Element.beam2t - Revision history

Fotis: /* Examples */

2007-05-29T13:25:19Z

‎Examples

Fotis: /* See also */

2007-05-28T22:53:49Z

‎See also

Fotis: /* Examples */

2007-05-28T22:48:41Z

‎Examples

Fotis: /* Examples */

2007-05-28T22:45:07Z

‎Examples

Fotis: /* Residual vector R */

2007-05-28T22:36:15Z

‎Residual vector R

Fotis: /* Examples */

2007-05-28T22:35:48Z

‎Examples

Fotis: /* Comments */

2007-05-28T22:35:25Z

‎Comments

Fotis: /* Examples */

2007-05-28T22:31:48Z

‎Examples

Fotis at 22:28, 28 May 2007

2007-05-28T22:28:00Z

Fotis: First submission.

2007-05-28T22:12:36Z

First submission.

New page

The following article describes the use of the <tt>element.beam2t</tt> command.
This command defines a simple, two-node elastic Timoshenko beam element.

=Syntax=
element.beam2t(id,node1,node2,mat,sec,rule)

where
{| border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-size: 95%;"
|- bgcolor="#CCCCCC" align="center"
!width="150"|Parameter
!width="150"|description
!width="150"|type
!width="150"|default
|-
|id || The elemental id ||align="center"| integer ||align="center"|-
|-
|node1-2|| The elemental [[node|nodes]] ||align="center"| integer ||align="center"|-
|-
|mat || The element [[material]] ||align="center"| integer ||align="center"|-
|-
|sec || The element [[section]] ||align="center"| integer ||align="center"|-
|-
|rule || The integration rule ||align="center"| integer ||align="center"|1
|}

=Examples=
Define a two-dimensional beam with id 1, start node 1, end node 2, material id 2, section id 3 and a two point Gauss integration rule.
element.beam2t(2,1,2,2,3,2)

=Comments=
[[Node]]s, [[material]] and [[section]] '''must''' be already defined.

=Theory=
The kinematic assumptions are those shown in Figure

=Implementation=

The element length and directional cosines are given as:
<pre>
L=sqrt((y2-y1)*(y2-y1)+(x2-x1)*(x2-x1));
c=(x2-x1)/L;
s=(y2-y1)/L;
</pre>

==Stiffness Matrix K==
The stiffness matrix K in global coordinates is given as:
<pre>
for k in GaussPoints:
xi=GaussPoints(k).xi
dx=GaussPoints(k).weight*0.5*L
for i=0..2:
for j=0..2:
Ni,dNi=shapeFunctions(i,xi)
Nj,dNj=shapeFunctions(j,xi)
K(3*i+0,3*j+0)+=(dNi*dNj*(C1*c*c+C3-c*c*C3) )*dx
K(3*i+0,3*j+1)+=(c*dNi*dNj*s*(-C3+C1) )*dx
K(3*i+0,3*j+2)+=(s*dNi*C3*Nj )*dx
K(3*i+1,3*j+0)+=(c*dNi*dNj*s*(-C3+C1) )*dx
K(3*i+1,3*j+1)+=(-dNi*dNj*(-c*c*C3-C1+C1*c*c))*dx
K(3*i+1,3*j+2)+=(-c*dNi*C3*Nj )*dx
K(3*i+2,3*j+0)+=(Ni*C3*dNj*s )*dx
K(3*i+2,3*j+1)+=(-Ni*C3*dNj*c )*dx
K(3*i+2,3*j+2)+=(dNi*C2*dNj+Ni*C3*Nj )*dx
</pre>
where,
<pre>
C1=E*A
C2=E*J
C3=a*G*A
</pre>

==Residual vector R==
The residual vector in global coordinates is given as:
R = facS*Fint - facG*Fgravity - facP*Fext
where <tt>facS</tt>, <tt>facG</tt> and <tt>facP</tt>, factors controlled by [[group.state]].
In local coordinates it yields,
<pre>
for k in GaussPoints:
xi=GaussPoints(k).xi
dx=GaussPoints(k).weight*0.5*L
for i=0..2:
Ni,dNi=shapeFunctions(i,xi)
R[3*i+0]+=facS*((dNi*sigma[0]) )*dx
R[3*i+1]+=facS*((dNi*sigma[2]) )*dx
R[3*i+2]+=facS*((dNi*sigma[1]-Ni*sigma[2]))*dx

R[0]-=facG*(0.5*b[0]*L);
R[1]-=facG*(0.5*b[1]*L);
R[3]-=facG*(0.5*b[0]*L);
R[4]-=facG*(0.5*b[1]*L);

R-=facP*Fext;
</pre>
and transforming it from local to global, one has
<pre>
for i=0..2:
d1=c*R[3*i+0]-s*R[3*i+1];
d2=s*R[3*i+0]+c*R[3*i+1];
R[3*i+0]=d1;
R[3*i+1]=d2;
</pre>

=Notes=
<references/>

=See also=
*[[node]] module.
*[[material]] module.
*[[section]] module.
*[[element]] module.
*[[User guide]].

[[Category:User guide]]

@@ Line 27: / Line 27: @@
 [[Image:Timoshenko_Example.png|thumb|350px|Shear locking effects in Timoshenko beam.]]
 Cantilever beam example, motivated by <ref>O.C. Zienkiewicz & R.L. Taylor, ''"The Finite Element Method for Solids and Structural Mechanics"'', 6th Edition, p307</ref>, demonstrating shear locking effects. Twenty elements are used for different span L to depth h ratios for a rectangular cross-section. The beams used are the Bernulli-Euler, a 2-node full integration (2 points) Timoshenko beam, a 2-node reduced integration (1 point) Timoshenko beam, a 3-node full integration (3 points) Timoshenko beam and a 3-node reduced integration (2 points) Timoshenko beam. The use of exact integration for the second one, leads to a solution which 'locks' as the beam becomes slender, whereas the reduced integration scheme shows no locking for the range plotted. The results are also given in the following table, where the example can be found [[Example:Timoshenko_Locking|here]].
 <br style="clear:both;"/>
   =====================================================
   |Ratio    |Tip Displacement (normalized)            |

@@ Line 25: / Line 25: @@
 =Examples=
-[[Image:Timoshenko_Example.png|thumb|450px|Shear locking effects in Timoshenko beam.]]
+[[Image:Timoshenko_Example.png|thumb|350px|Shear locking effects in Timoshenko beam.]]
-Cantilever beam example, motivated by <ref>O.C. Zienkiewicz & R.L. Taylor, ''"The Finite Element Method for Solids and Structural Mechanics"'', 6th Edition, p307</ref>, demonstrating shear locking effects. Twenty elements are used for different span L to depth h ratios for a rectangular cross-section. The beams used are the Bernulli-Euler, a 2-node full integration (2 points) Timoshenko beam, a 2-node reduced integration (1 point) Timoshenko beam, a 3-node full integration (3 points) Timoshenko beam and a 3-node reduced integration (2 points) Timoshenko beam. The use of exact integration for the second one, leads to a solution which 'locks' as the beam becomes slender, whereas the reduced integration scheme shows no locking for the range plotted. The results are also given in the following table, where the example can be found [[Link title]].
+Cantilever beam example, motivated by <ref>O.C. Zienkiewicz & R.L. Taylor, ''"The Finite Element Method for Solids and Structural Mechanics"'', 6th Edition, p307</ref>, demonstrating shear locking effects. Twenty elements are used for different span L to depth h ratios for a rectangular cross-section. The beams used are the Bernulli-Euler, a 2-node full integration (2 points) Timoshenko beam, a 2-node reduced integration (1 point) Timoshenko beam, a 3-node full integration (3 points) Timoshenko beam and a 3-node reduced integration (2 points) Timoshenko beam. The use of exact integration for the second one, leads to a solution which 'locks' as the beam becomes slender, whereas the reduced integration scheme shows no locking for the range plotted. The results are also given in the following table, where the example can be found [[Example:Timoshenko_Locking|here]].
   =====================================================
@@ Line 44: / Line 47: @@
   | 1000.000|  1.0000  0.0010  0.9990  0.9990  0.9990 |
   =====================================================
 =Comments=

@@ Line 25: / Line 25: @@
 =Examples=
-[[Image:Timoshenko_Example.png|thumb|350px|Shear locking effects in Timoshenko beam.]]
+[[Image:Timoshenko_Example.png|thumb|450px|Shear locking effects in Timoshenko beam.]]
-Cantilever beam example, motivated by <ref>O.C. Zienkiewicz & R.L. Taylor, ''"The Finite Element Method for Solids and Structural Mechanics"'', 6th Edition, p307</ref>, demonstrating shear locking effects.
+Cantilever beam example, motivated by <ref>O.C. Zienkiewicz & R.L. Taylor, ''"The Finite Element Method for Solids and Structural Mechanics"'', 6th Edition, p307</ref>, demonstrating shear locking effects. Twenty elements are used for different span L to depth h ratios for a rectangular cross-section. The beams used are the Bernulli-Euler, a 2-node full integration (2 points) Timoshenko beam, a 2-node reduced integration (1 point) Timoshenko beam, a 3-node full integration (3 points) Timoshenko beam and a 3-node reduced integration (2 points) Timoshenko beam. The use of exact integration for the second one, leads to a solution which 'locks' as the beam becomes slender, whereas the reduced integration scheme shows no locking for the range plotted. The results are also given in the following table, where the example can be found [[Link title]].
 <br style="clear:both;"/>

@@ Line 25: / Line 25: @@
 =Examples=
-[[Image:Timoshenko_Example.png|thumb|700px|Shear locking effects in Timoshenko beam.]]
+[[Image:Timoshenko_Example.png|thumb|350px|Shear locking effects in Timoshenko beam.]]
 Cantilever beam example, motivated by <ref>O.C. Zienkiewicz & R.L. Taylor, ''"The Finite Element Method for Solids and Structural Mechanics"'', 6th Edition, p307</ref>, demonstrating shear locking effects.

←Older revision		Revision as of 22:31, 28 May 2007
Line 25:		Line 25:

	=Examples=		=Examples=
		+	[[Image:Timoshenko_Example.png\|thumb\|700px\|Shear locking effects in Timoshenko beam.]]
		+	Cantilever beam example, motivated by <ref>O.C. Zienkiewicz & R.L. Taylor, ''"The Finite Element Method for Solids and Structural Mechanics"'', 6th Edition, p307</ref>, demonstrating shear locking effects.

		+	<br style="clear:both;"/>

	=Comments=		=Comments=

@@ Line 132: / Line 132: @@
 *[[section]] module.
 *[[element]] module.
 *[[User guide]].
 [[Category:User guide]]

@@ Line 89: / Line 89: @@
 	R[3*i+2]+=facS*((dNi*sigma[1]-Ni*sigma[2]))*dx
-R[0]-=facG*(0.5*b[0]*L);
+R[0]-=facG*(0.5*b[0]*L)
-R[1]-=facG*(0.5*b[1]*L);
+R[1]-=facG*(0.5*b[1]*L)
-R[3]-=facG*(0.5*b[0]*L);
+R[3]-=facG*(0.5*b[0]*L)
-R[4]-=facG*(0.5*b[1]*L);
+R[4]-=facG*(0.5*b[1]*L)
 R-=facP*Fext;
@@ Line 99: / Line 99: @@
 <pre>
 for i=0..2:
-     d1=c*R[3*i+0]-s*R[3*i+1];
+     d1=c*R[3*i+0]-s*R[3*i+1]
-     d2=s*R[3*i+0]+c*R[3*i+1];
+     d2=s*R[3*i+0]+c*R[3*i+1]
-     R[3*i+0]=d1;
+     R[3*i+0]=d1
-     R[3*i+1]=d2;
+     R[3*i+1]=d2
 </pre>

@@ Line 31: / Line 31: @@
 =Comments=
-[[Node]]s, [[material]] and [[section]] '''must''' be already defined.
+[[Node]]s, [[material]] and [[section]] '''must''' be already defined. This beam exhibits severe shear locking, and therefore it is recommended to adopt a reduced integration scheme.
 =Theory=

@@ Line 25: / Line 25: @@
 =Examples=
-Define a two-dimensional beam with id 1, start node 1, end node 2, material id 2, section id 3 and a two point Gauss integration rule.
-  element.beam2t(2,1,2,2,3,2)
 =Comments=
@@ Line 32: / Line 31: @@
 =Theory=
-The kinematic assumptions are those shown in Figure
+[[Image:Timoshenko_Kinematics.png|thumb|300px|Beam deformation including shear effect.]]
 =Implementation=