Foundations of Materials Science and Engineering
6th Edition
ISBN: 9781259696558
Author: SMITH
Publisher: MCG
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Chapter 15.10, Problem 22KCP
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Chapter 15 Solutions
Foundations of Materials Science and Engineering
Ch. 15.10 - Write the equation relating the energy of...Ch. 15.10 - Prob. 2KCPCh. 15.10 - Prob. 3KCPCh. 15.10 - Prob. 4KCPCh. 15.10 - Prob. 5KCPCh. 15.10 - Explain why metals absorb and/or reflect incident...Ch. 15.10 - Prob. 7KCPCh. 15.10 - Prob. 8KCPCh. 15.10 - Prob. 9KCPCh. 15.10 - Prob. 10KCP
Ch. 15.10 - Prob. 11KCPCh. 15.10 - Prob. 12KCPCh. 15.10 - Prob. 13KCPCh. 15.10 - Prob. 14KCPCh. 15.10 - Prob. 15KCPCh. 15.10 - Prob. 16KCPCh. 15.10 - What are the basic elements of an optical-fiber...Ch. 15.10 - Prob. 18KCPCh. 15.10 - Explain how optical fibers act as waveguides.Ch. 15.10 - Prob. 20KCPCh. 15.10 - Prob. 21KCPCh. 15.10 - Prob. 22KCPCh. 15.10 - Prob. 23KCPCh. 15.10 - Prob. 24KCPCh. 15.10 - Prob. 25KCPCh. 15.10 - Why are type I superconductors poor...Ch. 15.10 - Prob. 27KCPCh. 15.10 - Prob. 28KCPCh. 15.10 - Prob. 29KCPCh. 15.10 - Prob. 30KCPCh. 15.10 - Prob. 31KCPCh. 15.10 - Prob. 32AAPCh. 15.10 - A semiconductor emits green visible radiation at a...Ch. 15.10 - Prob. 34AAPCh. 15.10 - Calculate the reflectivity of ordinary light from...Ch. 15.10 - Prob. 36AAPCh. 15.10 - Prob. 37AAPCh. 15.10 - Prob. 38AAPCh. 15.10 - Prob. 39AAPCh. 15.10 - Prob. 40AAPCh. 15.10 - Prob. 41AAPCh. 15.10 - Prob. 42AAPCh. 15.10 - Prob. 43AAPCh. 15.10 - Prob. 44AAPCh. 15.10 - Prob. 45AAPCh. 15.10 - Prob. 46AAPCh. 15.10 - Prob. 47AAPCh. 15.10 - Prob. 48AAPCh. 15.10 - Prob. 49AAPCh. 15.10 - Prob. 50AAPCh. 15.10 - Prob. 51AAPCh. 15.10 - Prob. 52SEPCh. 15.10 - Prob. 53SEPCh. 15.10 - Prob. 54SEPCh. 15.10 - Prob. 55SEPCh. 15.10 - Prob. 56SEP
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- Problem 2 (55 pts). We now consider the FEM solution of Problem 1.(a) [5pts] Briefly describe the 4 steps necessary to obtain the approximate solution of thatBVP using the Galerkin FEM. Use the minimum amount of math necessary to supportyour explanations.(b) [20pts] Derive the weak form of the BVP.(c) [10pts] Assuming a mesh of two equal elements and linear shape functions, sketch byhand how you expect the FEM solution to look like. Also sketch the analytical solutionfor comparison. In your sketch, identify the nodal degrees of freedom that the FEMsolution seeks to find.(d) [10pts] By analogy with the elastic rod problem and heat conduction problem considered in class, write down the stiffness matrix and force vector for each of the twoelements considered in (c).(e) [10pts] Assemble the global system of equations, and verbally explain how to solve it.arrow_forwardAn aluminum rod of length L = 1m has mass density ρ = 2700 kgm3 andYoung’s modulus E = 70GPa. The rod is fixed at both ends. The exactnatural eigenfrequencies of the rod are ωexactn =πnLqEρfor n=1,2,3,. . . .1. What is the minimum number of linear elements necessary todetermine the fundamental frequency ω1 of the system? Discretizethe rod in that many elements of equal length, assemble the globalsystem of equations KU = ω2MU, and find the fundamentalfrequency ω1. Compute the relative error e1 = (ω1 − ωexact1)/ωexact1.Sketch the fundamental mode of vibration.arrow_forwardProblem 1 (65 pts, suggested time 50 mins). An elastic string of constant line tension1T is pinned at x = 0 and x = L. A constant distributed vertical force per unit length p(with units N/m) is applied to the string. Under this force, the string deflects by an amountv(x) from its undeformed (horizontal) state, as shown in the figure below.The PDE describing mechanical equilibrium for the string isddx Tdvdx− p = 0 . (1)(a) [5pts] Identify the BCs for the string and identify their type (essential/natural). Writedown the strong-form BVP for the string, including PDE and BCs.(b) [10pts] Find the analytical solution of the BVP in (a). Compute the exact deflectionof the midpoint v(L/2).(c) [15pts] Derive the weak-form BVP.(d) [5pts] What is the minimum number of linear elements necessary to compute the deflection of the midpoint?(e) [15pts] Write down the element stiffness matrix and the element force vector for eachelement.arrow_forward
- Problem 1 (35 pts). An elastic string of constant line tension1 T is pinned at x = 0 andx = L. A constant distributed vertical force per unit length p (with units N/m) is appliedto the string. Under this force, the string deflects by an amount v(x) from its undeformed(horizontal) state, as shown in the figure below.Force equilibrium in the string requires thatdfdx − p = 0 , (1)where f(x) is the internal vertical force in the string, which is given byf = Tdvdx . (2)(a) [10pts] Write down the BVP (strong form) that the string deflection v(x) must satisfy.(b) [2pts] What order is the governing PDE in the BVP of (a)?(c) [3pts] Identify the type (essential/natural) of each boundary condition in (a).(d) [20pts] Find the analytical solution of the BVP in (a).arrow_forwardProblem 2 (25 pts, (suggested time 15 mins). An elastic string of line tension T andmass per unit length µ is pinned at x = 0 and x = L. The string is free to vibrate, and itsfirst vibration mode is shown below.In order to find the frequency of the first mode (or fundamental frequency), the string isdiscretized into a certain number of linear elements. The stiffness and mass matrices of thei-th element are, respectivelyESMi =TLi1 −1−1 1 EMMi =Liµ62 11 2 . (2)(a) [5pts] What is the minimum number of linear elements necessary to compute the fundamental frequency of the vibrating string?(b) [20pts] Assemble the global eigenvalue problem and find the fundamental frequency ofvibration of the stringarrow_forwardI need part all parts please in detail (including f)arrow_forward
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