◼ written by Hakan Tanriover
2-D BEAM FORMULATION FROM SECTION 7.2 of "Non-Linear Finite Element Analysis of Solids and Structures" by M.A. Crisfield.
Cantilever beam subjected to end moment:
Elasticity modulus: elas
Crossection area: area
Moment of Inertia: Imom
Radius of gyration: rg
In[1]:=
elas = 2; area = 1; Imom = 1; rg = 1;
number of nodes: nnodes
number of nodes in an element: nelnode
number of elements: nelem
nodal degrees of freedom: ndof
element degrees of freedom: eldof
total degrees of freedom: totdof
node numbers for elements: elemnodes={left,right}
In[2]:=
nelem = 10
In[3]:=
nnodes = nelem + 1
In[4]:=
nelnode = 2
In[5]:=
ndof = 3
In[6]:=
eldof = ndof * nelnode
In[7]:=
totdof = ndof * nnodes
In[8]:=
beamlength = 1
In[9]:=
Do[elemnodes[i] = {i, i + 1}, {i, 1, nelem}]
initial coordinates of the nodes in the x and z direction:
In[10]:=
Do[coordx[i] = (i - 1) * (beamlength / nelem) , {i, 1, nnodes}]
In[11]:=
Do[coordz[i] = 0.0, {i, 1, nnodes}]
In[12]:=
In[14]:=
In[15]:=
Do[ncoordx[i] = coordx[i], {i, 1, nnodes}];
Do[ncoordz[i] = coordz[i], {i, 1, nnodes}];
initgeodat = Table[{coordx[i], coordz[i]}, {i, nnodes}]
ListPlot[initgeodat, Frame → True, PlotRange → {{-0.25`, 1.25`}, {-0.25`, 0.5`}},
Joined → True, PlotStyle → {Thickness[0.012`], Dashing[{0.02`}]}]
number of load steps: nst
In[16]:=
In[17]:=
nst = 40
Do[
"_____step load: df";
2
largerotbeam_2d.nb
df = 4 * Pi / 40;
"_____External Force";
Do[exf[i] = 0., {i, 1, totdof}];
exf[totdof] = st * df;
exforcevec = Array[exf, totdof];
Print["LOAD STEP:", st];
Do[
"_____tangent stiffness for elements";
Do[
"----node numbers of i and j";
nni = elemnodes[ne][[1]];
nnj = elemnodes[ne][[2]];
"-------length of element";
x21 = coordx[nnj] - coordx[nni];
z21 = coordz[nnj] - coordz[nni];
vecx21 = {x21, z21};
oldlengthel = Sqrt[vecx21.vecx21];
"check for different geometries";
beta0 = ArcSin[z21 / oldlengthel];
"----------let initial values be zero";
If[it ⩵ 1 && st ⩵ 1, u21[ne] = 0; w21[ne] = 0;
teta1[ne] = 0.; teta2[ne] = 0.;,
u21[ne] = u21[ne] + deltapvec[[nnj * ndof - 2]] - deltapvec[[nni * ndof - 2]];
w21[ne] = w21[ne] + deltapvec[[nnj * ndof - 1]] - deltapvec[[nni * ndof - 1]];
teta1[ne] = teta1[ne] + deltapvec[[nni * ndof]];
teta2[ne] = teta2[ne] + deltapvec[[nnj * ndof]];
];
d21 = {u21[ne], w21[ne]};
nvecx21 = vecx21 + d21;
newlengthel = Sqrt[nvecx21.nvecx21];
cosalpha = vecx21.(vecx21 + d21) / (newlengthel * oldlengthel);
sinalpha = (x21 * w21[ne] - z21 * u21[ne]) / (newlengthel * oldlengthel);
If[(sinalpha ≥ 0 && cosalpha ≥ 0),
largerotbeam_2d.nb
alpha = ArcSin[sinalpha]; Goto[alphas];
];
If[(sinalpha ≤ 0 && cosalpha ≥ 0),
alpha = 2 * Pi + ArcSin[sinalpha]; Goto[alphas]; "check!!!";
];
If[(sinalpha ≥ 0 && cosalpha ≤ 0),
alpha = ArcCos[cosalpha]; Goto[alphas];
];
If[(sinalpha ≤ 0 && cosalpha ≤ 0),
alpha = 2 * Pi - ArcCos[cosalpha]; Goto[alphas]; "check!!!";
];
Label[alphas];
"-------local nodal rotations at node 1 and 2";
tetaloc1 = teta1[ne] - alpha - beta0;
tetaloc2 = teta2[ne] - alpha - beta0;
"--------Local axial extension:uloc";
uloc = (2 / (newlengthel + oldlengthel)) * ((vecx21 + (d21 / 2)).d21);
nx21 = x21 + u21[ne];
nz21 = z21 + w21[ne];
cbeta = nx21 / newlengthel;
sbeta = nz21 / newlengthel;
r = {-cbeta, -sbeta, 0, cbeta, sbeta, 0};
z = {sbeta, -cbeta, 0, -sbeta, cbeta, 0};
Bmat = {{-cbeta, -sbeta, 0, cbeta, sbeta, 0},
{-sbeta / newlengthel, cbeta / newlengthel,
1, sbeta / newlengthel, -cbeta / newlengthel, 0},
{-sbeta / newlengthel, cbeta / newlengthel, 0,
sbeta / newlengthel, -cbeta / newlengthel, 1}};
Cl = (elas * area / oldlengthel) *
{{1, 0, 0}, {0, 4 * rg ^2, 2 * rg ^2}, {0, 2 * rg ^2, 4 * rg ^2}};
TBmat = Transpose[Bmat];
"--------First part of tangent stiffness matrix";
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largerotbeam_2d.nb
kt1 = TBmat.Cl.Bmat;
"---------Axial force:Nax";
Nax = (elas * area / oldlengthel) * uloc;
"--------Moments";
mom1 = (2 * elas * Imom / oldlengthel) * (2 * tetaloc1 + tetaloc2);
mom2 = (2 * elas * Imom / oldlengthel) * (tetaloc1 + 2 * tetaloc2);
"Local Internal force vector: qiloc";
qiloc = {Nax, mom1, mom2};
"Element Global Internal force vector:qi";
qie[ne] = TBmat.qiloc;
"--------z vector as matrix form";
zm = {{sbeta}, {-cbeta}, {0}, {-sbeta}, {cbeta}, {0}};
tzm = Transpose[zm];
"-------second part of stiffness";
ktb1 = (Nax / newlengthel) * (zm.tzm);
"--------r vector as matrix";
rm = {{-cbeta}, {-sbeta}, {0}, {cbeta}, {sbeta}, {0}};
trm = Transpose[rm];
"-------last part of stiffness";
ktb23 = ((mom1 + mom2) / newlengthel^2) * (rm.tzm + zm.trm);
"------element tangent stiffness matrix";
kte[ne] = kt1 + ktb1 + ktb23;
"___________END OF ELEMENT LOOP";
, {ne, 1, nelem}];
"_____Assembling Global tangent stiffness matrix";
largerotbeam_2d.nb
Do[kt[i, j] = 0., {i, 1, totdof}, {j, 1, totdof}];
For[e = 1, e ≤ nelem, e++,
Do[ kt[i + (e - 1) * ndof, j + (e - 1) * ndof] =
kte[e][[i, j]] + kt[i + (e - 1) * ndof, j + (e - 1) * ndof];
, {i, 1, eldof}, {j, 1, eldof}];
];
"_____Apply BC";
Do[kt[i, i] = 1, {i, 1, 3}];
Do[kt[1, i] = 0., {i, 2, totdof}];
Do[kt[2, i] = 0., {i, 3, totdof}];
Do[kt[3, i] = 0., {i, 4, totdof}];
Do[kt[i, 1] = 0., {i, 2, totdof}];
Do[kt[i, 2] = 0., {i, 3, totdof}];
Do[kt[i, 3] = 0., {i, 4, totdof}];
ktm = Table[kt[i, j], {i, totdof}, {j, totdof}];
iktm = Inverse[ktm];
"_____Assembly of Global Internal Force Vector";
Do[qi[i] = 0., {i, 1, totdof}];
For[e = 1, e ≤ nelem, e++,
Do[ qi[i + (e - 1) * ndof] = qie[e][[i]] + qi[i + (e - 1) * ndof]
, {i, 1, eldof}];
];
"_____Internal Force Vector";
"_____apply bc to force vector";
Do[qi[i] = 0., {i, 1, 3}];
qivec = Array[qi, totdof];
"_____Residual vector (qi-qe):qresvec";
qresvec = qivec - exforcevec;
"_____Convergence Check";
Normqresvec = Sqrt[qresvec.qresvec];
Normexforcevec = Sqrt[ exforcevec. exforcevec];
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largerotbeam_2d.nb
eps = 1 / 1000;
If[Normqresvec < (eps * Normexforcevec),
Print["ITERATION:", it, "___", "Normqresvec:", Normqresvec];
Break[]];
"_____Displacement vector:deltapvec";
deltapvec = -iktm.qresvec;
"______New coordinates";
Do[ncoordx[i] = ncoordx[i] + deltapvec[[i * ndof - 2]], {i, 1, nnodes}];
Do[ncoordz[i] = ncoordz[i] + deltapvec[[i * ndof - 1]], {i, 1, nnodes}];
Label[end];
, {it, 1, 20}];
geodat[st] = Table[{ncoordx[i], ncoordz[i]}, {i, nnodes}];
, {st, 1, nst}]
In[19]:=
ListPlot[Array[geodat, nst], Frame → True, PlotRange → {{-0.3, 1.2}, {-0.3, 1.2}},
PlotJoined → True, PlotStyle → {Dashing[{.02}]}, AspectRatio → 1]
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Out[19]=
0.4
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0.0
-0.2
-0.2
0.0
0.2
0.4
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0.8
1.0
1.2