Applied Statics and Strength of Materials (6th Edition)
6th Edition
ISBN: 9780133840728
Author: Limbrunner
Publisher: PEARSON
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Chapter 8, Problem 8.18P
Compute the radii of gyration with respect to the X-X and Y-Y centroidal axes for the built-up steel member of Problem 8.12 /.
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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.
An 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.
Chapter 8 Solutions
Applied Statics and Strength of Materials (6th Edition)
Ch. 8 - Calculate the moment of intertia with respect to...Ch. 8 - Calculate the moment of inertia of the triangular...Ch. 8 - A structural steel wide-flange section is...Ch. 8 - The concrete block shown has wall thicknesses of...Ch. 8 - A rectangle has a base of 6 in. and a height of 12...Ch. 8 - For the area of Problem 8.5 , calculate the exact...Ch. 8 - Check the tabulated moment of inertia for a 300610...Ch. 8 - For the cross section of Problem 8.3 , calculate...Ch. 8 - Calculate the moments of intertia with respect to...Ch. 8 - Calculate the moments of intertia with respect to...
Ch. 8 - The rectangular area shown has a square hole cut...Ch. 8 - For the built-up structural steel member shown,...Ch. 8 - Calculate the moments of inertia about both...Ch. 8 - Calculate the moment of inertia with respect to...Ch. 8 - For the two channels shown, calculated the spacing...Ch. 8 - Compute the radii of gyration about both...Ch. 8 - Two C1015.3 channels area welded together at their...Ch. 8 - Compute the radii of gyration with respect to the...Ch. 8 - Compute the radii of gyration with respect to the...Ch. 8 - Compute the radii of gyration with respect to the...Ch. 8 - Compute the radii of gyration with respect to the...Ch. 8 - Calculate the polar moment of inertia for a...Ch. 8 - Calculate the polar moment of inertia for a...Ch. 8 - For the areas (a) aid (b) of Problem 8.9 ,...Ch. 8 - For the following computer problems, any...Ch. 8 - For the following computer problems, any...Ch. 8 - For the following computer problems, any...Ch. 8 - For the cross section shown, calculate the moments...Ch. 8 - Calculate the moments of inertia of the area shown...Ch. 8 - For the cross-sectional areas shown, calculate the...Ch. 8 - For the cross-sectional areas shown, calculate the...Ch. 8 - Calculate the moments of intertia of the built-up...Ch. 8 - Calculate the moments of inertia about both...Ch. 8 - Calculate lx and ly of the built-up steel members...Ch. 8 - Calculate the least radius of gyration for the...Ch. 8 - A structural steel built-up section is fabricated...Ch. 8 - Calculate the polar moment of inertia for the...Ch. 8 - Determine the polar moment of inertia for the...Ch. 8 - Compute the radii of gyration with respect to the...Ch. 8 - Calculate the polar moment of inertia about the...Ch. 8 - The area of the welded member shown is composed of...
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- Problem 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_forwardProblem 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_forward
- I need part all parts please in detail (including f)arrow_forwardProblem 3 (10 pts, suggested time 5 mins). In class we considered the mutiphysics problem of thermal stresses in a rod. When using linear shape functions, we found that the stress in the rod is affected by unphysical oscillations like in the following plot E*(ux-a*T) 35000 30000 25000 20000 15000 10000 5000 -5000 -10000 0 Line Graph: E*(ux-a*T) MULT 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Arc length (a) [10pts] What is the origin of this issue and how can we fix it?arrow_forwardanswer the questions and explain all of it in words. Ignore where it says screencast and in class explanationarrow_forward
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