Plates vs Shells 1

[ekline]

What are the general differences between plates and shell elements? What is a good general handbook for FEA? I'd like something with an explanation of the different element formulations, solver theories, and potential pitfalls.

 

[Drej]

Search the archives, there's loads of info on this topic. See for example thread727-79633.




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[UcfSE]

Shells develop membrane forces, plates do not. When you find a book make sure it is from the point-of-view of your discipline. For instance, if you are mechanical engineering you probably would not want a book that concentrates on structural engineering applications.

 

[feadude]

plane stress elements are for in plane forces. plate elements are for out of plane forces. When there are both in plane and out of plane forces, then typically a shell element is used which is a plane stress and a plate element combined together. Even though the word "shell" in structural mechanics refers to membrane stresses in FEA shell elements normally can take both in plane and out of plane forces. The six degree of freedom "drilling" which is rotation about an axis perpendicular to the plane of the plate is tricky. Some use what is called a penalty factor formulation and others (mostly older FEA) ignor it.

 

[mtnengr]

The discussions brought forth by the participents todate are partially correct. I was always taught that the major differece between plates and shells is that Plates do not have curvature and Shells do. Both can be classified as purely membrane elements or elements that can support bending and membrane stress resultants. Both can support transverse shear forces.

Finite element formulations can consider representing a shell of revolution, a curved shell element or faceted plate elements. The first is described by a conic form of revolution. The second is a portion of a shell element that has two radii of curvature. The third is a flat plate with infinite radii of curvature. Various tri- quad- penti- planforms of the plate can be postulated and differing degrees of freedom can be considered for each class of element. Finite element formulations have been proposed to represent the displacement field within the element. Compatability is assured and equilibrium can be violated, but minimized using energy considerations.

The shell of revolution element can be represented by a single line along the length of the shell. The displacement field can be accurately defined when the user considers a specific shell theory, accounting for the various degrees of freedom. Both axisymmetric and non-axisymmetric responses can be considered. To represent the same shell of revolution with faceted plate elements, the circumferential geometry is approximated. The number of elements required to produce an accurate solution is dependent upon the loading condition and the number of elements required to represent the bending that takes place. A transformation matrix must be suppiled for each element accounting for the placement of the element along the shell's circumferential surface and how it is transformed into a global system. If one were to consider a curved shell element that has a finite planform in each direction, the compatability of the normals at the adjacent edges are assured.

I have found that for shell of revolution analysis, going back to basics and representing the governing equations from stress-strain relations, equilibrium and stress-displacement permits a far more accurate representation (a two-point boundary value problem). Solving these equations numerically generally eleminates all of the questions that arise in finite element formulations.

 

[mtnengr]

The response to date are partially correct.

I was taught that the major differece between a plate and a shell is that shells have curvature plates do not. Both can carry memberane, bending and shear stress resultants.

One has a choice as how to represent a shell. The shell can be considered as part of a conic that has two principle radii of curvature. The shell can be considered as a shell of revolution or as a shell segment.

Finite element methods formulate a stiffness based upon some displacement representation for the planform being considered. The shell of revolution is represent by a reference surface line following the axial curvature of a conic. The shell segment or the plate element has a two-dimensional planform. The geomtry of the assembled elements can be defined along the circumferential direction. The topography of this representation is dependent upon the shell being represented and the loading under investigation. Again displacement representations are developed that assure compatability but not equilibrium. The facated plate element requires an additional consideration or approximation by transforming the edges of adjacent elements to have compatable normals and then a proper global tranformation. Also, the approximation of curvature using flat elements can be a major problem.

With all of the above problems, I find it more convienent to start with the basic equations of equilibrium, strain-displacement and stress-strain relations and develop a two point boundary problem. Solve these equations numerically. Most of the issues between flat plates and curved elements go away.

Several references on this subject can be obtain at my website

http://www.volcano.net/~d.citerley

 

[pja]

The answers above were totally correct..the OP asked about the differences between plate ELEMENTS and shell ELEMENTS..classical plate elements don't consider the development of membrane stresses while shell elements do. Of course in real life plates develop membrane stresses..however in classical formulations these aren't considered and you can get away with 3 DOF per node.

 

[mtnengr]

When a response is made to the forums in Eng-Tips, one tries to be as brief as possible and yet accurate. Such is the case for this thread. My original response was based upon the assumption that the OP was trying to ascertain the difference between the use of plate elements or shell elements to represent a shell of revolution or shell segment. I was not precise enough in my discussion. Maybe we need a dictionary of terms to insure that everyone is on the same page.

I assumed that the definition of a PLATE was an element capable of supporting both in-plane and lateral loads. The PLATE definition also includes that the assumption of being flat. A SHELL is considered to be a PLATE with two dimensional curvature or as a complete shell of revolution that supports both in-plane and lateral loads. .

A discussion of plate finite elements, their formulation and corresponding kinematic and static displacements is presented in Table 3.1, Bathe, K. and E.L. Wilson, Numerical Methods in Finite Element Analysis, Prentice-Hall 1976. If one were to consider the use of these elements to emulate the response of a SHELL to any given set of loads, a plane stress and a plate bending finite element needs to be combined to represent a finite element PLATE. This requires 3 displacement variables at each node to be considered as degrees of freedom. The CQUAD or CTRIA series of isoparametric membrane-bending or plane strain quadrilateral or triangular plate elements used in MSC NASTRAN performs this task. ANSYS has similar elements. Neither set considers membrane-bending coupling.

Computer programs such as BOSOR4, BOSOR5 and STAGS, developed by Lockheed’s D. Bushnell and B. Almroth, et al, use specific shell theories to develop “finite elements” that incorporate the membrane-bending coupling. The first two use a conic description for defining a shell of revolution segment, while STAGS uses arbitrary shell segments to define the geometry. Similarly, ADINA uses a shell concept, but with a differing isoparametric field descriptions to arrive at a 2D SHELL element. Membrane-bending coupling is properly considered. For this class of element, six degrees of freedom at each node are often used.

A summary of computer codes , capable of performing the analysis requested by the OP, their references, capabilities and availability can be found in the paper: “Shell Analysis,” A Kalnins and L. Weingarten, Shock and Vibration Computer Programs—Reviews and Summaries, SVM-10, 1975 pp 507-525. By reading the references contained therein, one could develop an understanding of what is involved in performing a “proper” analysis and just how PLATE and SHELL elements are used.

I can not accept the use of plane stress elements alone to properly represent the response of a SHELL subject to arbitrary loadings.

 

[feadude]

one can use general purpose commercially available parabolic isoparametric shell elements that will automatically fit geometry because of extra nodes at mid sides irregardless of it being flat, single curvature or double curvature and get results within 1% of classical thories of shells of revolution. It would help to read structural mechanics and FEA textbooks and find out that classical definition of shell is not the same as shell element and original thread writer had asked a question about shell and plate elements not shells and plates. Big difference between the two.General purpose commercially available parabolic isoparametric shell element formulation can handle plate, shell geometry and in plane and out of plane loading as long as enough elements are used to represent the actual geometry within reasonable approximation such as curve fit, etc.

 

[pja]

Who said anything about using plane stress elements to represent shells? I don't think I read anywhere here where somebody suggested this. As feadude nicely stated most commercial codes use a shell element which combines a DKT type formulation for the bending behavior with some sort of isoparametric formulation for the membrane behavior.