references for 3-node beam elements ?

[bill5678]

i'm analysing a 1-Dimensional beam, which is subjected to a Uniformly Distributed Load (UDL).
i'm analysing the beam using Finite Element Analysis (FEA).
i'm using 1-Dimensional beam elements, that have 2 Degrees Of Freedom (DOF) at each node -- vertical displacement, and rotation. the expression for vertical displacement y (x), being a function of the position x along the length of the beam, is modelled using a (first-order) hermite polynomial interpolation expression.

i have a lot of references for the "standard" 2-node beam element. this element uses a cubic (i.e. power of 3) polynomial for y (x). it DOES NOT give the "exact" result when the UDL is applied.
but, i can't find any references for the 3-node beam element. this element uses a quintic (i.e. power of 5) polynomial for y (x). it DOES give the "exact" result when the UDL is applied.

so, my question is :- does anyone know of any references for 3-node beam elements ?

what i'd especially like are references that have explicit expressions for :-
1) the element shape functions N, in terms of x
2) the element strain-displacement vector B, in terms of x
3) the element stiffness matrix K
4) the element force vector F


any suggestions would be much appreciated.

 

[rb1957]

yes there are some out there (but not NASTRAN).

ASAS (Atkins Structural Analysis System) have two a 3 noded rod (FLA3) and a 3 noded beam (BAX3) to suit their 8 noded quads.

i know NASTRAN doesn't have one (although they have 8 noded quads).

i think other FEAs might, like ANSYS, probably not solidworks, ...

Quando Omni Flunkus Moritati

 

[trainguy]

Any reason why you don't just use a finer mesh with more 2-node beam elements?

tg

 

[IDS]

A UDL can be converted to point loads and moments applied to the beam segment ends that will give the "exact" deflected shape at the nodes, which is what the standard two node beam element does. The deflection between nodes will not be exact, but the difference will be negligible with even a small number of subdivisions.

Alternatively the spreadsheets below will give "exact" results for udls or trapezoidal loads:
Continuous beam analysis:

http://newtonexcelbach.wordpress.com/2012/09/06/continuous-beam-spreadsheet-with-units/

Frame analysis:

http://newtonexcelbach.wordpress.com/2013/03/17/3dframe-ver-1-03-and-frame4-ver-3-07/

The frame analysis programs use the standard 2 node beam to find node deflections, then the "exact" calculation to find deflections between nodes.

Of course, for any real structure the deviations from the "exact" solution due to effects such as shear lag and deviations of materials properties from those assumed will typically be at least an order of magnitude greater than differences between the 2-node beam results and the "exact" analysis.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[bill5678]

thanks for your replies

i'll give you a bit more info, about what i'm doing -- to give you an idea about what i'm after. i'll try NOT to use too much "technical jargon".

i'm working on a research study.
the study involves analysing the reliability of a 1-D beam.
the beam has fixed supports at both ends, has random bending stiffness EI, and has a random UDL P applied.
the beam is modelled using 32 equal size finite elements. this number is fixed -- i can't reduce the size of the elements and use a finer mesh. but, thanks for your suggestion to, anyway !

the "reliability" is described as the probability of :-
1) excessive vertical displacement, at the midspan
2) excessive bending moment, at the left support

during the reliability analysis, i have to generate random sample values of EI and P. i then use these sample values of EI and P in the finite element analysis. so, i'm writing my own finite element analysis code -- so the reliability analysis can "interact" with it. i'm not using commercial finite element analysis software. but, again, thanks for your suggestion to, anyway !

about the UDL P, and finite element analysis. i'm doing as you suggested -- converting the UDL P into "equivalent" nodal forces and moments.
also, there is a node at each of the points where i need to calculate the vertical displacement and bending moment.

also, i'm calculating the vertical displacement (U) and bending moment (M) from :-

a) F = KU, solve for U
b) M = KU + FEM, using element K, U, FEM (Fixed End Moments)
c) M = DBU = (EI)BU, using element D (stress-strain variable), B (strain-displacement vector), and U

note above, there are 2 different ways to calculate M.
because there is a UDL, the accuracy of (a), (b) and (c) depends on the number of nodes in each element.
for a 2-node element, (a) and (b) are exact, but (c) isn't.
for a 3-node element, (a), (b) and (c) are all exact.
so, i need to consider 2-node elements AND 3-node elements (so i can compare the results).

i have lots of books with info on 2-node elements. but, i don't have any books on 3-node elements.

so, given all of the above, does anyone know any books with info on 3-node elements ?

thanks again for your suggestions !

 

[IDS]

Why do you want to use FEA to find the moments and deflections in a 1D beam?

The continuous beam link I gave earlier will give an "exact" solution and also has links to more detailed background information.

Or why not use 2 node beams with method b), since you agree that is exact?

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[rb1957]

i'll try NOT to use too much "technical jargon". ... "jargon" ... bring it on, we can take it ! (heck, we invented most of it !)

funnily enough a google search for "three noded axial elements" found some hits that'd probably help ...




Quando Omni Flunkus Moritati

 

[bill5678]

thanks again for your replies

i'll give you even more info, about what i'm doing.

i'm working on a comparative research study.
the study involves analysing the reliability of a 1-D beam.
the reliability has previously been calculated by several authors. each author has a reliability analysis method that they developed. but, these different methods are supposed to give similar results. i'm simply trying to compare the results -- hence the "comparative study".

all of the authors modelled the beam using FEA. so, i also have to use FEA.

i said above that the bending moment (M) can be calculated by 2 different equations. so, i'm trying to compare the results using each of these 2 equations -- so i can show what gives the CORRECT answer, and what gives the INCORRECT answer. because, in this field of work, an INCORRECT answer (or, rather, knowing WHY the answer is INCORRECT) can be just as important as a CORRECT answer.

so, again :-
1) i have to use FEA
2) i need to consider 2-node elements AND 3-node elements (so i can compare the results)
3) i need to calculate the vertical displacement and bending moment, using the 3 equations i described

so, with that info, does anyone know any books with info on 3-node elements ?

also, thanks for the suggestion to do a google search for "three noded axial elements"

 

[rb1957]

i quite understnad what you are doing ... we used to call it a "patch" test. we used a single element with an easily calculated loading so we could see which element came closest to the theoretical results. did you real googling ? i've asked Atkins but no reply (and i'm not going to hound them).

Quando Omni Flunkus Moritati

 

[cooperDBM]

As the dimmest bulb in the room I could use a clarification. Are we not talking about a 2-D beam element which models lateral deflection in a plane? Isn't a 1-D beam element just a rod that models only axial strain?

 

[bill5678]

in reply to you :-

rb1957
1) that "patch test" sounds something like what i'm trying to do
2) yes, i tried google (as was suggested). i found a little info, but, i'm still looking

cooperDBM
1) no, you are definitely NOT a dim bulb ! this stuff (research) is highly specialised, and i have several year's experience. so, don't worry if you don't quite understand what i've tried to say above (half the time I don't even understand it )
2) also, no, i'm not talking about a 2-D beam element. i'm using a 1-D element (line element), that models displacement, rotation, and bending moment, but not axial force. but, let me know if you need any more clarification

 

[NattapolCha]

Hi I'm new.
I guess that I know what're you talking about. I follow the book of Eugenio Onate Example 1.4 p.18-20. The problem is shown as cantilever beam with uniform distributed load q. By using 2-nodes beam isoparametric formulation and cubic interpolation function. Nodal equilibrium is sastisfy and give the exact displacement and rotation at free end. But because of using cubic interpolation function ,Moment along 2 node is linear variation that 's not an exact distribution of curvature(require parabolic distribution). The exact moment occur only at the integration point (2 points) in the element. And sure refine mesh will improve solution of this problem.
I try to add middle node of beam element that have six degree of freedom on 3-nodes beam element. Then formulate stiffness matrix and equivalent nodal force vector. I notice that the midpoint-rotation of stiffness matrix is all zero ,so I can't find midpoint rotation of beam. Finally I ignore midpoint-rotation DOF and now the element is 3-nodes beam with 5 DOF. I formulate stiffness and equivalent nodal load again and can solving the exact midpoint deflection and free end deflection & rotation. With 4-th degree displacement shape function this time I get parabolic moment interpolation that lead to exact solution of moment throughout the beam w/o element refining.
Because the absent of midpoint-rotation DOF I think this formulation doesn't work for 3D frame analysis which require transformation of rotational axis(DOF) (that I ignore previously). Just like you I can't find any refrence of 3-node beam element formulation. I hope now you can find your answer and show me the way. Thank you