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```INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, VOL.
40, 1749—1765 (1997)
CALCULATION OF SHALLOW SHELL SUBJECT TO
INFLUENCE OF LOAD LOCAL EFFECT BY THE
DIFFERENCE TECHNIQUE ON IRREGULAR NETWORKS
TRAN DUC CHINH*
Department of Civil and Industrial Engineering, Hanoi ºniversity of Civil Engineering, Hanoi, Vietnam
SUMMARY
In this paper, a methodology for the calculation of shallow shell with positive Gauss radius, and boundaries
supported on the rectangular plane subject to influence of uniform load and concentrated forces has been
considered. In order to solve differential equations (written in the form that assumes deflections as
unknowns) of bending shallow shell theory, the author has used the finite difference technique on irregular
networks. A detailed algorithm has been formulated that enables to solve the problem by computer. By the
above algorithm, the author has obtained numerical results in the form of internal forces and deflections
diagrams. ( 1997 by John Wiley & Sons, Ltd.
local effect; finite difference technique; irregular networks; bending shallow shell theory; central derivatives; one-side derivatives
KEY WORDS:
1. INTRODUCTION
In fact, calculation of shallow shell by virtue of the roofs of industrial plant, market, supermarket,
etc. asks for too many new problems in content and method of calculation. These problems
include calculation of stiffened shell, shallow shell with folded surface, shell subject to influence of
non-uniform load, etc. Development of Computational Mechanics1 asks for solution of the above
problems by universal numerical method, including finite difference technique on irregular
networks.
The state of stress of shell in all the cases mentioned above is characterized by a strong local
effect. In order to clarify this effect, the mesh size should be narrowed at zones where local effect is
strong.
On the basis of methodology, the author has formulated an algorithm and a computer
program including three parts: formulation of equation set, solution of equations set and
calculation of internal forces and deflections.
By using this program, the state of stress for shell subject to load distributed by whatever law
but, symmetric to the centric axis of the shell middle surface may be identified.
* Ph.D., Department of Civil and Industrial Engg., Hanoi University of Civil Engineering
CCC 0029—5981/97/101749—17\$17.50
( 1997 by John Wiley & Sons, Ltd.
Revised 20 May 1996
1750
D. C. TRAN
2. EQUILIBRIUM EQUATION OF SHALLOW SHELL IN THE FORM OF
NON-DIMENSIONAL DEFLECTIONS
In order to use kinetic boundary conditions written for bending shallow shell, we can use
differential equations set of shallow shell with positive Gauss radius, and volume 1·5 times large
written by taking deflection as unknown. But its outstanding advantage that it satisfies deformation compatibility and boundary conditions exactly.
Overlooking impact of tangential loads, we have differential equations, of bending shallow
shell written in Cartesian co-ordinates as follows:2
L2uN
1!k L2uN
1#k L2vN
LwN
#
#
#(k #kk ) "0
x
y Lx
Lx2
2 Ly2
2 LxLy
L2vN
1!k L2vN
1#k L2uN
LwN
#
#
#(k #kk ) "0
y
x Ly
Ly2
2 Lx2
2 LxLy
(1)
LuN
D
LvN
p
+4wN #(k2#k2#2kk k )wN #(k #kk ) #(k #kk ) ! z"0
y
x y
x
y Lx
x
y
x Ly B
B
Each of the equations from (1) represents equilibrium condition of shallow shell written in the
form of a projection on respective co-ordinate axis Ox, Oy and Oz.
In set of Equations (1) k is the Poisson ratio, k and k are respective curvatures of the shell in
x
y
the direction of co-ordinate axes Ox and Oy, D"Eh3/12(1!k2) the flexural rigidity, E the
Young modulus of material, and h the shell thickness, B"Eh/(1!k2) the tensile (compression)
the resistance of shell rigidity, and p the load distributed on the shell surface (Figure 1).
z
For convenience, the set of equation (1) should be reproduced in non-dimensional form.
Calculation practice shows that normal deflection has the same order with shell thickness.
We introduce symbols
10uN
10vN
wN
u"
, v"
, w"
h
h
h
Using non-dimensional co-ordinates m"x/a, g"y/b in which a and b are the projection
length of boundary edges along axes Ox and Oy, respectively, then, shell curvatures are
k "10 ak , k "10 bk
m
x g
y
Figure 1. Geometry of shallow shell
INT. J. NUMER. METHODS ENG., VOL. 40: 1749—1765 (1997)
( 1997 by John Wiley & Sons, Ltd.
CALCULATION OF SHALLOW SHELL SUBJECT TO INFLUENCE OF LOAD LOCAL EFFECT
1751
In addition, the following symbols are given:
S
b"
b
ak
, c"Jab , j2" x
bk
a
y
Now, we have the set of equations (1) rewritten in the non-dimensional form as follows:
b2
L2u 1!k L2u 1#k L2v
Lw
#
#
#(k b2#kk ) "0
m
g
Lm2
2b2 Lg2
2 LmLg
Lm
A
A
B
1 L2v (1!k)b2 L2v 1#k L2u
k
Lw
#
#
# g #kk
"0
m Lg
b2
2
b2 Lg2
Lm2
2 LmLg
A
A
B
B
(2)
B
L4w
100h2
L4w
1 L4w
1
b4
#2
#
#k k j2# #2k w
m g
12c2
Lm4
Lm2 Lg2 b4 Lg4
j2
B A
Lu
k
Lv 100p c2(1!k2)
z
# k b2#kk
# g #kk
!
"0
m
g Lm
m Lg
b2
Eh2
In order to determine the state of shell stress, set of equation (2) should be solved with given
boundary conditions. As a result, we obtain deflections u, v and w.
Internal force and rotary deflection may be represented through u, v, w as follows:
(i) Normal resultants:
C
A
BD
C
A
BD
Eh2b
Lu
Lv
k
N"
#k # k # k w
x 10c(1!k2) Lm
m
Lg
b2 g
Lv
k
Eh2b
Lu
N"
#k # g #kk w
y 10c(1!k2) Lg
m
b2
Lm
(ii) Tangential force:
A
Lv
Eh2b
1 Lu
S"
#
20c(1#k) Lm b2 Lg
B
(iii) Bending moments in direction of coordinates axes:
A
A
b2 L2w
Eh4
k L2w
M "!
#
x
12(1!k2) c2 Lm2 b4 Lg2
B
b2 1 L2w
Eh4
L2w
M "!
#k
y
12(1!k2) c2 b4 Lg2
Lm2
B
(iv) Twisting moment:
L2w
Eh4
M "!
xy
12(1#k)c2 Lm·Lg
( 1997 by John Wiley & Sons, Ltd.
INT. J. NUMER. METHODS ENG., VOL. 40: 1749—1765 (1997)
1752
D. C. TRAN
(v) Elements of shearing force in direction of co-ordinate axes:
A
A
L3w
Eh4b
1 L3w
Q "!
b2
#
x
Lm3 b2 LmLg2
12(1!k2)c3
1 L3w
Eh4b
L3w
Q "!
#
y
12(1!k2)c3 b4 Lg3 Lm2Lg
B
B
(vi) Rotation of shell elements against co-ordinates:
A
A
B
B
hb Lw
k
h"
! m u
g
100
c Lm
h Lw
k
h"
! g v
m bc Lg
100
(vii) Rotation of shell elements surrounding the direction normal to the shell surface:
A
hb Lv
1 Lu
h"
!
z 20c Lm
b2 Lg
B
3. FORMULATION OF FINITE DIFFERENCE RELATIONSHIPS ON
IRREGULAR NETWORKS
Solving the equation set (2) by using finite difference technique, by which the equation set (2) is
reproduced as a set of (na#b) linear algebraic equations of which n is the number of network
nodes, a is the number of differential equations given by (2) and b the number of equations
describing boundary conditions.
Then, the derivative of the equation at each network node will be represented by the value of
the respective functions of the adjoining network nodes.
In most cases, generally, in order to solve the problem of shell theory, finite difference technique
on constant networks1~3 is used. At zones located near the contour, networks with narrowed
mesh size is used. This process is implemented after already basic equation set that has been
formulated for regular networks. Calculation practice shows that the magnitude of unknowns at
zones with narrowed mesh size does not affect unknowns obtained by solving the above basic
equation set. It is quite sure that classical approach to the regular networks will reduce accuracy
of the internal force and deflection (at zones with narrowed mesh size) in comparison with the
technique on irregular networks when formulating and solving the equation set (2) of the
problem.
Irregular networks4 are usually used to represent the state of stress at zones located near the
point where local load is situated on or near the shell boundary, etc.
Assume that the derivative of a certain function f at one point needs to be represented through
values of the function at adjoining points (Figure 2), and assume that s is the measurement of the
mesh size. If the network is formed at non-dimensional co-ordinates, s will be the relative mesh
size which is the ratio of the length of mesh size and respective length of measurement of the shell
in the plane.
Representing the function f at adjoining zone of point i in ¹aylor series:
A
B
\$
\$
p
3
p
p
2
+ S
+ S
+ S
i`k
i`k
i`k
f A\$ k/\$1
f @@@\$ · · ·
f \$ "f \$ k/\$1
f @\$ k/\$1
i
i
i p
i
i
3!
2!
1!
\$
A
INT. J. NUMER. METHODS ENG., VOL. 40: 1749—1765 (1997)
B
(3)
( 1997 by John Wiley & Sons, Ltd.
CALCULATION OF SHALLOW SHELL SUBJECT TO INFLUENCE OF LOAD LOCAL EFFECT
1753
Figure 2. Presentation of relative mesh size in the case, if the network is formed at non-dimensional co-ordinates
from this
s2
s3
"f ! s f @ # i f A! i f @@@#· · ·
i
i i
2! i
3! i
(4)
s2
s3
f "f #s
f @# i`1 f A# i`1 f @@@#· · ·
i`1
i
i`1 i
2! i
3! i
(5)
(s #s )2
(s #s )3
i`2 f A# i`1
i`2 f @@@#· · ·
f "f #(s #s ) f @# i`1
i
i
i`2
i
i`1
i`2 i
2!
3!
(6)
(s #s )2
(s #s )3
i~1 f A! i
i~1 f @@@#· · ·
f "f !(s #s ) f @# i
i~2
i
i
i~1 i
i
i
2!
3!
(7)
f
i~1
Considering the network node that is far from the shell boundary, value of derivatives may be
represented through the value of the respective functions at adjoining network node. Derivative
relationships formulated by this way is called central derivatives.
If the point in consideration belongs to the boundary or its adjoining zone, derivative value
may be represented by function value at the point located on one side of the point in consideration. The relationships of derivatives at this point is called one-side derivatives.
3.1. Formulation of central derivative representation
Subtracting (4) from (5) member by member, f @ may be obtained overlooking the value of
derivatives of secondary and tertiary order:
f !f
i~1
f @" i`1
i
s #s
i
i`1
Eliminating f @ from (4) and (5) we obtain
(8)
C
A
B
D
C
A
B
D
2
s
s
f A"
f ! 1# i`1 f ! i`1 f
(9)
i
i`1
i
s
s
s i~1
i
i
ss
1# i`1
i i`1
s
i
From (4) and (5), exact representation of f @ may be found:
i
1
s2
s2
f @"
f ! 1! i`1 f ! i`1 f
i
i`1
i
s2
s2 i~1
s
i
i
s
1# i`1
i`1
s
i
Taking note that the newly found representation of f @ changes by combining (9) and equations
i
(2) to obtain a set of algebraic equations with coefficients that are not symmetric against the main
diagonal that implies it does not obey the reciprocal law. Therefore, in this paper, we shall use only
representation of f @ according to (8).
( 1997 by John Wiley & Sons, Ltd.
A
B
A
B
INT. J. NUMER. METHODS ENG., VOL. 40: 1749—1765 (1997)
1754
D. C. TRAN
Formulas (8) and (9) may be written in the briefer form as follows:
i`1
f @" + A f ,
i
k k
k/i~1
i`1
f A" + B f
i
k k
k/i~1
(10)
of which,
k!i
A"
k
s (1#a )
i
i
2
B"
[a (i!k!1)i~k`1#(1#a )(k!i!1)k~i`1]
k
i
s2 a (1#a ) i~1
i i
i
s
s
s
a " i`1 , a " i`2 , a " i
i
i`1
i~1
s
s
s
i
i`1
i~1
According to the principle of derivation, the derivatives of third and fourth order contained in
(2) can be calculated:
i`1 k`1
i`1
+ AB f
f @@@"( f A )@" + A f A" +
k m m
i
k k
i
k/i~1 m/k~1
k/i~1
i`1 k`1
i`1
+ BB f
f IV"( f A )A" + B f A" +
k m m
i
k k
i
k/i~1 m/k~1
k/i~1
of which,
2
B "
[a
(k!m!1)k~m`1#(1#a
)(m!k!1)m~k`1]
m s2 a (1#a ) k~1
k~1
k k
k
Based on the expressions obtained, the value of derivatives at point ij can be found easily for
irregular networks generated in the direction of both axes co-ordinates (i is the number of
network lines in the direction of axis Ox and j is in the direction of axis Oy):
Lf
i`1
Lf
j`1
L2f
i`1
L2f
i`1 j`1
ij" + A f ,
ij" + A f ,
ij" + B f ,
ij " +
+ AAf ,
k k Lg
l il
k kj LmLg
k l kl
Lm
Lm2
k/i~1
l/j~1
k/i~1
k/i~1 l/j~1
L2f
j`1
L3f
i`1 k`1
L3f
j`1 l`1
ij" + B f ,
ij" +
ij" +
+ AB f ,
+ AB f ,
l il
k m mj
l n in
Lg2
Lm3
Lg3
l/j~1
k/i~1 m/k~1
l/j~1 n/l~1
i`1 j`1
L3f
i`1 j`1
L4f
i`1 k`1
L3f
ij " +
ij " +
ij" +
+ AB f ,
+ BA f ,
+ BB f ,
k
l
kl
k
l
kl
k m mj
LmLg2
Lm2Lg
Lm4
k/i~1 l/j~1
k/i~1 l/j~1
k/i~1 m/k~1
L4f
j`1 l`1
L4f
i`1 j`1
ij" +
ij " +
+ BB f ,
+ BB f
l
n
in
k l kl
Lg4
Lm2Lg2
l/j~1 n/l~1
k/i~1 l/j~1
(11)
Values of A , B , and B could be obtained from representations of A , B and B by substitul l
n
k k
m
ting index i in j, k in l and m in n.
INT. J. NUMER. METHODS ENG., VOL. 40: 1749—1765 (1997)
( 1997 by John Wiley & Sons, Ltd.
CALCULATION OF SHALLOW SHELL SUBJECT TO INFLUENCE OF LOAD LOCAL EFFECT
1755
3.2. Establishing representation of one-side derivatives
3.2.1. Case No 1. The number of points where derivatives are taken, will be minimum.
By eliminating f @ from (5) and (6), and overlooking all terms of the developing expression and
commencing from f @, we obtain
2
f A"
[ f a ! f (1# a )# f ]
i s2 a (1# a ) i i`1
i`1
i`1
i`2
i`1 i`1
i`1
For the derivative of first order we have the following representation:
f !f
i`1
f @" i`2
i
s a
i`1 i`1
If symbols
1
A "
(i!k)i~k`2
k1 s a
i`1 i`1
and
2
B "!
[(i!k#1)!(1#a )(k!i!2)k~1]
k1
i`1
s2 a (1#a )
i`1 i`1
i`1
are inserted, the derivatives of first and second orders will have the form
i`2
i`2
f @" + A f , f A" + B f
i
kl k
i
kl k
k/i
k/i
If derivation is made continuously, derivatives of third and fourth orders will be represented as
follows:
i`2 k`2
i`2 k`2
f @@@" + + A B f , f IV" + + B B f
i
kl ml m
kl ml m
k/i m/k
k/i m/k
of which B will be determined similarly to B substituting index i in k and k in m.
m1
k1
3.2.2. Case No 2. Similarly, f @ , f A , f @@@ and f IV will be found, of which
i i i
(k!i)k~i`2
2
A "!
, B "
[a (i!k!1)#(1#a )(i!k!2)i~k]
k2
k2 s2 a (1#a ) i~1
i`1
s
i~1
i~1 i~1
i~1
Representation similar to (11) of the one-side partial derivatives may be shown in both cases
but, the bound of totalization has to be changed according to the suitable directions and
according to the altered indexes of coefficients A and B.
If the one-side partial derivative has to be calculated at the network nodes located one unit far
from the shell contour, then, the partial derivatives of first and second orders will be determined
as the central derivatives and partial derivatives of third and fourth orders will be determined as
one-side derivatives, for example f @@@"+i`1 +k`2 A B f .
i
k/i~1 m/k k m1 m
Depending on boundary conditions, when solving various problems, there are many ways of
difference combinations for representations of central and one-side derivatives.
( 1997 by John Wiley & Sons, Ltd.
INT. J. NUMER. METHODS ENG., VOL. 40: 1749—1765 (1997)
1756
D. C. TRAN
4. EQUILIBRIUM EQUATIONS OF THE SHELL. BOUNDARY CONDITIONS AND
SYMMETRICAL CONDITIONS IN REPRESENTATION OF FINITE DIFFERENCES
By using (11) for the points of shell located far from the boundary, equations (2) can be rewritten
in the form
i`1
1!k j`1
1#k i`1 j`1
i`1
+ Bu #
+
+ A A v #(b2k #kk ) + A w "0
b2 + B u #
k kj
l il
k l kl
m
g
k kj
2b2
2
k/i~1
l/j~1
k/i~1 l/j~1
k/i~1
j`1
1#k i`1 j`1
1!k
k
j`1
1 i`1
+ Bv #
+
+ A A u # g #kk
+ A w "0
b2 + B v #
k kj
l il
k l kl
l il
m
2
b2
2
b2
k/i~1
l/j~i
k/i~1 l/j~1
l/j~1
(12)
A
A
B
B
A
B
i`1
k
j`1
1
(k b2#kk ) + A u # g #kk
+ A v # k k j2# # 2k w
m
g
k kj
l
il
m
g
m
ij
b2
j2
k/i~1
l/j~1
i`1 k`1
k`1 j`1
1 j`1 l`1
100h2
+ B B w #2 +
+ B Bw #
+
+ BB w
#
b4 +
k m mj
k l kl
l n in
12c2
b4
k/i~1 m/k~1
k/i~1 l/j~1
l/j~1 n/l~1
100c2(1!k2)
"
p
z
Eh2
A
B
In order to facilitate computer use, it is better to choose the following written ways for (12):
i`1 j`1
+
+ (A u #A v #A w )"0
uukl kl
uvkl kl
uwkl kl
k/i~1 l/j~1
i`1 j`1
+
+ (A u #A v #A w )"0
vukl kl
vvkl kl
vwkl kl
k/i~1 l/j~1
i`1 j`1
k`1 l`1
+
+
A u #A v # +
+ A
w "q
wukl kl
wvkl kl
wwkl kl
ij
k/i~1 l/j~1
m/k~1 n/l~1
A
(13)
B
of which
A
1#k
"A "
A AF
uvkl
vukl
k l
2
A "A "(k b2#kk )A d F
uwkl
wukl
m
g k 1
k
A "A " g #kk A c F
vwkl
wvkl
m l 1
b2
A
B
(1!k)
Bc F
A "B d b2#
l 1
uukl
k 1
2b2
(1!k)
B
b2B d F
A " l #
k 1
vvkl b2c
2
1
1
100h2
1
# j2#2k c d c d F#
d d B B b4# 2c d B B # c c B B F
A
"k k
1 1 2 2
1 2 k m
1 1 k l
wwmn
m g j2
12c2
b4 1 2 l n
A
B
c "1!(k!i)2,
1
A
d "1!(l!j)2, c "1! (m! k)2
1
2
100c2(1!k2)
d "1! (n!l)2, q "
p F
2
ij
zij
Eh2
B
(14)
F"1/4(s # s )(s # s )
i
i`1 j
j`1
INT. J. NUMER. METHODS ENG., VOL. 40: 1749—1765 (1997)
( 1997 by John Wiley & Sons, Ltd.
CALCULATION OF SHALLOW SHELL SUBJECT TO INFLUENCE OF LOAD LOCAL EFFECT
1757
All coefficients of equations set (12) will be multiplied by F to reduce the equations set into
canonical form.
It is necessary to emphasize that in case the shell will be subject to the action of concentrated
forces, the value of concentrated force P should be inserted in representation of q , instead of
zij
ij
product P F.
zij
For the network node in question, internal force can be rewritten in the form
(i) Normal resultants:
C
C
A
B D
B D
i`1
k j`1
k
+ Au #
+ A v # km# kg w
k kj b2
l il
ij
b2
k/i~1
l/j~1
1 j`1
i`1
kg
Eh2b
+ A v #k + A u #
N "
#kkm w
l il
k kl
yij 10c(1!k2) b2
ij
b2
k/j~1
k/i~1
(ii) Tangential force:
Eh2b
N "
xij 10c(1!k2)
Eh2b
S "
ij 20c(1#k)
C
A
i`1
1 j/1
+ Av#
+ Au
k k b2
l il
k/i~1
l/j~1
D
(iii) Bending moments:
C
C
D
i`1
k j`1
+ Bw #
+ Bw
k kj b4
l il
k/i~1
l/j~1
1 j`1
i`1
Eh4b2
+ B w #k + B w
M "!
l il
k kj
yij
12c2(1!k2) b4
l/j~1
k/i~1
(iv) Twisting moment:
Eh4b2
M "!
xij
12c2(1!k2)
D
Eh4
i`1 j`1
M "!
+
+ A Aw
xyij
k l kl
12c2(1#k)
k/i~1 l/j~1
(15)
(v) Shearing forces:
C
C
i`1 k`1
Eh4b
1 i`1 j`1
Q "!
+ AB w #
+
+ A Bw
b2 +
xij
k m mj b2
k l kl
12c2(1!k2)
k/i~1 m/k~1
k/i~1 l/j~1
1 j`1 l`1
i`1 j`1
Eh4b
+
+ AB w # +
+ AB w
Q "!
l n in
l k kl
yij
12c2(1!k2) b4
l/j~1 n/l~1
k/i~1 l/j~1
(vi) Rotations of shell element:
D
A
A
D
B
B
i`1
k
+ Aw ! m u
k kj 100 ij
k/i~1
j`1
k
h
+ Aw ! g v
h "
l il 100 ij
mij cb
l/j~1
(vii) Rotation around the normal direction of shell element:
hb
h "
gij
c
hb
h "
zij 20c
( 1997 by John Wiley & Sons, Ltd.
A
i`1
1 j`1
+ Av !
+ Au
k kj b2
l il
k/i~1
l/j~1
B
INT. J. NUMER. METHODS ENG., VOL. 40: 1749—1765 (1997)
1758
D. C. TRAN
Figure 3. Presentation of relative mesh size of the shell in plane for the case of hinge supporting edge
Boundary conditions written for the case of hinge supporting edge (Figure 3) have the form
u"0, v"0, w"0 when x"0, x"a, y"0, y"b.
If the number of line according to j on the axis m is considered as zero, and when g"1, it equals
to r, then u "v "w "u "v "w "0.
i0
i0
i0
ir
ir
ir
If the number of line (according to i on the axis g) equals to zero, and when m"1, it equals p,
then u "v "w "u "v "w "0.
0j
0j
0j
ir
ir
ir
The identical null condition of the bending moment taken for respective axes can be obtained
from the relation
L2w
i`1
1,j" + B w "0
k kj
La2
k/i~1
Assuming that s "s , then
i
i`1
L2w
w !2w !w
ij" 1,j
0,j
~1,j
La2
s2
i
or, if the former condition is considered, then w "!w
.
1,j
~1,j
Similarly w "!w
;w
"!w
; w "!w
.
i,1
i,~1 r~1,j
r`1,j i,p
i,p`1
In order to reduce coefficient matrix about 4 times for shells under symmetrical load having
rectangular plane of different curvatures in two directions, attention should be paid to the
property of symmetry relative to the central axis of the structure.
In order to show symmetrical conditions, the first and third equations of the equation set (13)
can be rewritten as follows:
(a)
b2(1#a )s
j j [u
!(1#a )u #a u
]
i`1,j
i ij
i i~1,j
2s a
i i
INT. J. NUMER. METHODS ENG., VOL. 40: 1749—1765 (1997)
( 1997 by John Wiley & Sons, Ltd.
CALCULATION OF SHALLOW SHELL SUBJECT TO INFLUENCE OF LOAD LOCAL EFFECT
1759
(1!k)s (1#a )
i
i [u
#
!(1#a )u #a u
]
ij`1
j ij
j ij~1
4b2s a
j j
1#k
[v
#v
!v
!v
]
#
i`1,j`1
i~1,j~1
i~1,j`1
i`1,j~1
8
(b)
1
# (k b2#kk )s (1#a )(w
!w
)"0
g j
j i`1,j
i~1,j
4 m
k
1
!u
)# g #kk
k k j2# #2k w #(k b2#kk )(u
ij
m
g i`1,j
i~1,j
m
m g
b2
j2
A
B
T
A
G
B
b4s (1#a )
1
j
j w
i`2,j
s3a
a (a #a )
i i
i`1 i
i`1
1#a
1
1#a
a #1
i`1#w
i# i
!w
#
i`1,j a a
ij a (a #a )
a
a #1
i i`1
i i
i`1
i
i~1
a #a
a
2
i~1
i`1# w
i`1
!w
#
i~1,j
i~2,j
a
a (1#a )
s s aa
i~1
i~2
i~1
ij i j
]Mw
!w
(1#a )! w
(1#a )#w
a
i`1,j`1
i`1,j
i
ij`1
i
i`1,j~1 j
#w
a #w (1#a )(1#a )! w
(1#a )a ! w
(1#a )a
i~1,j`1 i
ij
i
j
ij~1
i j
i~1,j
j i
1
1#a
s (1#a )
j`1
j w
]
!w
#w
a a N# i
ij`2 a (a #a )
ij`1 a a
i~1,j~1 i j
b4s3a
i`j j
j`1
j j`1
j j
1
1#a
a
a a
j# j`1
j`1 j`1#w
#w
#
!w
ij a (a #a )
ij~1 a
ij~2
a
1#a
j j
j`1
j
j~1
j~1
a
100p c2(1!k2)s s (1#a )(1#a )
j`1
z
ij
i
j "0
]
!
(16)
a (1#a )
4Eh2
j~2
j~1
In view of the fact that the shell and load have a symmetric property the networks also will
form also symmetrically: that means distance from network nodes (following different directions)
to the central axis will be repeated according to the mirror symmetric law:
](v
ij`1
!v
100h2
)#
ij~1
12j2
C
C
H
G
U
D
D
H
S "S
, S "S
, etc.
p@2
p@2`1
p@2
p@2`2
The number of network lines will be odd. It should be noted that at network nodes with same
distance to the symmetric axis, normal deflections and tangential deflections that are parallel to
the axis have the same values, while the tangential deflections in the direction that is orthogonal
to the axis will be equal in absolute value and opposite in sign. The symmetric condition may be
represented in the form of
w
"w
, w
"w
p@2~n
p@2`n
r@2~n
p@2`n
u
"!u
, u
"u
, u "0
p@2~n
p@2`n
r@2~n
r@2`n
p@2
v
"p
, v
"!v
, v "0
p@2~n
p@2`n
p@2~n
p@2`n
r@2
of which n is any integer.
For the nodes that do not lie on the axis, symmetric condition may be obtained easily by
replacing the said values of w in equations (16). Replacing these values in (16) when i"p/2, the
( 1997 by John Wiley & Sons, Ltd.
INT. J. NUMER. METHODS ENG., VOL. 40: 1749—1765 (1997)
1760
D. C. TRAN
first condition will be satisfied automatically; as for the second condition, coefficients of deflection
points (that do not lie on the axis) increase twice, while coefficients of deflection at the points that
belong to the symmetric axis, remain the same.
In order to obtain the validity of the reciprocal law, all coefficients should be divided by 2. We
can obtain the equations that are identical with those mentioned above by deduction according
to physical view also, but only if the network nodes that belong to the symmetric axis bear just
For the central network node ( j"r/2, i"p/2), coefficients of deflection at this point should be
divided by 4. As the result, for the network nodes belonging to the symmetric axis, equations in
the form of tangential deflection along the symmetric axis will be satisfied automatically, and
their formulation will be necessary. Therefore, we do not need to formulate equations for points
on the shell boundary.
If the shell has a square plane and identical curvature in two directions, the property of
symmetry to the diagonal of the equations set may be used also. In this case, it should be assumed
that the mesh size in both directions is unchanged. Then the normal deflections of points that are
symmetric to each other and to the diagonal will have equal values, on the one hand; and on the
other hand, the following relation will exist between tangential deflections in different directions:
w
"w
; v
"u
; u
"v
i,j`n
i`n,j
i`n,j
i,j`n
i`n,j
i,j`n
From the above arguments, formulation of equations for one of the tangential deflections at the
points on the diagonal will not be necessary, but in formulating the remaining equations, coefficients of deflections from the network nodes belonging to the diagonal should be divided by 2.
The complete equation set of the problem will have z "3(p!1)(r!1) equations; and if the
property of symmetry to the central axis is considered, the number of equations will decrease to
3pr p#r
z" !
4
2
In the case of symmetry to the diagonal ( p"r), the number of equations will be
z"3p( p/2#1)/(4!p), that means approximately (1/8) of the initial equation set of the problem.
5. RESOLUTION OF THE EQS. SET AND CALCULATION OF INTERNAL FORCES
In order to find non-dimensional deflection, any known method may be used to solve linear
algebraical equation set. Here, method of orthogonalization5 is used, following this we have
recurrent relation (for direct resolution) in the form of
i~1 a a
a "a ! + mk mi
ik
ik
a
m/1 mm
1 i~1
+ a b
b "b !
mi m
i
i
a
ii m/1
of which i is the number of the line of square matrix; k the number of column. Here, k'i means
considering only such triangle of the matrix, a is the coefficients of initial matrix; b the coefficients
of column matrix of load term; a and b are the intermediate parameters;
When solving conversely, we have
AB
AB
1 i`1
x "b !
+ a x
i
i
im m
a
ii m/z
INT. J. NUMER. METHODS ENG., VOL. 40: 1749—1765 (1997)
( 1997 by John Wiley & Sons, Ltd.
CALCULATION OF SHALLOW SHELL SUBJECT TO INFLUENCE OF LOAD LOCAL EFFECT
1761
It should be noted that when using the above formulas of orthogonalization method we shall
obtain two-dimension array of coefficients. Number of coefficients will be t"z2 .
In order to bring the matrix into one-dimensional form, we have to overlook coefficients of the
under-part of the matrix triangle. Then, the number of the term belonging to the diagonal is
i~1
s" + z!m#1
m/1
and the number of any term equals to l"s#k!i.
In this case, size of the coefficient array will be t"z(z#1)/2, which means that the calculated
volume of the array reduces approximately twice.
After finding non-dimensional deflections, the internal force value at any network nodes can be
determined according to (15). When doing so symmetric and boundary conditions of the problem
have to be taken into consideration.
6. SEQUENCE OF ESTABLISHING COEFFICIENT ARRAY OF
THE INITIAL MATRIX OF THE PROBLEM
For the initial matrix the coefficient array becomes minimum and the equation set takes the
canonical form such that the symmetric conditions of the set should be abided strictly in
accordance with the order of columns and lines of the array, and when coefficient array has been
formed, it should be changed into the one-dimensional form.
6.1. The case, when the shell has symmetric property to the central axis
We number the points of shell lying in the direction of co-ordinate axes (Figure 4). For
convenience, the points of shell boundary will be symbolized by the digit 2. Then, the points of
symmetric axis will take the symbol ( p/2#2) and (r/2#2).
For points inside the boundary of the fourth shell plane, we number in the following order: at
first, giving i"3 and keeping j variable; after that, giving i"4 and keeping j variable, etc.
Therefore, the number of any points at the shell will be s"j!2#(i!3)r/2.
Figure 4. Presentation of relative mesh size in the case, when the shell has symmetric property to the central axis
( 1997 by John Wiley & Sons, Ltd.
INT. J. NUMER. METHODS ENG., VOL. 40: 1749—1765 (1997)
1762
D. C. TRAN
Thus, at each network node we have three equations. For each point belonging to the
symmetric axis, we need only to formulate two equations (one equation of tangential deflection in
the direction of the symmetric axis is satisfied automatically. Hence, its formulation is not
necessary). For the central network node, only one equation will be necessary.
Equations will be written according to the order of the unknowns w, u, v.
Thus, responding to the order numbering of nodes as stated above, we obtain the numbers of
equations as follows:
(i) For the unknown w: s "d "3s !(i!1);
2
3
1
(ii) For the unknown u: s "d "3s !(i!2);
2
4
1
(iii) For the unknown v: s "d "3s !(i!3).
2
5
1
In this case, for points belonging to the line i"p/2#2, i.e., the symmetric axis is parallel to the
axis Ox ( j"r/2#2). Formulation of equation for the unknown v is not needed. Similarly, we can
determine the number of column of matrix coefficients for the equations. As pointed above, the
full quantity of equation will equal to
z"(3/4)pr!(p#r)/2.
After numbering all the coefficients, those of under triangular part of matrix may be overlooked.
6.2. ¹he case, when the Eqs. set has symmetric property to the diagonal.
Repeating the same procedure in this case, the number of any network node will take the form
A
B
i~3 p
s "j!2# +
!m
1
2
m/1
For the nodes belonging to the diagonal, we have i"j, and for the remaining nodes, we have
j'i. The number written for any network node (including the node belonging to the central
symmetric axis) will take the form
(i) For w: s "d "3s !2i #3;
2
3
1
(ii) For u: s "d "3s !2i #4;
2
4
1
(iii) For v: s "d "3s !2i #5.
2
5
1
If the network nodes belongs to the central symmetric axis j"p/2#2, formulation of
equations for the unknown v is not needed.
At the points of the diagonal,
(i) For w: s "d "3s !2i#4;
2
3
1
(ii) For u: s "d "3s !2i#5;
2
4
1
and formulation of equation for v is not needed.
At the central node formulation of equation for even u is not needed.
7. EXAMPLE 1: CALCULATION OF THE SHALLOW SHELL WITH RECTANGULAR
PLANE HINGE SUPPORTING JOINT ALONG THE BOUNDARY
Considering that the shell has two-way curvature (Figure 4) with its sizes in the plane are
a"18 m, b"24 m, and with curvatures are: k "1/24 (m~1), k "1/32 (m~1). As shown in
x
y
INT. J. NUMER. METHODS ENG., VOL. 40: 1749—1765 (1997)
( 1997 by John Wiley & Sons, Ltd.
CALCULATION OF SHALLOW SHELL SUBJECT TO INFLUENCE OF LOAD LOCAL EFFECT
1763
Figure 5. Diagrams of the internal forces and deflections for Example 1
Table I. Some results of internal forces and deflections at the central point of middle surface
of the shell (Example 2)
Int. forces and defl.
Cases
N (T/m)
1
N (T/m)
2
M (T/m)
Q (T/m)
¼ (mm)
Regular networks
!1·774
!0·80
—
0·445
!0·097
1·13
Irregular networks
!2·446
!0·81
—
0·447
!0·099
1·35
1%
0·5%
2·2%
20%
Error
37%
Figure 5, the displacement w is not zero along y"0, but is zero along x"0. For u, v, we have
u"v"0 along x"0 and along y"0. The number of mesh layers p"r"6, shell thickness is
h"3 cm, elastic module of material (concrete mark 300) E"3]106 T/m2, Poisson ratio k"0,
uniform distributed load is p "1 T/m2. In the problem, we use the relativistic mesh sizes of the
z
under mesh layer as follows: 0·01; 0·09; 0·4; 0·4; 0·09; 0·01.
8. EXAMPLE 2: CALCULATION OF THE SHALLOW SHELL ON RECTANGULAR
PLANE WITH LEVY’S TYPE BOUNDARY CONDITIONS
Considering that the shell has a two-way curvature (Figure 4) with its sizes in the plane are
a"18 m, b"24 m, and with curvatures are k "1/24 (m~1), k "1/32 (m~1). The number of
x
y
mesh layers p"r"6, shell thickness is h"3 cm, elastic module (concrete mark 300)
E"3]106 T/m2, Poisson ratio k"0, concentrated force P"2 T which is placed at the central
point of the middle surface of the shell. We use the relativistic mesh sizes of the under mesh layer
as follows: 0·01; 0·09; 0·4; 0·4; 0·09; 0·01.
( 1997 by John Wiley & Sons, Ltd.
INT. J. NUMER. METHODS ENG., VOL. 40: 1749—1765 (1997)
1764
D. C. TRAN
Figure 6. Diagrams of the internal forces and deflections for Example 2
.
INT. J. NUMER. METHODS ENG., VOL. 40: 1749—1765 (1997)
( 1997 by John Wiley & Sons, Ltd.
CALCULATION OF SHALLOW SHELL SUBJECT TO INFLUENCE OF LOAD LOCAL EFFECT
1765
Let the computer AT-386 programming by language FORTRAN-77. The results obtained of
the problem are diagrams of the internal forces and deflections given in the Figure 5 (Example 1)
and Figure 6 (Example 2).
Note: In Figure 6:
(1) the case of regular networks;
(2) the case of irregular networks.
Some numerical results of internal forces and deflections (at the central point, where the
concentrated force is placed) are given in Table I (for Example 2) for comparison.
9. CONCLUSIONS
Based on the obtained results of two examples, especially the second one, the following conclusions can be withdrawn:
(i) Classical approach to the regular networks will reduce accuracy of the internal forces and
deflections at the place where concentrated force are situated.
(ii) Difference technique on irregular networks is the universal numerical method, with the help
of which the load local effect can be evaluated effectively and exactly, especially at the place
where concentrated force are situated.
Problems, pertaining to the shallow shells with various boundary conditions, in particular the
Levy’s type, are treated to inllustrate the potentialities of this technique.
REFERENCES
1. D. V. Vainberg et al., ‘Algorithm and numerical problems of theory of plates and shell’, in ¹he Collection ¹heory of
plates and Shell, Academy of Science of Armenia, Erevan, 1964 (in Russian).
2. A. A. Nazarov, Fundamental ¹heory and Methods of Calculation of Shallow Shell, ed. I. A. Rabinovich, Moscow,
Strojizdat,1966 (in Russian).
3. N. P. Abovskii et al., ‘Calculation, of shallow shell by finite difference method in matrix form’, Krasnoiyarski ¹echnical
ºniversity, 1965 (in Russian).
4. B. Heinrich, Finite difference Methods on Irregular Networks. A Generalized Approach to Second Order Elliptic
Problems, Birkauser, Basel, 1987.
5. N. C. Berezin and N. P. Jydkov, Computational Methods, Vol. 1, Nauka, Moscow, 1966 (in Russian).
6. T. D. Chinh, Proc. ¼CCM-III, Vol. 1, Extended Abstracts, Chiba, Japan, August 1—5, 1994, pp. 151—152.
.
( 1997 by John Wiley & Sons, Ltd.
INT. J. NUMER. METHODS ENG., VOL. 40: 1749—1765 (1997)
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