Revision as of 01:35, 29 May 2007

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

Syntax

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

where

Examples

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.

Comments

Nodes, 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

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:

L=sqrt((y2-y1)*(y2-y1)+(x2-x1)*(x2-x1));
c=(x2-x1)/L;
s=(y2-y1)/L;

Stiffness Matrix K

The stiffness matrix K in global coordinates is given as:

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

where,

C1=E*A
C2=E*J
C3=a*G*A

Residual vector R

The residual vector in global coordinates is given as:

R = facS*Fint - facG*Fgravity - facP*Fext

where facS, facG and facP, factors controlled by group.state. In local coordinates it yields,

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;

and transforming it from local to global, one has

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;

Notes

<references/>

See also

• node module.
• material module.
• section module.
• element module.
• User guide.