UJOINT

Connection type UJOINT joins the position of two nodes and provides a universal constraint between their rotational degrees of freedom. Connection type UJOINT cannot be used in two-dimensional or axisymmetric analysis.

Related Topics
Connector elements
Connector element library
In Other Guides
*CONNECTOR BEHAVIOR
*CONNECTOR SECTION

ProductsAbaqus/StandardAbaqus/ExplicitAbaqus/CAE

Description

Figure 1. Connection type UJOINT.

Connection type UJOINT imposes kinematic constraints and uses local orientation definitions equivalent to combining connection types JOIN and UNIVERSAL.

The connector constraint forces and moments reported as connector output depend strongly on the order of the nodes and location of the nodes in the connector (see Connector behavior). Since the kinematic constraints are enforced at node b (the second node of the connector element), the reported forces and moments are the constraint forces and moments applied at node b to enforce the UJOINT constraint. Thus, in most cases the connector output associated with a UJOINT connection is best interpreted when node b is located at the center of the device enforcing the constraint. This choice is essential when moment-based friction is modeled in the connector since the contact forces are derived from the connector forces and moments, as illustrated below. Proper enforcement of the kinematic constraints is independent of the order or location of the nodes.

Friction

Predefined Coulomb-like friction in the UJOINT connection relates the kinematic constraint forces and moments in the connector to friction moments about the unconstrained rotations (about the two directions of the connection cross). The UJOINT connection type consists of four hinge-like connections placed at the four ends of the connection cross (see Figure 1) that generate frictional moments about the cross axes. The frictional moments in each of these hinges are computed in a fashion similar to the HINGE connection.

The constraint forces and moments are used first to compute a reaction force, Fr (the magnitude of the constraint forces enforcing the JOIN constraint), and a “twisting” constraint moment, Mtwist (the magnitude of the constraint moment enforcing the UNIVERSAL connection), as follows:

Fr=f12+f22+f32,
Mtwist=|m2|.

The two cross directions are given by e1a and e3b. The constraint moment, Mtwist, acts about an axis perpendicular to the connection cross given by ecross=e1a×e3b. Both Fr and Mtwist are considered to be applied at the center of the connection cross. The constraint moment, Mtwist, produces in each of the four hinges a bending-like moment about ecross:

Mtwisthinge=αtwistMtwist

and a transverse force in the cross plane

Ftwisthinge=βtwistMtwistLa,

where La represents a characteristic length of the cross arm between the center of the cross and the ends of the cross. The scaling factors αtwist and βtwist are nonlinear functions of the slenderness of the cross axes (the aspect ratio La/Rp, where Rp is the average radius of the four pins at the ends of the connection cross): they can be approximated by assuming the cross arm with rigid bodies for infinitely small aspect ratios, with Timoshenko beams for small aspect ratios (less than 20), and with Euler-Bernoulli beams for slender axes (large aspect ratios). Abaqus chooses the appropriate values automatically based on the user-specified geometric constants La and Rp. Figure 2 illustrates the evolution of the scaling factors as a function of the aspect ratio: as the aspect ratio approaches 0.0, αtwist approaches 0.0 and βtwist approaches 0.25; for large aspect ratios, αtwist approaches 0.125 and βtwist approaches 0.375.

Figure 2. Scaling factors in the UJOINT connection.

The constraint force, Fr, can be decomposed into axial forces along the two axes of the connection cross and a “bending” force perpendicular to the connection cross plane:

Faxial1=Fre1a,
Faxial3=Fre3b,and
Fbend=Frecross,

where

Fr=f1e1a+f2e2a+f3e3a.

Friction in the UJOINT connection is the superposition of four HINGE-like frictional effects due to rotations about the two cross axes. Since the rotations about the local 1- and 3-directions are the only possible relative motions in the connection, the frictional effects (CSM1 and CSM3) are formally written in terms of moments generated by tangential tractions and moments generated by contact forces. In the following equations subscript 1 refers to frictional effects about the local 1-direction, and subscript 3 refers to frictional effects about the local 3-direction. The frictional effects are written as follows:

Φ1=P1(f)-μMN10,    and
Φ3=P3(f)-μMN30,

where the potentials P1(f) and P3(f) represent the moment magnitudes of the frictional tangential tractions in the connector in directions tangent to the cylindrical surface on which contact occurs, MN1 and MN3 are the friction-producing normal moments on the same cylindrical surface, and μ is the friction coefficient. Frictional stick occurs in a particular direction if Φ1<0 or Φ3<0; and sliding occurs if Φ1=0 or Φ3=0, in which case the friction moments are μMN1 and μMN3.

The normal moments MN1 and MN3 are the sums of magnitude measures of force-producing connector moments, MC1=g1(f) and MC3=g3(f), and self-equilibrated internal contact moments (such as from a press-fit assembly), MC1int and MC3int, respectively:

MN1=2|MC1|+|MC1int|=2|g1(f)|+|MC1int|,    and
MN3=2|MC3|+|MC3int|=2|g3(f)|+|MC3int|.

The factor of two in the above equations comes from the fact that there are two hinges on each cross direction.

The moment magnitudes MC1 and MC3 are defined by summing the following contributions:

  • moment from axial forces, Faxialhinge1Ra and Faxialhinge3Ra, where Faxialhinge1=αaxialFaxial1, Faxialhinge3=αaxialFaxial3, and Ra is an average effective friction arm associated with the constraint force in the axial direction in each of the pins (if Ra is 0.0, Faxialhinge1 and Faxialhinge3 are ignored); and

  • moment from normal forces, Fn1Rp and Fn3Rp, where Fn1 and Fn3 are themselves sums of the following contributions:

    • transverse force contributions, Ftotalhinge1 (the magnitude of the total transverse force in the two hinges along the e1a-direction) and Ftotalhinge3 (the magnitude of the total transverse force in the two hinges along the e3b-direction):

      Ftotalhinge1=(Fbendhinge)2+(Ftwisthinge)2+(Ftransv1hinge)2,    and
      Ftotalhinge3=(Fbendhinge)2+(Ftwisthinge)2+(Ftransv3hinge)2,

      where Fbendhinge=Fbend4, Ftwisthinge is defined above, Ftransv1hinge=βaxialFaxial3, and Ftransv3hinge=βaxialFaxial1; and

    • force contributions from “bending,” Ftotalbend, obtained by scaling the total bending moment, Mtotalhinge (the magnitude of the total bending moment on each of the four hinges), by a length factor, as follows:

      Mtotalhinge=Mbendhinge2+Mtwisthinge2,
      Ftotalbend=2MtotalhingeLs,

      where Mbendhinge=18FbendLa, Mtwisthinge is defined above, and Ls represents a characteristic overlapping length between the pins and their sleeves. If Ls is 0.0, Mtotalhinge is ignored.

Thus,

MC1=g1(f)=Faxialhinge1Ra+Fn1Rp=Faxialhinge1Ra+(Ftotalhinge1+Ftotalbend)Rp=αaxialFaxial1Ra+Rp(Fbend4)2+(Ftwisthinge)2+(βaxialFaxial3)2+2RpLs(18FbendLa)2+(Mtwisthinge)2,    and
MC3=g3(f)=Faxialhinge3Ra+Fn3Rp=Faxialhinge3Ra+(Ftotalhinge3+Ftotalbend)Rp=αaxialFaxial3Ra+Rp(Fbend4)2+(Ftwisthinge)2+(βaxialFaxial1)2+2RpLs(18FbendLa)2+(Mtwisthinge)2.

The moment magnitudes of the frictional tangential tractions are P1(f)=|m1| and P3(f)=|m3|.

Summary

UJOINT
Basic, assembled, or complex: Assembled
Kinematic constraints: JOIN + UNIVERSAL
Constraint force and moment output: f1,f2,f3,m2
Available components: ur1,ur3
Kinetic force and moment output: m1,m3
Orientation at a: Required
Orientation at b: Optional
Connector stops: θ1minγθ1max
  θ3minγθ3max
Constitutive reference lengths: θ1ref,θ3ref
Predefined friction parameters: Required: Rp, La; optional: Ra, Ls, MC1int, MC3int
Contact moments for predefined friction: MC1, MC3