ELEMENTO RECTANGULAR SIMPLE MEJORADONodo central (5 nodos)

figura 1.1

Una manera de mejorar el comportamiento de un elemento finito, consiste en agregar a suexpansión para desplazamientos, una función que se anula sobre el contorno del elemento en la figura 1.1

Adoptaremos la siguiente aproximación para los desplazamientos:

Teniendo nuestra matriz [N] de funciones de forma y considerando el estado plano de tensiones podemosobtener nuestra matriz [B]

B1

a b⋅( )

b y−( )−

0

a x−( )−

0

a x−( )−

b y−( )−

b y−( )

0

x−

0

x−

b y−( )

y

0

x

0

x

y

y−

0

a x−( )

0

a x−( )

y−

π

acos π

x

a⋅

⋅ sin πy

b⋅

⋅

0

π

bsin π

x

a⋅

⋅ cos πy

b⋅

⋅

0

π

bsin π

x

b⋅

⋅ cos πy

b⋅

⋅

π

acos π

x

a⋅

⋅ sin πy

b⋅

⋅

:=a

DE

1 v2−

1

v

0

v

1

0

0

0

1 v−( )

2

⋅:=E

K BT D⋅ B⋅:= B

Sabemos que la matriz K está definida:

Sin embargo el programa no fué capaz de desarrollarla debido a la cantidad de operaciones y la extension de ella ya quecomo sabemos es una matriz de 10x10. Ante esta dificultad nos propusimos desarrollarla por partes ya que el programa sisería capaz de hacerlo de esta forma.El siguiente diagrama representa las sub matrices que se desarrollaron y qué posicion ocupan dentro de la matriz K

A continuación se presentan los algoritmos que utilizamos para hallar cada sub matriz.

Ke1 p

Ke1i j,

0

b

y

0

a

xKi j,

⌠⌡

d⌠⌡

d←

j 1 4..∈for

i 1 5..∈for

Ke1

:=

K

Ke1

E a2

a2

v⋅− 2 b2⋅+( )⋅

6 a⋅ b⋅ v2

1−( )⋅−

E

8 v⋅ 8−−

E a2

v⋅ a2− 4 b

2⋅+( )⋅

12 a⋅ b⋅ v2

1−( )⋅

E 3 v⋅ 1−( )⋅

8 v2⋅ 8−

−

E a2

a2

v⋅− 2 b2⋅+( )⋅

12 a⋅ b⋅ v2

1−( )⋅

E

8 v⋅ 8−−

E 2 a2⋅ b

2v⋅− b

2+( )⋅

6 a⋅ b⋅ v2

1−( )⋅−

E 3 v⋅ 1−( )⋅

8 v2⋅ 8−

E b2

v⋅ a2+ b

2−( )⋅

6 a⋅ b⋅ v2

1−( )⋅−

E

8 v⋅ 8−

E a2

v⋅ a2− 4 b

2⋅+( )⋅

12 a⋅ b⋅ v2

1−( )⋅

E 3 v⋅ 1−( )⋅

8 v2⋅ 8−

E a2

a2

v⋅− 2 b2⋅+( )⋅

6 a⋅ b⋅ v2

1−( )⋅−

E

8 v⋅ 8−

E a2

v⋅ a2− b

2+( )⋅

6 a⋅ b⋅ v2

1−( )⋅−

E 3 v⋅ 1−( )⋅

8 v2⋅ 8−

−

E b2

v⋅ a2+ b

2−( )⋅

6 a⋅ b⋅ v2

1−( )⋅−

E

8 v⋅ 8−

E 2 a2⋅ b

2v⋅− b

2+( )⋅

6 a⋅ b⋅ v2

1−( )⋅−

E 3 v⋅ 1−( )⋅

8 v2⋅ 8−

→

Ke2 p

Ke2i j 4−,

0

b

y

0

a

xKi j,

⌠⌡

d⌠⌡

d←

j 5 8..∈for

i 1 5..∈for

Ke2

:=

K

Ke2

E a2

a2

v⋅− 2 b2⋅+( )⋅

12 a⋅ b⋅ v2

1−( )⋅

E

8 v⋅ 8−

E a2

v⋅ a2− b

2+( )⋅

6 a⋅ b⋅ v2

1−( )⋅−

E 3 v⋅ 1−( )⋅

8 v2⋅ 8−

E a2

a2

v⋅− 2 b2⋅+( )⋅

6 a⋅ b⋅ v2

1−( )⋅−

E

8 v⋅ 8−

E 2 a2⋅ b

2v⋅− b

2+( )⋅

12 a⋅ b⋅ v2

1−( )⋅

E 3 v⋅ 1−( )⋅

8 v2⋅ 8−

−

E b2

v⋅ 4 a2⋅+ b

2−( )⋅

12 a⋅ b⋅ v2

1−( )⋅

E

8 v⋅ 8−−

E a2

v⋅ a2− b

2+( )⋅

6 a⋅ b⋅ v2

1−( )⋅−

E 3 v⋅ 1−( )⋅

8 v2⋅ 8−

−

E a2

a2

v⋅− 2 b2⋅+( )⋅

12 a⋅ b⋅ v2

1−( )⋅

E

8 v⋅ 8−−

E a2

v⋅ a2− 4 b

2⋅+( )⋅

12 a⋅ b⋅ v2

1−( )⋅

E 3 v⋅ 1−( )⋅

8 v2⋅ 8−

E b2

v⋅ 4 a2⋅+ b

2−( )⋅

12 a⋅ b⋅ v2

1−( )⋅

E

8 v⋅ 8−−

E 2 a2⋅ b

2v⋅− b

2+( )⋅

12 a⋅ b⋅ v2

1−( )⋅

E 3 v⋅ 1−( )⋅

8 v2⋅ 8−

−

→

Ke3 p

Ke3i j 8−,

0

b

y

0

a

xKi j,

⌠⌡

d⌠⌡

d←

j 9 10..∈for

i 1 1..∈for

Ke3

:=

K

Ke3 0

2 E⋅ a⋅ v⋅ 2 E⋅ a⋅− 4 E⋅ b⋅ v⋅ cosπ a⋅2 b⋅

2

1−

⋅+

π2

a2⋅ b⋅ π

2a

2⋅ b⋅ v2⋅−

→

Ke4 p

Ke4i 1− j 8−,

0

b

y

0

a

xKi j,

⌠⌡

d⌠⌡

d←

j 9 10..∈for

i 2 2..∈for

Ke4

:=

K

Ke42 E⋅

π2

a⋅ b⋅ v 1−( )⋅0

→

Ke5 p

Ke5i 2− j 8−,

0

b

y

0

a

xKi j,

⌠⌡

d⌠⌡

d←

j 9 10..∈for

i 3 3..∈for

Ke5

:=

K

Ke5 0

4 E⋅ b⋅ v⋅ sinπ a⋅2 b⋅

2

⋅ 2 E⋅ a⋅+ 2 E⋅ a⋅ v⋅−

π2

a2⋅ b⋅ π

2a

2⋅ b⋅ v2⋅−

→

Ke6 p

Ke6i 3− j 8−,

0

b

y

0

a

xKi j,

⌠⌡

d⌠⌡

d←

j 9 10..∈for

i 4 5..∈for

Ke6

:=

K

Ke6

2 E⋅

π2

a⋅ b⋅ v 1−( )⋅−

0

0

2 E⋅ a⋅ v⋅ 2 E⋅ a⋅− 4 E⋅ b⋅ v⋅ cosπ a⋅2 b⋅

2

1−

⋅+

π2

a2⋅ b⋅ π

2a2⋅ b⋅ v

2⋅−

→

Ke7 p

Ke7i 5− j,

0

b

y

0

a

xKi j,

⌠⌡

d⌠⌡

d←

j 1 4..∈for

i 6 8..∈for

Ke7

:=

K

Ke7

E

8 v⋅ 8−

E a2

v⋅ a2− b

2+( )⋅

6 a⋅ b⋅ v2

1−( )⋅−

E 3 v⋅ 1−( )⋅

8 v2⋅ 8−

E 2 a2⋅ b

2v⋅− b

2+( )⋅

12 a⋅ b⋅ v2

1−( )⋅

E 3 v⋅ 1−( )⋅

8 v2⋅ 8−

−

E b2

v⋅ 4 a2⋅+ b

2−( )⋅

12 a⋅ b⋅ v2

1−( )⋅

E 3 v⋅ 1−( )⋅

8 v2⋅ 8−

−

E a2

a2

v⋅− 2 b2⋅+( )⋅

12 a⋅ b⋅ v2

1−( )⋅

E

8 v⋅ 8−−

E b2

v⋅ 4 a2⋅+ b

2−( )⋅

12 a⋅ b⋅ v2

1−( )⋅

E

8 v⋅ 8−−

E 2 a2⋅ b

2v⋅− b

2+( )⋅

12 a⋅ b⋅ v2

1−( )⋅

→

Ke8 p

Ke8i 5− j 4−,

0

b

y

0

a

xKi j,

⌠⌡

d⌠⌡

d←

j 5 8..∈for

i 6 8..∈for

Ke8

:=

K

Ke8

E

8 v⋅ 8−−

E a2

v⋅ a2− 4 b

2⋅+( )⋅

12 a⋅ b⋅ v2

1−( )⋅

E 3 v⋅ 1−( )⋅

8 v2⋅ 8−

−

E 2 a2⋅ b

2v⋅− b

2+( )⋅

6 a⋅ b⋅ v2

1−( )⋅−

E 3 v⋅ 1−( )⋅

8 v2⋅ 8−

E b2

v⋅ a2+ b

2−( )⋅

6 a⋅ b⋅ v2

1−( )⋅−

E 3 v⋅ 1−( )⋅

8 v2⋅ 8−

E a2

a2

v⋅− 2 b2⋅+( )⋅

6 a⋅ b⋅ v2

1−( )⋅−

E

8 v⋅ 8−

E b2

v⋅ a2+ b

2−( )⋅

6 a⋅ b⋅ v2

1−( )⋅−

E

8 v⋅ 8−

E 2 a2⋅ b

2v⋅− b

2+( )⋅

6 a⋅ b⋅ v2

1−( )⋅−

→

Ke9 p

Ke9i 5− j 8−,

0

b

y

0

a

xKi j,

⌠⌡

d⌠⌡

d←

j 9 10..∈for

i 6 8..∈for

Ke9

:=

K

Ke9

2 E⋅

π2

a⋅ b⋅ v 1−( )⋅

0

2 E⋅

π2

a⋅ b⋅ v 1−( )⋅−

0

4 E⋅ b⋅ v⋅ sinπ a⋅2 b⋅

2

⋅ 2 E⋅ a⋅+ 2 E⋅ a⋅ v⋅−

π2

a2⋅ b⋅ π

2a

2⋅ b⋅ v2⋅−

0

→

Ke10 p

Ke10i 8− j,

0

b

y

0

a

xKi j,

⌠⌡

d⌠⌡

d←

j 1 4..∈for

i 9 10..∈for

Ke10

:=

K

Ke10

0

2 E⋅ a⋅ v⋅ 2 E⋅ a⋅− 4 E⋅ b⋅ v⋅ cosπ a⋅2 b⋅

2

1−

⋅+

π2

a2⋅ b⋅ π

2a2⋅ b⋅ v

2⋅−

2 E⋅

π2

a⋅ b⋅ v 1−( )⋅

0

0

4 E⋅ b⋅ v⋅ sinπ a⋅2 b⋅

2

⋅ 2 E⋅ a⋅+ 2 E⋅ a⋅ v⋅−

π2

a2⋅ b⋅ π

2a2⋅ b⋅ v

2⋅−

2 E⋅

π2

a⋅ b⋅ v 1−( )⋅−

0

→

Ke11 p

Ke11i 8− j 4−,

0

b

y

0

a

xKi j,

⌠⌡

d⌠⌡

d←

j 5 10..∈for

i 9 10..∈for

Ke11

:=

K

Ke11T

0

2 E⋅

π2

a⋅ b⋅ v 1−( )⋅

0

2 E⋅

π2

a⋅ b⋅ v 1−( )⋅−

π2

E⋅ a2

a2

v⋅− 2 b2⋅+( )⋅

8 a3⋅ b

3⋅ v2

1−( )⋅−

0

2 E⋅ a⋅ v⋅ 2 E⋅ a⋅− 4 E⋅ b⋅ v⋅ cosπ a⋅2 b⋅

2

1−

⋅+

π2

a2⋅ b⋅ π

2a2⋅ b⋅ v

2⋅−

0

4 E⋅ b⋅ v⋅ sinπ a⋅2 b⋅

2

⋅ 2 E⋅ a⋅+ 2 E⋅ a⋅ v⋅−

π2

a2⋅ b⋅ π

2a2⋅ b⋅ v

2⋅−

0

0

2 π2⋅ E⋅ a

2⋅ π2

E⋅ b2⋅+ π

2E⋅ b

2⋅ v⋅− π E⋅ a⋅ b⋅ sin2 π⋅ a⋅

b

⋅−

8 a3⋅ b

3⋅ 8 a3⋅ b

3⋅ v2⋅−

→

Ke

wi j, Ke1

i j, ←

i 1 5..∈for

j 1 4..∈for

wi j Ke7i 5− j, ←,

i 6 8..∈for

j 1 4..∈for

wi j, Ke10

i 8− j, ←

i 9 10..∈for

j 1 4..∈for

wi j, Ke2

i j 4−, ←

i 1 5..∈for

j 5 8..∈for

wi j, Ke8

i 5− j 4−, ←

i 6 8..∈for

j 5 8..∈for

wi j, Ke11

i 8− j 4−, ←

i 9 10..∈for

j 5 8..∈for

wi j, Ke3

i j 8−, ←

i 1 1..∈for

j 9 10..∈for

wi j, Ke4

i 1− j 8−, ←

i 2 2..∈for

j 9 10..∈for

wi j, Ke5

i 2− j 8−, ←

i 3 3..∈for

j 9 10..∈for

wi j, Ke6

i 3− j 8−, ←

i 4 5..∈for

j 9 10..∈for

wi j, Ke9

i 5− j 8−, ←

i 6 8..∈for

j 9 10..∈for

wi j, Ke11

i 8− j 8−, ←

i 9 10..∈for

j 9 10..∈for

w t⋅

:=

Ke1

Habiendo ya calculado todas las sub matricesprogramaremos una sub rutina paraensamblarlas en una matriz que llamaremos Ke(matriz de rigidez sin condensar)

A continuación mostraremos algunos elementos de la matriz Ke para demostrar que es correcto el procedimientohecho

Ke1 1,

E t⋅ a2

a2

v⋅− 2 b2⋅+( )⋅

6 a⋅ b⋅ v2

1−( )⋅−→

Ke2 2,

E t⋅ 2 a2⋅ b

2v⋅− b

2+( )⋅

6 a⋅ b⋅ v2

1−( )⋅−→

Ke3 3,

E t⋅ a2

a2

v⋅− 2 b2⋅+( )⋅

6 a⋅ b⋅ v2

1−( )⋅−→

Ke4 4,

E t⋅ 2 a2⋅ b

2v⋅− b

2+( )⋅

6 a⋅ b⋅ v2

1−( )⋅−→

Ke5 5,

E t⋅ a2

a2

v⋅− 2 b2⋅+( )⋅

6 a⋅ b⋅ v2

1−( )⋅−→

Ke7 7,

E t⋅ a2

a2

v⋅− 2 b2⋅+( )⋅

6 a⋅ b⋅ v2

1−( )⋅−→

Ke8 8,

E t⋅ 2 a2⋅ b

2v⋅− b

2+( )⋅

6 a⋅ b⋅ v2

1−( )⋅−→

Ke9 9, 0→

Ke10 10, 0→

Ahora procederemos a condensar la matriz a fin de obtener la matriz Keq de 8X8. Para alcanzar dicho objetivodesarrollaremos una rutina que nos permita obtener la matriz condensada.

Antes, definimos nuestro vector de cargas nodales

P NT Px

Py

⋅:=Px

P

Pxx1

a1−

⋅y1

b1−

⋅

Pyx1

a1−

⋅y1

b1−

⋅

Pxx1

a1−

⋅y1

b1−

⋅

Pyx1

a1−

⋅y1

b1−

⋅

Px x1⋅ y1⋅a b⋅

Py x1⋅ y1⋅a b⋅

Px y1⋅x1

a1−

⋅

b−

Py y1⋅x1

a1−

⋅

b−

Px sinπ x1⋅

a

⋅ sinπ y1⋅

b

⋅

Py sinπ x1⋅

a

⋅ sinπ y1⋅

b

⋅

→

Donde Px y Py son las componentes de una carga aplicada en un punto del elemento x1,y1

Kee r 0←

ri j, Ke

i j, ←

j 1 8..∈for

i 1 8..∈for

r

:=

Ke

Kei r 0←

ri j, Ke

i j 8+, ←

j 1 2..∈for

i 1 8..∈for

r

:=

Ke

Pe r 0←

ri

Pi

←

i 1 8..∈for

r

:=

P

Kie r 0←

ri j, Ke

i 8+ j, ←

j 1 8..∈for

i 1 2..∈for

r

:=

Ke

Kii r 0←

ri j, Ke

i 8+ j 8+, ←

j 1 2..∈for

i 1 2..∈for

r

:=

Ke

Pi r 0←

ri

Pi 8+←

i 1 2..∈for

r

:=

P

Keq Kee Kei Kii1−⋅ Kie⋅−:= Kee

Peq Pe Kei Kii1−⋅ Pi⋅−:= Pe

Finalmente mediante condensacion estática obtenemos la matriz Keq y Peq.No obstante como ya hemosmencionado el programa mathcad no nos permite visualizar la matriz Keq y Peq debido a la extension de ellas alusar notación algebraica, por este motivo sólo mostraremos algunos elementos de dichas matrices

Keq1 1,

E t⋅ a2

a2

v⋅− 2 b2⋅+( )⋅

6 a⋅ b⋅ v2

1−( )⋅−→ Keq

4 6, E t⋅ b

2v⋅ 4 a

2⋅+ b2−( )⋅

12 a⋅ b⋅ v2

1−( )⋅→ Keq

3 1, E t⋅ a

2v⋅ a

2− 4 b2⋅+( )⋅

12 a⋅ b⋅ v2

1−( )⋅→

Keq2 1,

2 E⋅ t2⋅ 2 E⋅ a⋅ v⋅ 2 E⋅ a⋅− 4 E⋅ b⋅ v⋅ cos

π a⋅2 b⋅

2

1−

⋅+

⋅

π2

a⋅ b⋅ v 1−( )⋅ 2 E⋅ a⋅ t⋅ 4 E⋅ b⋅ t⋅ v⋅ cosπ a⋅2 b⋅

2

⋅− 2 E⋅ a⋅ t⋅ v⋅− 4 E⋅ b⋅ t⋅ v⋅+

⋅

E t⋅8 v⋅ 8−

−→

Keq7 8,

E t⋅8 v⋅ 8−

t π2

a⋅ b⋅ π2

a⋅ b⋅ v⋅−( )⋅ 4 E⋅ b⋅ v⋅ sinπ a⋅2 b⋅

2

⋅ 2 E⋅ a⋅+ 2 E⋅ a⋅ v⋅−

⋅

π2

a⋅ b⋅ π2

a2⋅ b⋅ π

2a

2⋅ b⋅ v2⋅−( )⋅ v 1−( )⋅

−→

Keq6 8,

E t⋅ b2

v⋅ a2+ b

2−( )⋅

6 a⋅ b⋅ v2

1−( )⋅−→

Keq8 8,

E t⋅ 2 a2⋅ b

2v⋅− b

2+( )⋅

6 a⋅ b⋅ v2

1−( )⋅−→

Peq1 1, Px

x1

a1−

⋅y1

b1−

⋅Px sin

π x1⋅a

⋅ sinπ y1⋅

b

⋅ π2

a⋅ b⋅ π2

a⋅ b⋅ v⋅−( )⋅ 2 E⋅ a⋅ v⋅ 2 E⋅ a⋅− 4 E⋅ b⋅ v⋅ cosπ a⋅2 b⋅

2

1−

⋅+

⋅

2 E⋅ π2

a2⋅ b⋅ π

2a2⋅ b⋅ v

2⋅−( )⋅+→

Peq2 1, Py

x1

a1−

⋅y1

b1−

⋅2 E⋅ Py⋅ t⋅ sin

π x1⋅a

⋅ sinπ y1⋅

b

⋅ π2

a2⋅ b⋅ π

2a2⋅ b⋅ v

2⋅−( )⋅

π2

a⋅ b⋅ v 1−( )⋅ 2 E⋅ a⋅ t⋅ 4 E⋅ b⋅ t⋅ v⋅ cosπ a⋅2 b⋅

2

⋅− 2 E⋅ a⋅ t⋅ v⋅− 4 E⋅ b⋅ t⋅ v⋅+

⋅

+→

Peq3 1, Px

x1

a1−

⋅y1

b1−

⋅Px sin

π x1⋅a

⋅ sinπ y1⋅

b

⋅ π2

a⋅ b⋅ π2

a⋅ b⋅ v⋅−( )⋅ 4 E⋅ b⋅ v⋅ sinπ a⋅2 b⋅

2

⋅ 2 E⋅ a⋅+ 2 E⋅ a⋅ v⋅−

⋅

2 E⋅ π2

a2⋅ b⋅ π

2a2⋅ b⋅ v

2⋅−( )⋅+→

Peq4 1, Py

x1

a1−

⋅y1

b1−

⋅2 E⋅ Py⋅ t⋅ sin

π x1⋅a

⋅ sinπ y1⋅

b

⋅ π2

a2⋅ b⋅ π

2a

2⋅ b⋅ v2⋅−( )⋅

π2

a⋅ b⋅ v 1−( )⋅ 2 E⋅ a⋅ t⋅ 4 E⋅ b⋅ t⋅ v⋅ cosπ a⋅2 b⋅

2

⋅− 2 E⋅ a⋅ t⋅ v⋅− 4 E⋅ b⋅ t⋅ v⋅+

⋅

−→

Peq5 1,

Px x1⋅ y1⋅a b⋅

Px sinπ x1⋅

a

⋅ sinπ y1⋅

b

⋅ π2

a⋅ b⋅ π2

a⋅ b⋅ v⋅−( )⋅ 2 E⋅ a⋅ v⋅ 2 E⋅ a⋅− 4 E⋅ b⋅ v⋅ cosπ a⋅2 b⋅

2

1−

⋅+

⋅

2 E⋅ π2

a2⋅ b⋅ π

2a2⋅ b⋅ v

2⋅−( )⋅+→

Peq6 1,

Py x1⋅ y1⋅a b⋅

2 E⋅ Py⋅ t⋅ sinπ x1⋅

a

⋅ sinπ y1⋅

b

⋅ π2

a2⋅ b⋅ π

2a2⋅ b⋅ v

2⋅−( )⋅

π2

a⋅ b⋅ v 1−( )⋅ 2 E⋅ a⋅ t⋅ 4 E⋅ b⋅ t⋅ v⋅ cosπ a⋅2 b⋅

2

⋅− 2 E⋅ a⋅ t⋅ v⋅− 4 E⋅ b⋅ t⋅ v⋅+

⋅

+→

Peq7 1,

Px sinπ x1⋅

a

⋅ sinπ y1⋅

b

⋅ π2

a⋅ b⋅ π2

a⋅ b⋅ v⋅−( )⋅ 4 E⋅ b⋅ v⋅ sinπ a⋅2 b⋅

2

⋅ 2 E⋅ a⋅+ 2 E⋅ a⋅ v⋅−

⋅

2 E⋅ π2

a2⋅ b⋅ π

2a2⋅ b⋅ v

2⋅−( )⋅

Px y1⋅x1

a1−

⋅

b−→

Peq8 1,

Py y1⋅x1

a1−

⋅

b−

2 E⋅ Py⋅ t⋅ sinπ x1⋅

a

⋅ sinπ y1⋅

b

⋅ π2

a2⋅ b⋅ π

2a

2⋅ b⋅ v2⋅−( )⋅

π2

a⋅ b⋅ v 1−( )⋅ 2 E⋅ a⋅ t⋅ 4 E⋅ b⋅ t⋅ v⋅ cosπ a⋅2 b⋅

2

⋅− 2 E⋅ a⋅ t⋅ v⋅− 4 E⋅ b⋅ t⋅ v⋅+

⋅

−→

N

1x1

a−

1y1

b−

0

0

1x1

a−

1y1

b−

1x1

a−

1y1

b−

0

0

1x1

a−

1y1

b−

x1 y1⋅a b⋅

0

0

x1 y1⋅a b⋅

y1

b1

x1

a−

0

0

y1

b1

x1

a−

:=

x1

sin πx1

a⋅

sin πy1

b⋅

⋅

0

0

sin πx1

a⋅

sin πy1

b⋅

⋅