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Q1: 2 (15%) Consider a tapered bar of circular cross section. The length of the bar is 1m, and the radius varies as r(x)= 0.05–0.04x, where r and x are in meters. Assume Young's modulus E = 100 MPa. Both ends of the bar are fixed and F = 10,000 N is applied at the center. Determine the displacements, axial force distribution and the wall reactions using four elements of equal length. See Answer
Q2: The vertical column shown in the picture has both endsfixed. The elastic modulus E is uniform. Its cross-sectional area A(x) is uniform in the upper part and varieslinearly in the lower part. It carries its own weight, so thatthe load is proportional to the cross-sectional area: See Answer
Q3: 5 (20%) A traction Tx = 10 kN/m is applied at the top of the plane problem shown, with the bottom side is fixed and the top side is only allowed to move horizontally. E = 10GPAand v = 0.25. Use one plane element to calculate the stress and strain components for the case of (a) thickness t = 0.1 m and (b) thickness t = 10 m. See Answer
Q4:Q2. (10 points) The nodal coordinates of the triangular element are shown below. At the interior point P, the Y coordinate is 2.7 and N₂ = 0.4. Determine N₁, N3 at point P. Then determine the X coordinate at point P. Nodal coordinates are from nodes 1 to 3: (5,1) (2,6) (-2,-2)See Answer
Q5:6.13-1 Assume that the patch test is passed for the mesh in Fig. 6.13-1a. If all nine nodes : are assigned displacements consistent with a field of constant strain, what loads at the internal node should result from the calculation [K]{D], and why?See Answer
Q6:Consider a 4 node rectangular finite element given earlier in the class notes with x dimension a=3 cm and y dimension b-4 cm, and a thickness of 1 cm. For steel which has a mass density of 8 g/cm³, find the first row of the consistent mass matrix (8 terms) for the element and the first diagonal entry of the lumped mass matrix using the row-sum technique.See Answer
Q7:1. Find the analytical solution for the following tapered beam under an axial load, solving for total deflection 8. The initial diameter di = 0.75 in, the final diameter df = 0.125 in, the length L = 12 in, Young's Modulus E = 29(10)6 psi and the force F = 178 kips. Derive using real numbers and procedure in class. *Can use calculator for definite integral step. 2. Use numerical methods to solve for total deflection with 1, 2, 3 and 4 elements. Do calculations by hand and show all work! May treat as springs in series. Graph deflection vs. number of elements to show convergence on analytical solution. d. 14 F LL 7See Answer
Q8:For the beam below with fixed-fixed boundary conditions, do the following calculations by hand. Let k₁ = 480 #/in, k₂ = 310 #/in, k3 = 70 #/in, k4 = 90 #/in, k5 = 10 #/in, F₂ = 200 # and F3 = 500 #. 1. Solve for the displacements at nodes 1 through 4 (U₁, U₂, U3, and 4). 2. Solve for the reaction forces at both ends (R₁ and R4). R 1 F 2 TR ка www 42 3 K3 k 3 4. 4 3 mm K5 www 43 R 4See Answer
Q9:For the tapered aluminum beam below experiencing pure conduction heat transfer with k = 221 W/mK, d₁ = 30 cm, d₁0= 12 cm, L = 9 m, T₁ = 70°C, and T₁0 = 10°C. The beam is solid round tapered with diameter as a function of position d(x). There are 10 nodes on the beam equally spaced. Note: Cannot factor out A for global thermal matrix since it is constantly changing. Use average area for element (A₁ +Ai+1/2). 1. Calculate the temperature for nodes 1-10 [°C]. 2. Plot T vs. x. 3. Calculate the total heat transfer rate q [W]. Calculations: Matlab or Excel will be accepted but you must show all work and describe what the code does. If working in groups you still need to show that you did the work on your own! 0 T. 10 -ød(x) +See Answer
Q10:Please show all work Consider the continuously tapered beam under an axial load shown below. The initial diameter d₁ = 0.75 in, the final diameter df = 0.125 in, the length L = 12 in, Young's Modulus E = 29(10)6 psi and the force F = 178 kips. a. Use minimum total potential energy formulation to solve for total deflection & using 4 equally divided elements. b. Compare solution to analytical result. Note: Can use the derived formula from class. 1 d FSee Answer
Q11:Please show all work Consider the continuously tapered beam under an axial load shown below. The initial diameter d₁ = 0.75 in, the final diameter df = 0.125 in, the length L = 12 in, Young's Modulus E = 29(10)6 psi and the force F = 178 kips. a. Use minimum total potential energy formulation to solve for total deflection 8 using 4 equally divided elements. b. Compare solution to analytical result. Note: Can use the derived formula from class. To LSee Answer
Q12:For the tapered aluminum beam below experiencing pure conduction heat transfer with k = 221 W/mK, d₁ = 30 cm, d₁0= 12 cm, L = 9 m, T₁ = 70°C, and T₁0 = 10°C. The beam is solid round tapered with diameter as a function of position d(x). There are 10 nodes on the beam equally spaced. Note: Cannot factor out A for global thermal matrix since it is constantly changing. Use average area for element (A; +Ai+1/2). 1. Calculate the temperature for nodes 1-10 [°C]. 2. Plot T vs. x. 3. Calculate the total heat transfer rate q [W]. Calculations: Matlab or Excel will be accepted but you must show all work and describe what the code does. If working in groups you still need to show that you did the work on your own! 9 0 10 d; 10 -ød(x) +See Answer
Q13:Please show all work Consider the continuously tapered beam under an axial load shown below. The initial diameter d₁ = 0.75 in, the final diameter df = 0.125 in, the length L = 12 in, Young's Modulus E = 29(10)6 psi and the force F = 178 kips. a. Use minimum total potential energy formulation to solve for total deflection 8 using 4 equally divided elements. b. Compare solution to analytical result. Note: Can use the derived formula from class. To LSee Answer
Q14:Perform finite element analysis using SolidWorks Simulation on the attached file: Anchor Bracket 2-3.SLDPRT. Show the Von Mises stress as demonstrated in class. Note: split lines for hole are not necessary 30* the nowe pr right). Assume the following. • Material: • Mesh: . Fixture: . inum Fo Apply a Fixed (immovable) restraint on the inclined surface. External Load: 8600 N in the X-direction applied on the right, inside surface of the 16 mm diameter hole between user defined Split Lines. Determine the following: a. Use classical equations to compute stress at the inside (concave) surface and the outside (convex) surface of the anchor bracket at section B-B. Section B-B passes through the center of curvature of the curved beam and is considered to be a vertical line. Include a labeled free body diagram of the portion of the anchor bracket to the right of section B-B (show magnitude and direction of all reactions). b. Include a zoomed-in image of the hole so that the force Fx = 8600 N can clearly be seen to act between user specified Split Lines. c. Create a stress contour plot of von Mises stress in the anchor bracket. Include automatic labeling of maximum and minimum stress on this plot. 60 AISI 1010 Steel, hot rolled bar (use SI units) B In the Mesh property manager, select O Standard mesh; use the default mesh size. R40 I B 20 7 R100 30 पर 20 150 016 Surface is attached to a rigid frame (fixed).See Answer
Q15:Perform finite element analysis using SolidWorks Simulation on the attached file: Plate With Hole 3-1.SLDPRT Show the Von Mises stress as demonstrated in class. will be assigned or m Designates problems that introduce ne EXERCISE 1-Effect of Mesh Size at Hole Location supported (Fixed/immovable) at its left-end and subject to an axial, tensile force of 370 A rectangular bar with a centrally drilled hole is illustrated in Fig. E3-1. The bar is kN applied normal to its opposite end. The bar is made from 2018 aluminum alloy. Open the file: Plate With Hole 3-1. 210 • Material: Mesh: 210 Figure E3-1-Aluminum bar with central hole subject to an axial force. A geometric discontinuity is present in the form of the 40 mm diameter hole. (All dimensions in mm.) TTT 2018 Alloy aluminum (Use S.I. units) In the Mesh property manager, select a Ⓒ Standard mesh. High quality tetrahedral elements are to be used. Use three different meshes as specified in parts (a, c, and d) below. • Fixture: Fixed (Immovable) restraint applied to left end of the model. External Load: 370 kN applied normal to the right-end causing tension in the bar.See Answer
Q16:Problem 2: FEA Truss (20 Points) Consider the truss structure shown below. All truss members are considered linear elastic, with a rod diameter of 40 mm. The material is steel (E = 210 GPa). Place the origin of coordinate system at Node 1 with x to the right and y to the top. The structure is loaded with forces P1 = 20 kN at Node 7 and P2 = 10 kN at Node 5. Follow the same numbering scheme used for nodes and elements as show in the figure. The nodal coordinate positions in meters: b) c) d) e) 3 f) 10 KN 11 (10) 20 KN 60⁰ Node 1 2 3 4 5 6 7 XOONNTT6 X 2 2 TO Y 4.5 1.5 4.5 3 4.5 a) What is the total degree of freedom of the system? What is the size of the global stiffness matrix and force vector? 44 4.5 Determine the stiffness matrix for element 10. Assemble the stiffness matrix of element 10 into the global stiffness matrix (i.e., which rows and columns do these components go?) What are the known values of the global displacement vector d and force vector f? Explain the procedure to obtain reduced stiffness matrix. What is the size of the reduced stiffness matrix? Briefly describe the steps to solve the system once you have reduced the stiffness matrix. What are the unknowns that we solve for in FEA? How do we solve for the reaction forces?See Answer
Q17:Problem 3: FEA - BEAM (20 Points) Consider the beam fixed at both ends as shown in the diagram below. It is modeled using three elements with lengths L₁, L2, L3 and moment of inertia I1, I2, I3 respectively. The elastic modulus is same for all the beams. There is a distributed load Q on element 1 and a point load P acting at the center of element 2. Element 1 Element 2 Element 3 a) Draw a schematic diagram of the 3 elements with the loads and moments distributed along the nodes. Indicate the degree of freedom at each node. b) What is the total degree of freedom of the system? What is the size of the global stiffness matrix and force vector? c) Write down the elemental stiffness matrix for element 3. d) Assemble the stiffness matrix of elements 3 into the global stiffness matrix (i.e., which rows and columns do these components go?) e) What are the known values of the global displacement vector d and force vector f? f) Explain the procedure to obtain reduced stiffness matrix from global stiffness matrix for the given structure. What is the size of the reduced stiffness matrix? g) Write down the steps to perform a finite element structural analysis in Solidworks. Assume that you are given the CAD model.See Answer
Q18:Problem 4: FEA - BEAM (20 Points) Consider the beam fixed at one end as shown in the diagram below. It is divided into 2 elements of equal lengths. The moment of inertia of the beam is I, elastic modulus is E and the mass of the beam is M. The total length of the beam is L. There is a distributed load A (N/m) over element 2 and a moment B (N-m) at the free end. L/2 L/2 a) Draw a schematic diagram of the 2 elements with the loads and moments distributed along the nodes. Indicate the degree of freedom at each node. b) Write down the elemental stiffness matrix for each element. c) Assemble the element stiffness matrix into a global stiffness matrix. d) Write down the displacement vector and force vector. Substitute the relevant boundary conditions and loads. e) Find out the reduced stiffness matrix and reduced force vector. f) Consider the elemental mass matrix given below where m and I are the mass and length of the element respectively. B [1 0 0 m0 1²/12 0 1 20 0 Lo 0 0 1²/12] Assemble the global mass matrix using the elemental mass matrix [M] given above. g) Set up the eigenvalue problem to solve for the eigenvalues of the beam. [M] = 0 0 0See Answer
Q19:Problem 5: FEA Concepts (20 Points) Nodal Interpolation Consider a simple 3 node 2D triangular element with 2 DOFs for each node. The nodal displacements are U₁ = V₁ = V₂ = 0; U₂ = 4 mm, U3 = 6 mm, and V3 = 2 mm. E The shape functions are: |N₁ = UI N₂ N₂= V3 P (5,5) U3 U2 _a₁ +b₂x+c₁y 2A a₂ + b₂x+c₂y 24 a₂+bx+c₂y 24 where A is the area of triangle. Calculate the displacement at point P (5, 5). Node 1 2 3 a₁ = X₂Y₁X₂ Y₂ a₂ = X₂Y₁X₁Y₁ az = X₁ Y₂ - X₂Y₁ X Y 0 0 10 0 5 10 b₁ = y₁=Y₁ b₂ = y₁=Y₁ b₁ = y₁ - y₂ C₁ = X3 - X₂ C₂ = X₁ − X3 C₂ = x₂ −X₁See Answer
Q20:5. Practice Exercises 5.1. FSM Design and implement the following finite state machine with the rising edge of the clock with a 1-bit input that the next state depends on. The FSM does not have a reset button, and only when at state S3 and the input has the value of zero, your system will go back to S0. 0 5.2. 32-bit Adder start 1 S2 0 1 Figure 3. FSM diagram for task 5.1 1. Complete VHDL code. 2. Flow Summary and Successful Compilation. 3. RTL Viewer, 4. Power Consumption. 5. Maximum Frequency Analysis. 6. ModelSim simulation for the designs (except 32-bit Adder). S1 S4 Design and implement a 32-bit adder using Generate of smaller size bit adders. Analyze and evaluate your design. 6. Deliverables Practical demonstration of the sample exercises will be requested during the lab and instructor may ask for their code to be submitted on Moodle. Complete work of the practice exercises must be submitted on Moodle by the specified deadline. Provide the following for all designs:See Answer

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