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Analysis of arbitrary Mindlin plates or bridge decks by spline finite strip method
Conrpurm & Srrucrurn Vol. 54. No. I, pp. I I I-I 18. 1995 Copyright I’ 1994 Elsevier Saence Ltd Prinled in Great Bntain. All nghts reserved 0045.7949/95 $9.50 + 0.00
0045-7949(94)00306-8
ANALYSIS OF ARBITRARY MINDLIN PLATES OR BRIDGE DECKS BY SPLINE FINITE STRIP METHOD S. F. Ng and X. Chen Department
of Civil Engineering,
University
of Ottawa,
Ottawa,
Ontario,
Canada
KlN
9B4
(Received 20 May 1993) Abstract-A spline finite strip method for the analysis of arbitrary Mindhn plates is presented. The plate is first mapped into a square domain in the natural coordinate plane, by using the cubic serendipity shape function, and the mapped plate is discretised into a number of strips. The displacements of each strip are described by interpolation functions which are given as products of piecewise polynomials and E3 spline functions. The method, in which three independent reference quantities, W, 8,) 0,. are used to take into account the effects of transverse deformation, is applied to the analysis of single and multi-span fan-shaped bridge decks.
INTRODUCTION
finite strip method was first published in 1968 by Y. K. Cheung for the analysis of simply supported bridge deck structures. The method utilizes a series of orthogonal functions in the longitudinal direction, combined with the conventional finite element polynomial shape functions in the transverse direction, to simulate all the displacement components of the structure. Thus, the number of dimensions of the analysis is reduced by at least one. Consequently, the computer time, storage and input data are reduced significantly. Nevertheless, the use of this method can lead to difficulties. For instance, because the beam functions are continuously differentiable, it is difficult to use such functions to simulate the abrupt changes of bending moment at internal supports or at the vicinity of point forces. In order to overcome these difficulties, the mathematical tool called ‘B,-spline’ was used in 1982 as the longitudinal displacement function to form the sphne finite strips for the analysis of rectangular plates [I]. So far, the spline finite strip method has been successfully applied to a wide variety of problems [l-9]. However, the application of the method has been limited to thin plate analysis or restricted to the rectangular shape only. The object of the present study is to further extend the spline finite strip method to arbitrarily shaped, moderately thick plates, using the Mindhn plate theory. The efficiency and accuracy of this new numerical technique is demonstrated by several numerical examples.
The semi-analytical
discretization and numerical integration. The arbitrarily shaped plate or bridge deck is first mapped into a square domain, by suitable one-to-one transformation, and then discretization and analyses are carried out in the transformed plan, from which accurate results can be obtained for both thin and thick plates by using numerical integration. Assembly
transformations
In engineering structures, analysis of arbitrary shaped bridge decks are often curved along two sides, while the abutment/piers are skewed. Without loss of generality, a typical plan of such a bridge deck can be represented in Cartesian coordinates, as in Fig. l(a). By making use of the cubic serendipity shape function, the plate can be mapped from the Cartesian coordinate system into the natural coordinate 5 - q plane, as in Fig. I(b), that is:
x = f: N,(5, V)X,, ,=l
where (x,, yi) are coordinates of the ith control point (node) on the curved edges of the plate and N,(& q) are the shape functions given in Table 1. Other mappings may also be used but it has been established that this simple cubic serendipity shape function is sufficiently accurate for most purposes. In the Mindhn plate theory, allowance is made for the effect of transverse shear deformation [IO]. Therefore, in ad.dition to the lateral deflection, which is the sole reference quantity of the classical plate theory, the two rotations about the in-plane plate axes also
ANALYSIS OF ARBITRARY MINDLIN PLATE
The analysis of the arbitrary Mindlin plates may be carried out in three steps: coordinate transformation, CA.5 54 I--H
of two coordinate
111
S. F. Ng and X. Chen
112
? A
‘.
2.
/’
(-1, 1)
(1>1)
l
08
I
/
(-1, -113) 0
(1, l/3)
(-1, -l/3).
(1, -l/3)
.I
3.
/
/
I
Fig.
1. Co-ordinate
-5
I l6 (-1, -1)
transformation.
(a) Typical plan of flat discretization.
become independent reference quantities. Thus, the displacement field at any point of the plate can be expressed as:
(1, -1)
l
bridge.
(b) Transferred
The transformation the two coordinate
domain
for
between the first derivatives systems can be written as
ral
ral
Subsequent manipulations are based on relations stipulated by this equation. where u, u and w are the displacements in the X, y and z directions, respectively, and 0, and Q,. are the rotations contained in the zx and yz planes, respectively. In order to formulate the stiffness and load matrices, it is necessary to obtain the transformation between the first derivatives of 0,. O,., w of the two coordinate systems. Thus, we get:
ax
aw
aY
The total potential written as [I l]
aw
s
ay
&
x =;j+r,dxdy
energy
of the plate
- j$w
dx dy,
9
(I.31
ay
where [J] is the Jacobian matrix and matrix inverse matrix of [J], is given by
[r], the
(1.4)
of the Jacobian
matrix:
1J1= det[J] = J,, J22 - J2, J,,. Table
1. Shape
function
for a simplified element
cubic serendipity
N,(t. v) Corner node (1,4,5.8) M&side node (2.3.7.6)
(l/32)(1
(2.1)
in which, for simplicity, only the contribution of the distributed load of intensity q(x, y) has been considered, L and (T are, respectively, the generalized strain and stress vectors. From eqn 1.5, the generalized strain can be expressed as:
ao, ax
where 1JI is the determinant
can be
aw
x = z z ax =[Jl ax ax ay aw aw
&
as
Finite strip formulation for the analysis of arbitrary Mindlin plates
i-1I 1II i-1 aw
of
+ 65) (9$-J’= I’,+v-%) &=&I I (9/32)(i + 5,5 ) (1 - z2)= jl,;99,q) , t.= *I
ae,
ay
++z a0
OY
0,+g I)
O,.+o” ix,
ay
=
r,
0
[ 0
rh
1 5x8
Analysis of Mindlin plates
0
r21 l-22
(2.3)
1
0
(2.7)
0
D, and D,? are, respectively, the bending and shear contributions to the elasticity matrix D; E and v are the Young’s modulus and Poisson’s ratio respectively; h is the plate thickness and K is a shear correction factor used to account for the warping of the section [ 121.
ri2 1
6,
[ 11’
rcEh -p D,y-2(1+v)
where 0
113
and
(2.4) Displacement
The relationship between the generalized strains can be written as
stress and
In the context of the spline finite strip, the whole domain R in the 5 -q plane is partitioned into n strips along the t-axis and M sections along the n-axis, so that there are n x m subdomains and 3(m + 3)(n + 1) degrees of freedom for computation purposes (Fig. 2). In strip_& the displacement can be expressed as
(2.5) and, for an isotropic
D=
plate, D,
O-
0
D,,
function
(2.6)
F(59?I = W(5)1P(V)l{~~~
(3.1)
where
-
_
-
c-1, 111, ::
::
::
,, ,,
,,
,,
--
-_
r 4,
------_-T
,
(1, 1)
‘rn
“Tll-I
0
,I
0
0
0
0
0
0.
0
0
4,
0
0
0
0
0
0
0
0
0
0
I,
,,
0
0
I,
0
0
0
0
0
a,
,,
,,
,,
,,
0
0
0
0
0
,,
,,
,,
,,
t,
0
0
0
0
0
0
I,
0
0
(I
0
0
0
0
4
0
0
0
0
0
0
0
0
0
0
,
0
0
0
0
0
0
0
0
0
,,
,,
I,
4,
I,
4,
,I
0
0
0
0
,
-s
-4ririLijisili
C-l*
(1, -1)
Fig. 2. The discretisation
of the transformed
domain.
S. F. Ng and X. Chen
114
is the displacement function and N(t) is the shape function, Q(u) is the spline function and 6 is the displacement vector. For linear Mindlin spline strips, the displacements can be written as
In spline finite strip analysis, every loaded crosssection of point force and every supported crosssection should coincide with some longitudinal knot, in order to obtain satisfactory accuracy. Therefore, in some cases, unequal spacing between knots must be adopted. The unequally spaced B,-spline $,,,(q), with q,,, as the centre, is given by:
(3.2) where
N,=l-T,
N2=r
and
t=
(5 - 5,) (5 ,+, -5,)’
(3-3)
where
(3.8)
{e,},, {e,.}, and {w}, are vectors of rotation and deflections parameters along nodal line j. They are defined by PY), = 10,~. , I oyi“.
0,,_“, 10,),,,,+,I’
{e,),=[e,.,.~,,e,,,,,...o,~,~~,e,,”,+,i’ 14, = b,. - I>W,,O? w,,,,7W,.‘l? +II’. The spline represented
functions by
@(rl) =L#-1(V),
for
the
&(V)>‘.
whole
(3.4)
strip
can
be
.d?“(V),d+,,.,(I?)l~ (3.5)
Thus, the spiine finite strip method can be easily applied to problems with or without intermediate supports and with arbitrary loadings. In order to satisfy support conditions, such as free, simple and clamped ends at the two ends of the plate, the local splines at the ends immediately adjacent to the boundary point have to be modified in accordance with Table 2 [I]. A more convenient method of restraining the degrees of freedom by appropriate called the penalty function method, is springs, adopted in this paper to impose the general boundary conditions. For example, if the displacement w is prescribed to be zero at mode i, then the stiffness
in which each local spline is given as
(‘1
h3
+
-
L2)3
3h2(1 - V,,? I> + 3Nrl - 8,,? , Y - 3(9 - ?,?, , )’
~3+3~%L”+,-~)+3WI,,,+,-rl)~-3(%,,+,-~)3 (V,,,+ ? - II)’
:
0
tl,,,
2 d
?
d
rlw
- /
rl,,, 1G 17G i?,,, 11.Z G 4 G vl,,,+ , V., + I G V G rl,. + ? otherwise.
(3.6)
Analysis
of Mindlin
plates
local spline for end support
Table 2. Modified
Modified Boundary
Condition
Free end Simply supported Lamped end
matrix must stiffness:
be
[K]=a4
by
i
ri
the
new
if1
4 16
L1
adding
4
11
i-l i.
4 11
(3.9)
qf the st@ness matrix
By combining eqns 2.1 and 2.2 with the displacement function, eqn 3.2, and by utilizing the principle of minimum potential energy, a system of linear equivalent equations an be obtained by putting an - = 0
84
The stiffness
matrix
D% x 14= ss
hence K6 = F.
for one section
is
~~12’4,*~~1~~~~~1~~~~Tl5x8 x 1% xs,lJl d5 drl, (3. lo)
where B, the straindisplacement for the linear strip is given as
PI =
relationship
matrix
91 91-9-i 91 - l/290+9-,
The integration is carried out section-by-section using the (2 x 2) Gaussian integration scheme. The total stiffness matrix and load matrix are obtained by summing up the contribution of each section. Through similar procedures, we can derive all the necessary formulations for the quadratic and cubic Mindlin spline strips. Reduced integration and shear locking Shear locking is a well-known phenomenon for plates analyzed using a refined plate theory, such as the Mindlin plate theory. The problem is similar to that experienced with Mindlin plate finite elements. As the plate thickness becomes small, the influence of the shear terms tends to dominate and the numerical solution may yield unrealistically overstiff results (locking) unless some precautions are taken. The use of deliberate under-integration in evaluating the stiffness properties of isoparametric plate elements, based on a shear deformation theory, such as the Mindlin plate theory, is now common practice. A polynomial of degree 2n - 1 is integrated exactly by n-point Gauss quadrature. Use of more than n points will still produce the exact result. We consider the n-point Gauss quadrature as the ‘full’ numerical integration. If the function 4 = 4(t) is not a polynomial, Gauss quadrature is inexact, but becomes more accurate as more points are used. Table 3 shows the data of sampling points and weights. The foregoing linear transformation is sufficient to make arbitrary limit
N,(5).; 4(v) N,(r)#(?),,
0
0 0
4(5),c 4(v) N,(5) ml,,
0 0
0
0
0
N,(5),<407) N, (5)MI I,,,
0
0
0
0
wt),i d(rl) N2(5)4(v),,,
0 0
N,(5) 4(v)
0
0 0
N*(5) 4(v)
0
0
0
N,(5) 4(rl)
0
0
K(5) d(v)
0
0
0 0
N,(t),: rP(?) N,(5) 4(v),,
0 0
0
0
WC),; b@i) N,(5) $(v),q
Equation 3.11 can be expanded to a 8 x 24 matrix for each section, since we know there are four splines related to one section. The load vector can be obtamed readily, as in the standard finite element procedure, as
{F} =
90 90-49-I Eliminated
i-t1
The assembly of the stiffness of all point constraints to the structural stiffness matrix completes the treatment of the boundary. Formulation
9;
90
9-I Eliminated Eliminated
end
modified
i-l
9’1
local spline
(3.12)
0
0
(3.11)
1
changes. We can always consider a convenient reference interval, such as - 1 to + 1. In practice, the limit change is done automatically by isoparametric transformation. When using ‘full’ Gaussian numerical integration for the element based on the shear deformation theory, such as the Mindlin plate theory, efficient performance with accurate results is obtained for thick plates and shells but the results obtained are
116
S. F. Ng and X. Chen
Table 3. Sampling points and weights for Gauss quadrature over the interval 5 = - I to 5 = + I Order
n
I 2 3
Location
5, of sampling
kO.57735 +0.77495
0 02691 89626 66692 41482 0
Weight
factor
Moment along mid-span of fan-shaped bridge deck (mesh size 12 x 8)
W,
2 1 0.55555 55555 55555 0.88888 88888 88888
integration rules for matrices k, and k, for various Mindlin strip elements
Table 4. Gaussian
Gaussian Number
Strip element Linear Quadratic Cubic
Full(F) k, k, 2 3 4
Integration of integration Selective(S) k, k,
2 3 4
2 3 3
1 2 3
Rules
0.2 0.1’
7.0
” 7.5
” 8.5
8.0
1’ 9.5
9.0
points
” 12 12.5
13
IN C
Reduced(R) k,
k,
I
1 2 3
2 3
-*-+-*-*-* ” ” 10 10.5 11 11.5
B
0 Point load at A
Fig. 4. Moments
A
f Point load at B
*
Point load at C
along mid-span of fan-shaped (thickness = 0.168 in.).
bridge deck
NUMERICAL EXAMPLES
excessively stiff for plates with a thin geometry. A reduction in the order of numerical integrations leads to a considerable improvement in accuracy for thin plates, whether such reduction is ‘selective’ or ‘reduced’. Regarding the terminology used here, ‘full’ integration (n x n points over the element middle surface) means an order of integration sufficient to calculate the stiffness matrix exactly for a rectangular or parallelogrammic-shaped element; ‘selective’ integration implies the use of (n - 1) x (n - 1) integration points in evaluating the transverse shear stiffness contribution and n x n points in evaluating all other contributions; and ‘reduced’ integration implies the use of (n - 1) x (n - 1) points in evaluating all stiffness contributions. The number of integrating points required to exactly integrate the strip matrices obviously depends on the degree of the shape function polynomials of each particular strip. Table 4 gives the number of Gaussian integrating points needed for the full, reduced and selective integration of the bending stiffness matrix k, and the shear stiffness matrix
Example
1-fan
-shaped
bridge
deck
To illustrate the efficiency of the spline finite strip method presented in this paper, a 60” fan-shaped thin bridge deck (Fig. 3) is first analyzed. The arbitrary thin fan-shaped deck has internal and external radii of 177.8 mm (7 in.) and 330.2 mm (13 in.) respectively, the Young’s modulus and Poisson’s ratio are 6239.4 MPa (460000 lb/in.*) and 0.35 respectively; the original thickness of the plate is 4.267 mm (0.168 in.). Two additional deck thicknesses of 6.807 mm (0.268 in.) and 22.047 mm (0.868 in.) are also investigated here, in order to demonstrate that the spline finite strip method presented here can accurately solve both thin and moderately thick arbitrary shaped plates or bridge decks. The bridge was simply supported along its two ends. Concentrated loads were applied at three different locations (Points A,
E = 0.44 x lo6 ib/in2 h = 0.75 in
k. 12 E = 0.46 x lo6 Ib/in’
Y
Y
x
Fig. 3. Fan-shaped
bridge
deck model
(example
1).
Fig
5. Fan-shaped
continuous
bridge
model (example
2).
Analysis of Mindlin plates
117 M, along the symmetric axis AB 24 lb point load at A (mesh size 16 x8)
24 lb point load at B 20 18 e $
l6 14
4 12 ’ - 10 8
in
42
43
44
45
46
A
B
47
48
49
A
B Fig. 6. Transverse
+ Present method
[ 141
Reference deflection
across symmetrical
-o-
line AB (I).
B and C) and the mid-span deflections resulting from these loadings are tabulated in Tables 5 and 6 and compared with Cheung [8], Coull[13] and Sawko [ 141. Good agreements are found throughout. The distributions of the tangential moment across the mid-span (thickness = 0.168 in.) were also plotted in Fig. 4.
axis
M, along the symmetric axis AB 24 lb point load at B (mesh size 16 x8)
20 18 16 .9
14
4
12
”
10
z
8 6 4
42
43
44
45
46 in
47
48
49
B -
5C
A + Present method
Reference [ 141
Fig. 9. Longitudinal
moment (M,,) along the symmetric AB (2).
axis
M, along the symmetric axis AB 24 lb point load at A (mesh size 16 x 8) 6, 5
24 lb point load at A
4
5.0, 4.5
.9
4.0
$2
.E 3.5 :e
+ Present method
moment (kf,,) along the symmetric AB (1).
evident from these figures that the results of the spline finite strip method presented here compare extremely well with the results obtained by Sawko and Merriman [14].
continuous isotropic bridge
One of the most important advantages of the spline finite strip method, in comparison with the conventional finite strip method, is that the spline finite strip method can treat bridge decks with arbitrary intermediate supports and arbitrary point loads. A fanshaped continuous isotropic deck, originally designed and tested by Sawko and Merriman in 1971, is used here to demonstrate this particular versatility of the spline finite strip method. Details of this model are shown in Fig. 5. The 60” fan-shaped model was point-supported at eight nodes by columns which offered only vertical restraint along four radial lines. A point load was applied at two different locations (Points A and B) and the mid-span deflections and moments resulting from these loadings were compared with those from Sawko [14] in Figs 6-10. It is
[ 141
Reference
Fig. 8. Longitudinal
Example 2-Fan-shaped deck
50
in
3.0 1 2.5
,+‘+ ,+‘+
2.0
I+
1.5
F’+
1.01
’
1
42
43
I
44
I
45
I
I
I
I
46
47
48
49
I I
/“\
3
=
1
z-
0 -1
1:: 42
43
44
45
46
41
48
49
5’
in
50
in
q
,__&&+J
A
B A
B
-zFig. 7. Transverse
Reference
[ 141
deflection
+ Present method
across symmetrical
line AB (2).
Reference
Fig. 10. Transverse
[ 141
moment
+ Present method
(M,) along the symmetric AB.
axis
S. F. Ng and X Chen
118 CONCLUSION
A spline finite strip method to analyze moderately thick plates and bridge decks of arbitrary shapes, according to the Mindiin plate theory, is developed in this thesis. The reduced integration technique is used to eliminate ‘shear locking’ and the penalty function method is utilized to impose boundary condition at the end of strips and intermediate supports. Numerical examples have demonstrated that:
1. The spline finite method presented here can be applied successfully to analyze moderately plates of thickness-to-length ratios from 0.01 to 0.30. 2. The present method yields very accurate results for both thin and moderately thick rectangular and arbitrarily shaped bridge decks, subjected to both uniform and concentrated loads, and is much more efficient than the conventionally used finite element method, both in terms of data input and computation time required. 3. The method successfully overcomes the deficiencies of the conventional finite strip method in the treatment of patch loadings or intermediate supports. Further, the input data required is only a small fraction of that generally required by the conventional finite element method.
3. M. J. Chen, L. G. Tham, Y. K. Cheung, Spline finite strip for parallelogram plate. Con/: on Accuracy Estimutes and Adaotive Refinements in Finite Element Com-
purarions, Lisbon, Portugal, 1984, Vol. I, pp. 19-22. 955104. of irregular 4. J. L. Chen and K. P. Chong, Vibration
5
6
7
8
9
IO
11
REFERENCES
12
1. Y. K. Cheung, S.D Fan and C. Q. Wu, Spline finite strip in structure analysis. Proc. Inr. Conf. on Finite Element Method, Shanghai, 1982, 7044709.
13
2. Y. K. Cheung and S. C. Fan, Static analysis of right box girder bridges by spline finite strip method. Proc. Instn Cir. Engineers, Part 2, 1983, 75, June, pp. 311-323
14
plates by finite strip method with splined functions. Engineering Mechanics in CitGl Engineering, Proc. 5th Engineering Mechanics Div., AXE (Edited by A. P. Boresi and. K. P. Chong), Vol. 1. pp. 2566260 (1984). Y. K. Cheuna, L. G. Tham and W. Y. Li, Application of spline-finit&trip method in the analysis of curved slab bridge. Proc. instn Civ. Engineers, Part 2, 1986, 8 I, pp. 11 I-124. W. Y. Li, Y. K. Cheung and L. G. Tham, Spline finite strip analysis of general plates. J. Engng Mech. 112, 43354 (1986). Y. K. Cheung, L. G. Tham and W. Y. Li, Free vibration analysis of arbitrarily doubly curved shell by spline finite strip method. Research Report, University of Hong Kong (1989). Y. K. Cheung, L. G. Tham and W. Y. Li, Free vibration analysis of arbitrarily doubly curved shell by spline finite strip method. Research Report, University of Hong Kong (1989). L. G. Tham and H. Y. Szeto, Buckling analysis of arbitrarily shaped plates by sphne finite strip method. Comput. Struct. 36, 729-735 (1990). R. D. Mindlin, Influence of rotatory inertia and shear on flexurdl motions of isotropic elastic plates. J. appl. Mech. 37, 1031-1036. S. Timoshenko and S. Woinowsky-Krieger, Theory of Plate und Shells. 2nd Edn. McGraw-Hill, New York (I 959). S. Timoshenko, Strength of Materids. Van Nostrand, New York, 1955. A. Coull and P. C. Das, Analysis of curved bridge decks. Proc. Insm. Ck. engineers. 1967, 37. May, pp. 75-85. F. Sawko and P. A. Merriman. Finite element analysis by bridge curved in plan. Deaelopments in Bridge Design and Construction. Crosby Lock, Wood, Oxford (1971).
0045-7949(94)00306-8
ANALYSIS OF ARBITRARY MINDLIN PLATES OR BRIDGE DECKS BY SPLINE FINITE STRIP METHOD S. F. Ng and X. Chen Department
of Civil Engineering,
University
of Ottawa,
Ottawa,
Ontario,
Canada
KlN
9B4
(Received 20 May 1993) Abstract-A spline finite strip method for the analysis of arbitrary Mindhn plates is presented. The plate is first mapped into a square domain in the natural coordinate plane, by using the cubic serendipity shape function, and the mapped plate is discretised into a number of strips. The displacements of each strip are described by interpolation functions which are given as products of piecewise polynomials and E3 spline functions. The method, in which three independent reference quantities, W, 8,) 0,. are used to take into account the effects of transverse deformation, is applied to the analysis of single and multi-span fan-shaped bridge decks.
INTRODUCTION
finite strip method was first published in 1968 by Y. K. Cheung for the analysis of simply supported bridge deck structures. The method utilizes a series of orthogonal functions in the longitudinal direction, combined with the conventional finite element polynomial shape functions in the transverse direction, to simulate all the displacement components of the structure. Thus, the number of dimensions of the analysis is reduced by at least one. Consequently, the computer time, storage and input data are reduced significantly. Nevertheless, the use of this method can lead to difficulties. For instance, because the beam functions are continuously differentiable, it is difficult to use such functions to simulate the abrupt changes of bending moment at internal supports or at the vicinity of point forces. In order to overcome these difficulties, the mathematical tool called ‘B,-spline’ was used in 1982 as the longitudinal displacement function to form the sphne finite strips for the analysis of rectangular plates [I]. So far, the spline finite strip method has been successfully applied to a wide variety of problems [l-9]. However, the application of the method has been limited to thin plate analysis or restricted to the rectangular shape only. The object of the present study is to further extend the spline finite strip method to arbitrarily shaped, moderately thick plates, using the Mindhn plate theory. The efficiency and accuracy of this new numerical technique is demonstrated by several numerical examples.
The semi-analytical
discretization and numerical integration. The arbitrarily shaped plate or bridge deck is first mapped into a square domain, by suitable one-to-one transformation, and then discretization and analyses are carried out in the transformed plan, from which accurate results can be obtained for both thin and thick plates by using numerical integration. Assembly
transformations
In engineering structures, analysis of arbitrary shaped bridge decks are often curved along two sides, while the abutment/piers are skewed. Without loss of generality, a typical plan of such a bridge deck can be represented in Cartesian coordinates, as in Fig. l(a). By making use of the cubic serendipity shape function, the plate can be mapped from the Cartesian coordinate system into the natural coordinate 5 - q plane, as in Fig. I(b), that is:
x = f: N,(5, V)X,, ,=l
where (x,, yi) are coordinates of the ith control point (node) on the curved edges of the plate and N,(& q) are the shape functions given in Table 1. Other mappings may also be used but it has been established that this simple cubic serendipity shape function is sufficiently accurate for most purposes. In the Mindhn plate theory, allowance is made for the effect of transverse shear deformation [IO]. Therefore, in ad.dition to the lateral deflection, which is the sole reference quantity of the classical plate theory, the two rotations about the in-plane plate axes also
ANALYSIS OF ARBITRARY MINDLIN PLATE
The analysis of the arbitrary Mindlin plates may be carried out in three steps: coordinate transformation, CA.5 54 I--H
of two coordinate
111
S. F. Ng and X. Chen
112
? A
‘.
2.
/’
(-1, 1)
(1>1)
l
08
I
/
(-1, -113) 0
(1, l/3)
(-1, -l/3).
(1, -l/3)
.I
3.
/
/
I
Fig.
1. Co-ordinate
-5
I l6 (-1, -1)
transformation.
(a) Typical plan of flat discretization.
become independent reference quantities. Thus, the displacement field at any point of the plate can be expressed as:
(1, -1)
l
bridge.
(b) Transferred
The transformation the two coordinate
domain
for
between the first derivatives systems can be written as
ral
ral
Subsequent manipulations are based on relations stipulated by this equation. where u, u and w are the displacements in the X, y and z directions, respectively, and 0, and Q,. are the rotations contained in the zx and yz planes, respectively. In order to formulate the stiffness and load matrices, it is necessary to obtain the transformation between the first derivatives of 0,. O,., w of the two coordinate systems. Thus, we get:
ax
aw
aY
The total potential written as [I l]
aw
s
ay
&
x =;j+r,dxdy
energy
of the plate
- j$w
dx dy,
9
(I.31
ay
where [J] is the Jacobian matrix and matrix inverse matrix of [J], is given by
[r], the
(1.4)
of the Jacobian
matrix:
1J1= det[J] = J,, J22 - J2, J,,. Table
1. Shape
function
for a simplified element
cubic serendipity
N,(t. v) Corner node (1,4,5.8) M&side node (2.3.7.6)
(l/32)(1
(2.1)
in which, for simplicity, only the contribution of the distributed load of intensity q(x, y) has been considered, L and (T are, respectively, the generalized strain and stress vectors. From eqn 1.5, the generalized strain can be expressed as:
ao, ax
where 1JI is the determinant
can be
aw
x = z z ax =[Jl ax ax ay aw aw
&
as
Finite strip formulation for the analysis of arbitrary Mindlin plates
i-1I 1II i-1 aw
of
+ 65) (9$-J’= I’,+v-%) &=&I I (9/32)(i + 5,5 ) (1 - z2)= jl,;99,q) , t.= *I
ae,
ay
++z a0
OY
0,+g I)
O,.+o” ix,
ay
=
r,
0
[ 0
rh
1 5x8
Analysis of Mindlin plates
0
r21 l-22
(2.3)
1
0
(2.7)
0
D, and D,? are, respectively, the bending and shear contributions to the elasticity matrix D; E and v are the Young’s modulus and Poisson’s ratio respectively; h is the plate thickness and K is a shear correction factor used to account for the warping of the section [ 121.
ri2 1
6,
[ 11’
rcEh -p D,y-2(1+v)
where 0
113
and
(2.4) Displacement
The relationship between the generalized strains can be written as
stress and
In the context of the spline finite strip, the whole domain R in the 5 -q plane is partitioned into n strips along the t-axis and M sections along the n-axis, so that there are n x m subdomains and 3(m + 3)(n + 1) degrees of freedom for computation purposes (Fig. 2). In strip_& the displacement can be expressed as
(2.5) and, for an isotropic
D=
plate, D,
O-
0
D,,
function
(2.6)
F(59?I = W(5)1P(V)l{~~~
(3.1)
where
-
_
-
c-1, 111, ::
::
::
,, ,,
,,
,,
--
-_
r 4,
------_-T
,
(1, 1)
‘rn
“Tll-I
0
,I
0
0
0
0
0
0.
0
0
4,
0
0
0
0
0
0
0
0
0
0
I,
,,
0
0
I,
0
0
0
0
0
a,
,,
,,
,,
,,
0
0
0
0
0
,,
,,
,,
,,
t,
0
0
0
0
0
0
I,
0
0
(I
0
0
0
0
4
0
0
0
0
0
0
0
0
0
0
,
0
0
0
0
0
0
0
0
0
,,
,,
I,
4,
I,
4,
,I
0
0
0
0
,
-s
-4ririLijisili
C-l*
(1, -1)
Fig. 2. The discretisation
of the transformed
domain.
S. F. Ng and X. Chen
114
is the displacement function and N(t) is the shape function, Q(u) is the spline function and 6 is the displacement vector. For linear Mindlin spline strips, the displacements can be written as
In spline finite strip analysis, every loaded crosssection of point force and every supported crosssection should coincide with some longitudinal knot, in order to obtain satisfactory accuracy. Therefore, in some cases, unequal spacing between knots must be adopted. The unequally spaced B,-spline $,,,(q), with q,,, as the centre, is given by:
(3.2) where
N,=l-T,
N2=r
and
t=
(5 - 5,) (5 ,+, -5,)’
(3-3)
where
(3.8)
{e,},, {e,.}, and {w}, are vectors of rotation and deflections parameters along nodal line j. They are defined by PY), = 10,~. , I oyi“.
0,,_“, 10,),,,,+,I’
{e,),=[e,.,.~,,e,,,,,...o,~,~~,e,,”,+,i’ 14, = b,. - I>W,,O? w,,,,7W,.‘l? +II’. The spline represented
functions by
@(rl) =L#-1(V),
for
the
&(V)>‘.
whole
(3.4)
strip
can
be
.d?“(V),d+,,.,(I?)l~ (3.5)
Thus, the spiine finite strip method can be easily applied to problems with or without intermediate supports and with arbitrary loadings. In order to satisfy support conditions, such as free, simple and clamped ends at the two ends of the plate, the local splines at the ends immediately adjacent to the boundary point have to be modified in accordance with Table 2 [I]. A more convenient method of restraining the degrees of freedom by appropriate called the penalty function method, is springs, adopted in this paper to impose the general boundary conditions. For example, if the displacement w is prescribed to be zero at mode i, then the stiffness
in which each local spline is given as
(‘1
h3
+
-
L2)3
3h2(1 - V,,? I> + 3Nrl - 8,,? , Y - 3(9 - ?,?, , )’
~3+3~%L”+,-~)+3WI,,,+,-rl)~-3(%,,+,-~)3 (V,,,+ ? - II)’
:
0
tl,,,
2 d
?
d
rlw
- /
rl,,, 1G 17G i?,,, 11.Z G 4 G vl,,,+ , V., + I G V G rl,. + ? otherwise.
(3.6)
Analysis
of Mindlin
plates
local spline for end support
Table 2. Modified
Modified Boundary
Condition
Free end Simply supported Lamped end
matrix must stiffness:
be
[K]=a4
by
i
ri
the
new
if1
4 16
L1
adding
4
11
i-l i.
4 11
(3.9)
qf the st@ness matrix
By combining eqns 2.1 and 2.2 with the displacement function, eqn 3.2, and by utilizing the principle of minimum potential energy, a system of linear equivalent equations an be obtained by putting an - = 0
84
The stiffness
matrix
D% x 14= ss
hence K6 = F.
for one section
is
~~12’4,*~~1~~~~~1~~~~Tl5x8 x 1% xs,lJl d5 drl, (3. lo)
where B, the straindisplacement for the linear strip is given as
PI =
relationship
matrix
91 91-9-i 91 - l/290+9-,
The integration is carried out section-by-section using the (2 x 2) Gaussian integration scheme. The total stiffness matrix and load matrix are obtained by summing up the contribution of each section. Through similar procedures, we can derive all the necessary formulations for the quadratic and cubic Mindlin spline strips. Reduced integration and shear locking Shear locking is a well-known phenomenon for plates analyzed using a refined plate theory, such as the Mindlin plate theory. The problem is similar to that experienced with Mindlin plate finite elements. As the plate thickness becomes small, the influence of the shear terms tends to dominate and the numerical solution may yield unrealistically overstiff results (locking) unless some precautions are taken. The use of deliberate under-integration in evaluating the stiffness properties of isoparametric plate elements, based on a shear deformation theory, such as the Mindlin plate theory, is now common practice. A polynomial of degree 2n - 1 is integrated exactly by n-point Gauss quadrature. Use of more than n points will still produce the exact result. We consider the n-point Gauss quadrature as the ‘full’ numerical integration. If the function 4 = 4(t) is not a polynomial, Gauss quadrature is inexact, but becomes more accurate as more points are used. Table 3 shows the data of sampling points and weights. The foregoing linear transformation is sufficient to make arbitrary limit
N,(5).; 4(v) N,(r)#(?),,
0
0 0
4(5),c 4(v) N,(5) ml,,
0 0
0
0
0
N,(5),<407) N, (5)MI I,,,
0
0
0
0
wt),i d(rl) N2(5)4(v),,,
0 0
N,(5) 4(v)
0
0 0
N*(5) 4(v)
0
0
0
N,(5) 4(rl)
0
0
K(5) d(v)
0
0
0 0
N,(t),: rP(?) N,(5) 4(v),,
0 0
0
0
WC),; b@i) N,(5) $(v),q
Equation 3.11 can be expanded to a 8 x 24 matrix for each section, since we know there are four splines related to one section. The load vector can be obtamed readily, as in the standard finite element procedure, as
{F} =
90 90-49-I Eliminated
i-t1
The assembly of the stiffness of all point constraints to the structural stiffness matrix completes the treatment of the boundary. Formulation
9;
90
9-I Eliminated Eliminated
end
modified
i-l
9’1
local spline
(3.12)
0
0
(3.11)
1
changes. We can always consider a convenient reference interval, such as - 1 to + 1. In practice, the limit change is done automatically by isoparametric transformation. When using ‘full’ Gaussian numerical integration for the element based on the shear deformation theory, such as the Mindlin plate theory, efficient performance with accurate results is obtained for thick plates and shells but the results obtained are
116
S. F. Ng and X. Chen
Table 3. Sampling points and weights for Gauss quadrature over the interval 5 = - I to 5 = + I Order
n
I 2 3
Location
5, of sampling
kO.57735 +0.77495
0 02691 89626 66692 41482 0
Weight
factor
Moment along mid-span of fan-shaped bridge deck (mesh size 12 x 8)
W,
2 1 0.55555 55555 55555 0.88888 88888 88888
integration rules for matrices k, and k, for various Mindlin strip elements
Table 4. Gaussian
Gaussian Number
Strip element Linear Quadratic Cubic
Full(F) k, k, 2 3 4
Integration of integration Selective(S) k, k,
2 3 4
2 3 3
1 2 3
Rules
0.2 0.1’
7.0
” 7.5
” 8.5
8.0
1’ 9.5
9.0
points
” 12 12.5
13
IN C
Reduced(R) k,
k,
I
1 2 3
2 3
-*-+-*-*-* ” ” 10 10.5 11 11.5
B
0 Point load at A
Fig. 4. Moments
A
f Point load at B
*
Point load at C
along mid-span of fan-shaped (thickness = 0.168 in.).
bridge deck
NUMERICAL EXAMPLES
excessively stiff for plates with a thin geometry. A reduction in the order of numerical integrations leads to a considerable improvement in accuracy for thin plates, whether such reduction is ‘selective’ or ‘reduced’. Regarding the terminology used here, ‘full’ integration (n x n points over the element middle surface) means an order of integration sufficient to calculate the stiffness matrix exactly for a rectangular or parallelogrammic-shaped element; ‘selective’ integration implies the use of (n - 1) x (n - 1) integration points in evaluating the transverse shear stiffness contribution and n x n points in evaluating all other contributions; and ‘reduced’ integration implies the use of (n - 1) x (n - 1) points in evaluating all stiffness contributions. The number of integrating points required to exactly integrate the strip matrices obviously depends on the degree of the shape function polynomials of each particular strip. Table 4 gives the number of Gaussian integrating points needed for the full, reduced and selective integration of the bending stiffness matrix k, and the shear stiffness matrix
Example
1-fan
-shaped
bridge
deck
To illustrate the efficiency of the spline finite strip method presented in this paper, a 60” fan-shaped thin bridge deck (Fig. 3) is first analyzed. The arbitrary thin fan-shaped deck has internal and external radii of 177.8 mm (7 in.) and 330.2 mm (13 in.) respectively, the Young’s modulus and Poisson’s ratio are 6239.4 MPa (460000 lb/in.*) and 0.35 respectively; the original thickness of the plate is 4.267 mm (0.168 in.). Two additional deck thicknesses of 6.807 mm (0.268 in.) and 22.047 mm (0.868 in.) are also investigated here, in order to demonstrate that the spline finite strip method presented here can accurately solve both thin and moderately thick arbitrary shaped plates or bridge decks. The bridge was simply supported along its two ends. Concentrated loads were applied at three different locations (Points A,
E = 0.44 x lo6 ib/in2 h = 0.75 in
k. 12 E = 0.46 x lo6 Ib/in’
Y
Y
x
Fig. 3. Fan-shaped
bridge
deck model
(example
1).
Fig
5. Fan-shaped
continuous
bridge
model (example
2).
Analysis of Mindlin plates
117 M, along the symmetric axis AB 24 lb point load at A (mesh size 16 x8)
24 lb point load at B 20 18 e $
l6 14
4 12 ’ - 10 8
in
42
43
44
45
46
A
B
47
48
49
A
B Fig. 6. Transverse
+ Present method
[ 141
Reference deflection
across symmetrical
-o-
line AB (I).
B and C) and the mid-span deflections resulting from these loadings are tabulated in Tables 5 and 6 and compared with Cheung [8], Coull[13] and Sawko [ 141. Good agreements are found throughout. The distributions of the tangential moment across the mid-span (thickness = 0.168 in.) were also plotted in Fig. 4.
axis
M, along the symmetric axis AB 24 lb point load at B (mesh size 16 x8)
20 18 16 .9
14
4
12
”
10
z
8 6 4
42
43
44
45
46 in
47
48
49
B -
5C
A + Present method
Reference [ 141
Fig. 9. Longitudinal
moment (M,,) along the symmetric AB (2).
axis
M, along the symmetric axis AB 24 lb point load at A (mesh size 16 x 8) 6, 5
24 lb point load at A
4
5.0, 4.5
.9
4.0
$2
.E 3.5 :e
+ Present method
moment (kf,,) along the symmetric AB (1).
evident from these figures that the results of the spline finite strip method presented here compare extremely well with the results obtained by Sawko and Merriman [14].
continuous isotropic bridge
One of the most important advantages of the spline finite strip method, in comparison with the conventional finite strip method, is that the spline finite strip method can treat bridge decks with arbitrary intermediate supports and arbitrary point loads. A fanshaped continuous isotropic deck, originally designed and tested by Sawko and Merriman in 1971, is used here to demonstrate this particular versatility of the spline finite strip method. Details of this model are shown in Fig. 5. The 60” fan-shaped model was point-supported at eight nodes by columns which offered only vertical restraint along four radial lines. A point load was applied at two different locations (Points A and B) and the mid-span deflections and moments resulting from these loadings were compared with those from Sawko [14] in Figs 6-10. It is
[ 141
Reference
Fig. 8. Longitudinal
Example 2-Fan-shaped deck
50
in
3.0 1 2.5
,+‘+ ,+‘+
2.0
I+
1.5
F’+
1.01
’
1
42
43
I
44
I
45
I
I
I
I
46
47
48
49
I I
/“\
3
=
1
z-
0 -1
1:: 42
43
44
45
46
41
48
49
5’
in
50
in
q
,__&&+J
A
B A
B
-zFig. 7. Transverse
Reference
[ 141
deflection
+ Present method
across symmetrical
line AB (2).
Reference
Fig. 10. Transverse
[ 141
moment
+ Present method
(M,) along the symmetric AB.
axis
S. F. Ng and X Chen
118 CONCLUSION
A spline finite strip method to analyze moderately thick plates and bridge decks of arbitrary shapes, according to the Mindiin plate theory, is developed in this thesis. The reduced integration technique is used to eliminate ‘shear locking’ and the penalty function method is utilized to impose boundary condition at the end of strips and intermediate supports. Numerical examples have demonstrated that:
1. The spline finite method presented here can be applied successfully to analyze moderately plates of thickness-to-length ratios from 0.01 to 0.30. 2. The present method yields very accurate results for both thin and moderately thick rectangular and arbitrarily shaped bridge decks, subjected to both uniform and concentrated loads, and is much more efficient than the conventionally used finite element method, both in terms of data input and computation time required. 3. The method successfully overcomes the deficiencies of the conventional finite strip method in the treatment of patch loadings or intermediate supports. Further, the input data required is only a small fraction of that generally required by the conventional finite element method.
3. M. J. Chen, L. G. Tham, Y. K. Cheung, Spline finite strip for parallelogram plate. Con/: on Accuracy Estimutes and Adaotive Refinements in Finite Element Com-
purarions, Lisbon, Portugal, 1984, Vol. I, pp. 19-22. 955104. of irregular 4. J. L. Chen and K. P. Chong, Vibration
5
6
7
8
9
IO
11
REFERENCES
12
1. Y. K. Cheung, S.D Fan and C. Q. Wu, Spline finite strip in structure analysis. Proc. Inr. Conf. on Finite Element Method, Shanghai, 1982, 7044709.
13
2. Y. K. Cheung and S. C. Fan, Static analysis of right box girder bridges by spline finite strip method. Proc. Instn Cir. Engineers, Part 2, 1983, 75, June, pp. 311-323
14
plates by finite strip method with splined functions. Engineering Mechanics in CitGl Engineering, Proc. 5th Engineering Mechanics Div., AXE (Edited by A. P. Boresi and. K. P. Chong), Vol. 1. pp. 2566260 (1984). Y. K. Cheuna, L. G. Tham and W. Y. Li, Application of spline-finit&trip method in the analysis of curved slab bridge. Proc. instn Civ. Engineers, Part 2, 1986, 8 I, pp. 11 I-124. W. Y. Li, Y. K. Cheung and L. G. Tham, Spline finite strip analysis of general plates. J. Engng Mech. 112, 43354 (1986). Y. K. Cheung, L. G. Tham and W. Y. Li, Free vibration analysis of arbitrarily doubly curved shell by spline finite strip method. Research Report, University of Hong Kong (1989). Y. K. Cheung, L. G. Tham and W. Y. Li, Free vibration analysis of arbitrarily doubly curved shell by spline finite strip method. Research Report, University of Hong Kong (1989). L. G. Tham and H. Y. Szeto, Buckling analysis of arbitrarily shaped plates by sphne finite strip method. Comput. Struct. 36, 729-735 (1990). R. D. Mindlin, Influence of rotatory inertia and shear on flexurdl motions of isotropic elastic plates. J. appl. Mech. 37, 1031-1036. S. Timoshenko and S. Woinowsky-Krieger, Theory of Plate und Shells. 2nd Edn. McGraw-Hill, New York (I 959). S. Timoshenko, Strength of Materids. Van Nostrand, New York, 1955. A. Coull and P. C. Das, Analysis of curved bridge decks. Proc. Insm. Ck. engineers. 1967, 37. May, pp. 75-85. F. Sawko and P. A. Merriman. Finite element analysis by bridge curved in plan. Deaelopments in Bridge Design and Construction. Crosby Lock, Wood, Oxford (1971).