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. Received 2 October 1995 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 half of the load. 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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