacement 60 in. from the fixed end, the reactions at the fixed end, and the axial
When do stresses occur in a body made of a single material due to uniform temperature change in the body? Consider Problem 15.1 and also compare the solution to Example 15.1 in this chapter.
Consider two thermally incompatible materials, such as steel and aluminum, attached together as shown in Figure P15 -11. Will there be temperature-induced stress in each material upon uniform heating of both materials to the same temperature when the boundary conditions are simple supports (a pin
A bimetallic thermal control is made of cold-rolled yellow brass and magnesium alloy bars (Figure P15 -12). The bars are arranged with a gap of 0.005 in. between them at 72 °F. The brass bar has a length of 1.0 in. and a cross-sectional area of 0.10 in2, and the magnesium bar has a length of
For the plane stress element shown in Figure P15 -13 subjected to a uniform temperature drop of T = 50 °F, determine the thermal force matrix {fT}.Let E = 10 Ã— 106 psi v = 0.30, and Î± = 12.5 Ã— 10-6 (in./in.)/°F. The coordinates (in inches) are shown
For the plane stress element shown in Figure P15 -14 subjected to a uniform temperature rise of T = 50°C, determine the thermal force matrix {fT}. Let E = 70 GPa, v = 0.3, Î± = 23 Ã— 10-6 (mm/mm)/°C, and t = 5mm. The coordinates (in millimeters) are shown in the
For the plane stress element shown in Figure P15 -15 subjected to a uniform temperature rise of T = 50°F, determine the thermal force matrix {fT}. Let E = 30 Ã— 106 psi, v = 0.3, a = 7.0 Ã— 1026 (in./in.)/°F, and t = 1 in. The coordinates (in inches) are shown in the
For the plane stress element shown in Figure P15 -16 subjected to a uniform temperature drop of T = 20 °C, determine the thermal force matrix {fT}. Let E = 210 GPa, v = 0.25, and Î± = 12 Ã— 10-6 (mm/mm)/°C. The coordinates (in millimeters) are shown in the figure.
For the plane stress plate fixed along the left and right sides and subjected to a uniform temperature rise of 50Â°F as shown in Figure P15 -17, determine the stresses in each element. Let E = 10 Ã— 106 psi, v = 0.30, Î± = 12.5 Ã— 10-6 (in./in.)/Â°F, and t = 1/4 in. The coordinates (in
For the plane stress plate fixed along all edges and subjected to a uniform temperature decrease of 20°C as shown in Figure P15 -18, determine the stresses in each element. Let E = 210 GPa, v = 0.25, and Î± = 12 Ã— 10-6 (mm/mm)/°C. The coordinates of the plate are
If the thermal expansion coefficient of a bar is given by α = a0 (1 + x /L), determine the thermal force matrix. Let the bar have length L, modulus of elasticity E, and cross-sectional area A.
For the one-dimensional steel bar fixed at each end and subjected to a uniform temperature drop of T = 30°C as shown in Figure P15 -2, determine the reactions at the fixed ends and the stress in the bar. Let E = 200 GPa, A = 1 Ã— 10-2 m2, and a = 11.7 Ã— 10-6
Assume the temperature function to vary linearly over the length of a bar as T = t1 + t2 x; that is, express the temperature function as {T} = [N] {t}, where [N] is the shape function matrix for the two-node bar element. In other words, [N] = [1 - x/L x/L]. Determine the force matrix in terms of E,
Derive the thermal force matrix for the axisymmetric element of Chapter 9. [Also see Eq. (15.1.27).]
The square plate in Figure P15 -22 is subjected to uniform heating of 50 °F. Determine the nodal displacements and element stresses. Let the element thickness be t = 0.1 in., E = 30 Ã—106 psi, v = 0.33, and Î± = 10 Ã— 10-6 /°F. Then fix the left and
The square plate in Figure P15 -23 has element 1 made of steel with E = 30 Ã— 106 psi, v = 0.33, and Î± = 10 Ã— 10-6 /Â°F and element 2 made of a material with E = 15 Ã— 106 psi, v = 0.25, and Î± = 50 Ã— 10-6 /Â°F. Let the plate thickness be t = 0.1 in. Determine the nodal
Solve Problem 15.3 using a computer program.For the plane truss shown in Figure P15 -3, bar element 2 is subjected to a uniform temperature rise of T = 60 Â°F. Let E = 30 Ã— 106 psi, A = 2 in2, and a = 7.0 Ã— 10-6 (in./in.)/ Â°F. The lengths of the truss elements are shown in the figure.
Solve Problem 15.6 using a computer program.For the plane truss shown in Figure P15 -6, bar element 2 is subjected to a uniform temperature drop of T = 30°C. Let E = 70 GPa, A = 4 Ã— 10-2 m2, and a = 23 Ã— 10-6 (mm/mm)/°C. Determine the stresses in each bar and the displacement of
For the solid model of a fixture shown in Figure P15 -27, the inside surface of the hole is subjected to a temperature increase of 80°C. The right end surfaces are fixed. Determine the von Mises stresses throughout the fixture due to this temperature increase. What is the largest von Mises
For the plane truss shown in Figure P15 -3, bar element 2 is subjected to a uniform temperature rise of T = 60 °F. Let E = 30 Ã— 106 psi, A = 2 in2, and a = 7.0 Ã— 10-6 (in./in.)/ °F. The lengths of the truss elements are shown in the figure. Determine the stresses
For the plane truss shown in Figure P15-4, bar element 1 is subjected to a uniform temperature drop of 40°F. Let E = 30 Ã— 106 psi, A = 2 in2, and a = 7.0 Ã— 10-6 (in./in.)/°F. The lengths of the truss elements are shown in the figure. Determine the stresses in each
For the structure shown in Figure P15-5, bar element 1 is subjected to a uniform temperature rise of T = 40 °C. Let E = 200 GPa, A = 2 Ã— 10-2 m2, and a = 12 Ã— 10-6 (mm/mm)/°C. Determine the stresses in each bar.
For the plane truss shown in Figure P15 -6, bar element 2 is subjected to a uniform temperature drop of T = 30°C. Let E = 70 GPa, A = 4 Ã— 10-2 m2, and a = 23 Ã— 10-6 (mm/mm)/°C. Determine the stresses in each bar and the displacement of node 1.
For the bar structure shown in Figure P15 -7, element 1 is subjected to a uniform temperature rise of T = 30 Â°C. Let E = 210 GPa, A = 3 Ã— 10-2 m2, and a = 12 Ã— 10-6 (mm/mm)/Â°C. Determine the displacement of node 1 and the stresses in each bar.
A bar assemblage consists of two outer steel bars and an inner brass bar. The three-bar assemblage is then heated to raise the temperature by an amount T = 80Â°F. Let all cross-sectional areas be A = 2 in2 and L = 60 in., Esteel = 30 Ã— 10 psi Ebrass = 15 Ã— 106 psi, Î± = 6.5 Ã—
For the plane truss shown in Figure P15 -9, bar element 2 is subjected to a uniform temperature rise of T = 10 °C. Let E = 210 GPa, A = 12.5 cm2, and Î± = 12 Ã— 10-6/°C. What temperature change is needed in bars 1 and 3 to remove the stress due to the uniform
Determine the consistent-mass matrix for the one-dimensional bar discretized into two elements as shown in Figure P16-1. Let the bar have modulus of elasticity E, mass density (, and cross-sectional area A.Figure P16-1
For the beams shown in Figure P16 -11, determine the natural frequencies using first two and then three elements. Let E, (, I, and A be constant for the beams.Figure P16-11(a)(b)(c)(d)
Rework Problem 16.11 using a computer program with E = 3 Ã— 107 psi, ( = 0.00073 lb-s2 /in4, A = 1 in2, L = 100 in., and I = 0.0833 in4.Figure P16-13Refer to Problem 16.11, For the beams shown in Figure P16 -11, determine the natural frequencies using first two and then three elements.
For the beams in Figures P16 -13 and P16 -14 subjected to the forcing functions shown, determine the maximum deflections, velocities, and accelerations. Use a computer program. Use Î± = 3.00, Î² = 0.001, (t = 0.002 s.Figure P16-13,
For the rigid frames in P16 -16 subjected to the forcing functions shown, determine the maximum displacements, velocities, and accelerations. Use a computer program. Use Î² = 3.00, Î² = 0.001, (t = 0.002 s.Figure P16-16
For the rigid frame shown, a motor is located on the horizontal member at its center. The motor imparts a driving frequency of 2000 rpm (33.4 Hz) to the structure. Determine the first 3 natural frequencies and animate the associated modes of vibration on your computer. Is this driving frequency
A marble slab with k = 2W / (m ( oC), ( = 2500 kg / m3, and c = 800 W ( s / (kg ( oC) is 2 cm thick and at an initial uniform temperature of Ti = 200 oC. The left surface is suddenly lowered to 0 oC and is maintained at that temperature while the other surface is kept insulated. Determine the
A circular fin is made of pure copper with a thermal conductivity of k = 400 W / (m ( oC), h = 150 W / (m2 ( oC), mass density ( = 8900 kg / m3, and specific heat c = 375 J / (kg ( oC). The initial temperature of the fin is 25oC. The fin length is 2 cm and the diameter is 0.4 cm. The right tip of
For the one-dimensional bar discretized into three elements as shown in Figure P16-2, determine the lumped- and consistent-mass matrices. Let the bar properties be E, (, and A throughout the bar.Figure P16-2
For the one-dimensional bar shown in Figure P16-3, determine the natural frequencies of vibration, v's, using two elements of equal length. Use the consistent-mass approach. Let the bar have modulus of elasticity E, mass density (, and cross-sectional area A. Compare your answers to those obtained
For the one-dimensional bar shown in Figure P16-4, determine the natural frequencies of longitudinal vibration using first two and then three elements of equal length. Let the bar have E = 30 Ã— 106 psi, ( = 0.00073 lb-s2 / in4, A = 1 in2, and L = 60 in.Figure P16-4
For the spring-mass system shown in Figure P16-5, determine the mass displacement, velocity, and acceleration for five time steps using the central difference method. Let k = 2000 lb/ft and m = 2 slugs. Use a time step of (t = 0.03 s. You might want to write a computer program to solve this
For the spring-mass system shown in Figure P16-6, determine the mass displacement, velocity, and acceleration for five time steps using(a) The central difference method,(b) Newmark's time integration method, and(c) Wilson's method. Let k = 1200 lb/ft and m = 2 slugs.Figure P16-6
Rework Problems 16.7 and 16.8 using a computer program.Refer to Problem 16.7 and 16.8,1. For the bar shown in Figure P16-7, determine the nodal displacements, velocities, and accelerations for five time steps using two finite elements. Let E = 30 Ã— 106 psi, ( = 0.00073 lb-s2 / in4, A =
Solve Problem using matrices [A], [B], [C], [D], and {E} given bya. [A] + [B] b. [A] + [C] c. [A][C]T d. [D]{E} e. [D][C] f. [C][D]
Evaluate the following integral in explicit form:
The following integral represents the strain energy in a bar of length L and cross-sectional area A:Where and E is the modulus of elasticity. Show that dU/d{d} yields [k]{d}, where [k] is the bar stiffness matrix given by
Determine [A]-1 by the cofactor method.Solve Problems using matrices [A], [B], [C], [D], and {E} given by
Determine [D]-1 by the cofactor method.Solve Problems using matrices [A], [B], [C], [D], and {E} given by
Determine [B]-1 by row reductionSolve Problems using matrices [A], [B], [C], [D], and {E} given by
Determine [D]-1 by row reduction.Solve Problems using matrices [A], [B], [C], [D], and {E} given by
Show that ([A][B])T =[B]T [A]T by using
Find [T]-1 given thatand show that [T]-1 =[T]T and hence that [T] is an orthogonal matrix.
Given the matricesshow that the triple matrix product [X]T [A][X] is symmetric.
Determine the solution of the following simultaneous equations by Cramer's rule. 2x1 + 4x2 = 20 4x1 - 2x2 = 10
Determine the solution to the set of equations in problem B.1 by the inverse method.
Solve the following system of simultaneous equations by Gaussian elimination.2x1 - 4x2 - 5x3 = 62x2 + 4x3 = -11x1 - 1x2 + 2x3 = 2
Solve the following system of simultaneous equations by Gaussian elimination. 2x1 + 1x2 - 3x3 = 11 4x1 - 2x2 + 3x3 = 8 -2x1 + 2x2 - 1x3 = -6
Given the following relations: x1 = 3y1 - 2y2 ɀ1 = x1 + 2x2 x2 = 2y1 - y2 ɀ2 = 4x1 + 2x2 a. Write these relationships in matrix form. b. Express {z} in terms of {y}. c. Express {y} in terms of {z}.
Starting with the initial guess {X}T = [1 1 1 1 1] , perform five iterations of the Gauss-Seidel method on the following system of equations. On the basis of the results of these five iterations, what is the exact solution? 2x1 - 1x2 = - 1 -1x1 + 6x2 - 1x3 = 4 -2x2 + 4x3 - 1x4 = 4 - 1x3
Solve Problem B.1 by Gauss-Seidel iteration. Hint: Read the paragraphs on page 856.
Classify the solutions to the following systems of equations according to Section B.2 as unique, nonunique, or nonexistent.a.2x1 - 6x2 = 104x1 - 12x2 = 20b.6x1 + 3x2 = 92x1 + 6x2 = 12c.8x1 + 4x2 = 324x1 + 2x2 = 8d.1x1 + 1x2 + 1x3 = 12x1 + 2x2 + 2x3 = 23x1 + 3x2 + 3x3 = 3
Determine the bandwidths of the plane trusses shown in Figure PB -9. What conclusionscan you draw regarding labeling of nodes?
Determine the equivalent joint or nodal forces for the beam elements shown in Figurea.b.c.d.e.f.g.h.

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