Problem 1 (65 pts, suggested time 50 mins). An elastic string of constant line tension¹ T 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 amount v(x) from its undeformed (horizontal) state, as shown in the figure below. р v(x) X L The PDE describing mechanical equilibrium for the string is d dv dx dx - p=0. (1) (a) [5pts] Identify the BCs for the string and identify their type (essential/natural). Write down 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 deflection of 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 de- flection of the midpoint? (e) [15pts] Write down the element stiffness matrix and the element force vector for each element.

Problem 1 (65 pts, suggested time 50 mins). An elastic string of constant line tension¹ T 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 amount v(x) from its undeformed (horizontal) state, as shown in the figure below. р v(x) X L The PDE describing mechanical equilibrium for the string is d dv dx dx - p=0. (1) (a) [5pts] Identify the BCs for the string and identify their type (essential/natural). Write down 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 deflection of 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 de- flection of the midpoint? (e) [15pts] Write down the element stiffness matrix and the element force vector for each element.

Mechanics of Materials (MindTap Course List)

9th Edition

ISBN:9781337093347

Author:Barry J. Goodno, James M. Gere

Publisher:Barry J. Goodno, James M. Gere

Chapter2: Axially Loaded Members

Section: Chapter Questions

Problem 2.8.6P

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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).
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Subject:- physics
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