fem problems with solutions

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Basic FEM probles solved using ansys apdl

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    FEM Assignment 2014

    NIT-TRICHY

    FEM Assignment

    5th SemesterSubmitted by:

    NITINVARMAN

    111111060

    4/4/2014

    Software used: Ansys APDL

    This document contains an abstract about the work on Ansys to analyse different problems.

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    Problem 1An Aluminum cantilever beam shown in Figure 1 is subjected to a point load at the end with the

    other end fixed. Using one dimensional elements, discretize the model and perform the

    numerical analysis for the prediction of displacement von-mises stresses in the beam using

    ANSYS software. With beam theory validate the numerical results obtained in ANSYS.

    Figure 1

    Nomenclature:

    L =110m Length of beam

    b =10m Cross Section Base

    h =1 m Cross Section Height

    P=1000N Point Load

    E=70GPa Youngs Modulus of Aluminum at Room Temperature

    =0.33 Poissons Ratio of Aluminum

    Solution:

    Using Numerical Analysis in Ansys:

    The results obtained are

    Maximum Von-mises stress: 62.7KPa

    Maximum Deflection = 7.59mm

    Von mises Result:

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    Deflection result:

    Ansys Macro/NOPR

    KEYW,PR_SET,1

    KEYW,PR_STRUC,1

    KEYW,PR_THERM,0

    KEYW,PR_FLUID,0

    KEYW,PR_ELMAG,0

    KEYW,MAGNOD,0

    KEYW,MAGEDG,0

    KEYW,MAGHFE,0

    KEYW,MAGELC,0

    KEYW,PR_MULTI,0

    KEYW,PR_CFD,0

    /GO

    !*

    !*

    /PREP7

    K,1,0,0,0,

    K,2,110,0,0,LSTR, 1, 2

    !*

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    ET,1,BEAM188

    !*

    !*

    MPTEMP,,,,,,,,

    MPTEMP,1,0

    MPDATA,EX,1,,7e10

    MPDATA,PRXY,1,,0.33SECTYPE, 1, BEAM, RECT, , 0

    SECOFFSET, CENT

    SECDATA,10,1,0,0,0,0,0,0,0,0,0,0

    TYPE, 1

    MAT, 1

    REAL,

    ESYS, 0

    SECNUM, 1

    !*

    LMESH, 1

    LMESH, 1

    /UI,MESH,OFF

    ESIZE,10,0,

    LMESH, 1

    LCLEAR, 1

    LMESH, 1

    /UI,MESH,OFF

    FLST,2,1,3,ORDE,1

    FITEM,2,1

    !*

    /GO

    DK,P51X, ,0, ,0,ALL, , , , , ,

    FLST,2,1,3,ORDE,1

    FITEM,2,2

    !*

    /GO

    FK,P51X,FY,-1000

    FINISH

    /SOL

    /STATUS,SOLU

    SOLVE

    FINISH

    /POST1

    !*

    /EFACET,1

    PLNSOL, U,Y, 0,1.0

    !*

    PRNSOL,S,PRIN!*

    PRESOL,S,PRIN

    )/GOP

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    Theoretical validation:

    Von Mises Stress

    For Plane stress, Von Mises equivalent stress can be expressed as:

    = (x2

    - xy + y2

    + 3xy2)

    In the top most plane => xy=0, y=0.

    Therefore = x.

    Bending stress M(x) = Where I =b*h

    3/12 and c = h/2.

    M(x) = P ( x- L) x = 6P(x-L)/bh2 Maximum stress = 6P(L)/bh2=66 KPa

    Maximum Von Mises stress = 66 KPaDeflection:From Euler Bernoulli Equation

    () From previous derivations

    ( )

    Integrating once we get

    At the fixed end => x=0, y=0, Thus C1=0

    Integrating again we get

    EIy=P(x3/6Lx

    2/2)+C2

    Again at the fied end, y(0) = 0 Thus C2=0.

    Therefore deflection = y

    == (P/EI) * (x3/6Lx

    2/2) = Px

    2/6EI * (x-3L)

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    Maximum deflection is at x=L

    max= PL3/3EI = 7.61 mm.

    Thus the theoretical values and the Numerical analysis values obtained from Ansys are

    almost same.

    Problem 2:

    Determine the force in each member of the following truss. Indicate if the member is in tension

    or compression. The cross-sectional area of each member is 0.01 m2 and the Youngs modulus is

    200 X 109 N/m2.Figure 2 Plane Truss

    Solution:

    From the numerical analysis using Ansys, The forces on each nodes are given

    below

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    Figure 1 : Representation of the forces and constraints on links

    Figure 2 : Forces at different nodes

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    Element 2 (or member AB) applies a force of 1500 N in the x-direction and 800

    N in the negative y -direction on node 1 (or pin A). This means that the total

    force in AB is (15002+800

    2) = 1700 N . The resultant acts from A to B i.e. the

    member is pulling on pin A. So it must be in tension. Similarly, the force in

    Element 1 (AC) is 2000 N (tension) and in Element 3 (BC) is 2500 N(compression).

    The result is shown in the table below

    Link Identification Force(N) Tension / Compression

    AC 2000 Tension

    AB 1700 Tension

    BC 2500 Compression

    Ansys Macro:

    /NOPR

    KEYW,PR_SET,1

    KEYW,PR_STRUC,1

    KEYW,PR_THERM,0

    KEYW,PR_FLUID,0

    KEYW,PR_ELMAG,0

    KEYW,MAGNOD,0

    KEYW,MAGEDG,0

    KEYW,MAGHFE,0

    KEYW,MAGELC,0KEYW,PR_MULTI,0

    KEYW,PR_CFD,0

    /GO

    !*

    !*

    /PREP7

    !*

    ET,1,LINK180

    !*

    R,1,.01, ,0

    !*

    !*

    MPTEMP,,,,,,,,MPTEMP,1,0

    MPDATA,EX,1,,200e9

    MPDATA,PRXY,1,,

    N,1,0,2.8,,,,,

    N,2,1.50,2,,,,,

    N,3,0,0,,,,,

    FLST,2,2,1

    FITEM,2,1

    FITEM,2,3

    E,P51X

    FLST,2,2,1

    FITEM,2,2

    FITEM,2,3

    E,P51X

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    FLST,2,2,1

    FITEM,2,2

    FITEM,2,1

    E,P51X

    FLST,2,1,1,ORDE,1

    FITEM,2,1

    !*/GO

    D,P51X, , , , , ,UX,UY, , , ,

    FLST,2,1,1,ORDE,1

    FITEM,2,3

    !*

    /GO

    D,P51X, , , , , ,UX, , , , ,

    FLST,2,1,1,ORDE,1

    FITEM,2,2

    !*

    /GO

    F,P51X,FY,-2800

    FINISH

    /SOL

    /STATUS,SOLU

    SOLVE

    FINISH

    /POST1

    !*

    PRESOL,FORC

    )/GOP

    Problem 3 :

    Determine the nodal deflections, reaction forces, and stress for the truss

    system shown in Figure 3 below (E = 200GPa, A = 3250mm2).

    Solution:

    The numerical solution using ansys for the given problem is given below.

    The nodes, links and constraints given in Ansys are shown pictorially

    below:

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    The contour plot of the deflection is shown below:

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    The contour plot of stresses is shown below:

    The list of reaction forces are shown below:

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    Ansys Macro:

    /NOPR

    KEYW,PR_SET,1

    KEYW,PR_STRUC,1

    KEYW,PR_THERM,0

    KEYW,PR_FLUID,0

    KEYW,PR_ELMAG,0

    KEYW,MAGNOD,0

    KEYW,MAGEDG,0

    KEYW,MAGHFE,0

    KEYW,MAGELC,0

    KEYW,PR_MULTI,0

    KEYW,PR_CFD,0

    /GO

    !*

    !*

    /PREP7

    !*

    ET,1,LINK180

    !*

    R,1,.00325, ,0

    !*

    !*

    MPTEMP,,,,,,,,

    MPTEMP,1,0

    MPDATA,EX,1,,200e9MPDATA,PRXY,1,,

    N,1,,,,,,,

    N,2,1.8,3.118,,,,,

    N,3,3.6,0,,,,,

    N,4,5.4,3.118,,,,,

    N,5,7.2,0,,,,,

    N,6,9,3.118,,,,,

    N,7,10.8,0,,,,,

    FLST,2,2,1

    FITEM,2,1

    FITEM,2,2

    E,P51X

    FLST,2,2,1FITEM,2,2

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    FITEM,2,3

    E,P51X

    FLST,2,2,1

    FITEM,2,3

    FITEM,2,4

    E,P51X

    FLST,2,2,1FITEM,2,4

    FITEM,2,5

    E,P51X

    FLST,2,2,1

    FITEM,2,5

    FITEM,2,6

    E,P51X

    FLST,2,2,1

    FITEM,2,6

    FITEM,2,7

    E,P51X

    FLST,2,2,1

    FITEM,2,1

    FITEM,2,3

    E,P51X

    FLST,2,2,1

    FITEM,2,3

    FITEM,2,5

    E,P51X

    FLST,2,2,1

    FITEM,2,5

    FITEM,2,7

    E,P51X

    FLST,2,1,1,ORDE,1

    FITEM,2,1

    !*

    /GO

    D,P51X, , , , , ,UX,UY, , , ,

    FLST,2,1,1,ORDE,1

    FITEM,2,7

    !*

    /GO

    D,P51X, , , , , ,UY, , , , ,

    FLST,2,2,1

    FITEM,2,2

    FITEM,2,4

    E,P51X

    FLST,2,2,1FITEM,2,4

    FITEM,2,6

    E,P51X

    FLST,2,1,1,ORDE,1

    FITEM,2,1

    !*

    /GO

    F,P51X,FY,-280000

    FLST,2,1,1,ORDE,1

    FITEM,2,3

    !*

    /GO

    F,P51X,FY,-210000FLST,2,1,1,ORDE,1

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    FITEM,2,5

    !*

    /GO

    F,P51X,FY,-280000

    FLST,2,1,1,ORDE,1

    FITEM,2,7

    !*/GO

    F,P51X,FY,-360000

    FINISH

    /SOL

    /STATUS,SOLU

    SOLVE

    FINISH

    /POST1

    !*

    PRESOL,FORC

    PLDISP,0

    PLDISP,0

    !*

    !*

    /EFACET,1

    PLNSOL, U,Y, 0,1.0

    !*

    /EFACET,1

    PLNSOL, S,EQV, 0,1.0

    !*

    PLESOL, S,EQV, 0,1.0

    etable,saxl,ls

    pretab,saxl

    AVPRIN,0,1,

    ETABLE,1,S,EQV

    !*

    AVPRIN,0,1,

    ETABLE, ,S,X

    !*

    !*

    /EFACET,1

    PLNSOL, S,Y, 0,1.0

    !*

    PLESOL, S,INT, 0,1.0

    !*

    PLESOL, S,Y, 0,1.0

    !*

    PLESOL, S,X, 0,1.0!*

    PLESOL, S,EQV, 0,1.0

    GPLOT

    FINISH

    /SOL

    FINISH

    /PREP7

    )/GOP

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    Problem 4:

    The problem to be modeled in this example is a simple bicycle frame shown in

    the following figure. The frame is to be built of hollow aluminum tubing having

    an outside diameter of 25mm and a wall thickness of 2mm for the main part of

    the frame.

    Solution:

    The numerical analysis of the given problem is done using Ansys. The pictorial

    representation of the approach is shown below

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    The pink colour arrows in the picture indicates the constraints on the nodes

    The red colour arrows indicate the direction of forces on the nodes.

    The contour plot for the deformation is shown below:

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    From the above result we could infer that the maximum deformation of the

    nodes in the bicycle frame is around 0.13 mm

    The contour plot for the von-mises stress is shown below:

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    The forces on each element is shown below:

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    PRINT FORC ELEMENT SOLUTION PER ELEMENT

    ***** POST1 ELEMENT NODE TOTAL FORCE LISTING *****

    LOAD STEP= 1 SUBSTEP= 1

    TIME= 1.0000 LOAD CASE= 0

    THE FOLLOWING X,Y,Z FORCES ARE IN GLOBAL COORDINATES

    ELEM= 1 FX FY FZ MX MY MZ

    1 -402.73 280.34 0.14818E-11-0.25143E-09-0.24405E-09 5738.6

    3 402.73 -280.34 -0.14818E-11 0.25699E-09 0.24405E-09 -4228.4

    ELEM= 2 FX FY FZ MX MY MZ

    3 -402.73 280.34 0.14818E-11-0.25661E-09-0.24405E-09 4228.4

    4 402.73 -280.34 -0.14818E-11 0.26216E-09 0.24405E-09 -2718.1

    ELEM= 3 FX FY FZ MX MY MZ

    4 -402.73 280.34 0.14818E-11-0.26229E-09-0.24405E-09 2718.1

    5 402.73 -280.34 -0.14818E-11 0.26785E-09 0.24405E-09 -1207.9

    ELEM= 4 FX FY FZ MX MY MZ

    5 -402.73 280.34 0.14818E-11-0.26771E-09-0.24405E-09 1207.9

    6 402.73 -280.34 -0.14818E-11 0.27327E-09 0.24405E-09 302.36

    ELEM= 5 FX FY FZ MX MY MZ

    6 -402.73 280.34 0.14818E-11-0.27346E-09-0.24405E-09 -302.36

    7 402.73 -280.34 -0.14818E-11 0.27901E-09 0.24405E-09 1812.6

    ELEM= 6 FX FY FZ MX MY MZ7 -402.73 280.34 0.14818E-11-0.27893E-09-0.24405E-09 -1812.6

    8 402.73 -280.34 -0.14818E-11 0.28449E-09 0.24405E-09 3322.9

    ELEM= 7 FX FY FZ MX MY MZ

    8 -402.73 280.34 0.14818E-11-0.28453E-09-0.24405E-09 -3322.9

    9 402.73 -280.34 -0.14818E-11 0.29008E-09 0.24405E-09 4833.1

    ELEM= 8 FX FY FZ MX MY MZ

    9 -402.73 280.34 0.14818E-11-0.29024E-09-0.24405E-09 -4833.1

    10 402.73 -280.34 -0.14818E-11 0.29580E-09 0.24405E-09 6343.3

    ELEM= 9 FX FY FZ MX MY MZ

    10 -402.73 280.34 0.14818E-11-0.29578E-09-0.24405E-09 -6343.311 402.73 -280.34 -0.14818E-11 0.30134E-09 0.24405E-09 7853.6

    Ansys Macro:

    /NOPR

    KEYW,PR_SET,1

    KEYW,PR_STRUC,1

    KEYW,PR_THERM,0KEYW,PR_FLUID,0

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    KEYW,PR_ELMAG,0

    KEYW,MAGNOD,0

    KEYW,MAGEDG,0

    KEYW,MAGHFE,0

    KEYW,MAGELC,0

    KEYW,PR_MULTI,0

    KEYW,PR_CFD,0/GO

    !*

    !*

    /PREP7

    !*

    ET,1,PIPE288

    !*

    !*

    MPTEMP,,,,,,,,

    MPTEMP,1,0

    MPDATA,EX,1,,70000

    MPDATA,PRXY,1,,0.33

    !*

    !*

    SECTYPE,1,PIPE, ,1

    SECDATA,25,2,0,0,1,0,0,0,

    SECOFFSET,0,0,

    SECCONTROL,0,

    SOCEAN,0

    !*

    /REPLOT,RESIZE

    /REPLOT,RESIZE

    /REPLOT,RESIZE

    K,1,0,325,0,

    K,2,0,400,0,

    K,3,500,400,0,

    K,4,500,0,0,

    K,5,825,0,50,

    K,6,825,0,-50,

    LSTR, 1, 2

    LSTR, 2, 3

    LSTR, 3, 4

    LSTR, 4, 1

    /ANG,1,30,YS,1

    /REP,FAST

    LSTR, 6, 3

    LSTR, 5, 3

    LSTR, 6, 4LSTR, 4, 5

    /ANG,1,30,XS,1

    /REP,FAST

    GPLOT

    /ANG,1,-30,YS,1

    /REP,FAST

    /ANG,1,-30,YS,1

    /REP,FAST

    /ANG,1,-30,YS,1

    /REP,FAST

    /ANG,1,-30,YS,1

    /REP,FAST

    /ANG,1,-30,YS,1/REP,FAST

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    /ANG,1,-30,YS,1

    /REP,FAST

    /ANG,1,-30,YS,1

    /REP,FAST

    TYPE, 1

    MAT, 1

    REAL,ESYS, 0

    SECNUM, 1

    !*

    ESIZE,0,20,

    FLST,2,8,4,ORDE,2

    FITEM,2,1

    FITEM,2,-8

    LMESH,P51X

    /UI,MESH,OFF

    FLST,2,1,3,ORDE,1

    FITEM,2,2

    !*

    /GO

    DK,P51X, , , ,0,ALL, , , , , ,

    FLST,2,2,3,ORDE,2

    FITEM,2,5

    FITEM,2,-6

    !*

    /GO

    DK,P51X, , , ,0,UY,UZ, , , , ,

    FLST,2,1,3,ORDE,1

    FITEM,2,3

    !*

    /GO

    FK,P51X,FY,-600

    FLST,2,1,3,ORDE,1

    FITEM,2,4

    !*

    /GO

    FK,P51X,FY,-900

    /REPLOT,RESIZE

    FINISH

    /SOL

    /STATUS,SOLU

    SOLVE

    FINISH

    /POST1

    !*/EFACET,1

    PLNSOL, U,SUM, 0,1.0

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    Problem 5

    The Simple Conduction Example is constrained as shown in the following figure 5.

    Thermal conductivity (k) of the material is 10 W/m*C and the block is assumed to

    be infinitely long.

    Solution:

    The numerical analysis of the given problem is done using Ansys as

    shown below.

    The model with the thermal constraints on its faces is modelled as

    shown below:

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    The contour plot for the thermal deformation is shown below:

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    Ansys Macro:

    KEYW,PR_SET,1

    KEYW,PR_STRUC,0

    KEYW,PR_THERM,1

    KEYW,PR_FLUID,0

    KEYW,PR_ELMAG,0

    KEYW,MAGNOD,0

    KEYW,MAGEDG,0

    KEYW,MAGHFE,0

    KEYW,MAGELC,0

    KEYW,PR_MULTI,0

    KEYW,PR_CFD,0

    /GO

    /PREP7

    *SET,length,1.0

    *SET,height,1.0

    blc4,0,0,length, height

    ET,1, PLANE55

    MP,KXX,1,10 ESIZE,length/20 AMESH,ALL

    FINISH

    /SOLU

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    ANTYPE,0

    NSEL,S,LOC,Y,height D,ALL,TEMP,500

    NSEL,ALL

    NSEL,S,LOC,X,0

    NSEL,A,LOC,X,length

    NSEL,A,LOC,Y,0

    D,ALL,TEMP,100

    NSEL,ALL

    SOLVE

    FINISH

    /POST1

    PLNSOL,TEMP,,0,

    Problem 6:

    A distributed load of 1000 N/m (1 N/mm) will be applied to a solid steel beam

    with a rectangular cross section as shown in the figure below. The cross-

    section of the beam is 10mm x 10mm while the modulus of elasticity of the

    steel is 200GPa.

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    Solution:

    The numerical solution for the above problem is done using Ansys.

    The Finite element model with the forces and constraints is shown below:

    The contour plot

    for the

    deformation isshown below:

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    The contour plot for the von mises stress is shown below:

    Ansys Macro:

    KEYW,PR_SET,1

    KEYW,PR_STRUC,1

    KEYW,PR_THERM,0

    KEYW,PR_FLUID,0

    KEYW,PR_ELMAG,0

    KEYW,MAGNOD,0

    KEYW,MAGEDG,0

    KEYW,MAGHFE,0

    KEYW,MAGELC,0

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    KEYW,PR_MULTI,0

    KEYW,PR_CFD,0

    /GO

    K,1,0,0

    K,2,1000,0

    L,1,2

    ET,1,BEAM3

    R,1,100,833.333,10

    MP,EX,1,200000

    MP,PRXY,1,0.33

    ESIZE,100

    LMESH,ALL

    FINISH

    /SOLU

    ANTYPE,0

    DK,1,UX,0,,,UY

    DK,2,UY,0

    SFBEAM,ALL,1,PRES,1

    SOLVE

    FINISH

    )/GOP