5 Definition of structural elements
5.3. SPTP − Structural point properties
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SPTP
Item Description Unit Default
TYPE Type of property
SUPP Periphery of support (column head)
VOUT Periphery of haunches PNCH Periphery of punching
KPPX, KPPY etc. coupling condition (see explanation)
section number for SUPP/VOUT/PNCH coupled strutural point
VAL Property value (e.g. Plate thickness) 0.0
GRP Group number − 0
This record defines extended geometric properties or kinematic constraints or coupling conditions for a structural point. Geometric properties may be defined as a centred rectangle or as a general polygonal section by referencing a cross section in the database. A haunched region will be accounted for by a varying thickness for the generated mesh in the vicinity of the column head. In general there are multiple records of type SPTP are allowed for a single structural point.
For defining kinematic coupling conditions between structural points at parameter TYPE a number of special literals are provided which allow to fix one or multiple degrees of freedom:
The three conditions KPX0, KPY0 and KPZ0 however, do not satisfy mechanical equilibrium conditions as they do not consider the distance between the connec-ted structural points. For this reason, additional literals are provided which ac-count for the real distance between the points and which should be preferably used in most cases:
KPPX Connection of x displacement only (see formula 1) KPPY Connection of y displacement only (see formula 2) KPPZ Connection of z displacement only (see formula 3)
Couplings can also be defined in radial and tangential direction. Radial refers to the distance between the first and the connected point and tangential to all direc-tions perpendicular:
KPR Coupling of displacements in radial direction (see formula 18) KPT Coupling of displacements in tangential directions (see formula 19) KMR Coupling of rotations about the radial direction (see formula 18) KMT Coupling of rotations about all tangential directions (see formula 19) A number of additional literals are provided which allow to define special bound-ary conditions or to define a combination of the above mentioned relations:
KP = KPPX + KPPY + KPPZ
describes mechanically a rigid connection with hinged conditions at the reference node
KF = KP + KMX + KMY + KMZ
describes mechanically a rigid connection with clamped support at the reference node
KL = KP + KMT KQ = KP + KMR
KPEX Rotation about x−axis only (see formula 7 to 9) KPEY Rotation about y−axis only (see formula 10 to 12) KPEZ Rotation about z−axis only (see formula 13 to 15) KFEX Rotation about x−axis only (see formula 7 to 9) KFEY Rotation about y−axis only (see formula 10 to 12) KFEZ Rotation about z−axis only (see formula 13 to 15)
SYM Symmetry conditions about the mid−perpendicular ANTI Anti−symmetry conditions about the mid−perpendicular CYCL Cyclic symmetry conditions
INTE Interpolation of displacements only INTF Interpolation of all deformations
INTS Special form of the interpolation for mindlin plates
The mathematical formulas mentioned previously which are used for the calcula-tion of the coupled displacement condicalcula-tions are listed at the end of this seccalcula-tion.
With the exception of KPX, KPY and KPZ all coupling conditions satisfy the mech-anical equilibrium conditions by taking the real distances between the two con-nected points into account. Mechanically they act like infinitely stiff structural members and remain numerically stable when solving the finite element system.
Their primary application area is the formulation of boundary conditions for plates and shells and the modelling of stiff structural parts. As the kinematic constraints describe linear relationships, they are not capable to account for geometrically non−linear effects from second or third order theory.
List of kinematic displacement relations of couplings conditions
KPPX: ux = uxo + ϕyo ⋅ (z − zo) − ϕzo ⋅ (y − yo) (1) KPPY:
u
y= u
yo+ ϕ
zo ⋅(x − x
o) − ϕ
xo ⋅(z − z
o)
(2) KPPZ:u
z= u
zo+ ϕ
xo ⋅(y − y
o) − ϕ
yo ⋅(x − x
o)
(3)KP: KPPX + KPPY + KPPZ
KF additionally:
ϕ
x= ϕ
xo (4)ϕ
y= ϕ
yo (5)ϕ
z= ϕ
zo (6)KPEX:
u
y= u
yo− ϕ
xo ⋅(z − z
o)
(7)u
z= u
zo+ ϕ
xo ⋅(y − y
o) (8)
KFEX additionally:
ϕ
x= ϕ
xo (9)KPEY:
u
x= u
xo+ ϕ
yo ⋅(z − z
o)
(10)u
z= u
zo− ϕ
yo ⋅(x − x
o) (11)
KFEY additionally:
ϕ
y= ϕ
yo (12)KPEZ:
u
x= u
xo− ϕ
zo ⋅(y − y
o)
(13)u
y= u
yo+ ϕ
zo ⋅(x − x
o) (14)
KFEZ additionally:
ϕ
z= ϕ
zo (15)The conditions for fixed supports PR and PT and for coupling conditions KPR and KPT in radial and transversal directions respectively as well as their counterparts for moments are not explicitly but implicitly defined. The programs themselves create an appropriate explicit form.
PR:
u
t ⋅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−uo)t ⋅ n = 0
(ux−uxo)⋅dx + (uy−uyo)⋅dy + (uz−uzo)⋅dz = 0 (18)
KPT:
(u−u
o)
⋅n = 0
(ux* uxo)
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 included here:
SYM:
u
t ⋅n = − u
to ⋅n
ANTI:
u
t ⋅n = u
to ⋅n
In case of mesh refinement or in cases of stiff cross−girders there may arise a need for nodes that lie between two others and depend on them. This kind of de-pendency can be described by means of interpolating couplings INT?. The follow-ing picture shows a mesh with a so−called ’hangfollow-ing’ node which displacements can be calculated by interpolating the displacements of the two adjacent nodes:
INTE−couplings
The INTE−coupling is a constraint with special attributes. Herein, opposite to node to node couplings, one node (the middle node) is dependent on two other nodes. The displacements and rotations of the middle node are interpolated from the corresponding values of the adjacent nodes.
u0 = u1 · DD + u2 · (1−DD)
When the deflections of the outer nodes are somehow prescribed, e.g. fixed or provided with a certain stiffness, the deflection of the middle node is prescribed in the same way too. The coupling is rigid only when both nodes can not displace relatively to each other. A rigid body with three nodes must be described by means of two KP/KF couplings; the INTE−coupling can not be used in that case.
There are several variants of interpolation used by INTE−couplings, which are de-scribed in the following.
INTE
Displacements: linearly interpolated Rotations: not defined
Application: mesh refinements TALPA INTF
Displacements: linearly interpolated as in TYPE P
Rotations: “torsion” linearly interpolated, other rotations com−
puted from displacement differences divided by the respective node distances
Application: connection of beam elements onto disks stiff cross−girders between two supports
In the general three−dimensional case, if one draws the lines connecting the two nodes in the initial undeformed as well as in their deformed state, two rotational components are defined exactly by the secant angles of those. The third yet un-determined rotational component has the direction of the connecting line
(tor-sion), and it is normally interpolated. The general expression is very complicated;
however, INTE−couplings parallel to the axes of coordinates can be expressed by much simpler expressions, e.g.,
X = 0.
Y = d Z = 0.
results in:
ϕx = D uz / d ϕy = ϕy−m
ϕz = − D ux / d INTS
Displacements: quadratically interpolated Rotations: linearly interpolated
Application: mesh refinements of plates and shells
In mesh refinements of plates and shells there is a problem in coupling the transla-tional and rotatransla-tional degrees of freedom. Very poor elements function with a plain interpolation. Due to the peculiarities exhibited by the formulation of the SEPP/
ASE−elements, even in its simplest form, the INTE−conditions must be accord-ingly complicated. In case of regular elements by Kirchhoff’s theory for example, a cubic interpolation of the displacements and two of the rotations must be em-ployed. Mindlin elements also work with the so−called Kirchhoff constraints. In principle of course, translations and rotations are interpolated independently of one another, yet proper additional conditions are used to make sure that the shear force corresponds to the derivative of the moment.
A quadratic distribution of the bending deflection along with a linear distribution of the rotations can be accomplished through the introduction of an additional translational degree of freedom at the middle of an element’s side. This additional degree of freedom can be later eliminated. This method is also employed by V−
couplings. Although the formulation is consistent and leads to considerably better results than the older methods, it is not recommended unlimitedly. In particular, it should not be used with non−conforming elements.
The application of INTE in the direct vicinity of singularities is generally not recom-mended.
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The support of the slab may be done by different approaches, which can be selec-ted with the number of additional asterixes “*” at FIX and the CTRL option PSUP.
no Use the value of CTRL PSUP
* increase thickness for centre (CTRL PSUP 1)
= monolithic support
** do not increase thickness for centre (CTRL PSUP 0)
= hinged or elastic support
*** add kinematic constraints
(only for special purpose, CTRL PSUP 2 / 4)
The generation of such a mesh macro is currently only possible for supports within the slab and only if the central point is not to close to any other structural edge. If this is not the case, the point will become only a single node in the gener-ated FE−mesh. This behaviour may also be enforced with definition of CTRL PSUP −1.
See also: SPT