Code_Aster ®
Version
8.2
Titrate:
Constraints, efforts, forces and deformations
Date:
04/05/06
Author (S):
J.M. PROIX, Key P. MIALON
:
U2.01.05-C Page
: 1/16
Organization (S): EDF-R & D/AMA, MMC
Handbook of Utilization
U2.01 booklet: General concepts
Document: U2.01.05
Constraints, efforts, forces and deformations
Summary:
This document defines the sizes characterizing the constraints, the forces and the deformations inside one
structure in a calculation by finite elements in displacement and how that is translated in Code_Aster.
The expression of these sizes is given for the finite elements of mechanics: continuous medium 2D or 3D,
hulls and beams.
Handbook of Utilization
U2.01 booklet: General concepts
HT-62/06/004/A
Code_Aster ®
Version
8.2
Titrate:
Constraints, efforts, forces and deformations
Date:
04/05/06
Author (S):
J.M. PROIX, Key P. MIALON
:
U2.01.05-C Page
: 2/16
1 Statics
1.1 Constraints
The postulate of Cauchy is that the efforts of contacts exerted in a point by part of a medium
continuous on another depends only on the normal on the surface in this point delimiting the parts.
In accordance with this postulate, one calls vector forced, for the nonmicropolar mediums, F (N)
the vector which characterizes the forces of contact exerted through an element of surface dS of
normal N on part of a continuous medium [bib1].
It is shown [bib3], then, that the dependence in a fixed point of F compared to normal N is
linear and that there is a tensor which one calls tensor of the constraints such as:
F (N) = N
The unit of the constraints is N.m2 Pa.
For the whole of the structure “the state of constraint” is characterized by a field of tensor of
constraints which one more simply indicates by stress field.
1.2 Effort
With regard to the structures of beams or hulls, contrary to the case of the continuous medium, it
is necessary to note that:
·
only normal directions N of the cuts according to tangent space with the variety are
possible,
·
the characteristic sizes are obtained by integration in the section or the thickness
sizes defined for the continuous mediums.
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Constraints, efforts, forces and deformations
Date:
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1.2.1 Case of the discrete ones
The discrete ones are finite elements which can not have of a physical size. They are
represented by their matrix of stiffness. The efforts are obtained by the multiplication of this matrix
by the vector displacement:
F
D
= [K]
.
M
R
1.2.2 Case of the beams
One calls effort, the end cells (F, M) out of P, geometrical center of inertia of the section
straight line, of the torque resulting from the forces of contact exerted on the section [bib2].
With the preceding notations:
F
= F () ds
(NR)
M
= PM F
.
p
() ds (NR m)
P
For the beams whose cross-section is not regarded as rigid these end cells
are not sufficient: for example, for the beams taking of account the warping of
sections one is brought to consider an additional size of effort due to warping
(bimoment).
Multifibre beams (with local behavior 1D, connecting constraints to deformations, in one
certain number of points of the section) and the pipes (local behavior in plane constraints) are
comparable to elements of traditional beams with regard to the movement of fiber
average and torques of resulting efforts.
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Titrate:
Constraints, efforts, forces and deformations
Date:
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Author (S):
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:
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1.2.3 Case of the hulls
Either, a point P of a surface medium S thickness H, or DLL an element length on S, or N
normal directing the hull in this point.
Maybe, end cells in this point (F, M) of a torque resulting from the actions of contact
exerted through an element of surface dS = H DLL of tangent normal to S on part of S.
With the preceding notations:
+h/2
F (P)
=
F () dh
(NR)
- H/2
+h/2
M (P) =
PM F () dh (NR.)
m
- H/2
N
DLL
H
It is clear that M is in the tangent plan with S in P.
Either, NR (P) the projection of F (P) on the tangent level with S out of P, and or, T (P) its normal component
in this tangent plan.
In the same way that for the continuous mediums, one shows that there are two symmetrical tensors NR
and M, and a vector Q, defined in the tangent plan with S, such as:
F = NR
T = Q.
M = N M
(NR, M, Q) the efforts at the point P are called:
·
the tensor NR characterizes the membrane efforts,
·
the tensor M, bending moments,
·
the vector Q, efforts sharp.
Note:
·
There are no universal conventions on the denomination and the signs of these tensors.
In particular, the tensor of the bending moments is taken with a sign reverses in
the teaching of the ENPC and in practice of the French engineers of the civil engineering. Our
convention is used in the great codes of finite elements (ANSYS) and makes it possible to have it
even sign for a beam and a plate such as =.
·
For the curvilinear or surface material structures with nonlinear behavior, it is
necessary to relocate the stress field in the section or the thickness, but them
equilibrium equations always relate to the fields of effort. It is not necessary of
to go down again to the constraints to define L " state of stress ".
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Titrate:
Constraints, efforts, forces and deformations
Date:
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Author (S):
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:
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Links with the stress field
Under these conditions is a reference mark whose third component is carried by N, one has (, =1 or 2)
:
+h/2
NR
= NR
=
dh
- H/2
+h/2
M
= M
=
x3 dh
- H/2
+h/2
Q
=
dh
3
- H/2
1.3 Forces
nodal
One calls equivalent nodal force or more simply nodal force, a vector F which is it
representative of a linear form W (generally dependant on an energy) acting on fields of
displacement U (X) discretized by finite elements.
The fields of displacements U (X) are expressed starting from its nodal values which form a vector
Q and of the functions of form I (X) by:
(
U X) = Q (X)
I
I
I
Under these conditions:
W (U) = Q F
I
I
I
Note:
·
The concept of node here is very general and wants to say, in fact, carrying degree of freedom (that it
maybe of Lagrange or Hermite besides).
·
The concept of displacement is also very general and includes the concept of displacement
generalized including/understanding translations and rotations.
1.4
Representation of the fields
There are several ways of representing the fields in a modeling by finite elements:
·
for the continuous-current fields on all the field, one uses the values with the nodes, (CHAM_NO
of Aster)
(X) = (X)
I
I
I
one speaks then about constraints to the nodes or efforts with the nodes,
Note:
The effort or stress fields are generally discontinuous, if one them
represent way continuous it is only at ends of visualization.
·
for the discontinuous fields between the elements E, one uses the values in some then
points characteristic of the element (points of Gauss or nodes).
One speaks then about constraints to the nodes by elements or efforts with the nodes by elements,
or, of constraints at the points of Gauss or efforts to the nodes.
Handbook of Utilization
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Code_Aster ®
Version
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Titrate:
Constraints, efforts, forces and deformations
Date:
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Author (S):
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:
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In practice, for the discontinuous fields one uses:
·
representations with the nodes at ends of direct uses of the results (impression or
postprocessing of visualization),
·
at the points of Gauss (or in what holds place of it), to continue calculations requiring it
true “state of stress” in the element: geometrical rigidity, nodal force, calculations not
linear.
1.5
Sizes associated in Aster
1.5.1 SIGM_R
Size SIGM_R represents the “state of stress” of the structure, therefore it must have, with
minimum, components:
·
stress fields of the continuous mediums (in total reference mark):
SIXX SIYY SIZZ SIXY SIXZ SIYZ
·
fields of efforts of beam and discrete (in local reference mark with the beam, discrete):
NR
VY
VZ
MT
MFY
MFZ
·
for the beams with warping, it is necessary to add bimoment (necessarily in reference mark
room with fiber):
BX
·
fields of efforts of hull (necessarily in local reference mark on the surface):
NXX
NYY
NXY
MXX
MYY
MXY
QX
QY
Moreover, it is sometimes convenient to be able to directly exploit the fields of efforts of beam and
of discrete in the total reference mark:
FX
FY
FZ
MX
MY
MZ
It is also interesting to represent the components of a stress field on
elements of beams or hulls in the local reference mark. For that, one will use the same ones
components that in total reference mark, although confusion is possible. Into the future, one will introduce
a concept of reference mark of representation attached to the fields which will overcome the difficulty.
1.5.2 FORC_F and FORC_R
These sizes represent the forces applied to the structure to an interface.
For:
·
a continuous medium it is thus a vector of force,
·
a beam, a torque of forces,
·
a hull, a torque of forces.
This size must thus have the following components:
·
for a continuous medium:
FX
FY
FZ
·
more for the beams and the hulls:
MX
MY
MZ
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Date:
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Author (S):
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1.5.3 DEPL_R
Since in Aster,
·
a field can be attached only to only one size,
·
that methods finite elements mixed (mixing unknown of displacement type and
unknown factors of nodal the forces type) are not excluded,
·
that the dualisation of the boundary conditions results in having for unknown a vector
comprising variables of Lagrange which are nodal forces with the direction where one it specified
higher,
·
that it is necessary to be able to carry out any type of linear combination on
nodal forces,
·
that the classification of the unknown factors must be the same one as that of the second members,
the nodal forces (dual within the meaning of energy W of nodal displacements) have necessarily them
same components as displacements with knowing:
DX
DY
DZ
DRX
DRY
DRZ
more, for the beams with warping, bimoment: GRX.
1.6
Options of calculation
1.6.1 Calculation of the “state of stress”
1.6.1.1 Prefixes
:
SIEF_ELGA
They are the options which calculate the field representative of the “state of stress” and allow
to continue calculations (geometrical rigidity, nodal forces, etc.) in points of Gauss or it
who holds place of it. The prefix of these options is SIEF, because according to the elements, they calculate
constraints or of the efforts.
Option of calculation
Name
Calculation carried out 3D, 2D,
Beams:
Plates:
symbolic system of
COQUE_3D
POU_D_T
DKT
concept
Coques1D
POU_D_E
DST
RESULTAT
TUYAU
POU_D_TG
Q4G
Beams
POU_D_T_GD
multifibre Discrets
SIEF_ELGA_DEPL
idem
from one
(F, M)
field of
(NR, M, V)
displacement
in local reference mark
in local reference mark
in elasticity
linear
SIEF_ELGA_DEPL_C idem
from one
(F, M)
field of
(NR, M, V)
displacement
in local reference mark
(C)
in local reference mark
complex in
(C)
(C)
elasticity
linear
RAPH_MECA
SIEF_ELGA
into nonlinear
(F, M)
FULL_MECA
in local reference mark
These options thus calculate:
·
the stress field for the elements of continuous mediums 2D and 3D, and the elements with
local behavior
: COQUE_3D, hulls 1D (COQUE_AXIS,
COQUE_D_PLAN,
COQUE_C_PLAN), pipes, beams multifibre, in each “under-point” of integration
(layers in the thickness of the hulls, fibers, sectors angular and position in the thickness
for the pipes). The local reference mark of the plates and hulls is specific to each element,
·
the field of efforts for the beams (torque) and the plates (tensor) into linear.
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Date:
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1.6.2 Other representations of the state of stress
1.6.2.1 Prefixes
:
SIEF_ELNO
They are the options which calculate the field representative of the “state of stress” at ends
of exploitation (impression or postprocessing of visualization) to the nodes of the structure.
Option of calculation
Name
Calculation carried out
3D,
Beam, pipe,
Hull, plate
symbolic system
2D
multifibre beam,
of concept
Discrete
RESULTAT
SIEF_ELNO_ELGA
idem
by interpolation with
(F, M)
(NR, M, V)
nodes of the quantities
in local reference mark
at the points of Gauss
in local reference mark
“user” (*)
SIEF_ELNO_ELGA_C
idem
by interpolation with
(F, M) locates some (NR, M, V)
nodes of the quantities
in local reference mark
at the points of Gauss
(C)
room (C)
“user” (C)
(*) for the elements of plate and hull, the local reference mark is that definite starting from the data of the user
(key word ANGL_REP in AFFE_CARA_ELEM).
1.6.2.2 Prefixes
:
SIGM_ELNO
They are the options which calculate the stress fields whatever the modeling at ends
of exploitation (impression or postprocessing of visualization) to the nodes of the structure.
Option of calculation
Name
Calculation carried out
3D,
Beams
Hulls, plates
symbolic system of
2D
in 1 selected point
concept
in the thickness
RESULTAT
(inf, moy, sup)
SIGM_ELNO_DEPL
idem
starting from a field of
displacement in locates some local
in local reference mark
linear elasticity
6 components
6 components
SIGM_ELNO_DEPL_C idem
starting from a field of
displacement complexes (C) in local reference mark
in local reference mark
in linear elasticity
6 components 6 components
(C)
(C)
SIGM_ELNO_CART
idem in
components
total (Cartesian)
in total reference mark in total reference mark
starting from the field of
constraints in
local components
SIGM_ELNO_CART_C idem in
components
total complexes
(C)
in local reference mark
in local reference mark
(Cartesian) to leave
6 components 6 components
field of
(C)
(C)
constraints in
local components
complexes
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Note:
1) In this case, confusion is possible between the components in local reference mark and those in
total reference mark which bears the same name.
2) 6 components delivered in the local reference marks by the beams and the hulls
contain possibly null terms according to the models used. For the models
most standard:
-
three null terms for the beams,
-
two null terms for the hulls.
Thus, the stress field will be complete and, especially, it could be enriched each time
modeling will require it (beam with shearing, hull with pinching, etc…)
1.6.2.3 Prefixes
:
EFGE_ELNO
They are the options which calculate the efforts on the elements of beam or hull at ends
of exploitation (impression or postprocessing of visualization) to the nodes of the structure.
Option of calculation
Name
Calculation carried out
3D,
Beams, pipes,
Hulls,
symbolic system
2Ds beam multifibre, plates
of concept
Discrete
RESULTAT
EFGE_ELNO_DEPL
idem
starting from a field of
(F, M)
displacement in
not
(NR, M, V)
linear elasticity
in local reference mark
in local reference mark
EFGE_ELNO_DEPL_C idem
starting from a field of
(F, M)
displacement
not
(NR, M, V)
complex in elasticity
in local reference mark (C) in local reference mark
linear
(C)
EFGE_ELNO_CART
idem in
components
(F, M)
total
not
not
(Cartesian) to leave
in total reference mark
field of efforts in
local components
EFGE_ELNO_CART_C idem in
components
(F, M)
total complexes
not
not
(Cartesian) to leave
in total reference mark
field of efforts in
(C)
local components
complexes
1.6.3 Calculation of the nodal forces
1.6.3.1 Prefixes
:
FORC_NODA
The nodal forces are calculated starting from the “state of stress”, only one option is envisaged:
Option of calculation
Reference symbol Calcul carried out
3D
Beam
Hull
of concept
RESULTAT
FORC_NODA
idem
starting from a “SIEF_ELGA_ *”
F
(F, M)
(F, M)
Option REAC_NO of operator CALC_CHAM_NO carries out a call to FORC_NODA and withdrawn:
·
the loading in statics,
·
the loading, inertias and viscous in dynamics.
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2 Kinematics
2.1 Deformations
2.1.1 Medium
continuous
In this case, displacements of the structure are represented by a field of vector U to three
components in general.
The deformation (on the assumption of the small disturbances) is defined by the tensor of deformation
by (option EPSI_ELGA_DEPL and EPSI_ELNO_DEPL):
(
1
U)
ij
= (iu J + uj I)
2
,
,
One can want to calculate the “mechanical” deformation, i.e. by cutting off dilations
thermics (options EPME_ELGA_DEPL and EPME_ELNO_DEPL):
m
1
HT
ij (U) =
(iu J +uji) -
2
,
,
In the case of great displacements, the deformations of Green-Lagrange are (options
EPSG_ELGA_DEPL and EPSG_ELNO_DEPL):
1
E (U) = (U + U + U U
ij
I J
J I
K I K J)
2
,
,
,
,
S
To which have can want to cut off the thermal deformations (options EPMG_ELGA_DEPL and
EPMG_ELNO_DEPL):
1
E m
HT
ij (U) =
(U +u +u U
I J
J I
K I K J) -
2
,
,
,
,
2.1.2 Case of the beams
In the theories of traditional beams, each point P of the beam represents a section
straight line. They are thus, the end cells of the torque (T (S), (S)) of displacement of
presumedly rigid cross-section which characterizes the displacement of the point P to the curvilinear X-coordinate (S).
T is the translation of the center of inertia of the section, (S) the vector rotation of the section in it
not.
The application of the theorem of virtual work (cf [bib2]) naturally resulted in defining as
deformation the torque (,) derived from (T (S), (S)) compared to the curvilinear X-coordinate S:
D
= T +
ds
D
= ds
P (S)
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Let us pose then:
= +
L
T
= +
T
K
L is the longitudinal deflection,
T is the vector of the deformations of distortion (no one on the assumption of Navier-Bernoulli),
T is the deformation of torsion of the section,
K is the deformation of inflection.
Note:
For modelings of beam with taking into account of warping, kinematics is more
complicated to describe, but they lead however to concepts close to those
presented above.
2.1.3 Case of the hulls
We will limit ourselves here to the cases of the plates. Indeed, in the general case of the hulls:
·
space derivations use too complicated mathematical concepts for the framework
this document, [R3.07.04],
·
the hulls are very often modelled by elements of assembled plates.
In this case, in fact only the material normals are supposed to be rigid. Displacement
of these normals is thus represented by the end cells of a torque (T,). T is
translation of the point located on the average layer, the vector rotation of the normal in this point.
It is clear that the normal component of is null (in the case of nonmicropolar mediums). One
introduced, the vector L in the tangent plan defined by:
L = N
where N is the normal vector directing surface.
N
e2
e1
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Maybe, decomposition:
T = wn + C
C is tangent displacement,
W is the arrow.
In the same way that for the beams, the application of the theorem of virtual work (cf [bib2])
conduit to define as deformation the unit formed by the tensors E and K and the vector all
these sizes being defined in the tangent plan by:
1
E = (U
+ U
,
,)
2
1
K = (L
+ L
,
,)
2
= L + W
,
The deformation is thus defined by 7 realities.
E are the membrane deformations,
K are the opposite of the curvatures of the deformed average layer,
is the vector of deformation of distortion.
Note:
There still, there is no universal convention and the disparity of conventions is still
larger than for the tensors of efforts. The ENPC adopts a convention reverses for
tensor K for obvious geometrical ratios.
Link with the three-dimensional field of deformation
Under these conditions, one a:
= E + X
K
3
=
3
=
33
0.
2.2
Sizes associated in Aster
2.2.1 DEPL_R and DEPL_C
Sizes DEPL_R and DEPL_C have as components the degrees of freedom of modeling by
finite elements and thus do not have necessarily only the components of the fields of displacement
who are:
DX
DY
DZ
with which it is necessary to associate for the beams or the hulls:
DRX DRY
DRZ
For the hulls, we need the three components of the vector of rotation, because the equation with
finite elements can be expressed only in one total Cartesian reference mark.
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2.2.2 EPSI_R
Size EPSI_R represents the structural deformations, therefore it must have, at least,
components:
·
fields of deformations of the continuous mediums (in total reference mark):
EPXX EPYY EPZZ EPXY EPXZ EPYZ
·
fields of deformations of beam (in local reference mark with the beam):
EPXX GAXY GAXZ KY
KZ
GAT
·
fields of deformations of hull (necessarily in local reference mark on the surface)
EXX
EYY
EXY
KXX
KYY
KXY
GAX
GAY
2.3
Options of calculation
2.3.1 Prefixes: EPSI_ELGA_DEPL,
EPME_ELGA_DEPL,
EPSG_ELGA_DEPL,
EPMG_ELGA_DEPL
They are the options which calculate the fields of deformations at the points of integration of the elements.
Option of calculation
Reference symbol of Calcul carried out
3D Tuyaux,
Hulls,
concept RESULTAT
Beams multi_fibers plates
EPSI_ELGA_DEPL idem
starting from a field
not
of displacement in
in local reference mark
small deformations
6 components
EPSG_ELGA_DEPL idem
Tensor of Green-
not
not
Lagrange to be left
of a field of
displacement
EPME_ELGA_DEPL idem
starting from a field m not
not
of displacement and
of a field of
temperature in
small deformations
EPMG_ELGA_DEPL idem
Tensor of Green- m not
not
Lagrange to be left
of a field of
displacement and of one
field of
temperature
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2.3.2 Prefixes: EPSI_ELNO_DEPL,
EPME_ELNO_DEPL,
EPSG_ELNO_DEPL,
EPMG_ELNO_DEPL
They are the options which calculate the fields of deformations whatever the modeling at ends
of exploitation (impression or postprocessing of visualization) to the nodes of the structure.
Option of calculation
Reference symbol of Calcul carried out
3D Poutres,
Hulls, plates
concept RESULTAT
Pipes,
in 1 selected point
Beams multi_fibers in the thickness
(inf, moy, sup)
EPSI_ELNO_DEPL idem
starting from a field
in local reference mark:
of displacement in
in local reference mark
6 components
small deformations
6 components
EPSG_ELNO_DEPL idem
Tensor of Green-
not
not
Lagrange to be left
of a field of
displacement
EPME_ELNO_DEPL idem
starting from a field m not
not
of displacement and
of a field of
temperature in
small deformations
EPMG_ELNO_DEPL idem
Tensor of Green- m not
not
Lagrange to be left
of a field of
displacement and of one
field of
temperature
2.3.3 Prefixes
:
DEGE_ELNO
They are the options which calculate the deformations generalized on the elements of beam or of
hull at ends of exploitation (impression or postprocessing of visualization) to the nodes of
structure.
Option of calculation
Name
Calculation carried out
3D
Beams, Plaques beams,
symbolic system of
multifibre
Coques1D
concept
RESULTAT
DEGE_ELNO_DEPL
idem
starting from a field of
(,)
displacement into small not
(E, K,)
deformations
in local reference mark
in local reference mark
Handbook of Utilization
U2.01 booklet: General concepts
HT-62/06/004/A
Code_Aster ®
Version
8.2
Titrate:
Constraints, efforts, forces and deformations
Date:
04/05/06
Author (S):
J.M. PROIX, Key P. MIALON
:
U2.01.05-C Page
: 15/16
3 Bibliography
[1]
F. SIDOROFF: Run of mechanics of the solids Tome 1 E.C.L.
[2]
F. SIDOROFF: Run of mechanics of the solids Tome 2e.C.L.
[3]
C. TRUESDELL, W. NOLL: Encyclopedia off Physics volume III/3 - The non-linear Field
Theories off Mechanics Springer-Verlag, 1965.
Handbook of Utilization
U2.01 booklet: General concepts
HT-62/06/004/A
Code_Aster ®
Version
8.2
Titrate:
Constraints, efforts, forces and deformations
Date:
04/05/06
Author (S):
J.M. PROIX, Key P. MIALON
:
U2.01.05-C Page
: 16/16
Intentionally white left page.
Handbook of Utilization
U2.01 booklet: General concepts
HT-62/06/004/A
Outline document