Raft analysis and design-some practical

1

1. A. Hooper, MEng, PhD, DIC, CEng, MIStructE, MICE Ove Arup Partnership

Synopsis This paper describes the way in which the principles of soil-structure interaction have been applied in the analysis and design of rafts and other surface foundations. Various techniques of modelling the soil and structure are outlined, as is the method of coupling the foundation structure to a layered soil continuum. The general approach to design is also discussed, and summaries are given qf eight structural design projects ranging from simple strip footings to complex raft foundations.

Introduction Experience has shown that raft analysis is a subject-area where structural designers tend to flounder. In particular, it is not uncommon for project engineers to spend a great deal of time on a raft design and still not produce an efficient or economic solution. This, in turn, has often discouraged the use of rafts as a means of foundation support in favour of other systems which, although frequently less elegant and more expensive, are much easier to design.

Although general design is covered by the provisions of CP20041, for example, no detailed guidance is given for raft foundations. The main problem is, of course, how to represent the soil strata and how to couple the soil and structural models. It is one of a class of problems in what is usually referred to as soil-structure interaction, and a considerable body of work has been carried out on this important and fascinating subject2J. Most of the work has centered on developing analytical and numerical methods for the structural analysis of raft foundations, in some cases augmented by field observations; relatively little has been said about applications to structural design.

The objective of this paper is to outline the basis of foundation interaction analysis, and to illustrate its application to design practice by summarising the pertinent features associated with several foundation design projects.

Basis of interactive approach As structural engineers have progressively moved away from hand calculations to computer-based methods of structural analysis, it frequently occurs that raft foundations are modelled as a plate or grillage on linear elastic springs. This method can give good answers in certain cases, but its performance is problem-dependent. The method is not sufficiently general for practical purposes and can lead to grossly inaccurate results on the unsafe side. The main reason for this is that there is no interaction or cross-coupling between the springs. There are also significant problems associated with relating the spring stiffness to soil properties. Many other spring models of varying complexity have been developed over the past decades, but the presence of additional elements used to achieve cross-coupling makes it even more difficult to relate model parameters to soil behaviour.

A far more rational approach is to represent the soil by a layered elastic continuum, in which full account is taken of the interactive response through the soil. This approach has been found to give consistent and reliable results for a wide range of foundation interaction problems, and was used in the project work summarised herein4. The soil layers are assumed to be horizontal, and the calculation of vertical strain at any given point is based on a Boussinesq stress distribution together with the soil stiffness parameters at that point. The vertical layer displacements corresponding to these strains are then summed to give the surface displacements. This simplified method of dealing with layered strata gives satisfactory answers in most practical cases, and is closely analogous to the well-known Steinbrenner method of settlement calculation used for many years.

The Structural EngineerlVolume 62A/No. 8lAugust 1984

The determination of raft thickness is normally governed by the need to restrict the maximum shear stress in the concrete to an acceptable level and is generally undertaken as a hand calculation. The form of construction and geographical location of the structure may also influence the choice of raft thickness. Where material costs are high, for example, a cellular raft may be more economical than a monolithic raft. For basement foundations subject to an appreciable net reduction in vertical loading and to the hydrostatic pressures resulting from a high water table, some general thickening of the raft over the minimum structural dimension may be appropriate to give an additional margin of safety against uplift or flotation. For thinner rafts, it may be better to thicken the raft locally-e.g. beneath widely spaced columns so as to accommodate the localised shear forces while retaining a desirable degree of raft flexibility between columns. In all cases, of course, there should always be adequate space for fixing the steel reinforcement.

Having established the initial raft thickness, then the main object of the subsequent interaction analysis is to obtain realistic estimates of the soil and raft displacements, together with the raft bending moment field. Other useful results, such as raft contact pressures and shear forces, as well as the forces in any superstructure included in the foundation model, are also computed. The analysis may be repeated for rafts of other thicknesses, but this is not normally necessary.

It is usually the long-term structural response that is of principal concern, and the corresponding analysis relates to drained soil conditions. Where structures are founded on thick deposits of clay, the short-term response may also be of interest and the analysis is based on undrained soil properties. In either case, the computed flexural mode of the raft is governed by the classical theory of thin elastic plates, although a reduced Young’s modulus is used to take account of creep deformation of the concrete. Uncracked section properties are assumed but, if necessary, the effect of flexural cracking in the reinforced concrete can be included by means of a simple iterative procedures.6. In all cases, horizontal slip is allowed to occur between the soil and the base of the raft. It has been shown theoretically that the interfacial shear stresses required to prevent slip can have a profound effect in reducing computed raft bending moments7, but it is doubtful whether such stresses can be sustained in practice. It is therefore safer to neglect these shear stresses in design.

By including the stiffness of both the raft and any significant superstructure in the analysis, the computed differential settlements can be used directly with some confidence to assess the effect of raft distortion on architectural finishes or the redistribution of superstructure forces. Even with a fairly coarse finite element mesh, these results are likely to be sufficiently accurate for the necessary serviceability checks to be made. In practice, of course, some of the distortion and possible tilt will be built out during construction.

The method of dealing with computed raft bending moments is not always so straightforward, although no insurmountable problems have been encountered in practice. This is largely because the approach to design is made as simple as possible, i.e. the analysis is based on working loads and the required quantities of flexural reinforcement are based on the computed moments used in conjunction with CPl148. Alternatively, CP1 lo9 may be used, on the basis of appropriate values for the various partial safety factors. However, there is unlikely to be compatibility between the two methods in cases where the analysis is non-linear, and further work is required to resolve the matter. With either method, considerable care and experience is needed in the interpretation of computed moments, especially with regard to the density of the finite element mesh and the nature of the applied loading.

233

Reinforcement design is usually based on either the principal bending moments, or the orthogonal moments augmented by the twisting momentslo, depending on the structural layout and the pattern of computed results. For the particular plate bending finite elements used in the present work, these moments have been either the element centroid values or the average nodal values, noting that the latter will normally pick up higher peak values in areas of concentrated applied loading. In such cases, however, the actual raft bending moments will be lower than the theoretical values obtained from a classical elastic analysis because of the finite thickness of the structural members and because of local yield of the reinforced concrete in the vicinity of the re-entrant corners.

Whichever values of computed moment are chosen for design, their accuracy and reliability will greatly depend on the mesh density, as is the case for any stress analysis based on the finite element method. For example, where the raft is supporting a grid of columns, there is no possibility of computing a reasonably detailed bending moment field unless there are a sufficient number of raft finite elements between the columns. Where it is not possible to have such a fine mesh, then the results of the coarse mesh analysis must be followed by supplementary calculations to determine the local bending effects in these areas. This procedure is usually necessary for asymmetric foundations, where there are generally too few nodes available to model the structure in detail because of computer storage restrictions. Work is currently in progress to develop and test solution methods that enable larger problems to be solved at reasonable cost, thereby allowing the use of much finer meshes.

To some extent, the actual stresses in any raft are bound to depend on the method and sequence of construction; these effects are difficult to assess but probably modify only the detailed pattern of stresses rather than the broad structural response of the raft. In some cases, other factors affect the reinforcement detailing. For example, the requirements for watertight concrete in a basement or fluid containment structure may influence the size and spacing of steel reinforcementl1.12; similarly, for large concrete pours, where special provisions may be required to contain the shrinkage and temperature f0rces13~14. Where the structural geometry is particularly complex, as with multilevel rafts interconnected by vertical walls or inclined members of appreciable span, it may be advisable to specify equal top and bottom reinforcement based on the maximum computed bending moment, irrespective of whether this is shown by analysis to be sagging or hogging.

The development of more sophisticated design procedures is clearly possible, but whether these can be entirely justified in the light of the many approximations inherent in the analysis is open to question. It is essential to recognise the particular difficulties and uncertainties associated with foundation interaction analysis, and these should be reflected in a reasonably simple and conservative approach to design.

Outline of computational method The basic method of interactive raft analysis is simple, and the principal steps are:

-for any given raft plan layout, compute flexibility matrix of soil

-invert to give soil stiffness matrix -compute stiffness matrix of structure -add soil and structure stiffness matrices and solve for displacements

under given applied loading -from the displacements, compute moments, forces and rotations in

structure, together with interfacial contact pressures -where necessary, continue analysis into non-linear range by means

of iterative procedures, and compute corresponding forces and displacements

Thus by means of the hybrid technique of representing the structure by finite elements and the soil by surface or boundary elements, the 3-dimensional problem is essentially reduced to a 2-dimensional one to enable practical problems to be solved. Currently, a mainframe computer is required for the analysis, although the necessary space and storage requirements are likely to be available on the largest desktop computers in the near future.

Input data for the elastic soil model consist of little more than specifying the variation with depth of Young’s modulus and Poisson’s ratio. It is assumed that the soil is horizontally layered and is either isotropic, or transversely isotropic with a vertical axis of symmetry, the latter being particularly useful in modelling heavily overconsolidated clays where the horizontal stiffness typically is about double the vertical stiffness. It is further assumed that the soil layers are of uniform thickness

continuum

234

across the site, although numerical methods have been devised for dealing with inclined layers and other forms of horizontal heterogeneityls.

Suitable values of Poisson’s ratio may be assigned to various types of soil, and the elastic moduli may be estimated from the results of shear strength or penetration tests carried out during the site investigation. I t may also be possible to augment these results by means of more elaborate in situ tests using large diameter plates or pressuremeter apparatus. Where the reliability of the soil moduli is questionable, then the effects of using upper and lower bounding values may warrant investigation. It is normally found that, although the maximum computed settlements are directly proportional to these bounding values, the sensitivity of differential settlements - and hence bending moments - to the different soil moduli is very much lower.

Input data for the structural model take the usual form of nodal geometry, element topology, member properties, and applied loads, where the 2-dimensional elements used to model the raft, walls, and floor slabs, are based on the classical theory of thin elastic plates. This modelling can encompass raft foundations of virtually any plan shape and any distribution of vertical applied load, supported on a layered soil continuum. Pad and strip foundations, as well as groundslabs and pavement structures, can also be handled as they are simply particular forms of raft foundation. The presence of any superstructure (walls, columns, beams, slabs) can also be taken into account; this is an important consideration because, in many cases, the superstructure strongly influences raft behaviour. Extensive plotting facilities are essential to the analysis, both in checking input data and in presenting results in an intelligible and digestible manner.

Continuation of the analysis beyond the linear elastic range allows modelling of soil yield at the base of the raft and also separation of the contact surfaces arising from uplift forces. Piled rafts may be analysed by modelling the combined raft and pile group by a plain raft located at or near pile base level. Indeed, for large pile groups, interactive analyses based on discrete pile elements are impractical at present because of excessive computer storage requirements. Stepped rafts can be satisfactorily modelled only if the step size is small in comparison with the principal raft dimensions. Where this is not the case, the computed results will require careful post-processing to correct or make allowance for the stepped construction. It may be necessary to supplement such calculations by a plane strain analysis of a section through the step or, in extremis, by a fully 3-dimensional analysis.

Because the settlement profile for a raft of zero stiffness may be readily obtained during the initial part of the analysis, it is a simple matter to carry out a traditional settlement analysis at very low cost. Generally, differential settlements will be considerably overestimated, but a reasonably accurate value of maximum settlement will be obtained for a given soil stiffness profile. This part of the program has been adapted to operate on some of the larger desktop computers, and has been found particularly useful by those who, at least in the early stages of design, need to carry out only a simple settlement analysis.

Project summaries What follows is a brief description of eight project raft analyses taken in chronological order, compiled originally to provide general guidance on the scope and capabilities of foundation interaction analysisls. In each case, the results presented are confined to those directly relating to the principal objectives of the analysis. The foundations vary greately in size and complexity, from simple strip footings to large raft foundations, although the use of plain in situ reinforced concrete construction is assumed throughout. Some analyses were completed in a matter of days, while others were carried out over a period of many weeks.

In every case, the plate bending finite elements used to model the raft are either four-noded quadrilaterals or three-noded triangles, and the assumed elastic parameters for the concrete are E, = 15 GPa and v = 0.15. No iterative analyses to investigate flexural cracking were carried out in these examples, although non-linear analyses were performed in two cases to assess the effect of local yield of the soil on foundation behaviour. The vector plots of principal bending moment (Ml , M2) relate to the element centroids and arrowheads denote hogging (negative) moments.

The flexural response of a raft is largely governed by its relative stiffness, K, defined here as E, (1 - v2)t3/E(l - v,2)B3 for a rectangular (or circular) plan shape, where t and B denote the thickness and breadth (or radius), respectively, and E and W denote the isotropic elastic parameters of the soil. Where the soil is taken to be transversely isotropic, the vertical, horizontal and shear moduli are denoted by E,, Eh, and Gvh,

The Structural EngineerlVolume 62A/No. 8IAugust 1984

Paper: Hooper

5 Bays at 3-2m ‘/c 7 Bays at 3.2 m yc

(a) Plan

Fig 1. Silo raft .foundation,

l Joint ’

3200 3200 1140 3960 -

I I I i

‘50 Blinding Dimensions in mm

(b) Section A-A

respectively, while v,h and vh denote the Poisson’s ratios in the principal planes. In the present examples, the soil stiffness parameters relate solely to drained conditions.

A straightforward approach is generally adopted to take account of the stiffening effect of the superstructure. For flexible structures with widely spaced columns and few loadbearing walls and partitions, the superstructure stiffness may be either ignored in the raft analysis or included simply by adding the cumulative bending stiffness of the floorslabs to that of the raft (thickness t ) and then modelling this composite structure as a monolithic ‘equivalent raft’ of thickness T. If column shortening is taken into account, then lower values of T are obtained. Any such thickening of the raft naturally reduces differential settlements and increases bending moments, although the required moments in the actual raft are proportionately lower, and are obtained by factoring the computed values for the ‘equivalent raft’ by ( t / 7 ) 3 .

At the opposite extreme, the system of crosswalls and applied loading may be such that a realistic estimate of foundation behaviour can be obtained only by directly modelling the lower part of the superstructure, as well as the raft itself. This approach was adopted in one of the following examples, where the raft, basement walls, and groundfloor slab, were modelled as finite elements to represent the cellular substructure. Fortunately, this more elaborate form of structural modelling is needed only in a minority of practical cases.

On most occasions, the only significant additional stiffness is provided by loadbearing walls (e.g. core walls, crosswalls, basement retaining walls) and it is normally sufficient to model these walls and their floor connections by beam elements joined to the raft in the plan positions of the walls. It is usually necessary to consider only one or two storeys above raft level, and the simple rules devised to establish beam flange widths are given in Table 1. These effective flange widths depend on whether the floorslab is continuous on either side of the wall or only on one side: they also depend on the wall spacing and span (or the distance between points of contraflexure), so that the lowest value given in the table is used in the analysis. When calculating the second moment of area of the beam, the width of the raft section is taken to be the same as the flange width, although strictly the moment of the raft about its own neutral axis should be deducted from the total as it is already included in the plate bending finite element.

Douala

In only one case has it been possible to follow up the foundation analysis and design with settlement measurements. But some of the projects are located in parts of the world where the prospects of obtaining accurate and reliable settlement records over a period of years are extremely low. In other cases, the general economic climate has been such that clients understandably have not been prepared to pay for such measurements, and most professional institutions and associations have been short of funds. However, sufficiently good correlation has been obtained between the observed and computed behaviour of other foundations to justify a reasonable measure of confidence in the present approach, without necessarily succumbing to the utopian expectation that calculations can precisely model practical construction.

TABLE I-Effective flange widths .for beams used to model wa1l:floor system

L-beam I T-beam

b + 12d

B/2 B

b is the wall thickness d is the thickness of floorslab L is the effective span of wall B is the wall spacing

Silo foundations, Douala, Cameroun Structure. As part of a major extension to an existing brewery, the silo structure comprises a cellular raft foundation (Fig 1) supporting a 25m-high steel bin superstructure for malt storage. Each bin has a capacity of 100 t of malt, and is constructed from a plain steel hopper base and corrugated steel side-panels bolted on to steel stanchions. The raft is founded near the ground surface on sandy clay and silt strata, at a level well above the water table. The structure is to be built in two phases, separated by a period of several years, and the two component rafts are to be butt-jointed. The first phase was constructed during the 15-month period commencing October 1977. Design ana/ysis. Preliminary hand-calculations were carried out in order to estimate the most economic form of raft construction, having due regard to the isolated location of the site and the high cost of materials. Cellular construction of the form shown in Fig l(b) turned out to be most suitable, with fill material contained in the raft cells used to support the formwork for the upper concrete slab. The dimensions of the deeper raft section supporting the elevator tower were fixed because of machinery prerequisites, but the remaining raft section could be varied to suit structural design requirements. By using the known results for a uniformly loaded rectangular raft on an elastic layer”, it was possible to estimate the sensitivity of raft distortions and moments to variations in relative raft stiffness, and hence determine the optimum structural configuration. Hand checks were also made to ensure that shear stresses in the concrete were acceptable.

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1 2 3 4 5 6 7 %

B-

1 (Raft only)

I I

7-7

- .J

2

L ..I 5

Fig 2. Load cases .for Douala raft analysis

r- --- l

Grid line 1 2 3 4 5 6 7 8 01 1 I I I I 1

E 20 -+-

CI E T V = = = - - - -

60 I I

S

I I I l I I

r Average

Envelope :all load cases

Fig 3. Results .for beam strip along gridline B-B: Douala raft analysis

236

220kNm 220kNm f 220kNm 220 kNm - A

190kN 190kN I

4 1

Fig 4. Results .for machine strip .foundation

Having sized the raft in this way, a full interactive analysis was carried out for the purposes of detailed design. Taking the stage 1 raft, for example, the mesh (56 nodes, 42 rectangular elements) matched the basic grid layout of the cells, although the silo structure was subsequently extended by one bay (Fig 1). The cellular raft was reduced to the equivalent monolithic section, with thicker elements used to model the stiffer intake structure. The overall stiffening effect of the steelwork superstructure was estimated to be low, although the silo bins themselves were considered by the manufacturer to be rather sensitive to differential settlement. The soil strata were modelled by a 10m-thick upper layer (E = 8 MPa, U = 0-2) followed by a stiffer layer ( E = 24 MPa, U = 0-2) of equal thickness underlain by a rigid base.

Seven load cases were considered, so as to cover the worst anticipated combinations of full and empty silo bins (Fig 2). Plotting the results of these elastic analyses along selected orthogonal sections enabled the design envelopes of raft settlement, bending and twisting moment, and shear force, to be determined. These plots were prepared by hand at the time of the analysis, but would now be arranged as part of the post- processing of results by the computer. Results along the axis of symmetry of the raft are shown in Fig 3.

The local bending of the 3.2m square groundslabs spanning between the cross-beams was checked by hand, based on the raft contact pressures obtained in the main analysis. As the raft is moderately stiff relative to the soil ( K = 0.7) , computed pressures beneath the outer bays are typically double those on the inner bays. In view of possible consolidation of the fill material within the cellular raft, the corresponding top slab was designed to take its own self-weight in addition to the imposed loading.

No discernible effects of differential settlement on the silo structure have been observed. Indeed, the design settlement criteria were probably too severe because, at the end of construction, the bolted connections between the steel bins were left slack while the silos were filled and emptied. Only after this proof loading, and the consequent bedding-in of the joints and preloading of the soil, were the bolts fully tightened.

Machine .foundations, Popondetta, Papua New Guinea Structure. As part of the civil engineering works for a palm oil factory, the mill building contains a series of parallel strip foundations which support rotating machinery on plinths. Each strip is 14-6m long, lm

The Structural EngineerlVolume 62A/No. 8lAugust 1984

raper: nooper

E (a) Basement plan 8

Fig 5. Central Bank, Lagos

wide, and 0 -5m thick, increasing to a thickness of 1 3m at the plinths. At the centre of each plinth, the vertical design load is 90 kN, and a horizontal applied load of 80 kN acts at a distance of 1 -45m above the top of the plinth. Trial pits at the site confirmed the presence of a substantial thickness of medium dense sand. Construction took place during

Design analysis. The principal design requirement was to obtain realistic estimates of differential settlement along the length of the foundation in order to comply with shaft alignment tolerances. A secondary aim was to provide an independent check on the design bending moments obtained previously by hand calculation.

The foundation was modelled by 28 rectangular elements (58 nodes) spanning the full width, as no half-symmetry facilities were available at the time the analysis was carried out. Element properties varied to model the stepped cross-section, and each horizontal force was converted to an applied moment, based on a lever arm of 2.75m. The magnitude of these horizontal applied forces is small compared to the frictional forces that are capable of being mobilised on the base of the foundation. The soil model was taken as a 50m-thick homogeneous elastic stratum with E = 25 MPa and v = 0.3.

Computed values of settlement and longitudinal bending moment from an elastic analysis are shown in Fig 4. The average vertical contact pressure is only 25 kPa, and the computed differential settlements are well within the limiting value specified by the machinery manufacturer. The computed bending moments are also compatible with those obtained by hand calculation.

Central Bank, Lagos, Nigeria Structure. A proposed extension to an existing bank comprises a 32-storey office tower and three smaller buildings, each having several carpark floors. The 120m-high tower measures 60m x 30m in plan, and its design weight is 120 OOO t. There is a single-storey basement, where the layout of the internal walls is designed to provide security vaults and access corridors (Fig 5(a)). There is a ground level banking hall, and carparking on the first three floors is linked directly to the corresponding floors on the adjacent buildings. Detailed structural designs were prepared from December 1978 until May 1979, when the project was shelved.

The tower i s to be founded on a 2.5m-thick raft connected to a group of 400mm diameter prebored driven steel-cased piles at 1 -2m centres. There are 1200 such piles, extending to 37m below ground level. The soil strata consist of interbedded sand and clay layers, and the ground water level is close to the ground surface.

A significant feature of the superstructure which affects the foundation design is the truncation of the longitudinal spine walls at mezzanine level, leading to very high loads being transferred to the columns and gable walls at the edge of the raft, as well as to the central core (Fig 5(d)). Clearly, the substructure forms a vital link in this transfer of load as it will

1978-1979.

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-L L

Dense sand

Stiff clay

Dense to very dense sand

Hard clay

Very dense sand

E-MPa 0 200 400

(b) Layered soil model

(c) Substructure model

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ROOF LEVEL 32

PLANTROOM 31

30 29

28 27 26

25 2L

23 22

21 20 19 18 17

PLANTROOM 16 15 1 1 13 12

11 10

09 08 07 06 05

PLANTROOM OL 03 02

01

MEZZANINE

GROUNO FLOOR

BASEMENT

(d) Section A-A

1

act as a contiguous multicell box comprising the raft, the 900mm-thick outer walls, the 750mm-thick inner walls, and the 900mm-thick groundfloor slab. Furthermore, the wall structure above mezzanine level will provide additional restraint to substructure distortion. Design analysis. The principal aim of the foundation analysis was to identify in some detail the mode of load transfer through the substructure to the soil, and to check that the resulting stresses and displacements were commensurate with the reinforced concrete design. Preliminary elastic analyses were based on the 2.5m-thick raft, first unstiffened and then stiffened by beam elements, and gave a useful indication of the likely raft settlements and bending moments. Quarter-symmetry was assumed, and the piled raft was replaced by a plain raft located at pile base level, a simple, yet effective, approach which has proved to be very useful in the analysis of large piled raft foundations. The soil continuum beneath the

78

Spot values in mm

(a) Raft settlements

0 5 M N m l m --- Internal walls

(b) Principal raft bending moments

F

- I -

I I

I t

I I t-

I

+ I I +

t-+--Q

0 mMPa U

_ _ - internal walls

(c) Basement wall membrane stresses

Fig 6. Results .from Central Bank .foundation analysis

-*

raft is represented by a series of horizontal elastic layers whose moduli (Fig 5(b)) are mainly based on the shear strength and standard penetration test results given in the site investigation report. A Poisson’s ratio of 0 .2 is assumed for all strata. Analyses were also carried out to examine the effect of using different soil moduli, and of dispensing with the assumption of quarter-symmetry.

These initial results indicated that the raft moments and shear forces could be accommodated, albeit with some difficulty. The computed differential settlements of the unstiffened raft were unacceptably high, but this was to be expected in view of the relatively low raft stiffness ( K = 0 . l). The addition of beam elements substantially reduced these settlements, and also the corresponding raft bending moments, but this form of substructure modelling was felt to be oversimplified in this particular case. Accordingly, the full cellular action of the combined raft,

238 The Structural EngineerlVolume 62AINo. 8lAugust 1984

7 Bays at 4.8m '/c c

(a) Basement plan

Fig 7. Office tower, Lagos

basement walls and groundfloor slab was modelled by replacing the beam elements by vertical wall elements and horizontal floor elements capable of sustaining both bending and in-plane stresses. The finite element model (104 nodes, 78 plate elements, 38 wall elements) is shown in Fig 5(c), with the groundfloor slab omitted for clarity, and the vertical loads were applied at groundfloor level.

Results from the subsequent elastic analysis are partly summarised in Fig 6 , which illustrates the complex mode of sagging and hogging raft curvature. Because of the high average level of applied loading (573 kPa excluding raft self-weight), the analysis was continued into the non-linear region to examine the effect of local yield of the soil on the structural response of the foundation. This was achieved by specifying a limiting raft contact pressure of 900 kPa, a value based partly on intuition and partly on the anticipated behaviour of the pile group. The result was to decrease the transverse sagging curvature and to increase the longitudinal hogging curvature, especially towards each end of the foundation. However, it was considered that this hogging curvature would be much reduced in practice because of the additional restraining effect of the stiff superstructure walls.

Office to wer, Lagos, Nigeria Structure. One arm of this L-shaped building is a 23-storey tower block, 85m high, with corner core walls and a single-storey basement (Fig 7(a)). Most of the tower above fourth floor level consists of office accommodation: the remaining structure is a two-storey split-level carpark having plan dimensions of 20m x 30m.

The tower block foundation is a 1 5m-thick raft on a regular grid of 450mm diameter bored piles formed within layered sands and clays to a depth of 35m. There are 340 piles, and the spacing is generally 1.7m, decreasing to 1.2m in the lift core area. The founding level of these piles is the same as that for the bank building described previously, whose site is adjacent to that for the present structure. Construction commenced at the end of 1978. Design analysis. The objective of the analysis was to provide an independent check on the raft settlements and bending moments obtained in the local design office using other methods. The asymmetric raft foundation was modelled by 54 rectangular plate elements (70 nodes), with beam elements used to represent the stiffness of the substructure walls and groundfloor slab. The superstructure consists mainly of waffle floorslabs supported by widely spaced columns, and was not considered in the analysis to provide any additional flexural restraint to the raft. For calculation purposes, the raft was located at the level of the pile bases, and the layered soil model was the same as that used in the previous analysis of the bank building foundation.

Computed values of principal raft bending moment obtained from two elastic analyses, with and without the stiffening beam elements, are shown in Fig 7. These results indicate that the effect of the substructure is to change the distribution of raft bending moments without appreciably

The Structural Engineer/Volume 62AINo. 8/August 1984

\ 300mm outer walls

? l 2 MNm/m

(b) Raft moments without beam elements

changing the magnitude of the maximum moment. The maximum computed moments-including both the element centroid values shown and also the average nodal values-were rather smaller than those obtained in the main design. Ideally, asymmetric foundations of this type need to be modelled by a finer mesh in order to provide a sufficiently detailed moment field for reinforcement design.

Largely as a result of the low relative stiffness of the raft (K = 0.04) and the low average applied loading (347 kPa, including raft self-weight), compared with the previous bank tower, the specified limiting contact pressure of 900 kPa was exceeded at very few nodes, enabling the effect of local yield on raft behaviour to be neglected. The computed maximum

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Paper: Hooper

(a) Rail track section

Dimensions in mm Fig 8. Railway depot, Hona Kong

and minimum settlements were 42mm and 22mm for the unstiffened raft, and 41mm and 24mm for the stiffened raft. As a result of this analysis, the connections between the tower block and the carpark were redesigned to take account of the anticipated differential settlement between the two structures.

Railway depot, Hong Kong Structure. The depot is located at the northern end of the mass transit railway system and provides stabling and maintenance facilities for 30 eight-car trains. The six maintenance sidings, 200m long and spaced 7m apart, consist of continuous reinforced concrete rail track of stepped cross-section (Fig 8(a)). The steel rails are supported on concrete plinths at 1120mm centres. The track is founded on 3m of compacted fill overlying weathered rock, and was built during 1980. Design analysis. The objective of the soil-structure analysis was to obtain a realistic settlement profile of the track structure, together with design moments and shear forces for reinforcement detailing. This followed a separate analysis by others based on linear elastic spring supports which gave unacceptably high values of track settlement and bending moment.

The local flexural mode of this continuous track was examined by the quarter-symmetry modelling of a rectangular portion of track subjected to the vertical axle loads (240 kN on each axle) from one four-wheel bogie. The loads were assumed to be static, although some allowance was made for inertial and braking forces. In the finite element model (94 nodes, 95 elements), the element thicknesses across the section correspond to those shown in Fig 8(a), but the presence of the concrete plinths is ignored. The free boundaries of the model are sufficiently remote from the axle positions as to have little effect on the local bending in the heavily loaded areas. The soil is modelled as a Im-thick layer ( E = 2 MPa) overlying a 7m-thick layer whose elastic modulus varies linearly from 8.8 MPa at the top to 25.6 MPa at.the rigid base, with v = 0 .2 throughout.

Results of the elastic analysis are shown in Figs 8(b) and (c). The computed settlements were considered satisfactory, and the bending moment field formed the basis of the reinforced concrete design.

Housing development, Grangemouth, Scotland Structure. An extensive development of two-storey brickwork housing occupying a 0 . 7 ha site in Grangemouth, some 30 km west of Edinburgh, was built during 1982-1983, the start of construction having been postponed for 2 years for economic reasons. The site is located within the alluvial plain of the River Forth, and borehole records show at least 30m of soft clay and silt overlying stiff clay and dense granular material, with the water table about l m below ground level. Substantial variations in wall and roof height are reflected in the uneven foundation loading, which is mostly within the range 30-60 kN/m. Several alternative foundation schemes were examined during the initial stages of design, and a cellular raft construction turned out to be the most economic, provided that member sizes could be kept small, as indicated in Fig 9. Design analysis. In view of the thick deposit of compressible soil beneath the raft, the main foundation design problem was to limit differential settlements and hence avoid cracking of the unreinforced brickwork. The groundslab was modelled in the usual way (91 nodes, 67 plate elements),

k'

(b) Computed settlements

[120kN wheel loads1 I

c

. Free boundary

0 100 kNm/m U

4 Axle loads

(c) Computed principal bending moments

n Half-symmetry (a) Half-plan

Dwarf walls Joints omitted

Fig 9. Cellular raft, Grungemouth housing

The Structural EngineeriVolume 62A/No. 8IAugust 1984 240

raper: nooper

assuming half-symmetry of the structure, and the interconnecting system of dwarf walls was represented by a network of 113 beam elements having the appropriate stiffness properties (Fig 9(c)). No allowance was made for the stiffness of the brickwork superstructure, and the imposed loads were applied directly to the groundslab. The soil was modelled as a 40m-thick heterogeneous elastic layer underlain by a rigid stratum, and the variation of drained modulus with depth was estimated from the results of cone penetration tests carried out during the site investigation. The modulus was taken as 2 MPa to a depth of 6m below ground level, thence increasing linearly to 8-8 MPa at 40m depth. A Poisson's ratio of 0.25 was assumed throughout the layer.

The initial intention was to support the housing module on four separate rafts, butt-jointed to each other and with shear key connections. However, the first two foundation analyses showed that these joints could be eliminated, leading to a single continuous cellular raft. A third analysis showed that it was possible to replace the prestressed concrete groundfloor slabs with timber floors without increasing member sizes, thereby reducing costs still further and simultaneously satisfying an architectural preference for this type of flooring.

Some of the results of this third elastic analysis are shown in Fig 10. Maximum values of angular distortion are approximately 1/500 and occur in the end regions of the raft in the longitudinal direction. These values were deemed acceptable, especially as the in-plane stiffness of the brickwork will further reduce raft distortion, The principal moments in the groundslab were resolved into their orthogonal and twisting components for design purposes, with additional hand calculations carried out to check the two-way spanning of the groundslab between walls when subjected to the computed contact pressures. The bending moments carried by the dwarf walls, together with the corresponding plots of torque and shear force, provided the necessary information for detailing the wall reinforcement.

The foregoing interactive analysis demonstrates how raft foundations can provide a practical and economic means of supporting low-rise brickwork structures on poor ground, and suggests that they are particularly effective in reducing differential settlements. A programme of settlement measurements has been implemented in order to check the validity of the design, and to provide useful field data on actual structural behaviour.

Office tower, Cairo, Egypt Structure. The planned layout of this 23-storey structure, located in the commercial centre of Cairo, includes four levels of shopping, two plant- room floors, and 16 office floors. The 85m-high tower weighs 32 800 t (dead plus reduced live load, excluding raft self-weight), and there is a single-storey basement for carparking. The superstructure consists mostly

- 225 1 L5p-L .:.... ,.... .;--..< __._. ..,* :...(,.::.;..l ....j,...., ~........__ ? . ! . . I .::, ...-,.. L..:: ...... ' :.. ~ :ii: C'.. i.,.>.;:.: v.- ..:..... -- t 'Blinding concrete

Dimensions in mm

(b) Section A-A

Half-symmetry -.-.-.

l i l l i i Beam zlement Plate e p n t I I l u l n 1 ,,, II

I U 1 I U I O I (c) Finite element model

Unloaded raft --- Loaded raft

0 1 Q O v m r n

(a) Raft settlements

\ (b) Principal bending moments in groundslab

200mm&300mm internal walls

\

1 1-6m square

400mm outer

l i l n 0 0 5 1 MNm --I-.

(c) Bending moments in dwarf walls

Fig I O . Results of Grangemouth raft analysrs Fig 11. Basement plan .for qffice tower, Cairo

241 The Structural EngineerlVolume 62A/No. 8/August 1984

Paper: Hooper

of coffered floorslabs spanning between the octagonal corner core, peripheral columns, and four large internal columns, with precast concrete cladding to the facades. Detailed structural design took place during the first half of 1980, but the project was shelved later the same year.

The piled raft foundation is approximately 30m square, and the raft itself is 1 -6m thick (Fig 11). The 10m-long driven piles penetrate soft clay and silt, so that their enlarged bases are formed in the upper region of a thick layer of dense sand. There are 325 of these 520mm diameter end- bearing piles, spaced between 1 - 5m and 1 * 8m apart. The water table is within l m of ground level. Design analysis. The objective of the foundation analysis was to determine the pattern of raft settlements and bending moments resulting from the markedly asymmetric loading and to check the raft shear forces in the vicinity of the heavily loaded internal columns. These 1 -6m-square columns each carry a working load of 30 MN, and preliminary hand calculations indicated that a substantial amount of shear reinforcement would be necessary. But the raft thickness was not increased beyond 1 6m in order to minimise the extent of the dewatering operations during construction of the basement.

The whole raft was modelled (97 nodes, 87 plate elements) because of the absence of structural symmetry, and beam elements were used to represent the substructure walls: any additional stiffness contributed by the structure above groundfloor level was neglected. The piled raft was modelled as a plain raft located at pile base level on the surface of a 34m-thick sand layer underlain by a rigid base. The soil modulus at the upper surface was taken as 100 MPa, increasing approximately linearly to 530 MPa at 34m depth, and v = 0 - 2 was assumed throughout.

Two elastic analyses were carried out, with and without the stiffening beam elements. Some results for the latter case are shown in Fig 12, with the beam positions indicated on the bending moment plot. The deformed shape of the raft was similar in both analyses, and although the relative stiffness of the unstiffened raft is low (K = 0*02), the substructure walls appear to add little to the global raft stiffness. The principal difference was in the distribution of raft bending moments, although maximum raft moments were similar in both cases.

The analysis of the stiffened raft was continued into the non-linear range by specifying that raft contact pressures should not exceed the average contact pressure (356 kPa) by more than 50%. The result was to significantly increase raft settlements and bending moments in the corner area opposite to that containing the octagonal core, as shown in Fig 12. Changes of this magnitude were not anticipated prior to the analysis, despite the rather severe contact pressure limit and the absence of any load shedding along the pile shafts. They probably stem from the lack of a crosswall substructure, interconnected with the outer retaining walls, as this undoubtedly would have provided substantial stiffening to the relatively flexible raft. Had the project not been shelved in the final stages of design, this aspect would have been examined in more detail.

The design of the raft reinforcement was not entirely straightforward because the computed bending moment fields did not conform to clearly defined patterns. Finer mesh divisions in the finite element model would have helped in this respect, but the necessary computer storage facilities were unavailable. This shortcoming in the analysis applies to most asymmetric rafts subjected to heavy concentrated loads. Under these circumstances, a cautious approach is called for in design. Thus reinforcement quantities were based on maximum principal bending moments rather than -their orthogonal and twisting components, with shear links also specified with a generous margin of safety. In this connection, improvements in computer output could usefully be made as

Fig 12. Results.from Cairo raft analysis

242

(b) Settlements (non-linear analysis)

r-l--l-l

0 5MNm/m --- Beam positions

(c) Principal bending moments (elastic analysis)

1

L

" "#

0 5MNm/m --- Beam positions

(d) Principal bending moments (non-linear analysis)

The Structural EngineedVolume 62AINo. 8IAugust 1984

Paper: Hooper

(a) Finite element mesh 66m

5 3 5 4

O'

Nodes

Elements 48 47 46 45 44 43

I I I I I I

31 I I I I I I

(b) Typical results

Fig 13. Hammersmith raft analysis

detailed hand-checking for shear can be laborious and time consuming, especially when dealing with irregular mesh configurations.

Office development, London, England Structure. Part of the planned development .of a 2 .4 ha island site at Hammersmith in central London includes a new bus-rail interchange, an extensive office development, carparking areas, and a public library. There are four office blocks ranging from four to 10 stories high with single-storey basements. The irregular plan shape of one of the blocks is

The Structural EnnineerlVolume 62AlNo. 8lAugust 1984

shown in Fig 13(a). Apart from the service core, the superstructure is flexible, being mostly of floor-column construction. Preliminary schemes were prepared for both piled and raft foundations, but the developer- contractor strongly favoured the latter, chiefly for economy reasons. The basement is 3m deep and the 1 5m-thick raft is founded on gravel overlying London clay. The water table occurs in the gravel, although the pore water pressures within the clay stratum are not expected to be hydrostatic because of underdrainage. Design analysis. The objective of the analysis, carried out during the preliminary planning stage, was to examine the feasibility of using a large asymmetric raft foundation to support the variable-height office blocks. Settlements of the foundation and the surrounding ground were of particular interest, as were the approximate magnitudes of the raft bending moments.

The asymmetric raft mesh (99 nodes, 87 plate elements) shown in Fig 13(a) closely matches the plan shape of one of the four office buildings. Only eight additional beam elements were required to model the stiffness of the 200mm-thick basement walls. Design loads carried by the columns and core walls were applied as nodal forces and correspond to an average pressure of 156 kPa over the entire raft area. The soil strata were modelled as transversely isotropic elastic layers having a vertical axis of symmetry, with stiffness properties derived from previous retrospective analyses of buildings in London whose settlements have been measured. The assumed succession was 6m of gravel (E, = 100 MPa), 60m of London clay (E, = 25 + 22 MPa, where 2 is the depth (in m) below the top surface of the clay), and 20m of Woolwich and Reading Beds (E, = 200 MPa). The remaining parameters E h = 2. 3E,, G,, = 0 . 66E,, v,, = 0.1, and v,, = - 0 - 15, were used throughout.

Computed results along a transverse section of the raft passing through the core are shown in Fig 13(b) to illustrate the likely magnitude of settlements and bending moments. The settlement profiles show that the raft is essentially flexible because of its large plan area and the open layout of the superstructure. By extending the mesh to well outside the raft plan area and replacing the applied structural loads by the vector of raft reaction forces, the ground surface settlements in the surrounding area were computed. These settlements were acceptable in the design, as were the bending moments and the corresponding quantities of steel reinforcement. For detailed structural design, a finer mesh would be needed to give a more detailed picture of the bending moment field. However, the results of this preliminary analysis clearly demonstrated the feasibility of using a plain raft foundation.

243

Paper: Hooper Book reviews

Concluding remarks Analytical and numerical studies carried out over the past decade or so, together with field measurements in certain cases, have greatly improved our understanding of the interaction between structure and soil. One of the principal benefits to accrue from this work has been to put the structural analysis of rafts and similar foundations on a more rational basis. Used properly and sensibly, foundation interaction analysis can be cost effective in design practice because of savings in time and manpower: laborious hand-calculations are avoided, as is the use of oversimplified soil models.

Specific observations that have become evident during the course of the project work are: (a) The results of an interaction analysis often indicate that a plain raft foundation is to be preferred to some more costly alternative means of support, especially for building structures of up to moderate height. (b) In most practical cases, the stiffening effect of the superstructure on raft flexure can be adequately modelled by increasing the raft thickness or by using beam elements attached to the raft surface. Only in a minority of cases is i t necessary to use the more elaborate combination of wall, column and slab elements for the superstructure modelling. (c) For simple floor-column construction, superstructure effects are likely to be small. In contrast, the presence of loadbearing internal crosswalls and external retaining walls may significantly influence the magnitude and distribution of raft bending moments. At the same time, the additional loads taken by these walls may be considerable. In the case of basement retaining walls, for example, some thickening may be required to reduce shear stress to acceptable levels and additional reinforcement may be needed in the longitudinal direction specifically to accommodate in-plane forces. (d) Local yield of the soil can have a significant effect on the flexural response of raft foundations, although the best method of choosing a limiting contact pressure remains unresolved. In contrast, partial separation between raft and soil is rarely encountered in the design of building structures. (e) It is generally beneficial to base as much preliminary design work as possible on simple settlement analyses (zero raft stiffness) before entering into a full structural raft analysis. Computing costs are low using this simplified approach, even for fine meshes, and settlements outside the plan area of the foundation may be readily computed. (f) In some cases, particularly for asymmetric foundations, the maximum available mesh density has not been fine enough to give a sufficiently detailed pattern of raft bending moments for design purposes, necessitating an appreciable number of supplementary calculations. This problem can be overcome by significantly increasing the available number of nodes used to model the structure, and various solution methods are currently under investigation. (g) Sensible results for piled raft foundations incorporating a large group of piles have been obtained by the method of analysing a plain raft founded at or near pile base level. (h) In view of the inherent difficulties in modelling the interactive structural response of foundations, irrespective of the apparent degree of sophistication of the analysis, the conversion of computed moments and forces to reinforcement quantities should be approached with caution, and the methods used should be kept as simple as possible.

Acknowledgements The development work associated with the project analyses was funded by the Ove Arup Partnership. Various contributions were made by computer specialists over a period of several years and by the project design staff.

References 1.

2. 3.

4.

5 .

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

C P 2004 Foundations, London, British Standards Institution, 1972 Structure-soil interaction, London, Instn. Struct. Engrs, 1977 Hooper, J. A.: ‘Foundation interaction analysis’, Developments in Soil Mechanics--l (ed. C. R. Scott), London, Appl. Sci. Publn., 1978, ch. 5 , pp149-211 Hooper, J. A., and West, D. J.: ‘Structural analysis of a circular raft on yielding soil’, Proc. Instn. Civ. Engrs, 75, Pt. 2, June 1983,

Hooper, J. A.: ‘The effect of flexural cracking on differential raft settlements’, Proc. Instn. Civ. Engrs, 61, Pt. 2, September 1976,

Hooper, J. A.: ‘Analysis and design of a large raft foundation in Baghdad’, Proc. Instn. Civ. Engrs, 74, Pt. 1, November 1983,

Hooper, J. A.: ‘Non-linear analysis of a circular raft on clay’, Gkotechnique, 33, No. 1, 1983, ppl-20 C P l 14 The structural use of rein forced concrete in buildings: Part 2: Metric units, London, British Standards Institution, 1969 C P 110 The structural use q f concrete: Part I : Design, materials and workmanship, London, British Standards Institution, 1972 Wood, R. H.: ‘The reinforcement of slabs in accordance with a predetermined field of moments’, Concrete, 2, No. 2, 1968, pp69-76: discussion, 2, No. 8, pp319-321 BS 5337 The structural use qf concrete for retaining aqueous liquids, London, British Standards Institution, 1976 Guide to the design qf waterproof basements, London, Construction Industry Research and Information Association, 1978, guide 5 ‘Large pours for RC structures’, Proc. Concrete Society Symp, Birmingham, 1973 Birt, J. C . : Large concrete pours-a survey of current practice, London, Construction Industry Research and Information Association, 1974, report 49 Hooper, J . A.: ‘Interactive analysis of foundations on horizontally variable strata’, Proc. Insm. Civ. Engrs, 75, Pt. 2 September 1983, pp491-524 Hooper, J . A.: ‘Summary of raft design projects based on interactive foundation analysis, 1977-1980’, London, Ove Arup Partnership, April 1981 (unpublished report) Fraser, R. A., and Wardle, L. J.: ‘Numerical analysis of rectangular rafts on layered foundations’, Gkotechnique, 26, No.,

~ ~ 2 0 5 - 2 4 2

~ ~ 5 6 7 - 5 7 4

~ ~ 8 3 7 - 8 6 9

4, 1976, ~ ~ 6 1 3 - 6 3 0

Book reviews

Introductory structural C. K. Wang and C . G . Salmon Cliffs: Prentice-Hall, 1984) 591 .

analysis (Englewood pp, €28.45.

ISBN 0 13 501569 3. At an age of rapid developments in computer technology, when structural engineering students frequently misunderstand fundamental structural principles, i t is refreshing to see a new undergraduate text which reaffirms the importance of a proper understanding of statics, geometry, and the basic concepts. All in all, this well-presented and well-illustrated book can be warmly recommended to all engineering libraries.

The book concentrates on framed structures;

imperial units are used and loadings are based on north American practice. Chapters l to 12 cover the classical methods, including virtual work, moment area, Maxwell and Betti’s theorems, Muller-Breslau principle, Castigliano’s theorems, slope deflection, moment distribution, and the Williot-Mohr diagram which (despite its educational value) is often not taught these days. Chapters 13 to 20 cover the matrix methods, and here the reviewer would have welcomed further topics on computer application. Also, the stiffness method has been treated in a way that is specific to the type of structure under analysis; it would have been better to present a general technique and then show how this can be applied to different types-of structure. The authors have rightly devoted comparatively little space to the flexibility method; this reduced emphasis will be welcomed by many

teachers and students who are at times understandably worried by thoughts of ‘redundancies’ and ‘cuts’!

In the early chapters, it is not difficult to detect a Timoshenko flavour, which may be broadly taken as a sign of good quality. However, even Professor Timoshenko made the occasional mistakes, one of which is repeated in this 1984 book! Thus, on page 115, the authors explicitly qualify the virtual-work statement with the phrase ‘For a given elastic structure’, which is of course a mistake (see, for example, The Structural Engineer, vol. 61A, June 1983, ~ 1 7 5 ) . However, this and the earlier criticisms are minor compared with the value of another new book that reminds us of the importance of basic principles.

F. K. KONG

Continued on Page 248

244 The Structural EngineerlVolume 62A/No. 8IAugust 1984