The Solid ele ment is used to model three- dimensional solid struc tures.
Advanced Topics
• Over view
• Joint Con nec tivity
• De grees of Free dom
• Lo cal Co or di nate Sys tem
• Ad vanced Lo cal Co or di nate Sys tem
• Stresses and Strains
• Solid 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
171
Overview
The Solid ele ment is an eight node ele ment for mod el ing three dimensional struc -tures and sol ids. It is based upon an iso para met ric for mu la tion that in cludes nine op tional in com pati ble bend ing modes.
The in com pati ble bend ing modes sig nifi cantly im prove the bend ing be hav ior of the ele ment if the ele ment ge ome try is of a rec tan gu lar form. Im proved be hav ior is ex hib ited even with non- rectangular ge ome try.
Each Solid 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. Temp-erature- dependent, ani sotropic 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 faces; pore pres sure within the ele ment; and loads due to tem pera ture change.
An 2 x 2 x 2 nu meri cal in te gra tion scheme is used for the Solid. Stresses in the ele ment lo cal co or di nate sys tem are evalu ated at the in te gra tion points and ex trapo lated to the joints of the ele ment. An ap proxi 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 ele 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 ele ment ap -proxi 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 ele ment mesh.
Joint Connectivity
Each Solid ele ment has six quad ri lat eral faces, with a joint lo cated at each of the eight cor ners as shown in Figure 40 (page 173). It is im por tant to note the rela tive po si tion of the eight joints: the paths j1j2j3 and j5j6j7 should ap pear coun ter -clock wise when viewed along the di rec tion from j5 to j1. Mathe mati cally stated, the three vec tors:
• V12, from joints j1 to j2,
• V13, from joints j1 to j3,
• V15, from joints j1 to j5,
must form a posi tive tri ple prod uct, that is:
(V12 ´V13)×V15>0
172 Overview
The lo ca tions of the joints should be cho sen to meet the fol low ing geo met ric con di -tions:
• The in side an gle at each cor ner of the faces must be less than 180°. Best re sults will be ob tained when these an gles are near 90°, or at least in the range of 45° to 135°.
• The as pect ra tio of an ele ment should not be too large. This is the ra tio of the long est di men sion of the ele ment to its short est di men sion. Best re sults are ob -tained for as pect ra tios near unity, or at least less than four. The as pect ra tio should not ex ceed ten.
These con di tions can usu ally be met with ade quate mesh re fine ment.
Degrees of Freedom
The Solid 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. This ele ment con trib utes stiff ness to all of these trans la tional de grees of free dom.
Degrees of Freedom 173
j1
j2 j3
j4
j5
j6 j7
j8
Face 1
Face 2 Face 3
Face 4
Face 5 Face 6
Figure 40
Solid Element Joint Connectivity and Face Definitions
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
Each Solid ele ment has its own ele ment lo cal co or di nate sys tem used to de fine Ma te rial prop er ties, loads and out put. The axes of this lo cal sys tem are de noted 1, 2 and 3. By de fault these axes are iden ti cal to the global X, Y, and Z axes, re spec -tively. Both sys tems are right-handed co or di nate sys tems.
The de fault lo cal co or di nate sys tem is ad e quate for most sit u a tions. How ever, for cer tain mod el ing pur poses it may be use ful to use el e ment lo cal co or di nate sys tems that fol low the ge om e try of the struc ture.
For more in for ma tion:
• See Topic “Up ward and Hor i zon tal Di rec tions” (page 13) in Chap ter “Co or di -nate Sys tems.”
• See Topic “Ad vanced Lo cal Co or di nate Sys tem” (page 174) in this Chap ter.
Advanced Local Coordinate System
By de fault, the element lo cal 1-2-3 co or di nate sys tem is iden ti cal to the global X-Y-Z co or di nate sys tem, as de scribed in the pre vi ous topic. In cer tain mod el ing sit u a tions it may be use ful to have more con trol over the spec i fi ca tion of the lo cal co or di nate sys tem.
A va ri ety of meth ods are avail able to de fine a solidel e ment lo cal co or di nate sys tem. These may be used sep a rately or to gether. Lo cal co or di nate axes may be de fined to be par al lel to ar bi trary co or di nate di rec tions in an ar bi trary co or di nate sys -tem or to vec tors be tween pairs of joints. In ad di tion, the lo cal co or di nate sys -tem may be spec i fied by a set of three el e ment co or di nate an gles. These meth ods are de -scribed in the subtopics that fol low.
For more in for ma tion:
• See Chap ter “Co or di nate Sys tems” (page 11).
• See Topic “Lo cal Co or di nate Sys tem” (page 174) in this Chap ter.
174 Local Coordinate System
Reference Vectors
To de fine a solid-element lo cal co or di nate sys tem you must spec ify two ref er ence vec tors that are par al lel to one of the lo cal co or di nate planes. The axis ref er ence vec tor, Va, must be par al lel to one of the lo cal axes (I = 1, 2, or 3) in this plane and have a pos i tive pro jec tion upon that axis. The plane ref er ence vec tor, Vp, must have a pos i tive pro jec tion upon the other lo cal axis (j = 1, 2, or 3, but I ¹ j) in this plane, but need not be par al lel to that axis. Hav ing a pos i tive pro jec tion means that the pos i tive di rec tion of the ref er ence vec tor must make an an gle of less than 90°
with the pos i tive di rec tion of the lo cal axis.
To gether, the two ref er ence vec tors de fine a lo cal axis, I, and a lo cal plane, i-j.
From this, the pro gram can de ter mine the third lo cal axis, k, us ing vec tor al ge bra.
For ex am ple, you could choose the axis ref er ence vec tor par al lel to lo cal axis 1 and the plane ref er ence vec tor par al lel to the lo cal 1-2 plane (I = 1, j = 2). Al ter na tively, you could choose the axis ref er ence vec tor par al lel to lo cal axis 3 and the plane ref -er ence vec tor par al lel to the lo cal 3-2 plane (I = 3, j = 2). You may choose the plane that is most con ve nient to de fine us ing the pa ram e ter lo cal, which may take on the val ues 12, 13, 21, 23, 31, or 32. The two dig its cor re spond to I and j, re spec tively.
The de fault is value is 31.
Defining the Axis Reference Vector
To de fine the axis ref er ence vec tor, you must first spec ify or use the de fault val ues for:
• A co or di nate di rec tion axdir (the de fault is +Z)
• A fixed co or di nate sys tem csys (the de fault is zero, in di cat ing the global co or -di nate sys tem)
You may op tion ally spec ify:
• A pair of joints, axveca and axvecb (the de fault for each is zero, in di cat ing the cen ter of the el e ment). If both are zero, this op tion is not used.
For each element, the axis ref er ence vec tor is de ter mined as fol lows:
1. A vec tor is found from joint axveca to joint axvecb. If this vec tor is of fi nite length, it is used as the ref er ence vec tor Va
Advanced Local Coordinate System 175
2. Oth er wise, the co or di nate di rec tion axdir is eval u ated at the cen ter of the el e -ment in fixed co or di nate sys tem csys, and is used as the ref er ence vec tor Va
Defining the Plane Reference Vector
To de fine the plane ref er ence vec tor, you must first spec ify or use the de fault val ues for:
• A pri mary co or di nate di rec tion pldirp (the de fault is +X)
• A sec ond ary co or di nate di rec tion pldirs (the de fault is +Y). Di rec tions pldirs and pldirp should not be par al lel to each other un less you are sure that they are not par al lel to lo cal axis 1
• A fixed co or di nate sys tem csys (the de fault is zero, in di cat ing the global co or -di nate sys tem). This will be the same co or -di nate sys tem that was used to de fine the axis ref er ence vec tor, as de scribed above
You may op tion ally spec ify:
• A pair of joints, plveca and plvecb (the de fault for each is zero, in di cat ing the cen ter of the el e ment). If both are zero, this op tion is not used.
For each element, the plane ref er ence vec tor is de ter mined as fol lows:
1. A vec tor is found from joint plveca to joint plvecb. If this vec tor is of fi nite length and is not par al lel to lo cal axis I, it is used as the ref er ence vec tor Vp 2. Oth er wise, the pri mary co or di nate di rec tion pldirp is eval u ated at the cen ter of
the el e ment in fixed co or di nate sys tem csys. If this di rec tion is not par al lel to lo cal axis I, it is used as the ref er ence vec tor Vp
3. Oth er wise, the sec ond ary co or di nate di rec tion pldirs is eval u ated at the cen ter of the el e ment in fixed co or di nate sys tem csys. If this di rec tion is not par al lel to lo cal axis I, it is used as the ref er ence vec tor Vp
4. Oth er wise, the method fails and the anal y sis ter mi nates. This will never hap pen if pldirp is not par al lel to pldirs
A vec tor is con sid ered to be par al lel to lo cal axis I if the sine of the an gle be tween them is less than 10-3.
176 Advanced Local Coordinate System
Determining the Local Axes from the Reference Vectors
The pro gram uses vec tor cross prod ucts to de ter mine the lo cal axes from the ref er -ence vec tors. The three axes are rep re sented by the three unit vec tors V1, V2 and V3, re spec tively. The vec tors sat isfy the cross-prod uct re la tion ship:
V1=V2´V3
The lo cal axis Vi is given by the vec tor Va af ter it has been nor mal ized to unit length.
The re main ing two axes, Vj and Vk, are de fined as fol lows:
• If I and j per mute in a pos i tive sense, i.e., lo cal = 12, 23, or 31, then:
Vk =Vi ´Vp and Vj =Vk ´Vi
• If I and j per mute in a neg a tive sense, i.e., lo cal = 21, 32, or 13, then:
Vk =Vp ´Vi and Vj =Vi ´Vk
Advanced Local Coordinate System 177
V is parallel to axveca-axvecba V is parallel to plveca-plvecbp
V = V3 a
V = V x V2 3 p All vectors normalized to unit length.
V = V x V1 2 3
X Y
Z
Global
axveca
axvecb plveca
plvecb
Plane 3-1 j
V3 V2
V1
Va
Vp
Figure 41
Example of the Determination of the Solid Element Local Coordinate System Using Reference Vectors for local=31. Point j is the Center of the Element.
An ex am ple show ing the de ter mi na tion of the element lo cal co or di nate sys tem us -ing ref er ence vec tors is given in Figure 41 (page 177).