For the lay ered Sec tion prop erty, you de fine how the sec tion is builtup in the thick ness di rec tion. Any num ber of lay ers is al lowed, even a sin gle layer. Lay ers are lo -cated with re spect to a ref er ence sur face. This ref er ence sur face may be the mid dle sur face, the neu tral sur face, the top, the bot tom, or any other lo ca tion you choose.
By de fault, the ref er ence sur face con tains the el e ment nodes, al though this can be changed us ing joint off sets.
The thick-plate (Mindlin/Reissner) for mu la tion, which in cludes the ef fects of trans verse shear de for ma tion, is al ways used for bend ing be hav ior the lay ered shell.
The fol low ing eight pa ram e ters are spec i fied to de fine each layer, as il lus trated in Figure 37 (page 174.)
(1) Layer Name
The layer name is ar bi trary, but must be unique within a sin gle Sec tion. How ever, the same layer name can be used in dif fer ent Sec tions. This can be use ful be cause re sults for a given layer name can be plot ted si mul ta neously for el e ments hav ing dif fer ent Sec tions.
174 Section Properties
Thickness
Reference Surface
Distance Axis 1
Axis 3
Layer “A”
Layer “B”
Layer “C”
Layer “D”
Figure 37
Four-Layer Shell, Showing the Reference Surface, the Names of the Layers, and the Distance and Thickness for Layer “C”
(2) Layer Dis tance
Each layer is lo cated by spec i fy ing the dis tance from the ref er ence sur face to the cen ter of the layer, mea sured in the pos i tive lo cal-3 di rec tion of the el e ment. This value is called d in the ex am ples be low.
(3) Layer Thick ness
Each layer has a sin gle thick ness, mea sured in the lo cal-3 di rec tion of the el e ment.
For mod el ing rebar or ma te rial fi bers, you can spec ify a very thin “smeared” layer that has an equiv a lent cross-sec tional area. This value is called th in the ex am ples be low.
(4) Layer Type
You can choose be tween:
• Mem brane: Strains in the layer (e11,e22,g12) are com puted only from in-plane mem brane dis place ments, and stresses in the layer (s11,s22,s12) con trib ute only to in-plane mem brane forces (F11,F22,F12).
• Plate: Strains in the layer (e11,e22,g12,g13,g23) are com puted only from plate-bend ing ro ta tions and trans verse displacements, and stresses in the layer (s11,s22,s12,s13,s23) con trib ute only to platebend ing mo ments and trans -verse shear ing forces ( M11,M22,M12,V13,V23).
• Shell, which com bines mem brane and plate be hav ior: Strains in the layer (e11,e22,g12,g13,g23) are com puted from all dis place ments and platebend -ing ro ta tions, and stresses in the layer (s11,s22,s12,s13,s23) con trib ute to
a l l f o r c e s a n d p l a t e - b e n d i n g mo m e n t s (F11,F22,F12,M11,M22,M12,V13,V23).
In most ap pli ca tions, lay ers should use shell be hav ior. See shearwall mod el ing be low for an ex am ple of where you might want to sep a rate membrane and plate be -hav ior.
Im por tant Note: Mass and weight are com puted only for mem brane and shell lay ers, not for plate lay ers. This pre vents dou blecount ing when in de pend ent mem -brane and plate lay ers are used for the same ma te rial.
(5) Layer Number of Thickness In te gra tion Points
Ma te rial be hav ior is in te grated (sam pled) at a fi nite num ber of points in the thick -ness di rec tion of each layer. You may choose one to five points for each layer. The
Section Properties 175
lo ca tion of these points fol lows stan dard Guass in te gra tion pro ce dures. This value is called n in the ex am ples be low.
For a sin gle layer of lin ear ma te rial, one point in the thick ness di rec tion is ad e quate to rep re sent mem brane be hav ior, and two points will cap ture both mem brane and plate be hav ior. If you have mul ti ple lay ers, you may be able to use a sin gle point for thin ner lay ers.
Non lin ear be hav ior may re quire more in te gra tion points or more lay ers in or der to cap ture yield ing near the top and bot tom sur faces. Us ing an ex ces sive num ber of in te gra tion points can in crease anal y sis time. You may need to ex per i ment to find a bal ance be tween ac cu racy and com pu ta tional ef fi ciency.
(6) Layer Ma te rial
The ma te rial prop er ties for each layer are spec i fied by ref er ence to a pre vi ouslyde -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 be hav ior of the ma te rial de pends on the ma te rial com po nent be hav ior cho sen for the layer, as de scribed be low.
(7) Layer Ma te rial An gle
For orthotropic and uni ax ial ma te ri als, the ma te rial axes may be ro tated with re spect to the el e ment axes. Each layer may have a dif fer ent ma te rial an gle. For ex -am ple, you can model rebar in two or thogo nal di rec tions as two lay ers of uni ax ial ma te rial with ma te rial an gles 90° apart. This value is called ang in the ex am ples be -low. For fur ther in for ma tion, see topic “Sec tion Ma te rial An gle” above (page 173.) (8) Material Component Behavior
For each of the three mem brane stress com po nents (s11,s22,s12), you can choose whether the be hav ior is lin ear, non lin ear, or in ac tive. For a uni ax ial ma te rial, only the two com po nents (s11,s12) are sig nif i cant, since s22 = al ways. Ma te rial com -0 po nents are de fined in the ma te rial lo cal co or di nate sys tem, which de pends on the ma te rial an gle and may not be the same for ev ery layer.
If all three com po nents are lin ear (two for the uni ax ial ma te rial), then the lin ear ma te rial ma trix is used for the layer, ac cord ing to Equa tions (1) to (4) in Chap ter “Ma -te rial Prop er ties” (page 69). No-te that for anisotropic ma -te ri als, the shear cou pling terms in Equa tion (4) are ne glected so that the be hav ior is the same as given by Equa tion (3).
176 Section Properties
If one or more of the three com po nents is non lin ear or in ac tive, then all lin ear com po nents use an un cou pled iso tro pic lin ear stressstrain law, all non lin ear com po nents use the non lin ear stressstrain re la tion ship, and all in ac tive com po nents as -sume zero stress. The com po nents be come un cou pled, and be have as if Pois son’s ra tio is zero. The be hav ior is sum ma rized in the fol low ing ta ble:
Component Linear Nonlinear Inactive
s11 s11=e1×e11 Eqns. (5) s11 =0
s22 s22 =e1×e22 Eqns. (5) s22 =0
s12 s12 =e1×e12 Eqns. (6) s12 =0
Note that the lin ear equa tion for s12 is for an iso tro pic ma te rial with zero Poisson’s ra tio. See Chap ter “Ma te rial Prop er ties” (page 69) for Equa tions (5) and (6).
For a uni ax ial ma te rial, s22 = and s0 12 is half the value given in the ta ble above.
Trans verse shear be hav ior is al ways lin ear, and is con trolled by the cor re spond ing mo ment com po nents. For a layer of type Mem brane, the trans verse shear stresses (s13,s23) are both zero. For a layer of type Plate or Shell:
• s13 = if s0 11 is in ac tive, else s13 =g13×g13
• s23 = if s0 22 is in ac tive, else s23 =g23×g23 In ter ac tion Be tween Lay ers
Lay ers are de fined in de pend ently, and it is per mis si ble for lay ers to over lap, or for gaps to ex ist between the lay ers. It is up to you to de cide what is ap pro pri ate.
For ex am ple, when mod el ing a con crete slab, you can choose a sin gle layer to rep -re sent the full thick ness of con c-rete, and four lay ers to -rep -re sent -rebar (two near the top at a 90° an gle to each other, and two sim i lar lay ers at the bot tom.) These rebar lay ers would be very thin, us ing an equivalent thick ness to rep re sent the crosssec -tional area of the steel. Be cause the lay ers are so thin, there is no need to worry about the fact that the rebar layers over lap the con crete. The amount of ex cess con -crete that is con tained in the over lapped re gion is very small.
Lay ers are ki ne mat i cally con nected by the Mindlin/Reissner as sump tion that nor -mals to the ref er ence sur face re main straight af ter de for ma tion. This is the shell equiv a lent to the beam as sump tion that plane sec tions re main plane.
Section Properties 177
In te gra tion in the Plane
Force-de flec tion be hav ior is com puted by in te grat ing the stress-strain be hav ior through the thick ness and over the 1-2 plane of the el e ment. You can spec ify the num ber of in te gra tion points in the thick ness di rec tion of each layer as de scribed above.
For each of these thick ness lo ca tions, in te gra tion in the plane is per formed at the stan dard 2 x 2 Gauss points (co or di nates ±0.577 on a square of size ±1.0). Non lin -ear be hav ior is sam pled only at these points. This is equiv a lent to hav ing two fi bers, lo cated ap prox i mately at the ¼ and ¾ points, in each of the lo cal 1 and 2 di rec tions.
Plot ted or tab u lated stresses at lo ca tions other than the four Gauss points are in ter -po lated or ex trap o lated, and do not nec es sar ily rep re sent the sam pled non lin ear stresses. For this rea son, stresses at the joints may some times ap pear to ex ceed fail -ure stresses.
Example: Non lin ear Shear-Wall, “Realistic” Mod el ing
An im por tant ap pli ca tion for the lay ered shell el e ment is non lin ear shearwall mod -el ing, and it will serve as an ex am ple for other ap pli ca tions. Let's con sider an 18 inch (457 mm) thick ver ti cal wall, with two ver ti cal and two hor i zon tal lay ers of rebar hav ing 3 inch (76 mm) cover from both faces. The two hor i zon tal layers to -gether pro vide a 1% rebar area ra tio, and the two ver ti cal lay ers to -gether pro vide an area ra tio of 2%.
When mod el ing lin ear be hav ior, it is not usu ally nec es sary to in clude the rebar, but it is es sen tial for non lin ear be hav ior. In the sim plest case, the en tire wall sec tion will be con sid ered as non lin ear for both mem brane and bend ing be hav ior, lead ing to the most “re al is tic”, if not the most prac ti cal model. The re quires a lay ered sec -tion with five lay ers:
“Re al is tic” Shear-Wall Model
Layer Type Material th d ang n s11 s22 s12
1 Shell Conc 18.00 0. 0° 5 N N N
2 Shell Rebar 0.09 +6. 0° 1 N - N
3 Shell Rebar 0.09 -6. 0° 1 N - N
4 Shell Rebar 0.18 +6. 90° 1 N - N
5 Shell Rebar 0.18 -6. 90° 1 N - N
178 Section Properties
For the stress com po nents, “N” in di cates non lin ear, “L” in di cates lin ear, and “-”
in di cates in ac tive.
Note that for the rebar, s11 is al ways non lin ear. Ver ti cal rebar is de fined by set ting the ma te rial an gle to 90°, which aligns it with the shell lo cal2 axis. Hence the ver ti -cal rebar stress s11 cor re sponds to shell s22.
Also note that for the rebar, s12 is set to be non lin ear. This al lows the rebar to carry shear when the con crete cracks. This can taken to rep re sent dowel ac tion, al though no in for ma tion on ac tual dowel be hav ior is pres ent in the model, so it is only an ap prox i ma tion. You must use your en gi neer ing judge ment to de ter mine if this ap -proach is suit able to your needs. The most con ser va tive ap -proach is to set the rebar stress com po nent s12 to be in ac tive.
Example: Non lin ear Shear-Wall, “Practical” Mod el ing
The five-layer model above seems re al is tic, but pres ents many fail ure mech a nisms that may cloud the en gi neer ing in for ma tion re quired for per for mancebased de sign. When ever pos si ble, the sim plest model should be used to meet the en gi neer -ing goals. Do -ing this will make the anal y sis run faster and make the in ter pre ta tion of re sults eas ier.
With this in mind, a more prac ti cal model is pre sented be low, with only the ver ti cal mem brane stresses taken to be non lin ear. Such a model may be suit able for taller shear walls where col umn-like be hav ior gov erns:
“Prac ti cal” Shear-Wall Model
Layer Type Material th d ang n s11 s22 s12
1 Membr Conc 18.00 0. 0° 1 L N L
2 Membr Rebar 0.18 +6. 90° 1 N -
-3 Membr Rebar 0.18 -6. 90° 1 N -
-4 Plate Conc 16.00 0. 0° 2 L L L
In this model, only mem brane be hav ior is non lin ear, and only for the ver ti cal stress com po nent s22. This cor re sponds to rebar stress com po nent s11 when the ma te rial an gle is 90°.
It is gen er ally not nec es sary to in clude rebar for lin ear be hav ior, so the hor i zon tal rebar is omit ted, and the rebar shear stress com po nent s12 is set to be in ac tive.
Section Properties 179
Out-of-plane be hav ior is as sumed to be lin ear, so a sin gle con crete plate layer is used. The thick ness has been re duced to ac count for crack ing with out ex plicit non -lin ear mod el ing. Plate bend ing stiff ness is pro por tional to the cube of the thick ness.
Example: In-fill Panel
There are many ways to model an infill panel. Two ap proaches will be pre sented here, both in tended to rep re sent mem brane shear re sis tance only. The simplest is a sin gle layer of con crete ma te rial car ry ing only mem brane shear stress, as shown in the fol low ing model:
Infill Wall - Sim ple Shear Model
Layer Type Material th d ang n s11 s22 s12
1 Membr Conc 18.00 0. 0° 1 - - N
In the sec ond model, the con crete is as sumed to act as com pres sion struts along the two di ag o nals. For a square panel, these two struts would act at ma te rial an gles of
±45°, as shown in the fol low ing model:
Infill Wall - Com pres sion Strut Model
Layer Type Material th d ang n s11 s22 s12
1 Membr Conc 18.00 0. 45° 1 N -
-2 Membr Conc 18.00 0. -45° 1 N -
Other pos si bil i ties ex ist. For both mod els, there is no ver ti cal or hor i zon tal mem -brane stiff ness, and no plate-bend ing stiff ness. There fore, these mod els should only be used when the el e ment is com pletely sur rounded by frame or other sup port ing el e ments, and the elements should not be meshed.
Summary
As these ex am ples show, you have con sid er able flex i bil ity to cre ate lay ered shell sec tions to rep re sent a va ri ety of lin ear and non lin ear be hav ior. The sim plest model that ac com plishes the en gi neer ing goals should be used. Even when more com pli cated mod els may be war ranted, it is rec om mended to start with sim ple, mostly lin ear mod els, and in crease the level of com plex ity and nonlinearity as you gain ex pe -ri ence with your model and its be hav ior.
180 Section Properties