Line elements in 2D and 3D
BAR ELEMENT IN 2D (TRUSS, LINK)
This element can be loaded only in the axial direction, its orientation in 2D space in
relation to local (element) and global coordinate system shows Fig.3-1
v2
v1
yg
u1
y
L
x
α
xg
Fig.3-1 Bar element in 2D space
u2
In comparison with the simplest element Fig.3, there are two additional deformation
parameters v1, v2, not connected to any element stiffness. The stiffness matrix in
local element coordinate system is obtained from bar element in 1D, adding two
zero rows and columns, corresponding to parameters v1, v2 :
1

ES  0
k
L  1

0
0 1
0 0
0 1
0 0
0
0
0

0
.
Matrix of deformation parameters
 u1 
v 
δ   1 ,
u 2 
 
 v2 
The stiffness matrix in global coordinate system is then obtained by transformation
kg = TT. k . T , where T is a transformation matrix according to Fig.3-1
λ 0 
T

0 λ 
 cos
λ
 sin 
sin   cos(xxg ) cos(xyg ) 


cos  cos(yxg ) cos(yyg )
This element is used for solution of pin-jointed frame structures, or as a simple
model of cables and springs. In ANSYS it can be found under the name LINK1,
other details see ANSYS Online Manuals. Simple illustration of its usage can be
found in the Example 3.1
BAR ELEMENT IN 3D (TRUSS, LINK)
3D version of the previous element according to Fig.3-2 has six degrees of freedom
δ  u1, v1, w1, u2 , v2 , w2 
T
v1
w1
u1
v2
y
z
w2
u2
x
Fig.3-2 Bar element in 3D space
Its stiffness matrix in local coordinate system is created in by a similar procedure
1
0

ES  0
k

L  1
0

 0
0 0 1 0 0 
0 0 0 0 0 
0 0 0 0 0 

0 0 1 0 0 
0 0 0 0 0 

0 0 0 0 0 
,
and transformation to global coordinate system is done by the matrix T
λ 0 
T
 ,
0 λ 
cos(xxg ) cos(xy g ) cos(xz g ) 


λ  cos(yxg ) cos(yy g ) cos(yz g ) .
 cos(xz g ) cos(yz g ) cos(zz g ) 


Bar element in 3D is used for solution of pin-jointed frame structures, or as a simple model
of cables and springs. In ANSYS it can be found under the name LINK8 , other details see
ANSYS Online Manuals. An illustration of its usage can be found in the Example 3.2
BEAM ELEMENT
Beam is a line element with bending capability. Primary unknown function is the
deflection w. To assure convergence, there must be interelement continuity of deflection
and slope φ. Each node then has two degrees of freedom according to Fig. 3-3.
φ2
φ1
w1
w2
L
x
Fig.3-3 Beam element
Deflection is approximated by
w( x)  N.δ
, where
δT = | w1, φ1, w2, φ2 |
are the degrees of freedom
N = | N1 N2 N3 N4 |
cubic polynomial shape functions.
The stiffness matrix of beam element has the explicit form
12
k
EI
L3
Sym.
6L
4 L2
 12 6 L
 6 L 2 L2
12  6 L
4 L2
where I is the second moment of area, L length and E modulus of elasticity.
In this form, the element can be used only to model bending of straight beams. In
practical applications it must be combined with the bar capability as described in
the next paragraph.
GENERAL BEAM OR FRAME ELEMENT IN 2D
This element can be obtained as a combination of previous beam element (Fig. 3-3)
and the simplest 1D-bar element - see Fig.3-4. It has six degrees of freedom, three
in each node
δT = │ u1, w1, φ1, u2, w2, φ2 │,
and its stiffness matrix is derived from the matrices of the mentioned elements:
k
T
0
0
T
0
0
0
D
C
0
D
C
0
C
A
0
C
B
T
0
0
T
0
0
0
D
C
0
C
A
0
0
 D C
C
B
where T = ES/L, A = 4EI/L, B = 2EI/L, C = 6EI/L2, D = 12EI/L3.
φ2
φ1
u1
w1
u2
L
w2
Fig. 3-4 Frame element in 2D
Frame element in 2D is used for solution of frame structures with its members loaded by
bending moments and axial forces. In ANSYS it can be found under the name BEAM3, other
details see ANSYS Online Manuals.
GENERAL BEAM OR FRAME ELEMENT IN 3D
Straight element according to Fig.3-5 has twelve degrees of freedom, six in each node three displacements and three rotations.

δ  u1 , v1 , w1 ,  x1 ,  y1 ,  z1 , u2 , v2 , w2 ,  x 2 ,  y 2 ,  z 2

T
φy1
φz1
v1
w1
u1
φy2
φx1
v2
y
z
x
φz2
w2
u2
φx2
Fig.3-5 Frame element in 3D
The parameters u1, u2 are linked with the axial loading, with the stiffness matrix
according to simplest bar element. Parameters v1, φz1, v2, φz2 are linked with the bending
in the y direction, parameters w1, φy1, w2, φy2 are linked with the bending in the z
direction. Their stiffness matrix is created according to the beam element with
appropriate second area moment Iz, resp. Iy .
The rest of parameters, φx1, φx2 , correspond to torsion and using the analogy with
tension, the stiffness matrix can be written as
k
GI p  1  1
L  1 1 
where G is the shear modulus and Ip the polar moment of area.
The final stiffness matrix is completed from the partial ones in the same way as the
frame element matrix in 2D. The element is used for solution of general frame
structures in 3D with its members loaded by bending moments and axial forces and
torsion. In ANSYS it can be found under the name BEAM4, other details see ANSYS
Online Manuals.