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systems analysis and design using matlab

Introduction To Finite Element Analysis And Design 1st Edition Nam-Ho Kim, Bhavani V. Sankar - Solutions

29. The state of stress at a point is given by"80 20 40"a] = 20 60 10 MPa
40 10 20(a) Determine the strains using Young's modulus of 100 GPa and Poisson's ratio of 0.25.(b) Compute the strain energy density using these stresses and strains.(c) Calculate the principal stresses.(d) Calculate the principal strains from the strains calculated in (a).(e) Show that the
30. Consider the state of stress in problem 29 above. The yield strength of the material is 100 MPa.Determine the safety factors according to the following: (a) maximum principal stress criterion,(b) Tresca Criterion, and (c) von Mises criterion.
31. A thin-walled tube is subject to a torque T. The only non-zero stress component is the shear stress rA-y, which is given by tVj, = 10,000 T(Pa), where Tis the torque in N.m. When the yield strength cry = 300 MPa and the safety factor N = 2, calculate the maximum torque that can be applied
32. A thin-walled cylindrical pressure vessel with closed ends is subjected to an internal pressure p = 100 psi and also a torque T around its axis of symmetry. Determine T that will cause yield¬ing according to von Mises' yield criterion. The design requires a safety factor of 2. The nominal
33. A cold-rolled steel shaft is used to transmit 60 kW at 500 rpm from a motor. What should be the diameter of the shaft if the shaft is 6 m long and is simply supported at its end? The shaft also experiences bending due to a distributed transverse load of 200 N/m. Ignore bending due to the weight
34. For the stress matrix below, the two principal stresses are given as ai =2 and aj = —3, respec¬tively. In addition, two principal directions corresponding to the two principal stresses are also given below. The yield stress of the structure is given as ay = 4.5.2 0(a) Calculate the safety
35. The figure below shows a shaft of 1.5-tn diameter loaded by a bending moment Mz = 5,000 lb ¦ in, a torque T = 8,0001b • in, and an axial tensile force N = 6,0001b. If the material is ductile with the yielding stress cry = 40,000psi, determine the safety factor using (a) the maximum shear
36. A 20-mm-diameter rod made of a ductile material with a yield strength of 350 MPa is subject to a torque of T = 100 N • m and a bending moment of M = 150 N • m. An axial tensile force P is then gradually applied. What is the value of the axial force when yielding of the rod occurs?Solve the
37. A circular shaft of radius r in the figure has a moment of inertia I and polar moment of inertia J.The shaft is under torsion Tz in the positive z-axis and bending moment Mv in the positive x-axis.The material is mild steel with yield strength of 2.8 MPa. Use only the given coordinate system
38. A rectangular plastic specimen of size 100 x 100 x 10mm3 is placed in a rectangular metal cavity. The dimensions of the cavity are 101 x 101 x 9mm3. The plastic is compressed by a rigid punch until it is completely inside the cavity. Due to Poisson effect, the plastic also expands in the x and
1. Three rigid bodies, 2, 3, and 4, are connected by four springs, as shown in the figure. A horizon¬tal force of 1,000 N is applied on Body 4, as shown in the figure. Find the displacements of the three bodies and the forces (tensile/compressive) in the springs. What is the reaction at the wall?
2. Three rigid bodies, 2, 3, and 4, are connected by six springs as shown in the figure. The rigid walls are represented by 1 and 5. A horizontal force F3 = 1000N is applied on Body 3 in the direction shown in the figure. Find the displacements of the three bodies and the forces
3. Consider the spring-rigid body system described in Problem 2. What force F2 should be applied on Body 2 to keep it from moving? How will this affect the support reactions?Hint Impose the boundary condition 112 = 0 in the FEM and solve for displacements M3 and 114.Then, the force F2 will be the
4. Four rigid bodies, 1, 2, 3, and 4, are connected by four springs as shown in the figure. A hori¬zontal force of 1,000 N is applied on Body 1 as shown in the figure. Using FE analysis, (a) find the displacements of the two bodies (1 and 3), (2) find the element force (tensile/compressive)of
5. Determine the nodal displacements and reaction forces using the direct stiffness method. Calcu¬late the nodal displacements and element forces using the FE program.F, = 50 N *, = 50 N/cm k2 = 60 N/cm N1 N2 N3
6. In the structure shown below, rigid blocks are connected by linear springs. Imagine that only horizontal displacements are allowed. Write the global equilibrium equations [K]{Q} = {F}after applying displacement boundary conditions in terms of spring stiffness kdisplacement DOFs and applied loads
7. A structure is composed of two one-dimensional bar elements. When 10N force is applied to node 2, calculate displacement vector {Q}r = {uy, 112, 113} using the finite element method.
8. Use FEM to determine the axial force P in each portion, AB and BC, of the uniaxial bar. What are the support reactions? Assume E — 100 GPa, area of cross sections of the two portions AB and BC are, respectively, 10-4m2 and 2 x 10^4m2, and F = 10,000 N. The force F is applied at the cross
9. Consider a tapered bar of circular cross-section. The length of the bar is 1 m, and the radius varies as r(x) = 0.050 - 0.040a:, where r and x are in meters. Assume Young's modulus = 100MPa.Both ends of the bar are fixed, and F = 10,000 N is applied at the center. Determine the displace¬ments,
10. The stepped bar shown in the figure is subjected to a force at the center. Use FEM to determine the displacement at the center and reactions and flR.Assume: E = 100 GPa, area of cross sections of the three portions shown are, respectively, 10-4m2, 2 x 10~4m2, and 10~4m2, and F = 10,000N.Rl —(
11. Using the direct stiffness matrix method, find the nodal displacements and the forces in each element and the reactions.2000 lb/in 1500 lb/in.400 lb 500 lb/in 4 |
12. A stepped bar is clamped at one end and subjected to concentrated forces as shown.Note-. The node numbers are not in usual order!5kN 1 m m -2 m 1 m 2.7 Exercise 101 Assume: E = 100 GPa, section = 2 cm2.small area of cross-section = 1 cm2, and large area of cross-(a) Write the element stiffness
13. The uniaxial bar FE equation can be used for other types of engineering problems, if the proper analogy is applied. For example, consider the piping network shown in the figure. Each section of the network can be modeled using a FE. If the flow is laminar and steady, we can write the equations
14. For a two-dimensional truss structure, as shown in the figure, determine displacements of the nodes and normal stresses developed in the members using the direct stiffness method. Use E =30 x 106 N/cm2 and a diameter of the circular cross-section of 0.25 cm.^60N Pin joint 9 cm Roller support
15. For a two-dimensional truss structure, as shown in the figure, determine displacements of the nodes and normal stresses developed in the members using a FE program in Appendices. Use E = 30 x 106N/cm2 and a diameter of the circular cross section of 0.25 cm.
16. The truss structure shown in the figure supports force F at Node 2. FEM is used to analyze this structure using two truss elements as shown.iz, cl c(a) Compute the transformation matrix for Elements 1 and 2.(b) Compute the element stiffness matrices for both elements in the global coordinate
17. The truss structure shown in the figure supports the force F. FEM is used to analyze this struc¬ture using two truss elements as shown. Area of cross-section (for all elements) — A = 2 in2, Young's modulus = E = 30 x 106 psi. Both the elements are of equal length L = 10 ft.(a) Compute the
18. Use FEM to solve the plane truss shown below. Assume AE = 106N, L = 1 m. Determine the nodal displacements, forces in each element, and the support reactions.
19. The plane truss shown in the figure has two elements and three nodes. Calculate the 4 x 4 ele¬ment stiffness matrices. Show the row addresses clearly. Derive the final equations (after apply¬ing boundary conditions) for the truss in the form of [K]{Q}={F}. What are nodal displacements and the
20. Use FEM to solve the two plane truss problems shown in the figure below. Assume AE = 106 N, L = 1 m. Before solving the global equations [K]{Q} = {F}, find the determinant of [K]. Does [K] have an inverse? Explain your answer.
21. Determine the member force and axial stress in each member of the truss shown in the figure using one of FE analysis programs in the Appendix. Assume that Young's modulus is 104 psi and all cross-sections are circular with a diameter of 2 in. Compare the results with the exact solutions that
22. Determine the normal stress in each member of the truss structure. All joints are ball-joint and the material is steel, whose Young's modulus is E = 210 GPa.
23. The space truss shown has four members. Determine the displacement components of Node 5 and the force in each member. The node numbers are numbers in the circle in the figure. The dimensions of the imaginary box that encloses the truss are: 1 x 1 x 2 m. Assume AE = 106 N.The coordinates of the
24. The uniaxial bar shown below can be modeled as a one-dimensional truss. The bar has the following properties: L = 1 m, A = 10~4m2, E = 100 GPa, and a = IQ-A/0C. From the stressfree initial state, a force of 5,000 N is applied at Node 2 and the temperature is lowered by 100oC below the reference
25. In the structure shown below, the temperature of Element 2 is 100° C above the reference tem¬perature. An external force of 20,000 N is applied in the x-direction (horizontal direction) at Node 2. Assume E = lO'^a, A = 10~4m2, and a = 10~5/oC.20,000 N, ® , 1 m r|| 1 m(a) Write down the
26. Use FEA to determine the nodal displacements in the plane truss shown in Figure (a). The temperature of Element 2 is 100oC above the reference temperature, i.e., Ar'2' = 100oC. Com¬pute the force in each element. Show that the force equilibrium is satisfied at Node 3. Assume L = 1 m, AE = 107
27. Repeat Problem 26 for the new configuration with Element 5 added, as shown in Figure (b).
28. Repeat Problem 26 with an external force added at Node 3, as shown in Figure (c).
29. The properties of the members of the truss in the left side of the figure are given in the table.Calculate the nodal displacement and element forces. Show that force equilibrium is satisfied as
30. Repeat Problem 29 for the truss on the right side of the figure. Properties of Element 6 are same as those of Element 5, but AT = 0 "C.
31. The truss shown in the figure supports the force F = 2,000 N. Both elements have the same axial rigidity of AE = 107 N, thermal expansion coefficient of a = 10~6/OC, and length L = 1 ra.While the temperature at Element 1 remains constant, that of Element 2 is dropped by 100oC.(a) Write the 4 x
32. FEA was used to solve the truss problem shown below. The solution for displacements was obtained as 112 = 1 mm, V2 = —1 mm, i
33. Use EEM to solve the plane truss shown below. Assume AE = 106N, L = 1 m, a =20 x 10 6/C. The temperature of Element 1 is 100°C below the reference temperature, while Elements 2 and 3 are in the reference temperature. Determine the nodal displacements, forces
1. Use the Galerkin method to solve the following boundary value problem using (a) one-tei approximation and (b) two-term approximation. Compare your results with the exact soluti by plotting them on the same graph.d^u o^r+X"=0, 0
2. Solve the differential equation in Problem 1 using (a) two and (b) three finite elements. Use I local Galerkin method described in section 3.4. Plot the exact solution and two- and thri element solutions on the same graph. Similarly plot the derivative dujdx. Note: The boundi conditions are not
3. Using the Galerkin method, solve the following differential equation with the approximate s ution in the form of u(x) = c\x + C2A-2. Compare the approximate solution with the exact c by plotting them on a graph. Also compare the derivatives du/dxanddu/dx.
4. The one-dimensional heat conduction problem can be expressed by the following differential equation:d2T fc^r + !2 = 0, 0
5. Solve the one-dimensional heat conduction Problem 4 using the Rayleigh-Ritz method. For the heat conduction problem, the total potential can be defined as n1 fdT\2 2 \dxJ -QT dx Use the approximate solution f(x) = T\
6. Consider the following differential equation:d2u^ 2 ~f"f x — 0, 0
7. Solve the differential equation in Problem 6 for the following boundary conditions using Galer¬kin method:.m(0) = 1, ii(l) = 2 Assume the approximate solution as u{x) = 0o (x) + C\
8. Consider the following boundary value problem:Using equal-length two finite elements, calculate unknown u(x) and its derivative. Compare t finite element solution with the exact solution.
9. Consider the following boundary value problem:d ( du\ 2 m(1) =2^(2) = -i dxy ' 4(a) When two equal-length finite elements are used to approximate the problem, write interj lation functions and their derivatives.(b) Calculate the approximate solution using the Galerkin method.
10. The boundary value problem for a clamped-clamped beam can be written as^r~pW=0. OCjcCI vi;(0) = ve(l) = -y-(0) = ^y-(l) = 0: boundary conditions When a uniformly distributed load is applied, i.e., p{x) = po, calculate the approximate bei deflection w(x), using the Galerkin method, r::; Hint:
; 11. The boundary value problem for a cantilevered beam can be written as H o2{x) = C]X2 + C2X3. Solve for the boundary value problem using the Galerkin methi Compare the approximate solution to the exact solution by plotting the solutions on a graph.3
12. Repeat Problem 11 by assuming w(x) = ^c('0, (a') = c\x2 + C2X3 + C3X4 1=1
13. Consider a finite element with three nodes, as shown in the figure. When the solution is appro mated using u(x) = JVi (x)ui + ^2(x)ii2 + N3(x)u3, calculate the interpolation functk N[(x), Nzix), andA'3(x).Hint: Start with assumed solution in the following form: u(x) = cq + C]X + C2X2.If !r 12 3
14. A vertical rod of elastic material is fixed at both ends with constant cross-sectional area Young's modulus E, and height of L under the distributed load /per unit length. The verti deflection u(x) of the rod is governed by the following differential equation:Using three elements of equal
15. A bar component in the figure is under the uniformly distributed load q due to gravity. For linear elastic material with Young's modulus E and uniform cross-sectional area A, the govern¬ing differential equation can be written as
16. Consider a tapered bar of circular cross-section. The length of the bar is 1 m, and the radius varies as r(x) = 0.050 — 0.040x, where r and x are in meters. Assume Young's modulus = 100 MPa. Both ends of the bar are fixed, and a uniformly distributed load of 10,000 N/m is applied along the
17. A tapered bar with circular cross-section is fixed at ,r = 0, and an axial force of 0.3 x 106 N is applied at the other end. The length of the bar (L) is 0.3 m, and the radius varies as r(x) = 0.03 — O.OTx, where r and x are in meters. Use three equal-length finite elements to determine the
18. The stepped bar shown in the figure is subjected to a force at the center. Use FEM to determine the displacement field u(x), axial force distribution P(x), and reactions and Rr.Assume E = 100 GPa areas of cross-sections of the three portions shown are, respectively.10~4, 2 x 10~4 and 10~4 m2,
19. A bar shown in the figure is modeled using three equal-length bar elements. The total length of the bar is Lr = 1.5 m, and the radius of the circular cross-section is r = 0.1 m. When Young's modulus E = 207 GPa and distributed load q = 1,000 N/m, calculate displacement and stress using one of
20. Consider the tapered bar in Problem 17. Use the Rayleigh-Ritz method to solve the same prob¬lem. Assume the displacement in the form of u(x) = ag + aix + azx2. Compare the solutions for u(x) and Fix) with the exact solution given below by plotting them.«(*) = P{x) = EAix) — = F K ' jr&(0)
21. Consider the tapered bar in Problem 16. Use the Rayleigh-Ritz method to solve the same prob¬lem. Assume the displacement in the form of u(x) = (x — 1) (cix + C2X2).
1. Repeat Example 4.2 with the approximate deflection in the following form:V(x) = CiX2 + C2A'3 + C3X4. Compare the deflection curve with the exact solution.
2. The deflection of the simply supported beam shown in the figure is assumed as v(x) = cx{x - 1), where c is a constant. A force is applied at the center of the beam. Use the following properties: EI = 1000 N-m2 and L = 1 m. First, (a) show that the above approximate solution satisfies
3. Use the Rayleigh-Ritz method to determine the deflection v(x), bending moment M(x), and shear force Vy{x) for the beam shown in the figure. The bending moment and shear force are calculated from the deflection as M(x) = EId2v/dx2 and Vy{x) = -EltPv/dx?. Assume the dis¬placement as v(x) = eg +
4. The right end of a cantilevered beam is resting on an elastic foundation that can be represented by a spring with spring constant k = 1,000 N/m. A force of 1,000 N acts at the center of the beam as shown. Use the Rayleigh-Ritz method to determine the deflection v(x) and the force in the spring.
5. A cantilevered beam is modeled using one finite element. The nodal values of the beam element are given as Plot the deflection curve, bending moment, and shear force.
6. A simply supported beam with length L is under a concentrated vertical force —F at the center.When two equal-length beam elements are used, the finite element analysis yields the following nodal DOFs:f T?T^- ^{1Q !}Jr= \{ v'i ,e\u6vE2,6V2^ ^4638,}£ =/ ^' 0', ,1 06,0E,—I)Find the deflection
1. A simply supported beam with length L is under a uniformly distributed load — p. When two equal-length beam elements are used, the finite element analysis yields the following nodal DOFs:{1Q ,J}T =1 { f'l ,,'0 '1 ', 1\ )2' ,2024, El>3I ,3038} 4= 7(0?,-/^ ,-' ^',20,40,£-^/-J1.Find the
8. Consider a cantilevered beam with a Young's modulus E, moment of inertia /, height 2h, and length L. A couple mq is applied at the tip of the beam. One beam finite element is used to approximate the structure.(a) Calculate the tip displacement v and tip slope 0 using the finite element
9. The cantilever beam shown is modeled using one finite element. If the deflection at the nodes of a beam element are 0] = 0i = 0, and V2 = 0.01 m and slope 02 = 0, write the equation of the deformed beam v(s). In addition, compute the forces F2 and M2 acting on the beam to produce the above
10. Let a uniform cantilevered beam of length L be supported at the loaded end so that this end cannot rotate, as shown in the figure. For the given moment of inertia I, Young's modulus E, and applied tip load P, calculate the deflection curve v(x) using one beam element.
11. A cantilevered beam structure shown in the figure is under the distributed load. When q =l,000N/m, Lt = 1.5 m, E — 207GPa, and the radius of the circular cross-section r = 0.1m, solve the displacement of the neutral axis and stress on the top surface. Use three equal-length beam finite
13. In this chapter, we derived the beam finite element equation using the principle of minimum potential energy. However, the same finite element equation can be derived from the Galerkin method, as in Section 3.3. The governing differential equation of the beam is El^4= f(x), xe{0,L]where fix) is
14. Repeat the above derivation for the case of a cantilevered beam whose boundary condition is given by dv . . d1" „(0)=-(0)"v . . d v , s^L)=i^{L) = Q
15. Solve the simply supported beam problem in Example 4.9 using the MATLAB program in Appendix. You can use either distributed load capability in the program or equivalent nodal load. Plot the vertical displacement and rotation along the span of the beam. Compare them to the analytical solutions.
16. Consider a cantilevered beam with spring support at the end, as shown in the figure. Assume E = 100ksi, / = l.Oin4, L = lOin, k = 200lb/in, beam height h = lOin, and no gravity. The beam is subjected to a concentrated force F = 100 lb at the tip.(a) Using one beam element and one spring
17. A beam is clamped at the left end and on a spring at the right end. The right support is such that the beam cannot rotate at that end. Thus, the only active DOF is wz. A force F = 3,000 N acts downward at the right end as shown. The structure is modeled using two elements: one beam element and
18. A linearly varying distributed load is applied to the beam finite element of length L. The ma imum value of the load at the right side is qQ. Calculate "work equivalent" nodal forces a moment.
19. In general, a concentrated force can only be applied to the node. However, if we use the concept' 'work-equivalent'' load, we can convert the concentrated load within an element to correspoi ing nodal forces. A concentrated force P is applied at the center of one beam element of length
20. Use two equal-length beam elements to determine the deflection of the beam shown belc Estimate the deflection at Point B, which is at 0.5 m from the left support. EI = 1000 N-m2.0.5 m B1 m • 1 m ¦1,000 N
21. An external couple C2 is applied at Node 2 in the beam shown below. When EI = 105 N-r the rotations in radians at the three nodes are determined to be 9\ = —0.0S 02 = +0.05, 03 = -0.025(a) Draw the shear force and bending moment diagrams for the entire beam.(b) What is the magnitude of the
22. Two beam elements are used to model the structure shown in the figure. The beam is clam]to the wall at the left end (Node 1), supports a load P = 100N at the center (Node 2), ant simply supported at the right end (Node 3). Elements 1 and 2 are two-node beam element each of length 0.05m and the
23. Consider the clamped-clamped beam shown below. Assume that there are no axial forces acting on the beam. Use two elements to solve the problem, (a) Determine the deflection and slope at x = 0.5, 1, and 1.5 m; (b) Draw the bending moment and shear force diagrams for the entire beam; (c) What are
24. The frame shown in the figure is clamped at the left end and supported on a hinged roller at the right end. The radius of circular cross-section r = 0.05 m. An axial force P and a couple C act at the right end. Assume the following numerical values: L = 1 m, E — 80GPa, P= 15,000N, C=
25. A frame is clamped on the left side and inclined roller supported on the right side, as shown in the figure. A uniformly distributed force q is applied and the roller contact surface is assumed to be non-frictional. When one frame finite element is used to approximate the structure, the nodal
26. A circular ring of square cross-section is subjected to a pair of forces F = 10,000 N, as show in the figure. Use a finite element analysis program to determine the compression of the ring i.e., relative displacements of the points where forces are applied. Assume E = 70GPa. Detet mine the
27. The ring in Problem 26 can be solved using a smaller model considering the symmetry. Use til% model to determine the deflection and maximum force resultants. What are the appropriat boundary conditions for this model? Show that both models yield the same results
28. The frame shown in the figure is subjected to some forces at Nodes 2 and 3. The resultin displacements are given in the table below. Sketch the axial force, shear force, and bendin moment diagrams for Element 3. What are the support reactions at Node 4? Assume EI =1000 N-m2 and EA = 107 m2.
29. Solve the following frame structure using a finite element program in the Appendix. The frame is under a uniformly distributed load of q = 1000 N/m and has a circular cross-section with radius r = 0.1 m. For material property, Young's modulus E = 207 GPa. Plot the deformed geometry with an
30. A cantilevered frame (Element 1) and a uniaxial bar (Element 2) are joined at Node 2 using a bolted joint as shown in the figure. Assume there is no friction at the joint. The temperature of Element 2 is raised by 200°C above the reference temperature. Both of the elements have the same
31. The figure shown below depicts a load cell made of aluminum. The ring and the stem both have square cross-section: 0.1 x 0.1 m2. Assume the Young's modulus is 72 GPa. The mean radius of the ring is 0.05 m. In a load cell, the axial load is measured from the average of strains at points P,
32. In Problem 31, assume that load is applied eccentrically; i.e., the distance between the line of action of the applied force and the center line of load cell,e, is not equal to zero. Calculate the strain at P, Q, R, and S for e = 0.002 m. What is the average of these strains? Comment on the
1. Consider heat conduction in a uniaxial rod surrounded by a fluid. The left end of the rod is at Tq. g)The free stream temperature is 7°°. There is convective heat transfer across the surface of the rod as well over the right end. The governing equation and boundary conditions are as
2. Consider a heat conduction problem described in the figure. Inside the bar, heat is generalfrom a uniform heat source Qg = 10W/m3, and the thermal conductivity of the material k = 0.1 W/m/0C. The cross-sectional area A = 1 m2. When the temperatures at both ends a fixed to 0oC, calculate the
3. Repeat Problem 2 with Qg = 20x.
4. Determine the temperature distribution (nodal temperatures) of the bar shown in the figu using two equal-length finite elements with cross-sectional area of 1 m2. The thermal condu tivity is 10 W/m°C, The left side is maintained at 300oC. The right side is subjected to heat k by convection with

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