Variable node plate bending element for mat foundation analysis

ca45-7949/93 56.00 + 0.00 Pergamon Press Ltd

Computers & Strucrures Vol. 47. No. 3, pp. 371-381, 1993 Printed in Great Britain.

VARIABLE

NODE PLATE BENDING ELEMENT MAT FOUNDATION ANALYSIS

FOR

C. K. CHOI and H. S. KIM Department of Civil Engineering, Korea Advanced Institute of Science and Technology, Taejon 305-701, Korea (Received 5 May 1992)

Abstract-The variable node plate bending element, i.e. an element with one or two additional mid-side nodes is used effectively in the analysis of a mat foundation. The variable node elements can generate a nearly idea1 grid for mat foundation analysis in which more nodes are defined near the column location where the steep stress gradient is expected. The plate bending element used in this study is based on Mindlin/Reissner plate theory with substitute shear strain field. The nodal stresses of that element are obtained by local smoothing. The interaction between the soil and the mat foundation is modelled with

Winkler springs connected to nodal points in the mat model. The vertical stiffness of the soil material is represented in terms of a modulus of subgrade reaction and is computed in a similar way to the computation of consistent nodal force of uniform surface loading. Several mesh schemes were proposed and tested in order to find the most suitable scheme for mat foundation analysis.

INTRODUmION

VARIABLENODE PLATE BENDING FINITE ELEMENT

The mat on an elastic foundation is one of the foundation types commonly used where conventional footings or piles occupy most of the site and the design of these individual footings is no longer possible. Due to its continuous nature and integrity with the super-structure, a mat foundation prevents the independent differential column movement and thus improves overall performance of the structure. Among the methods used to analyse a mat foundation, discrete element methods such as the finite difference method (FDM), the finite grid method (FGM) and finite element method (FEM), where the mat is divided into elements by gridding, are common in practice [2]. A finite element analysis of mat foundation is based on the theory of flat plate bending with the mat being supported by the soil which is modelled in many cases using Winkler springs. The mat is modelled as a mesh of discrete elements interconnected at nodal points, and the Winkler springs are used as the soil-response model at each node. An ideal mat model would have a finer mesh near the column load location where the stress concentration may exist and a coarser mesh at some distant area from the concentrated loads where the stress gradient is reasonably easy. In this paper, the variable node plate bending element, i.e. an element with one or two additional mid-side nodes as shown in Fig. 1 is used to generate the nearly ideal mat analysis grid in which more nodes are defined near the column location. Some test results are presented to verify the effectiveness of using variable node elements for mat foundation analysis.

In many engineering circumstances, stress concentration phenomena occur at areas where abrupt changes of geometry exist, or at points under concentrated loading. To analyse such problems efficiently, it is usual to use a finer grid in the areas of high stress gradient and a coarser grid in other areas. These two heterogeneous mesh areas can be connected by using a variable node element side by side. The variable node plate bending element, which is very efficient for the local mesh refinement, is formulated with following shape functions [9] N,=N;-f(N,+N,),

N,=f(l-lcl)(l-tt)

N2=N;-f(N5+&),

N6=f(l+t)(l-Iql)

Nj=N;--f(Ne+N7),

N7=f(l-lQ)(l+~)

N.+=N;-f(N,+Ns),

N,=+(l

-t)(l

-I~I),

(1)

where N~=~(l+&~)(l+~iq),

fori=1,2,3,4.

(2)

In eqn (l), the shape functions for mid-side modes, N,, Ns, N7 and N, have nonzero values only when corresponding mid-side nodes exist. It will be acknowledged that these shape functions pertinent to mid-side nodes represent piecewise straight lines. A plate bending element based on isoparametric formulation tends to overestimate its shear stiffness for a thick plate. Such a defect increases as the order of shape function used in element stiffness formulation becomes lower. In order to use this type of

C.

312

Side

K. CHOI and H. S. KIM Table 1. Modified Gaussian quadrature of variable node element

3

7 3

i

Point

e,, rl,

WC)? Wrt,)

1 2 3 4

-0.788675 -0.211325 +0.211325 +0.788675

0.5 0.5 0.5 0.5

8 * Side

4

i

\

1

5 Side

1

2 1

l

:

Corner

X

:

Mid-Side

functions in a domain. These functionals are generally required to have a certain degree of inter-element continuity depending on terms in the functional. In many finite element fo~ulations, the quantities of primary engineering interest include derivatives of the function and in many instances, especially with lower order elements, these derivatives do not possess inter-element continuity. Therefore, the stresses are discontinuous between elements because of the nature of the assumed displacement variation. For the stress computation in a numerically integrated element such as an isoparametric element, experience has shown that while the integration points are the best stress sampling points, the nodes, which are the most useful output locations for stresses, appear to be the worst sampling points [14]. This phenomenon may indicate that interpolation functions tend to behave badly near the extremities of the interpolation region. It is therefore reasonable to expect that derivatives of the shape function (and hence stresses) sampled in the interior of elements would be more accurate than those sampled on the element peripheries. To solve such problems, the local stress smoothing procedure by the least square fit is used to obtain the smoothed stress field in this study [131. If the unknowns in the least-square problem are taken as the smoothed nodal stresses, let the smoothed function g(4, Q) be given at any point within an element by the expression piecewise

Node Node

Fig. 1. Variable node element.

more effectively in practice, the element should be modified to reduce its shear stiffness. For the proper element stiffness evaluation, several suggestions have been presented, namely (1) the addition of nonconforming displacement modes, (2) the (selectively) reduced integration, and (3) the construction of substitute shear strain fields. The plate bending element used in this study is based on Mindlin/Reissner plate theory with substitute shear strain fields [12]. The substitute shear strain polynomials for five-node elements, for example, are given in the Appendix [lo]. In addition to the modification of shear strains, another modification for evaluation of the stiffness of the variable node element is related to the numerical integration. It is common to evaluate the stiffness matrix of isoparametric elements by Gaussian quadrature. Such a numerical integration, however, cannot be applied directly over the entire domain of the variable node element due to the slope discontinuities in the displacement field. Thus, Gaussian quadrature is carried out over separate domains as shown in Fig. 2. Dashed lines represent boundaries of su~omains and 2 x 2 Gaussian quadrature is carried out over each subdomain [I 11. The coordinates and corresponding weight coefficients for the modified quadrature are listed in Table 1. element

NODAL STRESSES

Finite element anslysis is generally involved with the minimization of the functionaf defined in terms of

g(g,tl)=

C Njci,f i= In

(3)

where Ni is the smoothing shape function in eqn (1) at node i, cFiis the smoothed nodal stress at node i, and II is the number of nodes per element. The error in stresses at any point within the element is defined as e(T, rl) = ~$5 ~1 -g(L

SX

(4)

where the unsmoothed stresses ~(5, q) at any point within the element may be obtained by the usual stress-displacement relationship a([, tl) = PWl{G)‘, 5--node

6-node

7-node

Fig. 2. Gaussian quadrature of variable node element.

(9

where [D] is the elasticity matrix and [B] is the strain~spla~ent matrix and {Sje is the vector of the nodal displacements for the element.

373

Variable node plate bending element

The problem now becomes to find the smoothed nodal stresses 6,, (j2, . . . , cFpwhich minimizes the functional (6) where ne is the total number of elements. For II to be a minimum an a-.=0, 0,

fori=l,2

,...,

p.

(7)

Therefore, the smoothing matrix for each element, the form of which is similar to mass matrix, is given as ‘SfN,N,detJdcdq

* CORNER NODES + 2X.2 GAUSS POINTS Fig. 3. Local stress smoothing of four-node element.

..* jjN,N,,detJdeda’ where ui , a,, , uIII and eIy are the stresses at the Gauss points as defined in Fig. 3.

[Sl’ = $jN,N,detJdSdq

‘IV jjN,N,,detJd1;dq,

(8) and the associated right-hand vector’ is given as

side ‘smoothed force

where det J is the dete~inant of the Jacobian matrix. If the procedure outlined above is applied to each element separately, the following expression may be obtained [S]c((s) = (F)'.

fl0)

The average values of the smoothed comer stress of all the elements connected to the common mode are taken as the final nodal stresses. In the case of a four-node element it can be shown that stresses at corner nodes (7,) u2, u, and a, may be obtained from the expression

-- 1

1+Js2

-z 1

l-2 >

_;

1+yJ5

l_Js 2

-5

NUMERICAL TEST FOR VARIABLE NODE PLATE BENDING ELEMENTS

A moderately thin clamped square plate under a concentrated load at the centre is analysed in order to assess the performance of the variable node elements used in this paper. The plate to be tested and the mesh refined locally at the loading point are shown in Fig. 4 for one quarter of the plate. The distributions of vertical displacement (IV,) and bending moment (M,) along the horizontal X-axis are compared with the results obtained from a uniform 16 x 16 mesh composed with four-node elements, and are plotted in Fig. 5. The distribution of the vertical displacement obtained by the mesh with variable node elements is nearly identical to that of the 16 x 16 mesh, except that a small difference is identified at the loading point. Considering the large difference in the total degrees of freedom (DOF) used in the mesh with variable node elements (144 DOF) and that used in the 16 x 16 mesh (867DOF), the improvement achieved by the former is very significant. The maximum values of the analysis by the model with variable node elements and the uniform 16 x 16 model are shown in Table 2. From the analysis results obtained it is expected that the variable node plate bending elements can be

l-2 J5

cc.

=

-- 1 2

1

-; l+Y!Z 2 . i- 3 1

X

li 91

~111

=I"

G

TV

L/2

lc” p--

L/2

(11) E=3.0xl

06,

I/=0.3,

L=200.0,

t=2.5.

P=1000.0

Fig. 4. One quadrant of a clamp~~Isquare plate.

C. K. CHOIand H. S. KIM

374

200

*-Node --e--

16X16

Transition

Mesh

(867

Mesh

(144

dof)

60

70

80

doff

-300

-

4-Nods

--e--

Transition

16X16 Mesh

Mesh

(867

(144

dof)

dof)

-400 o

10 20

30

40

50

90 100

0

1020304050607080901 (b1

(a)

Fig. 5. Analysis results of square clamped plate along the X-axis. (a) Vertical displacement ( WZ) along the horizontal centre line. (b) Bending moment (A4,) along the horizontal centre line.

efficiently used in similar problems, foundation

such as the mat

analysis. MAT FOUNDATION MODELING

In order to obtain reliable information for practical problems of soil-foundation interaction, it becomes necessary to idealize the behaviour of the soil. The simplest type of idealized soil response assumes the linear elastic behaviour of the supporting soil medium. The linear elastic idealization of the supporting soil medium is usually represented by a mechanical or mathematical mode1 which exhibits the particular characteristics of soil behaviour. Several such idealizations have been developed [4]. The simplest mode1 of linear elastic behaviour of the supporting soil medium is generally attributed to Winkler [4]. He assumed that the surface displacement of the soil medium at every point is directly proportional to the stress applied to it at that point and completely independent of stresses or displacements at other points of the soil-foundation interface, even at the immediately neighbouring points. Winkler’s idealization of the soil medium can be physically represented as a system of closely spaced spring elements, each of which will be deformed by the stress applied directly to it while the neighbouring elements remain unaffected. The characteristic feature of this representation of the soil medium is its di~ntinuous behaviour of the surface displacement. Table 2. Maximum values of square clamped plate analysis

Elements Nodes Max. W’, Pos. M, Neg. &f, Pos. Q,

Variable node element

16 x 16 four-node

31 48 - 0.053295 92.96 -317.00 56.52

256 289 -0.052501 114.50 -311.70 56.54

The interaction of the compressible soil material with the mat foundation in this study is modelled with finite elastic springs connected to the nodal points in the model. The vertical stiffness of the soil material are represented in terms of a modulus of subgrade reaction. The spring stiffness at each node is computed in the similar way to the consistent nodal point forces ~r~s~nding to the surface loading and shown in Fig. 6 for four-, five-, six- and seven-node square elements. To simplify the placement of reinforcement bars, it is common practice to provide a uniform mesh of bars at the top and bottom of the mat in both directions. Thus, it is important in the mat foundation design to find maximum positive and negative bending moments and shear forces which generally occur at the column face and along the column line. For the finite element mesh for mat foundation analysis, it is desirable that the element should have an aspect ratio, length/width, of near unity (1.0) and sharp interior angles should be avoided. At locations in the mat where high rates of change of curvature are expected, a finer local mesh should be provided. Reasonably accurate results may be obtained whenever the element size does not exceed 150% of the mat thickness in the areas of angle curvature and low moment [S]. Keeping the above recommendations and the fact that the very fine mesh refinement is not possible in the mat fo~~tion design practice in mind, several mesh schemes of the basic mat foundation cell were proposed as shown in Fig. 7 and tested to find the most suitable scheme for the mat foundation analysis. PARAMETRIC STUDY OF THE MAT FOUNDATION MESH SCHEMES

The same mat example tested previously by Shukla [6] as shown in Fig. 8 is used to evaluate

Variable node plate bending element 0.25

0.25

0.25

I;_!

0.25

0.25

0.25

0.125

4-node

0.125

0.125

~~~0.25

1~~

0.25

375

0.25

0.125

0.125

0.0

0.25

0.0

~~10.25

0.25

0.0

0.125

-/-node

6-node

5-node

0.25

Fig. 6. Nodal spring constants of variable node elements.

Scheme-A (4EL. 17ND.)

Scheme-B (16EL. 33ND.)

Scheme-C (16EL. 41 ND.)

Fig. 7. Proposed mat foundation mesh schemes.

proposed mesh schemes. The mat is 51 x 51 ft (15.5 x 15.5 m) in dimension and symmetrically loaded so that only the lower left quarter needs to be tested. Other data given were E, = 3122 ksi (21.52 GPa), p = 0.15 (for concrete), k, = 28.4 k/ft3 (4467 kN/m3) and the mat thickness T = 2.Oft (0.61 m). In addition to the proposed mesh schemes, the meshes of 4 x 4 and 8 x 8 in a basic cell that consist of four-node elements only are analysed to compare the results. Since it is virtually impossible to get an exact solution of this mat example, the model shown in Fig. 9 that consists of Donea’s four-node elements, which are one-fourth of the column size, is taken as the reference model. In this reference model, column loads are not concentrated but ~st~buted over nine nodal points. The mesh layouts of

‘1

El

El

2OOkips

JOOkipt

\ 16”

1SOkips El

I

/ X 18” COLUMN

\

---@

-a

2OOkips Ei

--

X

Fig. 8. Dimensions and loading conditions of the example mat.

4 x 4, 8 x 8 and the proposed mesh schemes are shown in Fig. 9; the numbers of elements and nodes used in the tested meshes are given in Table 3. Results obtained by the proposed different mesh schemes along section A-A are plotted in Fig. 10. As observed in this figure, the performance of scheme A is not good enough to be used in mat analysis and results obtained by schemes B and C are nearly identical. Therefore, mid-side nodes along the column line in scheme C may be considered as surplus. Among the proposed mesh schemes, scheme B may be the most suitable for considering the results obtained and the number of DOF involved in the analysis. In addition to the numbers of elements and nodes in the tested models, the maximum vatues of tested models which will be used in the mat design are shown in Table 3. The performance of scheme B, which is selected as the optimal mesh scheme in this paper, is compared with that of other schemes as shown in Fig. 11. From this figure it is acknowledged that the stresses of scheme B at the loading point (X = 18.0 in Fig. Ila) are closer to those of reference model than 8 x 8 mesh model which uses more DOFs. The distributions along section R-B’, however, are rather closer to that of the 4 x 4 mesh model (Fig. 11~). The performance of scheme B along the column line, which is a more important area in mat foundation design, is highly improved by simply adding a few mid-side nodes to the 4 x 4 mesh model. Moments at the column faces are more meaningful than the values at the column centre and commonly used in the mat foundation design. The h4, values at the faces and the centre of the column loaded with

C. K.

CHOI

and H. S. KIM

4 X 4 MESH

SCHEME-A

SCHEME-H

8 X 8 MESH

REF. MESH

SCHEME-C

Fig. 9. Mesh models of the example mat.

300 kips are compared in Table 4 to show the significant difference of moments at different locations. From this table, it is observed that the column face

moment by scheme B is close enough to that of the reference model and scheme B can be effectively used in the mat foundation

analysis.

Table 3. The number of elements, nodes and maximum values of tested models

Elements Nodes Max. IV: Pos. lu, Neg. M, Pos. 0. Neg.

2:

Ref. mesh

4x4 mesh

1156 1225 - 0.05945 29.68 -42.02 34.75 - 35.85

49 64 -0.05797 27.41 - 14.58 13.64 - 14.65

8x8 mesh 196 225 -0.5902 28.74 -33.48 33.36 - 34.45

Scheme A

Scheme B

Scheme C

16 37 -0.05641 33.48 -25.25 13.62 - 14.01

49 80 -0.05819 25.84 -45.71 30.16 -31.09

49 92 -0.05837 28.62 -45.86 29.85 -30.85

Variable node plate bending element

377

-50b (4 40 30 20 10 0 -10 -20 -30

-50 I 0

I 5

I

10

I

15

I

20

I’

-40 0

25

4

8

12

@I

0

5

10

16

20

24

(b)

15

20

25

0

5

10

15

20

25

Cc)

w

Fig. 10. Analysis results of proposed mesh schemes. (a) Vertical displacement (W,) along section A-A’. (b) Bending moment (M,) along section A-A’. (c) Shear force (Q,) along _. ..,

Fig. 11. Performance comparisons of scheme B with 4 x 4 and 8 x 8 meshes. (a) Bending moment (&fv) along section A-A’. (b) Shear force ..~ (Q,) along section A-A’. (c) Bending moment (M,) along section B-B’.

SWIlOn A-A.

C. K. CHOI and

318

300 kips concentrated load Ref. (distributed load) Ref. (concentrated load) Scheme B

Column centre

Column face

- 42.02 -59.18 -45.71

-31.76 - 35.42 - 32.46t

t Extrapolated value. MAT FOUNDATION

FOR A 25STOREY

KIM

haunch girders are spaced at every 30 ft (9 m) on centres and run parallel to the narrow face of the building. Skip joists spaced at 6 ft (I .81 m) intervals span between the haunch girders. A 4-in (101.6-mm) thick concrete slab spanning between the skip joists completes the floor framing system. Lightweight concrete is used for floor framing members while normal weight concrete is employed for columns and shear walls. The mat layout and load condition are shown in Fig. 12(a) and a finite element model with scheme B which has additional side nodes at column and wall locations is shown in Fig. 12(b). The model with a 4 x 4 mesh shown in Fig. 12(c) is analysed to evaluate the performance of scheme B. The interior core walls could be modelled with beam elements of very large rigidities. The vertical deflection, bending moment

Table 4. M, at the column face and centre of under a Mesh model

H. S.

BUILDING

The analysis of the mat for a 25-storey reinforced concrete office building [15] is presented to show the efficiency of the variable node plate bending element for the mat foundation analysis. The floor framing for the building consists of a system of haunch girders running between the interior core walls and columns to the exterior. The

a-

9’S

-II (a)

Foundation

Mat

for

(b)

Mesh

by

(c)

Mesh

a

25-Story

Building

SCHEME-B

by

4X4

Fig. 12. Foundation mat for a 25-storey building and the finite element analysis model.

Variable node plate bending element

o+*

?

?

?

379

?

.

I

1

!

!

I i

I

-0.5-w -l.O-1.5-

! !

I

I

’

!

I

i

-2.o-2.5-

-3.0

i ,/** wf 1

I

! ! * _______ -*

i

1 -. ‘i_-___

i

I

I

1 k, =25pci Without Wall k, =I OOpci Without Wall

____._______ k, =25pci With Wall ___________ k, = 1 OOpci With Wall

(a)

300

200 Ii

k, -25pci Without Wall k, = 1 OOpci Without Wall

____________ k, =25pci With Wall __-k, = 1 OOpci With Wall

O-9

ih

k, =25pci Without Wall k, = 1 OOpci Without Wall

I

_“__________ k, =25pci With Wall .____-____ k, = 1OOpci With Wall

6)

Fig. 13. Analysis results of a 25-storey building mat by scheme B. (a) Vertical displacement (I+‘:) along section X-X‘. @) Bending moment (M,) along section X-X’. (c) Shear force (Q,) along section X-X’.

and shear force variation along X-X of the four analysis cases which are distinguished by two different values of subgrade reaction, namely 100 and 25 lb/id (743 and 185.75 kg/mm3), and by core wall modelling, i.e. with or without rigid beams, are compared in Fig. 13. These four analysis cases are modelled with scheme B. As expected, the mat experiences a larger deflection when supported on relatively more flexible springs. However, the general profiles of the variation of curvatures, which are the measures of bending moments in the mat, are very similar to each other case with different subgrade reaction. It is shown that the vertical deflection and bending moment of the plate under the core wall is considerably reduced.

The stress variation of scheme B along the D-D’ section is compared with that of the 4 x 4 model in Fig. 14. The value of subgrade reaction is 25 lb/in3 and the core wail is not considered. These two meshes have the same number of elements (360 elements) but have a different numbers of nodes, namely 531 nodes for scheme B and 403 nodes for the 4 x 4 mesh. From this figure, it can be seen that the performance of scheme B is superior to that of the 4 x 4 model, especially at loading points and mat edges. CONCLUSIONS

The use of the variable node plate bending element in mat foundation was investigated. From a clamped

380

C. K. CHOI and H. S.

KIM

SCHEME-B

_________

4x4

__----_--

4x4

(a) Bending moment (M,) along section D-D’

-400

SCHEME-B (b) Shear force (Q.)

along section D-D’

Fig. 14. Performance comparisons of scheme B with 4 x 4 in 25-storey building mat.

plate test it could be seen that this type of element could be efficiently used in mat foundation analysis and this was verified in the numerical example of the mat foundation for a 25storey reinforced concrete building. Local stress smoothing techniques were successfully applied to obtain the nodal stresses from the stresses at the Gauss points of the variable node elements. Among the proposed mesh schemes which include the variable node elements in modelling, scheme B was found to be optimum considering the analysis results obtained and the number of DOFs involved in the analysis. The performance of scheme B along the column lines, which is the most important area in mat foundation design, is significantly improved by simply adding a few mid-side nodes to the 4 x 4 uniform mesh model. The findings in this study have been implemented for practical applications in the mat foundation analysis routines of BUILDS-F [16], which is a foundation analysis and design computer program for building structures. REFERENCES 1.

R. B. Peck, W. E. Hanson and T. H. Thornburn, Foundation Engineering, 2nd Edn. John Wiley, New York (1974).

2. J. E. Bowles, Foundation Analysis and Design, 4th Edn. McGraw-Hill, New York (1988). 3. W. C. Teng, Foundation Design. Prentice-Hall, Englewood Cliffs, NJ (1962). 4. A. P. S. Selvadurai, Elastic Analysis of Soil-Foundation Interaction. Elsevier (1979). 5. J. E. Bowles, Mat design. AC1 Jnl 83, 1010-1017 (1986). 6. S. N. Shukla, A simplified method for design of mats on elastic foundations. ACI Jnl 81, 469475 (1984). 7. AC1 Committee 336, Suggested design procedures for combined footings and mats, (AC1 336.213-66) (Reaffirmed 1980). American Concrete Institute, Detroit (1966). ’ 8. S. C. Ball and J. S. Notch, Computer analysis/design of lame mat foundations. J. Struct. Ennna. __ ASCE 110, ll&-1191 (1984). 9. C. K. Choi and Y. M. Park, Compatible transition plate bending element for adaptive mesh refinement. In Proc. of the International Conference on Numerical Methods in Engineering: Theory and Applications (iWJMETA’90), Swansea, U.K. (1990). 10. Y. M. Park, Adaptive mesh h-refinement method using transition elements in plate bending problems. Ph.D. dissertation, KAIST, Seoul, Korea (1990). 11. A. K. Gupta, A finite element for transition from a fine to coarse grid. Int. J. Numer. Meth. Engng 12, 3545 (1978). 12. J. Donea and L. G. Lamain, A modified representation of transverse shear in Co quadrilateral plate elements. Comput. Meth. Appl. Mech. Engng 63, 183-207 (1987).

Variable node plate bending element 13. E. Hinton and J. S. Campbell, Local and global smoothing of discontinuous finite element functions using a least squares method. Int. J. Numer. Meth. Engng 8, 461480 (1974). 14. J. Barlow, Optima1 stress locations in finite element models. Int. J. Numer. Meth. Engng 10,243-251 (1976). 15. B. S. Taranath, Structural Analysis and Design ifTaU Buildinns. McGraw-Hill. New York (1988). 16. C. K. Choi and H. S. Kim, Integrated b&ding design system (part 4): foundation design. SEMR 90-03, Department of Civil Engineering, KAIST, Seoul, Korea (1990).

381

--au +rl)u -tu

+11;I,C)(Y*-Ys)f(82+85) (AlI

+v)(Y3-YJf(B3+8.4)

P,=fU

-5W4-w,)+$(1

+rw-

w*)

+ au - I5 IN% + w2- 2%) -fU

-tX.~-x,)f@,

- 81 + 0(x3 -4

+a,) fh + a,)

-:(l-lrI)(x,-x,)i(aI+a,) APPENDIX Pc=f(l

-q)(l

-l~I,c)(ws-w,)+(l+l5I,c)(%-%)

+t(l + rl)(w, - &) -f(l

--rl)(l -IrI,C)(x,-x,)f(a,+a,)

-31

+ v)(l + I5 I,c)(x2 - 4 % a2+a,)

-4

+tl)(x3-x4)5(a3+a4)

-tu

--1110 -15I,~)(Y,-Y,)f(B,+Bs)

-~1-IrI)(x2-xI)f(a2+a5) -fu

-5NY4-Y,)f(B,

+A)

-acl+t)(Y,-Y,)f(B1+B~) -$G -15l)(Y, -Ydf(B,

+Bs)

-au -15l)(Y*-Ys)f(B*+B~). where x,, yj are the Cartesian coordinates and w,, a, and /3, are the nodal DOF at node i.