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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) L The PDE describing mechanical equilibrium for the string is d dv T dx dx (ra) - - 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. (f) [15pts] Assemble the global system of equations and solve it for the midpoint deflection.

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) L The PDE describing mechanical equilibrium for the string is d dv T dx dx (ra) - - 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. (f) [15pts] Assemble the global system of equations and solve it for the midpoint deflection.

Mechanics of Materials (MindTap Course List)
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 (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) L The PDE describing mechanical equilibrium for the string is d dv T dx dx (ra) - - 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. (f) [15pts] Assemble the global system of equations and solve it for the midpoint deflection.
Transcribed Image Text: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) L The PDE describing mechanical equilibrium for the string is d dv T dx dx (ra) - - 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. (f) [15pts] Assemble the global system of equations and solve it for the midpoint deflection.
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