StaBIL-3.0
Short Description
Matlab...
Description
FACULTY OF ENGINEERING DEPARTMENT DEPARTMENT OF CIVIL ENGINEERING STRUCTURAL MECHANICS
KASTEELPARK KASTEELPARK ARENBERG 40 B-3001 LEUVEN
KATHOLIEKE UNIVERSITEIT LEUVEN
StaBIL: A FINITE ELEMENT TOOLBOX FOR MATLAB
VERSION 3.0 USER’S GUIDE
Report BWM-2014-***
David DOOMS, Miche JANSEN, Guido DE ROECK, Geert DEGRANDE, Geert LOMBAERT, Mattias SCHEVENELS, Stijn FRANC ¸ OIS September 2014
Contents
1 Functio unctions ns — By cate categor gory y
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1.1 Genera Generall functio functions ns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1.2 Postp Postproce rocessi ssing ng . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1.3 1.3
Dyna Dynami mics cs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1.4 1.4
Beam Be am funct functio ions ns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1.5 Truss functio functions ns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1.6 Genera Generall shel shelll functio functions ns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1.7 Shell4 Shell4 functio functions ns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1.8 Shell8 Shell8 functio functions ns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1.9 Nonline Nonlinear ar anal analysi ysiss functi functions ons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2 Tutor utoria iall
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2.1 Tutorial utorial:: static static anal analysi ysiss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2.2 Tutorial utorial:: dynamic dynamic analysi analysiss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2.2. 2.2.11
Eige Eigenm nmode odess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2.2.2
Modal superposition superposition:: time domain: domain: piecewise piecewise exact exact integrat integration ion . . . . . . . . .
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2.2.3 2.2 .3
Modal superpos superpositi ition: on: transform transform to freq frequen uency cy domai domain n. . . . . . . . . . . . . . .
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2.2.4 2.2 .4
Direct Direct time time integ integrat ration ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2.2.5 2.2 .5
Direct Direct solu solutio tion n in the frequenc frequency y domain domain . . . . . . . . . . . . . . . . . . . . . .
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2.2. 2.2.66
Compa Compari riso son n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2.3 Tutorial utorial:: shell analys analysis is . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2.4 Tutorial utorial:: nonline nonlinear ar anal analysi ysiss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3 Ex Exam ampl ples es
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3.1 3.1
Exam Exampl plee 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3.2 3.2
Exam Exampl plee 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
3.3 3.3
Exam Exampl plee 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
3.4 3.4
Exam Exampl plee 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
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Contents
3.5 3.5
Exam Exampl plee 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
3.6 3.6
Exam Exampl plee 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
3.7 3.7
Exam Exampl plee 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
4 Functions unctions — Alpha Alphabetica beticall list list
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Chapter 1
Functions — By category 1.1
General functions
getdof asmkm removedof addconstr tconstr nodalvalues elemloads accel elemforces elemdisp selectdof unselectdof selectnode reprow multdloads
Get the vector with the degrees of freedom of the model. Assemble stiffness and mass matrix. Remove DOF with Dirichlet boundary conditions equal to zero. Add constraint equations to the stiffness matrix and load vector. Return matrices to apply constraint equations. Construct a vector with the values at the selected DOF. Equivalent nodal forces. Compute the distributed loads due to an acceleration. Compute the element forces. Select the element displacements from the global displacement vector. Select degrees of freedom. Unselect degrees of freedom. Select nodes by location. Replicate rows from a matrix. Combine distributed loads.
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Functions — By category
getdof GETDOF
Get the vector with the degrees of freedom of the model.
DOF=getdof(Elements,Types) builds the vector with the labels of the degrees of freedom for which stiffness is present in the finite element model. Elements Types DOF
Element definitions [EltID TypID SecID MatID n1 n2 ...] Element type definitions {TypID EltName Option1 ... } Degrees of freedom (nDOF * 1)
See also DOF_TRUSS, DOF_BEAM, GETDOF.
removedof REMOVEDOF
Remove DOF with Dirichlet boundary conditions equal to zero.
DOF=removedof(DOF,seldof) removes the specified Dirichlet boundary conditions from the degrees of freedom vector. DOF seldof
Degrees of freedom (nDOF * 1) Dirichlet boundary conditions equal to zero
[NodID.dof]
See also GETDOF.
asmkm ASMKM
Assemble stiffness and mass matrix.
[K,M] = asmkm(Nodes,Elements,Types,Sections,Materials,DOF) K = asmkm(Nodes,Elements,Types,Sections,Materials,DOF) K = asmkm(Nodes,Elements,Types,Sections,Materials) assembles the stiffness and the mass matrix using the finite element method. Nodes Elements Types Sections Materials DOF K
Node definitions [NodID Element definitions [EltID Element type definitions {TypID Section definitions [SecID Material definitions [MatID Degrees of freedom (nDOF * 1) Stiffness matrix (nDOF * nDOF)
x y z] TypID SecID MatID n1 n2 ...] EltName Option1 ... } SecProp1 SecProp2 ...] MatProp1 MatProp2 ... ]
General functions
M
Mass matrix (nDOF * nDOF)
See also KE_TRUSS, KE_BEAM.
addconstr ADDCONSTR
Add constraint equations to the stiffness matrix and load vector.
[K,F]=addconstr(Constr,DOF,K,F) [K,F,M]=addconstr(Constr,DOF,K,[],M) [K,F,M]=addconstr(Constr,DOF,K,F,M) modifies the stiffness matrix, the mass matrix and the load vector according to the applied constraint equations. The dimensions of the stiffness matrix, the mass matrix and the load vector are kept the same. The resulting stiffness and mass matrix are not symmetric anymore. This function can be used as well to apply imposed displacements. Constr
DOF K F M
Constraint equation: Constant=CoefS*SlaveDOF+CoefM1*MasterDOF1+CoefM2*MasterDOF2+... [Constant CoefS SlaveDOF CoefM1 MasterDOF1 CoefM2 MasterDOF2 ...] Degrees of freedom (nDOF * 1) Stiffness matrix (nDOF * nDOF) Load vector (nDOF * nSteps) Mass matrix (nDOF * nDOF)
tconstr TCONSTR
Return matrices to apply constraint equations.
[T,Q0,MasterDOF]=tconstr(Constr,DOF) returns matrices to apply constraint equations to the stiffness and mass matrix and the load vector: Kr=T.’*K*T, Mr=T.’*M*T and Fr=T.’*(F-K*Q0). The original displacement vector is computed using U=T*Ur+Q0. Constr
DOF
Constraint equation: Constant=CoefS*SlaveDOF+CoefM1*MasterDOF1+CoefM2*MasterDOF2+... [Constant CoefS SlaveDOF CoefM1 MasterDOF1 CoefM2 MasterDOF2 ...] Degrees of freedom (nDOF * 1)
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4
Functions — By category
nodalvalues NODALVALUES
Construct a vector with the values at the selected DOF.
F=nodalvalues(DOF,seldof,values) constructs a vector with the values at the selected DOF. This function can be used to obtain a load vector, initial displacements, velocities or accelerations. DOF seldof values F
Degrees of freedom (nDOF * 1) Selected degrees of freedom [NodID.dof] (nValues * 1) Corresponding values [Value] (nValues * nSteps) Load vector (nDOF * nSteps)
See also ELEMLOADS.
elemloads ELEMLOADS
Equivalent nodal forces.
F=elemloads(DLoads,Nodes,Elements,Types,DOF) computes the equivalent nodal forces of a distributed load (in the global coordinate system). DLoads Nodes Elements Types DOF F
Distributed loads [EltID n1globalX n1globalY n1globalZ ...] Node definitions [NodID x y z] Element definitions [EltID TypID SecID MatID n1 n2 ...] Element type definitions {TypID EltName Option1 ... } Degrees of freedom (nDOF * 1) Load vector (nDOF * 1)
See also LOADS_TRUSS, LOADS_BEAM, NODALVALUES.
accel ACCEL
Compute the distributed loads due to an acceleration.
DLoads=accel(Accelxyz,Elements,Types,Sections,Materials) computes the distributed loads due to an acceleration. In order to simulate gravity, accelerate the structure in the direction opposite to gravity.
General functions
Accelxyz Elements Types Sections Materials DLoads
Acceleration [Ax Ay Az] (1 * 3) Element definitions [EltID TypID SecID MatID n1 n2 ...] Element type definitions {TypID EltName Option1 ... } Section definitions [SecID SecProp1 SecProp2 ...] Material definitions [MatID MatProp1 MatProp2 ... ] Distributed loads [EltID n1globalX n1globalY n1globalZ ...]
See also ELEMLOADS, ACCEL_BEAM, ACCEL_TRUSS.
elemforces ELEMFORCES
Compute the element forces.
[ForcesLCS,ForcesGCS] =elemforces(Nodes,Elements,Types,Sections,Materials,DOF,U,DLoads) [ForcesLCS,ForcesGCS] =elemforces(Nodes,Elements,Types,Sections,Materials,DOF,U) computes the element forces in the local (beam convention) and the global (algebraic convention) coordinate system. Nodes Elements Types Sections Materials DOF U DLoads ForcesLCS
Node definitions [NodID x y z] Element definitions [EltID TypID SecID MatID n1 n2 ...] Element type definitions {TypID EltName Option1 ... } Section definitions [SecID SecProp1 SecProp2 ...] Material definitions [MatID MatProp1 MatProp2 ... ] Degrees of freedom (nDOF * 1) Displacements (nDOF * 1) Distributed loads [EltID n1globalX n1globalY n1globalZ ...] Element forces in LCS (beam convention) [N Vy Vz T My Mz] (nElem * 12) ForcesGCS Element forces in GCS (algebraic convention) (nElem * 12) See also FORCES_TRUSS, FORCES_BEAM.
elemdisp ELEMDISP
Select the element displacements from the global displacement vector.
UeGCS = elemdisp(Type,NodeNum,DOF,U) selects the element displacements from the global displacement vector. Type
Element type e.g. ’beam’,’truss’, ...
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6
Functions — By category
M yj
x z
T j
j M zj
M yi
V zj
V zi
T i
N i
N j
V yj
y
i
M zi
V yi
Figure 1.1: Beam convention. NodeNum DOF U UeGCS
Node numbers (1 * nNodes) Degrees of freedom (nDOF * 1) Displacements (nDOF * 1) Elements displacements
See also ELEMFORCES, DOF_BEAM, DOF_TRUSS.
selectdof SELECTDOF
Select degrees of freedom.
L=selectdof(DOF,seldof) [L,I]=selectdof(DOF,seldof) L=selectdof(DOF,seldof,’Ordering’,ordering) creates the matrix to extract degrees of freedom from the global degrees of freedom by matrix multiplication. DOF seldof L I ordering
Degrees of freedom (nDOF * 1) Selected DOF labels (kDOF * 1) Selection matrix (kDOF * nDOF) Index vector (kDOF * 1) ’seldof’,’DOF’ or ’sorted’
Default: ’seldof’
General functions
Ordering of L and I similar as seldof, DOF or sorted
unselectdof UNSELECTDOF
Unselect degrees of freedom.
L=unselectdof(DOF,seldof) [L,I]=unselectdof(DOF,seldof) creates the matrix to unselect degrees of freedom from the global degrees of freedom. DOF seldof L I
Degrees of freedom (nDOF * 1) Unselected dof labels (ndof * 1) Selection matrix ((nDOF-ndof) * nDOF) Index vector ((nDOF-ndof) * 1)
See also SELECTDOF.
selectnode SELECTNODE
Select nodes by location.
Nodesel=selectnode(Nodes,x,y,z) Nodesel=selectnode(Nodes,xmin,ymin,zmin,xmax,ymax,zmax) selects nodes by location. Nodes Nodesel
Node definitions [NodID x y z] Node definitions of the selected nodes
reprow REPROW
Replicate rows from a matrix.
Matrix=reprow(Matrix,RowSel,nTime,RowInc) replicates the selected rows from a matrix a number of times and adds them below the existing rows. The k-th time the increment RowInc is added k times to the copied rows.
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8
Functions — By category
Matrix RowSel nTime RowInc
Matrix (nRow * nCol) Rows to be copied (1 * kRow) Number of times Increments that are added to the different columns of the copied rows (1 * nCol)
multdloads MULTDLOADS
Combine distributed loads.
DLoads=multdloads(DLoads_1,DLoads_2,...,DLoads_k) combines the distributed loads of multiple load cases into one 3D array. Each plane corresponds to one load case. DLoads_k
Distributed loads
DLoads
Distributed loads
See also ELEMLOADS.
[EltID n1globalX n1globalY n1globalZ ...] (nElem_k * 7) [EltID n1globalX n1globalY n1globalZ ...] (maxnElem * 7 * k)
Postprocessing
1.2
9
Postprocessing
plotnodes Plot the nodes. Plot the elements. plotelem plotdisp Plot the displacements. plotforc Plot the forces. plotlcs Plot the local element coordinate systems. plotstress Plot the stresses. Plot filled contours of stresses. plotstresscontourf animdisp Animate the displacements. getmovie Get the movie from a figure where an animation has been played. printdisp Display the displacements in the command window. printforc Display the forces in the command window.
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Functions — By category
plotnodes PLOTNODES
Plot the nodes.
plotnodes(Nodes) plots the nodes. Nodes
Node definitions
[NodID x y z]
plotnodes(...,ParamName,ParamValue) sets the value of the specified parameters. The following parameters can be specified: ’Numbering’ Plots the node numbers. Default: ’on’. ’GCS’ Plots the global coordinate system. Default: ’on’. ’Handle’ Plots in the axis with this handle. Default: current axis. Additional parameters are redirected to the PLOT3 function which plots the nodes. h = PLOTNODES(...) returns a struct h with handles to all the objects in the plot. See also PLOTELEM, PLOTDISP.
plotelem PLOTELEM
Plot the elements.
plotelem(Nodes,Elements,Types) plots the elements. Nodes Elements Types
Node definitions [NodID x y z] Element definitions [EltID TypID SecID MatID n1 n2 ...] Element type definitions {TypID EltName Option1 ... }
plotelem(...,ParamName,ParamValue) sets the value of the specified parameters. The following parameters can be specified: ’Numbering’ Plots the element numbers. Default: ’on’. ’GCS’ Plots the global coordinate system. Default: ’on’. ’Handle’ Plots in the axis with this handle. Default: current axis. Additional parameters are redirected to the PLOT3 function which plots the elements. h = PLOTELEM(...) returns a struct h with handles to all the objects in the plot. See also COORD_TRUSS, COORD_BEAM, PLOTDISP.
Postprocessing
11
plotdisp PLOTDISP
Plot the displacements.
plotdisp(Nodes,Elements,Types,DOF,U,DLoads,Sections,Materials) plotdisp(Nodes,Elements,Types,DOF,U,[],Sections,Materials) plotdisp(Nodes,Elements,Types,DOF,U) DispScal=plotdisp(Nodes,Elements,Types,DOF,U,DLoads,Sections,Materials) plots the displacements. If DLoads, Sections and Materials are supplied, the displacements that occur due to the distributed loads if all nodes are fixed, are superimposed. Nodes Elements Types DOF U DLoads
Sections Materials DispScal
Node definitions [NodID x y z] Element definitions [EltID TypID SecID MatID n1 n2 ...] Element type definitions {TypID EltName Option1 ... } Degrees of freedom (nDOF * 1) Displacements (nDOF * 1) Distributed loads [EltID n1globalX n1globalY n1globalZ ...] (use empty array [] when shear deformation (in beam element) is considered but no DLoads are present) Section definitions [SecID SecProp1 SecProp2 ...] Material definitions [MatID MatProp1 MatProp2 ... ] Displacement scaling
plotdisp(...,ParamName,ParamValue) sets the value of the specified parameters. The following parameters can be specified: ’DispScal’ Displacement scaling. Default: ’auto’. ’DispMax’ Mention maximal displacement. Default: ’on’. ’Undeformed’ Plots the undeformed mesh. Default: ’k:’. ’Handle’ Plots in the axis with this handle. Default: current axis. Additional parameters are redirected to the PLOT3 function which plots the deformations. [DispScal,h] = plotdisp(...) returns a struct h with handles to all the objects in the plot. See also DISP_TRUSS, DISP_BEAM, PLOTELEM.
plotforc PLOTFORC
Plot the forces.
plotforc(ftype,Nodes,Elements,Types,Forces,DLoads) plotforc(ftype,Nodes,Elements,Types,Forces) plots the forces (in beam convention).
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Functions — By category
ftype
Nodes Elements Types Forces DLoads
’norm’ Normal force (in the local x-direction) ’sheary’ Shear force in the local y-direction ’shearz’ Shear force in the local z-direction ’momx’ Torsional moment (around the local x-direction) ’momy’ Bending moment around the local y-direction ’momz’ Bending moment around the local z-direction Node definitions [NodID x y z] Element definitions [EltID TypID SecID MatID n1 n2 ...] Element type definitions {TypID EltName Option1 ... } Element forces in LCS (beam convention) [N Vy Vz T My Mz] (nElem * 12) Distributed loads [EltID n1globalX n1globalY n1globalZ ...]
plotforc(...,ParamName,ParamValue) sets the value of the specified parameters. The following parameters can be specified: ’ForcScal’ Force scaling. Default: ’auto’. ’Values’ Force values. Default: ’on’. ’Undeformed’ Plots the undeformed mesh. Default: ’k-’. ’Handle’ Plots in the axis with this handle. Default: current axis. Additional parameters are redirected to the PLOT3 function which plots the forces. [ForcScal,h] = plotforc(...) returns a struct h with handles to all the objects in the plot. See also FDIAGRGCS_BEAM, FDIAGRGCS_TRUSS.
plotlcs PLOTLCS
Plot the local element coordinate systems.
[h,vLCS] = plotlcs(Nodes,Elements,Types) [h,vLCS] = plotlcs(Nodes,Elements,Types,[],varargin) plotlcs(Nodes,Elements,Types,vLCS,varargin) Nodes Node definitions [NodID x y z] Elements Element definitions [EltID TypID SecID MatID n1 n2 ...] Types Element type definitions {TypID EltName Option1 ... } vLCS Element coordinate systems (nElem * 9) plotlcs(...,ParamName,ParamValue) sets the value of the specified parameters. The following parameters can be specified: ’GCS’ Plot the GCS. Default: ’on’. ’Undeformed’ Plots the undeformed mesh. Default: ’k-’. ’Handle’ Plots in the axis with this handle. Default: current axis.
Postprocessing
13
See also ELEMSTRESS
plotstress PLOTSTRESS
Plot the stresses.
plotstress(stype,Nodes,Elements,Types,Sections,Forces,DLoads) plotstress(stype,Nodes,Elements,Types,Sections,Forces) plots the stresses. stype
’snorm’ ’smomyt’
Nodes Elements Types Sections Forces
Normal stress due to normal force Normal stress due to bending moment around the local y-direction at the ’smomyb’ Normal stress due to bending moment around the local y-direction at the ’smomzt’ Normal stress due to bending moment around the local z-direction at the ’smomzb’ Normal stress due to bending moment around the local z-direction at the ’smax’ Maximal normal stress (normal force ’smin’ Minimal normal stress (normal force Node definitions [NodID x y z] Element definitions [EltID TypID SecID MatID Element type definitions {TypID EltName Option1 Section definitions [A ky kz Ixx Iyy Izz yt Element forces in LCS (beam convention) [N Vy Vz
DLoads
Distributed loads
top bottom top bottom and bending moment) and bending moment)
n1 n2 ...] ... } yb zt zb] T My Mz] (nElem * 12) [EltID n1globalX n1globalY n1globalZ ...]
plotstress(...,ParamName,ParamValue) sets the value of the specified parameters. The following parameters can be specified: ’StressScal’ Stress scaling. Default: ’auto’. ’Values’ Stress values. Default: ’on’. ’Undeformed’ Plots the undeformed mesh. Default: ’k-’. ’Handle’ Plots in the axis with this handle. Default: current axis. Additional parameters are redirected to the PLOT3 function which plots the stresses. [StressScal,h] = plotstress(...) returns a struct h with handles to all the objects in the plot. See also SDIAGRGCS_BEAM, SDIAGRGCS_TRUSS.
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Functions — By category
plotstresscontourf PLOTSTRESSCONTOURF
Plot filled contours of stresses.
plotstresscontourf(stype,Nodes,Elements,Types,S) plotstresscontourf(stype,Nodes,Elements,Types,S,DOF,U) plots stress contours (output from ELEMSTRESS). stype
’sx’ Normal stress (in the global/local x-direction) ’sy’ ’sz’ ’sxy’ Shear stress ’syz’ ’sxz’ Nodes Node definitions [NodID x y z] Elements Element definitions [EltID TypID SecID MatID n1 n2 ...] Types Element type definitions {TypID EltName Option1 ... } S Element stresses in GCS/LCS [sxx syy szz sxy syz sxz] (nElem * 72) plotstresscontourf(...,ParamName,ParamValue) sets the value of the specified parameters. The following parameters can be specified: ’location’ Location (top,mid,bot). Default: ’top’. ’GCS’ Plot the GCS. Default: ’on’. ’minmax’ Add location of min and max stress. Default: ’on’. ’colorbar’ Add colorbar. Default: ’on’. ’ncolor’ Number of colors in colormap. Default: 10. ’Handle’ Plots in the axis with this handle. Default: current axis. See also ELEMSTRESS.
animdisp ANIMDISP
Animate the displacements.
DispScal=animdisp(Nodes,Elements,Types,DOF,U) animates the displacements. Nodes Elements Types DOF U DispScal
Node definitions [NodID x y z] Element definitions [EltID TypID SecID MatID n1 n2 ...] Element type definitions {TypID EltName Option1 ... } Degrees of freedom (nDOF * 1) Displacements (nDOF * nSteps) Displacement scaling
ANIMDISP(...,ParamName,ParamValue) sets the value of the specified parameters. The following parameters can be specified:
Postprocessing
15
’DispScal’ ’Handle’ ’Fps’ ’CreateMovie’
Displacement scaling. Default: ’auto’. Plots in the axis with this handle. Default: current axis. Frames per second. Default: 12. Saves the movie in the userdata of the axis of the figure. Use getmovie to get the movie from the axis. Default: ’off’. ’Counter’ Displays the number of the frame for transient displacements. Default: ’on’. Additional parameters are redirected to the PLOTDISP function which plots the individual frames of the movie. See also GETMOVIE, PLOTDISP.
getmovie GETMOVIE
Get the movie from a figure where an animation has been played.
mov=getmovie(h) gets the movie from the userdata of the axis of a figure where an animation has been played. In order to save the movie in the userdata the animation should have been played using animdisp(...,’CreateMovie’,’on’). This function blocks the command prompt until the movie has become available. h mov
Handle of the axis e.g. gca Struct with the frames of the movie
The movie can be played with movie(gcf,mov). See also ANIMDISP, MOVIE.
printdisp PRINTDISP
Display the displacements in the command window.
printdisp(Nodes,DOF,U) displays the displacements in the command window. Nodes DOF U
Node definitions [NodID x y z] Degrees of freedom (nDOF * 1) Displacements (nDOF * 1)
See also PRINTFORC, PLOTDISP.
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Functions — By category
printforc PRINTFORC
Display the forces in the command window.
printforc(Elements,Forces) displays the forces in the command window. Elements Forces
Element definitions Element forces
See also PRINTDISP.
[EltID TypID SecID MatID n1 n2 ...] [N Vy Vz T My Mz] (nElem * 12)
Dynamics
1.3
17
Dynamics
eigfem msupt msupf cdiff newmark wilson ldlt
Compute the eigenmodes and eigenfrequencies of the finite element model. Modal superposition in the time domain. Modal superposition in the frequency domain. Direct time integration for dynamic systems - central diff. method. Direct time integration for dynamic systems - Newmark method Direct time integration for dynamic systems - Wilson-theta method LDL Gauss factorization of symmetric matrices.
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Functions — By category
eigfem EIGFEM
Compute the eigenmodes and eigenfrequencies of the finite element model.
[phi,omega]=eigfem(K,M,nMode) [phi,omega]=eigfem(K,M) computes the eigenmodes and eigenfrequencies of the finite element model. K M nMode phi omega
Stiffness matrix (nDOF * nDOF) Mass matrix (nDOF * nDOF) Number of eigenmodes and eigenfrequencies (default: all) Eigenmodes (in columns) (nDOF * nMode) Eigenfrequencies [rad/s] (nMode * 1)
msupt MSUPT
Modal superposition in the time domain.
x = MSUPT(omega,xi,t,pm,x1,y1,interp) calculates the modal displacements x(t) of a dynamic system with the eigenfrequencies omega and the modal damping ratios xi due to the modal excitation pm(t), given the initial conditions x1 and y1. The solution of the modal differential equations is performed by means of the piecewise exact method. Interp can be ’foh’ (first order hold) or ’zoh’ (zero order hold). In the first case the excitation is assumed to vary linearly within each time step while in the second case it is assumed to be constant: if t(k) {EltTypID EltName} Types= {1 beam ; 2 truss }; b=0.10;
Tutorial: static analysis
73
h=0.25; r=0.004; % Sections=[SecID A ky Sections= [1 b*h Inf 2 pi*r^2 NaN
kz Inf NaN
Ixx Iyy Izz 0 0 b*h^3/12 NaN NaN NaN
% Materials=[MatID E nu]; Materials= [ 1 30e6 0.2; 2 210e6 0.3]; % Elements=[EltID Elements= [1 2 3 4
TypID 1 1 1 2
SecID 1 1 1 2
MatID 1 1 1 2
yt h/2 NaN
yb h/2 NaN
zt b/2 NaN
zb] b/2; NaN];
% concrete % steel n1 1 3 5 1
n2 2 4 4 4
n3] 6; 6; 6; NaN];
% Check node and element definitions as follows: hold( on ); plotelem(Nodes,Elements,Types); title( Nodes and elements ); % Degrees of freedom % Assemble a column matrix containing all DOFs at which stiffness is % present in the model: DOF=getdof(Elements,Types); % Remove all DOFs equal to zero from the vector: % - 2D analysis: select only UX,UY,ROTZ % - clamp node 1 % - hinge at node 5 seldof=[0.03; 0.04; 0.05; 1.00; 5.01; 5.02]; DOF=removedof(DOF,seldof); % Assembly of stiffness matrix K K=asmkm(Nodes,Elements,Types,Sections,Materials,DOF); % Nodal loads: 5 kN horizontally on node 4. seldof=[4.01]; PLoad= [5]; % Assembly of the load vectors: P=nodalvalues(DOF,seldof,PLoad); % Distributed loads are specified in the global coordinate system % DLoads=[EltID n1globalX n1globalY n1globalZ ...] DLoads= [1 2 0 0 2 0 0]; P=P+elemloads(DLoads,Nodes,Elements,Types,DOF); % Constraint equations: Constant=Coef1*DOF1+Coef2*DOF2+ ... % Constraints=[Constant Coef1 DOF1 Coef2 DOF2 ...] Constr= [0 1 2.01 -1 3.01; 0 1 2.02 -1 3.02]; % Add constraint equations
74
[K,P]=addconstr(Constr,DOF,K,P); % Solve K * U = P U=K\P; % Plot displacements figure plotdisp(Nodes,Elements,Types,DOF,U,DLoads,Sections,Materials) % The displacements can be displayed as follows: printdisp(Nodes,DOF,U); % Compute element forces Forces=elemforces(Nodes,Elements,Types,Sections,Materials,DOF,U,DLoads); % The element forces can be displayed in a orderly table: printforc(Elements,Forces); % Plot element forces figure plotforc( norm ,Nodes,Elements,Types,Forces,DLoads) title( Normal forces ) figure plotforc( sheary ,Nodes,Elements,Types,Forces,DLoads) title( Shear forces ) figure plotforc( momz ,Nodes,Elements,Types,Forces,DLoads) title( Bending moments ) % Plot stresses figure plotstress( snorm ,Nodes,Elements,Types,Sections,Forces,DLoads) title( Normal stresses due to normal forces ) figure plotstress( smomzt ,Nodes,Elements,Types,Sections,Forces,DLoads) title( Normal stresses due to bending moments around z: top ) figure plotstress( smomzb ,Nodes,Elements,Types,Sections,Forces,DLoads) title( Normal stresses due to bending moments around z: bottom ) figure plotstress( smax ,Nodes,Elements,Types,Sections,Forces,DLoads) title( Maximal normal stresses ) figure plotstress( smin ,Nodes,Elements,Types,Sections,Forces,DLoads) title( Minimal normal stresses )
Tutorial
76
Tutorial
173714
−71 24
−271 −71
24
−271
173714
(a) normal forces 2325 −2325
0
0
760
−760
−8525
0
(b) b ending moments around z: top
8525
0
(c) b ending moments around z: b ottom 173714
−71 24
2325 −2325
173714
2254 2054
−71 24
785
−2597 −2397
−736
173714
173714
8549
(c) maximal
−271
−8501
(d) minimal
Figure 2.3: Normal stresses.
−271
Tutorial: dynamic analysis
2.2
77
Tutorial: dynamic analysis
A dynamic analysis is made of the frame that was treated in section 2.1. % StaBIL manual % Tutorial: dynamic analysis: model % Units: m, N L=4; H=4; nElemCable=8; % Nodes=[NodID X Y Z] Nodes= [1 0 0 0; 2 L 0 0]; Nodes=reprow(Nodes,1:2,4,[2 0 H/4 0]); Nodes= [Nodes; 11 0 H 0]; Nodes=reprow(Nodes,11,3,[1 L/4 0 0]); Nodes= [Nodes; 15 1 5 0]; % reference node Nodes= [Nodes; 16 0 0 0]; Nodes=reprow(Nodes,16,nElemCable,[1 L/nElemCable H/nElemCable 0]); % Check the node coordinates as follows: figure plotnodes(Nodes); % Element types -> {EltTypID EltName} Types= {1 beam }; b=0.10; h=0.25; r=0.004; % Sections=[SecID A ky Sections= [1 b*h Inf 2 pi*r^2 Inf
kz Inf Inf
Ixx Iyy Izz] 0 0 b*h^3/12; 0 0 pi*r^4/4];
% Materials=[MatID E nu]; Materials= [1 30e9 0.2 2500; 2 210e9 0.3 7850];
% concrete % steel
% Elements=[EltID TypID SecID MatID n1 n2 n3] Elements= [1 1 1 1 1 3 15; 2 1 1 1 2 4 15]; Elements=reprow(Elements,1:2,3,[2 0 0 0 2 2 0]); Elements=[ Elements; 9 1 1 1 11 12 15; 10 1 1 1 12 13 15; 11 1 1 1 13 14 15; 12 1 1 1 14 10 15; 13 1 2 2 16 17 15]; Elements=reprow(Elements,13,(nElemCable-1),[1 0 0 0 1 1 0]);
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Tutorial
% Check node and element definitions as follows: hold( on ); plotelem(Nodes,Elements,Types); title( Nodes and elements ); % Degrees of freedom DOF=getdof(Elements,Types); % Boundary conditions seldof=[0.03; 0.04; 0.05; 1.01; 1.02; 1.06; 2.01; 2.02; 16.01; 16.02]; DOF=removedof(DOF,seldof); % Assembly of stiffness matrix K [K,M]=asmkm(Nodes,Elements,Types,Sections,Materials,DOF); % DLoads=[EltID n1globalX n1globalY n1globalZ ...] DLoads=[1 2000 0 0 2000 0 0; 3 2000 0 0 2000 0 0; 5 2000 0 0 2000 0 0; 7 2000 0 0 2000 0 0]; b=elemloads(DLoads,Nodes,Elements,Types,DOF); % Spatial distribution, nodal (nDOF * 1) % Constraint equations: Constant=Coef1*DOF1+Coef2*DOF2+ ... % Constraints=[Constant Coef1 DOF1 Coef2 DOF2 ...] Constr= [0 1 9.01 -1 11.01; 0 1 9.02 -1 11.02; 0 1 10.01 -1 (16.01+nElemCable); 0 1 10.02 -1 (16.02+nElemCable)]; [K,b,M]=addconstr(Constr,DOF,K,b,M);
2.2.1
Eigenmodes
% StaBIL manual % Tutorial: dynamic analysis: eigenvalue problem % Units: m, N % Assembly of M and K tutorialdyna; % Eigenvalue problem nMode=12; [phi,omega]=eigfem(K,M,nMode); % Display eigenfrequenties disp( Lowest eigenfrequencies [Hz] ); disp(omega/2/pi); % Plot eigenmodes figure; plotdisp(Nodes,Elements,Types,DOF,phi(:,1), DispMax , off ) figure; plotdisp(Nodes,Elements,Types,DOF,phi(:,2), DispMax , off ) figure;
Tutorial: dynamic analysis
79
plotdisp(Nodes,Elements,Types,DOF,phi(:,5), DispMax , off ) figure; plotdisp(Nodes,Elements,Types,DOF,phi(:,8), DispMax , off ) figure; plotdisp(Nodes,Elements,Types,DOF,phi(:,11), DispMax , off ) figure; plotdisp(Nodes,Elements,Types,DOF,phi(:,12), DispMax , off ) % Animate eigenmodes figure; animdisp(Nodes,Elements,Types,DOF,phi(:,1)) title( Eigenmode 1 ) figure; animdisp(Nodes,Elements,Types,DOF,phi(:,2)) title( Eigenmode 2 ) figure; animdisp(Nodes,Elements,Types,DOF,phi(:,5)) title( Eigenmode 5 ) figure; animdisp(Nodes,Elements,Types,DOF,phi(:,8)) title( Eigenmode 8 ) figure; animdisp(Nodes,Elements,Types,DOF,phi(:,11)) title( Eigenmode 11 ) figure; animdisp(Nodes,Elements,Types,DOF,phi(:,12)) title( Eigenmode 12 )
2.2.2
Modal superposition: time domain: piecewise exact integration
% StaBIL manual % Tutorial: dynamic analysis: modal superposition: piecewise exact integration % Units: m, N % Assembly of M and K tutorialdyna; % Sampling parameters T=2.5; dt=0.002; N=T/dt; t=[0:N-1]*dt;
% % % %
% Eigenvalue analysis nMode=12; [phi,omega]=eigfem(K,M,nMode); xi=0.07;
% Number of modes to take into account % Calculate eigenmodes and eigenfrequencies % Constant modal damping ratio
% Excitation bm=phi. *b;
% Spatial distribution, modal (nMode * 1)
Time window Time step Number of samples Time axis
80
Tutorial
(a) Eigenmode 1
(b) Eigenmode 2
(b) Eigenmode 5
(c) Eigenmode 8
(d) Eigenmode 11
(e) Eigenmode 12
Figure 2.4: Results of example 1.
Tutorial: dynamic analysis
q=zeros(1,N); q((t>=0.50) & (t nodal displacements u=phi*x; % Nodal response (nDOF * N) % Figures figure; plot(t,x); title( Modal response (piecewise linear exact integration) ); xlabel( Time [s] ); xlim([0 4.1]) ylabel( Displacement [m kg^{0.5}] ); legend([repmat( Mode ,nMode,1) num2str([1:nMode]. )]); figure; c=selectdof(DOF,[9.01; 13.02; 17.02]); plot(t,c*u); title( Nodal response (piecewise linear exact integration) ); xlabel( Time [s] ); xlim([0 4.1]) ylabel( Displacement [m] ); legend( 9.01 , 13.02 , 17.02 ); % Movie figure; animdisp(Nodes,Elements,Types,DOF,u); % Display disp( Maximum modal response ); disp( max(abs(x),[],2)); disp( Maximum nodal response 9.01 13.02 17.02 ); disp( max(abs(c*u),[],2));
2.2.3
Modal superposition: transform to frequency domain
% StaBIL manual % Tutorial: dynamic analysis: direct method: frequency domain % Units: m, N % Assembly of M and K tutorialdyna; % Sampling parameters N=2048; dt=0.002; T=N*dt; F=N/T; df=1/T; t=[0:N-1]*dt;
% % % % % %
Number of samples Time step Period Sampling frequency Frequency resolution Time axis
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Tutorial
f=[0:N/2-1]*df; Omega=2*pi Omega=2*pi*f; *f;
% Pos Positiv itive e fre freque quenci ncies es cor corres respon pondin ding g to FFT [Hz [Hz] ] % Ide Idem m [ra [rad/ d/s] s]
% Eigenv Eigenvalue alue analysis nMode=12; [phi,omega]=eigfem [phi,omega]=eigfem(K,M,nMode); (K,M,nMode); xi=0.07;
% Numb Number er of mo mode des s to ta take ke in into to ac acco coun unt t % Calcula Calculate te eigenmodes eigenmodes and eigenf eigenfrequen requencies cies % Con Constan stant t mod modal al dam dampin ping g rat ratio io
% Excita Excitation tion bm=phi. *b; q=zeros q=zeros(1,N); (1,N); q((t>=0.50) q((t>=0.50) & (t nod nodal al dis displa placem cements ents u=phi*x; % No Noda dal l re resp spon onse se (n (nDO DOF F * N) % Fig Figure ures s figure; figure ; subplot(3,2,1); subplot (3,2,1); plot(t,q, plot (t,q, .- ); xlim([0 xlim([0 4.1]) ylim([0 ylim([0 1.2]); title( title ( Excitation Excitation time histor history y ); xlabel( xlabel ( Time [s] ); ylabel( ylabel ( Force [N/m] ); subplot(3,2,2); subplot(3,2,2); plot(f, plot (f,abs abs(Q)/F, (Q)/F, .- ); title( title ( Excitation Excitation frequen frequency cy conten content t ); xlabel( xlabel ( Frequency Frequency [Hz] ); ylabel( ylabel ( Force [N/m/ [N/m/Hz] Hz] ); subplot(3,2,4); subplot(3,2,4); plot(f, plot (f,abs abs(H), (H), .- ); title( title ( Modal transf transfer er functi function on ); xlabel( xlabel ( Frequency Frequency [Hz] ); ylabel( ylabel ( Displacement Displacement [m/N] ); legend([repmat( legend ([repmat( Mode ,nMode,1) num2str([1:nMode]. num2str([1:nMode]. )]); subplot(3,2,6); subplot(3,2,6); plot(f, plot (f,abs abs(X(:,1:N/2))/F, (X(:,1:N/2))/F, .- ); title( title ( Modal respon response se ); xlabel( xlabel ( Frequency Frequency [Hz] ); ylabel( ylabel ( Displacement Displacement [m kg^{0 kg^{0.5}/ .5}/Hz] Hz] ); subplot(3,2,5); subplot (3,2,5);
Tutorial: dynamic analysis
83
plot(t,x); plot(t,x); title( title ( Modal response (calculation (calculation in f-dom f-dom) ) ); xlabel( xlabel ( Time [s] ); xlim([0 xlim([0 4.1]) ylabel( ylabel ( Displacement Displacement [m kg^{0 kg^{0.5}] .5}] ); figure; figure; plot(t,x); plot (t,x); title( title ( Modal response (calculation (calculation in f-dom f-dom) ) ); xlabel( xlabel ( Time [s] ); xlim([0 xlim([0 4.1]) ylabel( ylabel ( Displacement Displacement [m kg^{0 kg^{0.5}] .5}] ); legend([repmat( legend ([repmat( Mode ,nMode,1) num2str([1:nMode]. num2str([1:nMode]. )]); figure; figure; c=selectdof c=selectdof(DOF (DOF,[9.01 ,[9.01; ; 13.02; 17.02]); 17.02]); plot(t,c*u); plot (t,c*u); title( title ( Nodal response (calculation (calculation in f-dom f-dom) ) ); xlabel( xlabel ( Time [s] ); xlim([0 xlim([0 4.1]) ylabel( ylabel ( Displacement Displacement [m] ); legend( legend ( 9.01 , 13.02 , 17.02 ); % Mo Movi vie e figure; figure ; animdisp(Nodes,Elements,Types,DOF,u); animdisp (Nodes,Elements,Types,DOF,u); % Dis Displa play y disp( disp ( Maximum Maximum modal respon response se ); disp( disp ( max max( (abs abs(x),[],2)); (x),[],2)); disp( Maximum disp( Maximum nod nodal al res respon ponse se 9.0 9.01 1 13. 13.02 02 17. 17.02 02 ); disp( disp ( max max( (abs abs(c*u),[],2)); (c*u),[],2));
2.2.4 2.2.4
Direc Directt tim time e int integr egrati ation on
% StaBIL StaBIL manual % Tutori Tutorial: al: dynamic analys analysis: is: direct time integration: integration: trapez trapezium ium rule % Un Unit its s: m, N % Assembly of M, K and C tutorialdyna; [phi,omega]=eigfem [phi,omega]=eigfem(K,M); (K,M); % Calcul Calculate ate eigenm eigenmodes odes and eigenf eigenfrequen requencies cies xi=0.07; % Dam Dampin ping g rat ratio io nModes=length nModes=length(K)(K)-size size(Constr,1); (Constr,1); C=M. *phi(:,1:nModes)*diag *phi(:,1:nModes)*diag(2*xi*omega(1:nModes))*phi(:,1:nModes). (2*xi*omega(1:nModes))*phi(:,1:nModes). *M; % Mo Moda dal l -> fu full ll da damp mpin ing g ma matr trix ix C % Sampli Sampling ng parame parameters ters T=2.5; dt=0.002; N=T/dt; t=[0:N-1]*dt; % Excita Excitation tion
% % % %
Time win Time window dow Time Ti me st step ep Number Num ber of sam sample ples s Time Ti me ax axis is
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Tutorial
Excitation time history
Excitation frequency content 0.1
1
0.08
] z H / m0.06 / N [ e 0.04 c r o F
] 0.8 m / N [ e 0.6 c r o F 0.4
0.02
0.2 0 0
1
2 Time [s]
3
0
4
40
50
] N0.6 / m [ t n e m0.4 e c a l p s 0.2 i D
0
0.15
0
10 −3
8
20 30 Frequency [Hz]
40
50
40
50
Modal response
x 10
] z H /
]
5 . 0
0.1
6
g k m [ t 4 n e m e c a 2 l p s i D
0.05
0
−0.05 0
20 30 Frequency [Hz]
0.8
Modal response (calculation in f−dom)
g k m [ t n e m e c a l p s i D
10
Modal transfer function
Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 Mode 6 Mode 7 Mode 8 Mode 9 Mode 10 Mode 11 Mode 12
5 . 0
0
1
2 Time [s]
3
4
0
0
10
20 30 Frequency [Hz]
Figure 2.5: Modal superposition in frequency domain. q=zeros q=zeros(1,N); (1,N); q((t>=0.50) q((t>=0.50) & (t=0.50) & (t t-dom Ud=[Ud, zeros(length(K),1), conj(Ud(:,end:-1:2))]; u=ifft(Ud,[],2); % Nodal response (nDOF * N) % Figures figure; c=selectdof(DOF,[9.01; 13.02; 17.02]); plot(t,c*u); title( Nodal response (direct method in f-dom) ); xlabel( Time [s] ); xlim([0 4.1]) ylabel( Displacement [m] ); legend( 9.01 , 13.02 , 17.02 ); % Movie figure; animdisp(Nodes,Elements,Types,DOF,u); % Display disp( Maximum nodal response 9.01 13.02 17.02 ); disp( max(abs(c*u),[],2));
2.2.6
Comparison
Computational cost in seconds Modal superposition Direct method
Time domain 0.167 0.213
Freq domain 0.147 0.789
Tutorial: dynamic analysis
87
−3
5
−3
x 10
5 9.01 13.02 17.02
4
] m [ t n e m e c a l p s i D
3
1
3 2 1
0
0
−1
−1
−2
0
0.5
1
1.5
2 2.5 Time [s]
3
3.5
4
(a) Modal superposition in time domain
−2 0
5 9.01 13.02 17.02
3
1
2 2.5 Time [s]
3
3.5
4
1
−1
−1
1
1.5
(b) Direct time integration
2 2.5 Time [s]
3
3.5
4
9.01 13.02 17.02
2
0
0.5
x 10
3
0
0
1.5
4
] m [ t n e m e c a l p s i D
2
−2
1
−3
x 10
4 ] m [ s t n e m e c a l p s i d l a d o N
0.5
(b) Modal superposition in frequency domain
−3
5
9.01 13.02 17.02
4
] m [ t n e m e c a l p s i D
2
x 10
−2 0
0.5
1
1.5
2 2.5 Time [s]
(c) Direct method in frequency domain
Figure 2.7: Nodal response: comparison.
3
3.5
4
88
2.3
Tutorial
Tutorial: shell analysis
A barrel vault roof subjected to its self weight is analysed. The curved edges are simply supported and the straight edges are free. Due to symmetry only a quarter of the roof is modelled and symmetry boundary conditions are applied.
Edge 2: simply supported
Edge 1: symmetry conditions
Edge 3: free boundary
zy x
Edge 4: symmetry conditions
Figure 2.8: 8 × 8 mesh of a quarter of a cylindrical roof. % A BARREL VAULT ROOF SUBJECTED TO ITS SELF WEIGHT % Reference: COOK, CONCEPTS AND APPL OF F.E.A., 2ND ED., 1981, PP. 284-287. %% Parameters R=25; L=50; t=0.25; theta = 40*pi/180; E = 4.32*10^8; nu = 0; rho = 36.7347; N = 8;
% % % % % % % %
Radius of cylindrical roof Length of cylindrical roof Thickness of roof Angle of cylindrical roof Youngs modulus Poisson coefficient Density Number of elements
%% Mesh % Lines Line1 = Line2 = Line3 =
= [x1 y1 z1;x2 y2 z2;...] [0 0 0;0 0 L/2]; [R*sin(theta*(0:0.1:1). ) R*cos(theta*(0:0.1:1). )-R L*ones(11,1)/2]; [R*sin(theta) R*cos(theta)-R L/2; R*sin(theta) R*cos(theta)-R 0];
Tutorial: shell analysis
Line4 = [R*sin(theta*(1:-0.1:0). ) R*cos(theta*(1:-0.1:0). )-R zeros(11,1)]; % Specify element type for mesh Materials = [1 E nu rho]; Sections = [1 t]; Types = {1 shell8 }; % Mesh the area between lines 1,2,3,4 with N * N elements of type shell8, % section number 1 and material number 1. [Nodes,Elements,Edge1,Edge2,Edge3,Edge4] = ... makemesh(Line1,Line2,Line3,Line4,N,N,Types(1,:),1,1); % Check mesh: figure; plotnodes(Nodes, numbering , off ); hold( on ) plotelem(Nodes,Elements,Types, numbering , off ); title( Nodes and elements ); %% Assemble stiffness matrix % Select all dof: DOF = getdof(Elements,Types); % Apply boundary conditions: % - Line1 and Line4: symmetry condition % - Line2: simply supported sdof = [Edge1+0.01;Edge1+0.06;Edge1+0.05;Edge2+0.02;Edge2+0.01; Edge2+0.06;Edge4+0.03;Edge4+0.04;Edge4+0.05]; DOF = removedof(DOF,sdof); % Assemble K: K = asmkm(Nodes,Elements,Types,Sections,Materials,DOF); %% Solution % Apply gravitational acceleration and determine equivalent nodal forces: DLoads=accel([0 9.8 0],Elements,Types,Sections,Materials); P=elemloads(DLoads,Nodes,Elements,Types,DOF);
% Solve K * U = P: U = K\P; % Plot displacements: figure; plotdisp(Nodes,Elements,Types,DOF,U) title( Displacements ) % Check target displacement: TP1 = selectdof(DOF,intersect(Edge3,Edge4)+0.02); Utp1 = TP1*U ratio_u = -Utp1/0.3016 %% Stress
89
90
% Determine element stress in global and local(element) coordinate system: [SeGCS,SeLCS,vLCS]=elemstress(Nodes,Elements,Types,Sections,Materials,DOF,U); % print stress: printstress(Elements,SeGCS) % plot stress contour: figure; plotstresscontour( sx ,Nodes,Elements,Types,SeGCS, location , bot ) title( sx in gcs (element solution) ) % plot filled contours: figure; plotstresscontourf( sx ,Nodes,Elements,Types,SeGCS, location , bot ) title( sx in gcs (element solution) ) figure; plotstresscontour( sx ,Nodes,Elements,Types,SeLCS, location , bot ) title( sx in lcs ) % plot lcs for shell elements figure; plotlcs(Nodes,Elements,Types,vLCS) title( local coordinate system ) % Calculate nodal solution % SnGCS: stress arranged per element % SnGCS2: stress arranged per node [SnGCS,SnGCS2]=nodalstress(Nodes,Elements,Types,SeGCS); figure; plotstresscontour( sx ,Nodes,Elements,Types,SnGCS, location , bot ); title( sx in gcs (nodal solution) ) % Stress ratios: ratio_sz = SnGCS2(intersect(Edge3,Edge4),16)/358420 ratio_st = SnGCS2(intersect(Edge4,Edge1),14)/(-213400) %% Shell forces % Shell forces (element solution): [FeLCS]=elemshellf(Elements,Sections,SeLCS); figure; plotshellfcontour( my ,Nodes,Elements,Types,FeLCS) title( my (element solution) ) % Nodal solution: [FnLCS,FnLCS2]=nodalshellf(Nodes,Elements,Types,FeLCS); figure; plotshellfcontour( my ,Nodes,Elements,Types,FnLCS) title( my (nodal solution) ) %% Principal stress % Principal stresses: [Spr,Vpr]=principalstress(Elements,SnGCS);
Tutorial
Tutorial: shell analysis
figure; plotstresscontour( s1 ,Nodes,Elements,Types,Spr, location , bot ); title( s1 ) % plot principal stresses: figure; plotprincstress(Nodes,Elements,Types,Spr,Vpr) title( principal stress directions )
91
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Tutorial
zy x
(a) Local coordinate systems
(b) Displacements
5
0.0599
5
x 10
3.5841
−0.1401
3.2082
−0.3401
2.8323
−0.5401
2.4564
−0.7401
2.0804
−0.9401
1.7045
−1.1401
1.3286
−1.3401
0.9527 zy
−1.5401 x −1.7401 −1.9401
σxx
zy
0.5768 x
MIN
0.2008
MIN
−0.1751
MAX
−2.1401
(c)
x 10
−0.551
at bottom of shell in GCS (element solution)
(d)
σxx
MAX
at bottom of shell in LCS (element solution)
2079.965 1881.9048 1683.8446 1485.7844 1287.7242 1089.6639 891.6037 693.5435 zy
495.4833 x 297.4231 99.3629
zy
MAX
x
MIN
−98.6973
(e)
my
in LCS (nodal solution)
(f ) Principal stresses at top of shell
Figure 2.9: Cylindrical roof: results.
Tutorial: nonlinear analysis
2.4
93
Tutorial: nonlinear analysis
This tutorial performs a geometric nonlinear analysis of an arc loaded by a point force. Several methods are used to compute the critical buckling load. clear %% GEOMETRIC NONLINEAR ARC % (Example based on p268 of Rin = 100; Rout = 103; alpha = 60*pi/180; E = 40000; nu = 0.2; m = 6; n = 150; Types = {1 plane4 }; Sections = [1 1]; Materials = [1 E nu];
Nonlinear Finite Element Methods, Wriggers (2008)) % Inner radius arc % Outer radius arc % Arc angle % Young s modulus % Poisson s coefficient % number of elements in radial direction % number of elements in tangential direction % {EltTypID EltName} % [SecID t] % [MatID E nu]
%% MESH npoints = 100; theta = linspace(-alpha/2+pi/2,alpha/2+pi/2,npoints). ; [Xin,Yin] = pol2cart(flipud(theta),Rin*ones(npoints,1)); [Xout,Yout] = pol2cart(theta,Rout*ones(npoints,1)); Yin = Yin-Rin*cos(alpha/2); Yout = Yout-Rin*cos(alpha/2); Z = zeros(npoints,1); Line1 = [Xin(end) Yin(end) 0;Xout(1) Yout(1) 0]; Line2 = [Xout Yout Z]; Line3 = [Xout(end) Yout(end) 0;Xin(1) Yin(1) 0]; Line4 = [Xin Yin Z]; [Nodes,Elements,Edge1,Edge2,Edge3,Edge4] = makemesh(Line1,Line2,Line3,Line4,n,m,Types(1,:),1,1); figure; plotelem(Nodes,Elements,Types, numbering , off ); title( Elements ); %% DOF DOF = getdof(Elements,Types); sdof = [Edge1((end-1)/2+1);Edge3((end-1)/2+1)]; sdof = [sdof+0.01;sdof+0.02]; DOF = removedof(DOF,sdof); %% LOAD pdof = Edge2((end-1)/2+1)+0.02; Lp = selectdof(DOF,pdof); pmax = 1000; P = -pmax*Lp. ; %% LINEAR ANALYSIS K = asmkm(Nodes,Elements,Types,Sections,Materials,DOF); U = K\P; %% Newton Raphson Options.method = nr ; Options.nloadincrem = 50;
% Nonlinear solver: Newton-Raphson % Number of load increments
Tutorial: nonlinear analysis
95
hold on; plot(-Lp*Ual,lambda_al*pmax); plot(-Lp*Ual(:,alstability.nonpositivepivots>0),lambda_al(alstability.nonpositivepivots>0)*pmax, x- ) xlabel( U ); ylabel( P );axis tight; ax = axis; plot([ax(1),ax(2)],[lambda(1) lambda(1)], k-- ); plot(-Lp*Unlcr,lambdanlcr*pmax, rx ); plot(-Lp*Ual2,lambda_al2*pmax, r-- ) legend( Stable path , Unstable path , Linear critical load ,... Nonlinear critical load , Bifurcated path ); title( Load-displacement curve ); hold off; %% ANIMATE LOAD DISPLACEMENT CURVE nstep = size(Ual,2); figure for istep = [(1:10:nstep),nstep] clf subplot(2,1,1) plotdisp(Nodes,Elements,Types,DOF,Ual(:,istep), dispscal ,1, dispmax , off ); subplot(2,1,2) plot(-Lp*Ual(:,1:istep),pmax*lambda_al(:,1:istep), .- ); xlabel( U ); ylabel( P ); axis(ax); pause(0.1) end
96
Tutorial
1000
Stable path Unstable path Linear critical load Nonlinear critical load Bifurcated path
800 600 400 200 P
y z x
0 −200 −400 −600 −800 0
(a) Mesh
Maximal displacement: 8.76453
(c) Linear small displacements
5
10
15 U
(b) Nonlinear load-displacement curve
Maximal displacement: 30.7559
(d) Nonlinear displacements
Figure 2.10: Arc: results.
20
25
30
Chapter 3
Examples
98
3.1
Examples
Example 1
A frame is clamped clamped at node 1 and has a hinge hinge at node 8. An intern internal al hinge is presen presentt at nodes 2 and 3. At node 4 a point load load of −20 kN is applied and at node 7 a bending b ending momen momentt of −1 kNm is applied. applied. At node 8 a vertical displacement of −0.02m is imposed. The frame consists consists of rectangular rectangular concrete beams (Young’s (Young’s modulus E modulus E = = 30 × 106 kN kN//m2 and Poisson coefficient ν coefficient ν = = 0.2) with a height of 0. 0 .4m and a width of 0. 0 .2 m. 2 3
4
2
3
5 4 6
5
7
9
1
6
y 1z
x
8
Figure 3.1: Example 1: nodes, elements, loads and boundary conditions. % StaBIL StaBIL manual % Ex Exam ampl ple e 1 % Un Unit its s: m, kN % Nod Nodes es=[ =[No NodI dID D Nodes= [1 [1 2 3 4 5 6 7 8 9
X 0 0 0 4 6 6 7 6 2
Y 0 4 4 4 4 3 3 0 2
Z] 0; 0; 0; 0; 0; 0; 0; 0; 0; 0];
% ref refere erence nce nod node e
% Che Check ck the nod node e coo coordin rdinate ates s as fol follow lows: s: figure plotnodes(Nodes); plotnodes (Nodes); % Ele Elemen ment t typ types es -> {El {EltTy tTypID pID EltName} EltName} Type Types= s= {1 beam }; b=0.20; h=0.40; % Sections=[SecID A Sections= [1 [1 b*h
ky Inf
kz In I nf
Ixx Iyy Izz] 0 0 b*h^3/12];
Example 1
99
% Mat Materi erials als=[M =[MatI atID D E nu] nu]; ; Materials= [1 30e6 0.2]; % Ele Elemen ments ts=[E =[EltI ltID D Elements= [1 2 3 4 5 6
TypID Typ ID 1 1 1 1 1 1
SecID Sec ID 1 1 1 1 1 1
MatID Mat ID 1 1 1 1 1 1
% con concre crete te n1 1 3 4 5 6 6
n2 2 4 5 6 7 8
n3] 9; 9; 9; 9; 9; 9];
% Che Check ck ele elemen ment t def defini initio tions ns as fol follow lows: s: hold( hold ( on ); plotelem(Nodes,Elements,Types); plotelem (Nodes,Elements,Types); title( title ( Nodes and ele element ments s ); % Deg Degree rees s of fre freedo edom m % Ass Assemb emble le a col column umn matrix containi containing ng all DOF DOFs s at whi which ch sti stiffn ffness ess is % pr pres esen ent t in th the e mo mode del: l: DOF=getdof DOF=getdof(Elements,Types); (Elements,Types); % Re Remo move ve al all l DO DOFs Fs eq equal ual to ze zero ro fr from om th the e ve vect ctor or: : % - 2D analys analysis is: : se selec lect t on only ly UX,UY UX,UY,R ,ROT OTZ Z % - cla clamp no n ode 1 % - re remo move ve the horizo horizont ntal al displ displac acem emen ent t at no node de 8 seldof=[0.03 seldof=[0.03; ; 0.04; 0.05; 1.00; 8.01]; DOF=removedof DOF=removedof(DOF,seldof); (DOF,seldof); % Ass Assemb embly ly of sti stiffn ffness ess matrix matrix K K=asmkm K=asmkm(Nodes,Elements,Types (Nodes,Elements,Types,Sections,Materials ,Sections,Materials,DOF); ,DOF); % No Noda dal l lo load ads: s: -2 -20 0 kN vert vertic ical ally ly on no node de 4 % -1 kNm on node 7. seldof=[4.02 seldof=[4.02; ; 7.06]; PLoad=[-20; PLoad=[-20; -1]; % Ass Assemb embly ly of the load vec vector tors: s: P=nodalvalues P=nodalvalues(DOF,seldof,PLoad); (DOF,seldof,PLoad); % Constr Constraint aint equati equations ons: : Consta Constant= nt=Coef1 Coef1*DOF1 *DOF1+Coef2 +Coef2*DOF2 *DOF2+ + ... % Constra Constraint ints s=[C =[Cons onstan tant t Coe Coef1 f1 DOF1 DOF1 Coe Coef2 f2 DOF2 DOF2 ...] ...] % Imp Impose osed d dis displa placem cement ent -0.02 m ver vertic ticall ally y in nod node e 8. Constr= [0 1 2.01 -1 3.01; 0 1 2.02 -1 3.02; -0.02 1 8 . 02 Na N aN NaN]; % Add con constr strain aint t equ equatio ations ns [K,P]=addconstr [K,P]=addconstr(Constr,DOF,K,P); (Constr,DOF,K,P); % Solve K * U = P U=K\P; % Plot displacements displacements figure plotdisp(Nodes,Elements,Types,DOF,U) plotdisp (Nodes,Elements,Types,DOF,U)
100
% The dis displa placem cement ents s can be dis displa played yed as fol follow lows: s: printdisp(Nodes,DOF,U); printdisp (Nodes,DOF,U); % Com Comput pute e ele elemen ment t for forces ces Forces=elemforces Forces=elemforces(Nodes,Elements,Types (Nodes,Elements,Types,Sections,Materials ,Sections,Materials,DOF,U); ,DOF,U); % Th The e el elem emen ent t fo forc rces es ca can n be di disp spla laye yed d in a or orde derl rly y ta tabl ble: e: printforc(Elements,Forces); printforc (Elements,Forces); % Plo Plot t ele elemen ment t for forces ces figure plotforc( plotforc ( norm ,Nodes,Elements,Types,Forces) title( title ( Normal forces ) figure plotforc( plotforc ( sheary ,Nodes,Elements,Types,Forces) title( title ( Shear forces ) figure plotforc( plotforc ( momz ,Nodes,Elements,Types,Forces) title( title ( Bending moment moments s )
Examples
Example 1
101
−8.823 3.485
−11.177 3.485
3.485 0.000 −11.177 0.000
−8.823
(a) Displacements
(b) Normal forces −11.177
3.485 8.823
3.485
(c) Shear forces
−11.177
−11.177
−3.485 8.823
0.000 0.000
−0.000 −3.485 −3.485 −0.000
−3.485
12.939 12.939 35.293 35.293
−13.939
(d) Bending moments
Figure 3.2: Results of example 1.
−1.000 −1.000 10.454 9.454
0.000
102
3.2
Examples
Example 2 7
y 1z
x
2
1
2
3
3
4
4
5
5
6
Figure 3.3: Example 2: nodes, elements, loads and boundary conditions. % StaBIL manual % Example 2 % Units: m, kN % Nodes=[NodID Nodes= [1 2 3 4 5 6 7
X Y Z] 0 0 0.5 0 1 0 0 -0.25 1 -0.25 1.25 0 0.3 0.4
0; 0; 0; 0; 0; 0; 0];
% reference node
% Check the node coordinates as follows: figure plotnodes(Nodes); % Element types -> {EltTypID EltName} Types= {1 beam ; 2 truss }; b=0.02; h=0.05; A=0.0002; % Sections=[SecID A Sections= [1 b*h 2 A
ky Inf NaN
kz Inf NaN
% Materials=[MatID E nu]; Materials= [1 210e6 0.3];
Ixx Iyy Izz] 0 0 b*h^3/12; NaN NaN NaN];
% steel
% Elements=[EltID TypID SecID MatID n1 n2 n3] Elements= [1 1 1 1 1 2 7; 2 1 1 1 2 3 7;
Example 2
103
3 4 5
2 2 2
2 2 2
1 1 1
1 3 3
4 5 6
NaN; NaN; NaN];
% Check element definitions as follows: hold( on ); plotelem(Nodes,Elements,Types); title( Nodes and elements ); % Degrees of freedom % Assemble a column matrix containing all DOFs at which stiffness is % present in the model: DOF=getdof(Elements,Types); % Remove all DOFs equal to zero from the vector: % - 2D analysis: select only UX,UY,ROTZ % - clamp nodes 4, 5, 6 seldof=[0.03; 0.04; 0.05; 4.00; 5.00; 6.00]; DOF=removedof(DOF,seldof); % Assembly of stiffness matrix K K=asmkm(Nodes,Elements,Types,Sections,Materials,DOF); % Distributed loads are specified in the global coordinate system % DLoads=[EltID n1globalX n1globalY n1globalZ ...] DLoads= [1 0 0 0 0 -0.02 0; 2 0 -0.02 0 0 0 0;]; P=elemloads(DLoads,Nodes,Elements,Types,DOF); % Solve K * U = P U=K\P; % Plot displacements figure plotdisp(Nodes,Elements,Types,DOF,U,DLoads,Sections,Materials) title( Displacements ) % The displacements can be displayed as follows: printdisp(Nodes,DOF,U); % Compute element forces Forces=elemforces(Nodes,Elements,Types,Sections,Materials,DOF,U,DLoads); % The element forces can be displayed in an orderly table: printforc(Elements,Forces); % Plot element forces figure plotforc( norm ,Nodes,Elements,Types,Forces,DLoads) title( Normal forces ) figure plotforc( sheary ,Nodes,Elements,Types,Forces,DLoads) title( Shear forces )
104
Examples
figure plotforc( momz ,Nodes,Elements,Types,Forces,DLoads) title( Bending moments )
−5000e−6
0e−6
0e−6
−5000e−6
(a) Displacements
−5000e−6
0e−6
−5000e−6
(b) Normal forces 5000e−6
0e−6
0e−6
−5000e−6
(c) Shear forces
−1667e−6 −1667e−6
(d) Bending moments
Figure 3.4: Results of example 2.
0e−6
0e−6
Example 3
3.3
105
Example 3
The frame consists of rectangular concrete beams (Young’s modulus E = 30 × 106 kN/m2 and Poisson coefficient ν = 0.2) with a height of 0.2m and a width of 0.1 m.
6 8
11 10
12 3 5
15
7
14
7
9 13
2
4 8
6
10
2
3
z y 9 x
5
1
4 1 Figure 3.5: Example 3: nodes, elements, loads and boundary conditions. % StaBIL manual % Example 3 % Units: m, kN % Nodes=[NodID Nodes= [1 2 3 4 5 6 7 8 9 10
X 2.5 2.5 2.5 2.5 1.25 1.25 0 0 0 0
Y 0 5 5 0 0 5 0 5 0 5
Z] 0; 0; 2.5; 2.5; 3.75; 3.75; 2.5; 2.5; 0; 0];
% reference node
% Check the node coordinates as follows: figure
106
Examples
plotnodes(Nodes); % Element types -> {EltTypID EltName} Types= {1 beam }; b=0.1; h=0.2; % Sections=[SecID A Sections= [1 b*h
ky Inf
kz Inf
Ixx 0.229*h/b^3
% Materials=[MatID E nu]; Materials= [1 30e6 0.2]; % Elements=[EltID Elements= [1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
TypID 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
SecID 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
Iyy h*b^3/12
Izz] b*h^3/12];
% concrete MatID 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
n1 1 2 10 9 1 2 10 9 3 3 8 5 4 7 7
n2 2 10 9 1 4 3 8 7 4 6 6 6 5 5 8
n3] 3; 3; 8; 5; 3; 4; 7; 8; 2; 10; 10; 8; 9; 9; 10];
% Check node and element definitions as follows: hold( on ); plotelem(Nodes,Elements,Types); title( Nodes and elements ); DOF=getdof(Elements,Types); % Remove all DOFs equal to zero from the vector: % clamp nodes 1, 2, 9 and 10 seldof=[1.00; 2.00; 9.00; 10.00]; DOF=removedof(DOF,seldof); % Assembly of stiffness matrix K K=asmkm(Nodes,Elements,Types,Sections,Materials,DOF); % Nodal loads: load case 1: 6 kN horizontally on nodes 4, 7, 5 % load case 2: 15 kN horizontally on nodes 7 seldof=[4.02; 7.02; 5.02]; PLoad=[6 0; 6 15; 6 0]; % Assembly of the load vectors: P=nodalvalues(DOF,seldof,PLoad);
Example 3
% Solve K * U = P U=K\P; % Plot displacements figure plotdisp(Nodes,Elements,Types,DOF,U) title( Displacements ) % The displacements can be displayed as follows: printdisp(Nodes,DOF,U(:,1)); printdisp(Nodes,DOF,U(:,2)); % Compute element forces Forces=elemforces(Nodes,Elements,Types,Sections,Materials,DOF,U); % The element forces can be displayed in a orderly table: printforc(Elements,Forces(:,:,1)); printforc(Elements,Forces(:,:,2)); % Plot element forces figure plotforc( norm ,Nodes,Elements,Types,Forces(:,:,1)) title( Normal forces ) figure plotforc( sheary ,Nodes,Elements,Types,Forces(:,:,1)) title( Shear forces along the y-axis ) figure plotforc( shearz ,Nodes,Elements,Types,Forces(:,:,1)) title( Shear forces along the z-axis ) figure plotforc( momx ,Nodes,Elements,Types,Forces(:,:,1)) title( Torsional moments ) figure plotforc( momy ,Nodes,Elements,Types,Forces(:,:,1)) title( Bending moments around the y-axis ) figure plotforc( momz ,Nodes,Elements,Types,Forces(:,:,1)) title( Bending moments around the z-axis ) figure plotforc( norm ,Nodes,Elements,Types,Forces(:,:,2)) title( Normal forces ) figure plotforc( sheary ,Nodes,Elements,Types,Forces(:,:,2)) title( Shear forces along the y-axis ) figure plotforc( shearz ,Nodes,Elements,Types,Forces(:,:,2)) title( Shear forces along the z-axis )
107
108
Examples
figure plotforc( momx ,Nodes,Elements,Types,Forces(:,:,2)) title( Torsional moments ) figure plotforc( momy ,Nodes,Elements,Types,Forces(:,:,2)) title( Bending moments around the y-axis ) figure plotforc( momz ,Nodes,Elements,Types,Forces(:,:,2)) title( Bending moments around the z-axis )
Figure 3.6: Displacements for load cases 1 and 2.
Example 3
109
(a) Normal forces
(b) Torsional moments
(c) Shear forces along z
(d) Shear forces along y
(e) Bending moments around y
(f) Bending moments around z
Figure 3.7: Results for load case 1 of example 3.
110
Examples
(a) Normal forces
(b) Torsional moments
(c) Shear forces along z
(d) Shear forces along y
(e) Bending moments around y
(f) Bending moments around z
Figure 3.8: Results for load case 2 of example 3.
Example 4
3.4
111
Example 4
The frame consists of rectangular concrete beams (Young’s modulus E = 35 × 109 N/m2 and Poisson coefficient ν = 0.2) with a height of 0.5m and a width of 0.2 m and concrete columns of 0.3m by 0.2 m.
205 206
307 105
106 203
303 5
3 3
206
203
103
308 106
204
205 305 304 204 6 201 104 210 306 105310 301 309 201 104 202 101 1104
1 10 z y 1 x
6 103
5 101
302 4
2
102
202
102
2 Figure 3.9: Example 4: nodes, elements, loads and boundary conditions. % StaBIL manual % Example 4 % Units: m, kN % Types={EltTypID EltName} Types= {1 beam ; 2 truss }; % Sections bCol=0.2; % Column section width hCol=0.3; % Column section height bBeam=0.2; % Beam section width hBeam=0.5; % Beam section height Atruss=0.001;
112
% Sections=[SecID Sections= [1 2 3
Examples
A hCol*bCol hBeam*bBeam Atruss
% Materials=[MatID E Materials= [1 35e9 2 210e9
ky Inf Inf NaN
nu 0.2 0.3
kz Inf Inf NaN
Ixx 0.196*bCol^3*hCol 0.249*bBeam^3*hBeam NaN
Iyy bCol^3*hCol/12 bBeam^3*hBeam/12 NaN
Izz hCol^3*bCol/12 hBeam^3*bBeam/12 NaN
rho]; 2500; %concrete 7850]; %steel
L=5; H=3.5; B=4; % Nodes=[NodID X Y Z] Nodes= [1 0 0 0; 2 L 0 0] Nodes=reprow(Nodes,1:2,2,[2 0 0 H]) Nodes=[Nodes; 10 2 0 2] Nodes=reprow(Nodes,1:7,2,[100 0 B 0])
% reference node
figure plotnodes(Nodes); % Elements=[EltID TypID SecID MatID Elements=[ 1 1 1 1 2 1 1 1 Elements=reprow(Elements,1:2,1,[2 0 Elements=[ Elements; 5 1 2 1 6 1 2 1 Elements=reprow(Elements,1:6,2,[100 Elements=[ Elements; 301 2 3 2 302 2 3 2 Elements=reprow(Elements,19:20,1,[2 Elements=reprow(Elements,19:22,1,[4 Elements=[ Elements; 309 2 3 2 310 2 3 2
n1 n2 1 3 2 4 0 0 2
n3] 10; 10]; 2 0])
3 4 10; 5 6 10]; 0 0 0 100 100 100]) 3 103 NaN; 4 104 NaN]; 0 0 0 2 2 0]) 0 0 0 100 100 0]) 4 106 NaN; 6 104 NaN];
hold( on ); plotelem(Nodes,Elements,Types); title( Nodes and elements ); % Plot elements in different colors in order to check the section definitions figure plotelem(Nodes,Elements(find(Elements(:,3)==1),:),Types, r ); hold( on ); plotelem(Nodes,Elements(find(Elements(:,3)==2),:),Types, g ); plotelem(Nodes,Elements(find(Elements(:,3)==3),:),Types, b ); title( Elements: sections ) % Degrees of freedom DOF=getdof(Elements,Types);
yt hCol/2 hBeam/2 NaN
y h h N
Example 4
% Boundary conditions: hinges seldof=[ 1.01; 1.02; 1.03; 2.01; 2.02; 2.03; 101.01; 101.02; 101.03; 102.01; 102.02; 102.03; 201.01; 201.02; 201.03; 202.01; 202.02; 202.03;]; DOF=removedof(DOF,seldof); % Assembly of stiffness matrix K K=asmkm(Nodes,Elements,Types,Sections,Materials,DOF); % Loads % Own weight DLoadsOwn=accel([0 0 9.81],Elements,Types,Sections,Materials); % Wind load % DLoads=[EltID n1globalX n1globalY n1globalZ ...] DLoadsWind =[1 0 0 0 0 1500 0; 2 0 0 0 0 1500 0; 3 0 1500 0 0 1500 0; 4 0 1500 0 0 1500 0]; DLoads= multdloads(DLoadsOwn,DLoadsWind); P=elemloads(DLoads,Nodes,Elements,Types,DOF); % Solve K * U = P U=K\P; figure plotdisp(Nodes,Elements,Types,DOF,U(:,1),DLoads(:,:,1),Sections,Materials) title( Displacements: own weight ) figure plotdisp(Nodes,Elements,Types,DOF,U(:,2),DLoads(:,:,2),Sections,Materials) title( Displacements: wind ) % Compute forces [ForcesLCS,ForcesGCS]=elemforces(Nodes,Elements,Types,Sections,Materials,DOF,U,DLoads); % Compute reaction forces for load case 1 Freac=reaction(Elements,ForcesGCS(:,:,1),[1.03; 2.03; 101.03; 102.03; 201.03; 202.03]) % Plot element forces for load case 1 figure plotforc( norm ,Nodes,Elements,Types,ForcesLCS(:,:,1),DLoads(:,:,1)) title( Normal forces: Own weight ) figure plotforc( sheary ,Nodes,Elements,Types,ForcesLCS(:,:,1),DLoads(:,:,1)) title( Shear forces along y: Own weight ) figure plotforc( shearz ,Nodes,Elements,Types,ForcesLCS(:,:,1),DLoads(:,:,1)) title( Shear forces along z: Own weight ) figure plotforc( momx ,Nodes,Elements,Types,ForcesLCS(:,:,1),DLoads(:,:,1))
113
114
title( Torsional moments: Own weight ) figure plotforc( momy ,Nodes,Elements,Types,ForcesLCS(:,:,1),DLoads(:,:,1)) title( Bending moments around y: Own weight ) figure plotforc( momz ,Nodes,Elements,Types,ForcesLCS(:,:,1),DLoads(:,:,1)) title( Bending moments around z: Own weight ) % Plot element forces for load case 2 figure plotforc( norm ,Nodes,Elements,Types,ForcesLCS(:,:,2),DLoads(:,:,2)) title( Normal forces: Wind ) figure plotforc( sheary ,Nodes,Elements,Types,ForcesLCS(:,:,2),DLoads(:,:,2)) title( Shear forces along y: Wind ) figure plotforc( shearz ,Nodes,Elements,Types,ForcesLCS(:,:,2),DLoads(:,:,2)) title( Shear forces along z: Wind ) figure plotforc( momx ,Nodes,Elements,Types,ForcesLCS(:,:,2),DLoads(:,:,2)) title( Torsional moments: Wind ) figure plotforc( momy ,Nodes,Elements,Types,ForcesLCS(:,:,2),DLoads(:,:,2)) title( Bending moments around y : Wind ) figure plotforc( momz ,Nodes,Elements,Types,ForcesLCS(:,:,2),DLoads(:,:,2)) title( Bending moments around z: Wind ) % Load combinations % Safety factors gamma_own=1.35; gamma_wind=1.5; % Combination factors psi_wind=1; % Load combination U_UGT=gamma_own*U(:,1)+gamma_wind*psi_wind*U(:,2); Forces_UGT=gamma_own*ForcesLCS(:,:,1)+gamma_wind*psi_wind*ForcesLCS(:,:,2); DLoads_UGT(:,1)=DLoads(:,1,1) DLoads_UGT(:,2:7)=gamma_own*DLoads(:,2:7,1)+gamma_wind*psi_wind*DLoads(:,2:7,2); figure plotdisp(Nodes,Elements,Types,DOF,U_UGT,DLoads_UGT,Sections,Materials) printdisp(Nodes,DOF,U_UGT); printforc(Elements,Forces_UGT); % Plot element forces figure plotforc( norm ,Nodes,Elements,Types,Forces_UGT,DLoads_UGT) title( Normal forces: UGT ) figure plotforc( sheary ,Nodes,Elements,Types,Forces_UGT,DLoads_UGT)
Examples
Example 4
title( Shear forces along y: UGT ) figure plotforc( shearz ,Nodes,Elements,Types,Forces_UGT,DLoads_UGT) title( Shear forces along z: UGT ) figure plotforc( momx ,Nodes,Elements,Types,Forces_UGT,DLoads_UGT) title( Torsional moments: UGT ) figure plotforc( momy ,Nodes,Elements,Types,Forces_UGT,DLoads_UGT) title( Bending moments around y: UGT ) figure plotforc( momz ,Nodes,Elements,Types,Forces_UGT,DLoads_UGT) title( Bending moments around z: UGT ) % Plot stresses figure plotstress( snorm ,Nodes,Elements,Types,Sections,Forces_UGT,DLoads_UGT) title( Normal stresses due to normal forces ) figure plotstress( smomyt ,Nodes,Elements,Types,Sections,Forces_UGT,DLoads_UGT) title( Normal stresses due to bending moments around y: top ) figure plotstress( smomyb ,Nodes,Elements,Types,Sections,Forces_UGT,DLoads_UGT) title( Normal stresses due to bending moments around y: bottom ) figure plotstress( smomzt ,Nodes,Elements,Types,Sections,Forces_UGT,DLoads_UGT) title( Normal stresses due to bending moments around z: top ) figure plotstress( smomzb ,Nodes,Elements,Types,Sections,Forces_UGT,DLoads_UGT) title( Normal stresses due to bending moments around z: bottom ) figure plotstress( smax ,Nodes,Elements,Types,Sections,Forces_UGT,DLoads_UGT) title( Maximal normal stresses ) figure plotstress( smin ,Nodes,Elements,Types,Sections,Forces_UGT,DLoads_UGT) title( Minimal normal stresses )
115
116
Examples
(a) Displacements
(b) Normal forces
(c) Torsional moments
(d) Shear forces along z
(e) Shear forces along y
(f) Bending moments around y
(g) Bending moments around z
Figure 3.10: Results for load case 1 of example 4.
Example 4
117
(a) Displacements
(b) Normal forces
(c) Torsional moments
(d) Shear forces along z
(e) Shear forces along y
(f) Bending moments around y
(g) Bending moments around z
Figure 3.11: Results for load case 2 of example 4.
118
Examples
(a) Displacements
(b) Normal forces
(c) Torsional moments
(d) Shear forces along z
(e) Shear forces along y
(f) Bending moments around y
(g) Bending moments around z
Figure 3.12: Results for load combination UGT of example 4.
Example 4
119
(a) normal forces
(b) bending moments around y: top
(c) bending moments around y: bottom
(d) bending moments around z: top
(e) bending moments around z: bottom
(f) maximal
(g) minimal
Figure 3.13: Normal stresses for load combination UGT of example 4.
120
3.5
Examples
Example 5
In this example the eigenfrequencies of a simply supported rectangular plate are calculated using shell4 and compared with the theoretical solution. % dynamic plate problem (dkt element) % parameters Lx = 10; Ly = 10; t = 1; E = 200000; nu = 0.3; rho = 7000; m = 20; n = 20; Types = {1 shell4 }; Sections = [1 t]; Materials = [1 E nu rho];
% % % % % % % % % % %
length x-direction length y-direction thickness plate E-modulus Poisson-coeff. mass density number of elements in x-direction number of elements in y-direction {EltTypID EltName} [SecID t] [MatID E nu]
% mesh Line1 Line2 Line3 Line4
= = = =
[0 0 0;Lx 0 0]; [Lx 0 0;Lx Ly 0]; [Lx Ly 0;0 Ly 0]; [0 Ly 0;0 0 0];
[Nodes,Elements,Edge1,Edge2,Edge3,Edge4] = makemesh(Line1,Line2,Line3,Line4,n,m,Types(1,:),1,1); figure; plotnodes(Nodes); figure; plotelem(Nodes,Elements,Types); % DOFs (simply supported plate) DOF = getdof(Elements,Types); sdof = [0.01;0.02;0.06;[Edge1;Edge2;Edge3;Edge4]+0.03;[Edge1;Edge3]+0.05;[Edge2;Edge4]+0.04]; DOF = removedof(DOF,sdof); % K & M [K,M] = asmkm(Nodes,Elements,Types,Sections,Materials,DOF); % eigenmodes nMode = 10; [phi,omega] = eigfem(K,M,nMode); figure; animdisp(Nodes,Elements,Types,DOF,phi(:,1)); % analytical solution [mm,nn] = meshgrid((1:m),(1:n)); aomega = sqrt(E*t^2/(12*(1-nu^2)*rho))*((mm*pi/Lx).^2+(nn*pi/Ly).^2);
Example 5
121
aomega = reshape(aomega,numel(aomega),1); aomega = sort(aomega); ratio = omega./aomega(1:nMode)
(a) 0.1268 Hz
(b) 0.3170 Hz
(c) 0.3170 Hz
(d) 0.5050 Hz
Figure 3.14: The first four eigenmodes of a thin plate. f FEM[Hz] 0.1268 0.3170 0.3170 0.5050 0.6355 0.6355 0.8202 0.8202 1.0848 1.0848
f analytical[Hz] 0.1270 0.3176 0.3176 0.5082 0.6352 0.6352 0.8258 0.8258 1.0799 1.0799
f FEM/f analytical 0.9983 0.9982 0.9982 0.9939 1.0004 1.0004 0.9932 0.9932 1.0046 1.0046
Table 3.1: Eigenfrequencies of a thin plate.
122
Examples
3.6
Example 6
A thin disk with a circular hole is subjected to uniaxial tension. The stress concentration around the hole is examined. % Stress concentration around a circular hole in a disk. r = 1; % Radius of hole [m] L = 20; % Width of plate [m] p = 1; % Uniform pressure [N/m^2] n = 40; % Mesh parameter (nElem = m*n) m = 20; % Mesh parameter Types = {1 shell8 }; % {EltTypID EltName} Sections = [1 1]; % [SecID t] Materials = [1 200000 0]; % [MatID E nu]
% define lines Line1 = [r 0 0;L/2 0 0]; Line2 = [L/2 0 0;L/2 L/2 0;0 L/2 0]; Line3 = [0 L/2 0;0 r 0]; Line4 = [r*sin((0:5). *pi/10) r*cos((0:5). *pi/10) zeros(6,1)]; [Nodes,Elements,Edge1,Edge2,Edge3,Edge4] = makemesh(Line1,Line2,Line3,Line4,... m,n,Types(1,:),Sections(1,1),Materials(1,1), L2method , linear ); % Check mesh: figure; plotnodes(Nodes, numbering , off ); hold( on ) plotelem(Nodes,Elements,Types, numbering , off ); title( Nodes and elements ); % Select all dof: DOF = getdof(Elements,Types); % Apply boundary conditions: % - Line1 and Line3: symmetry condition seldof = [0.03;0.04;0.05;0.06;Edge1+0.02;Edge3+0.01]; DOF = removedof(DOF,seldof); % Assemble K: K = asmkm(Nodes,Elements,Types,Sections,Materials,DOF); % C P U
Apply load to upper edge (equivalent nodal forces): = selectdof(DOF,Edge2(1:(end-1)/2+1)+0.02); = C . *[1/6; repmat([2/3; 1/3],((length(Edge2)-1)/2-2)/2,1);2/3; 1/6]*p*L/m; = K\P;
% Calculate stress in cylindrical global coordinate system [SeGCS]=elemstress(Nodes,Elements,Types,Sections,Materials,DOF,U, gcs , cyl ); % Get nodal stress from element solution [SnGCS] = nodalstress(Nodes,Elements,Types,SeGCS); % plot results figure;
Example 6
123
plotstresscontourf( sx ,Nodes,Elements,Types,SnGCS) title( \sigma_{r} ) figure; plotstresscontourf( sy ,Nodes,Elements,Types,SnGCS) title( \sigma_{\theta} ) figure; plotstresscontourf( sxy ,Nodes,Elements,Types,SnGCS) title( \sigma_{r\theta} )
(a)
σθ
(c)
σrθ
(b)
σr
Figure 3.15: Results.
124
3.7
Examples
Example 7
The geometrical nonlinear response of a truss is examined. clear; %% Geometric nonlinear truss with multiple snap-throughs % based on example on p281 of Non-linear Modeling and Analysis of Solids % and Structures, Krenk (2008) h = 1; % Height truss ll = 1.697; % Lower length hl = 1.414; % Upper length w = 2; % Width Sections = [1 1]; Materials = [1 1 1 1]; Types = {1 truss }; Nodes = [(1:3). ll*(-1:1). zeros(3,2)]; Nodes = reprow(Nodes,(1:3),1,[3 0 w 0]); Nodes = [Nodes; (7:9). hl*(-1:1). (w/2)*ones(3,1) h*ones(3,1)]; Elements = [1 2;2 3;4 5;5 6;1 4;2 5;3 6; 1 7;2 8;3 9;4 7;5 8;6 9; 1 8;3 8;4 8;6 8;7 8;8 9]; nElem = size(Elements,1); Elements = [(1:nElem). ,ones(nElem,3),Elements]; figure; plotnodes(Nodes); hold on; plotelem(Nodes,Elements,Types); hold off; title( Nodes and Elements ); % BOUNDARY CONDITIONS DOF = getdof(Elements,Types); seldof = [(1:6)+0.01,(1:6)+0.02,(1:6)+0.03]. ; DOF = removedof(DOF,seldof); % POINT LOAD seldof = (7:9)+0.03; L = selectdof(DOF,seldof); P = L . *[-1.5;-1;-1.5]; % SELECT VERTICAL DISPLACEMENT MID SPAN seldof = 8.03; Lw = selectdof(DOF,seldof); seldof = 9.03; Lv = selectdof(DOF,seldof); % LINEAR ANALYSIS K = asmkm(Nodes,Elements,Types,Sections,Materials,DOF); U = K\P; % LOAD CONTROL (NEWTON-RAPHSON) Options.method = nr ;
% nonlinear solver: Newton-Raphson
Example 7
125
nloadincrem = ceil(1/0.03); % number of load increments Options.nloadincrem = nloadincrem; [Unr,K,lambda_nr] = nlsolver(Nodes,Elements,Types,Sections,Materials,DOF,P,[],Options); % ARC-LENGTH CONTROL Options.arclength = on ; Options.maxoiter = 3000; % maximum number of arc-length steps Options.initdl = 0.05; % Initial arc-length [Ual,K,lambda_al] = nlsolver(Nodes,Elements,Types,Sections,Materials,DOF,P,[],Options); % PLOT RESULTS figure plotdisp(Nodes,Elements,Types,DOF,U, dispscal ,1); title( Linear small displacements ) figure plotdisp(Nodes,Elements,Types,DOF,Unr(:,end), dispscal ,1); title( Nonlinear large displacements ) v=axis; figure subplot(2,2,1) plotdisp(Nodes,Elements,Types,DOF,[Ual(:,end),Unr(:,end)], dispscal ,1, dispmax , off ); subplot(2,2,2) plot(-Lw*Ual,lambda_al,-Lw*Unr,lambda_nr); axis([-0.5 2.5 -0.1 0.1]);xlabel( w );ylabel( P ); subplot(2,2,3) plot(-Lv*Ual,lambda_al,-Lv*Unr,lambda_nr); axis([-0.5 2.5 -0.1 0.1]);xlabel( v );ylabel( P ); subplot(2,2,4) plot(-Lw*Ual,-Lv*Ual,-Lw*Unr,-Lv*Unr); axis([-0.5 2.5 0 2]);xlabel( w );ylabel( v ); % ANIMATE LOAD DISPLACEMENT CURVE nstep = size(Ual,2); figure for istep = 1:nstep clf subplot(2,2,1) plotdisp(Nodes,Elements,Types,DOF,Ual(:,istep), dispscal ,1, dispmax , off ); axis(v); subplot(2,2,2) plot(-Lw*Ual(:,1:istep),lambda_al(:,1:istep), .- ); axis([-0.5 2.5 -0.1 0.1]);xlabel( w );ylabel( P ); subplot(2,2,3) plot(-Lv*Ual(:,1:istep),lambda_al(:,1:istep), .- ); axis([-0.5 2.5 -0.1 0.1]);xlabel( v );ylabel( P ); subplot(2,2,4) plot(-Lw*Ual(:,1:istep),-Lv*Ual(:,1:istep), .- ); axis([-0.5 2.5 0 2]);xlabel( w );ylabel( v ); pause(0.1) end
126
Examples
7 11
4
18 16
8 5
8 3
12 19 5
14
9 17
9
1 z y 2 x
1
4 13 6
15
6 10 7
2 3
Maximal displacement: 2.85617
(a) Elements
(b) Nonlinear displacements
0.1 0.08 0.06 0.04 0.02 λ
0 −0.02 −0.04 −0.06 −0.08 −0.1 −0.5
0
0.5
1 w
1.5
2
2.5
(c) Vertical load-displacement curve
Figure 3.16: Results.
Chapter 4
Functions — Alphabetical list
accel.tex accel_beam.tex accel_shell4.tex accel_shell8.tex accel_truss.tex addconstr.tex animdisp.tex asmkm.tex b_shell8.tex cdiff.tex coord_beam.tex coord_shell4.tex coord_shell8.tex coord_truss.tex disp_beam.tex disp_shell4.tex disp_shell8.tex disp_truss.tex dispgcs2lcs_beam.tex dispgcs2lcs_truss.tex dof_beam.tex dof_shell4.tex dof_shell8.tex dof_truss.tex eigfem.tex elemdisp.tex elemforces.tex elemloads.tex elempressure.tex elemshellf.tex elemstress.tex fdiagrgcs_beam.tex fdiagrgcs_truss.tex fdiagrlcs_beam.tex
p. p. p. p. p. p. p. p. p. p. p. p. p. p. p. p. p. p. p. p. p. p. p. p. p. p. p. p. p. p. p. p. p. p.
4 24 51 61 33 3 14 2 59 19 26 54 62 35 26 55 62 35 27 36 22 48 58 32 18 5 5 4 39 39 39 28 36 28
128
forces_beam.tex forces_truss.tex forceslcs_beam.tex forceslcs_truss.tex gaussq.tex getdof.tex getmovie.tex grid_shell4.tex grid_shell8.tex ke_beam.tex ke_dkt.tex ke_shell4.tex ke_shell8.tex ke_truss.tex kelcs_beam.tex kelcs_shell4.tex kelcs_truss.tex kenl_kbeam.tex kenl_plane4.tex kenl_truss.tex ldlt.tex lincrpoint.tex loads_beam.tex loads_shell4.tex loads_shell8.tex loads_truss.tex loadslcs_beam.tex loadslcs_shell4.tex makemesh.tex msupf.tex msupt.tex nedloadlcs_beam.tex nelcs_beam.tex newmark.tex nlincrpoint.tex nlincrpoint2.tex nlsolver.tex nodalshellf.tex nodalstress.tex nodalvalues.tex patch_shell4.tex patch_shell8.tex plotdisp.tex plotelem.tex plotforc.tex plotlcs.tex plotnodes.tex plotprincstress.tex plotshellfcontour.tex plotshellfcontourf.tex
Functions — Alphabetical list
p. p. p. p. p. p. p. p. p. p. p. p. p. p. p. p. p. p. p. p. p. p. p. p. p. p. p. p. p. p. p. p. p. p. p. p. p. p. p. p. p. p. p. p. p. p. p. p. p. p.
24 34 25 34 40 2 15 56 63 22 50 48 58 32 22 49 32 70 69 69 20 67 23 52 61 33 24 52 41 18 18 26 25 19 67 68 66 42 42 4 56 63 11 10 11 12 10 43 43 44
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