Ho mo ge neous ma te rial prop er ties are used for the non-lay ered Mem brane, Plate, and Shell sec tion types. The fol low ing data needs to be spec i fied.
Section Thickness
Each ho mo ge neous Sec tion has a con stant mem brane thick ness and a con stant bend ing thick ness. The mem brane thick ness, th, is used for cal cu lat ing:
• The mem brane stiff ness for full-shell and pure-mem brane Sec tions
• The el e ment vol ume for the el e ment self-weight and mass cal cu la tions The bend ing thick ness, thb, is use for cal cu lat ing:
• The plate-bend ing and trans verse-shear ing stiffnesses for full-shell and pure-plate Sec tions
Nor mally these two thick nesses are the same and you only need to spec ify th. How -ever, for some ap pli ca tions, you may wish to ar ti fi cially change the mem brane or plate stiff ness. For this pur pose, you may spec ify a value of thb that is dif fer ent from th. For more de tailed con trol, such as rep re sent ing cor ru gated or orthotropic con struc tion, the use of prop erty mod i fi ers is better. See Topic “Prop erty Modifiers” (page 181.)
Thickness Formulation
Two thick ness for mu la tions are avail able, which de ter mine whether or not trans -verse shear ing de for ma tions are in cluded in the plate-bend ing be hav ior of a plate or shell el e ment:
Section Properties 171
• The thick-plate (Mindlin/Reissner) for mu la tion, which in cludes the ef fects of trans verse shear de for ma tion
• The thinplate (Kirchhoff) for mu la tion, which ne glects trans verse shear ing de -for ma tion
Shearing de for ma tions tend to be im por tant when the thick ness is greater than about onetenth to onefifth of the span. They can also be quite sig nif i cant in the vi -cin ity of bend ing-stress con cen tra tions, such as near sud den changes in thick ness or sup port con di tions, and near holes or re-en trant cor ners.
Even for thinplate bend ing prob lems where shear ing de for ma tions are truly neg li -gi ble, the thick-plate for mu la tion tends to be more ac cu rate, al though some what stiffer, than the thin-plate for mu la tion. How ever, the ac cu racy of the thick-plate for mu la tion is more sen si tive to large as pect ra tios and mesh dis tor tion than is the thin-plate for mu la tion.
It is gen er ally rec om mended that you use the thick-plate for mu la tion un less you are us ing a dis torted mesh and you know that shear ing de for ma tions will be small, or un less you are try ing to match a the o ret i cal thin-plate so lu tion.
The thick ness for mu la tion has no ef fect upon mem brane be hav ior, only upon plate-bend ing be hav ior.
Section Material
The ma te rial prop er ties for each Sec tion are spec i fied by ref er ence to a pre vi -ously-de fined Ma te rial. The ma te rial may be iso tro pic, uni ax ial, or orthotropic. If an anisotropic ma te rial is cho sen, orthotropic prop er ties will be used. The ma te rial prop er ties used by the Shell Sec tion are:
• The moduli of elas tic ity, e1, e2, and e3
• The shear modulus, g12, g13, and g23
• The Pois son’s ra tios, u12, u13, and u23
• The co ef fi cients of ther mal ex pan sion, a1 and a2
• The mass den sity, m, for com put ing el e ment mass
• The weight den sity, w, for com put ing Self-Weight and Grav ity Loads
The prop er ties e3, u13, and u23 are con densed out of the ma te rial ma trix by as sum -ing a state of plane stress in the el e ment. The re sult -ing, mod i fied val ues of e1, e2, g12, and u12 are used to com pute the mem brane and plate-bend ing stiffnesses.
172 Section Properties
The shear moduli, g13 and g23, are used to com pute the trans verse shear ing stiff -ness if the thick-plate for mu la tion is used. The co ef fi cients of ther mal ex pan sion, a1 and a2, are used for mem brane ex pan sion and ther mal bend ing strain.
All ma te rial prop er ties (ex cept the den si ties) are obtained at the ma te rial tem per a -ture of each in di vid ual el e ment.
See Chap ter “Ma te rial Prop er ties” (page 69) for more in for ma tion.
Section Material Angle
The ma te rial lo cal co or di nate sys tem and the el e ment (Shell Sec tion) lo cal co or di -nate sys tem need not be the same. The lo cal 3 di rec tions al ways co in cide for the two sys tems, but the ma te rial 1 axis and the el e ment 1 axis may dif fer by the an gle a as shown in Figure 36 (page 173). This an gle has no ef fect for iso tro pic ma te rial prop er ties since they are in de pend ent of ori en ta tion.
See Topic “Lo cal Co or di nate Sys tem” (page 70) in Chap ter “Ma te rial Prop er ties”
for more in for ma tion.
Section Properties 173
3 (Element, Material) a
a
1 (Element) 1 (Material)
2 (Element)
2 (Material)
Figure 36
Shell Section Material Angle