On modified boundary conditions
for the free edge of a shell
A. M. A. van der Heijden
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M OOn modified boundary conditions
for the free edge of a shell
Proefschrift t e r verkrijging
van de graad van d o c t o r in de technische w e t e n s c h a p p e n
aan de Technische H o g e s c h o o l Delft, o p gezag v a n de rector magnificus prof. ir. L. Huisman,
v o o r een commissie aangewezen door het college van dekanen t e verdedigen op
woensdag 29 September 1976 t e 14.00 uur d o o r
Arnoldus Maria Antonius van der Heijden
werktuigkundig ingenieur, geboren te Eindhoven
o -^
O tN> BIBLIOTHEEK TU Delft P 1142 4300 C 267026Delft University Press / 1976
Dit proefschrift is goedgekeurd door de promoter PROF. DR. IR. W. T. KOITER
CONTENTS
page
INTRODUCTION 6 1. ON SHELL THEORY WITH MODIFIED BOUNDARY CONDITIONS
FOR THE FREE EDGES 12 1.1 The equations of shell theory with m.b.c. 12
1.2 The Prager-Synge theorem 22 1.3 Improved error estimate 26
1.3.1 Preliminaries 27 1.3.2 Error estimate 28 1.4 On the calculation of the primary stresses at a
free edge of a shell 34 2. APPLICATION OF COMPLEX FUNCTIONS FOR THE CASE OF
BENDING OF A FLAT PLATE WITH A HOLE 37
2.1 The governing equations 37 2.2 Application of the theory to the bending of an
infinite plate with an elliptic hole 45 2.3 Discussion of the results for the stresses at
the edge 58 2.4 Comparison between our theory and Reissner's
theory for a plate with a circular hole 79
3. OTHER APPLICATIONS 86 3.1 Perforated plates 86 3.2 The cylindrical shell weakened by a circular hole 88
APPENDICES
I. THE THREE-DIMENSIONAL EQUATIONS FOR SHELLS 93 II. ON THE DETERMINATION OF THE FUNCTIONS (j)°(z)
AND ijj°(z) 97 III. SOME CALCULATIONS FOR THE PLATE WITH AN
ELLIPTICAL HOLE 100
page
111.1 The determination of the modified boundary
conditions 100
111.2 The evaluation of (ti*(c;in) 102
III. 3 On the evaluation of dl/d^ 109
IV. ADDITIONAL RESULTS FOR THE PLATE WITH THE
ELLIPTIC HOLE 112
REFERENCES 114
SUMMARY 124
SAMENVATTING 128
INTRODUCTION
By a plate or more generally a shell we understand a piece of solid matter contained in the narrow space between two
(curved) surfaces which are parallel or almost parallel to each other. Their distance is the shell thickness h, which is supposed to be small compared with other dimensions of the shell, in particular, with its radii of curvature. The sur-face which halves the shell thickness everywhere is called the middle surface and serves in the stress analysis the same purpose as the axis of a beam.
Interest in the construction of a linear theory for the extensional and flexural deformation of plates (from the
three-dimensional equations of linear elasticity) dates back to the early part of the nineteenth century. Starting from the supposition that all the quantities which occur can be expanded in powers of the distance from the middle surface, POISSON (1828) and CAUCHY (1828) both derived equations of equilibrium and free vibrations which hold when the displace-ment of the middle surface is directed at right angles to the plane of the plate. Much controversy has arisen concerning POISSON's boundary conditions. These express that the
resul-tant forces and couples applied at the edge (three components in plate bending) must be equal to the forces and couples arising from the internal stresses. KIRCHHOFF (1850) proved that three boundary conditions are too many for the
order equation obtained by POISSON, and that two conditions are sufficient for the complete determination of the
deflexion. He showed also that the two requirements of POISSON dealing with the twisting moment and with the shearing force must be replaced by one boundary condition. His method rests on the assumptions:
(1) normals to the undeformed middle surface remain normal to the deformed middle surface;
(2) changes in length of these normals may be neglected; (3) the state of stress is approximately plane and
paral-lel to the middle surface which itself remains unstretched.
These assumptions enabled him to express the potential energy of the bent plate in terms of the curvatures produced in its middle surface. The equations of motion and boundary condi-tions were then deduced by the principle of virtual work. The physical significance of this reduction in the number of boundary conditions has been explained by KELVIN and TAIT
(1883) by an appeal to SAINT V E N A N T ' S principle.
Later FRIEDRICHS (1950) showed, in a discussion of the boundary conditions in plate theory on the basis of the classical three-dimensional theory of elasticity, that the KIRCHHOFF-boundary conditions are rigorously valid for plates. KOITER (1963) showed, by a direct consideration of the equations of equilibrium and the boundary conditions in the theory of shells, that the KIRCHHOFF-boundary conditions are rigorously valid, although incomplete if the effect of transverse shear deformations cannot be neglected. His dis-cussion is independent of properties of the material.
The problem of shells was first attacked from the point of view of the three-dimensional equations of linear elasticity by H. ARON (1874). His analysis contained some questionable
approximations. The first adequate linear theory of elastic thin shells was given by A.E.H. LOVE (1888) and is also de-scribed in detail in LOVE's famous treatise on the mathemat-ical theory of elasticity. LOVE employs certain special assumptions analogous to KIRCHHOFF's; in the current litera-ture on shell theory, these assumptions are sometimes refer-red to as the KIRCHHOFF - LOVE assumptions. This theory, which has come to be known as LOVE's first approximation, despite shortcomings, has since occupied a position of prominence. Since engineering applications of the theory were hardly in sight, the subject of shell theory remained largely dormant until the late twenties of the present century. A signifi-cant exception is the development of a manageable theory of axisymmetric deformations of shells of revolution by
H. REISSNER and E. 1-IEISSNER (1913).
General interest in shell theory was re-awakened around 1930 by the needs of concrete shell roof design in civil engineering, and by the advent of monocoque aircraft struc-tures. The excellent monograph by W. FLUGGE "Statik und Dynamik der Schalen", published in 1934, admirably served the purpose to introduce engineers to the complicated and often perplexing problems of shell analysis. Like most research on shell theory in the pre-World-War-2 years, this book concen-trated on special problems for simple configurations, thus evading the complexity of LOVE's general formulation. Notable contributions to the development and simplification of the theory for circular cylindrical shells during the nineteen-thirties are also due to L.H. DONNELL.
Revival of interest in the general theory of thin elastic shells occurred first in the Soviet Union, around 1940, where significant contributions are due to A.L. GOL'DENVEIZER, A.I. LUR'E, Kh.M. MUSHTARI, W.S. WLASSOW, V.V. NOVOZHILOV and
many others. Early authors on the general aspects of shell theory in the United States were J.L. SYNGE, W.Z. CHIEN and E. REISSNER, followed in the nineteen-fifties by P.M. NAGHDI, J.L. SANDERS, B, BUDIANSKY and many others. Many of these writers gave contributions in the field of non-linear shell
theory.
In recent years the so-called "asymptotic approach" has attracted considerable attention. We mention in particular J.N. GOODIER, K.O. FRIEDRICHS and R.F. DRESSLER, A.E. GREEN and A.L. GOL'DENVEIZER. This approach which begins with the three-dimensional equations of linear elasticity, roughly speaking, may be described as follows.
First appropriate scaling of the coordinates is introduced and then each component of the stress tensor and the displace-ment vector is developed in an asymptotic power series of a characteristic small parameter. Introducing these expansions into the equations and boundary conditions of the theory of elasticity, and collecting terms with the same powers of the small parameter, one obtains sets of equations and boundary conditions which do not contain the small parameter. These partial problems are two-dimensional problems and are in principle much easier to solve than the three-dimensional problem. For example, in the problem of bending of a plate these two-dimensional problems correspond with the classical bending theory of plates, a problem of edge torsion and a problem of plane strain in a semi-infinite strip orthogonal to the edge and to the middle surface.
It was the asymptotic approach that enabled FRIEDRICHS, GOL'DENVEIZER and others to analyse the influence of edge effects, and the interaction between the edge effects and the solution of the interior problems. GOL'DENVEIZER's analysis further showed that an improved solution may be obtained in
the interior of the shell without increasing the order of the equations, but simply by modifying the boundary conditions. Explicit expressions for the modified boundary conditions were given solely for plates. For shells with free edges
explicit expressions for modified boundary conditions were given by KOITER and VAN DER HEIJDEN. (Their analysis is not based on an asymptotic approach.) Although the asymptotic approach has proved in the last decade to be a very powerful tool, it has one drawback namely it is only a formal approach. There remains an important piece of work to be done, namely
the furnishing of the mathematical proofs of the asymptotic series expansions with respect to a small parameter. Such a proof requires the determination or the assessment of the order of magnitude of the remainder terms of the expansions. Once the proof of the correctness of the series expansions has been furnished, the degree of absolute accuracy of the asymptotic approximations will at the same time be quantitat-ively determined. This also means that herewith the rigorous foundation of the two-dimensional, asjrmptotic theory of shells on the three-dimensional, linear theory of elasticity will be completed.
The problem of establishing a firm foundation of shell theory is the problem of showing that two-dimensional shell theory is consistent with three-dimensional elasticity theory. The criteria employed for assessing the validity of shell theory are error estimates, either for the differential equa-tions or, what is even more important, for the soluequa-tions. For the linear theory of thin elastic shells KOITER and DANIELSON have shown recently that when the edge tractions are distribu-ted over the shell thickness in accordance with the require-ments implied by shell theory, the root mean square error of the three-dimensional stress field predicted by linear shell
is of order e^ = h^/L^ + h/R, where R is the smallest princi-pal radius of curvature of the midsurface and L is the
smallest local wave length of the deformation pattern on this surface. In the case of a more "irregular" distribution of the edge tractions the resulting root mean square errors, as well as the local errors in the interior domain are, in gen-eral, much larger, of order e* = h/L + h/R, whereas the local errors in the edge zone are even of order unity.
In this thesis we will show that by application of the modified boundary conditions for a shell with free edges the root mean square errors in the interior domain are reduced to e^ = \P-I\P- + h/R. Further we refer to [27] where it was shown that also in the "cornerpoints" at the free edge of a shell the primary stresses are determined with a formal error of 0(£^), once the solution of the shell problem with modified boundary conditions is known. We shall evaluate the primary stresses in the "corner points" of a plate with an elliptical hole, loaded by bending couples at infinity. The theory developed in [27] was applied by DIJKSMAN to the problem of a cilindrical shell with a circular hole, loaded by torsional couples at its endfaces, and by NIEUWENHUIS to the problem of perforated plates loaded by bending couples respectively
twisting couples at infinity. Their results will be discussed in chapter 3.
C H A P T E R 1
ON SHELL T H E O R Y W I T H M O D I F I E D B O U N D A R Y C O N D I T I O N S F O R T H E F R E E E D G E S
1.1 T H E E Q U A T I O N S OF SHELL T H E O R Y W I T H M . B . C . -*• a
Let r(x ) , o = 1,2, denote the radius vector from a fixed origin in space to a generic point on the middle surface of the undeformed shell as a vector valued function of the pair of Gaussian coordinates. The tangential base vectors are ->• ->•
a = r, , where the comma preceeding the subscript a denotes a a r o • • . . ot partial differentiation with respect to the coordinate x .
-»• ->-e B
The reciprocal base is defined by a .a = 6 . The covariant a a
and contravariant metric tensors of the middle surface are -»•-»• J ag ->-a ->-e „,
given by a . = a .a. and a = a .a . These tensors are em-ag a 3
ployed in lowering and raising indices of surface tensors. The determinant of the covariant metric tensor is denoted by a, the covariant alternating tensor by £ . The normal to the
•* 1 aS -*°'^ ->
m i d d l e surface is defined b y n = — e a x a.. T h e second 2 a & f u n d a m e n t a l tensor is specified b y b „ = n . r , „. C o v a r i a n t ag ag ct surface d i f f e r e n t i a t i o n w i t h r e s p e c t to a c o o r d i n a t e x is d e n o t e d b y a n a d d i t i o n a l s u b s c r i p t a p r e c e e d e d by a v e r t i c a l s t r o k e . A l l the d e r i v a t i v e s i n t h e a n a l y s i s a r e a s s u m e d to be c o n t i n u o u s .
A p o i n t in shell space is i d e n t i f i e d b y its distance z to the m i d d l e surface and by the s u r f a c e c o o r d i n a t e s of its p r o j e c t i o n on the m i d d l e s u r f a c e . T h e shell faces z = ± -r- h ,
where h is the constant shell thickness, are surfaces
paral-lel to the middle surface. The coordinate z is orthonormal to
the surface coordinates, and the components g . of the spatial
cxp
metric tensor g.. (with determinant g) reduce to a . at the
ij ag
middle surface. The edge of the shell is assumed to be a ruled
surface formed by normals to the middle surface along an edge
curve on this surface. Let v denote the unit vector in the
tangent plane, normal to the edge curve and positive outwards.
The positive sense on the edge curve is defined by the
tan-- > • tan-- > • tan-- > •
gential unit vector t = n xv. The normal curvature tc, . , and
the geodetic torsion and curvature </^-, and K , . of the edge
curve are introduced by the relations
" ' s = ^ n ) ^ " ^ ( t ) ^ '
^'s = ^ ( t ) " " ^ v ) ^ '
^ ( t ) = - \ g ^ ^ •
t , = K - , n - K . V , s ( v ) ( n ) K , , = b „ t " t ^ ( v ) ag ' ( 1 . 1 . 1 )A deformation of the middle surface is described by the
two-dimensional displacement field
u(x ) = u a + w n , (1.1.2)
where the surface vector u and the invariant w are functions
a
of the Gaussian surface coordinates. Apart from a rigid body
displacement, the deformation is specified completely by the
differences between the first and second fundamental tensors
a -, b . in the deformed configuration and the corresponding
ag ag
° f o
tensors a ., b . in the undeformed configuration. We shall
ag ag
°
therefore employ the strain measures
^ag = i (^ag - ^ g ) ' PaB = ^a6 " ^ B ' *^ ('•'•^^
The associated expressions in terms of the displacement
com-*' In the literature the tensor p is sometimes denoted by
p^g (cf. e.g. [44]).
ponents are [45]
^ag = i ^\|3 ^ ''g|a^ "^ag^' ^^''•''^
Pag = ^|ag - ^ X g ^ ^ ^a\|g "^ ^g"K|a * ^a|g "K" ^'•'•^^
In shell theory we consider virtual deformations of the
KIRCHHOFF-LOVE type, in which normals to the undeformed middle
surface move to normals of the deformed middle surface
with-out any change in length. They are
aompleteZy
specified by
arbitrary variations of the first and second fundamental
ten-sors, subject of course to the conditions of compatibility
implied by (1.1.4) and (1.1.5). The internal virtual work per
unit area of the middle surface in a KIRCHHOFF-LOVE type
vir-tual deformation is therefore specified by an invariant
ex-pression
ag . ^ ag .
ag ag
( Y R fyR
where n and m are symmetric tensors of stress resultants
and stress couples respectively.
The external loads on the shell faces are reduced to
statically equivalent loads acting on the middle surface,
along the lines described in detail by NAGHDI [62]. For the
sake of brevity we omit body forces, and we assume that the
reduction of the loads introduces no surface couples. The
latter assumption implies that the tangential load vectors on
the two faces are parallel, and approximately equal in
magni-tude. The reduced tangential loads per unit area of the middle
a
surface are described by a surface vector p , and the reduced
normal load per unit area is a surface invariant p. The loads
on the shell edge are likewise reduced to statically
equival-ent line loads along the edge curve on the middle surface, a
force N and a couple M, both per unit arc length, where the
couple vector lies in the tangent plane. We write
N = N, . V + N. ,"t + Q n = N°'a + Q n , (1.1.6)
(v) (t) a
M = M, ,v + M. ,t = e „ M'^ a " . (1.1.7)
(v) (t) ag
The physical significance of N, ,, N. , and Q are
respect-ively the normal, tangential and transverse loads per unit
length, whereas M, , and M, , represent the torsional and
^ (v) (t)
bending moments per unit length.
The virtual work of the external loads in an arbitrary
KIRCHHOFF-LOVE type virtual deformation is now the sum of two
terms, an integral over the middle surface, and a line
inte-gral along the edge curve
J [p" 5u + p6w] dS + /
[N" 6 U + Q 6 W + M"6(J)] ds, (1.1.8)
S " 3S " "
-••
where <|) is the rotation vector defined by
->• fiR -*• -*•
(
|
) = e "^ (|) a^ + Qn, (1.1.9)
4. = w, + b"" u , n = 4- e"*^ u„| . (1.1.10)
*a a a K 2 g|a
By standard methods we now obtain the equations of
equi-librium from the principle of virtual work [45].
, ga ^ , a gK. ^ , a gK I ^ a „
/, , ,,\
(n + b ^ m ) | g + b ^ m | g + P = 0 , (1.1.11)
ag 1 ag , ag . /1 • i -,\
- m - + c „ m + b „ n + p = 0 . (1.1.12)
lag ag ag
These equations are in complete agreement with NAGHDI's
equations [62, (5.36)], obtained by integration of the three
dimensional equations of equilibrium over the shell
thick-ness.
The principal of virtual work also yields the dynamic
boundary conditions
Tn^" + 2 b^^^] V V. = N, , - K, . M. ., (1.1.13)
L K J a g (v) (t) (v)'
p . 2 b « m ^ ^ ] V g = \ t ) ^ <(v) V ) ' ('•••'^^
"""'la^g ^ ('""Vg^'s = % ) , s - Q' ('•'••5)
m°'^v V. = - M, ,, (1.1.16)
a g (t)
The equilibrium equations (1.1.11), (1.1.12) and the boundary
conditions (1.1.13) - (1.1.16) are fully exact in the
follow-ing sense: they ensure the overall equilibrium of any shell
element of finite thickness h, bounded by the shell faces and
by the ruled surface described by the normals to the middle
surface through a closed curve of infinitesimal length on
this surface. This follows from the fact that the virtual
deformations allow for all six degrees of freedom of such a
shell element of thickness h, in the motion of this element
as a rigid body.
The approximate nature of shell theory enters the picture
at the stage where the constitutive relations are introduced
fy ft CiR
between the stress resultants and stress couples n , m on
the one hand, and the middle surface deformation y o> P n on
' ag ag
the other hand. In the theory of elastic shells the
consti-tutive equations are easily obtained from the expression for
the elastic energy per unit area of the middle surface.
Starting from the assumption of an approximately plane state
of stress in the shell, KOITER [40] used qualitative
argu-ments to obtain the approximate expression for the elastic
energy per unit area of the middle surface
V ( Y . P ) = 2 T T ^
,, ^ a g ^ a g h 2 , s o e ^ a g ]
(l-v)YgY^+VY3Y^ + y2 i ( l - v ) p g p ^ + vp^pgi
( 1 . 1 . 1 7 )
where E denotes YOUNG's modulus and v is POISSON's ratio. In elastic bodies the internal virtual work equals the increment of the stored elastic energy. The constitutive equations associated with (1.1.17) are therefore obtained in the form
ag Eh r,, . ag ag KI /, , ION
n = -jr;j2 I ('~^)Y + va Y,^ . (1.1.18)
ag Eh^ r,, , ag ^ ag K] /, , ,QN
™ ° 12(l-v2) [('"^)P •" ^^ P j " (1.1.19)
Employing JOHN's rigorous and concrete pointwise estimates for all the stresses and all their derivatives in the interior domain (and in the absence of surface loads) [34], KOITER and SIMMONDS [47] showed rigorously that the constitutive equa-tions (1.1.18) and (1.1.19) contain relative errors of order e^, which implies that also the elastic energy per unit area of the middle surface has inherent relative errors of order
£ 2 .
In general, however, near edges, and near places where the load changes rapidly, the assumption of plane stress does not hold and in the vicinity of such places the expressions
(1.1.18), (1.1.19) will not hold, not even approximately. In practice we hardly ever know the precise distribution of the actual edge tractions over the thickness, or the similar displacement distribution. The exception occurs only in the case of a free edge. In that case we are ensured that the actual dynamic boundary conditions of the three-dimensional theory are specified exactly by zero edge tractions. Let us consider this case more closely. In the case of a fvee edge
the right-hand members of (1.1.13) - (1.1.16) are all spec-ified to be zero. This does not imply, however, that the edge tractions would also be zero. On the contrary, the
non-ag
tangential edge tractions of the same order of magnitude as occur in the interior shell domain. These tangential edge tractions are only absent in problems where the edge twisting moment happens to be zero. If we wish to consider the case of a really traction-free edge, if perhaps again only approxi-mately, we have to investigate at least the effect of a removal of edge tractions of the same order as the internal stresses.
We remove the linear distribution of tangential edge stresses, associated with the twisting couple, by applying the negative of these stresses as additional external loads. The resulting modification of the primary internal stress distribution is quite simple, if we are dealing with the straight edge of a flat plate and with a constant value of the twisting couple along this edge. In this case the addi-tional stresses due to the removal of the twisting couple decay exponentially with increasing distance to the edge, and they are negligible already at a distance equal to a few times the plate thickness. If we assume that the internal stress distribution in the plate (before the removal of the edge twisting couple) is a uniform torsion, the removal of the edge stresses reduces the primary stresses in the edge zone of a depth of the order of magnitude of the plate thickness, and the result is a decrease of the elastic strain energy. This phenomenon is indeed well-known in the theory of torsion of a bar with a thin rectangular cross-section (width b, height h << b ) , whose torsional rigidity is given by [55]
\ = 3 ^'^^^ 1 - ^ A ^ . 0 ( ^ e - ^ / 2 h j ] ^ (J ^ , 2 0 ) where the numerical constant A is
384
A = ^ I (2k-l)~5 = 1.26049. (1.1.21)
^ k=l
This constant may also be expressed in terms of the RIEMANN zeta function 00 C(n) = 2''(2''-l)"^ I (2k-l)"'' (1.1.22) k=l resulting in A 372 ,,. A = ^ f 5 - ^ ( 5 ) . (1.1.23) Expression (1.1.20) for the torsional rigidity may be
interpreted as follows. The removal of the twisting couple along the longitudinal edges of the thin bar or plate results in a reduction of the torsional rigidity of a magnitude -pr AGh'* at each edge, and the associated reduction in elastic
energy is -rrr AGh'*a)2 per unit length along each edge, where u 12
is the specific twist in uniform torsion of the bar.
The situation is, of course, more complicated, if we are dealing with a thin shell with a curved edge and a variable primary twisting moment along this edge. Nevertheless, if the shell is thin (h/R << 1 ) , if the edge curvature is small
|h K I << 1 ) , and if the edge twisting moment varies slowly along the edge, the most important effect of the removal of the twisting moment along the edge will again be a reduction of the torsional strain energy by an amount -TT AGh'*u)2 per unit length of the edge, where w is now the local, slowly varying twist at the edge. We are thus led to consider a modified shell problem in which the elastic strain energy of classical shell theory is reduced by the line integral of yr- AGh'*w2 along the free edge. It will be convenient for the
integral by a slightly modified edge twist o) , given by
•-' = ^ g ^"^' ^ ^(t) \ g ^"^' - 2^(v) \ g ^"^'- (••'•2^)
Such a modification, adding terms of the order of magnitude
of the product of a middle surface strain and a shell
curva-ture, does not affect the accuracy of the elastic energy if
it is applied to the entire shell [40], and it is a fortiori
permissible in the line integral. Our modified shell theory
is thus based on the
subtraction
of a line integral of
vir-tual work along the free shell edge
/
J
Ahm*6u*ds, (1.1.25)
3S
where m* = -7- Gh^oj* is the twisting couple associated with the
b
edge twist.
After f a i r l y lengthy a l g e b r a [48] the following r e s u l t i s
o b t a i n e d for the modified boundary c o n d i t i o n s a t a free edge
[n + 2 b m ] v v „ - - ^ [ K , , A h m*],
- - T - K , , < ,. A h m * = 0,
'- K -' a g 2 '- (v) •' s 2 ( t ) (n)
( 1 . 1 . 2 6 )
[ n ^ " + 2 b V n t V = 0, ( 1 . 1 . 2 7 )
•- < -' a g
" ' " ^ l a ^ g ^ ( " ' " \ S ) ' s 4 ( ^ n ) ^ ^ " ' * ) ' s - i ^ ( v ) ^ t ) ^^™* = ° '
( 1 . 1 . 2 8 )
m'^^^v. + I ( A h m * ) , ^ = 0. ( 1 . 1 . 2 9 )
a ps / sOne easily observes from a comparison with (1.1.13)
-(1.1.16), that only one of the four KIRCHHOFF conditions
remains unchanged, namely (1.1.14) for the tangential edge
1 «
loads. The terms involving the couple y A h m ' may, after
removal to the right-hand member, be interpreted a fictitious
edge loads, applied to the shell in the conventional theory
with KIRCHHOFF boundary conditions, namely a force
N* = N* V + Q* n and a couple M* = M* . t, both per unit length, where the components of the fictitious edge loads are specified by
^(v) = 1 t ^ v ) ^^'°*]'s ' i ^(t)^(n) ^ h - * ' ('-^-^O)
^*
= 7 I^^(n) ^^°'*^'s " {^(v)^(t) ^^'"*' (1.1.31)
^*(t) " 1 (Ahm*),g. (1.1.32)
These fictitious edge loads, acting along a smooth closed free edge curve, are self-equilibrated. The simple form of our modified boundary conditions in terms of fictitious edge
loads suggests that they may perhaps be obtained in a simpler way. The clue to this alternative approach is to be found in L A M B ' S ingenious interpretation of the KIRCHHOFF conditions
[52], from which we arrive at the concept of fictitious "edge beams". These "edge beams" have to behave such that the load
they transmit to the shell are in accordance with the require-ments of shell theory. For details of this more elementary derivation of the modified boundary conditions we refer to [27,47]. The boundary conditions (1.1. 26) - (1.1.29) are non-homogeneous in the sense that they contain terms with a dif-ferent order of magnitude. In fact, the additional terms in-volving the twisting couple y A h m * are of order h/L times the primary terms. (L is the wave length of the deformation pattern, h/L << 1.) The term m* may be written in the form m* = m(l+0(e*)), where m denotes the edge twisting couple corresponding with the classical theory.
In the derivation of the modified boundary conditions, terms of 0(e2) were neglected compared to unity, and it is therefore more appropriate to write the boundary conditions in the following form
[n^«+2 b%^^]v^Vg={[<(^) Ahm],^+1<(^^K^^^ A h m , (1.1.33)
[n^" + 2 b«m^^]t V, = 0, (1.1.34)
•- K -" a g
a g i ^ . a g ^ , Ir . u l ^.1 A U
m + (m V t„), = - -T K, , A h m ,
+ -^
K. \</^\
A h m ,
'a ^ a g"s 2'- (n)
•' s 2
(v) (t)
(1.1.35)
m^^v^v. = - I (Ahm), , (1.1.36)
a p z s
where the terms on the right hand sides may be interpreted as
fictitious edge loads, which are at most of order E* times
the primary loads on the shell.
The solution of the equations (1.1.11) and (1.1.12) under
the boundary conditions (1.1.33) - (1.1.36) may now be
ob-tained in the following way. Construct the solution of the
equations of equilibrium under the homogeneous boundary
con-ditions (1.1.33) - (1.1.36) (the right-hand members equal to
zero), the right hand members of the boundary conditions may
then be calculated, and then construct the solution of the
equilibrium equations under the inhomogeneous boundary
condi-tions (1.1.33) - (1.1.36). The solution of the shell problem
is then given by the sum of the solutions of the auxiliary
problems.
Although we only have to solve two shell problems, it turns
out that the analytical treatment of the problem with the
inhomogeneous boundary conditions is in general much more
difficult to handle than the problem according to classical
shell theory. This is mostly due to the appearance of
deriva-tives with respect to the arc length along the edge in the
right-hand members of the boundary conditions.
1.2 THE PRAGER-SYNGE THEOREM
First we shall consider some basic error estimates from
the linear theory of elasticity.
Let T denote an arbitrary distribution of the symmetric stress tensor in a three-dimensional elastic body. The asso-ciated complementary elastic energy is a positive definite homogeneous quadratic functional C. [T] of this stress field and is given by
C 2 [ T ( X \ Z ) ]
= ^ jji ^(1+v) T^^ T.j - V
TJT^ldV, (1.2.1)
where the integration proceeds over the entire volume of the body.Let a denote the unique actual solution for the stress distribution in the body for the linear boundary value prob-lem of the theory of elasticity, and let u denote the associ-ated displacement field. The existence of this solution is ensured under some weak conditions on the shape of the body and the boundary data, and the displacement field is also unique, except for a possible additional rigid body displace-ment field in the absence of kinematic boundary conditions
[cf. 59].
Let u denote any kinematically admissible displacement field, i.e. a displacement field which is continuous and piecewise continuously differentiable, and which satisfies the kinematic boundary conditions. Let a denote the stress field associated with u by the (linear) strain-displacement relations and HOOKE's law.
'^
. . .Let a denote any statically admissible stress distribution, i.e. a piecewise continuous and continuously differentiable distribution of the synmietric stress tensor which satisfies the equations of equilibrium in the interior of the body as well as the dynamic boundary conditions on its surface.
For the actual solution a of the boundary value problem and for any pair of a statically admissible stress distribution a,
and a stress distribution 5, associated by HOOKE's law with a
ii ^
kinematically admissible displacement field u, PRAGER and SYNGE have established the basic equality [67]
cjo - 1 (5 + 0)1 = -I- cfe-ol
2L>
2 ^ V ] 4 2L'\, \j
(1.2.2)1 ,^ '^
This equation implies that — (a + a) may be regarded as an 2 'Ki '\i
"approximation" to the actual solution, if the complementary energy associated with the stress difference a -a is "small".
In this case we may also consider each of the stress fields o or a as an "approximation", in view of the inequal-ities
(1.2.3)
(1.2.4) which are a consequence of (1.2.2). In other words, the root mean square error of the "approximate" solutions for the
1 /" "^^ ^ '^
stress distribution -r (a + a), a or a may be estimated in
Z '\, -v •%, ^
terms of the root mean square value of the stress difference
a -a. In the case of inequality (1.2.4) we have moreover
a -a (1.2.5)
where the right-hand member represents the elastic energy as a functional of the displacement field u - u which vanishes on the part of the surface where the displacement vector is specified. This implies that if the complementary energy of the stress difference a - o is made sufficiently small, then the elastic energy of the displacement field u - u will be small. It follows that u is a good approximation to the actual displacement field u.
Let n and m denote the maximum values of the tensor of stress resultants n . and the tensor of stress couples m „
ag ap respectively. The wave length L of shell theory is the
largest number such that
I l ^ n i | . m | l ^ n i l ^ i "
' V . K I - L ' l'^ag,Kl^i:' l\g,6K ! - ; [ ? ' ''"ag.6Kl-^' (1.2.6) where commas denote partial differentiation with respect to
the surface coordinates. We define a generalised minimum radius of curvature R as the largest number such that
|b J < -^ , |b - I < -V • (1.2.7)
' ag ' R ' ag,K ' R-^Further we introduce a small parameter z^ defined by h h2
= 2 = ^ + ^ . (1.2.8) The constitutive equations (1.1.18) and (1.1.19) are known
to involve errors of order e^, which implies that also the complementary elastic energy corresponding with a solution of the equations of shell theory involves errors which are in general at least of order z^ . In fact the error is larger when the edge tractions in the dynamic boundary conditions
for the three dimensional problem are not specified in accord-ance with the statically admissible stress distribution derived from shell theory [44] i.e. an approximately linear distribution of the stresses a . and an approximately
para-ap
bolic distribution of the stresses a _ over the thickness of a3
the shell.
We shall not dwell upon the influence of irregular geo-metric boundary conditions because in this work we have con-fined ourselves to the special case of free edges.
case of regular edge conditions the root mean square error of the solution of shell theory is of order z^, Regular edge conditions are defined here as prescribed edge tractions statically equivalent to the prescribed loads (1.1.13)
-(1.1.16), and distributed over the thickness conforming to the requirements of shell theory. For irregutaz' boundary condi-tions, such as the absence of the twisting moment along a free edge, which does occur along this edge under the bound-ary conditions (1.1.13)- (1.1.16), the error in the interior of the shell is of order e* = h/L+ h/R. In the next section we shall show that this error of order e* is reduced to an error of order E2=h2/L2 + h/R in the interior domain of the shell by applying the modified boundary conditions (1.1.33) -(1.1.36). The interior domain is defined here as the part of the shell of width 0(L) adjacent to the edge with the excep-tion of a boundary zone near the free edge of a width of the order of magnitude of the shell thickness.
1.3 IMPROVED ERROR ESTIMATE
In paragraph 1.1 we have discussed the modified boundary conditions for the free edge of a shell. We shall use these conditions in the form given by the expressions (1.1.26) -(1.1.29), and we shall show that the solution of the shell problem under these conditions plus two rapidly decaying stress fields corresponding with a state of torsion and a state of plane strain at the edge has a relative error of order e^. This implies that the solution of the shell problem under the modified boundary conditions yields the stress
dis-tribution in the interior domain of the shell with a relative error not exceeding E^.
1.3.1 Preliminaries
Let us denote the part of the shell between the edge and a distance of order L (L as defined in (1.2.6)) from the edge by D. The volume of this part of the shell is denoted by V, and is of order hL/K , where K is the normal curvature of
n n
the edge. In the discussion of the error estimate we shall make use of the L„-norm. Let x be a tensorfield defined on D,
then the L„-norm of T on D is defined by
||T|| = ( Id T2 dV
)K
(1.3.1.1)
D
This norm of a stress field T is closely related to the com-plementary elastic energy C_ [x] (C„ as defined in (1.2.1))
lllll^ =.fJ7 T ^ J x . . d V < - j ^ C 2 [ T ] if 0 ^ V < ^.(1.3.1.2)
In the next paragraph we shall have to deal with states of stress that are rapidly decaying with increasing distance from the edge. It is therefore convenient to introduce geo-detic coordinates on the middle surface in the edge zone, with lines x'^ = constant parallel to the edge curve and orthogonal trajectories as lines x^ = constant. We select the arc length a along the edge curve as the coordinate x^ = s, and the negative distance of the parallel curve to the edge curve as the coordinate x^ = x. The positive direction of x^ at the edge curve is therefore the unit normal vector v in the tangent plane, and the positive direction of the s-coor-dinate is the unit tangent vector t. For these coors-coor-dinates the length of line element follows from
and the only non-vanishing CHRISTOFFEL symbols of the second
kind are
r' = - la r2 = -J_a r2 = _i_a
22 2 22,1' 12
2a.^^
22,1' 22
2&^^
22,2*
(1.3.1.4)
The metric tensor in the edge zone is specified by
g,, = l. g,2 = 0, g22=l+2x<^, gi3 = g23 = 0, g33 = 1
(1.3.1.5)
with an error of order e2. From (1.3.1.4) and (1.3.1.5)
follows
4 = -^' 4 = ^' 4 = ° ('•^•'•^)
with an error of order 0(£2/L).
1.3.2 Error estimate
In order to be able to use the PRAGER-SYNGE theorem we
have to construct a statically admissible stress field, i.e.
a stress field which satisfies the equilibrium equations and
the boundary conditions, and a stress field corresponding
with a kinematically admissible displacement field, such
that the difference between the stress fields is
suffi-ciently small.
KOITER [45] and DANIELSON [6] have shown that the stress
field
s°'e(,)=in«^-l|zm'^^ + 0(£2;).
s « ( z ) = - f n « ^ l 3 . ^ ( ^ - , ) m ' ^ ^ | ^ . 0 ( £ 2 ; ) , (,.3.2.1)
3 3 3 ( . ) = , 3 3 ( _ o ) . | i n ^ ^ | ^ 3 - f (|^-,)m«^|^^.0(£2;)
where s = max s in D, is an exact solution of the equilib-rium equations. The pseudo-stress tensor s is defined by s = /g/a a ^, where a is the stress tensor of CAUCHY.
ry fi ClPi
Here n and m follow from the solution of the shell prob-lem under the modified boundary conditions. Further they showed that the stress field (1.3.2.1) corresponds with the stress field obtained from the kinematically admissible displacement field given by
/ \ / ,_i< \ n ( 1 + v ) gl
u ( z ) = u (w, + b u ) z 3 „• m k z
-a
o
a a K Eh a'g
" 2Eh L^
2(l+v)n L - vn„ ^ ' a ' g g'a z2 - i -^ Eh3 2(l+v)m'^L- vm^ a'g g ' a_z3
U3(z) = w - g ^ m ^ z2, (1.3.2.2) the difference being again 0(e2s).
However, the stress field (1.3.2.1) does not satisfy the dynamic boundary conditions of the three-dimensional theory of elasticity for the free edge. In order to overcome this shortcoming we shall, guided by our knowledge that the main cause of errors near the edge is the state of torsion at the edge, add a stress field that balances the couples and forces at the edge, again with errors of 0(£2s). This pro-cess is nepro-cessarily one of trial and error, and we may also be guided in our attempts by available knowledge due to GOLDENVEIZER and KOLOS [l5, 5 0 ] , on the similar problem in
the case of flat plates. In fact, it will appear that the problem in question for shell theory is no more difficult than in plate theory.
If the edge were straight and subjected to a constant torsional edge moment m to be removed, the associated additional displacement in the edge zone would be specified by the well-known solution of the torsion problem
Uj = 0, U2 = -g- *,, , u^ = 0 , (1.3.2.3)
where
^
is the plane harmonic function
8 V (-1)^ j2k+l 11 . 2k+l 3 ,, . o ,s
* " 74" 2 (2k+l)'t ^''P ' ~ h ~
'"^^
^^^ ~ h ~ '''' • (1.3.2.4)
k=o
We shall try and modify the displacements (1.3.2.3) in such a
way that the associated kinematically admissible stress
field satisfies the edge conditions for the stresses to be
removed as well as the equations of equilibrium to the
approximation required for our purpose. Edge tractions of
order e2s and body forces of order e2g/h are admissible
errors because their removal in their turn will not involve
internal stresses of a larger order then E 2 S , since the
error loads in question are confined to an edge zone of depth
0(h).
In cases of a variable twisting moment m along the edge,
the displacement field (1.3.2.3) implies transverse normal
stresses s^^ of order (m/hL), larger than admissible. If we
wish to retain the simple assumption u = 0, known to be
valid in plate theory [l5, 5 0 ] , we have to replace the
assumption u = 0 by u = -5m, <|)/G.
In the case of a curved edge (K ^ 0 ) , the distribution
of shear stresses s-^2 associated with the displacement
com-ponent u- violates the required linear distribution over the
thickness along the edge. This defect may be repaired by
adding to u terms of type 6m<j>, X K / G or 6m(t)K /G. We shall
2
I
n
n
include both types of correction in order to be able also to
satisfy the equilibrium equations to the desired
approxima-tion. Our
final assumption for the displacement field
is
therefore
6m,
6m r 1
*' "2 = "G" L*'l^''l*'l^^n^'=2^^J'
U3 = 0, (1.3.2.5)
where c, and c„ are two constants connected by the relation Cj + C 2 - 2 = 0.
It is now a simple matter to evaluate the kinematically admissible stress distribution associated with (1.3.2.5). The result is -12m, 41,, + 0 s"'l
[hR L2j'
12 = 6m*,,,
[I-(C,-2)XKJ- 0 [ ^ . ^ ] .
s22 = 12m,^4>., + 0 1 : ^ + " ' hR L 2 (1.3.2.6) ,13 _ 6m, (|), + 0M'
s23 = 6m[{l + (c,-2)xKj*,,3+ C 2 K ^ * . 3 ] + 0 [^ + ^ ] ,= 33 = Q fjL +
_5L1We now introduce these expressions in the left-hand members of the equilibrium equations given by (1.19) in appendix I, in order to get a condition for c and c„ such that the stress field (1.3.2.6) is a statically admissible stress distribution. Noticing that the expressions (1.19) for this stress field may be written in the form
Tl/eN = gll + sl2 + cl3 + 0 f_E_ _ m j | L (s; s,j + s,2 + s,^ + U ^ + ^L2J
L^(s) = s,- + s 2 3 . 3 < ^ s - - 0 ^ ^ . ^ m m (1.3.2.7)
L 3 ( S ) = s,13 + s23 . 0 ^^,^ . ^^^,
. . + (j>»oT = 0 we e a s i l y o b t a i n
L"(s)
f_m_ , m "I
[h2R * h i J j '
(i = 1 , 2 , 3 ) ,
( 1 . 3 . 2 . 8 )
provided c, and c satisfy the relation c C 2 + 1
0.
Solving c and c. from the two equations for these quantities 1 3
we obtain c = — , c = -r-. The stress field sjl = -12 m,^(^,,j + 0 m m hR "" L 2
6m(l) , , ( 1 - I XK^) + 0
Sl2 t s22 tsJ3 = -6m,^*,3 . 0
m _m_ hR •" L2 12 m, ((!,, + 0 s 1 m m (1.3.2.9)323 = em[(l-|xK^),,,3.|K^*,3].0 [^.^-j,
fm m^
IhR^p-J
s33 = 0thus satisfies all our requirements.
At X = 0 we obtain by summation of the FOURIER series
.J=.-6[l-i|i)..,.o[^]
,12
12
h ? " " " O h R ^ L ?
(1.3.2.10)
At x^ = ± h/2 the tractions are zero with an error of order £2s. After superposition of the stress fields (1.3.2.1) and (1.3.2.9) the remaining edge tractions at the edge x = 0 are
.11 = = 12 m
• s N
z
" 1
,13
0, s12
0, (1.3.2.11)with an error of 0(£2s). We now remove the edge tractions s^^ by applying loads with the opposite sign. The corresponding stress distribution follows from the plane strain problem for a strip X < 0, -h/2 < x3 l h/2, with boundary conditions
s l l = - 1 2 m ,
( A ^ - ( J ) , J ,sl3 = o o n x = 0 ,
(1.3.2.12)
g31 = g33 = 0 on x3 = + h/2.
We shall not solve this problem explicitly (for an explicit
solution of this problem we refer to [3l]), but we shall use
the property that the solution of this problem decays
expo-nentially with x -*• - •». Let us denote the solution of this
problem by s^^, s33 s-'^^ and s22 = v(sl'^
+s^^).
Introduction
*' " ^ p ' p ' p P ' p
V
of this stress field into the equilibrium equations yields
^'^§p) = ° (hl?^ h ^ ] ' (i= 1.2,3). (1.3.2.13)
The total stress field is now given by
s = s + s + s , (1.3.2.14)
~t ~p
and fulfils all the requirements for the application of the
PRAGER-SYNGE theorem. As a consequence of this theorem we
obtain the following estimate
l|s-i||p = 0(E2||s||p), (1.3.2.15)
where S denotes the actual solution according to the
three-dimensional linear theory of elasticity, and s is the stress
field defined in (1.3.2.14).
In other words each component
of the stress field s has a root mean square error not
exceeding 0 ( E 2 S ) .Making use of the fact that the stress fields s and s
~t -p
decay exponentially with x -*•-«> we obtain the following
result:
The
solution
of shell theory with modified boundary
condi-tions has relative error of order £2 in the
interior
of the
shell.
First we notice that it is possible to increase the accuracy of the solution of the shell problem (in the interior of the shell) by modifying only the boundary conditions, without increasing the order of the differential equations.
Secondly we note that our result establishes the validity of the second step of GOL'DENVEIZER's asymptotic theory for plates with a free edge. Although it will be possible to establish his results of other boundary conditions along the
same lines as we employed in our work, it will be less
meaningful because in practice we hardly know what the actual boundary conditions are.
1.4 ON THE CALCULATION OF THE PRIMARY STRESSES AT A FREE EDGE OF A SHELL
In the previous section we have shown that near the edge the state of stress differs considerably from the state of stress predicted by the classical theory of shells, and also from the state of stress predicted by shell theory with modi-fied boundary conditions. In order to obtain the stress dis-tribution with a relative error of 0(£2) in the edge zone we would have to solve the so-called "torsion problem" and the much more complicated "plane strain problem". However, if we
restrict ourselves to the determination of the normal stresses on a cross-section in the "corner points", the intersection of the faces with the free edge, we can evade the solution of GOL'DENVEIZER*s corresponding "plane strain problem" (cf. e.g.
[32]). At the "corner points", which are always in a state of (approximately) plane stress, even if the edge tractions do not vanish, the stress contribution of the (approximate) plane strain problem, which is formulated in terms of the edge loads to be removed, may be calculated directly from
the normal edge tractions at these points.
This contribution has to be added to the membrane and
bending stresses of the solution of the shell problem under
modified boundary conditions, and the (uni-axial) warping
stress associated with non-uniform torsion. For details of
this analysis we refer to [27]. The results of this theory
may be represented by the general formula
n _ 6m
a^^(edge, z = + ^h) =
—^ +
— ^ ±
/•i^ , 84
,^.
1116
.^.
(1+v) -3- c(3) - V - - 5 - c(5)
h
(1.4.1)
where 5(5) is RIEMANN's zeta function. The first two terms
represent the stresses of shell theory with
modified
boundary
conditions.
The first term between the brackets represents the uni-axial
stress arising from constrained warping in non-uniform
tor-sion, and the last term within the brackets is due to the
plane strain problem.
Although we expect that the edge stresses determined from
(1.4.1) will possess an error of 0 ( E 2 ) we are still in need
of a rigorous proof. It seems to us that this proof will be
much more complicated than the proof for the accuracy of the
stress distribution in the interior of the shell, since we
now require point-wise error estimates. Point-wise error
estimates for the stresses have only been obtained for the
simple axisymmetric torsion problem for shells of revolution
in linear elasticity theory [33].
There remains the question whether the stresses calculated
from (1.4.1) will be the maximum values of the stresses at
the edge. According to shell theory the bending stresses are
distributed linearly over the thickness, and will thus attain
their maximum values for z = ± h/2. The remaining states of stress may attain their maximum value in the interval -h/2 i z ^ h/2, but if the bending state of stress is suffi-ciently significant it can hardly be expected that the maxi-mum attained in an interior point z (-h/2 < z < h/2) will be
larger than the edge maximum including the bending stress. We may therefore safely assume that in those cases where the bending stresses are important, the stresses obtained from
(1.4.1) will also be the most important stresses at the edge.
CHAPTER 2
APPLICATION OF COMPLEX FUNCTIONS FOR THE CASE OF BENDING OF A FLAT PLATE WITH A HOLE
2.1 THE GOVERNING EQUATIONS
First we notice that for the special case of bending of a flat plate and in the absence of surface loads the equation
(1.1.12) reduces to the differential equation
m^^l^g = 0, (2.1.1) to be solved under the modified boundary conditions
m°'^v v^ = - Ah(m ^v^t^), , (2.1.3) a g ag s
^ r* ft
where - m .v t = m* is the twisting moment at the edge of the plate calculated according to classical plate theory, i.e. the solution of (2.1.1) with the homogeneous boundary conditions (2.1.2) and (2.1.3), and K is the curvature of the edge.
Equation (2.1.1) is equivalent to the equation
M w = 0, (2.1.A) where w(x ,x2) is the normal displacement of a point (xl,x2)
in the middle surface and A is the Laplacian operator.
In the following we assume in a point in the middle of the plate a right-handed, orthogonal cartesian coordinate system xyz*, with orientation such that the x-y-plane is the middle
surface of the plate. For the coordinate perpendicular to the middle surface we introduce z* in order to make a distinction between this coordinate and a complex variable z = x + i y . The origin of the coordinate system is chosen within the hole in the plate.
Analogously to the plane elastic problems, a formulation in terms of two analytic functions (p(z) and tp(z), is possible. This method of MUSKHELISHVILI was first applied to the bending problem of a thin plate by LECHNIZKI [53], and will now be applied to find solutions of (2.2.1) under the boundary condi-tions (2.1.2), (2.1.3).
In the following we will use the conformal mapping of the region outside the hole on to the interior of the unit circle Y. Let z = co(?) be the function that determines this conformal mapping. With the aid of z = a)(c) = to(p e ) we switch from
the cartesian coordinates x,y to the orthogonal curvilinear coordinates p = constant, 6 = constant. The purpose of applying conformal mapping is that one of the curvilinear coordinate lines p = constant coincides with the edge of the hole. At the edge of the unit circle we have C = a.
The expressions for the modified boundary conditions read 186 ^^oe
M = - - 7 5 ^(5) h - g f ^ E M*(s), (2.1.5)
where K ^ is the curvature of the edge of the hole. This last u
expression may also be replaced by the. equivalent condition s s
J Q d s + M g = J P(s)ds + C = F(s) + C, (2.1.7) o o
where C is a real constant.
The expression for the deflexion w, in terms of the ana-lytical functions (t),(c) and X,(?) is
w = Re [ ^ ) ((.,(<;) + X,(i;) ] , (2.1.8) where Re [ ] stands for the real part of the complex
func-tion between the brackets, and a bar above a funcfunc-tion denotes the complex conjugate of that function. Except for a constant the function X,(?) = X(z) is determined by
^ {X(z)} = iKz) = 4'i(?). (2.1.9) The expressions for the bending and twisting couples, and the
transverse shear forces are
M^ = - D {(l+v)[<|,'(z) +V(z) ] + "4^ [ z <l>"(z) +z ^"(z) ]
+ -*y^ [ * ' ( z ) + ; | 7 u ) ] }
My = - D {(l+v)[*'(z) + ^ ^ ) ] - -L}i [ i ^"(z) +z f ^ ) ]
- - ^ [;|j'(z) +^7(7) ] } ,
M^y = - i D - ^ i ^ [ z f ( z ) + * ' ( z ) - z <j,"(z)-*'(z) ] ,
Q^ = - 2D [V'(z) +*"(z) ] ,
Qy = - 2 iD [ ^ . " ( z ) - ^ ^ ) ] , (2.1.10)
where D is the bending stiffness defined byFh3
° = T Y r p ^ ' (2.1.11)
E is Y O U N G ' S modulus, v is POISSON's ratio and h is the plateIt proves to be more convenient to employ the following
com-plex combinations of these quantities
M - M + 2i M = 2D (1-v) Q z 4)"(z)
+ i>'(.z)
]
y X xy
M + M = - 2D (1+v) [ (|)'(z) + (|)'(z) ]
X y
Q - i Q = - 4D <l)"(z). (2.1.12)
X y
The corresponding expressions for our curvilinear system read
M^-M^ +2i M^^ = ^Dd^vK^
[_^U)^'U)*r^U)],
(2.1.13)
p2 w'((;)
M +M„ = - 2(l+v) D [*(?)+$(?) ] , (2.1.14)
P Owhere *(?) is defined by
«(?) = -J-T-r . (2.1.16)
E x p r e s s i n g M , M „ and Q i n teirras of M , M , M , Q and
p p e ^p x y x y x Q we may r e w r i t e ( 2 . 1 . 5 ) and ( 2 . 1 . 7 ) t o y i e l d M c o s 2 ( n , x ) + M c o s 2 ( n , y ) + 2 M c o s ( n , x ) c o s ( n , y ) = M* ( s ) , X y xy ( 2 . 1 . 1 7 ) (M - M ) c o s ( n , x ) c o s ( n , y ) + M | c o s 2 ( n , x ) - cos2 ( n , y ) } + y X xy s / (Q dy - Q dx) = F ( s ) + C, ( 2 . 1 . 1 8 ) o ^ w h e r e c o s ( n , x ) = 4 ^ and c o s ( n , y ) = - - ^ . ( 2 . 1 . 1 9 ) a s a s
F o l l o w i n g LECHNIZKI [53] we s h a l l d e r i v e from ( 2 . 1 . 1 7 ) and
(2.1.18) two more convenient boundary conditions. Multiplying
(2.1.17) by cos(n,x) and (2.1.18) by cos(n,y) and subtracting
the equations obtained, we arrive at
s
M dy - (M -f Q dy-Q dx) dx = M*dy + (F+C) dx. (2.1.20)
X xy •' ^x y
o •'
Multiplying (2.1.17) by cos(n,y) and (2.1.18) by cos(n,x) and
adding the equations obtained, we find
s
(M + f (Q dy-Q dx)) dy-M dx = - M*dx + (F + C) dy. (2.1.21)
xy •' X y yThese two (real) equations may now be combined to one
(com-plex) equation, i.e. the two boundary conditions (2.1.5) and
(2.1.6) will be reduced to a single
complex
condition.
Multi-plying (2.1.20) by i and subtracting (2.1.21) we find
i M dy - i M (dx-idy) + i ( / Q dy-Q dx) (dx+idy) +
X xy X y
+M dx = (M* +iF)(dx+idy) + iC(dx+idy). (2.1.22)
Introducing the expressions (2.1.11) into this equation and
using the relations
dx+idy = dz, dx-idy = dz , (2.1.23)
we obtain
- D (3+v) ((>'(z)dz+D(l-v) (|)'(z) dz+D(l-v){z cfi" (z) + i|;' (z) } dz =
= (M*+iF)dz + i Cdz. (2.1.24)
Integration of this equation yields the following functional
equation
s
- D(3+v) <j.(z) +D(l-v) [z (j)'(z) + i|;(z)] = / (M* + iF)dz + iCz + C ,
o
w h e r e C is a complex constant. For simply-connected regions and for an infinite plate w i t h a hole the constants C and C m a y , without loss of generality, b e chosen to b e zero [73]. At
the edge the following functional equation holds
< (t>j(a) + u)(o) w h e r e and 3+v 1-v '|(CT) + ii/,(a) ^ ^ ''2 (2.1.26)
^1 ^ ^^2 = 5 0 ^
J {M*(z) + i F(z)}dz z=a (2.1.27) To determine the functions <[),(?) and ii.iz) we proceed as follows: calculate the deflexion W (x,y) for the plate without a hole. From this solution we may calculate the bendingmoments M , M , the torsional couple M and the transverse x' y' ^ xy
0 0 . . . .
shear forces Q , Q along the contour C which coincides with X y
the edge of the hole. Further we may calculate the two corre-sponding analytic functions tj) (z) and ^jj (z) (see appendix II).
For a plate with a free edge we may write M = M° + M*, M = M° + M*, M = H°__ + M' Q = Q° + Q*. Q X X ^x ^y y O A Q + Q y y xy xy
*
xy (2.1.28)Here M * , M * , M * , Q * and Q * are the edge loads to b e applied
x y x y x y
to m a k e the edge f r e e . The corresponding holomorphic functions are (|)*(z) and ^* (z). The functions <\>(z) and i|'(z) w h i c h charac-terize the state of stress in the plate w i t h a hole are n o w given by
(l>(z) (z) + <t>*(z); ip(z) = ^ (z) + i|j*(z), (2.1.29)
Introducing the notation
^°(z)
= <^°[to(o] =
^^^\o;
^°(z) = /[oi(?)] =
^^^\o;
<(>*(z) = (fr*[a)(c)] =
^U)
; ^*(z) = 4'*[«(?)] = r ( ? ) ,
(2.1.30)
we may write for the functions (2.1.29)
<fj(?) = / ' \ ? ) + * ^ ( 0 ; ^,(0 =^('^(0+>l'^(?). (2.1.31)
since the influence of the hole on the stress distribution in
the plate decreases with increasing distance from the edge of
the hole, the functions cf) (5) and tp (5) may be represented as
^o(^)=
K
n
n"
". ^"' ^JO = I K ^''•
(2.1.32)
n=l
(Remember we have mapped the outer region of the plate on to
the interior of the unit circle.)
Substitution of the expressions (2.1.31) into the
func-tional equation (2.1.26) leads to
K^Ja)
+ ^
^ $^(a) + *^(a)
w'(a)
'r''2
,^(l)(a).ii(£^<^0)'(,),^(l)(,)
a)'(a)
(2.1.33)
From the conjugate expression of (2.1.26) we obtain
M(q)
*o(«) ^ ^ % K^'^'' K^'^
f, - if2 -
, , ( ! ) ( , ) . £ i ^ , ( 0 ' ( , ) , , ( . ) ( , )
(2.1.34)
The functions * (c) and
\h
(c) are obtained from CAUCHY's
inte-o inte-o
*^(0
1 r *0<°> , Y (2.1.35)1
"^J"^
*,(0 = 2 ^ J 4rr- da,
Ya-c
(2.1.36)where Y is the contour of the unit circle.
Multiplying the functional equations (2.1.33) and (2.1.34) by T^—: and integrating along the contour Y we obtain the
•^ 2TTI O-I; 6 6 6
f u n c t i o n a l e q u a t i o n s
1 r a)(a)
do
Kit (c) +i:r-r 4> (a) = ir-^ f
^ o ^ ^ " ^ 2TTI J , , . ^o^ -^ o - c 2Tri •' a - ? Y t«)'(a) ^ Y
( 2 . 1 . 3 7 )
»o''' * l i i / T ^
Y "'(a)
Here the function f is given by
, , . do 1 c c da f = f , + i f 2 - K(|) '.(1)/ V w(a) ( O ) + -'• a)'(a)
Y
.(')'(a)+*(') (2.1.38)(a)
(2.1.39)and f is the conjugate function of f.
Equation (2.1.37) may be reduced to a FREDHOLM integral equation of the second kind. However if z = io(c) is a rational function, ij) (i;) may be obtained from (2.1.37) directly. Once (k (?) is a known function tp (?) may be obtained from (2.1.38).
o o
Although it is possible to solve the functional equations directly, it is more appropriate to solve them in two steps. First solve the equations with f 0. From these sol-utions f, and f- may be calculated. Then solve the functional
. . (1) equations with known f and f remembering that now $ (a) =
ij/^'^o) = 0, so that (j)|(5) = <)>^(?) and .J;j(c) = i>JO- If we denote the sum of these solutions by (})(c) respectively ip(5)
and put
4j(C)
<t'(0
a)'(0 '
(2.1.40)
we obtain
M (edge) = - 2(l+v)D [ $ (a) + * (a) ] - M (edge) . (2.1.41)
The edge stresses a in the "corner points" are now given by
a (edge, z = ± h/2) =
+ -^Mg(edge) + (1+v)
845(3) „ 11165(5)
1 ^l^pQ^edge)
h ai
(2.1.42)
2.2 APPLICATION OF THE THEORY TO THE BENDING OF AN INFINITE
PLATE WITH AN ELLIPTIC HOLE
We consider an infinite plate of thickness h with an
ellip-tical hole. The principal axes of the ellips coincide with the
X and y axes of the cartesian coordinate system, the length
of the principal axes is 2a respectively 2b. The middle
sur-face of the plate coincides with z* = 0. The plate is loaded
at infinity such that for
• • 2z*
0 = - ^ T, o = X = 0,
X h y xy
000 = a = x = 0 . X y xy(2.2.1)
The bending moments at infinity are then given by
|x|-^»M = M = J
~ -
Tz*dz* = -^ Th2,
|y| H. oo M y = 0.
(2.2.2)
the plane with an elliptical hole on to the interior of the unit circle is given by
z = 0.(5) = R ( I + m5 ) , (2.2.3)
where
T) a+b a-b / o o / \
^ = ^ - ' °^ = 7rb • (2.2.4) As mentioned in section 2.1 we will try to solve the equations
(2.1.27) and (2.1.28) in two steps. First we solve the problem with f = f = 0, which is equivalent to solving the problem
. (1) according to the classical plate theory. The functions if (?)
and ip (5) are given by (see appendix II) •
^^'^^^^ = - 4 ( ^ ( ? ^ " ' ^ ) ' (2.2.5)
'^^'^(^^ = - 2 ( T ^ ( | " " ' ^ )• (2.2.6) Substitution of these functions into expression (2.1.39)
yields
£, , MR(m-l) , 1 , /o o T\
^(°) =
~J0=^
( ° - a )• (2-2-7)
so that (on the unit circle'.)
- M R ( m - l ) / I s rn o Q\
f = - 2 r P ^ ( 0 - " )• (2-2-8)
The right-hand sides of the equations (2.1.37) and (2.1.38)are now given by
(2.2.9) 2^1 J 0-5 r e s p e c t i v e l y 1 r I da
2.i
I '
a-5
=
46
MR(m-l) , 2 ( 1 - V ) D ' ^ ' . MR(m-l) . 2 ( l - v ) D "-• (2.2.10)Remembering the power series expansion (2.1.32), we may
evalu-ate the second term on the left-hand side of equation
(2.1.37)
1 f 1 1+ma
1
r
1 l+ma^ r - -n+1 da _
•T—r
I — 5— ) n a a = 0,
2TTI -" a m-a2 ^ n a-c
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