Questions and Answers of Systems Analysis And Design Using MATLAB

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
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
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
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
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
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
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,
; 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
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),
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
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
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.
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) =
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
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
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
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:
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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,
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
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
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.,
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
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
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
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
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
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
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
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
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
5. The nodal temperatures of heat conduction problem are given in the one-dimensional figu]Calculate the temperature at jc = 0.2 using: (a) two two-node elements and (b) one three-no element.| , 0.5
6. In order to solve one-dimensional steady-state heat transfer problem, one element with thre nodes is used. The shape functions and the conductivity matrix before applying boundary cc ditions are
7. The one-dimensional heat chamber in the figure is modeled using one element with three nodes.There is a uniform heat source inside the wall generating Q = 300 W/m3. The thermal conduc¬tivity of
8. Consider heat conduction in a uniaxial rod surrounded by a fluid. The right end of the rod is attached to a wall and is at temperature Tr. One half of the bar is insulated as indicated. The free
9. A well-mixed fluid is heated by a long iron plate of conductivity k = 12 W/m/0C and thickness t = 0.12m. Heat is generated uniformly in the plate at the rate Qg = 5,000 W/m3. If the surface
10. A cooling fin of square cross-sectional area A = 0.25 x 0.25 m2, length L = 2m, and conduc¬tivity k = 10 W/m/°C extends from a wall maintained at temperature Tw = 100oC. The surface convection
11. Find the heat transfer per unit area through the composite wall in the figure. Assume onedimensional heat flow and there is no heat flow between B and C. The thermal conductivities are kA =
12. Consider a wall built up of concrete and thermal insulation. The outdoor temperature is T0 = — 170C, and the temperature inside is Tj = 20oC. The wall is subdivided into three ele¬ments. The
1. Repeat Example 6.2 with the following element connectivity:Element 1: 1-2-4 Element 2: 2-3^1 Does the different element connectivity change the results?
2. Solve Example 6.2 using one of the finite element programs given in the Appendix.
3. Using two CST elements, solve the simple shear problem depicted in the figure and determ whether the CST elements can represent the simple shear condition accurately. Material proj ties are given
4. Solve Example 6.4 using one of finite element programs in the Appendix.
5. A structure shown in the figure is modeled using one triangular element. Plane strain assui tion is used.(a) Calculate the strain-displacement matrix [B],(b) When nodal displacements are given by
6. Calculate the shape function matrix [N] and strain-displacement matrix [B] of the triangular ancj element shown in the figure.:lethe
7. The coordinate of the nodes and corresponding displacements in a triangular element are given in the table. Calculate the displacement u and v and strains gvv, Eyj,, and at the centroid of the
8. A2mx2mx 1 mm square plate with E = 70 GPa and v = 0.3 is subjected to a uniformly dis¬tributed load as shown in Figure (a). Due to symmetry, it is sufficient to model one quarter of the plate
9. A beam problem under the pure bending moment is solved using CST finite elements, as shown in the figure. Assume E = 200 GPa and v = 0.3. The thickness of the beam is 0.01 m. To simu¬late the
10. For a rectangular element shown in the figure, displacements at four nodes are given by{i
11. Six rectangular elements are used to model the cantilevered beam shown in the figure. Sketch the graph of ffxv along the top surface that a finite element analysis would yield. There is no need
12. A rectangular element as shown in the figure is used to represent a pure bending problem. Due to the bending moment M, the element is deformed as shown in the figure with displacement{q} = {«!,
14. A uniform beam is modeled by two rectangular elements with thicknessb. Qualitatively, and without performing calculations, plot axx and Tly along the top edge from A to C, as predicted by FEA.
15. A beam problem under the pure bending moment is solved using five rectangular finite ele¬ments, as shown in the figure. Assume E = 200GPaand v = 0.3 are used. The thickness of the beam is 0.01
16. The quadrilateral element shown in the figure has the nodal displacement of {u\, vi, 112, V2, U3, V3, 114, V4} = {-1, 0, -1, 0, 0, 1, 0, 1}.(a) Find the (s, t) reference coordinates of point A
18. A quadrilateral element in the figure is mapped into the reference element.(a) A point P has a coordinate (x, y) = (Vi, y) in the physical element and (s, t) = (-Vi, t) in the parent element.
19. Consider the plane stress four-node element shown below. Its global node numbers are shown in the figure. The coordinates of the nodes in the global x-y coordinate systems is shown next to each
20. A linearly varying pressure p is applied along the edge of the four-node element shown in the figure. The finite element method converts the distributed force into an equivalent set of nodal
21. Determine the Jacobian matrix for the following isoparametric elements. If the temperature at the nodes of both elements are {Ti, T2, T3, T4} = {100, 90, 80, 90}, compute the tempera¬ture at the
22. Integrate the following function using one-point and two-point numerical integration (Gauss quadrature). The exact integral is equal to 2. Compare the accuracy of the numerical integration with
23. A six-node finite element as shown in the figure is used for approximating the beam problem.(a) Write the expressions of displacements u(x,y) and v(x,y) in terms of polynomials with un¬known
24. Consider a quadrilateral element shown in the figure below. The nodal temperatures of the ele¬ment are given as {jTi , Ti, Ts, T^} = {80, 40, 40, 80}.(a) Compute the expression of the
1. Determine the height of the beam in Example 8.1 when the load factor of X = 2.0 is used with the failure stress of 40 ksi.
2. Determine the height of the beam in Example 8.1 so that the safety margin is 10 ksi with the failure stress of 40 ksi.
3. A two-dimensional truss shown in the figure is made of aluminum with Young's modulus E =80GPa and failure stress ay = 150 MPa. Determine the minimum cross-sectional area of each member so that the
4. Consider a stepped beam modeled using two beam elements. The cross-sections are circular.Use Young's modulus E = 80GPa, yield stress ay = 250MPa, and L = 1 m. When F2 = F3 =1,000N, calculate the
5. A cantilever beam of length 1 m is subjected to a uniformly distributed load p(x) = p0 =12,000 N-m and a clockwise couple 5,000 N-m at the tip. The load factors for the distributed load and couple
6. The frame shown in the figure is clamped at the left end and supported on a hinged roller at the right end. An axial force P and a couple C act at the right end. The load factor for the axial
7. All members of the truss shown in the figure initially have a circular cross-section with diame¬ter of 2 in. Using one of the finite element analysis programs in Appendix, calculate the mini¬mum
8. Consider a two-bar structure in the figure with Young's modulus E = 100GPa, yield stress ay = 250MPa, and F = 10,000N. Design variables are bi = area of section AB and bi =area of section BC.
9. Repeat Problem 8 with the initial design ofb) = 2 x 10 4 m2 and bi = 1 x 10 4 m2. Compare the results with those of Problem 8.
10. For the clamped beam shown in the figure, two design variables are defined, as /i = bi and h = b2. Using the finite element method and sensitivity analysis, calculate the sensitivity of the
11. Calculate the sensitivity of the vertical displacement my in Problem 10 using forward finite dif- ference method with perturbation size 1%. Compare the results with the exact sensitivity.
12. Consider a simply supported beam of length L = 1 m subjected to a uniformly distributed trans¬verse load p0 = 100 N-m. The cross-section is rectangular with width w = 0.01 m and height h = 0.02

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