NODE − Nodal Coordinates and Constraints

3 Input Description

3.8. NODE − Nodal Coordinates and Constraints

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NODE

Item Description Dimension Default

NO X Y Z FIX NREF DX DY DZ COOR Node number X−coordinate Y−coordinate

Z−coordinate (omitted by 2−D−systems) Node constraints

Node number of reference node

Directions for couplings or polar bound ary conditions

(DZ not necessary for 2−D−system) System of coordinates CA Cartesian coordinates CY Cylindrical coordinates SP Spherical coordinates − m/* m/* m/* LIT − m/* m/* m/* LIT ! 0 0 0 * − * * * CA Remarks

Coordinates or constraints for all the nodes can be defined as often as one likes with MESH, IMES, CUBE, TRAN, MIRR or NODE. The last input is valid at any time. Only support conditions can be modified by RESTART; couplings, however, can not be partially redefined, thus in RESTART either all couplings and dependent boundary conditions (PR, PT, MR and MT) must be input again or none at all. When only the constraints or certain coordinates are being modified, a − (default value) must be input for the rest of the coordi nates.

The nodes need not be numbered in a consecutive order. Coordinates

Input coordinate systems

The input values for y or z by CY or SP are interpreted as angles in degrees. The default system of coordinates is CA. Any definition holds for all following nodes until a new explicit definition is given. While in CA mode, one can switch to cylindrical coordinates for certain nodes through the use of negative node numbers for them.

Regardless of the input mode, the coordinates of the nodes are immediately converted to Cartesian ones and they are the only ones used thereafter. As an example, the following definitions of coordinates are equivalent:

NODE 1 12 45 30 COOR SP NODE 1 6 45 10.392 COOR CY NODE 1 4.243 4.243 10.392 COOR CA NODE −1 6 45 10.392 COOR CA

If a reference node is defined, all coordinates are considered to be relative to those of the reference node.

An earlier defined node can be translated with respect to its old position, if a reference node with the same node number is input. The input of a coupling is not allowed in such case.

Beispiel: KNOT 15 1 2 KREF 15

Der Knoten 15 wird relativ zu seiner bisherigen Lage um 1 Meter in X− und um 2 Meter in Y−Richtung verschoben.

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Example: NODE 15 1 2 NREF 15

Node 15 is translated 1 m in the X− and 2 m in the Y−direction with respect to its previous position.

It is impossible to specify couplings to a reference node and absolute coordi nates in the same record. It is best, in principle, first to define all the nodal coordinates and then all the couplings (without coordinates).

Nodal constraints

All the constraints of a node can be described by any combination of the follow ing literals (limited up to 8 characters). Any degree of freedom not included in a 2−D system gets fixed. The default constraint is the value defined by FIXS in SYST.

PX Constraint of displacement in x PY Constraint of displacement in y PZ Constraint of displacement in z PR Constraint of radial displacement PT Constraint of tangential displacement MX Contstraint of rotation about x

MY Contstraint of rotation about y MZ Contstraint of rotation about z

MR Contstraint of rotation about radial direction MT Contstraint of rotation about tangential direction MB Constraint of warping XP = PY + PZ YP = PX + PZ ZP = PX + PY PP = PX + PY + PZ XM = MY + MZ YM = MX + MZ ZM = MX + MY MM = MX + MY + MZ + MB FREE = Deletion of all constraints F = PP + MM

DEL = Node will be deleted

usefull for auxiliary nodes, which should not appear in the graphs nor the results.

A boundary condition on a symmetry or an anti−symmetry axis can be defined by PRMT or PTMR, respectively, if the direction of the coupling is defined per pendicular to the axis. A direction must be defined in case of PR, PT, MR, MT by means of DX, DY, DZ or the reference node.

Support conditions can be also defined in relation to another node (reference node). The following input is therefore allowed only in conjunction with the parameter NREF. Combinations with other literals are not allowed. Opposite to constraints, coupling conditions can not be subsequently overwritten; addi tional couplings, however, can be defined so long as no multiple definition oc curs.

KPX Coupling of x−displacement only (ux = uxo)

KPY Coupling of y−displacement only (u_y = u_yo) KPZ Coupling of z−displacement only (u_z = u_zo) KPR Coupling of radial displacement

KPT Coupling of tangential displacements

KMX Coupling of rotation about the x−axis (ϕ_x = ϕ_xo) KMY Coupling of rotation about the y−axis (ϕ_y = ϕ_yo) KMZ Coupling of rotation about the z−axis (ϕ_z = ϕ_zo) KMR Coupling of rotations about the radial direction

KMT Coupling of rotations about the tangential directions KP Articulated connection to rigid body at the reference node KPPX Connection of x displacement only (flexible yz−plane) KPPY Connection of y displacement only (flexible xz−plane) KPPZ Connection of z displacement only (flexible xy−plane) KPEX Rotation about x−axis only (flexible, rigid yz−disk) KPEY Rotation about y−axis only (flexible, rigid xz−disk) KPEZ Rotation about z−axis only (flexible, rigid xy−disk) KL = KP + KMT

KQ = KP + KMR

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KFEX Rotation about x−axis only (flexible, rigid yz−disk) KFEY Rotation about y−axis only (flexible, rigid xz−disk) KFEZ Rotation about z−axis only (flexible, rigid xy−disk) SYM Symmetry conditions about the mid−perpendicular

ANTI Anti−symmetry conditions about the mid−perpendicular CYCL Cyclic symmetry conditions

Coupling conditions describe infinitely stiff elements and special boundary conditions which are numerically stable. Their application area is the for mulation of boundary conditions for plates and shells and the modelling of very stiff structural parts. General kinematic constraints can be defined using the records KINE and INTE. Kinematic constraints can not take care of any non−linear geometric analysis.

Kinematic conditions of couplings

KPPX: u_x = u_xo + ϕ_yo ⋅ (z − z_o) − ϕ_zo ⋅ (y − y_o) (1) KPPY:

u

= u

_yo

+ ϕ

_zo ⋅

(x − x

) − ϕ

_xo ⋅

(z − z

)

(2) KPPZ:

u

= u

_zo

+ ϕ

_xo

⋅ (y − y

) − ϕ

_yo

⋅ (x − x

)

(3) KP: KPPX + KPPY + KPPZ KF additionally:

ϕ

= ϕ

_xo (4)

ϕ

= ϕ

yo (5)

ϕ

= ϕ

zo (6) KPEX:

u

= u

_yo

− ϕ

_xo ⋅

(z − z

)

(7)

u

= u

_zo

+ ϕ

_xo ⋅

(y − y

)

(8)

KFEX additionally:

ϕ

= ϕ

_xo (9) KPEY:

u

= u

_xo

+ ϕ

_yo ⋅

(z − z

)

(10)

u

= u

_zo

− ϕ

_yo ⋅

(x − x

)

(11)

KFEY additionally:

ϕ

= ϕ

_yo (12) KPEZ:

u

= u

_xo

− ϕ

_zo

⋅ (y − y

)

(13)

u

= u

_yo

+ ϕ

_zo ⋅

(x − x

)

(14)

KFEZ additionally:

ϕ

= ϕ

_zo (15)

The conditions PR and PT, KPR and KPT as well as their counterparts for mo ments are not explicitly but implicitly defined. The programs themselves create an appropriate explicit form.

PR:

u

⋅

n = 0

u

_x ⋅

dx + u

_y ⋅

dy + u

_z ⋅

dz = 0

(16) PT:

u

⋅

n = 0

ux dx +uy dy + uz dz

(17)

KPR: (u−u_o)t ⋅ n = 0

(u_x−u_xo)⋅dx + (u_y−u_yo)⋅dy + (u_z−u_zo)⋅dz = 0 (18)

KPT:

(u−u

)

⋅

n = 0

(u_x* u_xo) dx +ǒuy* uyoǓ dy +(uz* uzo) dz

(19)

The symmetry and anti−symmetry conditions are given in the following equations in vectorial form. A presentation by their components is not in cluded here:

SYM:

u

t ⋅

n = − u

t_o ⋅

n

ANTI:

u

t ⋅

n = u

t_o ⋅

n

The directional or differential vector n = (dx,dy,dz) is built from the differ ences of the node coordinates. These coordinate differences can also be speci fied explicitly by means of DX, DY and DZ.

Certain degrees of freedom that have been coupled can be constrained again with input of a constraint after the coupling condition.

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