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	!width="150"|default		!width="150"|default
	|-		|-
−	|id                || The elemental id              ||align="center"| integer  ||align="center"|-	+	id                || The elemental id              ||align="center"| integer  ||align="center"|-
	|-		|-
	|dof<sub>1-2</sub> || The elemental [[node|nodes]]  ||align="center"| integer  ||align="center"|-		|dof<sub>1-2</sub> || The elemental [[node|nodes]]  ||align="center"| integer  ||align="center"|-
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	=Comments=		=Comments=
	[[Node]]s, [[material]] and [[section]] '''must''' be already defined.		[[Node]]s, [[material]] and [[section]] '''must''' be already defined.
		+
		+	=Implementation=
		+	==Length and directional cosines==
		+	<pre>
		+	L=sqrt((y2-y1)*(y2-y1)+(x2-x1)*(x2-x1));
		+	cosX[0]=(x2-x1)/L;
		+	cosX[1]=(y2-y1)/L;
		+	</pre>
		+
		+	==Gravity load vector b==
		+
		+	==Stiffness Matrix K==
		+	<pre>
		+	K(0,0)= K1;  K(0,1)= K2;  K(0,2)= K4;  K(0,3)=-K1;  K(0,4)=-K2;  K(0,5)= K4;
		+	K(1,0)= K2;  K(1,1)= K3;  K(1,2)= K5;  K(1,3)=-K2;  K(1,4)=-K3;  K(1,5)= K5;
		+	K(2,0)= K4;  K(2,1)= K5;  K(2,2)= K6;  K(2,3)=-K4;  K(2,4)=-K5;  K(2,5)= K7;
		+	K(3,0)=-K1;  K(3,1)=-K2;  K(3,2)=-K4;  K(3,3)= K1;  K(3,4)= K2;  K(3,5)=-K4;
		+	K(4,0)=-K2;  K(4,1)=-K3;  K(4,2)=-K5;  K(4,3)= K2;  K(4,4)= K3;  K(4,5)=-K5;
		+	K(5,0)= K4;  K(5,1)= K5;  K(5,2)= K7;  K(5,3)=-K4;  K(5,4)=-K5;  K(5,5)= K6;
		+	</pre>
		+	where,
		+	<pre>
		+	K1=C1*c*c+C2*s*s;
		+	K2=(C1-C2)*c*s;
		+	K3=C1*s*s+C2*c*c;
		+	K4=-C3*s;
		+	K5=C3*c;
		+	K6=4*E*J/L;
		+	K7=2*E*J/L;
		+	C1=E*A/L;
		+	C2=12*E*J/(L*L*L);
		+	C3=6*E*J/(L*L);
		+	</pre>
		+
		+	==Residual vector R==
		+	Residual vector 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 this yields,
		+	<pre>
		+	R[0]=facS*(  KN*u[0]  - KN*u[3])                      - facG*( 0.5*b[0]*L);
		+	R[1]=facS*(  K12*u[1] + K6*u[2] - K12*u[4] + K6*u[5]) - facG*( 0.5*b[1]*L);
		+	R[2]=facS*(  K6*u[1]  + K4*u[2] - K6*u[4]  + K2*u[5]) - facG*( b[1]*L*L/12.);
		+	R[3]=facS*(- KN*u[0]  + KN*u[3])                      - facG*( 0.5*b[0]*L);
		+	R[4]=facS*(- K12*u[1] - K6*u[2] + K12*u[4] - K6*u[5]) - facG*( 0.5*b[1]*L);
		+	R[5]=facS*(  K6*u[1]  + K2*u[2] - K6*u[4]  + K4*u[5]) - facG*(-b[1]*L*L/12.);
		+	R-=facP*Fext
		+	</pre>
		+	and transforming it from local to global,
		+	<pre>
		+	R0=c*R[0]-s*R[1];
		+	R1=s*R[0]+c*R[1];
		+	R3=c*R[3]-s*R[4];
		+	R4=s*R[3]+c*R[4];
		+	R[0]=R0;
		+	R[1]=R1;
		+	R[3]=R3;
		+	R[4]=R4;
		+	</pre>
	=See also=		=See also=

---

Revision as of 13:28, 14 February 2007

The following article describes the use of the element.beam2e command. This command defines a simple, two-node Euler beam member.

Contents 1 Syntax 2 Examples 3 Comments 4 Implementation 4.1 Length and directional cosines 4.2 Gravity load vector b 4.3 Stiffness Matrix K 4.4 Residual vector R 5 See also

Syntax

element.beam2e(id,dof1,dof2,mat,sec)

where

id || The elemental id ||align="center"| integer ||align="center"|-

Parameter	description	type	default
dof1-2	The elemental nodes	integer	-
mat	The element material	integer	-
sec	The element section	integer	-

Examples

 element.beam2e(2,1,2,2,3)

Comments

Nodes, material and section must be already defined.

Implementation

Length and directional cosines

L=sqrt((y2-y1)*(y2-y1)+(x2-x1)*(x2-x1));
cosX[0]=(x2-x1)/L;
cosX[1]=(y2-y1)/L;

Gravity load vector b

Stiffness Matrix K

K(0,0)= K1;  K(0,1)= K2;   K(0,2)= K4;   K(0,3)=-K1;   K(0,4)=-K2;   K(0,5)= K4;
K(1,0)= K2;  K(1,1)= K3;   K(1,2)= K5;   K(1,3)=-K2;   K(1,4)=-K3;   K(1,5)= K5;
K(2,0)= K4;  K(2,1)= K5;   K(2,2)= K6;   K(2,3)=-K4;   K(2,4)=-K5;   K(2,5)= K7;
K(3,0)=-K1;  K(3,1)=-K2;   K(3,2)=-K4;   K(3,3)= K1;   K(3,4)= K2;   K(3,5)=-K4;
K(4,0)=-K2;  K(4,1)=-K3;   K(4,2)=-K5;   K(4,3)= K2;   K(4,4)= K3;   K(4,5)=-K5;
K(5,0)= K4;  K(5,1)= K5;   K(5,2)= K7;   K(5,3)=-K4;   K(5,4)=-K5;   K(5,5)= K6;

where,

K1=C1*c*c+C2*s*s;
K2=(C1-C2)*c*s;
K3=C1*s*s+C2*c*c;
K4=-C3*s;
K5=C3*c;
K6=4*E*J/L;
K7=2*E*J/L;
C1=E*A/L;
C2=12*E*J/(L*L*L);
C3=6*E*J/(L*L);

Residual vector R

Residual vector is given as

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

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

R[0]=facS*(  KN*u[0]  - KN*u[3])                      - facG*( 0.5*b[0]*L);
R[1]=facS*(  K12*u[1] + K6*u[2] - K12*u[4] + K6*u[5]) - facG*( 0.5*b[1]*L);
R[2]=facS*(  K6*u[1]  + K4*u[2] - K6*u[4]  + K2*u[5]) - facG*( b[1]*L*L/12.);
R[3]=facS*(- KN*u[0]  + KN*u[3])                      - facG*( 0.5*b[0]*L);
R[4]=facS*(- K12*u[1] - K6*u[2] + K12*u[4] - K6*u[5]) - facG*( 0.5*b[1]*L);
R[5]=facS*(  K6*u[1]  + K2*u[2] - K6*u[4]  + K4*u[5]) - facG*(-b[1]*L*L/12.);
R-=facP*Fext

and transforming it from local to global,

R0=c*R[0]-s*R[1];
R1=s*R[0]+c*R[1];
R3=c*R[3]-s*R[4];
R4=s*R[3]+c*R[4];
R[0]=R0;
R[1]=R1;
R[3]=R3;
R[4]=R4;

See also

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