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    HW 2 Solution

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    SS
    This document contains instructions for solving multiple problems involving structural analysis using the finite element method. The key points are: 1) Solve a beam problem with two elements by writing the global stiffness matrix after applying boundary conditions and assembling the element stiffness matrices. 2) Determine the interpolation functions and equivalent nodal loads for a three-node bar element subjected to a uniform distributed load. 3) Use the Rayleigh-Ritz method to analyze a beam with specified boundary conditions, distributed load, and end moment, expressing the displacement as a cubic polynomial and determining the coefficients. 4) Additional problems are listed to solve involving finite element analysis of beams and trusses.

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    © All Rights Reserved
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    HW 2 Solution

    Uploaded by

    SS
    44 views5 pages
    This document contains instructions for solving multiple problems involving structural analysis using the finite element method. The key points are: 1) Solve a beam problem with two elements by writing the global stiffness matrix after applying boundary conditions and assembling the element stiffness matrices. 2) Determine the interpolation functions and equivalent nodal loads for a three-node bar element subjected to a uniform distributed load. 3) Use the Rayleigh-Ritz method to analyze a beam with specified boundary conditions, distributed load, and end moment, expressing the displacement as a cubic polynomial and determining the coefficients. 4) Additional problems are listed to solve involving finite element analysis of beams and trusses.

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    Hw 2 Solution

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    This document contains instructions for solving multiple problems involving structural analysis using the finite element method. The key points are: 1) Solve a beam problem with two elements by writing the global stiffness matrix after applying boundary conditions and assembling the element stiffness matrices. 2) Determine the interpolation functions and equivalent nodal loads for a three-node bar element subjected to a uniform distributed load. 3) Use the Rayleigh-Ritz method to analyze a beam with specified boundary conditions, distributed load, and end moment, expressing the displacement as a cubic polynomial and determining the coefficients. 4) Additional problems are listed to solve involving finite element analysis of beams and trusses.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    44 views5 pages

    HW 2 Solution

    Uploaded by

    SS
    This document contains instructions for solving multiple problems involving structural analysis using the finite element method. The key points are: 1) Solve a beam problem with two elements by writing the global stiffness matrix after applying boundary conditions and assembling the element stiffness matrices. 2) Determine the interpolation functions and equivalent nodal loads for a three-node bar element subjected to a uniform distributed load. 3) Use the Rayleigh-Ritz method to analyze a beam with specified boundary conditions, distributed load, and end moment, expressing the displacement as a cubic polynomial and determining the coefficients. 4) Additional problems are listed to solve involving finite element analysis of beams and trusses.

    Copyright:

    © All Rights Reserved
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    Download as pdf or txt
    of 5

    1.

    Solve Problem 2.5-3 using two beam elements. Write matrix equation after applying
    boundary conditions
    At hinge, no rotational DOFs are connected. Thus, the global DOF should be
    {QS } = {v1, 1, v2 , 2(1), 2(2), v3, 3 }T . Since Node 1 and Node 3 are clamped, the free DOFs are

    {Q} = {v2, 2(1), 2(2) }T


    For Element 1 (only for free DOFs),
    EI 12 6a v2
    [ k(1) ] = 3z
    2 (1)
    a 6a 4a 2
    For Element 2 (only for free DOFs),
    EI 12 6b v2
    [ k(1) ] = 3 z
    2 (2)
    b 6b 4b 2
    After assembly, the matrix equation becomes
    12(a 3 + b 3 ) 6ab 3 6a 3b v
    P

    EI z

    3
    2 3
    (1)

    4a b
    0 2 = 0
    6ab

    a 3b 3

    3

    0

    0
    4a 3b 2 2(2)
    6a b


    2. Consider a bar element with three nodes, as shown in the figure. When the solution is approximated
    by u(x ) = N 1(x )u1 + N 2 (x )u2 + N 3 (x )u 3 , calculate interpolation functions N 1(x ), N 2 (x ), N 3 (x ) .
    When a distributed load q0 is uniformly distributed on the element, calculate work-equivalent nodal
    forces.
    u2

    u1
    1

    u3

    2
    x

    0.5 m

    3
    0.5 m

    Solution:
    Since three nodes are available, we can use second-order approximation of the solution:
    u ( x) = a0 + a1 x + a2 x 2
    By imposing three nodal solutions, we have

    u (0) = u1 = a0
    u ( 12 ) = u2 = a0 + 12 a1 + 14 a2
    u (1) = u3 = a0 + a1 + a2

    By solving a1, a2, and a3 with respect to the nodal solution, we have
    a0 = u1
    a1 = 3u1 + 4u2 u3
    a2 = 2u1 4u2 + 2u3

    Thus, from the interpolation relation, we have


    u ( x ) = (1 3 x + 2 x 2 )u1 + 4( x x 2 )u2 + ( x + 2 x 2 )u3
    = N1u1 + N 2u2 + N 3u3

    where

    N1 = 1 3 x + 2 x 2
    N2 = 4x 4x2
    N3 = x + 2x2
    The work-equivalent load can be obtained from the expression of potential of applied load as
    1
    1
    1

    q
    u
    (
    x
    )d
    x
    =
    u
    q
    N
    (
    x
    )d
    x
    +
    u
    q
    N
    (
    x
    )d
    x
    +
    u
    q
    N 3 (x )dx

    1
    0 0
    1
    2
    0 0
    2
    3
    0 0
    0 0

    = 16 q 0u1 + 23 q 0u2 + 16 q 0u 3

    V =

    Thus, the work-equivalent load becomes

    q0
    {F} = 4

    3. Use the Rayleigh-Ritz method to determine the deflection v(x), bending moment M(x), and shear
    force Vy(x) for the beam shown in the figure. Assume EI = 1,000 N-m2, L = 1 m, and p0 = 100 N/m, and
    C = 100 N-m. The displacement is expressed as v(x ) = c0 + c1x + c2x 2 + c3x 3 . Make sure the
    displacement boundary conditions are satisfied a priori. Hint: Potential energy of a couple is calculated as
    V = Cdv / dx , where the rotation is calculated at the point of application of the couple.

    p0
    y
    C

    x
    L
    Solution:

    v(x ) = c0 + c1x + c2x 2 + c3x 3

    dv
    = c1 + 2c2x + 3c3x 2
    dx

    Displacement boundary condition: v(0) = 0, v (0) = 0

    c0 = c1 = 0

    v(x ) = c2x 2 + c3x 3


    Now we only need to determine c2 and c3.

    dv
    = 2c2x + 3c3x 2
    dx

    d 2v

    = 2c2 + 6c3x

    dx 2

    d 3v
    dx 3

    = 6c3

    Potential energy:

    1
    EI
    0 2

    1 EI
    2

    d 2v 2
    L
    dv

    dx 2 dx 0 vpydx dx (1)C

    ( 2c2

    + 6c3x ) dx

    L
    0

    (c2x 2 + c3x 3 )( 100x )dx ( 2c2 + 3c3 )C


    1

    0 2000 (c22 + 6c2c3x + 9c32x 2 )dx + 0 ( 100c2x 3 + 100c3x 4 )dx 100 ( 2c2 + 3c3 )
    = 2000 ( c22 + 3c2c3 + 3c32 ) + ( 25c2 + 20c3 ) 100 ( 2c2 + 3c3 )
    = 2000 ( c22 + 3c2c3 + 3c32 ) 175c2 280c3
    =


    = 0 2000 ( 2c2 + 3c3 ) 175 = 0

    c2

    = 0 2000 ( 3c2 + 6c3 ) 280 = 0

    c3

    c2 = 0.035

    c = 0.005833

    3
    Final results:

    v(x ) = 0.035x 2 + 0.005833x 3


    M (x ) = EI

    d 2v
    dx 2

    = 1000 ( 2 0.035 + 6 0.005833x ) = 70 + 35x

    Vy (x ) = EI

    4. Solve problem 4.5-8.

    d 3v
    dx 3

    = 1000(6 0.005833) = 35

    5. A space frame structure as shown in the figure consists of 25 truss members. All members have the
    same circular cross-sections with diameter d = 0.5 in. At nodes 1 and 2, a constant force F = 60,000 lb is
    applied in the ydirection. Four nodes (7, 8, 9, and 10) are fixed on the ground. The frame structure is
    made of a steel material whose properties are Youngs modulus E = 3107 psi, Poissons ratio = 0.3.
    Calculate displacements of all nodes and stress of all members using finite element software. Provide a
    plot that shows labels for elements and nodes along with boundary conditions. Provide deformed
    geometry of the structure and a table of stress in each element.

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