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An infinite element theory
An infinite element theory John F. Brotchie Division of Building Research, Australia 3190 (Received January 1983)
CSIRO, PO Box 56, Graham
Road, Highett,
Victoria.
A technique for solution of plane problems in mathematical physics or mechanics is presented. The method was developed initially for elastic plate problems with transverse load concentrations but is here extended to other two-dimensional problems. Draw-down problems with a number of wells in a porous medium, heat-flow problems in a plate with a distributed source, and with discrete sinks, or membrane displacements under concentrated forces, are considered in particular. The technique is to treat the two-dimensional medium as infinite: each source or sink is treated separately and equilibrated by an auxiliary source function. Superposition allows the effects of the separate sources and sinks to be added. The residual effect of the auxiliary source, after superposition, is negligible so that the summed effects of the individual sources and sinks gives the required solution. The method is approximate but the error may be made arbitrarily small. It is simple and well suited to the class of problems above, and to either manual or computer based analyses. Examples of its use are given. Key words: mathematical multiple singularities
Introduction The finite element theory is widely known and used. This paper discussesan alternative, infinite element theory. AJI infinite element differs from finite elements in that it is without external boundaries or with these boundaries at infinity. The type of problem to which it is suited also differs, in that it may involve multiple singularities or large fluctuations in a function or its derivatives over small areas of an extensive or infinite, two dimensional solution space. Two-dimensional isotropic elements of infinite extent are considered in particular. Unlike finite elements they cannot be arranged contiguously in one plane. On the other hand, if they are assumed to be identical and linear in behaviour, these behaviours can be superposed. They are easy to locate; a definition of their centres (and direction) is enough. Their centres do not have to coincide, and it is often of advantage if they do not.
Method Consider initially a thin plate in the linear eiastic range with concentrated transverse loading. Consider the concentrated loads Pi to each act on a different infinite plate element i. Since there is no finite boundary, there are no finite boundary forces to equilibrate the applied load Pt. This equilibrium can be provided in other artificial ways 0307-904X/83/03153-04/$03.00 0 1983 Butterworth &Co.
(Publishers)
Ltd.
model,
plane problems,
mathematical
physics,
and a convenient means in the case of a plate is to apply an equilibrating auxiliary force to its surface proportional to the displacement &. Thus each separate loading Pi on the plate is then equilibrated by an artificial or auxiliary reaction of intensity X@i,i.e.: Pi =
s
X$i da
0
in which $i (=&(A)) is the displacement of the plate under load Pi and the auxiliary equilibrating force system (which may be provided by a liquid of density X or springs of modulus A). Similarly, each individual reaction Ri on the plate is equilibrated by an auxiliary reaction of intensity X#i. If all loads and actual reactions are superposed, the net displacement is I#J= C#i + C$i and the net auxiliary real‘ tion is A$. The parameter h may be so chosen 111. ‘!/I IlG+ll and hence: ~~h~dUl*llh~idUl.~llij 0
0
Appl.
Math.
Modelling,
1983,
Vol.
7, June
153
Infinite
element
theory:
J. F. Brotchie
Thus after superposition, the residual effect of the auxiliary (artificial) reaction is negligible, and the sum of the superposed displacements, I$, approximates the actual displacement in the plate to any degree of accuracy required. Shears and moments may then be obtained by the same superposition procedure. This technique is particularly suited to the accurate analysis of displacements, moments and shears in internal panels of flat plate structures and has been utilizedIm to produce tables of these at various locations in the panel, under various concentrated or uniform loadings and various panel shapes. The same techniques may be applied to other twodimensional problems. The mathematical basis is as follows: consider initially the problem of a thin elastic plate under transverse loads.
the solid body or fluid pressure head or surface level in the porous medium. For a membrane in a uniform tensile field, equation (7) again applies, in which e is the membrane tension, and $I the transverse displacements under transverse loading f. The source function f is again considered to be comprised of individual source Pi and sinks Rj SO that:
f da= CPi + ZRi (8) s The addition of the artificial sink &#Iin this case gives the modified equation: eV’@-h#=f
(9)
Superposition may again be utilized to allow each source Pi (or sink Rj) to be applied separately and to be equili-
Formulation
brated by the artificial sink function h#i (or X$). Superposition of the separate potentials #i and $j gives:
The plate equation may be written in the biharmonic form4 :
Again X is so chosen that:
dV=V=@ = f
#=C@i+
dV2V2#+h@=f
(2)
where:
f da = ZPi + ERj
(4)
0
Each load Pi and reaction Ri may then be analysed separately and their separate effects superposed. The equations for $i for various loading distributions are derived elsewhere.2’4 Two only are given here. The equation for r#+at a distance ri from a point load Pi at the origin i is given by: @i = Pi(/‘/2md)
kei(xi)
(5)
where kei(xi) is a Bessel Kelvin function of the order zero,’ Xi is a dimensionless radial ordinate = ri/l where ri is the radial ordinate, and I = (d/X)“4 is the ‘radius of relative stiffness’ of the plate. Under a uniform load of intensity1 over the whole plate: $i
(6)
= fix
Various loading conditions may be obtained by superposition from the two above. Superposition of all loads and reactions then gives the actual behaviour in the plate. The error due to the residual term X$ may be made arbitrarily small by choosing X small (or I large) or may be corrected for by considering the effects of the net load h@(see below). Other classical two-dimensional problems may be handled by this same technique. Consider the classof plane problems, such as two-dimensional steady-state flows of heat through solids or of fluids through a porous medium, which is governed by the Poisson equation: eV2@= f
(7)
in which e is a flow characteristic of the medium, f is a source function, and $I a potential such as temperature in
154
Appl.
Math.
(10)
(1)
in which d is the plate stiffness, f is the applied load, V* is the Laplace operator and $I the displacement normal to the plate. Addition of the artificial reaction gives:
I
C@j
Modelling,
1983,
Vol.
7, June
11911 < II9ill
and hence the net effect of the artificial sink is negligible after superposition as before. Under a single point source of magnitude Pi on the infinite element, the solution of equation (9) is given by: 6
=
(PiDre)
Kdxi)
(11)
in which Ke(Xj) is a modified Bessel function of zero order,’ xi = r,/l is the nondimensional radial ordinate from the point source i, ri is the actual radial ordinate from the point source i, and I = (e/h) 1’2where e is the flow characteristic (or uniform tensile membrane force). Under a uniform source f, the solution is given by: @i = flh (12) The required solution 4 is again given by equation (10). Where the sinks Rj are of unknown magnitude, these may be determined from solution of equation (IO) for known values of @Jat various points j. The technique has particular application where the sources and sinks are discrete and concentrated at points (of singularity in $Jand its derivatives) - or uniform - and continuous. Draw-down problems in wells, heat flow problems in plates, and membrane displacement problems in a uniform tensile field, may each be treated in this way. The mixed harmonic and biharmonic equation:
V2V2@+ V2#= f is only rarely met, e.g., in large deflection theory of plates4 (combined flexure and membrane action) or other refined theories in physics but may be handled in the same way. The technique may also be used with finite external boundaries, using collocation to satisfy external boundary conditions.6
Examples Plate j7exure
The technique above was developed initially for plate deflection problems 3,4 but its broader applications are con-
lntlnite
sidered here. Examples of the use of the technique for plate problems are given in the literature.‘12 In the case of plates, the solution technique has been programmed for computer solution using equations (3), (5) and (6) and the resulting tables and contour plots of displacements, moments and shears in internal panels of flat plate structures under various loading conditions and panel geometries are presented. A typical plot of panel moments for a square panel under uniform loading is shown in Figure 1. Moments in a 1S-to-l rectangular panel under uniform loading are shown in Figure 2. A more extensive set of tables’ using this technique have been produced recently.
element
theory:
J. F. Brotchle
-0.4 -0
3
-0.2
-01 0
-cY
01 02 0.3 0.4
Draw-down An example of its extension to a draw-down problem with a number of (oil or water) wells in a porous medium is now considered. The liquid is being pumped from a number of wells arranged in a square grid of spacing S. A quantity Q is being taken from each well, of radius 0.02s. The draw-down profile is required. The draw down surface is obtained from equations (lo), (11) and (12). The parameter 1 is selected arbitrarily at a value sufficiently large to reduce the error to an acceptable amount (see later), e.g. set I = S, the well spacing, in calculating xi for use in equation (11). Including only the closest four wells in equation (lo), the surface level 4 at a point midway between four wells is given by:
0 v
6 0
I I 0 0 $AidW-0
I 0
I p
I 0
f
0 P NuA”‘OI-4
?
P
0
?
-0.4 -0.3 -0
2
-01 0
-‘Y
0.1 0.2 0.3 0.4
4 = (Q/2ne)
4Ko( 1/d2)
= 0.415 Q/e
At the edge of the well (radius 0.025’) the draw-down level 4, including only the closest four wells in equation (10) is qb= Q/e(0.641+ 0.069 +0.067 +0.067) = 0.844Q/e. The differential draw-down is then $I = (0.844 -0.415)Q/e = 0.429Q/e. If the next closest 100 wells are included, the estimated differential draw-down in 0.414 Q/e. Greater accuracy
tx
Column radius 0.01 L contour interval Figure 2 Moments M, and My in a 1: 1.5 panel load (V = 0, p = 1, taken from reference 2)
uniform
. requires a large value of I (e.g. 1 = 10 or more times the well spacing) and inclusion of additional wells. The draw-down at other points in the field may be similarly determined, allowing draw-down curves to be plotted in similar form to the moments in Figure 1. The technique is applicable to computer-based solution in which the function Kc(x) is generated, or to hand calculation as above using tables’ of the function Kc(x). The parameter 1 or X is chosen to provide the required trade-off between error level due to the functions X$Jand rate of damping of the functions K,(x) with increasing x, and hence the number of terms that must be included in the summation over i and J’in equation (10).
Membrane
Figure 1 Moment My is a square internal panel of a flat plate structure under uniform load (Poisson’s ratio Y = 0, panel load p = 1, contour interval 0.01, taken from reference 2)
0.01 under
displacements
A homogeneous infinite membrane under uniform tension c is supported on circular columns, arranged in a square grid, and loaded at the centre of one square panel. The columns are at spacing S and are of radius O.O2S, and the load Q has radius 0.02s also. Membrane displacement under the load Q and the reactions Pi at the columns is given by equations (10) and (11). Equating this displacement to zero at the columns allows the reactions Pi to be determined. If only the closest four columns are considered and 1 is given the value 2S, displacements at the columnsare#=O=O.l94Q/e+ l.lSlPJe: giving Pi = 0.169Q
Appl.
Math. Modelling,
1983, Vol. 7, June
155
Infinite
element
theory:
J. F. Brotchie
I = 8s gives Pi = 0.206Q and 4 = 0.637Q/~
(3) and (10). Even fewer columns may be considered when only differential displacements are required, or when only one panel is loaded, as in the examples above. Hence a practical effect of the auxiliary reaction is to allow the series summation of equations (3) and (10) to be truncated (by replacing the actual coupling of loads and reactions which is not reflected in the mathematical solutions of equations (1) and (7) by a coupling with the auxiliary source).
I= 16SgivesPi=0.215Qand#=0.638Q/e
Conclusions
and displacement at the centre of the panel under the ring load Q is: #=0.751Q/e-0.778x(O.l69Q/e) = 0.619Q,‘c Setting I = 4S gives Pi = 0.192Q and $ = 0.633Q/e and For greater accuracy, a larger value of I and additional column reactions Pi should be included.
Error estimate The estimated error in displacement due to the residual A@,in the case of the uniformly loaded plate with columns arranged in a square grid (example l), is: S4 < 0.0008
I4
IV,
in which S is the column spacing and W0the displacement of the panel centre. The estimated relative error in maximum moments is of the same order. For the case of a uniformly loaded membrane with column supports in a square grid, or the equivalent drawdown problem of example 7, the estimated maximum error in displacement is of the order of: r2n YZ O.OlOF or <0.03% W0 in which S is again the column (or wall’) spacing and IV0 the differential displacement at the panel centre. The relative error is larger for the single load of example 3, as both the reaction and the displacement are affected in this case. These errors may each be made arbitrarily small by choice of a suitably large value for 1. Displacements @iin equations (5) and (11) are significant onlv over a radius -Tiz 5, so that columns outside this radius may be neglected in the summations of equations
156 Appl. Math. Modelling, 1983, Vol. 7, June
The technique presented here is particularly useful for discrete sources and sinks in an extensive or infinite medium, e.g. as in the case of oil or water wells in a sedimentary basin, discrete heat sources or sinks in a continuous plate, or concentrated transverse loads or reactions on a tensile membrane or on a thin elastic plate. This class of stress concentration or singularity problem, in which # and/or its derivatives may vary substantially over small areas in the regions of the singular points, is well suited to the technique described above and not so well suited to finite element methods. The singularity itself is avoided by selecting points close to but not at the singular point, or by considering sources and sinks of small but finite area. The technique was developed initially for plate deflection problems and is generalized here to other two-dimensional problems in mathematical physics of the above class.
References Bares, R. ‘Tables for the analysis of plates, slabs and diaphragms based on the elastic theory’, Macdonald and Evans, Plymouth, UK, 1979 Brotchie, J. F. and Wynn, A. ‘Elastic deflections and moments in an internal panel of a flat plate structure’, CSIRO AUG. Div. Building Rex Tech. Paper 4 (Second Series), 1975 Brotchie, J. F. Proc. Amer. Concr. hr. 1957, 54, 31 Brotchie, J. F. ‘On elastic plastic behaviour of flat plate structures’, D.Eqq Thesis, University of California, Berkeley, 1961 Jahnke, E., Emde, F. and Ltiseh, F. ‘Tables of higher functions’, Teubner Verlagsgesellschaft, Stuttgart, 1960 Robinson, N. I. and Brotchie, J. F. ‘Plate analysis using collocation at the boundary’, CSIRO Ausr. Div. Building Res. Tech. Paper 25,197O
CSIRO, PO Box 56, Graham
Road, Highett,
Victoria.
A technique for solution of plane problems in mathematical physics or mechanics is presented. The method was developed initially for elastic plate problems with transverse load concentrations but is here extended to other two-dimensional problems. Draw-down problems with a number of wells in a porous medium, heat-flow problems in a plate with a distributed source, and with discrete sinks, or membrane displacements under concentrated forces, are considered in particular. The technique is to treat the two-dimensional medium as infinite: each source or sink is treated separately and equilibrated by an auxiliary source function. Superposition allows the effects of the separate sources and sinks to be added. The residual effect of the auxiliary source, after superposition, is negligible so that the summed effects of the individual sources and sinks gives the required solution. The method is approximate but the error may be made arbitrarily small. It is simple and well suited to the class of problems above, and to either manual or computer based analyses. Examples of its use are given. Key words: mathematical multiple singularities
Introduction The finite element theory is widely known and used. This paper discussesan alternative, infinite element theory. AJI infinite element differs from finite elements in that it is without external boundaries or with these boundaries at infinity. The type of problem to which it is suited also differs, in that it may involve multiple singularities or large fluctuations in a function or its derivatives over small areas of an extensive or infinite, two dimensional solution space. Two-dimensional isotropic elements of infinite extent are considered in particular. Unlike finite elements they cannot be arranged contiguously in one plane. On the other hand, if they are assumed to be identical and linear in behaviour, these behaviours can be superposed. They are easy to locate; a definition of their centres (and direction) is enough. Their centres do not have to coincide, and it is often of advantage if they do not.
Method Consider initially a thin plate in the linear eiastic range with concentrated transverse loading. Consider the concentrated loads Pi to each act on a different infinite plate element i. Since there is no finite boundary, there are no finite boundary forces to equilibrate the applied load Pt. This equilibrium can be provided in other artificial ways 0307-904X/83/03153-04/$03.00 0 1983 Butterworth &Co.
(Publishers)
Ltd.
model,
plane problems,
mathematical
physics,
and a convenient means in the case of a plate is to apply an equilibrating auxiliary force to its surface proportional to the displacement &. Thus each separate loading Pi on the plate is then equilibrated by an artificial or auxiliary reaction of intensity X@i,i.e.: Pi =
s
X$i da
0
in which $i (=&(A)) is the displacement of the plate under load Pi and the auxiliary equilibrating force system (which may be provided by a liquid of density X or springs of modulus A). Similarly, each individual reaction Ri on the plate is equilibrated by an auxiliary reaction of intensity X#i. If all loads and actual reactions are superposed, the net displacement is I#J= C#i + C$i and the net auxiliary real‘ tion is A$. The parameter h may be so chosen 111. ‘!/I IlG+ll and hence: ~~h~dUl*llh~idUl.~llij 0
0
Appl.
Math.
Modelling,
1983,
Vol.
7, June
153
Infinite
element
theory:
J. F. Brotchie
Thus after superposition, the residual effect of the auxiliary (artificial) reaction is negligible, and the sum of the superposed displacements, I$, approximates the actual displacement in the plate to any degree of accuracy required. Shears and moments may then be obtained by the same superposition procedure. This technique is particularly suited to the accurate analysis of displacements, moments and shears in internal panels of flat plate structures and has been utilizedIm to produce tables of these at various locations in the panel, under various concentrated or uniform loadings and various panel shapes. The same techniques may be applied to other twodimensional problems. The mathematical basis is as follows: consider initially the problem of a thin elastic plate under transverse loads.
the solid body or fluid pressure head or surface level in the porous medium. For a membrane in a uniform tensile field, equation (7) again applies, in which e is the membrane tension, and $I the transverse displacements under transverse loading f. The source function f is again considered to be comprised of individual source Pi and sinks Rj SO that:
f da= CPi + ZRi (8) s The addition of the artificial sink &#Iin this case gives the modified equation: eV’@-h#=f
(9)
Superposition may again be utilized to allow each source Pi (or sink Rj) to be applied separately and to be equili-
Formulation
brated by the artificial sink function h#i (or X$). Superposition of the separate potentials #i and $j gives:
The plate equation may be written in the biharmonic form4 :
Again X is so chosen that:
dV=V=@ = f
#=C@i+
dV2V2#+h@=f
(2)
where:
f da = ZPi + ERj
(4)
0
Each load Pi and reaction Ri may then be analysed separately and their separate effects superposed. The equations for $i for various loading distributions are derived elsewhere.2’4 Two only are given here. The equation for r#+at a distance ri from a point load Pi at the origin i is given by: @i = Pi(/‘/2md)
kei(xi)
(5)
where kei(xi) is a Bessel Kelvin function of the order zero,’ Xi is a dimensionless radial ordinate = ri/l where ri is the radial ordinate, and I = (d/X)“4 is the ‘radius of relative stiffness’ of the plate. Under a uniform load of intensity1 over the whole plate: $i
(6)
= fix
Various loading conditions may be obtained by superposition from the two above. Superposition of all loads and reactions then gives the actual behaviour in the plate. The error due to the residual term X$ may be made arbitrarily small by choosing X small (or I large) or may be corrected for by considering the effects of the net load h@(see below). Other classical two-dimensional problems may be handled by this same technique. Consider the classof plane problems, such as two-dimensional steady-state flows of heat through solids or of fluids through a porous medium, which is governed by the Poisson equation: eV2@= f
(7)
in which e is a flow characteristic of the medium, f is a source function, and $I a potential such as temperature in
154
Appl.
Math.
(10)
(1)
in which d is the plate stiffness, f is the applied load, V* is the Laplace operator and $I the displacement normal to the plate. Addition of the artificial reaction gives:
I
C@j
Modelling,
1983,
Vol.
7, June
11911 < II9ill
and hence the net effect of the artificial sink is negligible after superposition as before. Under a single point source of magnitude Pi on the infinite element, the solution of equation (9) is given by: 6
=
(PiDre)
Kdxi)
(11)
in which Ke(Xj) is a modified Bessel function of zero order,’ xi = r,/l is the nondimensional radial ordinate from the point source i, ri is the actual radial ordinate from the point source i, and I = (e/h) 1’2where e is the flow characteristic (or uniform tensile membrane force). Under a uniform source f, the solution is given by: @i = flh (12) The required solution 4 is again given by equation (10). Where the sinks Rj are of unknown magnitude, these may be determined from solution of equation (IO) for known values of @Jat various points j. The technique has particular application where the sources and sinks are discrete and concentrated at points (of singularity in $Jand its derivatives) - or uniform - and continuous. Draw-down problems in wells, heat flow problems in plates, and membrane displacement problems in a uniform tensile field, may each be treated in this way. The mixed harmonic and biharmonic equation:
V2V2@+ V2#= f is only rarely met, e.g., in large deflection theory of plates4 (combined flexure and membrane action) or other refined theories in physics but may be handled in the same way. The technique may also be used with finite external boundaries, using collocation to satisfy external boundary conditions.6
Examples Plate j7exure
The technique above was developed initially for plate deflection problems 3,4 but its broader applications are con-
lntlnite
sidered here. Examples of the use of the technique for plate problems are given in the literature.‘12 In the case of plates, the solution technique has been programmed for computer solution using equations (3), (5) and (6) and the resulting tables and contour plots of displacements, moments and shears in internal panels of flat plate structures under various loading conditions and panel geometries are presented. A typical plot of panel moments for a square panel under uniform loading is shown in Figure 1. Moments in a 1S-to-l rectangular panel under uniform loading are shown in Figure 2. A more extensive set of tables’ using this technique have been produced recently.
element
theory:
J. F. Brotchle
-0.4 -0
3
-0.2
-01 0
-cY
01 02 0.3 0.4
Draw-down An example of its extension to a draw-down problem with a number of (oil or water) wells in a porous medium is now considered. The liquid is being pumped from a number of wells arranged in a square grid of spacing S. A quantity Q is being taken from each well, of radius 0.02s. The draw-down profile is required. The draw down surface is obtained from equations (lo), (11) and (12). The parameter 1 is selected arbitrarily at a value sufficiently large to reduce the error to an acceptable amount (see later), e.g. set I = S, the well spacing, in calculating xi for use in equation (11). Including only the closest four wells in equation (lo), the surface level 4 at a point midway between four wells is given by:
0 v
6 0
I I 0 0 $AidW-0
I 0
I p
I 0
f
0 P NuA”‘OI-4
?
P
0
?
-0.4 -0.3 -0
2
-01 0
-‘Y
0.1 0.2 0.3 0.4
4 = (Q/2ne)
4Ko( 1/d2)
= 0.415 Q/e
At the edge of the well (radius 0.025’) the draw-down level 4, including only the closest four wells in equation (10) is qb= Q/e(0.641+ 0.069 +0.067 +0.067) = 0.844Q/e. The differential draw-down is then $I = (0.844 -0.415)Q/e = 0.429Q/e. If the next closest 100 wells are included, the estimated differential draw-down in 0.414 Q/e. Greater accuracy
tx
Column radius 0.01 L contour interval Figure 2 Moments M, and My in a 1: 1.5 panel load (V = 0, p = 1, taken from reference 2)
uniform
. requires a large value of I (e.g. 1 = 10 or more times the well spacing) and inclusion of additional wells. The draw-down at other points in the field may be similarly determined, allowing draw-down curves to be plotted in similar form to the moments in Figure 1. The technique is applicable to computer-based solution in which the function Kc(x) is generated, or to hand calculation as above using tables’ of the function Kc(x). The parameter 1 or X is chosen to provide the required trade-off between error level due to the functions X$Jand rate of damping of the functions K,(x) with increasing x, and hence the number of terms that must be included in the summation over i and J’in equation (10).
Membrane
Figure 1 Moment My is a square internal panel of a flat plate structure under uniform load (Poisson’s ratio Y = 0, panel load p = 1, contour interval 0.01, taken from reference 2)
0.01 under
displacements
A homogeneous infinite membrane under uniform tension c is supported on circular columns, arranged in a square grid, and loaded at the centre of one square panel. The columns are at spacing S and are of radius O.O2S, and the load Q has radius 0.02s also. Membrane displacement under the load Q and the reactions Pi at the columns is given by equations (10) and (11). Equating this displacement to zero at the columns allows the reactions Pi to be determined. If only the closest four columns are considered and 1 is given the value 2S, displacements at the columnsare#=O=O.l94Q/e+ l.lSlPJe: giving Pi = 0.169Q
Appl.
Math. Modelling,
1983, Vol. 7, June
155
Infinite
element
theory:
J. F. Brotchie
I = 8s gives Pi = 0.206Q and 4 = 0.637Q/~
(3) and (10). Even fewer columns may be considered when only differential displacements are required, or when only one panel is loaded, as in the examples above. Hence a practical effect of the auxiliary reaction is to allow the series summation of equations (3) and (10) to be truncated (by replacing the actual coupling of loads and reactions which is not reflected in the mathematical solutions of equations (1) and (7) by a coupling with the auxiliary source).
I= 16SgivesPi=0.215Qand#=0.638Q/e
Conclusions
and displacement at the centre of the panel under the ring load Q is: #=0.751Q/e-0.778x(O.l69Q/e) = 0.619Q,‘c Setting I = 4S gives Pi = 0.192Q and $ = 0.633Q/e and For greater accuracy, a larger value of I and additional column reactions Pi should be included.
Error estimate The estimated error in displacement due to the residual A@,in the case of the uniformly loaded plate with columns arranged in a square grid (example l), is: S4 < 0.0008
I4
IV,
in which S is the column spacing and W0the displacement of the panel centre. The estimated relative error in maximum moments is of the same order. For the case of a uniformly loaded membrane with column supports in a square grid, or the equivalent drawdown problem of example 7, the estimated maximum error in displacement is of the order of: r2n YZ O.OlOF or <0.03% W0 in which S is again the column (or wall’) spacing and IV0 the differential displacement at the panel centre. The relative error is larger for the single load of example 3, as both the reaction and the displacement are affected in this case. These errors may each be made arbitrarily small by choice of a suitably large value for 1. Displacements @iin equations (5) and (11) are significant onlv over a radius -Tiz 5, so that columns outside this radius may be neglected in the summations of equations
156 Appl. Math. Modelling, 1983, Vol. 7, June
The technique presented here is particularly useful for discrete sources and sinks in an extensive or infinite medium, e.g. as in the case of oil or water wells in a sedimentary basin, discrete heat sources or sinks in a continuous plate, or concentrated transverse loads or reactions on a tensile membrane or on a thin elastic plate. This class of stress concentration or singularity problem, in which # and/or its derivatives may vary substantially over small areas in the regions of the singular points, is well suited to the technique described above and not so well suited to finite element methods. The singularity itself is avoided by selecting points close to but not at the singular point, or by considering sources and sinks of small but finite area. The technique was developed initially for plate deflection problems and is generalized here to other two-dimensional problems in mathematical physics of the above class.
References Bares, R. ‘Tables for the analysis of plates, slabs and diaphragms based on the elastic theory’, Macdonald and Evans, Plymouth, UK, 1979 Brotchie, J. F. and Wynn, A. ‘Elastic deflections and moments in an internal panel of a flat plate structure’, CSIRO AUG. Div. Building Rex Tech. Paper 4 (Second Series), 1975 Brotchie, J. F. Proc. Amer. Concr. hr. 1957, 54, 31 Brotchie, J. F. ‘On elastic plastic behaviour of flat plate structures’, D.Eqq Thesis, University of California, Berkeley, 1961 Jahnke, E., Emde, F. and Ltiseh, F. ‘Tables of higher functions’, Teubner Verlagsgesellschaft, Stuttgart, 1960 Robinson, N. I. and Brotchie, J. F. ‘Plate analysis using collocation at the boundary’, CSIRO Ausr. Div. Building Res. Tech. Paper 25,197O