Local effects in the analysis of structures

336 Pages·1985·11.028 MB·English

by  Pierre Ladevèze

STUDIES IN APPLIED MECHANICS
1. Mechanicsand Strength of Materials (Skalmierski)
2. Nonlinear Differential Equations (Fuclk and Kufner)
3. Mathematical Theory of Elasticand Elastico-Plastic Bodies
An Introduction (Necas and Hlavacek)
4. Variational, Incremental and Energy Methods in Solid Mechanics
and Shell Theory (Mason)
5. Mechanicsof Structured Media,Parts A and B (Selvadurai, Editor)
6. Mechanicsof Material Behavior (Dvorak and Shield, Editors)
7. Mechanicsof Granular Materials: New Models and Constitutive Relations
(Jenkinsand Satake, Editors)
8. Probabilistic Approach to Mechanisms (Sandler)
9. Methods of Functional Analysisfor Application in Solid Mechanics (Mason)
10. Boundary Integral Equation Methods in Eigenvalue Problemsof Elastodynamics
and Thin Plates (Kitahara)
11. Mechanicsof Material Interfaces (Selvadurai and Voyiadjis, Editors)
12. Local Effects in the Analysis ofStructures (Ladeveze, Editor)
STUDIES IN APPLIED MECHANICS 12
Local Effects in the
Analysis of Structures
Edited by
Pierre Ladeveze
Laboratoire de Mécanique et Technologie (E.N.S.E. T./Université Paris 6/C.N.R.S.),
Cachan, France
ELSEVIER
Amsterdam — Oxford — New York — Tokyo 1985
ELSEVIER SCIENCE PUBLISHERS B.V.
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Library of Congress Cataloging-in-Publication Data
Main entry under title:
Local effects in the analysis of structures.
(Studies in applied mechanics ; 12)
Selection of papers presented at the EUR0MECH
Colloquium "Inclusion of Local Effects in the Analysis
of Structures," held Sept. 11-14, 1984 at Laboratoire
de mécanique et technologie, Cachan, France.
Bibliography: p.
1. Structures, Theory of—Congresses. 2. Stress
concentration—Congresses. I. Ladev^ze, Pierre,
1945- . II. EUR0MECH Colloquium "Inclusion of Local
Effects in the Analysis of Structures" (1984 :
Laboratoire de mécanique et technologie, Cachan,
France) III. Series.
TA645.L63 1985 624.1f71 85-13150
ISBN 0-444-42520-9 (U.S.)
ISBN 0-444-42520-9 (Vol. 12)
ISBN 0-444^1758-3 (Series)
© Elsevier Science Publishers B.V., 1985
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Printed in The Netherlands
V
F O R E W O RD
At the present time, the Inclusion of Local Effects in the Analysis of Structu-
res is undoubtedly a question of prime importance for Engineering Design. The
classical computational approaches are not readily adapted to take into account
the local effects - appropriate treatments are necessary. This book attempts to
provide an introduction to and a survey of the specific computational methods.
It begins with the various theories which allow to separate and then to deter-
mine the local and global effects. Chapter 2 discusses edge effects for compo-
site structures. Chapter 4 deals with general numerical methods, especially for
effects due to large local variations of geometry. Chapter 3 concerns some dyna-
mic problems - it is an opening towards non-conventional local effects in Struc-
tural Mechanics.
VI
The papers, for a part, have been presented at the EUROMECH Colloquium
"Inclusion of Local Effects in the Analysis of Structures" which has been held
on September 11-14, 1984 in CACHAN (France) - LABORATOIRE DE MECANIQUE ET TECH-
NOLOGIE .
The Scientific Committee included :
P. LADEVEZE
R. OHAYON
M. PREDELEANU
E. SANCHEZ-PALENCIA
N.Q. SON.
3
ON SAINT-VENANT'S PRINCIPLE IN ELASTICITY
P. LADEVEZE
Laboratoire de Mécanique et Technologie, E.N.S.E.T./Université PARIS 6/C.N.R.S.,
61, Avenue du Président Wilson - 94230 CACHAN (France)
The Saint-Venant's Principle is considered from the angle of its present
practical interest in Structural Mechanics, especially for composite structures.
The interior large wavelength effect must be separated from the edge or extremi-
ty effects with a small wavelength in order to be computed. In a more precise
way, it is a theorem which expresses conditions ensuring localization of dis-
placements and stresses. This point of view is built on certain characteristic
properties of the solutions and not on the properties of zero resultant-moment
loadings. Moreover, and this is an important point, the diameter 0 of the beam
or the thickness 2h of the plate are not considered as small parameters. The
approach has nothing to do with asymptotic methods.
Beams and plates are studied as far as possible from a unitary point of
view. The problem is restricted to the so-called Saint-Venant Problem for
which the non-zero loadings and displacements are only prescribed on the edges
of the structure. Additional loadings on the lateral surface of the beam or on
the faces of the plate essentially only modify the interior effect. This effect
is now well-known [6] [15] [16] [17] [18] [19] [20] [32].
This paper starts by proposing several major properties for the solutions
which are localized or not. They are inferred from the particularities of the
geometry. The localization concepts are stated precisely. The corresponding
solutions decrease exponentially as a function of the distance to the edges
such that the decaying length is 0(0) for beams and 0(h) for plates. In fact,
the Saint-Venant Principle which characterizes such solutions expresses the
orthogonality to the interior large wavelength solutions. Some auxiliary pro-
blems are used to write this condition in terms of the data on the edges for
any boundary conditions.
The splitting up of the solution into the interior effect and the edge
effects is a central problem. Several results concerning the existence and the
unicity of such a splitting up are presented.
4
1. GENERALITIES - HYPOTHESES
1.1. Beams
Cylindrical beams are considered (figure 1). The domain Ω is defined by
Ω = {M = X + t.N ; t e]0,L[;Xe S}
where S denotes the cross-section, t is the N- coordinate. The boundary of the
cross-section is supposed as having the usual regularity, namely corner points
can occur.
Jk=L·
Figure 1
The end cross-sections are denoted by S and S, . The Hooke tensor IK is
0
taken to be constant on lines parallel to the center line.
1.2. Plates
The thickness is constant. It is supposed that the material is homogeneous
on planes parallel to the middle surface Σ. The domain is defined by
Ω = {M = m + Nz , m 6 Σ , ze ]-h,h[}
and the boundary of the middle surface is supposed as having the usual regulari-
ty, namely corner points can occur.
i N=N
3
>
Figure 2
1.3. Basic problem
Our aim is to study end effects, so we shall restrict the paper to this
specific problem which, by analogy with the beam theory, will be designated as
5
the Saint-Venant Problem. It is characterized by the following assumptions :
- the body-forces are zero
- the lateral surfaces of the beam and the upper and lower surfaces of the plate
are free.
In other words, the non-zero forces and displacements are only prescribed
on the end surfaces S and S..
0
Moreover, we will consider only physically admissible solutions namely
those with finite energy.
Remarks
- This framework contains two important particular cases :
. composite beams with homogeneous layers parallel to the center line
. composite plates with homogeneous layers parallel to the middle surface.
- The boundary conditions on the end cross-section can be of any type.
Notations
The scalar product of the vectors V,W will be written VW. π denotes both the
orthogonal projection on the cross-section for the beams and the orthogonal
projection on the middle surface for the plates.
1.4. Saint-Venant's solutions
They are large wavelength "solutions" of the Saint-Venant Problem. They
verify all the equations except for the boundary conditions on S and S, .
0
- Beams
The Saint-Venant solutions can be written as follows :
U* = A T* + B M* + V + Q X
t t
c* NL = A0 T* + B°M*
where
- T*,M* : resultant-moment of normal stress vector
- V, Ω : vectors which are constant related to X-coordinates
- A,B,A°B0 : linear operators which are constant related to t.
- Plates
For the sake of simplicity, the Saint-Venant solution is written for homoge-
neous isotropic materials. It is the classical Kirchhoff-Love solution :
TTU* = grad Γ-ζ W + 3λ + 4μ z3 Ají] + u + — -— z 2 grad ω
L 6(λ 2μ) m -I 8(λμ) m
+ +
6
N U* W + ^ z2 - 2η2(λ+μ) Δ W - λζ
m
λ+2μ 4(λ+μ)
with W(m), u(m), w(m) such as :
Δ Δ W = 0 - ^- gradfdiv u) + 2μ div (ÏÏE(U)ÏÏ) = 0
m m μ m m m
λ+2
di vu =-Aí2lL·
ω
m 4(λ+μ)
1.5. Interior and exterior problems
The beam and the plate described by the figures (1) and (2) have in fact
two edges. Therefore, it is natural to introduce :
- the interior problem related to an end section S
0
So
So Figure 3
the exterior problem related to an end section S
0
XSo
S^
Figure U
7
These problems are particular Saint-Venant Problems. It should be recalled
that the solutions have to lead to finite energy. "+" shall denote quantities
connected with the interior problem, "-" those connected with the exterior one.
For plates, the following coordinates system is introduced :
C. ,t €]-OO,-HX>[ is a family of closed curves such that :
"Ct = Cn
t=0
- the domain interior to C. decreases with t and tends to zero with t -> °°
- the domain exterior to C.i decreases with (-f). The interior diameter of C.,
tends to infinity with (-f).
JHîLJ"^
QN
3
Figure 5
For starshaped domains, it is possible to use
I Onu i
t = -log ' tl
|0m|
0
where nu, m are homothetic points.
Moreover, the t-section is defined by S = C x]-h,h[. So, a plate is mapped on
t t
an abstract beamwith a variable "cross-section", t is the coordinate generating
this different "cross-sections".
With these notations, beams and plates can be studied using the same termi-
nology.
2. INTERIOR AND EXTERIOR PROBLEMS - BASIC PROPERTIES
2.1. Semi-groups IR ,IR~
- Beams
Let D be the space of displacement values at the cross-section S
(D = [H 2 (S)]3). S is the chosen reference cross-section and U a given dis-
0 0
placement belonging to D.