Statics and Mechanics of Materials (5th Edition)
5th Edition
ISBN: 9780134382593
Author: Russell C. Hibbeler
Publisher: PEARSON
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Textbook Question
Chapter 5.5, Problem 3PP
In each ease, identify any two-force members, and then draw the free-body diagrams of each member of the frame.
Prob. P5-3
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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 5 Solutions
Statics and Mechanics of Materials (5th Edition)
Ch. 5.3 - In each ease, calculate the support reactions and...Ch. 5.3 - Identify the zero-force members in each truss....Ch. 5.3 - Determine the force in each member of the truss...Ch. 5.3 - Determine the force in each member of the truss...Ch. 5.3 - Determine the force in each member of the truss...Ch. 5.3 - Determine the greatest load P that can be applied...Ch. 5.3 - Identify the zero-force members in the truss....Ch. 5.3 - Determine the force in each member of the truss...Ch. 5.3 - Determine the force in each member of the truss...Ch. 5.3 - Determine the force in each member of the truss...
Ch. 5.3 - Determine the force in each member of the truss...Ch. 5.3 - Determine the force in each member of the truss...Ch. 5.3 - Determine the force in each member of the truss,...Ch. 5.3 - Determine the force in each member of the truss,...Ch. 5.3 - Determine the force in each member of the truss...Ch. 5.3 - Determine the force in each member of the truss in...Ch. 5.3 - Members AB and BC can each support a maximum...Ch. 5.3 - Members AB and BC can each support a maximum...Ch. 5.3 - Determine the force in each member of the truss...Ch. 5.3 - If the maximum force that any member can support...Ch. 5.3 - Determine the force in each member of the truss...Ch. 5.3 - Determine the force in each member of the truss...Ch. 5.3 - Determine the force in each member of the truss...Ch. 5.3 - Determine the force in each member of the truss...Ch. 5.4 - Determine the force in members BC, CF, and FE and...Ch. 5.4 - Determine the force in members LK, KC, and CD of...Ch. 5.4 - Determine the force in members KJ, KD, and CD of...Ch. 5.4 - Determine the force in members EF, CF, and BC of...Ch. 5.4 - Determine the force in members GF, GD, and CD of...Ch. 5.4 - Determine the force in members DC, HI, and JI of...Ch. 5.4 - Determine the force in members DC, HC and HI of...Ch. 5.4 - Determine the force in members ED, EH, and GH of...Ch. 5.4 - Determine the force in members HG, HE, and DE of...Ch. 5.4 - Determine the force in members CD, HI, and CH of...Ch. 5.4 - Determine the force in members CD, CJ, KJ, and DJ...Ch. 5.4 - Prob. 22PCh. 5.4 - The Howe truss is subjected to the loading shown....Ch. 5.4 - The Howe truss is subjected to the loading shown....Ch. 5.4 - Determine the force in members EF, CF, and BC, and...Ch. 5.4 - Determine the force in members AF, BF, and BC, and...Ch. 5.4 - Prob. 27PCh. 5.4 - Determine the force in members BC, BE, and EF of...Ch. 5.4 - Prob. 29PCh. 5.4 - Determine the force in members CD, CF, and CG and...Ch. 5.4 - Determine the force developed in members FE, EB,...Ch. 5.5 - In each ease, identify any two-force members, and...Ch. 5.5 - F5-13. Determine the force P needed to hold the...Ch. 5.5 - Determine the horizontal and vertical components...Ch. 5.5 - If a 100-N force is applied to the handles of the...Ch. 5.5 - Determine the horizontal and vertical components...Ch. 5.5 - Determine the force P required to hold the 100-lb...Ch. 5.5 - In each case, determine the force P required to...Ch. 5.5 - Determine the force P required to hold the 50-kg...Ch. 5.5 - Determine the force P required to hold the 150-kg...Ch. 5.5 - Determine the reactions at the supports A, C, and...Ch. 5.5 - Determine the resultant force at pins A, B, and C...Ch. 5.5 - Determine the reactions at the supports at A, E,...Ch. 5.5 - The wall crane supports a load of 700 lb....Ch. 5.5 - The wall crane supports a load of 700 lb....Ch. 5.5 - Determine the horizontal and vertical components...Ch. 5.5 - Determine the force in members FD and DB of the...Ch. 5.5 - Determine the force that the smooth 20-kg cylinder...Ch. 5.5 - The three power lines exert the forces shown on...Ch. 5.5 - The pumping unit is used to recover oil. When the...Ch. 5.5 - Determine the force that the jaws J of the metal...Ch. 5.5 - Prob. 47PCh. 5.5 - Prob. 48PCh. 5.5 - Prob. 49PCh. 5.5 - Determine the force created in the hydraulic...Ch. 5.5 - The hydraulic crane is used to lift the 1400-lb...Ch. 5.5 - Determine force P on the cable if the spring is...Ch. 5.5 - Prob. 53PCh. 5.5 - Prob. 54PCh. 5.5 - Prob. 55PCh. 5.5 - Determine the force P on the cable if the spring...Ch. 5.5 - Prob. 57PCh. 5.5 - Prob. 58PCh. 5.5 - Prob. 59PCh. 5.5 - Prob. 60PCh. 5.5 - The platform scale consists of a combination of...Ch. 5 - All the problems solutions must include FBDs....Ch. 5 - Determine the force in each member of the truss...Ch. 5 - Determine the force in member GJ and GC of the...Ch. 5 - Determine the force in members GF, FB, and BC of...Ch. 5 - Prob. 5RPCh. 5 - Determine the horizontal and vertical components...Ch. 5 - Prob. 7RPCh. 5 - Determine the resultant forces at pins B and C on...
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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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