calculo 4
DESCRIPTION
en este imforme se habla de las matrices en un plano tetraedro mediante el software mapleTRANSCRIPT
ELEMENTO TETRADRICO LINEAL
UNIVERSIDAD PRIVADA DEL NORTE
FACULTAD DE INGENIERIA
ESCUELA ACADEMICO PROFESIONAL DE INGENIERIA DE MINASCURSO: CLCULO IVCICLO: V
TRABAJO:ELEMENTO TETRAEDRIO LINEALDOCENTE:HUAMN ROJAS EVERALUMNOS: N K K Kcalculo 4
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ELEMENTO TETRADRICO LINEAL (SLIDO) 46
INTRODUCCION:
OBJETIVOS:
EL ELEMENTO TETRADRICO LINEAL (SLIDO)
El elemento tetradrico lineal (slido) es un elemento finito tridimensional con ambas coordenadas locales y globales. Se caracteriza por funciones de forma lineales. Tambin se le llama el tetraedro tensin constante. El elemento tetradrico lineal tiene un mdulo de elasticidad E y la relacin de Poisson . Cada tetraedro lineal tiene cuatro nodos con tres grados de libertad en cada nodo, como se muestra en la Fig. 15.1. Las coordenadas globales de los cuatro nodos se designan por (x1, y1, z1), (x2, y2, z2), (x3, y3, z3), y (x4, y4, z4). La numeracin de los nodos de cada elemento es muy importante - usted debe numerar los nodos de tal manera que el volumen del elemento es positivo. Se aconseja comprobar realmente esto usando la funcin TetrahedronElementVolume MATLAB que est escrito especficamente para este propsito. En este caso la matriz de rigidez elemento est dada por:
[k] = V [B]T [D][B]
DondeVeselvolumendelelementodadopor
Elelemento(slido)tetradricolineal ylamatriz[B]estdadapor.
LasfuncionesdeformaN1,N2,N3yN4estndadaspor:
Donde 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3 estn dada por:
Lamatriz[D]estdadapor
Est claro que el elemento tetradrico lineal tiene doce grados de libertad - tres en cada nodo. En consecuencia, para una estructura con n nodos, la matriz de rigidez global K ser de tamao 3n 3n (ya que tenemos tres grados de libertad en cada nodo). El K matriz de rigidez global est montado al hacer llamadas a la funcin TetrahedronAssemble MATLAB que est escrito especficamente para este propsito. Este proceso se ilustra en detalle en los ejemplos.Una vez obtenida la matriz de rigidez global K tenemos la siguiente estructura ecuacin:
Donde T es el vector global de desplazamiento nodal y F es la fuerza vector nodal global. En este paso las condiciones de contorno se aplican manualmente a los vectores u y F. Entonces la matriz se resuelve mediante la separacin y eliminacin de Gauss.Finalmente, una vez se encuentran los desplazamientos y reacciones desconocidas, el vector de tensin se obtiene para cada elemento de la siguiente manera:
Donde es el vector de la tensin en el elemento (del tamao de 6 x 1 ) y u es el 12 1 elemento vector de desplazamiento . El vector est escrito para cada elemento como {} = [x y z xy yz zx] T.FUNCIONES USADAS DE MATLAB
Las cinco funciones de MATLAB utilizados para el elemento tetradrico lineal (slido) son:
TetrahedronElementVolume (x1, y1, z1, x2, y2, z2, x3, y3, z3, x4, y4, z4): Esta funcin devuelve el volumen elemento dado las coordenadas del nodo de primera (x1, y1, z1), las coordenadas del segundo nodo (x2, y2, z2), las coordenadas de la tercera nodo (x3, y3, z3), y las coordenadas del cuarto nodo (x4, y4, z4).
TetrahedronElementStiffness (E, NU, x1, y1, z1, x2, y2, z2, x3, y3, z3, x4, y4, z4): Esta funcin calcula la matriz de elemento de rigidez para cada tetraedro lineal con mdulo de elasticidad E, la relacin de Poisson NU, y las coordenadas (x1, y1, z1) para el nodo en primer lugar, (x2, y2, z2) para el segundo nodo, (x3, y3, z3) para el tercer nodo, y (x4, y4, z4) para el cuarto nodo. Devuelve el 12 12 matriz de rigidez elemento k.
TetrahedronAssemble (K, k, i, j, m, n): Esta funcin ensambla la matriz k elemento de rigidez del tetraedro lineal unirse a los nodos i, j, m, y n en la matriz de rigidez global K. Devuelve el 3n 3n matriz global de rigidez K cada vez que un elemento se monta.
TetrahedronElementStresses (E, NU, x1, y1, z1, x2, y2, z2, x3, y3, z3, x4, y4, z4): Esta funcin calcula el elemento subraya usando el mdulo de elasticidad E, coeficiente de Poisson NU, las coordenadas (x1, y1, z1) para el nodo en primer lugar, (x2, y2, z2) para el segundo nodo, (x3, y3, z3) para el tercer nodo, y (x4, y4, z4) para el cuarto nodo , y el vector de elemento de desplazamiento u. Devuelve el vector de esfuerzo para el elemento.
TetrahedronElementPStresses (sigma): Esta funcin calcula las tres tensiones principales para el elemento utilizando el elemento de tensin vector sigma. Devuelve un vector de 3 1 en la forma [sigma1sigma2sigma3]T, donde sigma1, sigma2 y sigma3 son los esfuerzos principales para el elemento. Esta funcin no devuelve los principales ngulos.
LISTADO DEL CDIGO FUENTE DE MATLAB
Para cada funcin tenemos:
Funcin y=TetrahedronElementVolume(x1, y1, z1, x2, y2, z2, x3, y3, z3, x4, y4, z4)% TetrahedronElementVolume Esta funcin devuelve el volumen% Del elemento tetradrico lineal% Cuyo primer nodo tiene coordenadas% (X1, y1, z1), segundo nodo tiene% Coordenadas (x2, y2, z2), tercer nodo% Tiene coordenadas (x3, y3, z3), yCuarto nodo% tiene coordiantes% (X4, y4, z4).xyz = [y1 z1 1 x1; Z2 y2 x2 1; Z3 y3 1 x3; 1 x4 y4 z4];y = det (xyz) / 6;
funcin y=TetrahedronElementStiffness(E, NU, x1, y1, z1, x2, y2, z2, x3, y3, z3, x4, y4, z4)% TetrahedronElementStiffness Esta funcin devuelve el elemento% Matriz de rigidez para un linealElemento tetradrico% (slido) con% Mdulo de elasticidad E,% Ratio de NU, coordenadas de Poisson% Del primer nodo (x1, y1, z1),% Coordenadas del segundo nodo% (X2, y2, z2), coordenadas del% Tercer nodo (x3, y3, z3), y% Coordenadas del cuarto nodo% (X4, y4, z4).% El tamao del elemento de rigidez% Matriz es de 12 x 12.XYZ=[1x 1y1z1;1x 2y2z2;1x 3y3z3z41x 4y4];V = det(xyz)/6;mbeta1 = [1 y2 z2 ; 1 y3 z3 ; 1 y4 z4];mbeta2 = [1 y1 z1 ; 1 y3 z3 ; 1 y4 z4];mbeta3 = [1 y1 z1 ; 1 y2 z2 ; 1 y4 z4];mbeta4 = [1 y1 z1 ; 1 y2 z2 ; 1 y3 z3];mgamma1 = [1 x2 z2 ; 1 x3 z3 ; 1 x4 z4];mgamma2 = [1 x1 z1 ; 1 x3 z3 ; 1 x4 z4];mgamma3 = [1 x1 z1 ; 1 x2 z2 ; 1 x4 z4];mgamma4 = [1 x1 z1 ; 1 x2 z2 ; 1 x3 z3];mdelta1 = [1 x2 y2 ; 1 x3 y3 ; 1 x4 y4];mdelta2 = [1 x1 y1 ; 1 x3 y3 ; 1 x4 y4];mdelta3 = [1 x1 y1 ; 1 x2 y2 ; 1 x4 y4];mdelta4 = [1 x1 y1 ; 1 x2 y2 ; 1 x3 y3];beta1 = 1*det(mbeta1);beta2 = det(mbeta2);beta3 = 1*det(mbeta3);beta4 = det(mbeta4);gamma1 = det(mgamma1);gamma2 = 1*det(mgamma2);gamma3 = det(mgamma3);gamma4 = 1*det(mgamma4);delta1 = 1*det(mdelta1);delta2 = det(mdelta2);delta3 = 1*det(mdelta3);delta4 = det(mdelta4);B1 = [beta1 0 0 ; 0 gamma1 0 ; 0 0 delta1 ;gamma1 beta1 0 ; 0 delta1 gamma1 ; delta1 0 beta1];B2 = [beta2 0 0 ; 0 gamma2 0 ; 0 0 delta2 ;gamma2 beta2 0 ; 0 delta2 gamma2 ; delta2 0 beta2];B3 = [beta3 0 0 ; 0 gamma3 0 ; 0 0 delta3 ;gamma3 beta3 0 ; 0 delta3 gamma3 ; delta3 0 beta3];B4 = [beta4 0 0 ; 0 gamma4 0 ; 0 0 delta4 ;gamma4 beta4 0 ; 0 delta4 gamma4 ; delta4 0 beta4];B = [B1 B2 B3 B4]/(6*V);D = (E/((1+NU)*(12*NU)))*[1NU NU NU 0 0 0 ;NU 1NU NU 0 0 0 ;NU NU 1NU 0 0 0 ;0 0 0 (12*NU)/2 0 0 ; 0 0 0 0 (1 2*NU)/2 0 ; 0 0 0 0 0 (12*NU)/2];y = V*B*D*B;
funcin y = TetrahedronAssemble(K,k,i,j,m,n)
% TetrahedronAssemble esta funcin ensambla la rigidez del elemento% matriz k del elemento lineal tetradrico (slido) % con nodos i, j, m y n en la% matriz de rigidez global K.% Esta funcin devuelve la rigidez global % matriz K despus de la matriz de rigidez del elemento% k est montado.K(3*i2,3*i2) = K(3*i2,3*i2) + k(1,1);K(3*i2,3*i1) = K(3*i2,3*i1) + k(1,2);K(3*i2,3*i) = K(3*i2,3*i) + k(1,3);K(3*i2,3*j2) = K(3*i2,3*j2) + k(1,4);K(3*i2,3*j1) = K(3*i2,3*j1) + k(1,5);K(3*i2,3*j) = K(3*i2,3*j) + k(1,6);K(3*i2,3*m2) = K(3*i2,3*m2) + k(1,7);K(3*i2,3*m1) = K(3*i2,3*m1) + k(1,8);K(3*i2,3*m) = K(3*i2,3*m) + k(1,9);K(3*i2,3*n2) = K(3*i2,3*n2) + k(1,10);K(3*i2,3*n1) = K(3*i2,3*n1) + k(1,11);K(3*i2,3*n) = K(3*i2,3*n) + k(1,12);K(3*i1,3*i2) = K(3*i1,3*i2) + k(2,1);K(3*i1,3*i1) = K(3*i1,3*i1) + k(2,2);K(3*i1,3*i) = K(3*i1,3*i) + k(2,3);K(3*i1,3*j2) = K(3*i1,3*j2) + k(2,4);K(3*i1,3*j1) = K(3*i1,3*j1) + k(2,5);K(3*i1,3*j) = K(3*i1,3*j) + k(2,6);K(3*i1,3*m2) = K(3*i1,3*m2) + k(2,7);K(3*i1,3*m1) = K(3*i1,3*m1) + k(2,8);K(3*i1,3*m) = K(3*i1,3*m) + k(2,9);K(3*i1,3*n2) = K(3*i1,3*n2) + k(2,10);K(3*i1,3*n1) = K(3*i1,3*n1) + k(2,11);K(3*i1,3*n) = K(3*i1,3*n) + k(2,12);K(3*i,3*i2) = K(3*i,3*i2) + k(3,1);K(3*i,3*i1) = K(3*i,3*i1) + k(3,2);K(3*i,3*i) = K(3*i,3*i) + k(3,3);K(3*i,3*j2) = K(3*i,3*j2) + k(3,4);K(3*i,3*j1) = K(3*i,3*j1) + k(3,5);K(3*i,3*j) = K(3*i,3*j) + k(3,6);K(3*i,3*m-2) = K(3*i,3*m2) + k(3,7);K(3*i,3*m1) = K(3*i,3*m1) + k(3,8);K(3*i,3*m) = K(3*i,3*m) + k(3,9);K(3*i,3*n2) = K(3*i,3*n2) + k(3,10);K(3*i,3*n1) = K(3*i,3*n1) + k(3,11);K(3*i,3*n) = K(3*i,3*n) + k(3,12);K(3*j2,3*i2) = K(3*j2,3*i2) + k(4,1);K(3*j2,3*i1) = K(3*j2,3*i1) + k(4,2);K(3*j2,3*i) = K(3*j2,3*i) + k(4,3);K(3*j2,3*j2) = K(3*j2,3*j2) + k(4,4);K(3*j2,3*j1) = K(3*j2,3*j1) + k(4,5);K(3*j2,3*j) = K(3*j2,3*j) + k(4,6);K(3*j2,3*m2) = K(3*j2,3*m2) + k(4,7);K(3*j2,3*m1) = K(3*j2,3*m1) + k(4,8);K(3*j2,3*m) = K(3*j2,3*m) + k(4,9);K(3*j2,3*n2) = K(3*j2,3*n2) + k(4,10);K(3*j2,3*n1) = K(3*j2,3*n1) + k(4,11);K(3*j2,3*n) = K(3*j2,3*n) + k(4,12);K(3*j1,3*i2) = K(3*j1,3*i2) + k(5,1);K(3*j1,3*i1) = K(3*j1,3*i1) + k(5,2);K(3*j1,3*i) = K(3*j1,3*i) + k(5,3);K(3*j1,3*j2) = K(3*j1,3*j2) + k(5,4);K(3*j1,3*j1) = K(3*j1,3*j1) + k(5,5);K(3*j1,3*j) = K(3*j1,3*j) + k(5,6);K(3*j1,3*m2) = K(3*j1,3*m2) + k(5,7);K(3*j1,3*m1) = K(3*j1,3*m1) + k(5,8);K(3*j1,3*m) = K(3*j1,3*m) + k(5,9);K(3*j1,3*n2) = K(3*j1,3*n2) + k(5,10);K(3*j1,3*n1) = K(3*j1,3*n1) + k(5,11);K(3*j1,3*n) = K(3*j1,3*n) + k(5,12);K(3*j,3*i2) = K(3*j,3*i2) + k(6,1);K(3*j,3*i1) = K(3*j,3*i1) + k(6,2);K(3*j,3*i) = K(3*j,3*i) + k(6,3);K(3*j,3*j2) = K(3*j,3*j2) + k(6,4);K(3*j,3*j1) = K(3*j,3*j1) + k(6,5);K(3*j,3*j) = K(3*j,3*j) + k(6,6);K(3*j,3*m2) = K(3*j,3*m2) + k(6,7);K(3*j,3*m1) = K(3*j,3*m1) + k(6,8);K(3*j,3*m) = K(3*j,3*m) + k(6,9);K(3*j,3*n2) = K(3*j,3*n2) + k(6,10);K(3*j,3*n1) = K(3*j,3*n1) + k(6,11);K(3*j,3*n) = K(3*j,3*n) + k(6,12);K(3*m2,3*i2) = K(3*m2,3*i2) + k(7,1);K(3*m2,3*i1) = K(3*m2,3*i1) + k(7,2);K(3*m2,3*i) = K(3*m2,3*i) + k(7,3);K(3*m2,3*j2) = K(3*m2,3*j2) + k(7,4);K(3*m2,3*j1) = K(3*m2,3*j1) + k(7,5);K(3*m2,3*j) = K(3*m2,3*j) + k(7,6);K(3*m2,3*m2) = K(3*m2,3*m2) + k(7,7);K(3*m2,3*m1) = K(3*m2,3*m1) + k(7,8);K(3*m2,3*m) = K(3*m2,3*m) + k(7,9);K(3*m2,3*n2) = K(3*m2,3*n2) + k(7,10);K(3*m2,3*n1) = K(3*m2,3*n1) + k(7,11);K(3*m2,3*n) = K(3*m2,3*n) + k(7,12);K(3*m1,3*i2) = K(3*m1,3*i2) + k(8,1);K(3*m1,3*i1) = K(3*m1,3*i1) + k(8,2);K(3*m1,3*i) = K(3*m1,3*i) + k(8,3);K(3*m1,3*j2) = K(3*m1,3*j2) + k(8,4);K(3*m1,3*j1) = K(3*m1,3*j1) + k(8,5);K(3*m1,3*j) = K(3*m1,3*j) + k(8,6);K(3*m1,3*m2) = K(3*m1,3*m2) + k(8,7);K(3*m1,3*m1) = K(3*m1,3*m1) + k(8,8);K(3*m1,3*m) = K(3*m1,3*m) + k(8,9);K(3*m1,3*n2) = K(3*m1,3*n2) + k(8,10);K(3*m1,3*n1) = K(3*m1,3*n1) + k(8,11);K(3*m1,3*n) = K(3*m1,3*n) + k(8,12);K(3*m,3*i2) = K(3*m,3*i2) + k(9,1);K(3*m,3*i1) = K(3*m,3*i1) + k(9,2);K(3*m,3*i) = K(3*m,3*i) + k(9,3);K(3*m,3*j2) = K(3*m,3*j2) + k(9,4);K(3*m,3*j1) = K(3*m,3*j1) + k(9,5);K(3*m,3*j) = K(3*m,3*j) + k(9,6);K(3*m,3*m2) = K(3*m,3*m2) + k(9,7);K(3*m,3*m1) = K(3*m,3*m1) + k(9,8);K(3*m,3*m) = K(3*m,3*m) + k(9,9);K(3*m,3*n2) = K(3*m,3*n2) + k(9,10);K(3*m,3*n1) = K(3*m,3*n1) + k(9,11);K(3*m,3*n) = K(3*m,3*n) + k(9,12);K(3*n2,3*i2) = K(3*n2,3*i2) + k(10,1);K(3*n2,3*i1) = K(3*n2,3*i1) + k(10,2);K(3*n2,3*i) = K(3*n2,3*i) + k(10,3);K(3*n2,3*j2) = K(3*n2,3*j2) + k(10,4);K(3*n2,3*j1) = K(3*n2,3*j1) + k(10,5);K(3*n2,3*j) = K(3*n2,3*j) + k(10,6);K(3*n2,3*m2) = K(3*n2,3*m2) + k(10,7);K(3*n2,3*m1) = K(3*n2,3*m1) + k(10,8);K(3*n2,3*m) = K(3*n2,3*m) + k(10,9);K(3*n2,3*n2) = K(3*n2,3*n2) + k(10,10);K(3*n2,3*n1) = K(3*n2,3*n1) + k(10,11);K(3*n2,3*n) = K(3*n2,3*n) + k(10,12);K(3*n1,3*i2) = K(3*n1,3*i2) + k(11,1);K(3*n1,3*i1) = K(3*n1,3*i1) + k(11,2);K(3*n1,3*i) = K(3*n1,3*i) + k(11,3);K(3*n1,3*j2) = K(3*n1,3*j2) + k(11,4);K(3*n1,3*j1) = K(3*n1,3*j1) + k(11,5);K(3*n1,3*j) = K(3*n1,3*j) + k(11,6);K(3*n1,3*m2) = K(3*n1,3*m2) + k(11,7);K(3*n1,3*m1) = K(3*n1,3*m1) + k(11,8);K(3*n1,3*m) = K(3*n1,3*m) + k(11,9);K(3*n1,3*n2) = K(3*n1,3*n2) + k(11,10);K(3*n1,3*n1) = K(3*n1,3*n1) + k(11,11);K(3*n1,3*n) = K(3*n1,3*n) + k(11,12);K(3*n,3*i2) = K(3*n,3*i2) + k(12,1);K(3*n,3*i1) = K(3*n,3*i1) + k(12,2);K(3*n,3*i) = K(3*n,3*i) + k(12,3);K(3*n,3*j2) = K(3*n,3*j2) + k(12,4);K(3*n,3*j1) = K(3*n,3*j1) + k(12,5);K(3*n,3*j) = K(3*n,3*j) + k(12,6);K(3*n,3*m2) = K(3*n,3*m2) + k(12,7);K(3*n,3*m1) = K(3*n,3*m1) + k(12,8);K(3*n,3*m) = K(3*n,3*m) + k(12,9);K(3*n,3*n2) = K(3*n,3*n2) + k(12,10);K(3*n,3*n1) = K(3*n,3*n1) + k(12,11);K(3*n,3*n) = K(3*n,3*n) + k(12,12);y = K;
funcin y = TetrahedronElementStresses(E, NU, x 1, y1, z1, x 2, y2, z2, x 3, y3, z3, x 4, y4, z4, u)% TetrahedronElementStresses esta funcin devuelve el elemento% vector de tensin de forma linealelemento (slido) tetradrico % con% mdulo de elasticidad E, Cociente de Poisson % NU, coordenadas% del primer nodo (x 1, y1, z1), coordenadas % del segundo nodo% (x 2, y2, z2), coordenadas de la% tercer nodo (x 3, y3, z3), coordenadas % del cuarto nodo% (x 4, y4, z4) y el desplazamiento del elemento% vector u.% El tamao de la tensin del elemento% vector es 6 x 1xyz = [1 x1 y1 z1 ; 1 x2 y2 z2 ; 1 x3 y3 z3 ; 1 x4 y4 z4];V = det(xyz)/6;mbeta1 = [1 y2 z2 ; 1 y3 z3 ; 1 y4 z4];mbeta2 = [1 y1 z1 ; 1 y3 z3 ; 1 y4 z4];mbeta3 = [1 y1 z1 ; 1 y2 z2 ; 1 y4 z4];mbeta4 = [1 y1 z1 ; 1 y2 z2 ; 1 y3 z3];mgamma1 = [1 x2 z2 ; 1 x3 z3 ; 1 x4 z4];mgamma2 = [1 x1 z1 ; 1 x3 z3 ; 1 x4 z4];mgamma3 = [1 x1 z1 ; 1 x2 z2 ; 1 x4 z4];mgamma4 = [1 x1 z1 ; 1 x2 z2 ; 1 x3 z3];mdelta1 = [1 x2 y2 ; 1 x3 y3 ; 1 x4 y4];mdelta2 = [1 x1 y1 ; 1 x3 y3 ; 1 x4 y4];mdelta3 = [1 x1 y1 ; 1 x2 y2 ; 1 x4 y4];mdelta4 = [1 x1 y1 ; 1 x2 y2 ; 1 x3 y3];beta1 = 1*det(mbeta1);beta2 = det(mbeta2);beta3 = 1*det(mbeta3);beta4 = det(mbeta4);gamma1 = det(mgamma1);gamma2 = 1*det(mgamma2);gamma3 = det(mgamma3);gamma4 = 1*det(mgamma4);delta1 = 1*det(mdelta1);delta2 = det(mdelta2);delta3 = 1*det(mdelta3);delta4 = det(mdelta4);B1 = [beta1 0 0 ; 0 gamma1 0 ; 0 0 delta1 ;gamma1 beta1 0 ; 0 delta1 gamma1 ; delta1 0 beta1];B2 = [beta2 0 0 ; 0 gamma2 0 ; 0 0 delta2 ;gamma2 beta2 0 ; 0 delta2 gamma2 ; delta2 0 beta2];B3 = [beta3 0 0 ; 0 gamma3 0 ; 0 0 delta3 ;gamma3 beta3 0 ; 0 delta3 gamma3 ; delta3 0 beta3];B4 = [beta4 0 0 ; 0 gamma4 0 ; 0 0 delta4 ;gamma4 beta4 0 ; 0 delta4 gamma4 ; delta4 0 beta4];B = [B1 B2 B3 B4]/(6*V);D = (E/((1+NU)*(12*NU)))*[1NU NU NU 0 0 0 ; NU 1NU NU 0 0 0 ;NU NU 1NU 0 0 0 ; 0 0 0 (12*NU)/2 0 0 ; 0 0 0 0 (12*NU)/2 0 ;0 0 0 0 0 (12*NU)/2];y = D*B*u;
funcin y = TetrahedronElementPStresses(sigma)% TetrahedronElementPStresses esta funcin devuelve los trestensiones % principales del elemento%dado el vector de tensin del elemento.% Los ngulos principales no son devueltos.s1 = sigma (1) + sigma(2) + sigma(3);s2 = sigma(1)*sigma(2) + sigma(1)*sigma(3) + sigma(2)*sigma(3) sigma(4)*sigma(4) sigma(5)*sigma(5) sigma(6)*sigma(6);ms3 = [sigma(1) sigma(4) sigma(6) ; sigma(4) sigma(2) sigma(5) ;sigma(6) sigma(5) sigma(3)];s3 = det(ms3);y = [s1; s2 ; s3];
EJEMPLO DE APLICCION:
Considere la placa delgada se somete a una carga uniformemente distribuida. Utilice cinco elementos tetradricos lineales para resolver este problema. Dado E = 210 GPa, = 0,3, t = 0.025 m, y w = 3000 kN / m2, determine:
1. la matriz de rigidez global para la estructura.2. los desplazamientos en los nodos 3, 4, 7 y 8.3. Las reacciones en los nodos 1, 2, 5 y 6.4. las tensiones en cada elemento.5. los esfuerzos principales para cada elemento.
Desratizacin de placa delgada en Cinco lineal Tetraedros
SOLUCIN:
Utilice los seis pasos descritos en el Cap. 1 para resolver este problema usando el elemento lineal tetradrico.
PASO 1 - discretizar el dominio:: Subdividimos la placa en cinco elementos lineales tetradricos solamente para fines ilustrativos. Ms elementos deben ser utilizados con el fin de obtener resultados fiables. As, el dominio se subdivide en cinco elementos y ocho nodos. La fuerza total debido a la carga distribuida se divide por igual entre los nodos 3, 4, 7, y 8 en la proporcin 1: 2: 2: 1. Esta relacin se obtiene considerando que los nodos 4 y 7 soportan cargas a partir de dos elementos cada uno de los nodos, mientras 3 y 8 soportan cargas de cada uno de los elementos. Las unidades utilizadas en los clculos de MATLAB son kN y metro.
Tabla. Elemento de conectividad para el ejemplo:
Nmero de elementoNodo iNodo jNodo mNodo n
11246
21437
36571
46784
51647
PASO 2 - Escribiendo el Elemento Rigidez Matrices:Las cinco matrices de elemento de rigidez k1, k2, k3, k4 y k5 se obtienen al colocar la funcin TetrahedronElementStiffness en el MATLAB. Cada matriz tiene un tamao de 12 12. E=210e6
E =210000000
NU=0.3
NU =
0.3000
k1=TetrahedronElementStiffness(E,NU,0,0,0,0.025,0,0,0.025,0.5,0,0.025,0,0.25)
k1 =
1.0e+008 *
Columns 1 through 72.355800-2.35580.05050.10100
00.673100.0337-0.67310-0.0337
000.67310.06730-0.67310
-2.35580.03370.06732.3642-0.0841-0.1683-0.0017
0.0505-0.67310-0.08410.68570.00840.0337
0.10100-0.6731-0.16830.00840.69830
0-0.03370-0.00170.033700.0017
-0.0505000.0505-0.0059-0.00500
0000-0.0034-0.00170
00-0.0673-0.006700.06730
0000-0.0067-0.00340
-0.1010000.1010-0.0050-0.02360
Columns 8 through 12-0.0505000-0.1010
00000
00-0.067300
0.05050-0.006700.1010
-0.0059-0.00340-0.0067-0.0050
-0.0050-0.00170.0673-0.0034-0.0236
00000
0.00590000.0050
00.001700.00340
000.006700
00.003400.00670
0.00500000.0236
k2=TetrahedronElementStiffness(E,NU,0,0,0,0.025,0.5,0,0,0.5,0,0,0.5,0.25)
k2 =
1.0e+008 *
Columns 1 through 7
0.0017000-0.03370-0.0017
00.00590-0.0505000.0505
000.00170000
0-0.050502.355800-2.3558
-0.03370000.673100.0337
000000.6731-0.0673
-0.00170.05050-2.35580.0337-0.06732.3642
0.0337-0.00590.00340.0505-0.67310-0.0841
00.0050-0.0017-0.10100-0.67310.1683
000000.0673-0.0067
00-0.00340000
0-0.005000.101000-0.1010
Columns 8 through 120.03370000
-0.00590.005000-0.0050
0.0034-0.00170-0.00340
0.0505-0.1010000.1010
-0.67310000
0-0.67310.067300
-0.08410.1683-0.00670-0.1010
0.6857-0.00840-0.00670.0050
-0.00840.6983-0.06730.0034-0.0236
0-0.06730.006700
-0.00670.003400.00670
0.0050-0.0236000.0236
k3=TetrahedronElementStiffness(E,NU,0.025,0,0.25,0,0,0.25,0,0.5,0.25,0,0,0)
k3 =
1.0e+008 *Columns 1 through 7
2.355800-2.3558-0.05050.10100
00.67310-0.0337-0.673100.0337
000.67310.06730-0.67310
-2.3558-0.03370.06732.36420.0841-0.1683-0.0017
-0.0505-0.673100.08410.6857-0.0084-0.0337
0.10100-0.6731-0.1683-0.00840.69830
00.03370-0.0017-0.033700.0017
0.050500-0.0505-0.00590.00500
00000.0034-0.00170
00-0.0673-0.006700.06730
0000-0.00670.00340
-0.1010000.10100.0050-0.02360
Columns 8 through 12
0.0505000-0.1010
00000
00-0.067300
-0.05050-0.006700.1010
-0.00590.00340-0.00670.0050
0.0050-0.00170.06730.0034-0.0236
00000
0.0059000-0.0050
00.00170-0.00340
000.006700
0-0.003400.00670
-0.00500000.0236
k4=TetrahedronElementStiffness(E,NU,0.025,0,0.25,0,0.5,0.25,0.025,0.5,0.25,0.025,0.5,0)
k4 =
1.0e+008 *
Columns 1 through 7
0.00170000.03370-0.0017
00.005900.050500-0.0505
000.00170000
00.050502.355800-2.3558
0.03370000.67310-0.0337
000000.6731-0.0673
-0.0017-0.05050-2.3558-0.0337-0.06732.3642
-0.0337-0.0059-0.0034-0.0505-0.673100.0841
0-0.0050-0.0017-0.10100-0.67310.1683
000000.0673-0.0067
000.00340000
00.005000.101000-0.1010
Columns 8 through 12
-0.03370000
-0.0059-0.0050000.0050
-0.0034-0.001700.00340
-0.0505-0.1010000.1010
-0.67310000
0-0.67310.067300
0.08410.1683-0.00670-0.1010
0.68570.00840-0.0067-0.0050
0.00840.6983-0.0673-0.0034-0.0236
0-0.06730.006700
-0.0067-0.003400.00670
-0.0050-0.0236000.0236
k5=TetrahedronElementStiffness(E,NU,0,0,0,0.025,0,0.25,0.025,0.5,0,0,0.5,0.25)
k5 =
1.0e+008 *
Columns 1 through 71.18210.04210.0841-1.18040.0084-0.0841-1.1754
0.04210.34280.0042-0.0084-0.3370-0.0008-0.0421
0.08410.00420.3492-0.08410.0008-0.3475-0.0168
-1.1804-0.0084-0.08411.1821-0.04210.08411.1737
0.0084-0.33700.0008-0.04210.3428-0.0042-0.0084
-0.0841-0.0008-0.34750.0841-0.00420.34920.0168
-1.1754-0.0421-0.01681.1737-0.00840.01681.1821
-0.0421-0.3361-0.00080.00840.33020.00420.0421
0.01680.0008-0.3256-0.01680.00420.3239-0.0841
1.17370.00840.0168-1.17540.0421-0.0168-1.1804
-0.00840.3302-0.00420.0421-0.33610.00080.0084
-0.0168-0.00420.32390.0168-0.0008-0.32560.0841
Columns 8 through 12-0.04210.01681.1737-0.0084-0.0168
-0.33610.00080.00840.3302-0.0042
-0.0008-0.32560.0168-0.00420.3239
0.0084-0.0168-1.17540.04210.0168
0.33020.00420.0421-0.3361-0.0008
0.00420.3239-0.01680.0008-0.3256
0.0421-0.0841-1.18040.00840.0841
0.3428-0.0042-0.0084-0.33700.0008
-0.00420.34920.0841-0.0008-0.3475
-0.00840.08411.1821-0.0421-0.0841
-0.3370-0.0008-0.04210.34280.0042
0.0008-0.3475-0.08410.00420.3492
PASO 3 - Montaje de la matriz Global de Rigidez:Dado que la estructura tiene ocho nodos , el tamao de la matriz de rigidez global es de 24 24 .Por lo tanto para obtener K primero establecimos una matriz cero de tamao 24 24 a continuacin, hacer cinco llamadas a la funcin TetrahedronAssemble MATLAB ya que tenemos cinco elementos en la estructura. Cada llamada a la funcin reunir a un elemento. Los siguientes son los comandos de MATLAB. El resultado final se muestra slo despus de que el quinto elemento tiene ha montado.
K=zeros(24,24);
K=TetrahedronAssemble(K,k1,1,2,4,6);
K=TetrahedronAssemble(K,k2,1,4,3,7);
K=TetrahedronAssemble(K,k3,6,5,7,1);
K=TetrahedronAssemble(K,k4,6,7,8,4);
K=TetrahedronAssemble(K,k5,1,6,4,7)
K =
1.0e+008 *
Columns 1 through 7
3.54630.04210.0841-2.35580.05050.1010-0.0017
0.04211.02850.00420.0337-0.673100.0505
0.08410.00421.04750.06730-0.67310
-2.35580.03370.06732.3642-0.0841-0.16830
0.0505-0.67310-0.08410.68570.00840
0.10100-0.6731-0.16830.00840.69830
-0.00170.050500002.3642
0.0337-0.00590.0034000-0.0841
00.0050-0.00170000.1683
-1.1754-0.1262-0.0168-0.00170.03370-2.3558
-0.1262-0.3361-0.00080.0505-0.0059-0.00500.0337
0.01680.0008-0.32560-0.0034-0.0017-0.0673
-0.006700.10100000
0-0.00670.00500000
0.06730.0034-0.02360000
-1.1804-0.0084-0.2524-0.006700.06730
0.0084-0.33700.00080-0.0067-0.00340
-0.2524-0.0008-0.34750.1010-0.0050-0.02360
1.17370.00840.0168000-0.0067
-0.00840.3302-0.01260000
-0.0168-0.01260.3239000-0.1010
0000000
0000000
0000000
Columns 8 through 14
Columns 15 through 210.0673-1.18040.0084-0.25241.1737-0.0084-0.0168
0.0034-0.0084-0.3370-0.00080.00840.3302-0.0126
-0.0236-0.25240.0008-0.34750.0168-0.01260.3239
0-0.006700.1010000
00-0.0067-0.0050000
00.0673-0.0034-0.0236000
0000-0.00670-0.1010
00000-0.00670.0050
0000-0.06730.0034-0.0236
01.1737-0.00840.0168-1.18040.00840.2524
00.00840.33020.0126-0.0084-0.33700.0008
0-0.01680.01260.32390.2524-0.0008-0.3475
-0.1683-2.3558-0.03370.0673-0.0017-0.05050
-0.0084-0.0505-0.67310-0.0337-0.00590.0034
0.69830.10100-0.673100.0050-0.0017
0.10103.5463-0.04210.0841-1.17540.12620.0168
0-0.04211.0285-0.00420.1262-0.3361-0.0008
-0.67310.0841-0.00421.0475-0.01680.0008-0.3256
0-1.17540.1262-0.01683.5463-0.0421-0.0841
0.00500.1262-0.33610.0008-0.04211.02850.0042
-0.00170.0168-0.0008-0.3256-0.08410.00421.0475
0-0.0017-0.05050-2.3558-0.0337-0.0673
0-0.0337-0.0059-0.0034-0.0505-0.67310
00-0.0050-0.0017-0.10100-0.6731
Columns 22 through 24
Paso 4 - la aplicacin de las condiciones de contorno:La matriz para esta estructura se puede escribir utilizando la matriz de rigidez global obtenido en la etapa anterior. Las condiciones de contorno para este problema se dan como:
U1x = U1y = U1z = U2x = U2y = U2z = 0U5x = U5y = U5z = U6x = U6y = U6z = 0F3x = 0, F3y = 3.125, F3z = 0F4x = 0, F4y = 6.25, F4z = 0F7x = 0, F7y = 6.25, F7z = 0F8x = 0, F8y = 3.125, F8z = 0
A continuacin insertamos las condiciones anteriores en la ecuacin matricial para esta estructura (no se muestra aqu) y proceder al paso de la solucin a continuacin.
PASO 5 - Solucin de las ecuaciones:Resolviendo el sistema de ecuaciones resultante ser realizada por particin (manualmente) y la eliminacin de Gauss (con MATLAB). Primero dividimos la ecuacin resultante extrayendo los sub-matrices en filas 7 a 12, 19 a 24 filas y columnas 7 a 12, columnas 19 a 24. Por lo tanto, obtenemos la siguiente ecuacin sealando que los nmeros se muestran slo a dos cifras decimales aunque MATLAB realiza los clculos utilizando al menos cuatro cifras decimales.
Se obtiene la solucin del sistema anterior usando MATLAB como sigue. Tenga en cuenta que el operador barra invertida "\" se utiliza para la eliminacin de Gauss.
K = [K (7: 12,7: 12) K (7: 12,19: 24); K (19: 24,7: 12) K (19: 24,19: 24)]
k =
1.0e+008 *
Columns 1 through 7
2.3642-0.08410.1683-2.35580.0337-0.0673-0.0067
-0.08410.6857-0.00840.0505-0.673100
0.1683-0.00840.6983-0.10100-0.6731-0.0673
-2.35580.0505-0.10103.54630.0421-0.0841-1.1804
0.0337-0.673100.04211.0285-0.0042-0.0084
-0.06730-0.6731-0.0841-0.00421.04750.2524
-0.00670-0.0673-1.1804-0.00840.25243.5463
0-0.00670.00340.0084-0.3370-0.0008-0.0421
-0.10100.0050-0.02360.25240.0008-0.3475-0.0841
000-0.00670-0.1010-2.3558
0000-0.0067-0.0050-0.0505
000-0.0673-0.0034-0.0236-0.1010
Columns 8 through 120-0.1010000
-0.00670.0050000
0.0034-0.0236000
0.00840.2524-0.00670-0.0673
-0.33700.00080-0.0067-0.0034
-0.0008-0.3475-0.1010-0.0050-0.0236
-0.0421-0.0841-2.3558-0.0505-0.1010
1.02850.0042-0.0337-0.67310
0.00421.0475-0.06730-0.6731
-0.0337-0.06732.36420.08410.1683
-0.673100.08410.68570.0084
0-0.67310.16830.00840.6983
f=[0 ; 3.125 ; 0 ; 0 ; 6.25 ; 0 ; 0 ; 6.25 ; 0 ; 0 ;3.125 ;0]
f = 03.1250 0 06.2500 0 06.2500 0 03.1250 0
u=k\f
u =
1.0e-005 *
-0.00040.60820.0090-0.01270.60780.00560.01270.6078-0.00560.00040.6082-0.0090
Ahora est claro que el desplazamiento horizontal a lo largo de la direccin y en ambos nodos 3 y 8 es 0,6082 m, y el desplazamiento horizontal a lo largo de la direccin y en ambos nodos 4 y 7 es 0,6078 m. Estos resultados se comparan con el resultado de aproximadamente 0.7mobtained en los ejemplos anteriores y problemas en el cap. 11 a travs de 14 usando otros elementos.
PASO 6 - Post-procesamiento:En este paso, se obtienen las reacciones en los nodos 1, 2, 5, y 6, y las tensiones en cada elemento usando MATLAB como sigue. Primero creamos el vector de desplazamiento nodal U global, entonces calculamos la fuerza nodal vector F.
U=[0;0;0;0;0;0;u(1:6);0;0;0;0;0;0;u(7:12)]
U =
1.0e-005 *
0 0 0 0 0 0-0.0004 0.6082 0.0090-0.0127 0.6078 0.0056 0 0 0 0 0 0 0.0127 0.6078-0.0056 0.0004 0.6082-0.0090
F=K*U
F =
-31.3296-5.3492 -9.328630.7045 -4.0258 -3.0777 0.0000 3.1250 0 0.0000 6.2500 0.0000-30.7045-4.0258 3.077731.3296-5.3492 9.3286 0.0000 6.2500 0.0000 0.0000 3.1250 0.0000
Las reacciones de la fuerza a lo largo de las tres direcciones estn claramente aparecen arriba. Obviamente equilibrio de fuerzas se satisface para este problema. Siguiente hemos creado el elemento de desplazamiento nodal vectores u1, u2, u3, u4, y u5 entonces calculamos las tensiones de elemento sigma1, sigma2, sigma3, Sigma4 y sigma5 haciendo llamadas a los funcin de MATLAB TetrahedronElementStresses.
u1=[U(1) ; U(2) ; U(3) ; U(4) ; U(5) ; U(6) ; U(10) ;U(11) ; U(12) ; U(16) ; U(17) ; U(18)]
u1 =
1.0e-005 *
0 0 0 0 0 0-0.0127 0.6078 0.0056 0 0 0
u2=[U(1) ; U(2) ; U(3) ; U(10) ; U(11) ; U(12) ; U(7) ;U(8) ;U(9) ; U(19) ; U(20) ; U(21)]
u2 =
1.0e-005 *
0 0 0-0.0127 0.6078 0.0056-0.0004 0.6082 0.0090 0.0127 0.6078-0.0056
u3=[U(16) ; U(17) ; U(18) ; U(13) ; U(14) ; U(15) ;U(19) ; U(20) ; U(21) ; U(1) ; U(2) ; U(3)]
u3 =
1.0e-005 *
0 0 0 0 0 0 0.0127 0.6078-0.0056 0 0 0
u4=[U(16) ; U(17) ; U(18) ; U(19) ; U(20) ; U(21) ;U(22) ; U(23) ; U(24) ; U(10) ; U(11);U(12)]u4 =
1.0e-005 *
0 0 0 0.0127 0.6078-0.0056 0.0004 0.6082-0.0090-0.0127 0.6078 0.0056
u5=[U(1) ; U(2) ; U(3) ; U(16) ; U(17) ; U(18) ;U(10) ; U(11); U(12) ; U(19) ; U(20) ; U(21)]
u5 =
1.0e-005 *
0 0 0 0 0 0-0.0127 0.6078 0.0056 0.0127 0.6078-0.0056
sigma1=TetrahedronElementStresses(E,NU,0,0,0,0.025,0,0,0.025,0.5,0,0.025,0,0.25,u1)
sigma1 =
1.0e+003 *
1.4728 3.4365 1.4728-0.0205 0.0090 0 sigma2=TetrahedronElementStresses(E,NU,0,0,0,0.025,0.5,0,0,0.5,0,0,0.5,0.25,u2)
sigma2 =
1.0e+003 *
0.0064 2.7694 0.7102-0.0129 0.0134-0.0704
sigma3=TetrahedronElementStresses(E,NU,0.025,0,0.25,0,0,0.25,0,0.5,0.25,0,0,0,u3)
sigma3 =
1.0e+003 *
1.4728 3.4365-0.0090 0
sigma4=TetrahedronElementStresses(E,NU,0.025,0,0.25,0,0.5,0.25,0.025,0.5,0.25,0.025,0.5,0,u4)
sigma4 =
1.0e+003 *
0.0064 2.7694 0.7102 0.0129-0.0134-0.0704
sigma5=TetrahedronElementStresses(E,NU,0,0,0,0.025,0,0.25,0.025,0.5,0,0,0.5,0.25,u5)
sigma5 =
1.0e+003 *
0.00962.79410.79450.00000.00000.2204
Por lo tanto es evidente que las tensiones normales y a lo largo de la direccin y en elementos 1, 2, 3, 4, y 5 son 3.4365MPa (traccin), 2.7694MPa (traccin), 3.4365MPa (traccin), 2.7694MPa (traccin), y 2.7941MPa (traccin), respectivamente. Es claro que las tensiones en el enfoque de la direccin y de cerca el valor correcto de 3 MPa (traccin) .Siguiente calculamos los esfuerzos principales para cada elemento, haciendo llamadas a la funcin MATLAB TetrahedronElementPStresses funcin.
s1=TetrahedronElementPStresses(sigma1)
s1 =
1.0e+009 *
0.00000.01237.4534
s2=TetrahedronElementPStresses(sigma2)
s2 =
1.0e+006 *
0.00351.9839-1.2296
s3=TetrahedronElementPStresses(sigma3)
s3 =
1.0e+009 *
0.00000.01237.4534
s4=TetrahedronElementPStresses(sigma4)
s4 =
1.0e+006 *
0.00351.9839-1.2296
s5=TetrahedronElementPStresses(sigma5)s5 =
1.0e+008 *
0.00000.0221-1.1431
Los esfuerzos principales en las tres direcciones se muestran claramente por encima de cada uno de los cinco elementos en este ejemplo.
DESARROLLO EN MATLAB:>> E=210e6
E =
210000000
>> NU=0.3
NU =
0.3000
>> k1=TetrahedronElementStiffness(E,NU,0,0,0,0.025,0,0,0.025,0.5,0,0.025,0,0.25)
k1 =
1.0e+08 *
Columns 1 through 7 2.3558 0 0 -2.3558 0.0505 0.1010 0 0 0.6731 0 0.0337 -0.6731 0 -0.0337 0 0 0.6731 0.0673 0 -0.6731 0 -2.3558 0.0337 0.0673 2.3642 -0.0841 -0.1683 -0.0017 0.0505 -0.6731 0 -0.0841 0.6857 0.0084 0.0337 0.1010 0 -0.6731 -0.1683 0.0084 0.6983 0 0 -0.0337 0 -0.0017 0.0337 0 0.0017 -0.0505 0 0 0.0505 -0.0059 -0.0050 0 0 0 0 0 -0.0034 -0.0017 0 0 0 -0.0673 -0.0067 0 0.0673 0 0 0 0 0 -0.0067 -0.0034 0 -0.1010 0 0 0.1010 -0.0050 -0.0236 0Columns 8 through 12 -0.0505 0 0 0 -0.1010 0 0 0 0 0 0 0 -0.0673 0 0 0.0505 0 -0.0067 0 0.1010 -0.0059 -0.0034 0 -0.0067 -0.0050 -0.0050 -0.0017 0.0673 -0.0034 -0.0236 0 0 0 0 0 0.0059 0 0 0 0.0050 0 0.0017 0 0.0034 0 0 0 0.0067 0 0 0 0.0034 0 0.0067 0 0.0050 0 0 0 0.0236
>> k2=TetrahedronElementStiffness(E,NU,0,0,0,0.025,0.5,0,0,0.5,0,0,0.5,0.25)
k2 =
1.0e+08 *
Columns 1 through 7 0.0017 0 0 0 -0.0337 0 -0.0017 0 0.0059 0 -0.0505 0 0 0.0505 0 0 0.0017 0 0 0 0 0 -0.0505 0 2.3558 0 0 -2.3558 -0.0337 0 0 0 0.6731 0 0.0337 0 0 0 0 0 0.6731 -0.0673 -0.0017 0.0505 0 -2.3558 0.0337 -0.0673 2.3642 0.0337 -0.0059 0.0034 0.0505 -0.6731 0 -0.0841 0 0.0050 -0.0017 -0.1010 0 -0.6731 0.1683 0 0 0 0 0 0.0673 -0.0067 0 0 -0.0034 0 0 0 0 0 -0.0050 0 0.1010 0 0 -0.1010
Columns 8 through 12 0.0337 0 0 0 0 -0.0059 0.0050 0 0 -0.0050 0.0034 -0.0017 0 -0.0034 0 0.0505 -0.1010 0 0 0.1010 -0.6731 0 0 0 0 0 -0.6731 0.0673 0 0 -0.0841 0.1683 -0.0067 0 -0.1010 0.6857 -0.0084 0 -0.0067 0.0050 -0.0084 0.6983 -0.0673 0.0034 -0.0236 0 -0.0673 0.0067 0 0 -0.0067 0.0034 0 0.0067 0 0.0050 -0.0236 0 0 0.0236
>> k3=TetrahedronElementStiffness(E,NU,0.025,0,0.25,0,0,0.25,0,0.5,0.25,0,0,0)
k3 =
1.0e+08 *
Columns 1 through 7 2.3558 0 0 -2.3558 - 0.0505 0.1010 0 0 0.6731 0 -0.0337 - 0.6731 0 0.0337 0 0 0.6731 0.0673 0 -0.6731 0 -2.3558 -0.0337 0.0673 2.3642 0.0841 -0.1683 -0.0017 -0.0505 -0.6731 0 0.0841 0.6857 -0.0084 -0.0337 0.1010 0 -0.6731 -0.1683 -0.0084 0.6983 0 0 0.0337 0 -0.0017 -0.0337 0 0.0017 0.0505 0 0 -0.0505 -0.0059 0.0050 0 0 0 0 0 0.0034 -0.0017 0 0 0 -0.0673 -0.0067 0 0.0673 0 0 0 0 0 -0.0067 0.0034 0 -0.1010 0 0 0.1010 0.0050 -0.0236 0
Columns 8 through 12 0.0505 0 0 0 -0.1010 0 0 0 0 0 0 0 -0.0673 0 0 -0.0505 0 -0.0067 0 0.1010 -0.0059 0.0034 0 -0.0067 0.0050 0.0050 -0.0017 0.0673 0.0034 -0.0236 0 0 0 0 0 0.0059 0 0 0 -0.0050 0 0.0017 0 -0.0034 0 0 0 0.0067 0 0 0 -0.0034 0 0.0067 0 -0.0050 0 0 0 0.0236
>> k4=TetrahedronElementStiffness(E,NU,0.025,0,0.25,0,0.5,0.25,0.025,0.5,0.25,0.025,0.5,0)
k4 =
1.0e+08 *
Columns 1 through 7 0.0017 0 0 0 0.0337 0 -0.0017 0 0.0059 0 0.0505 0 0 -0.0505 0 0 0.0017 0 0 0 0 0 0.0505 0 2.3558 0 0 -2.3558 0.0337 0 0 0 0.6731 0 -0.0337 0 0 0 0 0 0.6731 -0.0673 -0.0017 -0.0505 0 -2.3558 -0.0337 -0.0673 2.3642 -0.0337 -0.0059 -0.0034 -0.0505 -0.6731 0 0.0841 0 -0.0050 -0.0017 -0.1010 0 -0.6731 0.1683 0 0 0 0 0 0.0673 -0.0067 0 0 0.0034 0 0 0 0 0 0.0050 0 0.1010 0 0 -0.1010
Columns 8 through 12 -0.0337 0 0 0 0 -0.0059 -0.0050 0 0 0.0050 -0.0034 -0.0017 0 0.0034 0 -0.0505 -0.1010 0 0 0.1010 -0.6731 0 0 0 0 0 -0.6731 0.0673 0 0 0.0841 0.1683 -0.0067 0 -0.1010 0.6857 0.0084 0 -0.0067 -0.0050 0.0084 0.6983 -0.0673 -0.0034 -0.0236 0 -0.0673 0.0067 0 0 -0.0067 -0.0034 0 0.0067 0 -0.0050 -0.0236 0 0 0.0236
>> k5=TetrahedronElementStiffness(E,NU,0,0,0,0.025,0,0.25,0.025,0.5,0,0,0.5,0.25)
k5 =
1.0e+08 *
Columns 1 through 7
1.1821 0.0421 0.0841 -1.1804 0.0084 -0.0841 -1.1754 0.0421 0.3428 0.0042 -0.0084 -0.3370 -0.0008 -0.0421 0.0841 0.0042 0.3492 -0.0841 0.0008 -0.3475 -0.0168 -1.1804 -0.0084 -0.0841 1.1821 -0.0421 0.0841 1.1737 0.0084 -0.3370 0.0008 -0.0421 0.3428 -0.0042 -0.0084 -0.0841 -0.0008 -0.3475 0.0841 -0.0042 0.3492 0.0168 -1.1754 -0.0421 -0.0168 1.1737 -0.0084 0.0168 1.1821 -0.0421 -0.3361 -0.0008 0.0084 0.3302 0.0042 0.0421 0.0168 0.0008 -0.3256 -0.0168 0.0042 0.3239 -0.0841 1.1737 0.0084 0.0168 -1.1754 0.0421 -0.0168 -1.1804 -0.0084 0.3302 -0.0042 0.0421 -0.3361 0.0008 0.0084 -0.0168 -0.0042 0.3239 0.0168 -0.0008 -0.3256 0.0841
Columns 1 through 7 1.1821 0.0421 0.0841 -1.1804 0.0084 -0.0841 -1.1754 0.0421 0.3428 0.0042 -0.0084 -0.3370 -0.0008 -0.0421 0.0841 0.0042 0.3492 -0.0841 0.0008 -0.3475 -0.0168 -1.1804 -0.0084 -0.0841 1.1821 -0.0421 0.0841 1.1737 0.0084 -0.3370 0.0008 -0.0421 0.3428 -0.0042 -0.0084 -0.0841 -0.0008 -0.3475 0.0841 -0.0042 0.3492 0.0168 -1.1754 -0.0421 -0.0168 1.1737 -0.0084 0.0168 1.1821 -0.0421 -0.3361 -0.0008 0.0084 0.3302 0.0042 0.0421 0.0168 0.0008 -0.3256 -0.0168 0.0042 0.3239 -0.0841 1.1737 0.0084 0.0168 -1.1754 0.0421 -0.0168 -1.1804 -0.0084 0.3302 -0.0042 0.0421 -0.3361 0.0008 0.0084 -0.0168 -0.0042 0.3239 0.0168 -0.0008 -0.3256 0.0841
>> K=zeros(24,24)
K =
Columns 1 through 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Columns 1 through 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
>> K=TetrahedronAssemble(K,k1,1,2,4,6)
K =
1.0e+08 *
Columns 1 through 7 2.3558 0 0 -2.3558 0.0505 0.1010 0 0 0.6731 0 0.0337 -0.6731 0 0 0 0 0.6731 0.0673 0 -0.6731 0 -2.3558 0.0337 0.0673 2.3642 -0.0841 -0.1683 0 0.0505 -0.6731 0 -0.0841 0.6857 0.0084 0 0.1010 0 -0.6731 -0.1683 0.0084 0.6983 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.0337 0 -0.0017 0.0337 0 0 -0.0505 0 0 0.0505 -0.0059 -0.0050 0 0 0 0 0 -0.0034 -0.0017 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.0673 -0.0067 0 0.0673 0 0 0 0 0 -0.0067 -0.0034 0 -0.1010 0 0 0.1010 -0.0050 -0.0236 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Columns 8 through 14
0 0 0 -0.0505 0 0 0 0 0 -0.0337 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.0017 0.0505 0 0 0 0 0 0.0337 -0.0059 -0.0034 0 0 0 0 0 -0.0050 -0.0017 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0017 0 0 0 0 0 0 0 0.0059 0 0 0 0 0 0 0 0.0017 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0034 0 0 0 0 0 0.0050 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Columns 15 through 21
0 0 0 -0.1010 0 0 0 0 0 0 0 0 0 0 0 -0.0673 0 0 0 0 0 0 -0.0067 0 0.1010 0 0 0 0 0 -0.0067 -0.0050 0 0 0 0 0.0673 -0.0034 -0.0236 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0050 0 0 0 0 0 0.0034 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0067 0 0 0 0 0 0 0 0.0067 0 0 0 0 0 0 0 0.0236 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Columns 22 through 24
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
>> K=TetrahedronAssemble(K,k2,1,4,3,7)
K =
1.0e+08 *