The Plane el e ment is used to model plane-stress and plane-strain be hav ior in two-di men sional sol ids. The Plane el e ment/ob ject is one type of area ob ject. De - pending on the type of sec tion prop er ties you as sign to an area, the ob ject could also be used to model shell and axisymmetric solid be hav ior. These types of el e ments are dis cussed in the pre vi ous and fol low ing Chap ters.
Advanced Topics
• Over view
• Joint Con nec tivity • De grees of Free dom • Lo cal Co or di nate Sys tem • Stresses and Strains • Sec tion Prop erties • Mass
• Self- Weight Load • Grav ity Load
• Sur face Pres sure Load • Pore Pres sure Load
• Tem pera ture Load • Stress Out put
Overview
The Plane el e ment is a three- or four-node el e ment for mod el ing two-di men sional sol ids of uni form thick ness. It is based upon an isoparametric for mu la tion that in - cludes four op tional in com pat i ble bend ing modes. The el e ment should be pla nar; if it is not, it is for mu lated for the pro jec tion of the el e ment upon an av er age plane calculated for the el e ment.
The in com pat i ble bend ing modes sig nif i cantly im prove the bend ing be hav ior of the el e ment if the el e ment ge om e try is of a rect an gu lar form. Im proved be hav ior is ex hib ited even with non-rect an gu lar ge om e try.
Struc tures that can be mod eled with this el e ment in clude: • Thin, pla nar struc tures in a state of plane stress • Long, pris matic struc tures in a state of plane strain
The stresses and strains are as sumed not to vary in the thick ness di rec tion. For plane-stress, the el e ment has no out-of-plane stiff ness. For plane-strain, the el e - ment can sup port loads with anti-plane shear stiff ness.
Each Plane 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 ter pret ing out put. Temperature- dependent, or tho tropic ma te rial prop er ties are al lowed. Each ele ment may be loaded by grav ity (in any di - rec tion); sur face pres sure on the side faces; pore pres sure within the ele ment; and loads due to tem pera ture change.
An 2 x 2 nu mer i cal in te gra tion scheme is used for the Plane. Stresses in the el e ment lo cal co or di nate sys tem are eval u ated at the 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 stresses 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 the fi nite el 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.
Joint Connectivity
The joint con nec tiv ity and face def i ni tion is iden ti cal for all area ob jects, i.e., the Shell, Plane, and Asolid elements. See Topic “Joint Con nec tiv ity” (page 129) in Chap ter “The Shell El e ment” for more in for ma tion.
The Plane el e ment is in tended to be pla nar. If you de fine a four-node el e ment that is not pla nar, an av er age plane will be fit through the four joints, and the pro jec tion of the el e ment onto this plane will be used.
Degrees of Freedom
The Plane ele ment ac ti vates the three trans la tional de grees of free dom at each of its con nected joints. Ro ta tional de grees of free dom are not ac ti vated.
The plane-stress el e ment con trib utes stiff ness only to the de grees of free dom in the plane of the ele ment. It is nec es sary to pro vide re straints or other sup ports for the trans la tional de grees of free dom that are nor mal to this plane; oth er wise, the struc - ture will be un sta ble.
The plane-strain el e ment mod els anti-plane shear, i.e., shear that is nor mal to the plane of the el e ment, in ad di tion to the in-plane behavior. Thus stiff ness is cre ated for all three translational de grees of freedom.
See Topic “De grees of Free dom” (page 29) in Chap ter “Joints and De grees of Free - dom” for more in for ma tion.
Local Coordinate System
The el e ment lo cal co or di nate sys tem is iden ti cal for all area ob jects, i.e., the Shell, Plane, and Asolid el e ments. See Topics “Lo cal Co or di nate Sys tem” (page 132) and “Ad vanced Lo cal Co or di nate Sys tem” (page 133) in Chap ter “The Shell El e ment” for more in for ma tion.
Stresses and Strains
The Plane ele ment mod els the mid- plane of a struc ture hav ing uni form thick ness, and whose stresses and strains do not vary in the thick ness di rec tion.
Plane-stress is ap pro pri ate for struc tures that are thin com pared to their pla nar di - men sions. The thick ness nor mal stress (s33) is as sumed to be zero. The thick ness nor mal strain (e33) may not be zero due to Pois son ef fects. Trans verse shear stresses (s12, s13) and shear strains (g12, g13) are as sumed to be zero. Dis place - ments in the thick ness (lo cal 3) di rec tion have no ef fect on the el e ment.
Plane-strain is ap pro pri ate for struc tures that are thick com pared to their pla nar di - men sions. The thick ness nor mal strain (e33) is as sumed to be zero. The thick ness nor mal stress (s33) may not be zero due to Pois son ef fects. Trans verse shear stresses (s12, s13) and shear strains (g12, g13) are de pend ent upon displacements in the thick ness (lo cal 3) di rec tion.
See Topic “Stresses and Strains” (page 69) in Chap ter “Ma te rial Prop er ties” for more in for ma tion.
Section Properties
A Plane Sec tion is a set of ma te rial and geo met ric prop er ties that de scribe the cross-sec tion of one or more Plane el e ments. Sec tions are de fined in de pend ently of the Plane el e ments, and are as signed to the area ob jects.
Section Type
When de fin ing an area sec tion, you have a choice of three ba sic el e ment types: • Plane (stress or strain) – the sub ject of this Chap ter, a two-di men sional solid,
with translational de grees of free dom, ca pa ble of sup port ing forces but not mo - ments.
• Shell – shell, plate, or mem brane, with translational and ro ta tional de grees of free dom, ca pa ble of sup port ing forces and mo ments. This el e ment is cov ered in Chap ter “The Shell El e ment” (page 127).
• Asolid – axisymmetric solid, with translational de grees of free dom, ca pa ble of sup port ing forces but not mo ments. This el e ment is cov ered in Chap ter “The Asolid El e ment” (page 159).
For Plane sec tions, you may choose one of the fol low ing sub-types of be hav ior: • Plane stress
• Plane strain, in clud ing anti-plane shear
Material Properties
The ma te rial prop er ties for each Plane el e ment are spec i fied by ref er ence to a pre vi - ously-de fined Ma te rial. Orthotropic prop er ties are used, even if the Ma te rial se - lected was de fined as anisotropic. The ma te rial prop er ties used by the Plane el e - ment are:
• The moduli of elas tic ity, e1, e2, and e3 • The shear modu lus, g12
• For plane-strain only, the shear moduli, 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, a2, and a3 • 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, u23, and a3 are not used for plane stress. They are used to com pute the thick ness-nor mal stress (s33) in plane strain.
All ma te rial prop er ties (ex cept the den si ties) are ob tained 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 erties” (page 67) for more in for ma tion.
Material Angle
The ma te rial lo cal co or di nate sys tem and the el e ment (Plane Section) 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 37 (page 154). 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 68) in Chap ter “Ma te rial Prop erties” for more in for ma tion.
Thickness
Each Plane Sec tion has a uni form thick ness, th. This may be the ac tual thick ness, par tic u larly for plane-stress el e ments; or it may be a rep re sen ta tive por tion, such as a unit thick ness of an in fi nitely-thick plane-strain el e ment.
The el e ment thick ness is used for cal cu lat ing the el e ment stiff ness, mass, and loads. Hence, joint forces com puted from the el e ment are pro por tional to this thick ness.