The 12th International Conference of International Association for Computer Methods and Advances in Geomechanics (IACMAG) 1-6 October, 2008 Goa, India

A Parametric Study on Raft Foundation

G. S. Kame, S. K. Ukarande MGM College of Engineering & Technology, Navi Mumbai, India K. Borgaonkar, V. A. Sawant Dept. of Civil Engineering, Indian Institute of Technology Roorkee, Roorkee, India

Keywords: raft, Winkler foundation, thick plate

ABSTRACT: The Finite element analysis adopted for raft is based on the classical theory of thick plates resting on Winkler foundation that accounts for the transverse shear deformation of the plate. Four noded, isoparametric rectangular elements with three degree of freedom per node are considered in the development of finite element formulation. Independent bilinear shape functions are assumed for displacement and rotational degrees of freedom. Effect of raft thickness, soil modulus and load pattern on the response is considered.

1 Introduction A raft or mat foundation is a large concrete slab used to interface columns in several lines with the base soil. It may occupy the entire foundation area or only a part of it. A raft foundation may be used where the base soil has a low bearing capacity and/or the column loads are so large that more than 50 percent of the area is covered by conventional spread footings. The methods available for analysis of such rafts are, Rigid beam analysis (conventional method) and Non-rigid or Elastic method. Rigid beam analysis can be used when the settlements are small. This is the simplest approach. It assumes that mat is infinitely rigid with negligible flexural deflection and the soil is a linear elastic material. It also assumes the soil bearing pressure is uniform across the bottom of the footing if only concentric axial loads are present or it varies linearly across the footing if eccentric or moment loads are present. Although this type of analysis is appropriate for spread footing, it does not accurately model mat foundations. Mats are not truly rigid, so the settlement beneath the columns will be greater than that beneath unloaded areas. These differential settlements will cause variations in the soil bearing pressure and corresponding changes in the flexural stresses in the mat. Non-rigid or Elastic method involves plates or beams on elastic foundations, plates or beams on elastic half space (elastic continuum), Readymade closed form solutions by elastic theory and, Discrete element methods, where the mat is divided into elements by grids. The Discrete element method includes, Finite Difference Method, Finite Element Method (FEM), and Finite Grid Method (FGM). Finite element analysis is the most accurate way of analyzing the raft in which raft can be considered as plate resting on elastic foundation. The soil below the raft is treated as either Winkler foundation or elastic continuum. Noorzaei et al (1991) used beam element, plate element and brick element to model space frame, raft and soil in the analysis. A detailed parametric study of the effect of variation in raft thickness on space frame-raft soil system had presented. In the present paper Finite element analysis of raft is presented by modelling raft as thick plate resting on Winkler foundation wherein effect of raft thickness, soil modulus and load pattern on the response is considered.

2 Finite element formulation for raft The finite element method transforms the problem of plates on elastic foundation into a computer-oriented procedure of matrix structural analysis. The plate (raft) is idealized as a mesh of finite elements interconnected only at the nodes (corners), and the soil may be modeled as a set of isolated springs (Winkler foundation). The Finite element analysis adopted for raft is based on the classical theory of thick plates resting on Winkler foundation (Figure 1) that accounts for the transverse shear deformation of the plate. The formulation is based on the assumptions that deflections are small compared with the thickness of plate, and that a normal to the middle surface of the undeformed plate remains straight, but not necessarily normal to the middle surface of deformed

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plate. The stresses normal to the middle surface are considered negligible.

Figure 1 Structural idealisation of raft and supporting soil

Figure 2 Rectangular plate element

Four noded, isoparametric rectangular elements with three degree of freedom per node (the transverse displacement w, rotation about x-axis θx, and rotation about y-axis θy) are considered in the development of finite element formulation. Independent bilinear shape functions are assumed for displacement and rotational degrees of freedom. The strain energy of an isotropic, linear elastic plate including transverse shear deformation effects can be expressed as,

[ ][ ][ ] [ ][ ] [ ] ∫∫∫ −+=AA

sT

Ab

T dAyxqwdADdAHDHU ),(21

21 γγ …(1)

[ ] [ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−=

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

∂

∂−

∂∂

∂

∂−

∂∂

=

2100

0101

)1(12,

2

3

νν

ν

νθθ

θ

θ

EtD

xy

y

x

H b

yx

y

x

…(2)

[ ] [ ] ⎥⎦

⎤⎢⎣

⎡+

=

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

+∂∂

−∂∂

=1001

)1(2,

νκ

θ

θγ EtD

xwyw

s

x

y

…(3)

In the above equations, q is the intensity of load per unit area, E is the Young’s modulus, ν is the Poisson’s ratio, κ is the shear correction factor, t is the plate thickness and A is the element area. Using independent shape functions, the nodal variables w, θx, and θy can be written in matrix form as,

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{ } [ ]{ }

{ } [ ]444333222111

4321

4321

4321

000000000000000000000000

yxyxyxyxT

y

x

wwwwdwith

dNdNNNN

NNNNNNNNw

θθθθθθθθ

θθ

=

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

…(4)

[ ] [ ]{ } [ ] [ ]{ }

[ ] [ ] 4,10

0

0

00

00

,

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

∂∂

−∂∂

=

⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

∂∂

−∂∂

∂∂

−

∂∂

=

==

iN

xN

Ny

N

Band

xN

yN

yN

xN

B

whichindBanddBH

ii

ii

is

ii

i

i

ib

sb γ

…(5)

From which the equation of strain energy can be expressed in a simplified form as follows.

{ } [ ] [ ] [ ]{ }

{ } [ ] [ ][ ]{ } { } [ ]∫ ∫∫ ∫

∫ ∫

− −− −

− −

−+

=

1

1

1

1

1

1

1

1

1

1

1

1

21

21

ηξηξ

ηξ

ddJqNdddJdBDBd

ddJdBDBdU

Tw

Tss

Ts

T

bbT

bT

…(6)

[ ] [ ]00000000 4321 NNNNNwhere w = The above equation represents the strain energy due to bending and transverse shear deformation of plate and potential energy due to applied external load. The total strain energy of the raft-foundation system can be obtained by adding the strain energy Uf of the foundation. By assuming full contact between raft and soil-subgrade, the strain energy of foundation with modulus of subgrade reaction ks can be written as,

{ } [ ] [ ]{ }∫ ∫∫− −

==1

1

1

121

21 ηξddJdNkNddAwkwU ws

Tw

T

As

Tf

…(7) The total strain energy Ut of the plate foundation system is given by addition of two.

ft UUU += …(8)

By equating first variation of total strain energy to zero, the force-deflection equation for plate-foundation element can be expressed as,

[ ] [ ] [ ][ ]{ } { }Qdkkk fsb =++ …(9)

where

[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]

[ ] [ ] [ ] { } [ ]∫ ∫∫ ∫

∫ ∫∫ ∫

− −− −

− −− −

==

==

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

;

;

ηξηξ

ηξηξ

ddJqNQddJNkNk

ddJBDBkddJBDBk

Twws

Twf

ssT

ssbbT

bb

…(10)

3 Validation A computer program is developed in FORTRAN 90 based on above formulation. For validation of the program, simply supported square plate with dimension a × a subjected to uniformly distributed load q is considered. Central deflections obtained from the finite element analysis are compared with the analytical solutions available for thin plates. Central deflection wmax of the plate is converted in non-dimensional form wn, as follows

42

3max

)1(12 qaEtw

wn μ−=

Non- dimensional Deflection for square plates with dimensions 7.62m ×7.62m and thickness ranging from

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0.1524m to 0.533m are reported in Table 1.

Table 1 Non-Dimensional Deflection

Plate thickness t (m) 0.1524 0.2286 0.3048 0.3810 0.4572 0.5334 Non-Dimensional Deflection wn 0.00445 0.00862 0.01294 0.01694 0.02055 0.02379

Analytical value of Non-Dimensional Deflection for thin square plate is 0.00406, and computed value of Non-Dimensional Deflection for 0.1524m thick plate is found to be 0.00445, which is in close agreement with standard solution. This provides necessary validation check for the developed computer program.

4 Parametric Study In the present investigations it is aimed to study the effect of following parameters on the rectangular raft foundation using finite element analysis. 1. Thickness of the raft (0.45m, 0.9m, 1.5m) 2. Modulus of subgrade reaction (40000, 100000, 200000, 400000)kN/m3, 3. Column loads (three load patterns LP-I, LP-II, LP-III as indicated in Figure 3) For specified three load patterns, raft thickness and soil modulus are varied to study their effect on the response. For load pattern I and II, raft with dimension 10m×10m are considered, where as 26m ×14m is employed in load pattern III. Deflection and bending moments in the raft are devised to compare the response.

Figure 3 Load Patterns considered in the analysis

4.1 Load pattern I A typical deflection pattern at different positions in the raft is presented in Figure 4a, and the deflection along central line for different thickness are compared in Figure 4b (Soil Modulus 40000). As expected, central line is deflected more as compare with second, but response to raft thickness is complex. With increase in the thickness deflection curves are flatter and central deflections are increased at higher thickness for Soil Modulus 40000. Maximum deflections for different thickness and soil modulus are compared in Table 2. Maximum deflections are found to be decreasing with increase in raft thickness and soil modulus. Figure 5 presents typical moment diagram at different position for 0.45m thickness and comparison of moment along centre line for all thickness. Bending moment under the load is hogging moment. Effect of thickness on bending moment is negligible with marginal increase positive moment at smaller thickness.

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Table 2 Comparison of Maximum Deflection

Maximum Deflection(mm) for modulus Raft

Thickness(m) 40000 100000 200000 400000

0.45 3.72 1.78 1.120 0.765

0.9 3.64 1.56 0.867 0.517

1.5 3.88 1.60 0.842 0.460

Figure 4 Deflected Shape (Es 40000, LP-I)

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Figure 5 Moment diagram (Es40000, LP-I)

4.2 Load pattern II Deflected shape and bending moment diagram along central line of raft for all thicknesses are compared in Figure 6 and 7 for soil modulus 100000 kN/m3. Effect of Soil Modulus and thickness on maximum deflection is compared in Table 3. Maximum deflections are observed to be decreasing with increase in soil modulus. Effect of raft thickness is quite complex. At lower soil modulus, deflections are increasing with raft thickness and at higher modulus reverse trend is observed. Similar comparisons for bending moment are reported in Table 4 and 5. Positive bending moments are increasing with raft thickness and negative bending moments are reduced with raft thickness. When effect of soil modulus is considered, it is found that positive bending moments are decreasing at higher modulus, and negative bending moments are increasing with soil modulus.

Table 3 Effect of Soil Modulus and thickness on Deflection

Maximum Deflection in mm

Soil Modulus Thickness

0.45m Thickness

0.9m Thickness

1.5m 40000 2.75 2.87 3.15

100000 1.15 1.19 1.28 200000 0.62 0.612 0.65 400000 0.383 0.318 0.333

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Table 4 Effect of Thickness (LP-II)

Max. positive BM Max. negative BM

Thickness Second Quarter Centre Second Quarter Centre 0.45 75.9555 41.6505 63.269 -72.041 0 -71.439 0.9 92.0945 68.9105 85.623 -67.5455 0 -64.405 1.5 98.9725 81.8925 97.377 -65.9105 0 -62.008

Table 5 Effect of Soil Modulus (LP-II)

Max. negative BM Max. positive BM

Es Edge Second Quarter Centre Edge Second Quarter Centre 100000 -40.023 -74.932 0 -74.286 68.976 60.7645 27.7125 49.439 200000 -49.754 -84.134 0 -81.643 58.971 51.363 19.895 41.698 400000 -53.469 -87.599 -0.321 -83.846 50.428 43.531 14.2355 35.831

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4.3 Load pattern III Similar variations in deflected shape and bending moment diagram for three thicknesses are compared in Figure 8 and 9 for soil modulus 100000 kN/m3. Effect of thickness on maximum deflection and bending moments are compared in Table 7. Maximum deflections are observed to be decreasing with increase in raft thickness. Maximum positive bending moments are increasing with raft thickness, whereas negative bending moments are more or less constant for given loading combination.

Table 7 Comparison of maximum values for all thickness LP-III

Maximum Thickness

(m) Deflection

(mm) Positive BM

(kNm) Negative BM

(kNm)

0.45 13.00 179.675 -298.153

0.9 11.14 197.511 -294.255

1.5 10.11 219.049 -296.448

5 Conclusions A parametric study on raft foundations is presented wherein effect of raft thickness, soil modulus and load pattern

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on the response is considered. At lower soil modulus, deflections are increasing with raft thickness and at higher modulus trend is reverse. Positive bending moments are increasing with raft thickness and negative bending moments are reduced with raft thickness. When effect of soil modulus is considered, it is found that positive bending moments are decreasing at higher modulus, and negative bending moments are increasing with soil modulus.

6 References Noorzaei J., Viladkar M. N., Godbole p. N. 1991. Soil-structure interaction of space frame-raft-soil syatem: A parametric study.

Computers & Structures, 40(5), 1235-1247.

Zienkiewicz O. C. 1968. The Finite Element Method in Structural & Continuum Mechanics, McGraw Hill, London.

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