Statics and Mechanics of Materials
2nd Edition
ISBN: 9780073398167
Author: Ferdinand P. Beer, E. Russell Johnston Jr., John T. DeWolf, David Mazurek
Publisher: McGraw-Hill Education
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Textbook Question
Chapter 2.4, Problem 57P
Determine (a) the x, y, and z components of the 750-N force, (b) the angles
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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 2 Solutions
Statics and Mechanics of Materials
Ch. 2.1 - 2.1 and 2.2 Determine graphically the magnitude...Ch. 2.1 - 2.1 and 2.2 Determine graphically the magnitude...Ch. 2.1 - Two structural members B and C are bolted to...Ch. 2.1 - Two structural members B and C are bolted to...Ch. 2.1 - The 300-lb force is to be resolved into components...Ch. 2.1 - The 300-lb force is to be resolved into components...Ch. 2.1 - A trolley that moves along a horizontal beam is...Ch. 2.1 - A disabled automobile is pulled by means of two...Ch. 2.1 - Two forces are applied as shown to a hook support....Ch. 2.1 - A disabled automobile is pulled by means of two...
Ch. 2.1 - A trolley that moves along a horizontal beam is...Ch. 2.1 - For the hook support shown, determine by...Ch. 2.1 - The cable stays AB and AD help support pole AC....Ch. 2.1 - Solve Prob. 2.4 by trigonometry.Ch. 2.1 - For the hook support of Prob. 2.9, determine by...Ch. 2.2 - 2.16 and 2.17 Determine the x and y components of...Ch. 2.2 - 2.16 and 2.17 Determine the x and y components of...Ch. 2.2 - 2.18 and 2.10 Determine the x and y components of...Ch. 2.2 - 2.18 and 2.19 Determine the x and y components of...Ch. 2.2 - Member BD exerts on member ABC a force P directed...Ch. 2.2 - Member BC exerts on member AC a force P directed...Ch. 2.2 - Cable AC exerts on beam AB a force P directed...Ch. 2.2 - The hydraulic cylinder BD exerts on member ABC a...Ch. 2.2 - Prob. 24PCh. 2.2 - Determine the resultant of the three forces of...Ch. 2.2 - Determine the resultant of the three forces of...Ch. 2.2 - Determine the resultant of the three forces of...Ch. 2.2 - For the collar loaded as shown, determine (a) the...Ch. 2.2 - Prob. 29PCh. 2.2 - A hoist trolley is subjected to the three forces...Ch. 2.2 - For the post loaded as shown, determine (a) the...Ch. 2.3 - Two cables are tied together at C and are loaded...Ch. 2.3 - 2.33 and 2.34 Two cables are tied together at C...Ch. 2.3 - Prob. 34PCh. 2.3 - Prob. 35PCh. 2.3 - Prob. 36PCh. 2.3 - Two forces of magnitude TA=8 kips and TB=15 kips...Ch. 2.3 - Prob. 38PCh. 2.3 - Prob. 39PCh. 2.3 - Two forces P and Q are applied as shown to an...Ch. 2.3 - Prob. 41PCh. 2.3 - A sailor is being rescued using a boatswains chair...Ch. 2.3 - For the cables of Prob. 2.32, find the value of a...Ch. 2.3 - Prob. 44PCh. 2.3 - Prob. 45PCh. 2.3 - Prob. 46PCh. 2.3 - Two cables tied together at C are loaded as shown....Ch. 2.3 - Collar A is connected as shown to a 50-1b load and...Ch. 2.3 - Cogar A is connected as shown to a 50-lb load and...Ch. 2.3 - A movable bin and its contents have a combined...Ch. 2.3 - A 600 lb crate is supported by several...Ch. 2.3 - Prob. 52PCh. 2.3 - A 200-kg crate is to be supported by the...Ch. 2.3 - Prob. 54PCh. 2.3 - Prob. 55PCh. 2.4 - Determine (a) the x, y, and z components of the...Ch. 2.4 - Determine (a) the x, y, and z components of the...Ch. 2.4 - The end of the coaxial cable AE is attached to the...Ch. 2.4 - The end of the coaxial cable AE is attached to the...Ch. 2.4 - A gun is aimed at a point A located 35 east of...Ch. 2.4 - Prob. 61PCh. 2.4 - Determine the magnitude and direction of the force...Ch. 2.4 - Prob. 63PCh. 2.4 - Prob. 64PCh. 2.4 - Prob. 65PCh. 2.4 - Prob. 66PCh. 2.4 - Prob. 67PCh. 2.4 - Prob. 68PCh. 2.4 - Prob. 69PCh. 2.4 - In order to move a wrecked truck, two cables are...Ch. 2.4 - In order to move a wrecked truck, two cables are...Ch. 2.4 - Prob. 72PCh. 2.4 - Prob. 73PCh. 2.4 - Knowing that the tension is 425 lb in cable AB and...Ch. 2.4 - Knowing that the tension is 510 lb in cable AB and...Ch. 2.4 - A frame ABC is supported in part by cable DBE that...Ch. 2.4 - For the plate of Prob. 2.68, determine the...Ch. 2.4 - The boom OA carries a load P and is supported by...Ch. 2.4 - For the boom and loading of Prob. 2.78, determine...Ch. 2.5 - A container is supported by three cables that are...Ch. 2.5 - A container is supported by three cables that are...Ch. 2.5 - A crate is supported by three cables as shown....Ch. 2.5 - A crate is supported by three cables as shown....Ch. 2.5 - A crate is supported by three cables as shown....Ch. 2.5 - A 1600-lb crate is supported by three cables as...Ch. 2.5 - Three wires are connected at point D, which is...Ch. 2.5 - Prob. 87PCh. 2.5 - A rectangular plate is supported by three cables...Ch. 2.5 - A rectangular plate is supported by three cables...Ch. 2.5 - Prob. 90PCh. 2.5 - Solve Prob. 2.90, assuming that a fiend is helping...Ch. 2.5 - Prob. 92PCh. 2.5 - Prob. 93PCh. 2.5 - Prob. 94PCh. 2.5 - Prob. 95PCh. 2.5 - Prob. 96PCh. 2.5 - Prob. 97PCh. 2.5 - Prob. 98PCh. 2.5 - Prob. 99PCh. 2.5 - Prob. 100PCh. 2.5 - Prob. 101PCh. 2.5 - Prob. 102PCh. 2.5 - Solve Prob. 2.102 assuming that y=275mm.Ch. 2 - Two structural members A and B are bolted to a...Ch. 2 - Determine the x and y components of each of the...Ch. 2 - The hydraulic cylinder BC exerts on member AB a...Ch. 2 - Prob. 107RPCh. 2 - Knowing that a=20, determine the tension (a) in...Ch. 2 - Prob. 109RPCh. 2 - Prob. 110RPCh. 2 - Prob. 111RPCh. 2 - Prob. 112RPCh. 2 - Prob. 113RPCh. 2 - A transmission tower is held by three guy wires...Ch. 2 - Prob. 115RP
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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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