Applied Statics and Strength of Materials (6th Edition)
6th Edition
ISBN: 9780133840544
Author: George F. Limbrunner, Craig D'Allaird, Leonard Spiegel
Publisher: PEARSON
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Chapter 4, Problem 4.2P
Sketch free-body diagram for the members shown.
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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 4 Solutions
Applied Statics and Strength of Materials (6th Edition)
Ch. 4 - and 4.2 Sketch free-body diagram for the members...Ch. 4 - Sketch free-body diagram for the members shown.Ch. 4 - A steel cylinder having a mass of 120 kgis...Ch. 4 - A 50-lb block is supported by a pin support and a...Ch. 4 - A cylinder weighing 200 lb is supported on an...Ch. 4 - A weight W is supported by a flexible cable and an...Ch. 4 - The ladder shown is supported by a smooth...Ch. 4 - What horizontal force F applied at the center of...Ch. 4 - Calculate the force in cable AB and the angle (...Ch. 4 - Calculate the horizontal force F that should be...
Ch. 4 - Calculate the reactions of the two smooth inclined...Ch. 4 - Calculate the force in each cable for the...Ch. 4 - Three members of a truss intersect at joint B as...Ch. 4 - Four concurrent forces in equilibrium act at point...Ch. 4 - The beam shown carries vertical concentrated...Ch. 4 - Find the reactions at A and B for the beam shown....Ch. 4 - A simply supported beam spans 10 m. The beam...Ch. 4 - The beam shown carries vertical loads. Calculate...Ch. 4 - Calculate the reaction at each support for the...Ch. 4 - Calculate the reactions at A and B for the beam...Ch. 4 - Calculate the reactions at A and B for the beam...Ch. 4 - A 12-ft simple beam is supported at each end. It...Ch. 4 - The beam shown carries vertical loads as...Ch. 4 - Determine the reactions for the beam shown. The...Ch. 4 - Calculate the reaction at each support for the...Ch. 4 - Calculate the wall reactions for the cantilever...Ch. 4 - Determine the reactions at supports A and B of the...Ch. 4 - A mass M of 300 kg is supported by a boom, as...Ch. 4 - Rework Problem 4.28 assuming that point D has been...Ch. 4 - Calculate the force in the tie rod BC and the...Ch. 4 - The davit shown is used in pairs for...Ch. 4 - For the following computer problems, any...Ch. 4 - For the following computer problems, any...Ch. 4 - For the following computer problems, any...Ch. 4 - For the structure shown, draw free-body diagram...Ch. 4 - A 1200-lb load is supported by a cable that runs...Ch. 4 - For the pin-connected frame shown, sketch a...Ch. 4 - For the concurrent force system shown, calculate...Ch. 4 - A strut having a mass of 40 kg/m is supported by a...Ch. 4 - Calculate the reaction at each support for the...Ch. 4 - Calculate the reaction at each support for the...Ch. 4 - A beam supports a nonuniformly distributed load as...Ch. 4 - Calculate the reactions at each support for the...Ch. 4 - Compute reactions at each support for the beam...Ch. 4 - A rod of uniform cross section weighs 4 lb/ft and...Ch. 4 - A 12-ft-long weightiness member supports two...Ch. 4 - A uniform rod AB, having a weight of 5.00 lb and a...Ch. 4 - The plastic barrel tent anchor of Problem 2.11...Ch. 4 - Compute the reactions at A and B for the bracket...Ch. 4 - The truss shown is supported by a pin at A and a...Ch. 4 - Find the reactions at supports A and B for the...Ch. 4 - Find the reactions at supports A and B for the...Ch. 4 - Determine the reactions at A and B for the truss...Ch. 4 - A 40-ft ladder weighing 130 lb is pin-connected to...Ch. 4 - The frame shown is pin-connected at point A and...Ch. 4 - Prob. 4.56SPCh. 4 - A horizontal beam is pin-connected to a wall at...Ch. 4 - Calculate the force in the cable for the structure...Ch. 4 - The Thenard shutter dam shown was originally...Ch. 4 - An inclined railway can be used to lift heavy...Ch. 4 - Two cylinders are supported in a box, as shown....
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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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