Fotis: /* Syntax */

2007-05-28T22:54:50Z

‎Syntax

Fotis: /* See also */

2007-05-28T22:54:24Z

‎See also

Fotis at 22:53, 28 May 2007

2007-05-28T22:53:16Z

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The following article describes the use of the <tt>element.beam3t</tt> command.
This command defines a quadratic, three-node elastic Timoshenko beam element.

=Syntax=
element.beam2t(id,node1,node2,node1,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"|-
|-
|node1-3|| 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"|2
|}

=Examples=
[[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]].

 

=====================================================
|Ratio |Tip Displacement (normalized) |
-----------------------------------------------------
| | EULER |TIM2F | TIM2R | TIM3F | TIM3R |
=====================================================
| 2.000| 1.0000 1.2349 1.2390 1.2400 1.2400 |
| 3.000| 1.0000 1.0964 1.1057 1.1067 1.1067 |
| 4.000| 1.0000 1.0426 1.0590 1.0600 1.0600 |
| 5.000| 1.0000 1.0120 1.0374 1.0384 1.0384 |
| 10.000| 1.0000 0.9144 1.0086 1.0096 1.0096 |
| 20.000| 1.0000 0.7076 1.0014 1.0023 1.0023 |
| 50.000| 1.0000 0.2776 0.9994 1.0000 1.0000 |
| 100.000| 1.0000 0.0876 0.9991 0.9994 0.9994 |
| 500.000| 1.0000 0.0038 0.9990 0.9990 0.9990 |
| 1000.000| 1.0000 0.0010 0.9990 0.9990 0.9990 |
=====================================================

=Comments=
[[Node]]s, [[material]] and [[section]] '''must''' be already defined. The middle node, namely node 3, should be in the middle of the element. This beam is a straight one.

=Theory=
[[Image:Timoshenko_Kinematics.png|thumb|300px|Beam deformation including shear effect.]]
The kinematic assumptions are those shown in the figure on the right.
 

=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..3:
for j=0..3:
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..3:
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..3:
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.
*[[element.beam2t]] module.
*[[User guide]].

[[Category:User guide]]

@@ Line 3: / Line 3: @@
 =Syntax=
-  element.beam2t(id,node<sub>1</sub>,node<sub>2</sub>,node<sub>1</sub>,mat,sec,rule)
+  element.beam2t(id,node<sub>1</sub>,node<sub>2</sub>,node<sub>3</sub>,mat,sec,rule)
 where

@@ Line 132: / Line 132: @@
 *[[section]] module.
 *[[element]] module.
-*[[element.beam2t]] module.
+*[[element.beam2t]].
 *[[User guide]].
 [[Category:User guide]]

Element.beam3t - Revision history

Fotis: /* Syntax */

Fotis: /* See also */

Fotis at 22:53, 28 May 2007