Statics and Mechanics of Materials (5th Edition)
5th Edition
ISBN: 9780134382593
Author: Russell C. Hibbeler
Publisher: PEARSON
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Textbook Question
Chapter 2, Problem 6RP
Express F1 and F2 as Cartesian
Prob. R2–6
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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 (5th Edition)
Ch. 2.3 - In each case, construct the parallelogram law to...Ch. 2.3 - In each case, show how to resolve the force F into...Ch. 2.3 - Determine the magnitude of the resultant force...Ch. 2.3 - Determine the magnitude of the resultant force....Ch. 2.3 - Determine the magnitude of the resultant force and...Ch. 2.3 - Resolve the 30-lb force into components along the...Ch. 2.3 - Resolve the force into components acting along...Ch. 2.3 - Prob. 6FPCh. 2.3 - If = 60 and F = 450 N, determine the magnitude of...Ch. 2.3 - If the magnitude of the resultant force is to be...
Ch. 2.3 - Determine the magnitude of the resultant force FR...Ch. 2.3 - Determine the magnitudes of the two components of...Ch. 2.3 - Solve Prob. 24 with F = 350 lb. 24. Determine the...Ch. 2.3 - Determine the magnitude of the resultant force FR...Ch. 2.3 - Resolve the force F1 into components acting along...Ch. 2.3 - Resolve the force F2 into components acting along...Ch. 2.3 - If the resultant force acting on the support is to...Ch. 2.3 - Determine the magnitude of the resultant force and...Ch. 2.3 - If = 60, determine the magnitude of the resultant...Ch. 2.3 - Determine the angle for connecting member A to...Ch. 2.3 - The force acting on the gear tooth is F = 20 lb....Ch. 2.3 - The component of force F acting along line aa is...Ch. 2.3 - Force F acts on the frame such that its component...Ch. 2.3 - Force F acts on the frame such that its component...Ch. 2.3 - If F1 = 30 lb and F2 = 40 lb, determine the angles...Ch. 2.3 - Determine the magnitude and direction of FA so...Ch. 2.3 - Determine the magnitude of the resultant force...Ch. 2.3 - Prob. 20PCh. 2.3 - If the resultant force of the two tugboats is 3...Ch. 2.3 - If FB = 3 kN and = 45, determine the magnitude of...Ch. 2.3 - If the resultant force of the two tugboats is...Ch. 2.4 - Resolve each force into its x and y components....Ch. 2.4 - F28. Determine the magnitude and direction of the...Ch. 2.4 - Prob. 9FPCh. 2.4 - Prob. 10FPCh. 2.4 - Prob. 11FPCh. 2.4 - Determine the magnitude of the resultant force and...Ch. 2.4 - Determine the magnitude of the resultant force and...Ch. 2.4 - Prob. 25PCh. 2.4 - Prob. 26PCh. 2.4 - Determine the magnitude of the resultant force and...Ch. 2.4 - Prob. 28PCh. 2.4 - Determine the magnitude of the resultant force...Ch. 2.4 - Prob. 30PCh. 2.4 - Prob. 31PCh. 2.4 - Prob. 32PCh. 2.4 - Determine the magnitude of the resultant force and...Ch. 2.4 - Prob. 34PCh. 2.4 - Prob. 35PCh. 2.4 - Determine the magnitude of the resultant force and...Ch. 2.4 - Determine the magnitude and direction of the...Ch. 2.6 - Sketch the following forces on the x, y, z...Ch. 2.6 - In each case, establish F as a Cartesian vector,...Ch. 2.6 - Show how to resolve each force into its x, y, z...Ch. 2.6 - Determine the coordinate direction angles of the...Ch. 2.6 - Prob. 14FPCh. 2.6 - Prob. 15FPCh. 2.6 - Prob. 16FPCh. 2.6 - Prob. 17FPCh. 2.6 - Determine the resultant force acting on the hook....Ch. 2.6 - The force F has a magnitude of 80 lb. Determine...Ch. 2.6 - The bolt is subjected to the force F, which has...Ch. 2.6 - Determine the magnitude and coordinate direction...Ch. 2.6 - Prob. 41PCh. 2.6 - Prob. 42PCh. 2.6 - Express each force in Cartesian vector form and...Ch. 2.6 - Prob. 44PCh. 2.6 - Determine the magnitude and coordinate direction...Ch. 2.6 - Determine the magnitude and coordinate direction...Ch. 2.6 - Prob. 47PCh. 2.6 - Determine the magnitude and coordinate direction...Ch. 2.6 - Prob. 49PCh. 2.6 - Prob. 50PCh. 2.6 - Prob. 51PCh. 2.6 - Determine the magnitude and coordinate direction...Ch. 2.6 - Prob. 53PCh. 2.6 - Prob. 54PCh. 2.6 - Determine the magnitude and coordinate direction...Ch. 2.8 - In each case, establish a position vector from...Ch. 2.8 - In each case, express F as a Cartesian vector. (a)...Ch. 2.8 - Prob. 19FPCh. 2.8 - Determine the length of the rod and the position...Ch. 2.8 - Prob. 21FPCh. 2.8 - Express the force as a Cartesian vector. Prob....Ch. 2.8 - Prob. 23FPCh. 2.8 - Prob. 24FPCh. 2.8 - Determine the length of the connecting rod AB by...Ch. 2.8 - Express force F as a Cartesian vector; then...Ch. 2.8 - Express each force as a Cartesian vector, and then...Ch. 2.8 - If F = {350i 250j 450k} N and cable AB is 9 m...Ch. 2.8 - The 8-m-long cable is anchored to the ground at A....Ch. 2.8 - The 8-m-long cable is anchored to the ground at A....Ch. 2.8 - Express each of the forces in Cartesian vector...Ch. 2.8 - If FB = 560 N and FC = 700 N, determine the...Ch. 2.8 - If FB = 700 N, and FC = 560 N, determine the...Ch. 2.8 - The plate is suspended using the three cables...Ch. 2.8 - Prob. 66PCh. 2.8 - Determine the magnitude and coordinate direction...Ch. 2.8 - Prob. 68PCh. 2.8 - The load at A creates a force of 60 lb in wire AB....Ch. 2.8 - Determine the magnitude and coordinate direction...Ch. 2.9 - In each case, set up the dot product to find the...Ch. 2.9 - In each case, set up the dot product to find the...Ch. 2.9 - Determine the angle between the force and the...Ch. 2.9 - Determine the angle between the force and the...Ch. 2.9 - Determine the angle between the force and the...Ch. 2.9 - Determine the projected component of the force...Ch. 2.9 - Find the magnitude of the projected component of...Ch. 2.9 - Determine the components of the force acting...Ch. 2.9 - Prob. 31FPCh. 2.9 - Prob. 71PCh. 2.9 - Determine the magnitudes of the components of F =...Ch. 2.9 - Determine the angle between BA and BC. Probs. 273Ch. 2.9 - Determine the magnitude of the projected component...Ch. 2.9 - Prob. 75PCh. 2.9 - Determine the magnitude of the projection of the...Ch. 2.9 - Determine the angle between the pole and the wire...Ch. 2.9 - Determine the magnitude of the projection of the...Ch. 2.9 - Determine the magnitude of the projected component...Ch. 2.9 - Prob. 80PCh. 2.9 - Determine the angle between the two cables. Prob....Ch. 2.9 - Determine the projected component of the force...Ch. 2.9 - Determine the angles and between the flag pole...Ch. 2.9 - Determine the magnitudes of the components of F...Ch. 2.9 - Prob. 85PCh. 2.9 - Determine the angle between the pipe segments BA...Ch. 2.9 - If the force F = 100 N lies in the plane DBEC,...Ch. 2.9 - Determine the magnitudes of the components of the...Ch. 2.9 - Determine the magnitudes of the projected...Ch. 2.9 - Determine the magnitude of the projected component...Ch. 2.9 - Two cables exert forces on the pipe. Determine the...Ch. 2.9 - Determine the angle between the two forces. Prob....Ch. 2 - Determine the magnitude of the resultant force FR...Ch. 2 - Resolve the force into components along the u and...Ch. 2 - Determine the magnitude of the resultant force...Ch. 2 - The cable exerts a force of 250 lb on the crane...Ch. 2 - The cable attached to the tractor at B exerts a...Ch. 2 - Express F1 and F2 as Cartesian vectors. Prob. R26Ch. 2 - Determine the angle between the edges of the...Ch. 2 - Determine the projection of the force F along the...
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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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