A Lo cal Con straint does not have its own lo cal co or di nate sys tem. The con straint equa tions are writ ten in terms of con strained joint lo cal co or di nate sys tems, which may dif fer. The axes of these co or di nate sys tems are de noted 1, 2, and 3.
Selected Degrees of Freedom
For each Lo cal Con straint you may spec ify a list, ldofs, of up to six de grees of free -dom in the joint lo cal co or di nate sys tems that are to be con strained. The de grees of free dom are in di cated as U1, U2, U3, R1, R2, and R3.
Constraint Equations
The con straint equa tions re late the dis place ments at any two con strained joints (sub scripts I and j) in a Lo cal Con straint. These equa tions are ex pressed in terms of the trans la tions (u1, u2, and u3) and the ro ta tions (r1, r2, and r3), all taken in joint lo cal co or di nate sys tems. The equa tions used de pend upon the se lected de grees of free -dom and their signs. Some im por tant cases are described next.
Axisymmetry
Axisymmetry is a type of sym me try about a line. It is best de scribed in terms of a cy lin dri cal co or di nate sys tem hav ing its Z axis on the line of sym me try. The struc -ture, load ing, and dis place ments are each said to be axisymmetric about a line if they do not vary with an gu lar po si tion around the line, i.e., they are in de pend ent of the angular coordinate CA.
To en force axisymmetry us ing the Lo cal Con straint:
• Model any cy lin dri cal sec tor of the struc ture us ing any axisymmetric mesh of joints and el e ments
• As sign each joint a lo cal co or di nate sys tem such that lo cal axes 1, 2, and 3 cor -re spond to the co or di nate di -rec tions +CR, +CA, and +CZ, -re spec tively
• For each axisymmetric set of joints (i.e., hav ing the same co or di nates CR and CZ, but dif fer ent CA), de fine a Lo cal Con straint us ing all six de grees of free -dom: U1, U2, U3, R1, R2, and R3
• Re strain joints that lie on the line of sym me try so that, at most, only ax ial trans -la tions (U3) and ro ta tions (R3) are per mit ted
The cor re spond ing con straint equa tions are:
62 Local Constraint
u1j = u1i
u2j = u2i
u3j = u3i
r1i = r1j
r2i = r2j
r3i = r3j
The nu meric sub scripts re fer to the cor re spond ing joint lo cal co or di nate systems.
Cyclic symmetry
Cy clic sym me try is an other type of sym me try about a line. It is best de scribed in terms of a cy lin dri cal co or di nate sys tem hav ing its Z axis on the line of sym me try.
The struc ture, load ing, and dis place ments are each said to be cy cli cally sym met ric about a line if they vary with an gu lar po si tion in a re peated (periodic) fashion.
To en force cy clic sym me try us ing the Lo cal Con straint:
• Model any num ber of ad ja cent, rep re sen ta tive, cy lin dri cal sec tors of the struc -ture; de note the size of a sin gle sec tor by the an gle q
• As sign each joint a lo cal co or di nate sys tem such that lo cal axes 1, 2, and 3 cor -re spond to the co or di nate di -rec tions +CR, +CA, and +CZ, -re spec tively
• For each cy cli cally sym met ric set of joints (i.e., hav ing the same co or di nates CR and CZ, but with co or di nate CA dif fer ing by mul ti ples of q), de fine a Lo cal Con straint us ing all six de grees of free dom: U1, U2, U3, R1, R2, and R3.
• Re strain joints that lie on the line of sym me try so that, at most, only ax ial trans -la tions (U3) and ro ta tions (R3) are per mit ted
The cor re spond ing con straint equa tions are:
u1j = u1i
u2j = u2i
u3j = u3i
r1i = r1j
r2i = r2j
r3i = r3j
Local Constraint 63
The nu meric sub scripts re fer to the cor re spond ing joint lo cal co or di nate systems.
For ex am ple, sup pose a struc ture is com posed of six iden ti cal 60° sec tors, iden ti -cally loaded. If two ad ja cent sec tors were mod eled, each Lo cal Con straint would ap ply to a set of two joints, ex cept that three joints would be con strained on the sym me try planes at 0°, 60°, and 120°.
If a sin gle sec tor is mod eled, only joints on the sym me try planes need to be con -strained.
Symmetry About a Point
Sym me try about a point is best de scribed in terms of a spher i cal co or di nate sys tem hav ing its Z axis on the line of sym me try. The struc ture, load ing, and dis place ments are each said to be sym met ric about a point if they do not vary with an gu lar po si tion about the point, i.e., they are in de pend ent of the an gu lar co or di nates SB and SA.
Ra dial trans la tion is the only dis place ment component that is permissible.
To en force sym me try about a point us ing the Lo cal Con straint:
• Model any spher i cal sec tor of the struc ture us ing any sym met ric mesh of joints and el e ments
• As sign each joint a lo cal co or di nate sys tem such that lo cal axes 1, 2, and 3 cor -re spond to the co or di nate di -rec tions +SB, +SA, and +SR, -re spec tively
• For each sym met ric set of joints (i.e., hav ing the same co or di nate SR, but dif -fer ent co or di nates SB and SA), de fine a Lo cal Con straint us ing only de gree of freedom U3
• For all joints, re strain the de grees of free dom U1, U2, R1, R2, and R3
• Fully re strain any joints that lie at the point of sym me try The cor re spond ing con straint equa tions are:
u3j = u3i
The nu meric sub scripts re fer to the cor re spond ing joint lo cal co or di nate systems.
It is also pos si ble to de fine a case for sym me try about a point that is sim i lar to cy clic sym me try around a line, e.g., where each octant of the struc ture is iden ti cal.
64 Local Constraint
Welds
A Weld can be used to con nect to gether dif fer ent parts of the struc tural model that were de fined us ing sep a rate meshes. A Weld is not a sin gle Con straint, but rather is a set of joints from which the pro gram will au to mat i cally gen er ate mul ti ple Body Con straints to con nect to gether coincident joints.
Joints are con sid ered to be co in ci dent if the dis tance be tween them is less than or equal to a tol er ance, tol, that you spec ify. Set ting the tol er ance to zero is per mis si -ble but is not recommended.
One or more Welds may be de fined, each with its own tol er ance. Only the joints within each Weld will be checked for co in ci dence with each other. In the most com mon case, a sin gle Weld is de fined that con tains all joints in the model; all co in ci dent groups of joints will be welded. How ever, in sit u a tions where struc tural dis -con ti nu ity is de sired, it may be nec es sary to pre vent the weld ing of some co in ci dent joints. This may be fa cil i tated by the use of multiple Welds.
Figure 12 (page 65) shows a model de vel oped as two sep a rate meshes, A and B.
Joints 121 through 125 are as so ci ated with mesh A, and Joints 221 through 225 are as so ci ated with mesh B. Joints 121 through 125 share the same lo ca tion in space as Joints 221 through 225, re spec tively. These are the in ter fac ing joints be tween the two meshes. To con nect these two meshes, a sin gle Weld can be de fined con tain ing all joints, or just joints 121 through 125 and 221 through 225. The pro gram would
Welds 65
Mesh B 221
222
223 224
125 123 124
122 121
Mesh A
225
Figure 12
Use of a Weld to Connect Separate Meshes at Coincident Joints
gen er ate five Body Con straints, each con tain ing two joints, re sult ing in an integrated model.
It is per mis si ble to in clude the same joint in more than one Weld. This could re sult in the joints in dif fer ent Welds be ing con strained to gether if they are co in ci dent with the com mon joint. For ex am ple, sup pose that Weld 1 con tained joints 1,2, and 3, Weld 2 con tained joints 3, 4, and 5. If joints 1, 3, and 5 were co in ci dent, joints 1 and 3 would be con strained by Weld 1, and joints 3 and 5 would be con strained by Weld 2. The pro gram would cre ate a sin gle Body Con straint con tain ing joints 1, 3, and 5. One the other hand, if Weld 2 did not con tain joint 3, the pro gram would only gen er ate a Body Con straint con tain ing joint 1 and 3 from Weld 1; joint 5 would not be constrained.
For more in for ma tion, see Topic “Body Con straint” (page 51) in this Chap ter.