EBK STATICS AND MECHANICS OF MATERIALS
EBK STATICS AND MECHANICS OF MATERIALS
5th Edition
ISBN: 8220102955295
Author: HIBBELER
Publisher: PEARSON
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Chapter 8.4, Problem 9P
To determine

Find the elongation of the bar.

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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.
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.
Problem 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).

Chapter 8 Solutions

EBK STATICS AND MECHANICS OF MATERIALS

Ch. 8.4 - The material for the 50-mm-long specimen has the...Ch. 8.4 - If the elongation of wire BC is 0.2 mm after the...Ch. 8.4 - A tension test was performed on a steel specimen...Ch. 8.4 - Data taken from a stressstrain test for a ceramic...Ch. 8.4 - Data taken from a stressstrain test for a ceramic...Ch. 8.4 - Prob. 4PCh. 8.4 - The stress-strain diagram for a steel alloy having...Ch. 8.4 - Prob. 6PCh. 8.4 - The rigid beam is supported by a pin at C and an...Ch. 8.4 - The rigid beam is supported by a pin at C and an...Ch. 8.4 - Prob. 9PCh. 8.4 - The stressstrain diagram for an aluminum alloy...Ch. 8.4 - The stressstrain diagram for an aluminum alloy...Ch. 8.4 - Prob. 12PCh. 8.4 - A bar having a length of 5 in. and cross-sectional...Ch. 8.4 - The rigid pipe is supported by a pin at A and an...Ch. 8.4 - The rigid pipe is supported by a pin at A and an...Ch. 8.4 - Prob. 16PCh. 8.4 - The rigid beam is supported by a pin at C and an...Ch. 8.4 - Prob. 18PCh. 8.4 - Prob. 19PCh. 8.6 - A 100 mm long rod has a diameter of 15 mm. If an...Ch. 8.6 - A solid circular rod that is 600 mm long and 20 mm...Ch. 8.6 - Prob. 15FPCh. 8.6 - Prob. 16FPCh. 8.6 - The acrylic plastic rod is 200 mm long and 15 mm...Ch. 8.6 - The plug has a diameter of 30 mm and fits within a...Ch. 8.6 - The elastic portion of the stress-strain diagram...Ch. 8.6 - The elastic portion of the stress-strain diagram...Ch. 8.6 - The brake pads for a bicycle tire arc made of...Ch. 8.6 - The lap joint is connected together using a 1.25...Ch. 8.6 - The lap joint is connected together using a 1.25...Ch. 8.6 - Prob. 27PCh. 8.6 - The shear stress-strain diagram for an alloy is...Ch. 8.6 - Prob. 29PCh. 8 - The elastic portion of the tension stress-strain...Ch. 8 - Prob. 2RPCh. 8 - Prob. 3RPCh. 8 - Prob. 4RPCh. 8 - Prob. 5RPCh. 8 - Prob. 6RPCh. 8 - The stress-strain diagram for polyethylene, which...Ch. 8 - The pipe with two rigid caps attached to its ends...Ch. 8 - Prob. 9RPCh. 8 - Prob. 10RP
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Material Properties 101; Author: Real Engineering;https://www.youtube.com/watch?v=BHZALtqAjeM;License: Standard YouTube License, CC-BY