Bar elements

Bar elements are available for one, two, and three dimensional analysis.

One dimensional bar elements bar1e Compute element matrix

bar1s Compute normal force

bar1we Compute element matrix for bar element with elastic support bar1ws Compute normal force for bar element with elastic support

Two dimensional bar elements bar2e Compute element matrix

bar2s Compute normal force

bar2ge Compute element matrix for geometric nonlinear element

bar2gs Compute normal force and axial force for geometric nonlinear ele-ment

Three dimensional bar elements bar3e Compute element matrix

bar3s Compute normal force

One dimensional bar element bar1e

Purpose:

Compute element stiffness matrix for a one dimensional bar element.

x

(x₁) (x2)

u1 u2

EA

x

Syntax:

Ke=bar1e(ex,ep) [Ke,fe]=bar1e(ex,ep,eq)

Description:

bar1e provides the element stiffness matrix Ke for a one dimensional bar element.

The input variables

ex = [ x₁ x₂ ] ep = [ E A ]

supply the element nodal coordinates x₁ and x₂, the modulus of elasticity E, and the cross section area A.

The element load vector fe can also be computed if uniformly distributed load is applied to the element. The optional input variable

eq = q_¯x 

then contains the distributed load per unit length, q_¯x.

x x

q

_x

Theory:

The element stiffness matrix ¯K^e, stored in Ke, is computed according to K¯^e= D_EA

L

 1 −1

−1 1



where the axial stiffness D_EA and the length L are given by D_EA = EA; L = x₂− x1

The element load vector ¯f_l^e, stored in fe, is computed according to

¯f_l^e= q_¯xL 2

 1 1



bar1s One dimensional bar element

Purpose:

Compute normal force in a one dimensional bar element.

x

bar1s computes the normal force in the one dimensional bar element bar1e.

The input variables ex and ep are defined in bar1e and the element nodal displace-ments, stored in ed, are obtained by the function extract. If distributed load is applied to the element, the variable eq must be included. The number of evaluation points for normal force and displacement are determined by n. If n is omitted, only the ends of the bar are evaluated.

The output variables

es =

contain the normal force, the displacement, and the evaluation points on the local

¯

x-axis. L is the length of the bar element.

Theory:

The nodal displacements in local coordinates are given by

¯

The displacement u(¯x) and the normal force N (¯x) are computed from u(¯x) = N¯a^e+ u_p(¯x)

One dimensional bar element bar1s

where

N = 1 ¯x C⁻¹= 1− _L^¯x _L^¯x

B = 0 1 C⁻¹ = 1 L

−1 1 

u_p(¯x) =− q_¯x D_EA

x¯² 2 − L¯x

2



N_p(¯x) =−q_¯x^x¯−L 2



in which D_EA, L, and q_¯x are defined in bar1e and C⁻¹ =

 1 0

−_L¹ _L¹



bar1we One dimensional bar element with elastic support

Purpose:

Compute element stiffness matrix for a one dimensional bar element with elastic support.

x

(x1) (x₂)

u₁ EA u₂

x

kx

Syntax:

Ke=bar1we(ex,ep) [Ke,fe]=bar1we(ex,ep,eq)

Description:

bar1we provides the element stiffness matrix Ke for a one dimensional bar element with elastic support. The input variables

ex = [ x₁ x₂ ] ep = [ E A k_¯x ]

supply the element nodal coordinates x₁ and x₂, the modulus of elasticity E, the cross section area A and the stiffness of the axial springs k_¯x.

The element load vector fe can also be computed if uniformly distributed load is applied to the element. The optional input variable

eq = q_¯x 

then contains the distributed load per unit length, q_¯x.

x x

q

_x

Theory:

The element stiffness matrix ¯K^e, stored in Ke, is computed according to K¯^e= ¯K^e₀+ ¯K^e_s

K¯^e₀ = D_EA L

 1 −1

−1 1



K¯^e_s= k_¯xL

 ₁

3 1

1 6

6 1

3



where the axial stiffness D_EA and the length L are given by D_EA = EA; L = x₂− x1

The element load vector ¯f_l^e, stored in fe, is computed according to

¯f^e= q_¯xL^ 1 ^

One dimensional bar element with elastic support bar1ws

Purpose:

Compute normal force in a one dimensional bar element with elastic support.

x

bar1ws computes the normal force in the one dimensional bar element bar1we.

The input variables ex and ep are defined in bar1we and the element nodal displace-ments, stored in ed, are obtained by the function extract. If distributed load is applied to the element, the variable eq must be included. The number of evaluation points for normal force and displacement are determined by n. If n is omitted, only the ends of the bar are evaluated.

The output variables

es =

contain the normal force, the displacement, and the evaluation points on the local

¯

x-axis. L is the length of the bar element.

Theory:

The nodal displacements in local coordinates are given by

¯

The displacement u(¯x) and the normal force N (¯x) are computed from u(¯x) = N¯a^e+ u_p(¯x)

N (¯x) = D_EAB¯a^e+ N_p(¯x)

bar1ws One dimensional bar element with elastic support

where

N = 1 ¯x C⁻¹= 1− _L^¯x _L^¯x

B = 0 1 C⁻¹ = 1 L

−1 1 

u_p(¯x) = k_¯x D_EA

¯x²−L¯x

2 ¯x³−L²¯x 6



C⁻¹¯a^e− q_¯x D_EA

x¯² 2 −L¯x

2



N_p(¯x) = k_¯x ^2¯x−L₂ ^3¯x2₆^−L2C⁻¹¯a^e− q¯x



¯ x− ^L₂^ in which D_EA, L, k_¯x and q_¯x are defined in bar1we and

C⁻¹ =

 1 0

−_L¹ _L¹



Two dimensional bar element bar2e

Purpose:

Compute element stiffness matrix for a two dimensional bar element.

E, A

x y

(x₂,y₂)

(x₁,y₁) x

u₁ u₂

u₃ u₄

Syntax:

Ke=bar2e(ex,ey,ep) [Ke,fe]=bar2e(ex,ey,ep,eq) Description:

bar2e provides the global element stiffness matrix Ke for a two dimensional bar ele-ment.

The input variables ex = [ x₁ x₂ ]

ey = [ y₁ y₂ ] ep = [ E A ]

supply the element nodal coordinates x₁, y₁, x₂, and y₂, the modulus of elasticity E, and the cross section area A.

The element load vector fe can also be computed if uniformly distributed axial load is applied to the element. The optional input variable

eq = q_¯x 

then contains the distributed load per unit length, q_¯x. Theory:

The element stiffness matrix K^e, stored in Ke, is computed according to K^e= G^T K¯^e G

where

K¯^e= D_EA L

 1 −1

−1 1



G =

 n_x¯x n_{y ¯x} 0 0 0 0 n_x¯x n_{y ¯x}



bar2e Two dimensional bar element

x

q

_x

y x

where the axial stiffness D_EA and the length L are given by D_EA = EA; L =



(x₂− x₁)²+ (y₂− y₁)²

and the transformation matrix G contains the direction cosines n_x¯x = x₂− x₁

L n_{y ¯x} = y₂− y₁ L

The element load vector f_l^e, stored in fe, is computed according to f_l^e= G^T f¯_l^e

where

¯f_l^e= q_¯xL 2

 1 1



Two dimensional bar element bar2s

Purpose:

Compute normal force in a two dimensional bar element.

x

bar2s computes the normal force in the two dimensional bar element bar2e.

The input variables ex, ey, and ep are defined in bar2e and the element nodal dis-placements, stored in ed, are obtained by the function extract. If distributed loads are applied to the element, the variable eq must be included. The number of evalua-tion points for secevalua-tion forces and displacements are determined by n. If n is omitted, only the ends of the bar are evaluated.

The output variables

es =

contain the normal force, the displacement, and the evaluation points on the local

¯

x-axis. L is the length of the bar element.

Theory:

The nodal displacements in global coordinates a^e= [ u₁ u₂ u₃ u₄ ]^T

are also shown in bar2e. The transpose of a^e is stored in ed.

bar2s Two dimensional bar element

The nodal displacements in local coordinates are given by

¯

a^e= Ga^e

where the transformation matrix G is defined in bar2e.

The displacement u(¯x) and the normal force N (¯x) are computed from u(¯x) = N¯a^e +u_p(¯x)

N (¯x) = D_EAB¯a^e +N_p(¯x) where

N =

1 ¯x C⁻¹=

1− _L^¯x _L^¯x

B = 0 1 C⁻¹ = 1 L

−1 1 

u_p(¯x) = −_D^q_EA^{x¯ ¯x}₂² − ^L¯₂^x

N_p(¯x) = −q¯x



¯ x−^L₂^

where D_EA, L, q_¯x are defined in bar2e and C⁻¹ =

 1 0

−_L¹ _L¹



Two dimensional bar element bar2ge

Purpose:

Compute element stiffness matrix for a two dimensional geometric nonlinear bar.

E, A

bar2ge provides the element stiffness matrix Ke for a two dimensional geometric nonlinear bar element.

The input variables ex = [ x₁ x₂ ]

ey = [ y₁ y₂ ] ep = [ E A ]

supply the element nodal coordinates x₁, y₁, x₂, and y₂, the modulus of elasticity E, and the cross section area A. The input variable

Qx = [ Q_¯x ]

contains the value of the axial force, which is positive in tension.

Theory:

The global element stiffness matrix K^e, stored in Ke, is computed according to K^e= G^T K¯^e G

bar2ge Two dimensional bar element

G =

⎡

⎢⎢

⎢⎣

n_x¯x n_{y ¯x} 0 0 n_x¯y n_{y ¯y} 0 0 0 0 n_x¯x n_{y ¯x} 0 0 n_x¯y n_{y ¯y}

⎤

⎥⎥

⎥⎦

where the axial stiffness D_EA and the length L are given by D_EA = EA; L =



(x₂− x₁)²+ (y₂− y₁)²

and the transformation matrix G contains the direction cosines n_x¯x = n_{y ¯y} = x₂− x1

L n_{y ¯x} =−nx¯y = y₂− y1

L

Two dimensional bar element bar2gs

Purpose:

Compute axial force and normal force in a two dimensional bar element.

x

bar2gs computes the normal force in the two dimensional bar elements bar2g.

The input variables ex, ey, and ep are defined in bar2ge and the element nodal displacements, stored in ed, are obtained by the function extract. The number of evaluation points for section forces and displacements are determined by n. If n is omitted, only the ends of the bar are evaluated.

The output variable Qx contains the axial force Q_¯x and the output variables

es =

contain the normal force, the displacement, and the evaluation points on the local

¯

x-axis. L is the length of the bar element.

Theory:

The nodal displacements in global coordinates are given by a^e= [ u₁ u₂ u₃ u₄ ]^T

The transpose of a^e is stored in ed. The nodal displacements in local coordinates are given by

¯

a^e= Ga^e

bar2gs Two dimensional bar element

where the transformation matrix G is defined in bar2ge. The displacements associ-ated with bar action are determined as

¯ a^e_bar =

 u¯₁

¯ u₃



The displacement u(¯x) and the normal force N (¯x) are computed from u(¯x) = N¯a^e_bar

N (¯x) = D_EAB¯a^e_bar where

N = 1 ¯x C⁻¹= 1− _L^¯x _L^¯x

B = 0 1 C⁻¹ = 1 L

−1 1 

where D_EA and L are defined in bar2ge and C⁻¹ =

 1 0

−_L¹ _L¹



An updated value of the axial force is computed as Q_¯x = N (0)

Three dimensional bar element bar3e

Purpose:

Compute element stiffness matrix for a three dimensional bar element.

E, A

(x₁,y₁,z₁)

(x₂,y₂,z₂)

z

x

y x

u₁ u₂ u₃

u₄ u₅

u₆

Syntax:

Ke=bar3e(ex,ey,ez,ep) [Ke,fe]=bar3e(ex,ey,ez,ep,eq) Description:

bar3e provides the global element stiffness matrix Ke for a three dimensional bar element.

The input variables ex = [ x₁ x₂ ] ey = [ x₁ x₂ ] ez = [ y₁ y₂ ]

ep = [ E A ]

supply the element nodal coordinates x₁, y₁, z₁, x₂, y₂, and z₂, the modulus of elasticity E, and the cross section area A.

The element load vector fe can also be computed if uniformly distributed axial load is applied to the element. The optional input variable

eq = q_¯x 

then contains the distributed load per unit length, q_¯x. Theory:

The element stiffness matrix K^e, stored in Ke, is computed according to K^e= G^T K¯^e G

where

K¯^e= D_EA L

 1 −1

−1 1



G =

 n_x¯x n_{y ¯x} n_{z ¯x} 0 0 0 0 0 0 n_x¯x n_{y ¯x} n_{z ¯x}



bar3e Three dimensional bar element

where the axial stiffness D_EA and the length L are given by D_EA = EA; L =



(x₂− x1)²+ (y₂− y1)²+ (z₂− z1)² and the transformation matrix G contains the direction cosines

n_x¯x = x₂− x1

L n_{y ¯x} = y₂− y1

L n_{z ¯x} = z₂− z1

L

The element load vector f_l^e, stored in fe, is computed according to f^e_l = G^T f¯^e_l

where

¯f_l^e= q_¯xL 2

 1 1



Three dimensional bar element bar3s

Purpose:

Compute normal force in a three dimensional bar element.

N

bar3s computes the normal force in a three dimensional bar element bar3e.

The input variables ex, ey, and ep are defined in bar3e and the element nodal displace-ments, stored in ed, are obtained by the function extract. The number of evaluation points for section forces and displacements are determined by n. If n is omitted, only the ends of the bar are evaluated.

The output variables

es =

contain the normal force, the displacement, and the evaluation points on the local

¯

x-axis. L is the length of the bar element.

Theory:

The nodal displacements in global coordinates are given by a^e= [ u₁ u₂ u₃ u₄ u₅ u₆ ]^T

The transpose of a^e is stored in ed.

Three dimensional bar element bar3s

The nodal displacements in local coordinates are given by

¯

a^e= Ga^e

where the transformation matrix G is defined in bar3e.

The displacement u(¯x) and the normal force N (¯x) are computed from u(¯x) = N¯a^e +u_p(¯x)

N (¯x) = D_EAB¯a^e +N_p(¯x) where

N =

1 ¯x C⁻¹=

1− _L^¯x _L^¯x

B = 0 1 C⁻¹ = 1 L

−1 1 

u_p(¯x) = −_D^q_EA^{x¯ ¯x}₂² − ^L¯₂^x

N_p(¯x) = −q¯x



¯ x−^L₂^

where D_EA, L, q_¯x are defined in bar3e and C⁻¹ =

 1 0

−_L¹ _L¹



In document Division of Structural Mechanics (Page 54-73)