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Summary 

<table><tr><td colspan="2">TRANSLATOR</td></tr><tr><td>Basic, assembled, or complex:</td><td>Assembled</td></tr><tr><td>Kinematic constraints:</td><td>SLOT + ALIGN</td></tr><tr><td>Constraint force and moment output:</td><td> $f_{2}, f_{3}, m_{1}, m_{2}, m_{3}$ </td></tr><tr><td>Available components:</td><td> $u_{1}$ </td></tr><tr><td>Kinetic force and moment output:</td><td> $f_{1}$ </td></tr><tr><td>Orientation at  $a$ :</td><td>Required</td></tr><tr><td>Orientation at  $b$ :</td><td>Optional</td></tr><tr><td>Connector stops:</td><td> $l_{1}^{min} \leq l \leq l_{1}^{max}$ </td></tr><tr><td>Constitutive reference lengths:</td><td> $l_{1}^{ref}$ </td></tr><tr><td>Predefined friction parameters:</td><td>Optional:  $R_{r}, L, F_{C}^{int}$ </td></tr><tr><td>Contact force for predefined friction:</td><td> $F_{C}$ </td></tr></table>

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# 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.

![](images/page-662_4b1e36ce0ccb74ccd4fd26b8d14acdf00245c4956e5e839236fad3a011dcccaa.jpg)

<details>
<summary>text_image</summary>

e₁ᵃ
e₃ᵇ
e₂ᵃ
e₂ᵇ
</details>

Figure 31.1.5–32 Connection type UJOINT.

# Description

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,” Section 31.2.1). 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 31.1.5–32) 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.

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The constraint forces and moments are used first to compute a reaction force, $F _ { r }$ (the magnitude of the constraint forces enforcing the JOIN constraint), and a “twisting” constraint moment, $M _ { t w i s t }$ (the magnitude of the constraint moment enforcing the UNIVERSAL connection), as follows:

$$
F _ {r} = \sqrt {f _ {1} ^ {2} + f _ {2} ^ {2} + f _ {3} ^ {2}},
$$

$$
M _ {t w i s t} = | m _ {2} |.
$$

The two cross directions are given by ${ \bf e } _ { 1 } ^ { a }$ and $\mathbf { e } _ { 3 } ^ { b }$ . The constraint moment, $M _ { t w i s t }$ , acts about an axis perpendicular to the connection cross given by $\mathbf { e } _ { c r o s s } = \mathbf { e } _ { 1 } ^ { a } \times \mathbf { e } _ { 3 } ^ { b }$ . Both $F _ { r }$ and $M _ { t w i s t }$ are considered to be applied at the center of the connection cross. The constraint moment, $M _ { t w i s t }$ , produces in each of the four hinges a bending-like moment about $\mathbf { e } _ { c r o s s }$ :

$$
M _ {t w i s t} ^ {h i n g e} = \alpha_ {t w i s t} M _ {t w i s t}
$$

and a transverse force in the cross plane

$$
F _ {t w i s t} ^ {h i n g e} = \beta_ {t w i s t} \frac {M _ {t w i s t}}{L _ {a}},
$$

where $L _ { a }$ represents a characteristic length of the cross arm between the center of the cross and the ends of the cross. The scaling factors $\alpha _ { t w i s t }$ and $\beta _ { t w i s t }$ are nonlinear functions of the slenderness of the cross axes (the aspect ratio $L _ { a } / R _ { p }$ , where $R _ { p }$ 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 $L _ { a }$ and $R _ { p }$ . Figure 31.1.5–33 illustrates the evolution of the scaling factors as a function of the aspect ratio: as the aspect ratio approaches $0 . 0 , \alpha _ { t w i s t }$ approaches 0.0 and $\beta _ { t w i s t }$ approaches 0.25; for large aspect ratios, $\alpha _ { t w i s t }$ approaches 0.125 and $\beta _ { t w i s t }$ approaches 0.375. The constraint force, $F _ { r }$ , can be decomposed into axial forces along the two axes of the connection cross and a “bending” force perpendicular to the connection cross plane:

$$
F _ {a x i a l 1} = \mathbf {F} _ {r} \cdot \mathbf {e} _ {1} ^ {a},
$$

$$
F _ {a x i a l 3} = \mathbf {F} _ {r} \cdot \mathbf {e} _ {3} ^ {b}, \mathrm{and}
$$

$$
F _ {b e n d} = \mathbf {F} _ {r} \cdot \mathbf {e} _ {c r o s s},
$$

where

$$
\mathbf {F} _ {r} = f _ {1} \mathbf {e} _ {1} ^ {a} + f _ {2} \mathbf {e} _ {2} ^ {a} + f _ {3} \mathbf {e} _ {3} ^ {a}.
$$

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![](images/page-664_bf2fe17ec9231b16bd05a6a27bade9fc2f8b1904200d90dcede82084d05f04f7.jpg)

<details>
<summary>line</summary>

| Aspect ratio | α_axial | β_twist | α_twist | β_axial |
| ------------ | ------- | ------- | ------- | ------- |
| 0            | 0.25    | 0.25    | 0.25    | 0.00    |
| 10           | 0.45    | 0.35    | 0.10    | 0.05    |
| 20           | 0.48    | 0.37    | 0.12    | 0.08    |
| 30           | 0.49    | 0.38    | 0.12    | 0.09    |
| 40           | 0.49    | 0.38    | 0.12    | 0.09    |
| 50           | 0.49    | 0.38    | 0.12    | 0.09    |
</details>

Figure 31.1.5–33 Scaling factors in the UJOINT connection.

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:

$$
\Phi_ {1} = \mathrm{P} _ {1} (\mathbf {f}) - \mu \mathrm{M} _ {\mathrm{N} _ {1}} \leq 0, \quad \mathrm{and}
$$

$$
\Phi_ {3} = \mathrm{P} _ {3} (\mathbf {f}) - \mu \mathrm{M} _ {\mathrm{N} _ {3}} \leq 0,
$$

where the potentials $\mathrm { P _ { 1 } } ( \mathbf { f } )$ and $\mathrm { P _ { 3 } } ( \mathbf { f } )$ represent the moment magnitudes of the frictional tangential tractions in the connector in directions tangent to the cylindrical surface on which contact occurs, $\mathrm { M } _ { \mathrm { N } _ { 1 } }$ and $\mathrm { M } _ { \mathrm { N } _ { 3 } }$ are the friction-producing normal moments on the same cylindrical surface, and $\mu$ is the friction coefficient. Frictional stick occurs in a particular direction if $\Phi _ { 1 } < 0$ or $\Phi _ { 3 } ~ < ~ 0 .$ ; and sliding occurs if $\Phi _ { 1 } = 0$ or $\Phi _ { 3 } = 0$ , in which case the friction moments are $\mu \mathrm { { M } _ { N _ { 1 } } }$ and $\mu \mathrm { { M _ { N _ { 3 } } } }$ .

The normal moments $\mathrm { M } _ { \mathrm { N } _ { 1 } }$ and $\mathrm { M } _ { \mathrm { N } _ { 3 } }$ are the sums of magnitude measures of force-producing connector moments, $\mathrm { M } _ { \mathrm { C } _ { 1 } } = g _ { 1 } ( \mathbf { f } )$ and $\mathrm { { M } } _ { \mathrm { { C } _ { 3 } } } = g _ { 3 } ( \mathbf { f } )$ , and self-equilibrated internal contact moments (such as from a press-fit assembly), $\mathrm { M } _ { \mathrm { C _ { 1 } } } ^ { \mathrm { i n t } }$ and $\mathrm { M } _ { \mathrm { C _ { 3 } } } ^ { \mathrm { i n t } }$ , respectively:

$$
\mathrm{M} _ {\mathrm{N} _ {1}} = 2 | \mathrm{M} _ {\mathrm{C} _ {1}} | + | \mathrm{M} _ {\mathrm{C} _ {1}} ^ {\mathrm{int}} | = 2 | g _ {1} (\mathbf {f}) | + | \mathrm{M} _ {\mathrm{C} _ {1}} ^ {\mathrm{int}} |, \quad \mathrm{and}
$$

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$$
\mathrm{M} _ {\mathrm{N} _ {3}} = 2 | \mathrm{M} _ {\mathrm{C} _ {3}} | + | \mathrm{M} _ {\mathrm{C} _ {3}} ^ {\mathrm{int}} | = 2 | g _ {3} (\mathbf {f}) | + | \mathrm{M} _ {\mathrm{C} _ {3}} ^ {\mathrm{int}} |.
$$

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

The moment magnitudes $\mathrm { M } _ { \mathrm { C } _ { 1 } }$ and $\mathrm { { M } _ { C _ { 3 } } }$ are defined by summing the following contributions:

moment from axial forces, $F _ { a x i a l 1 } ^ { h i n g e } R _ { a }$ and $F _ { a x i a l ~ 3 } ^ { h i n g e } R _ { a }$ , where $\begin{array} { r c l } { F _ { a x i a l \ 1 } ^ { h i n g e } } & { = } & { \alpha _ { a x i a l } F _ { a x i a l 1 } } \end{array}$ , constraint force in the axial direction in each of the pins (if Fhinge $\begin{array} { l c l } { F _ { a x i a l \ 3 } ^ { h i n g e } } & { = } & { \alpha _ { a x i a l } F _ { a x i a l 3 } } \end{array}$ , and is an average effective friction arm associated with the $R _ { a }$ $R _ { a }$ is $0 . 0 , F _ { a x i a l 1 } ^ { h i n g e }$ and $F _ { a x i a l 3 } ^ { h i n g e }$ are ignored); and   
• moment from normal forces, $F _ { n 1 } R _ { p }$ and $F _ { n 3 } R _ { p }$ , where $F _ { n 1 }$ and $F _ { n 3 }$ are themselves sums of the following contributions:

transverse force contributionhinges along the -direction) s, $F _ { t o t a l . 1 } ^ { h i n g e }$ $F _ { t o t a l 3 } ^ { h i n g e }$ magnitude of the total transverse force in the two(the magnitude of the total transverse force in the two hinges along the $\mathbf { e } _ { 3 } ^ { b } { \mathrm { - } } \mathrm { d i r e c t i o n } )$ :

$$
F _ {t o t a l 1} ^ {h i n g e} = \sqrt {(F _ {b e n d} ^ {h i n g e}) ^ {2} + (F _ {t w i s t} ^ {h i n g e}) ^ {2} + (F _ {t r a n s v 1} ^ {h i n g e}) ^ {2}}, \quad \mathrm{and}
$$

$$
F _ {t o t a l 3} ^ {h i n g e} = \sqrt {(F _ {b e n d} ^ {h i n g e}) ^ {2} + (F _ {t w i s t} ^ {h i n g e}) ^ {2} + (F _ {t r a n s v 3} ^ {h i n g e}) ^ {2}},
$$

where $\begin{array} { r } { F _ { b e n d } ^ { h i n g e } = \frac { F _ { b e n d } } { 4 } , F _ { t w i s t } ^ { h i n g e } } \end{array}$ Fbend 4 is defined above, $F _ { t r a n s v 1 } ^ { h i n g e } = \beta _ { a x i a l } F _ { a x i a l 3 }$ , and $F _ { t r a n s v 3 } ^ { h i n g e } =$ $\beta _ { a x i a l } F _ { a x i a l 1 } ;$ and

– ontributions from “bending,” (the magnitude of the total b $F _ { t o t a l } ^ { b e n d }$ , obtained by scaling the total bending moment, moment on each of the four hinges), by a length $M _ { t o t a l } ^ { h i n g e }$

$$
M _ {t o t a l} ^ {h i n g e} = \sqrt {M _ {b e n d} ^ {h i n g e ^ {2}} + M _ {t w i s t} ^ {h i n g e ^ {2}}},
$$

$$
F _ {t o t a l} ^ {b e n d} = 2 \frac {M _ {t o t a l} ^ {h i n g e}}{L _ {s}},
$$

where $M _ { b e n d } ^ { h i n g e } \ = \ \textstyle { \frac { 1 } { 8 } } F _ { b e n d } L _ { a } , \ M _ { t w i s t } ^ { h i n g e }$ is defined above, and $L _ { s }$ represents a characteristic overlapping length between the pins and their sleeves. If $L _ { s }$ is $0 . 0 , M _ { t o t a l } ^ { h i n g e }$ gninge is ignored.

Thus,

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$$
\begin{array}{l} \mathrm{M} _ {\mathrm{C} _ {1}} = g _ {1} (\mathbf {f}) = F _ {a x i a l 1} ^ {h i n g e} R _ {a} + F _ {n 1} R _ {p} \\ = F _ {a x i a l 1} ^ {h i n g e} R _ {a} + (F _ {t o t a l 1} ^ {h i n g e} + F _ {t o t a l} ^ {b e n d}) R _ {p} \\ = \alpha_ {a x i a l} F _ {a x i a l 1} R _ {a} + R _ {p} \sqrt {(\frac {F _ {b e n d}}{4}) ^ {2} + (F _ {t w i s t} ^ {h i n g e}) ^ {2} + (\beta_ {a x i a l} F _ {a x i a l 3}) ^ {2}} \\ + \frac {2 R _ {p}}{L _ {s}} \sqrt {(\frac {1}{8} F _ {b e n d} L _ {a}) ^ {2} + (M _ {t w i s t} ^ {h i n g e}) ^ {2}}, \quad \mathrm{and} \\ \end{array}
$$

$$
\begin{array}{l} \mathrm{M} _ {\mathrm{C} _ {3}} = g _ {3} (\mathbf {f}) = F _ {a x i a l 3} ^ {h i n g e} R _ {a} + F _ {n 3} R _ {p} \\ = F _ {a x i a l 3} ^ {h i n g e} R _ {a} + (F _ {t o t a l 3} ^ {h i n g e} + F _ {t o t a l} ^ {b e n d}) R _ {p} \\ = \alpha_ {a x i a l} F _ {a x i a l 3} R _ {a} + R _ {p} \sqrt {(\frac {F _ {b e n d}}{4}) ^ {2} + (F _ {t w i s t} ^ {h i n g e}) ^ {2} + (\beta_ {a x i a l} F _ {a x i a l 1}) ^ {2}} \\ + \frac {2 R _ {p}}{L _ {s}} \sqrt {(\frac {1}{8} F _ {b e n d} L _ {a}) ^ {2} + (M _ {t w i s t} ^ {h i n g e}) ^ {2}}. \\ \end{array}
$$

The moment magnitudes of the frictional tangential tractions are $\mathrm { P } _ { 1 } ( \mathbf { f } ) = | m _ { 1 } |$ and $\mathrm { P _ { 3 } } ( \mathbf { f } ) = | m _ { 3 } |$ .

Summary

UJOINT 
<table><tr><td>Basic, assembled, or complex:</td><td>Assembled</td></tr><tr><td>Kinematic constraints:</td><td>JOIN + UNIVERSAL</td></tr><tr><td>Constraint force and moment output:</td><td> $f_1, f_2, f_3, m_2$ </td></tr><tr><td>Available components:</td><td> $ur_1, ur_3$ </td></tr><tr><td>Kinetic force and moment output:</td><td> $m_1, m_3$ </td></tr><tr><td>Orientation at a:</td><td>Required</td></tr><tr><td>Orientation at b:</td><td>Optional</td></tr><tr><td>Connector stops:</td><td> $\theta_1^{min} \leq \gamma \leq \theta_1^{max}$  $\theta_3^{min} \leq \gamma \leq \theta_3^{max}$ </td></tr><tr><td>Constitutive reference lengths:</td><td> $\theta_1^{ref}, \theta_3^{ref}$ </td></tr><tr><td>Predefined friction parameters:</td><td>Required:  $R_p, L_a$ ; optional:  $R_a, L_s, M_{C_1}^{int}, M_{C_3}^{int}$ </td></tr><tr><td>Contact moments for predefined friction:</td><td> $M_{C_1}, M_{C_3}$ </td></tr></table>

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# UNIVERSAL

Connection type UNIVERSAL provides a connection between two nodes where the rotations are fixed about one local direction and free about two others. Connection type UNIVERSAL provides the rotational part of a UJOINT connection. Connection type UNIVERSAL cannot be used in two-dimensional or axisymmetric analysis.

![](images/page-667_69dc805452e61a369552bb120bb560a8ac1cd3ab43d8d096cf2df60a1450bb31.jpg)

<details>
<summary>text_image</summary>

e₁ᵃ
e₂ᵃ
a
b
e₃ᵇ
e₂ᵇ
e₁ᵃ ⊥ e₃ᵇ
</details>

Figure 31.1.5–34 Connection type UNIVERSAL.

# Description

A UNIVERSAL connection constrains the rotation about the shaft directions at two nodes. The shaft directions at nodes a and b are ${ \bf e } _ { 2 } ^ { a }$ and $\mathbf { e } _ { 2 } ^ { b } ,$ respectively. A UNIVERSAL connection requires that local direction ${ \bf e } _ { 1 } ^ { a }$ be perpendicular to $\mathbf { e } _ { 3 } ^ { b }$ . This single constraint is written

$$
\mathbf {e} _ {1} ^ {a} \cdot \mathbf {e} _ {3} ^ {b} = 0.
$$

This constraint is equivalent to constraining the second Cardan angle to be zero in a Cardan angle parameterization of the local directions at node b relative to those at node a. If the initial orientation directions at node b do not satisfy the above constraint condition, the universal constraint will hold the second Cardan angle fixed at its initial value.

The constraint moment imposed by the UNIVERSAL connection is

$$
\bar {\bf m} = m _ {2} \left(\cos \alpha {\bf e} _ {2} ^ {a} + \sin \alpha {\bf e} _ {3} ^ {a}\right).
$$

A UNIVERSAL connection allows two free rotational components of relative motion between two nodes. The first and third Cardan angles that position local directions at node b relative to those at node a are

$$
\alpha = - \tan^ {- 1} \left(\frac {\mathbf {e} _ {2} ^ {a} \cdot \mathbf {e} _ {3} ^ {b}}{\mathbf {e} _ {3} ^ {a} \cdot \mathbf {e} _ {3} ^ {b}}\right) \quad \mathrm{and} \quad \gamma = - \tan^ {- 1} \left(\frac {\mathbf {e} _ {1} ^ {a} \cdot \mathbf {e} _ {2} ^ {b}}{\mathbf {e} _ {1} ^ {a} \cdot \mathbf {e} _ {1} ^ {b}}\right).
$$

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The two available components of relative motion for the UNIVERSAL connection, $u r _ { 1 }$ and $u r _ { 3 }$ , are the changes in the two unconstrained Cardan angles when the second Cardan angle is fixed. Therefore,

$$
u r _ {1} = \alpha - \alpha_ {0} \quad \text { and } \quad u r _ {3} = \gamma - \gamma_ {0}  ,
$$

where $\alpha _ { 0 }$ and $\gamma _ { 0 }$ are the initial Cardan angles. The connector constitutive rotations are

$$
u r _ {1} ^ {m a t} = \alpha - \theta_ {1} ^ {r e f} \quad \mathrm{and} \quad u r _ {3} ^ {m a t} = \gamma - \theta_ {3} ^ {r e f}.
$$

The kinetic moment in the UNIVERSAL connection is

$$
\mathbf {m} _ {u n i v e r s a l} = m _ {1} \mathbf {e} _ {1} ^ {a} + m _ {3} \mathbf {e} _ {3} ^ {b}.
$$

# Friction

When used by itself, there is no predefined Coulomb-like friction in the UNIVERSAL connection. However, when the UNIVERSAL connection is used in combination with the JOIN connection type, the predefined friction is the same as the UJOINT connection.

# Summary
<table><tr><td colspan="2">UNIVERSAL</td></tr><tr><td>Basic, assembled, or complex:</td><td>Basic</td></tr><tr><td>Kinematic constraints:</td><td> $\mathbf{e}_{1}^{a} \cdot \mathbf{e}_{3}^{b} = 0$ </td></tr><tr><td>Constraint moment output:</td><td> $m_{2}$ </td></tr><tr><td>Available components:</td><td> $ur_{1}, ur_{3}$ </td></tr><tr><td>Kinetic moment output:</td><td> $m_{1}, m_{3}$ </td></tr><tr><td>Orientation at a:</td><td>Required</td></tr><tr><td>Orientation at b:</td><td>Optional</td></tr><tr><td>Connector stops:</td><td> $\theta_{1}^{min} \leq \alpha \leq \theta_{1}^{max}$ , $\theta_{3}^{min} \leq \gamma \leq \theta_{3}^{max}$ </td></tr><tr><td>Constitutive reference angles:</td><td> $\theta_{1}^{ref}, \theta_{3}^{ref}$ </td></tr><tr><td>Predefined friction parameters:</td><td>None</td></tr><tr><td>Contact force for predefined friction:</td><td>None</td></tr></table>

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# WELD

Connection type WELD provides a fully bonded connection between two nodes.

![](images/page-669_f874ffed6a36bfaf0512a149d44a4714e385ed8427ad03acf9a7c98ca7f61d42.jpg)

<details>
<summary>natural_image</summary>

Illustration of a blue rectangular block connected to a red cylindrical tube with yellow circular ends (no text or symbols)
</details>

![](images/page-669_c18aaa57530aa512758b3bf8b01617f5fb72be5ce6c275c9151c34e1f8332dc5.jpg)

<details>
<summary>text_image</summary>

e₂ᵃ, e₂ᵇ
↑
e₁ᵃ, e₁ᵇ
a, b
e₃ᵃ, e₃ᵇ
</details>

Figure 31.1.5–35 Connection type WELD.

# Description

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

# Summary

WELD 

<table><tr><td>Basic, assembled, or complex:</td><td>Assembled</td></tr><tr><td>Kinematic constraints:</td><td>JOIN + ALIGN</td></tr><tr><td>Constraint force and moment output:</td><td> $f_{1}, f_{2}, f_{3}, m_{1}, m_{2}, m_{3}$ </td></tr><tr><td>Available components:</td><td>None</td></tr><tr><td>Kinetic force and moment output:</td><td>None</td></tr><tr><td>Orientation at a:</td><td>Optional</td></tr><tr><td>Orientation at b:</td><td>Optional</td></tr><tr><td>Connector stops:</td><td>None</td></tr><tr><td>Constitutive reference lengths and angles:</td><td>None</td></tr><tr><td>Predefined friction parameters:</td><td>None</td></tr><tr><td>Contact force for predefined friction:</td><td>None</td></tr></table>

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