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Chang et al.: Finite Difference Analysis of Vertically Loaded Raft Foundation Based on the Plate Theory with Boundary Concern 135
FINITE DIFFERENCE ANALYSIS OF VERTICALLY LOADED RAFT
FOUNDATION BASED ON THE PLATE THEORY WITH BOUNDARY
CONCERN
Der-Wen Chang 1, Hsin-Wei Lien
2, and Tzuyu Wang 2
ABSTRACT
This paper introduces the finite difference formulations for the deformations of a surface raft foundation under vertically
uniform loads. The thin-plate theory and central difference scheme were adopted to derive the equations in which the finite raft was
affected by boundary conditions. With the use of approximate soil springs underneath the raft, none-uniform settlements of the
foundation can be analyzed through the newly proposed method. A computer program WERAFT-S is suggested accordingly.
Solutions of the analysis were examined with the three-dimensional finite element analysis on numerical models. It was found that
the foundation settlements of the newly proposed analysis can be rational by modeling the soils underneath raft as the axial forced
elements providing that their stiffness were calibrated carefully prior to the analysis. If the Lysmer’s analog model was adopted for
the soil springs, much smaller foundation settlements would be resulted at the corners since the analog model is more adequate for
rigid footing analysis. Applicability and details of the newly proposed solution are introduced.
Key words: Raft foundation, finite difference analysis, thin plate theory, boundary condition, vertical loads.
1. INTRODUCTION
The load-settlement behaviors of the shallow foundations can
be modeled as two-dimensional or three-dimensional problems.
Both modeling can be done by the Finite Element Analysis (FEA).
For more simplified solutions of the two-dimensional model, the
foundation can be treated as a one-dimensional beam on a series
of soil springs. Such analysis is termed as Beam on Elastic
Foundation or Winkler foundation where the soil springs can be
elastic or inelastic. This type of solution is applicable when the
length-to-width ratio (L/W) of the foundation (where L is the
length, W is the width) exceeds 10. The plane-strain condition is
assumed in this case. Such analysis has been discussed for decades
(Biot 1937; Mathews 1958; Bowles 1977; Ting and Mockry 1984;
Jones 1997; Chen 1998; Tomlinson and Boorman 2001; Dinev
2012; Chiou et al. 2016; Chang et al. 2016). Among the available
analyses, finite difference approach (FDA) was suggested on foun-
dation settlements caused by statically structural loads (Bowles
1977; Tomlinson and Boorman 2001). A more recent application
can be found in Chang et al. (2016) on seismic responses of a rec-
tangular piled raft foundation subjected to horizontal ground exci-
tations. A recent report comparing various solutions of the beam
analysis can be found in Omer and Arbabi (2015).
The three-dimensional analysis can be done by simply
treating the foundation as a two-dimensional raft (or mat) resting
upon the soil springs. Complexities are evolved in deriving the
governing equations of the foundation’s deformations.
Nevertheless, analytical solutions in this regard have been
suggested by Timoshenko and Krieger (1959), Vlasov and
Leontev (1966), and Kukreti and Ko (1992). Owing to the
complexities involved the mathematical expressions, these
solutions were difficult to be used by common engineers. An
approximate solution of the two-dimensional raft based on a series
of strip footings has been suggested by Poulos (1991). The limits
of the strip-footing solution were modified by Poulos (1994) with
another type of approximate numerical solutions for plate on soil
continuums. Boundary integral solution was adopted in such
analysis.
The finite difference solution of the two-dimensional infinite
plate was initially suggested by Bowles (1977). The boundary
effects of the finite raft were not discussed in such analysis. Recent
discussions on design and analysis of the raft foundation can be
found in Gupta (1997) and Hemsley (1998), again the finite plate
solutions of FDA were not available until then. It should be
pointed out that the boundary effects of the foundation are
relatively important when the foundation became more flexible.
On the other hand, FDA such as the FLAC program (Itasca
2017) is based on examining the stress continuities at the material
particles. Applying them to the nodes of a structural system
requires considerable amount of iterations to ensure the
equilibriums. Besides FDA, available solution such as Finite-Grid
Analysis (FGA) has been suggested as well to solve the problem.
The FGA is another type of numerical methods based on simpli-
fied structural elements (beam-column or lumped mass) where
nodal forces are applied at the grids. Examples of FGA can be
found in Bowles (1996), and this type of analysis has been adopted
popularly in many studies on raft and piled raft foundations. It
should be noted that the FGA is different from the FDA. Figure
1(a) depicts respectively the beam and mat treatments of the raft
foundations. This study proposed a three-dimensional FDA based
on the governing equations of a surface raft foundation subjected
to vertically static loads. The boundary effects were included in
such analysis. The proposed analysis is able to calculate the
differential settlements of the raft where the flexibility of the
foundation is encountered.
Journal of GeoEngineering, Vol. 13, No. 3, pp. 135-147, September 2018 http://dx.doi.org/10.6310/jog.201809_13(3).5
Manuscript received December 27, 2017; revised June 22, 2018; accepted June 25, 2018.
1* Professor (corresponding author), Department of Civil Engineering,
Tamkang University, New Taipei City, Taiwan 25137, R.O.C. (e-mail:
[email protected]). 2 Graduate student, Department of Civil Engineering, Tamkang
University, New Taipei City, Taiwan 25137, R.O.C.
136 Journal of GeoEngineering, Vol. 13, No. 3, September 2018
2. THEORY AND EQUATIONS
Theory of Plate can be found in many textbooks of Structural
Mechanics. The theory can be categorized for thin plate and thick
plate. In general, if the thickness of the plate (D) is less than ten
percent of the width (W) of plate, it can be regarded as thin-plate.
The Kirchhoff-Love classical plate theory is restricted to thin
plates, whereas the Mindlin-Reissner plate theory is applicable to
thick plates. The distinction between the thick plate theory and thin
plate theory is that the in-plane shear strains are considered in the
thick plate theory. For most of the large size raft foundations where
thickness is less than the order of a tenth of width of the foundation,
the classical plate theory can be applied. According to Timoshenko
and Woinowsky-Krieger (1959), the governing equation of verti-
cal displacements of the thin plate subjected to vertically uniform
load (q) and point load (P) can be written as follows,
4 4 4 2 2
4 2 2 4 3 3
2 12 (1 ) 12 (1 )
( )
w w w q P
x x y y ED ED x y
− −+ + = +
(1)
where w is the vertical displacement of the raft, and E are
respectively the Poisson’s ratio and Young’s Modulus of raft, D is
thickness of the raft, and x and y are the spatial variables. For a raft
foundation resting on the ground surface as shown in Fig. 1(b), the
moments and shear forces should be vanished at edge of the
foundation. By looking at the foundation with plan view, the top
and bottom edges of the raft where y = constant, Mx (bending
moment rotating at the x-direction) and Vy (vertical shear force at
the surface normal to the y-direction) along the edge will become
zero. They can be written as follows,
2 2 2 2( / / ) 0xM B w y w x= − + = (2)
3 3
3 2
(2 )0y
w wV B
y y x
− = − + =
(3)
where B is the expression of ED3/(12(1 − 2)). Similarly, at the left
and right edges of the raft where x = constant, the boundary
conditions are My = 0 and Vx = 0. The corresponding equations are:
2 2 2 2( / / ) 0yM B w x w y= − + = (4)
Fig. 1 Modeling of raft foundation: (a) beam and plate models,
(b) boundary conditions of a surface raft foundation on
elastic half-space
3 3
3 2
(2 )0x
w wV B
x x y
− = − + =
(5)
For the governing equation of the raft foundation with the
influences of soil reactions, a number of models can be used. One
can refer to the summary made by Gazetas (1991) on static and
dynamic impedance functions of soils. For simplicity, the axial-
force-element (AFE) spring model was adopted in this study to
simulate the ground of an elastic half-space, in which the soil
spring constant Ks is simply calculated as EsAs/l where Es is the
Young’s Modulus of the soil, As is the effective area of the soils
underneath the nodes, and l is the length of the soil spring. For a
surface foundation, the influences of the soil reactions on the raft
can be written as follows,
4 4 4 * 2 2
4 2 2 4 3 3
2 12 (1 ) 12 (1 )
( )
w w w q P
x x y y ED ED x y
− − + + = +
*q P
B B x y= +
(6)
where q* = q –ΣKswk /Ar = q – (Es/l)ΣAsk wk /Ar; wk is foundation
settlement at the k th node, Ask is the area of soil spring under the
k th node, and Ar is the total area of the raft which equals to ΣArk,
where Ark stands for the area of raft at the k th node. Now defining
𝑞𝑘∗ as the modified load allocated at the k th node, for simplicity,
𝑞𝑘∗ can be approximated as q − (Es / l )wk(Ask /Ark), where (Ask /Ark)
is called as the area ratio at the k th node. Note that the above equa-
tion assumes that the soil reactions underneath the raft are uni-
formly distributed. This assumption matches closely well with the
flexible foundations where the raft size is relatively large. For
smaller raft foundation that behaves more rigidly, this assumption
is invalid.
3. FINITE DIFFERENCE EQUATIONS
Using the central-difference formulas, the resulting
formulations for the nodes at a surface foundation can be
expressed in Eqs. (7) ~ (31). The orientation and categories of
these nodal points are shown in Fig. 2. Details of the derivations
can be found in Lien (2018). Examples of the derivations are
presented in Appendix for top-edge nodes, inner top-edge nodes,
top-left corner nodes, top-left corner right-hand-side (RHS) nodes,
top-left corner down-side (DS) nodes, and inner top-left corner
nodes. Note that the distances between the nodes in x- and y-
directions are kept the same (i.e., x = y = s) for simplicity of the
expressions. In addition, the point load P applied at arbitrary nodes
of the raft can be taken as an extra uniform load applied to that
node within the area which is equal to xy. Therefore, the fol-
lowing equations are derived without the point load expression.
Figure 3 shows the nodal points used and the fictitious points en-
countered in the derivations. Figure 4 depicts the allocations of the
corresponding equations.
General nodes:
, 2 1, 1 , 1 1, 1 2, 1,2 8 2 8i j i j i j i j i j i jw w w w w w+ − + + + + − −+ − + + −
, 1, 2, 1, 1 , 120 8 2 8i j i j i j i j i jw w w w w+ + − − −+ − + + −
*4
1, 1 , 22 i j i j
qw w s
B+ − −+ + = (7)
Chang et al.: Finite Difference Analysis of Vertically Loaded Raft Foundation Based on the Plate Theory with Boundary Concern 137
Fig. 2 Layout of the discrete nodes of a vertically loaded raft
foundation
Fig. 3 Nodal relations for derivations of the discrete FD for-
mulas: (a) general node, (b) edge node, (c) nearest cor-
ner edge node, (d) inner edge node, (e) corner node, and
(f) inner corner node
Fig. 4 Allocations of the corresponding equations of the
proposed analysis
Top-edge nodes:
2 2 22, 1, ,
2 21, 2,
1, 1 , 1
( 1) (4 4 8) ( 6 8 16)
(4 4 8) ( 1)
( 2 4) (4 12)
i j i j i j
i j i j
i j i j
w w w
w w
w w
− −
+ +
− − −
− + + + − + − − +
+ + − + − +
+ − + + −
*4
1, 1 , 2( 2 4) 2i j i j
qw w s
B+ − −+ − + + = (8)
Bottom-edge nodes:
2 2 22, 1, ,
2 21, 2,
1, 1 , 1
( 1) (4 4 8) ( 6 8 16)
(4 4 8) ( 1)
( 2 4) (4 12)
i j i j i j
i j i j
i j i j
w w w
w w
w w
− −
+ +
− + +
− + + + − + − − +
+ + − + − +
+ − + + −
*4
1, 1 , 2( 2 4) 2i j i j
qw w s
B+ + +− + =+ + (9)
Left-edge nodes:
2 2 2, 2 , 1 ,
2 2, 1 , 2
1, 1 1,
( 1) (4 4 8) ( 6 8 16)
(4 4 8) ( 1)
( 2 4) (4 12)
i j i j i j
i j i j
i j i j
w w w
w w
w w
− −
+ +
+ − +
− + + + − + − − +
+ + − + −
+ −
+
+ − +
*4
1, 1 2,( 2 4) 2i j i j
qw w s
B+ + ++ − + + = (10)
Right-edge nodes:
2 2 2, 2 , 1 ,
2 2, 1 , 2
1, 1 1,
( 1) (4 4 8) ( 6 8 16)
(4 4 8) ( 1)
( 2 4) (4 12)
i j i j i j
i j i j
i j i j
w w w
w w
w w
− −
+ +
− − −
− + + + − + − − +
+ + − + − +
+ − + + −
*4
1, 1 2,( 2 4) 2i j i j
qw w s
B− + −+ − + + = (11)
Top-left corner right-hand-side (RHS) node:
2 21, ,
2 21, 2,
1, 1 , 1
(2 4 6) ( 5 8 15)
(4 4 8) ( 1)
( 2 4) (4 12)
i j i j
i j i j
i j i j
w w
w w
w w
−
+ +
− − −
+ − + − − +
+ + − + − +
+ − + +
−
*4
1, 1 , 2( 2 4) 2i j i j
qw w s
B+ − −− + =+ + (12)
138 Journal of GeoEngineering, Vol. 13, No. 3, September 2018
Top-left corner down-side (DS) node:
2 2 2, 2 , 1 ,
2, 1 1, 1
1, 1, 1
( 1) (4 4 8) ( 5 8 15)
(2 4 6) ( 2 4)
(4 12) ( 2 4)
i j i j i j
i j i j
i j i j
w w w
w w
w w
− −
+ + −
+ + +
− + + + − + − − +
+ + − + − +
+
− + − +
*4
2,2 i j
qw s
B++ = (13)
Bottom-left corner RHS node:
2 21, ,
2 21, 2,
1, 1 , 1
(2 4 6) ( 5 8 15)
(4 4 8) ( 1)
( 2 4) (4 12)
i j i j
i j i j
i j i j
w w
w w
w w
−
+ +
− + +
+ − + − − +
+ + − + − +
+ − + −
+
*4
1, 1 , 2( 2 4) 2i j i j
qw w s
B+ + ++ − + + = (14)
Bottom-left corner up-side (US) node:
2 2, 1 ,
2 2, 1 , 2
1, 1 1,
(2 4 6) ( 5 8 15)
(4 4 8) ( 1)
( 2 4) (4 12)
i j i j
i j i j
i j i j
w w
w w
w w
−
+ +
+ − +
+ − + − − +
+ + − + − +
+ − + + −
*4
1, 1 2,( 2 4) 2i j i j
qw w s
B+ + ++ − + + = (15)
Top-right corner left-hand-side (LHS) node:
2 2 22, i 1, ,
21, 1, 1
, 1 1, 1
( 1) (4 4 8) ( 5 8 15)
(2 4 6) ( 2 4)
(4 12) ( 2 4)
i j j i j
i j i j
i j i j
w w w
w w
w w
− −
+ − −
− + −
− + + + − + − − +
+ + − + − +
+ − +
+
−
*4
, 22 i j
qw s
B−+ = (16)
Top-right corner DS node:
2 2 2, 2 , 1 ,
2, 1 1, 1
1, 1, 1
( 1) (4 4 8) ( 5 8 15)
(2 4 6) ( 2 4)
(4 12) ( 2 4)
i j i j i j
i j i j
i j i j
w w w
w w
w w
− −
+ − −
− − +
− + + + − + − − +
+ + − + − +
+ − + − +
*4
2,2 i j
qw s
B−+ = (17)
Bottom-right corner LHS node:
2 2 22, 1, ,
21, 1, 1
, 1 1, 1
( 1) (4 4 8) ( 5 8 15)
(2 4 6) ( 2 4)
(4 12) ( 2 4)
i j i j i j
i j i j
i j i j
w w w
w w
w w
− −
+ − +
+ + +
− + + + − + − − +
+ + − + − +
+ − + − +
*4
, 22 i j
qw s
B++ = (18)
Bottom-right corner US node:
2 2, 1 ,
2 2, 1 , 2
1, 1 1,
(2 4 6) ( 5 8 15)
(4 4 8) ( 1)
( 2 4) (4 12)
i j i j
i j i j
i j i j
w w
w w
w w
−
+ +
− − −
+ − + − − +
+ + − + − +
+ − + + −
*4
1, 1 2,( 2 4) 2i j i j
qw w s
B− + −+ − + + = (19)
Inner top-edge nodes:
1, 1 , 1 1, 1
2, 1, , 1,
2, 1, 1 , 1
( 2) (2 6) ( 2)
8 19 8
2 8
i j i j i j
i j i j i j i j
i j i j i j
w w w
w w w w
w w w
− + + + +
− − +
+ − − −
− + + − + − +
+ −
−
+
+ −
+
*4
1, 1 , 22 i j i j
qw w s
B+ − −+ + = (20)
Inner bottom-edge nodes:
1, 1 , 1 1, 1
2, 1, , 1,
2, 1, 1 , 1
( 2) (2 6) ( 2)
8 19 8
2 8
i j i j i j
i j i j i j i j
i j i j i j
w w w
w w w w
w w w
− − − + −
− − +
+ − + +
− + + − + − +
+ −
+ −
+ −
+
*4
1, 1 , 22 i j i j
qw w s
B+ + ++ + = (21)
Inner left-edge nodes:
1, 1 1, 1, 1
, 2 , 1 , , 1
, 2 1, 1 1,
( 2) (2 6) ( 2)
8 19 8
2 8
i j i j i j
i j i j i j i j
i j i j i j
w w w
w w w w
w w w
− + − − −
+ + −
− + + +
− + + − + − +
+ − + −
+ + −
*4
1, 1 2,2 i j i j
qw w s
B+ − ++ + = (22)
Inner right-edge nodes:
1, 1 1, 1, 1
, 2 , 1 , , 1
, 2 1, 1 1,
( 2) (2 6) ( 2)
8 19 8
2 8
i j i j i j
i j i j i j i j
i j i j i j
w w w
w w w w
w w w
+ + + + −
+ + −
− − + −
− + + − + − +
+ − + −
+ + −
*4
1, 1 2,2 i j i j
qw w s
B− − −+ + = (23)
Top-left corner:
5 4 3 2
,4 3 2
5 4 3 2
1,4 3 2
5 4 3 2
2,4 3 2
5 4 3 2
, 14 3 2
5
4 20 32 16 4 4
4 4 1
4 20 32 16 4 4
4 4 1
2 10 16 8 2 2
4 4 1
4 20 32 16 4 4)
4 4 1
2
+
+
+
+
i j
i j
i j
i j
w
w
w
w
+
+
−
− + − − +
− − +
− + − + + −+
− − +
− + − − ++
− − +
− + − + + −+
− − +
−+
4 3 2 *4
, 24 3 2
10 16 8 2 2
4 4+ 1i j
qw s
B−
+ − − +=
− − +
(24)
Chang et al.: Finite Difference Analysis of Vertically Loaded Raft Foundation Based on the Plate Theory with Boundary Concern 139
Bottom-left corner:
5 4 3 2
,4 3 2
5 4 3 2
1,4 3 2
5 4 3 2
2,4 3 2
5 4 3 2
, 14 3 2
5
4 20 32 16 4 4
4 4 1
4 20 32 16 4 4
4 4 1
2 10 16 8 2 2
4 4 1
4 20 32 16 4 4
4 4 1
2
+
1
+
+
+
i j
i j
i j
i j
w
w
w
w
+
+
+
− + − − +
− − +
− + − + + −+
− − +
− + − − ++
− − +
− + − + + −+
−
− +
−+
4 3 2 *4
, 24 3 2
0 16 8 2 2
4 4+ 1i j
qw s
B+
+ − − +=
− −
+
(25)
Top-right corner:
5 4 3 2
,4 3 2
5 4 3 2
1,4 3 2
5 4 3 2
2,4 3 2
5 4 3 2
( , 1)4 3 2
5
4 20 32 16 4 4
4 4 1
4 20 32 16 4 4
4 4 1
2 10 16 8 2 2
4 4 1
4 20 32 16 4 4
4 4 1
2
+
+
+
+
i j
i j
i j
i j
w
w
w
w
−
−
−
− + − − +
− − +
− + − + + −+
− − +
− + − − ++
− − +
− + − + + −+
− −
+
+
4 3 2 *
4( , 2)4 3 2
10 16 8 2 2
4 4 1+i j
qw s
B−
− + − − +=
− − +
(26)
Bottom-right corner:
5 4 3 2
,4 3 2
5 4 3 2
1,4 3 2
5 4 3 2
2,4 3 2
5 4 3 2
, 14 3 2
5
4 20 32 16 4 4
4 4 1
4 20 32 16 4 4
4 4 1
2 10 16 8 2 2
4 4 1
4 20 32 16 4
+
+
+
4
4 4 1
2 1
+
i j
i j
i j
i j
w
w
w
w
−
−
+
− + − − +
− − +
− + − + + −+
− − +
− + − − ++
− − +
− + − + + −+
− −
+
−+
4 3 2 *4
, 24 3 2
0 16 8 2 2
4 4+ 1i j
qw s
B+
+ − − +=
− −
+
(27)
Inner top-left node:
1, 1 , 1 1, 1
2, 1, , 1,
*4
1, 1 , 1 1, 1 , 2
( 2) (2 6) ( 2 2)
8 18 (2 6)
2 8 ( 2)
i j i j i j
i j i j i j i j
i j i j i j i j
w w w
w w w w
qw w w w s
B
+ + + − +
+ + −
+ − − − − −
− + + − + − +
+ − + + −
+ − + − + + =
(28)
Inner bottom-left node:
1, 1 , 1 1, 1
2, 1, , 1,
*4
1, 1 , 1 1, 1 , 2
( 2) (2 6) ( 2 2)
8 18 (2 6)
2 8 ( 2)
i j i j i j
i j i j i j i j
i j i j i j i j
w w w
w w w w
qw w w w s
B
+ − − − −
+ + −
+ + + − + +
− + + − + − +
+ − + + −
+ − + − + +
=
(29)
Inner top-right node:
1, 1 , 1 1, 1
2, 1, , 1,
*4
1, 1 , 1 1, 1 , 2
( 2) (2 6) ( 2 2)
8 18 (2 6)
2 8 ( 2)
i j i j i j
i j i j i j i j
i j i j i j i j
w w w
w w w w
qw w w w s
B
− + + + +
− − +
− − − + − −
− + + − + − +
+ − + + −
+ − + + =
+ −
(30)
Inner bottom-right node:
1, 1 , 1 1, 1
2, 1, , 1,
*4
1, 1 , 1 1, 1 , 2
( 2) (2 6) ( 2 2)
8 18 (2 6)
2 8 ( 2)
i j i j i j
i j i j i j i j
i j i j i j i j
w w w
w w w w
qw w w w s
B
− − − + −
− − +
− + + + + +
− + + − + − +
+ − + + −
+ − + − =
+ +
(31)
With the above 25 discrete equations derived from the governing
equation via the influences of the boundary conditions, one can
establish a set of independent equations for the nodes allocated at
the raft. Matrix analysis is required to solve for the nodal
displacements of the raft. It is necessary to point out that when
calculating the soil reactions at the nodes along the edges, the area
ratio (Ask/Ark) of the AFE springs can be represented by a value, n,
defined as the ratio of the length for the soils underneath the nodes
and the spacing distance (which is kept as 1.0 m) between adjacent
nodes. For the nodes at the corners, the area ratio of the AFE spring
would be equal to n2. The standard length and width of the soils
under a single node are kept as 1 meter for simplicity. Figure 5
depicts the area of the soil springs for the nodes at the edge and the
corner of the foundation. Should the raft locate in a certain depth
from the ground surface, the boundary conditions need to be
modified. A computer program WERAFT-S (Lien 2018) was
thereafter suggested for static behaviors of the surface raft on
elastic soils.
Fig. 5 Schematic layout of the area ratio (Ask/Ark) in calculating
soil reactions for the nodes
140 Journal of GeoEngineering, Vol. 13, No. 3, September 2018
4. COMPARISONS WITH 3D FEM SOLUTIONS
A numerical example is presented to show the validations of
the proposed solutions. A square surface raft foundation resting on
an elastic half-space is assumed with the width (W) of 26 m and
the thickness (D) of 1m. Young’s modulus (E), Poisson’s ratio (),
and unit weight () of the concrete raft foundation are respectively
as 3 × 104 MPa, 0.15 and 24 kN/m3. The soils underneath the
foundation are assumed with the shear wave velocity (Vs) of 150
m/s, Poisson’s ratio (s) 0.4, and unit weight (s) of 19 kN/m3. Note
that Vs2 × s/g = Gs, where g is acceleration of the gravity, Gs is the
shear modulus of the soil. Es can be calculated as 2Gs(1 + s). An
uniform load of 100 kPa is applied on top of the foundation.
In order to establish a fundamental solution to validate the
program WERAFT-S, three-dimensional (3D) FEA solutions of
the numerical model were obtained from Midas-GTS (Midas
2017). It should be noted that 3D FDA or FEA has been adopted
in many research studies to verify the simplified analysis, and the
Midas-GTS analysis have been used to model the complicated
foundation behaviors (Poulos 1994, 2001; Abderlrazaq et al. 2011).
Essential boundaries (i.e., roller and hinge) and mostly eight-node
brick elements were used to generate the structural system. Figure
6 depicts the discrete mesh used in the FE analysis.
Figure 7 shows that the foundation settlements from FE
analysis will gradually become stable by increasing the FE zone
from 60 to 200 m (note that the length, width and thickness of the
zone were kept the same for the analytical zone). The solutions
became stable when the analytical zone was increased to 200 m ×
Fig. 6 Discrete mesh and boundaries of the finite element
analysis (a) plan view (b) analytical zone (c) essential
boundaries (Note: 1 m3 8-node cubic elements and 64 m3
8-node cubic elements were the largest solid elements
used for raft and soils, respectively. Rollers were used at
sides and bottom, which formed the 2-dimensional hinges
along the edges and 3-dimensional hinges at the corners
of bottom plane)
200 m × 200 m. Therefore, the following study was conducted
using the optimized core. It can be seen that the differential
settlements were resulted in the analysis. In general, the center
exerts the largest settlement, whereas the corner has the smallest
settlement. The soil mass involved in the FE analysis seems to
have significant influence on foundation settlements due to the as-
sumed linear elasticity. Validation of the WERAFT-S analysis was
conducted to compare with the FE solutions. The influences of the
area ratio (Ask/Ark = n) of the edge node, the shear wave velocity
(Vs) and the Poisson’s ratio (s) of soil, the thickness of raft (D),
and the load intensity (q) were studied to find the optimal length
(l) of soil spring.
Figure 8 shows the effects of area ratio (n) at the edge and
length of the AFE springs (l) from WERAFT-S analysis on
normalized settlements (w/D) at center, edge, and corner of the
foundation. Note that x and y are kept as one meter in this study.
Therefore the raft is modeled with 729 (i.e., 27 × 27) nodal points.
The horizontal dash lines represent the settlements from FE
analysis. Note that when n = 2.5 and the length of spring (l)
approximates 21 m, it seems to provide compatible solutions with
the FE analysis. The matches were found sensitive to the
settlement occurred at the corner. As the area ratio increases from
2 to 3, the optimal length of the spring at the corner would increase
rapidly from 17 m to 25 m, where the corresponding ones at the
edge and the center are varying around 18 ~ 21 m and 21 m,
respectively.
Fig. 7 The influence of the FE analytical zone dimensions on the
foundation settlements
Fig. 8 The effect of area ratio of edge nodes and length of underneath soil spring on foundation settlements:
(a) n = 2, (b) n = 2.5, and (c) n = 3
Chang et al.: Finite Difference Analysis of Vertically Loaded Raft Foundation Based on the Plate Theory with Boundary Concern 141
5. PARAMETRIC STUDIES ON AFE SPRINGS
Figure 9 shows the effects of shear wave velocity (Vs) of the
soil with the length of AFE springs (l) to the solutions of the
normalized settlement (w/D). Again, the horizontal dash lines are
settlements from FE analysis. Note that the area ratio for the edge
nodes is 2.5 and the area ratio for the corner nodes is 6.25. It can
be found that the matching of the foundation settlements is
significantly dependent of the length of AFE springs. As the shear
wave velocity of the soil decreases from 180 m/s to 120 m/s, the
optimal length of the soil springs will increase approximately in
the range of 16 m ~ 28 m. This implies that when simulating the
raft foundation on relatively soft soils, the optimal length of AFE
springs needs to be longer. In this case, the optimal length of AFE
spring is much smaller than the effective thickness (2 × foundation
width) of a square mat. The reason behind this is that the stresses
are constant along the AFE spring while the real ground stresses
due to the surface loading are gradually decreased with the depth.
Figure 10 shows the effects of Poisson’s ratio of the soil.
Again, the area ratio for the edge nodes is 2.5 and the area ratio for
the corner node is 6.25. Again, the influences of the Poisson’s ratio
were found similar to the shear wave velocity of the soil since the
Young’s Modulus of the soil would be increased accordingly. The
optimal lengths of AFE springs were found decreasing from
approximately 24 m to 18 m when the Poisson’s ratio increases
from 0.3 to 0.5.
Figure 11 shows the effects of thickness of raft (D) in a range
of 0.8 ~ 1.2 m where the area ratio for soils at the edge nodes was
kept at 2.5. It can be seen that the optimal length of the soil spring
for the match is around 21 m regardless of the raft thickness. The
thickness of raft rarely affects the optimal length of the AFE spring.
Figure 12 shows the effects of load intensity (q). For n = 2.5,
it can be found that the effect of load intensity seems relatively
unimportant. As the foundation settlement increases linearly with
the load intensity, the matching of the appropriate length of soil
springs is nearly the same. This is mainly due to the linear elastic
assumption of the soil. If nonlinear soil behavior was encountered,
the assessment should be examined further.
As a result, it can be found that the solutions of WERAFT-S
with the applications of AFE springs are highly dependent of
following parameters: (1) The area ratio of the soil at the edge of
foundation and (2) The optimal length of AFE spring. It seems that
for n = 2.5 and l/W approximates 0.66 ~ 1.1, compatible solutions
can be found. Apart from that, if the size of a raft foundation was
infinitely large or the foundation was rigid enough, Eq. (7) with
the assumption that the foundation settlements were approxi-
mately the same regardless of the locations could provide rational
estimation. However, if differential settlements of the foundations
became significant, the boundary effects must be taken into ac-
count carefully.
(a) Vs = 120 m/s (b) Vs = 150 m/s (c) Vs = 180 m/s
Fig. 9 The effects of shear wave velocity of soil and length of underneath soil spring on foundation settlements
(a) s = 0.3 (b) s = 0.4 (c) s = 0.5
Fig. 10 The effects of Poisson’s ratio of soil and length of underneath soil spring on foundation settlements
142 Journal of GeoEngineering, Vol. 13, No. 3, September 2018
(a) D = 0.8 m (b) D = 1.0 m (c) D = 1.2m
Fig. 11 The effects of raft thickness and length of underneath soil spring on foundation settlements
(a) q = 100 kPa (b) q = 200 kPa (c) q = 300 kPa
Fig. 12 The effect of load intensity and length of underneath soil spring on foundation settlements
Three-dimensional plots on varying shear wave velocity of
the soil (Vs) and he length of soil spring (l) for settlements of the
raft with standard parameters and n = 2.5 are revealed in Fig. 13.
The inclined blue plane is formed by the solutions from
WERAFT-S, whereas the red plane is formed by FEA. Note that
the matching lines of these planes can be seen from Fig. 13. The
projections of the matching lines at the Vs-l plane are shown in Fig.
14. Curve fitting equations from the regression analysis are
obtained for the proposed analysis. Figure 15 depict the contour
plots of the foundation settlements obtained from WERAFT-S and
FEA. The standard numerical example is referred. Appropriate
length of the soil springs was kept at 21 m while the area ratio (n)
for the nodes at the edges was kept as 2.5. It is obvious that good
agreements between the solutions can be achieved.
Fig. 13 Three-dimensional plot for effects of shear wave velocity of soil and length of the soil spring on settlements of the raft foundation
(standard model parameters and n = 2.5 for nodes along the edge)
Chang et al.: Finite Difference Analysis of Vertically Loaded Raft Foundation Based on the Plate Theory with Boundary Concern 143
(a) Center (b) Edge (c) Corner
Fig. 14 Fitting curves and projections of matching lines for shear wave velocity of the soil versus length
of soil spring underneath the raft foundation
(a) (b)
Analysis Raft displacement (cm)
Center Edge Corner
Midas-GTS 1.88 1.31 0.97
WERAFT-S 1.85 1.44 0.96
(c)
Fig. 15 Plan view on contours of the foundation settlements obtained on the standard numerical model:
(a) FEA, (b) WERAFT, and (c) Displacement values
6. OBSERVATIONS ON LYSMER’S ANALOG
SPRINGS
If the Lysmer’s analog model (Lysmer and Richart 1966) was
used for the soil springs, where soil stiffness underneath the raft
Ks can be modeled as 4Gsro/(1 − s) and ro is the equivalent radius
of the foundation, by computing the total spring constant Ks and
averaging it to all the nodes, the results of the foundation
settlements were able to show. It can be found that the foundation
settlements calculated from the Lysmer’s analog are rational for Vs
at 150 m/s. Correspondent contour plot of the settlements is shown
in Fig. 16. For Vs at 120 m/s and 180 m/s, the deviations of the
computed foundation settlements with the FEA ones were found
significant. It is believed to be caused by the nature of Lysmer’s
analog, which was initially proposed for rigid footing. To achieve
much closer solutions, a modified coefficient R is used to multiply
with the soil stiffness for the predictions. R = 0.7, 1.0, and 1.2 are
respectively used for Vs = 120, 150, and 180 m/s. In comparison,
the results using AFE springs were shown in Table 1 with optimal
lengths, L = 30, 21, and 12 m for Vs = 120, 150, and 180 m/s, re-
spectively. Although the results obtained from the Lysmer’s ana-
log can be improved, it seems that settlements at the corner are still
smaller than FEM ones. The comparisons indicate that the use of
discrete soil springs really needs further attentions since they
might provide wrong predictions to the real continuums.
144 Journal of GeoEngineering, Vol. 13, No. 3, September 2018
Table 1 Foundation settlements obtained from Midas-GTS
analysis and WERAFT-S analysis using AFE springs
and Lysmer’s analog springs
Shear wave
velocity,
Vs
Analysis Midas-GTS
WERAFT-S
w/ AFE spring
WERAFT-S
w/ Lysmer’s analog spring
Location Settlement (cm)
120 m/s
Center 4.23 4.03 4.04
Edge 3.00 3.18 3.13
Corner 2.25 2.27 2.00
150 m/s
Center 1.88 1.86 1.86
Edge 1.31 1.37 1.36
Corner 0.97 0.92 0.79
180 m/s
Center 1.06 1.11 1.08
Edge 0.75 0.79 0.76
Corner 0.56 0.51 0.42
Fig. 16 Contour plot of foundation settlements from WERAFT-
S analysis using Lysmer’s analog springs
7. CONCLUSIONS
This paper presents a newly proposed finite difference (FD)
analysis WERAFT-S on surface raft foundation under vertical
loads using the thin-plate theory. The effects of boundary con-
ditions were considered in deriving the FD equations of the raft
foundation settlements. Axial force elements (AFE) were
suggested for soil springs underneath the raft. Solutions of the
proposed analysis were verified on a numerical raft foundation
(L × W × D = 26 m × 26 m × 1 m) resting on an elastic half-space
of the soils (where Vs = 150 m/s and s = 0.4). Vertically uniform
load of 100 kPa was assumed at top of the foundation. Validation
of the analysis was made with those from three-dimensional finite
element analysis using Midas-GTS program. It was found that the
area ratio (Ask/Ark) equals to 2.5 for edge nodes of the foundation
(for corner nodes, it would be 6.25) is adequate for comparisons.
The length of the AFE springs at 12 m ~ 30 m can provide
agreeable solutions with the FEA when the shear wave velocity of
the soils is in a range of 180 m/s ~ 120 m/s. Fitting functions for
the optimal length of AFE spring with respect to the shear wave
velocity of the soil were thus suggested. The lysmer’ analog model
was also studied, it was found that the model has some limits for
foundation settlements at the corners, and the adequateness of the
solutions would be dependent of the soil stiffness. Since nearly all
the existing foundation springs are developed for rigid foundation,
to use the constant soil stiffness underneath a flexible foundation
needs to be very careful. It should be pointed out that if the raft
foundation is rigid enough, then the solution proposed by Bowles
(1977) can be used assuming that the foundation settlement is
uniform. But if a raft foundation is relatively large and the
foundation became more flexible, where the boundary effects
cannot be ignored, then the proposed analysis can be useful.
Vertical loads such as uniform, non-uniform, and point loads can
be taken into account in the analysis with pre-calculations.
External bending moment applied with the column load on the
foundation can be simulated by shifting the column load with an
offset distance which is equal to the moment divided by the
column load. For more realistic solution considering inelastic soil
behaviors under the foundation, one could use more rigorous soil
spring model. In that case, the capacity and serviceability
performance of the raft foundation will be able to estimate.
ACKNOWLEDGMENTS
This paper is a part of the research study supported through
the grant MOST 106-2221-E-032-025-MY2 of Ministry of
Science and Technology (MOST) in ROC whose funding is
greatly appreciated.
NOTATIONS
As Area of the soil underneath a node of the raft (m2)
Ar Total area of the raft (m2)
Ark Area of the raft at the k th node (m2)
Ask Area of the soil underneath the k th node (m2)
B Parameter equals to ED3/(12(1 − 2)) (kN-m)
D Thickness of the raft foundation (m)
E Young’s modulus of the raft (kPa)
Es Young’s modulus of the soil (kPa)
g Acceleration of the gravity (= 9.81 m/s2)
Gs Shear modulus of the soil (kPa)
Ks Spring constant of the soil (kN/m)
L Length of the raft foundation (m)
l Length of the soil spring (m)
Mx Bending moment rotating along x-direction (kN-m)
My Bending moment rotating along y-direction (kN-m)
n Area ratio for nodes at the edge of raft foundation
n2 Area ratio for nodes at the corner of raft foundation
P Point load applied to the raft foundation (kN)
q Uniform load applied to the raft foundation (kPa)
q* Uniform load intensity subtracting the soil reactions (kPa)
𝑞𝑘∗ Modified load intensity applied at the k th node (kPa)
R Modified coefficient
ro Equivalent radius of the foundation (m)
Chang et al.: Finite Difference Analysis of Vertically Loaded Raft Foundation Based on the Plate Theory with Boundary Concern 145
s Equalized nodal spacing distance (m)
Vs Shear wave velocity of the soil (m/s)
Vx Vertical shear force acting on a surface with normal in x-di-
rection (kN)
Vy Vertical shear force acting on a surface with normal in y-di-
rection (kN)
W Width of the raft foundation (m)
w Settlement of the raft foundation (m)
wk Foundation settlement at the k th node
x Spatial variable in x-direction (m)
y Spatial variable in y-direction (m)
x Nodal spacing distance in x-direction (m)
y Nodal spacing distance in y-direction (m)
Unit weight of the raft foundation (kN/m3)
s Unit weight of the soil (kN/m3)
Poisson’s ratio of the raft
s Poisson’s ratio of the soil
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Bowles, J.E. (1977). Foundation Analysis and Design, 2nd Ed., McGraw-Hill Companies, Inc.
Bowles, J.E. (1996). Foundation Analysis and Design, 5th Ed.,
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Chen, C.N. (1998). “Solution of beam on elastic foundation by DQEM,” Journal of Engineering Mechanics, ASCE, 124(12), 1381-1384. https://doi.org/10.1061/(ASCE)0733-9399(1998) 124:12(1381)
Chiou, J.S., Lin, H.S., Yeh, F.Y., and Sung, Y.C. (2016). “Plastic settlement evaluation of embedded railroads under repeated train loading.” Journal of GeoEngineering, TGS, 11(2), 97-107.
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Dinev, D. (2012). “Analytical solution of beam on elastic foundation by singularity functions.” Engineering MECHANICS, 19(6), 381-392.
Gazetas, G. (1991). Foundation Engineering Handbook, Chapter 15, Foundation Vibrations, Fang, H.Y., Ed., Springer, 553-593.
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Hemsley, J.A. (1998). Elastic Analysis of Raft Foundations, ICE Pub-
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Itasca (2017). FLAC Version 8.0, Itasca Consulting Group. Inc.
Jones, M. (1997). Analysis of Beams on Elastic Foundations: Using
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Kukreti, A.R. and Ko, M.G. (1992). “Analysis of rectangular plate resting on an elastic half space using an energy approach.”
Applied Mathematical Modeling, 16(7), 338-356.
Lien, H.W. (2018). Finite Difference Analysis of Piled Raft Founda-tions under Vertically Loads, Master Thesis, Dept. of Civil
Engineering, Tamkang University, Taiwan (in Chinese).
Lysmer, J. and Richart, F.E. (1966). “Dynamic response of footing to vertical loading.” Journal of Soil Mechanics and Foundation Division, ASCE, 92(1), 65-91.
Mathews, P.M. (1958). “Vibrations of a beam on elastic foundation.” Journal of Applied Mathematics and Mechanics, 38(3-4), 105-115. https://doi.org/10.1002/zamm.19590390105
Midas (2017). Midas GTS NX User Manual, Midas IT Co.
Omer, J.R. and Arbabi, A. (2015). “Evaluation of finite element, finite difference and elasticity methods for hypothetical raft foundations installed on layered strata.” Geotechnical and Geological Engineering, 33(4), 1129-1140. http://doi.org/10. 1007/s10706-015-9867-7
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APPENDIX
The Central Difference Formula (CDF) of Eq. (6) neglecting
P and assuming that x = y = s can be derived for any arbitrary
point of (i, j) where i and j respectively denote the nodal order in
x- and y-directions as follows,
, 2 1, 1 , 1 1, 1 2, 1,
, 1, 2, 1, 1 , 1
2 8 2 8
20 8 2 8
i j i j i j i j i j i j
i j i j i j i j i j
w w w w w w
w w w w w
+ − + + + + − −
+ + − − −
+ − + + −
+ − + + −
*4
1, 1 , 22 i j i j
qw w s
B+ − −+ + = (A1)
Eq. (A1) is the same as Eq. (7) which has 13 unknown
displacements of the foundation. It is applicable to all the nodal
points which are not affecting by the boundaries. Now consider the
nodal points along top edge of the raft where Mx and Vy are both
vanished, the CDF of Eqs. (2) and (3) can be expressed as follows,
, 1 , , 1 1, , 1,2 2
1[ 2 2 ]] [ 0x i j i j i j i j i j i jM w w w w w w
S S+ − + −− + + − +
=
(A2)
, 2 , 1 , 1 , 23
1, 1 , 1 1, 1 1, 13
1[ 2 2 ]
2
2[ 2
2
y i j i j i j i j
i j i j i j i j
V w w w wS
w w w wS
+ + − −
− + + + + − −
= − + −
−− + −
+
146 Journal of GeoEngineering, Vol. 13, No. 3, September 2018
, 1 1, 12 ] 0i j i jw w− + −+ − = (A3)
A.1 Derivations for Top Edge Nodes
For any nodes denoted as (i, j) at the top edge applying Eq.
(A2), the following relations can be obtained.
, 1 1, , 1, , , 1[ 2 ] 2i j i j i j i j i j i jw w w w w w+ + − −= − − + + − (A4)
1, 1 , 1, 2, 1, 1, 1[ 2 ] 2i j i j i j i j i j i jw w w w w w− + − − − − −= − − + + −
(A5)
1, 1 2, 1, , 1, 1, 1[ 2 ] 2i j i j i j i j i j i jw w w w w w+ + + + + + −= − − + + −
(A6)
Substituting Eqs. (A4) ~ (A6) into the CDF expression of Eq. (A3),
one can achieve Eq. (A7)
2 2, 2 2, 1,
2 2, 1,
22, 1, 1
( 2 ) (4 8 4)
( 6 12 12) (4 8 4)
( 2 ) ( 2 4)
i j i j i j
i j i j
i j i j
w w w
w w
w w
+ − −
+
+ − −
= − + + − −
+ − + + + − −
+ − + + − +
, 1 1, 1 , 2(4 12) ( 2 4)i j i j i jw w w− + − − + − + − + + (A7)
Substituting Eqs. (A4) ~ (A7) into Eq. (A1), the governing
equation such as Eq. (8) at the nodal points along top edge of the
raft can be found. Note that only nine nodal displacements are left
in the equation.
A.2 Derivations for Top-Left Corner RHS Node
For right-hand-side (RHS) node (i, j) near to the top-left
corner, the node of (i − 2, j) is a fictitious point, therefore, the
following procedure is taken for the replacements. Applying Mx =
0 at the top-left corner node (i − 1, j), the following equation is
valid from Eq. (A4).
1, 1 , 1, 2, 1, 1, 1[ 2 ] 2i j i j i j i j i j i jw w w w w w− + − − − − −= − − + + −
, 1, 2, 1, 1(2 2)i j i j i j i jw w w w− − − −= − + + − − (A8)
Similarly at the left edge of the raft, My = 0 yields that
1, , 1 , , 1 , 1,[ 2 ] 2i j i j i j i j i j i jw w w w w w− + − += − − + + − (A9)
1, 1 , 2 , 1 , , 1 1, 1[ 2 ] 2i j i j i j i j i j i jw w w w w w− + + + + + += − − + + −
(A10)
1, 1 , , 1 , 2 , 1 1, 1[ 2 ] 2i j i j i j i j i j i jw w w w w w− − − − − + −= − − + + −
(A11)
Therefore, at the top-left corner where the node is (i − 1, j),
the following equation is valid.
2, , 1, 1, 1 1, 1(2 2)i j i j i j i j i jw w w w w− − − + − −= − + + − − (A12)
Substituting Eq. (A8) into Eq. (A12), one can find the
following relations.
2, , 1,2i j i j i jw w w− −= − + (A13)
Substituting Eq. (A13) into Eq. (8), Eq. (12) can be obtained.
A.3 Derivations for Top-Left Corner DS Node
For left-hand-side (LHS) node near to the top-right corner,
one can adopt Eq. (12) and change the first subscripts i − 1, i + 1
and i + 2 to i + 1, i − 1 and i − 2. This will result in Eq. (16). For
top-left corner down-side (DS) node, Eq. (16) can be applied to it
by replacing i with j and j with i. After the replacements, also note
that i − 2 and i − 1 need to be replaced by i + 2 and i + 1. This will
result in Eq. (13).
A.4 Derivations for Inner Top Edge nodes
For inner top-edge nodes, the node of (i, j + 2) is the fictitious
point. One can apply the edge node relation such as Eq. (A4) at the
edge nodes denoted by (i, j + 1), which leads to the following
equation,
, 2 1, 1 , 1 1, 1 , 1 ,[ 2 ] 2i j i j i j i j i j i jw w w w w w+ + + + − + += − − + + −
(A14)
Substituting Eq. (A14) into Eq. (A1), one can obtain Eq. (20).
A.5 Derivations for Top-Left Corner Node
Note that Eq. (A13) is also valid if the top-left corner node
became (i, j), therefore Eq. (A13) can be rewritten as follows,
1, 1, ,2i j i j i jw w w− += − + (A15)
Substituting Eqs. (A11) and (A15) into Eq. (A7), the following
equation can be obtained,
2 2, 2 2, ,
2 22, , 1
( 2 ) (4 8 4)
( 2 ) ( 4 8 4)
i j i j i j
i j i j
w w w
w w
+ −
+ −
= − + + − +
+ − + + − + −
2, 2(2 4 1) i jw −+ − + (A16)
Similarly, the relations shown in Eq. (A15) can be adopted
when considering My = 0 at the corner node (i, j), one could obtain
the following equation,
, 1 , 1 ,2i j i j i jw w w+ −= − + (A17)
Substituting Eqs. (A9) ~ (A11) into the equation of Vx = 0, the
following equation can be achieved.
2 22, , 2 , 1
2 2, , 1
2, 2 1, 1
( 2 ) (4 8 4)
( 6 12 12) (4 8 4)
( 2 ) ( 2 4)
i j i j i j
i j i j
i j i j
w w w
w w
w w
− − −
+
+ + −
= − + + − −
+ − + + + − −
+ − ++ + −
1, 1, 1 2,(4 12) ( 2 4)i j i j i jw w w+ + + + + − + − + + (A18)
Again, combining Eqs. (A9) ~ (A11) and Eq. (A18) with Eq. (A1),
one can obtain Eq. (10). Substituting Eqs. (A6) and (A17) into Eq.
(18), the following equation is obtained,
Chang et al.: Finite Difference Analysis of Vertically Loaded Raft Foundation Based on the Plate Theory with Boundary Concern 147
2 22, , 2 ,
2 2, 2 1,
( 2 ) (4 8 4)
( 2 ) ( 4 8 4)
i j i j i j
i j i j
w w w
w w
− −
+ +
= − + + − +
+ − + + − + −
22,(2 4 1) i jw ++ − + (A19)
Now, combining Eq. (A16) with Eq. (A19) would result in
4 3 2
2, ,4 3 2
2
1,4 3 2
4 3 2
2,4 3 2
4 16 16 4
4 4 1
4 8 4
4 4 1
4 6 4 1
4 4
+
+
1+
i j i j
i j
i j
w w
w
w
−
+
+
− + − +=
− − +
− + −+
− − +
− + − ++
− − +
4 3 2
, 14 3 2
4 3 2
, 24 3 2
4 16 20 8
4 4 1
2 8 10 4
+
+4 4 1
i j
i j
w
w
−
−
− + − +
− − +
− + − ++
− − +
(A20)
Oppositely by substituting Eq. (A19) into Eq. (A16), the following
equation is obtained.
4 3 2
, 2 ,4 3 2
2
, 14 3 2
4 3 2
, 24 3 2
4 3 2
1,4 3 2
4 16 16 4
4 4 1
4 8 4
4 4 1
4 6 4 1
4 4 1
4 16
+
+
+
+
20 8
4 4 1
i j i j
i j
i j
i j
cw w
w
w
w
+
−
−
+
− + − +=
− − +
− + −+
− − +
− + − ++
− − +
− + −+
− − +
4 3 2
2,4 3 2
2 8 10 4
4+ 4 1i jw +
− + − ++
− − +
(A21)
Combining Eqs. (A11), (A15) with Eq. (A20), the governing
equation at the top-left corner can be derived as follows,
5 4 3 2
,4 3 2
4 3
1,4 3 2
4 3
2,4 3 2
4 20 32 16 4 4
4 4 1
4 8 8 4
4 4 1
2 4 4 2
+
+
4 4 1+
i j
i j
i j
w
w
w
+
+
− + − − +
− − +
− + −+
− − +
− + − ++
− − +
5 4 3 2
, 14 3 2
8 36 56 32
+
4
4 4 1i jw −
− + − + −+
− − +
5 4 3 2 *4
, 24 3 2
4 18 28 16 2
4 4 1+i j
qw s
B−
− + − ++ =
− − + (A22)
With the similar procedures combining Eqs. (A6), (A17) with Eq.
(A21), the governing equation at the corner can be written as:
5 4 3 2
,4 3 2
4 3
, 14 3 2
4 3
, 24 3 2
5 4 3 2
1,4 3 2
4 20 32 16 4 4
4 4 1
4 8 8 4
4 4 1
2 4 4 2
4 4 1
8 36 56 32 4
4 4 1
+
+
+
+
i j
i j
i j
i j
w
w
w
w
−
−
+
− + − − +
− − +
− + −+
− − +
− + − ++
− − +
− + − + −+
− −
+
5 4 3 2 *4
2,4 3 2
4 18 28 16 2
4+ 4 1i j
qw s
B+
− + − ++ =
− − + (A23)
Note that because Eqs. (A22) and (A23) are both applicable at
top-left corner, therefore the solution is suggested averaging the
two equations for the nodal displacement at the corner, thus Eq.
(24) is suggested.
A.6 Derivations for Inner Top-Left Corner Node
For inner node (i, j) nearest to the top-left corner, Mx = 0 can
be applied to the edge node at (i, j + 1), My = 0 are applied to the
edge node at (i − 1, j). Therefore, the following equations can be
achieved.
, 2 1, 1 , 1 1, 1 , 1 ,[ 2 ] 2i j i j i j i j i j i jw w w w w w+ + + + − + += − − + + −
(A24)
2, 1, 1 1, 1, 1 1, ,[ 2 ] 2i j i j i j i j i j i jw w w w w w− − + − − − −= − − + + −
(A25)
Substituting Eqs. (A24) and (A25) into Eq. (A1), Eq. (28) is thus
obtained.
