The Shell el e ment is a type of area ob ject that is used to model mem brane, plate, and shell be hav ior in pla nar and three-di men sional struc tures. The shell ma te rial may be ho mo ge neous or lay ered through the thick ness. Ma te rial nonlinearity can be con sid ered when us ing the lay ered shell.
Basic Topics for All Users
• Over view
• Joint Con nec tiv ity
• Edge Constraints
• De grees of Free dom
• Lo cal Co or di nate Sys tem
• Sec tion Prop er ties
• Mass
• Self-Weight Load
• Uni form Load
• Sur face Pres sure Load
• In ter nal Force and Stress Out put
157
Advanced Topics
• Ad vanced Lo cal Co or di nate Sys tem
• Prop erty Mod i fi ers
• Joint Off sets and Thick ness Overwrites
• Grav ity Load
• Tem pera ture Load
Overview
The Shell ele ment is a three- or four- node for mu la tion that com bines mem brane and plate- bending be hav ior. The four- joint ele ment does not have to be pla nar.
Each Shell el e ment has its own lo cal co or di nate sys tem for de fin ing Ma te rial prop -er ties and loads, and for in t-er pret ing out put. Tem p-er a ture-de pend ent, orthotropic ma te rial prop er ties are al lowed. Each el e ment may be loaded by grav ity and uni -form loads in any di rec tion; sur face pres sure on the top, bot tom, and side faces; and loads due to strain and tem per a ture change.
A four-point nu mer i cal in te gra tion for mu la tion is used for the Shell stiff ness.
Stresses and in ter nal forces and mo ments, in the el e ment lo cal co or di nate sys tem, are eval u ated at the 2-by-2 Gauss in te gra tion points and ex trap o lated to the joints of the el e ment. An ap prox i mate er ror in the el e ment stresses or in ter nal forces can be es ti mated from the dif fer ence in val ues cal cu lated from dif fer ent el e ments at tached to a com mon joint. This will give an in di ca tion of the ac cu racy of a given fi niteel e -ment ap prox i ma tion and can then be used as the ba sis for the se lec tion of a new and more ac cu rate fi nite el e ment mesh.
Struc tures that can be mod eled with this el e ment in clude:
• Floor sys tems
• Wall sys tems
• Bridge decks
• Three-di men sional curved shells, such as tanks and domes
• De tailed mod els of beams, col umns, pipes, and other struc tural mem bers Two dis tinct for mu la tions are avail able: ho mog e nous and lay ered.
158 Overview
Homogeneous
The ho mo ge neous shell com bines in de pend ent membrane and plate be hav ior.
These be hav iors be come cou pled if the el e ment is warped (nonpla nar.) The mem -brane be hav ior uses an iso para met ric for mu la tion that in cludes trans la tional plane stiff ness com po nents and a “drill ing” ro ta tional stiff ness com po nent in the di rec tion nor mal to the plane of the ele ment. See Tay lor and Simo (1985) and Ibra -him be go vic and Wil son (1991). In-plane dis place ments are qua dratic.
Plate-bend ing be hav ior in cludes two-way, out-of-plane, plate ro ta tional stiff ness com po nents and a translational stiff ness com po nent in the di rec tion nor mal to the plane of the el e ment. You may choose a thinplate (Kirchhoff) for mu la tion that ne glects trans verse shear ing de for ma tion, or a thickplate (Mindlin/Reissner) for mu -la tion which in cludes the ef fects of trans verse shear ing de for ma tion. Out-of-p-lane dis place ments are cu bic.
For each ho mo ge neous Shell el e ment in the struc ture, you can choose to model pure-mem brane, pure-plate, or full-shell be hav ior. It is gen er ally rec om mended that you use the full shell be hav ior un less the en tire struc ture is pla nar and is ad e -quately re strained.
Layered
The lay ered shell al lows any num ber of lay ers to be de fined in the thick ness di rec -tion, each with an in de pend ent lo ca -tion, thick ness, be hav ior, and ma te rial. Ma te rial be hav ior may be non lin ear.
Mem brane de for ma tion within each layer uses a strain-pro jec tion method (Hughes, 2000.) In-plane dis place ments are qua dratic. Un like for the ho mo ge neous shell, the
“drill ing” de grees of free dom are not used, and they should not be loaded. These ro ta tions nor mal to the plane of the el e ment are only loosely tied to the rigidbody ro -ta tion of the el e ment to pre vent in s-ta bil ity.
For bend ing, a Mindlin/Reissner for mu la tion is used which al ways in cludes tran sverse shear de for ma tions. Outofplane dis place ments are qua dratic and are con -sis tent with the in-plane displacements.
The lay ered Shell usu ally rep re sents full-shell be hav ior, al though you can con trol this on a layerbylayer basis. Un less the lay er ing is fully sym met ri cal in the thick -ness di rec tion, mem brane and plate be hav ior will be cou pled.
Overview 159
160 Overview
Axis 1
Axis 1 Axis 3
Axis 3 Axis 2
Axis 2
j1
j1
j2
j2 j3
j3
j4
Face 1
Face 1 Face 3
Face 3 Face 4
Face 2
Face 2 Face 6: Top (+3 face)
Face 5: Bottom (–3 face)
Face 6: Top (+3 face) Face 5: Bottom (–3 face)
Four-node Quadrilateral Shell Element
Three-node Triangular Shell Element
Figure 31
Area Element Joint Connectivity and Face Definitions
Joint Connectivity
Each Shell el e ment (and other types of area ob jects/el e ments) may have ei ther of the fol low ing shapes, as shown in Figure 31 (page 160):
• Quad ri lat eral, de fined by the four joints j1, j2, j3, and j4.
• Tri an gu lar, de fined by the three joints j1, j2, and j3.
The quad ri lat eral for mu la tion is the more ac cu rate of the two. The tri an gu lar ele -ment is only rec om mended for lo ca tions where the stresses do not change rap idly.
The use of large tri an gu lar el e ments is not rec om mended where in-plane (membrane) bend ing is sig nif i cant. The use of the quad ri lat eral ele ment for mesh -ing vari ous geo me tries and tran si tions is il lus trated in Figure 32 (page 162), so that tri an gu lar el e ments can be avoided altogether.
Edge con straints are also avail able to cre ate tran si tions be tween mis-matched meshes with out us ing dis torted el e ments. See Subtopic “Edge Con straints” (page 163) for more in for ma tion.
The joints j1 to j4 de fine the cor ners of the ref er ence sur face of the shell el e ment.
For the ho mo ge neous shell this is the mid-sur face of the el e ment; for the lay ered shell you choose the lo ca tion of this sur face rel a tive to the ma te rial layers.
You may op tion ally as sign joint off sets to the el e ment that shift the ref er ence sur -face away from the joints. See Topic “Joint Off sets and Thick ness Overwrites”
(page 183) for more in for ma tion.