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beam1, beam1t

Purpose

Element functions for a 12 DOF beam element. beam1t is a 2 node beam with pretension available for non-linear cable statics and dynamics.

Description

beam1

In a model description matrix, element property rows for beam1 elements follow the format

 [n1 n2 MatID ProID nR 0 0 EltID p1 p2 x1 y1 z1 x2 y2 z2]

where


n1,n2node numbers of the nodes connected
MatIDmaterial property identification number
ProIDelement section property identification number
nr 0 0number of node not in the beam direction defining bending plane 1 in this case {v} is the vector going from n1 to nr. If nr is undefined it is assumed to be located at position [1.5 1.5 1.5].
vx vy vzalternate method for defining the bending plane 1 by giving the components of a vector in the plane but not collinear to the beam axis. If vy and vz are zero, vx must be negative or not an integer. MAP=beam1t('map',model) returns a normal vector MAP giving the vector used for bending plane 1. This can be used to check your model.
p1,p2pin flags. These give a list of DOFs to be released (condensed before assembly). For example, 456 will release all rotation degrees of freedom. Note that the DOFS are defined in the local element coordinate system.
x1,...optional components in global coordinate system of offset vector at node 1 (default is no offset)
x2,...optional components of offset vector at node 2

Isotropic elastic materials are the only supported (see m_elastic). p_beam describes the section property format and associated formulations.

Failure to define orientations is a typical error with beam models. In the following example, the definition of bending plane 1 using a vector is illustrated.

  cf=feplot(femesh('test2bay'));
  % Map is in very variable direction due to undefined nr
  % This is only ok for sections invariant by rotation
  beam1t('map',cf.mdl);fecom('view3'); 
  
  % Now define generator for bending plane 1
  i1=feutil('findelt eltname beam1',cf.mdl); % element row index
  cf.mdl.Elt(i1,5:7)=ones(size(i1))*[-.1 .9 0]; % vx vy vz
  beam1t('map',cf.mdl);fecom('view2'); 

beam1 adds secondary inertia effects which may be problematic for extremely short beams and beam1t may then be more suitable.

beam1t

For the bending part, this element solves

  ρ A ( w −Ω2 w) + 
2
∂ x2



E Iy
2 w
∂ x2



−  
∂ 
∂ x



 T
∂ w
∂ x



f = 0      (9.1)

with boundary conditions in transverse displacement

wgiven or  F=T
∂ w
∂ x
EIy
3 w
∂ x3
    (9.2)

and rotation

∂ w
∂ x
given or  M=EIy
2 w
∂ x2
    (9.3)

This element has an internal state stored in a InfoAtNode structure where each column of Case.GroupInfo{7}.data gives the local basis, element length and tension [bas(:);L;ten]. Initial tension can be defined using a .MAP field in the element property.

This is a simple example showing how to impose a pre-tension :

 model=femesh('TestBeam1 divide 10');
 model=fe_case(model,'FixDof','clamp',[1;2;.04;.02;.01;.05]);
 model.Elt=feutil('SetGroup 1 name beam1t',model);
 d1=fe_eig(model,[5 10]); 
 model=feutil('setpro 112',model,'MAP', ...
   struct('dir',{{'1.5e6'}},'lab',{{'ten'}}));
 d2=fe_eig(model,[5 10]); 

 figure(1);plot([d2.data./d1.data-1]);
 xlabel('Mode index');ylabel('Frequency shift');

Strains in a non-linear Bernoulli Euler section are given by

є11=


∂ u
∂ x
+
1
2



∂ w0
∂ x
2





z
2 w0
∂ x2
    (9.4)

See also

p_beam, m_elastic, fe_mk, feplot


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