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Find the FEM solution of this problem by following the steps below. 1. Write appropriate BCs for this BVP 2. Starting from the strong BVP derived the weak BVP. 3. Divide the rod into two elements of equal length (mesh), assume piecewise linear u(x) and u(x) to express the weak BVP as sum of matrix terms. Identify the stiffness matrix and force vector of each element within this matrix expression. 4. Assemble the stiffness matrices and force vectors into a global system of equations. 5. Apply essential BCs and solve for the nodal displacements. 6. Sketch the FEM solution and the analytical solution of the strong BVP. An elastic bar of the length L and cross section area A is rigidly attached to the ceiling of a room, and it supports a mass M. Due to the acceleration of gravity g the rod deforms vertically. The deformation of the rod is measured by the vertical displacement u(x) governed by the following equations: dx (σ(x)) + b(x) = 0 PDE σ(x) = Edx du Hooke's law (1) b(x) = gp= body force per unit volume where E is the constant Young's modulus, p is the density, and σ(x) the axial stress in the rod. g * I u(x) L 2

Find the FEM solution of this problem by following the steps below. 1. Write appropriate BCs for this BVP 2. Starting from the strong BVP derived the weak BVP. 3. Divide the rod into two elements of equal length (mesh), assume piecewise linear u(x) and u(x) to express the weak BVP as sum of matrix terms. Identify the stiffness matrix and force vector of each element within this matrix expression. 4. Assemble the stiffness matrices and force vectors into a global system of equations. 5. Apply essential BCs and solve for the nodal displacements. 6. Sketch the FEM solution and the analytical solution of the strong BVP. An elastic bar of the length L and cross section area A is rigidly attached to the ceiling of a room, and it supports a mass M. Due to the acceleration of gravity g the rod deforms vertically. The deformation of the rod is measured by the vertical displacement u(x) governed by the following equations: dx (σ(x)) + b(x) = 0 PDE σ(x) = Edx du Hooke's law (1) b(x) = gp= body force per unit volume where E is the constant Young's modulus, p is the density, and σ(x) the axial stress in the rod. g * I u(x) L 2

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
Chapter7: Analysis Of Stress And Strain
Section: Chapter Questions
Problem 7.6.13P: A solid spherical ball of magnesium alloy (E = 6.5 × l0-6 psi, v = 0.35) is lowered into the ocean...
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Find the FEM solution of this problem by following the steps below. 1. Write appropriate BCs for this BVP 2. Starting from the strong BVP derived the weak BVP. 3. Divide the rod into two elements of equal length (mesh), assume piecewise linear u(x) and u(x) to express the weak BVP as sum of matrix terms. Identify the stiffness matrix and force vector of each element within this matrix expression. 4. Assemble the stiffness matrices and force vectors into a global system of equations. 5. Apply essential BCs and solve for the nodal displacements. 6. Sketch the FEM solution and the analytical solution of the strong BVP.
Transcribed Image Text:Find the FEM solution of this problem by following the steps below. 1. Write appropriate BCs for this BVP 2. Starting from the strong BVP derived the weak BVP. 3. Divide the rod into two elements of equal length (mesh), assume piecewise linear u(x) and u(x) to express the weak BVP as sum of matrix terms. Identify the stiffness matrix and force vector of each element within this matrix expression. 4. Assemble the stiffness matrices and force vectors into a global system of equations. 5. Apply essential BCs and solve for the nodal displacements. 6. Sketch the FEM solution and the analytical solution of the strong BVP.
An elastic bar of the length L and cross section area A is rigidly attached to the ceiling of a room, and it supports a mass M. Due to the acceleration of gravity g the rod deforms vertically. The deformation of the rod is measured by the vertical displacement u(x) governed by the following equations: dx (σ(x)) + b(x) = 0 PDE σ(x) = Edx du Hooke's law (1) b(x) = gp= body force per unit volume where E is the constant Young's modulus, p is the density, and σ(x) the axial stress in the rod. g * I u(x) L 2
Transcribed Image Text:An elastic bar of the length L and cross section area A is rigidly attached to the ceiling of a room, and it supports a mass M. Due to the acceleration of gravity g the rod deforms vertically. The deformation of the rod is measured by the vertical displacement u(x) governed by the following equations: dx (σ(x)) + b(x) = 0 PDE σ(x) = Edx du Hooke's law (1) b(x) = gp= body force per unit volume where E is the constant Young's modulus, p is the density, and σ(x) the axial stress in the rod. g * I u(x) L 2
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