References for Equivalent Nodal Force Computations

[Celt83]

Is anyone aware of any worked examples for the computations of Equivalent Nodal Forces for partial loading normal to a shell element?
I have a few of the books noted in the FAQ here that I have started working through namely:
Zienkiewicz - Volumes 1 and 2
Finite element procedures by K J Bathe
A First Course in the Finite Element Method by Daryl L Logan
Programming the Finite Element Method by Smith and Griffiths

I admittedly need to work on my calculus so don't have a full grasp yet on the body and traction force integrals, but none of the books appeared to directly address loadings of this type and in general don't provide many examples for the equivalent nodal forces.

A secondary question is for the body and traction force integrals are these typically done based on the actual element geometry or the isoperimetric formulation? Seems you could use the isoperimetric shape functions and jacobian and just need to work out the load function change of variable from x,y to eta,xi (for many cases this would appear to be a direct substitution).

I'm making a thing:

http://www.thestructuraltoolbox.com

(It's no Kootware and it will probably break but it's alive!)

 

[FEA way]

Check also "Concepts and Applications of Finite Element Analysis" by Cook.

 

[Celt83]

Thanks for the recommendation FEA way, have been eyeing a few copies of that one on Ebay for a few weeks.

I'm making a thing:

http://www.thestructuraltoolbox.com

(It's no Kootware and it will probably break but it's alive!)

 

[NRP99]

There might be some information in "The finite element analysis of shells by Chapelle and Bathe". You can refer any FEA of shell book which may have information you need.

 

[centondollar]

The integrals are done using reference elements, as you mentioned, and often using isoparametric mapping, i.e., polynomial order of the element (e.g., linear) is the same order as the polynomial order of the element (linear representing piecewise linear, e.g., triangle, shapes). In practice, often a 1D integration scheme is used in two orthogonal directions in the reference element.

To express the integral of a typical element using a reference element, you will need the Jacobian (for scaling when changing "dxdy" to "dXidEta") and also to transform the gradient using the chain rule (in 2 dimensions); then, choose a quadrature scheme and implement it.

Note that Reissner-Mindlin plates suffer from shear locking unless a rather complicated mathematical trick (not just reduced integration as for a Timoshenko beam) is applied, so I would advice to start from the classical plate model - or better yet, to let commercial software provide a reliable and general plate solver.

 

[Celt83]

Thanks NRP99, I'll give that text a look.

Thanks for the explanation centondollar.

I'm making a thing:

http://www.thestructuraltoolbox.com

(It's no Kootware and it will probably break but it's alive!)

 

[Celt83]

FEA Way: thanks again for that recommendation the Cook book has an excellent section on the equivalent nodal forces was able to grab a copy of the third edition for under $5 off of Abe Books.

I'm making a thing:

http://www.thestructuraltoolbox.com

(It's no Kootware and it will probably break but it's alive!)