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

Fig.3-1 Bar element in 2D space

α xg

yg

yx

u1

v1

v2u2

L

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 :

Matrix of deformation parameters

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

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

.

0000

0101

0000

0101

L

ESk ,

2

2

1

1

v

u

v

u

δ

λ0

0λT

)cos()cos(

)cos()cos(

cossin

sincos

gg

gg

yyyx

xyxx

λ

BAR ELEMENT IN 3D (TRUSS, LINK)

3D version of the previous element according to Fig.3-2 has six degrees of freedom

Fig.3-2 Bar element in 3D space

Twvuwvu 222111 ,,,,,δ

v1w1

u1

u2

v2

w2x

y

z

Its stiffness matrix in local coordinate system is created in by a similar procedure

and transformation to global coordinate system is done by the matrix T

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

,

000000

000000

001001

000000

000000

001001

L

ESk

λ0

0λT , .

)cos()cos()cos(

)cos()cos()cos(

)cos()cos()cos(

ggg

ggg

ggg

zzyzxz

yzyyyx

xzxyxx

λ

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.

Fig.3-3 Beam element

Deflection is approximated by , where

are the degrees of freedom

cubic polynomial shape functions.

w1 w2L

φ1 φ2

x

δN.)( xw

δT = | w1, φ 1, w2, φ 2 |

N = | N1 N2 N3 N4 |

The stiffness matrix of beam element has the explicit form

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 │,

2

22

3

4.

612

264

612612

LSym

L

LLL

LL

L

EI

k

w1 w2L

φ1 φ2

u1 u2

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.

and its stiffness matrix is derived from the matrices of the mentioned elements:

where T = ES/L, A = 4EI/L, B = 2EI/L, C = 6EI/L2, D = 12EI/L3.

ACBC

CDCD

TT

BCAC

CDCD

TT

00

00

0000

00

00

0000

k

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.

Fig.3-5 Frame element in 3DThe 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

Tzyxzyx wvuwvu 222222111111 ,,,,,,,,,,, δ

v1w1

u1 φx1

φy1

φz1

u2 φx2

φy2

v2

w2φz2x

y

z

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.

11

11

L

GI pk