336.2R-88 Suggested Analysis and Design Procedures for ... · ANALYSIS AND DESIGN OF COMBINED...

ACI 336.2R-88(Reapproved 2002)

Clyde N. Baker, Steven C. BallJoseph E. BowlesJoseph P. ColacoXI. T. Davisson

This report dealsa single column ofootings and matstural aspects of thbe eliminated andrelation of the twotural elements. Tconditions. The re

Keywords: concretesfoundations: reinforcstructural design.

ACI CommitCommentaries ning, executinspecifications. in the Projectare desired tobe phrased inProject Docu

Suggested Analysis and Design Procedures forCombined Footings and Mats

Reported by ACI Committee 336

Edward J. Ulrich Shyam N. ShuklaChairman Secretary

Jr. John A. Focht, Jr.M. GaynorJohn P. GnaedingerFritz Kramrisch

Hugh S. Lacy.Jim LewisJames S. NotchIngvar Schousboe

with the design of foundations carrying more thanf wall load. These foundations are called combined. Although it is primarily concerned with the struc-e design, considerations of soil mechanics cannot the designer should focus on the important inter- fields in connection with the design of such struc-

his report is limited to vertical effects of all loadingport excludes slabs on grade.

; earth pressure: footings: foundations; loads (forces); mated concrete; soil mechanics; stresses; structural analysis;

John F. SeidenstickerBruce A. SuprenantJagdish S. SyalJohn J. Zils

CONTENTSChapter 1 -General, p. 336.2R-2

1.1-Notation1.2-Scope1.3-Definitions and loadings1.4-Loading combinations1.5-Allowable pressure1.6-Time-dependent considerations1.7-Design overview

Chapter 2-Soil structure interaction, p. 336.2R-42.1-General2.2-Factors to be considered2.3-Investigation required to evaluate variable factors

Chapter 3-Distribution of soil reactions,p. 336.2R-6

3.1-General3.2-Straight-line distribution3.3-Distribution of soil pressure governed by modulus of subgrade

reaction

tee Reports, Guides, Standard Practices, andare intended for. guidance in designing, plan-g, or inspecting construction and in preparingReference to these documents shall not be made Documents. If items found in these documents be part of the Project Documents they should mandatory language and incorporated into thements.

336.2R

Chapter 4-Combined footings, p. 336.2R-74. l-Rectangular-shaped footings4.2-Trapezoidal or irregularly shaped footings4.3-Overturning calculations

Chapter 5-Grid foundations and strip footingssupporting more than two columns, p. 336.2R-8

5.l-General5.2-Footings supporting rigid structures5.3-Column spacing5.4-Design procedure for flexible footings5.5-Simplified procedure for flexible footings

Chapter 6-Mat foundations, p. 336.2R-96.1-General6.2-Finite difference method6.3-Finite grid method6.4-Finite element method6.5-Column loads6.6-Symmetry6.7-Node coupling of soil effects6.8-Consolidation settlement6.9-Edge springs for mats6.10-Computer output6.11-Two-dimensional or three-dimensional analysis6.12--Mat thickness6.13-Parametric studies6. 4-Mat foundation detailing/construction

Chapter 7-Summary, p. 336.2R-20

This report supercedes ACI 336.2R-66 (Reapproved 1980).Copyright 0 2002, American Concrete Institute.All rights reserved including rights of reproduction and use in any form or

by any means, including the making of copies by any photo process, or by anyelectronic or mechanical device, printed, written, or oral, or recording for soundor visual reproduction or for use in any knowledge or retrieval system or de-vice, unless permission in writing is obtained from the copyright proprietors.

-1

336.2R-2 MANUAL OF CONCRETE PRACTICE

Chapter 8-References, p. 336.2R-218.1 -Specified and/or recommended references8.2-Cited references

CHAPTER 1-GENERAL1.1-Notation

The following dimensioning notation is used: F =force; e= length; Q = dimensionless.

A =b =B =Bm =Bp =c =

D =Do =Df =

base area of footing, e2width of pressed edge, l

Dst =

e =ei =

E =

Ec =E’ =

Es =Fvh =

G =hw =H =Hci =A H =I =

IB =IF =Iw =

i =

J =kp =ksi =k'si =

ks =

foundation width, or width of beam column element, 4mat width, Pplate width, !distance from resultant of vertical forces to overturning edgeof the base, !dead load or related internal moments and forces, Fdead load for overturning calculations, Fthe depth Df should be the depth of soil measured adjacentto the pressed edge of the combined footing or mat at thetime the loads being considered are appliedstage dead load consisting of the unfactored dead load of thestructure and foundation at a particular time or stage ofconstruction, Feccentricity of resultant of all vertical forces, Peccentricity of resultant of all vertical forces with respect tothe x- and y-axes (ex and ey, respectively), !vertical effects of earthquake simulating forces or related in-ternal moment or force, Fmodulus of elasticity of concrete, F/e2modulus of elasticity of the materials used in the superstruc-ture, F/l?’soil modulus of elasticity, F/e2vertical effects of lateral loads such as earth pressure, waterpressure, fill pressure, surcharge pressure, or similar lateralloads, Fshear modulus of concrete, F/e’height of any shearwalls in structure, esettlement of foundation or point, !consolidation (or recompression) settlement of point i, lmagnitude of computed foundation settlement, t’plan moment of inertia of footing (or mat) about any axisx(Ix) or y(Iy) , f”moment of inertia of one unit width of the superstructure, t”moment of inertia per one unit width of the foundation, t”base shape factor depending on foundation shape and flexi-bility, e4vertical displacement of a node, t!torsion constant for finite grid elements, e4coefficient of subgrade reaction from a plate load test, F/l3

coefficient of subgrade reaction contribution to node i, F/E’revised coefficient of subgrade reaction contribution to nodei, F/4”, see Section 6.8q/6 = coefficient (or modulus) of vertical subgrade reac-tion; generic term dependent on dimensions of loaded area,F/f”

kvl =

K =Kr =L =

Ls =

Lst =

M ' =ME =

basic value of coefficient of vertical subgrade reaction for asquare area with width B = 1 ft, F/4’spring constant computed as contributory node area x ks, F/lrelative stiffness factor for foundation, Qlive load or related internal moments and forces produced bythe load, Fsustained live loads used to estimate settlement, F. A typicalvalue would be 50 percent of all live loads.stage service live load consisting of the sum of all unfac-tored live loads at a particular stage of construction, Fbending moment per unit length, F loverturning moment about base of foundation caused by anearthquake simulating force, F 4

MF =

Mw =

Mo =

MR =n =P =q =

qa =qu =

qult=

qi =

_̂ q =

Rv =

R v min

S =

SR =tw =W =

Xi =

Z =Z' =

s ==

; =

p =v =

c =

Y=

overturning moment about base of foundation, caused by Fvhloads, F foverturning moment about base of foundation, caused bywind loads, blast, or similar lateral loads, F llargest overturning moment about the pressed edge or cen-troid of the base, F !resultant resisting moment, F !exponent used to relate plate kp to mat ks, Qany force acting perpendicular to base area, Fsoil contact pressure computed or actual, F/P2allowable soil contact pressure, F/P2unconfined (undrained) compression strength of a cohesivesoil, F/e2ultimate soil bearing capacity; a computed value to allowcomputation of ultimate stregth design moments and shearsfor the foundation design, also used in overturning calcula-tions, F/e2actual or computed soil contact pressure at a node point asfurnished by the mat analysis. The contact pressures areevaluated by the geotechnical analysis for compatibility withqa and foundation movement, F/f2average increase in soil pressure due to unit surface contactpressure, F/f2resultant of all given design loads acting perpendicular tobase area, Fleast resultant of all forces acting perpendicular to base areaunder any condition of loading simultaneous with the over-turning moment, Fsection modulus of mat plan area about a specified axis; Sxabout x-axis; Sy about y-axis, t3stability ratio (formerly safety factor), Qthickness of shearwalls, evertical effects of wind loads, blast, or similar lateral loads,Fthe maximum deflection of the spring at node i as a linearmodel, Pfoundation base length or length of beam column element, !footing effective length measured from the pressed edge tothe position at which the contact pressure is zero, evertical soil displacement, k’torsion constant adjustment factor, Qfooting stiffness evaluation factor defined by Eq. (5-3), l/t?Poisson’s ratio, Qdistance from the pressed edge to Rv min (see Fig. 4-l and 4-2,esummation symbol, Qunit weight of soil, F/e’

1.2-ScopeThis report addresses the design of shallow founda-

tions carrying more than a single column or wall load.Although the report focuses on the structural aspects ofthe design, soil mechanics considerations are vital andthe designer should include the soil-structure interac-tion phenomenon in connection with the design ofcombined footings and mats. The report excludes slabs-on-grade.

1.3-Definitions and loadingsSoil contact pressures acting on a combined footing

or mat and the internal stresses produced by themshould be determined from one of the load combina-tions given in Section 1.3.2, whichever produces themaximum value for the element under investigation.Critical maximum moment and shear may not neces-sarily occur with the largest simultaneously applied loadat each column.

ANALYSIS AND DESIGN OF COMBINED FOOTINGS AND MATS 336.2R-3

1.3.1 DefinitionsCoefficient of vertical subgrade reaction ks-Ratio be-

tween the vertical pressure against the footing or matand the deflection at a point of the surface of con-tact

k, = q/6

Combined footing-A structural unit or assembly ofunits supporting more than one column load.

Contact pressure q-Pressure acting at and perpendic-ular to the contact area between footing and soil,produced by the weight of the footing and all forcesacting on it.

Continuous footing-A combined footing of prismaticor truncated shape, supporting two or more columnsin a row.

Grid foundation-A combined footing, formed by in-tersecting continuous footings, loaded at the inter-section points and covering much of the total areawithin the outer limits of assembly.

Mat foundation-A continuous footing supporting anarray of columns in several rows in each direction,having a slablike shape with or without depressionsor openings, covering an area of at least 75 percentof the total area within the outer limits of the assem-bly.

Mat area-Contact area between mat foundation andsupporting soil.

Mat weight-Weight of mat foundation.Modulus of subgrade reaction-See coefficient of ver-

tical subgrade reaction.Overburden-Weight of soil or backfill from base of

foundation to ground surface. Overburden should bedetermined by the geotechnical engineer.

Overturning-The horizontal resultant of any combi-nation of forces acting on the structure tending torotate the structure as a whole about a horizontalaxis.

Pressed edge-Edge of footing or mat along which thegreatest soil pressure occurs under the condition ofoverturning.

Soil stress-strain modulus-Modulus of elasticity of soiland may be approximately related (Bowles 1982) tothe coefficient of subgrade reaction by the equation

Es = k~(1-p2)/.

Soil pressure-See contact pressure.Spring constant-Soil resistance in load per unit de-

flection obtained as the product of the contributoryarea and k,. See coefficient of vertical subgrade re-action.

Stability ratio (SR)-Formally known as safety factor,it is the ratio of the resisting moment MR to the over-turning moment Mo.

Strip footing: See continuous footing definition.Subgrade reaction: See contact pressure and Chapter 3.Surcharge: Load applied to ground surface above the

foundation.

1.3.2 Loadings-Loadings used for design shouldconform to the considerations and factors in Chapter 9of ACI 318 unless more severe loading conditions arerequired by the governing code, agency, structure, orconditions.

1.3.2.1 Dead loads-Dead load D consisting of thesum of:

a. Weight of superstructure.b. Weight of foundation.c. Weight of surcharge.d. Weight of fill occupying a known volume.

1.3.2.2 Live loads-Live load L consisting of thesum of:

a. Stationary or moving loads, taking into accountallowable reductions for multistory buildings or largefloor areas, as stated by the applicable building code.

b. Static equivalents of occasional impacts.Repetitive impacts at regular intervals, such as those

caused by drop hammers or similar machines, and vi-bratory excitations, are not covered by these designrecommendations and require special treatment.

1.3.2.3 Effects of lateral loads-Vertical effects oflateral loads Fvh such as:

a. Earth pressure.b. Water pressure.c. Fill pressure, surcharge pressure, or similar.d. Differential temperature, differential creep and

shrinkage in concrete structures, and differential settle-ment.

Vertical effects of wind loads, -blast, or similar lat-eral loads W.

Vertical effects of earthquake simulating forces E.Overturning moment about base of foundation,

caused by earthquake simulating forces ME.Overturning moment about base of foundation,

caused by FVH loads MF.Overturning moment about foundation base, caused

by wind loads, blast, or similar lateral loads Mw.Dead load for overturning calculations Do, consist-

ing of the dead load of the structure and foundationbut including any buoyancy effects caused by partspresently submerged or parts that may become sub-merged in the future. The influence of unsymmetricalfill loads on the overturning moments Mo, as well as theresultant of all vertical forces Rv min, shall be investi-gated and used if found to have a reducing effect on thestability ratio SR.

Service live load Ls, consisting of the sum of all un-factored live loads, reasonably reduced and averagedover area and time to provide a useful magnitude forthe evaluation of service settlements. Also called sus-tained live load.

Stage dead load Dst, consisting of the unfactoredDead Load of the structure and foundation at a partic-ular time or stage of construction.

Stage service live load Lst, consisting of the sum of allunfactored Live Loads up to a particular time or stageof construction, reasonably reduced and averaged overarea and time, to provide a useful magnitude for theevaluation of settlements at a certain stage.


1.4-Loading combinationsIn the absence of conflicting code requirements, the

following conditions should be analyzed in the designof combined footings and mats.

1.4.1 Evaluation of soil pressure-Select the combi-nations of unfactored (service) loads which will pro-duce the greatest contact pressure on a base area ofgiven shape and size. The allowable soil pressure shouldbe determined by a geotechnical engineer based on ageotechnical investigation.

Loads should be of Types D, L, Fvh, W, and E as de-scribed in Section 1.3.2, and should include the verticaleffects of moments caused by horizontal components ofthese forces and by eccentrically (eccentric with regardto the centroid of the area) applied vertical loads.

a. Consider buoyancy of submerged parts where thisreduces the stability ratio or increases the contact pres-sures, as in flood conditions.

b. Obtain earthquake forces using the applicablebuilding code, and rational analysis.

1.4.2 Foundation strength design-Although the al-lowable stress design according to the Alternate DesignMethod (ADM) is considered acceptable, it is best todesign footings or mat foundations based on theStrength Design Method of ACI 318. Loading condi-tions applicable to the design of mat foundations aregiven in more detail in Chapter 6.

After the evaluation of soil pressures and settlement,apply the load factors in accordance with Section 9.2 ofACI 318.

1.4.3 Overturning-Select from the several applica-ble loading combinations the largest overturning mo-ment Mo as the sum of all simultaneously applicableunfactored (service) load moments (MF, Mw, and ME)and the least unfactored resistance moment MR result-ing from Do and Fvh to determine the stability ratio SRagainst overturning in accordance with the provisionsof Chapter 4.

1.4.4 Settlement-Select from the combinations ofunfactored (service) loads, the combination which willproduce the greatest settlement or deformation of thefoundation, occurring either during and immediatelyafter the load application or at a later date, dependingon the type of subsoil. Loadings at various stages ofconstruction such as D, Dst, and Lst should be evalu-ated to determine the initial settlement, long-term set-tlement due to consolidation, and differential settle-ment of the foundation.

1.5-Allowable pressureThe maximum unfactored design contact pressures

should not exceed the allowable soil pressure, qa. qashould be determined by a geotechnical engineer.Where wind or earthquake forces form a part of theload combination, the allowable soil pressure may beincreased as allowed by the local code and in consulta-tion with the geotechnical engineer.

1.6-Time-dependent considerationsCombined footings and mats are sensitive to time-

dependent subsurface response. Time-dependent con-

siderations include (1) stage loading where the initialload consists principally of dead load; (2) foundationsettlement with small time dependency such as mats onsand and soft carbonate rock; (3) foundation settle-ment which is time-dependent (usually termed consoli-dation settlements) where the foundation is sited overfine-grained soils of low permeability such as silt andclay or silt-clay mixtures; (4) variations in live loading;and (5) soil shear displacements. These five factors mayproduce time dependent changes in the shears and mo-ments.

1.7-Design overviewMany structural engineers analyze and design mat

foundations by computer using the finite elementmethod. Soil response can be estimated by modelingwith coupled or uncoupled “soil springs.” The springproperties are usually calculated using a modulus ofsubgrade reaction, adjusted for footing size, tributaryarea to the node, effective depth, and change of mod-ulus with depth. The use of uncoupled springs in themodel is a simplified approximation. Section 6.7 con-siders a simple procedure to couple springs within theaccuracy of the determination of subgrade response.The time-dependent characteristics of the soil response,consolidation settlement or partial-consolidation settle-ment, often can significantly influence the subgrade re-action values. Thus, the use of a single constant mod-ulus of subgrade reaction can lead to misleading re-sults.

Ball and Notch (1984), Focht et al. (1978) and Ban-avalkar and Ulrich (1984) address the design of matfoundations using the finite element method and time-dependent subgrade response. A simplified method,using tables and diagrams to calculate moments, shears,and deflections in a mat may be found in Bowles(1982), Hetenyi (1946), and Shukla (1984).

Caution should be exercised when using finite ele-ment analysis for soils. Without good empirical results,soil springs derived from values of subgrade reactionmay only be a rough approximation of the actual re-sponse of soils. Some designers perform several finiteelement analyses with soil springs calculated from arange of subgrade moduli to obtain an adequate de-sign.

CHAPTER 2-SOIL STRUCTURE INTERACTION2.1-General

Foundations receive loads from the superstructurethrough columns, walls, or both and act to transmitthese loads into the soil. The response of a footing is acomplex interaction of the footing itself, the super-structure above, and the soil. That interaction maycontinue for a long time until final equilibrium is es-tablished between the superimposed loads and the sup-porting soil reactions. Moments, shears, and deflec-tions can only be computed if these soil reactions canbe determined.


2.2-Factors to be consideredNo analytical method has been devised that can eval-

uate all of the various factors involved in the problemof soil-structure interaction and allow the accurate de-termination of the contact pressures and associatedsubgrade response. Simplifying assumptions must bemade for the design of combined footings or mats. Thevalidity of such simplifying assumptions and the accu-racy of any resulting computations must be evaluatedon the basis of the following variables.

2.2.1 Soil type below the footing-Any method ofanalyzing a combined footing should be based on a de-termination of the physical characteristics of the soillocated below the footing. If such information is notavailable at the time the design is prepared, assump-tions must be made and checked before construction todetermine their validity. Consideration must be given tothe increased unit pressures developed along the edgesof rigid footings on nongranular soils and the oppositeeffect for footings on granular soils. The effect ofembedment of the footing on pressure variation mustalso be considered.

2.2.2 Soil type at greater depths-Consideration oflong-term consolidation of deep soil layers should beincluded in the analysis of combined footings and mats.Since soil consolidation may not be complete for anumber of years, it is necessary to evaluate the behav-ior of the foundations immediately after the structureis built, and then calculate and superimpose stressescaused by consolidation.

2.2.3 Size of footing-The effect of the size of thefooting on the magnitude and distribution of the con-tact pressure will vary with the type of soil. This factoris important where the ratio of perimeter to area of afooting affects the magnitude of contact pressures, suchas in the case of the increased edge pressure, Section2.2.1, and the long-term deformation under load, Sec-tion 2.2.2. The size of the footing must also be consid-ered in the determination of the subgrade modulus. SeeSection 3.3.

2.2.4 Shape of footing-This factor also affects theperimeter-to-area ratio. Generally, simple geometricforms of squares and rectangles are used. Other shapessuch as trapezoids, octagons, and circles are employedto respond to constraints dictated by the superstructureand property lines.

2.2.5 Eccentricity of loading-Analysis should in-clude consideration of the variation of contact pres-sures from eccentric loading conditions.

2.2.6 Footing stiffness-The stiffness of the footingmay influence the deformations that can occur at thecontact surface and this will affect the variation ofcontact pressures (as will be seen in Fig. 3.1). If a flex-ible footing is founded on sand and the imposed load isuniformly distributed on top of the footing, then thesoil pressure is also uniformly distributed. Since the re-sistance to pressure will be smallest at the edge of thefooting, the settlement of the footing will be larger atthe edges and smaller at the center. If, however, thefooting stiffness is large enough that the footing can be

considered to act as a rigid body, a uniform settlementof the footing occurs and the pressure distribution mustchange to higher values at the center where the resis-tance to settlement is greater and lower values at theperimeter of the footing where the resistance to settle-ment is lower.

For nongranular soils, the stiffness of the footing willaffect the problem in a different manner. The settle-ment of a relatively flexible footing supported on a claysoil will be greatest at the center of the footing al-though the contact soil pressure is uniform. This oc-curs because the distribution of soil pressure at greaterdepths has a higher intensity under the center of thefooting. If the footing may be considered to act as arigid body, the settlement must be uniform and the unitsoil pressures are greater at the edge of the footing.

2.2.7 Superstructure stiffness-This factor tends torestrict the free response of the footing to the soil de-formation. Redistribution of reactions occur within thesuperstructure frame as a result of its stiffness, whichreduces the effects of differential settlements. This fac-tor must be considered together with Section 2.2.6 toevaluate the validity of stresses computed on the basisof foundation modulus theories. Also, such redistribu-tion may increase the stresses in elements of the super-structure.

2.2.8 Modulus of subgrade reaction-For smallfoundations [B less than 5 ft (1 .5 m)], this soil propertymay be estimated on the basis of field experimentswhich yield load-deflection relationships, or on the ba-sis of known soil characteristics. Soil behavior is gen-erally more complicated than that which is assumed inthe calculation of stresses by subgrade reaction theo-ries. However, provided certain requirements and limi-tations are fulfilled, sufficiently accurate results can beobtained by the use of these theories. For mat founda-tions, this soil property cannot be reliably estimated onthe basis of field plate load tests because the scale ef-fects are too severe.

Sufficiently accurate results can be obtained usingsubgrade reaction theory, but modified to individuallyconsider dead loading, live loading, size effects, and theassociated subgrade response. Zones of different con-stant subgrade moduli can be considered to provide amore accurate estimate of the subgrade response ascompared to that predicted by a single modulus ofsubgrade reaction. A method is described in Ball andNotch (1984) and Bowles (1982), and case histories aregiven in Banavalkar and Ulrich (1984) and Focht et al.(1978). Digital computers allow the designer to use matmodels having discrete elements and soil behavior hav-ing variable moduli of subgrade reaction. The modulusof subgrade reaction is addressed in more detail in Sec-tion 3.3 and Chapter 6.

2.3-Investigation required to evaluate variablefactors

Methods are available to estimate the influence ofeach of the soil structure interaction factors listed inSection 2.2. Desired properties of the structure and the


Z’=Jq-e( ) (3-4)

combined footing can be chosen by the design engi-neer. The designer, however, must usually accept thesoil as it exists at the building site, and can only rely oncareful subsurface exploration and testing, and geo-technical analyses to evaluate the soil properties affect-ing the design of combined footings and mats. In someinstances it may be practical to improve soil properties.Some soil improvement methods include: dynamic con-solidation, vibroflotation, vibroreplacement, surcharg-ing, removal and replacement, and grouting.

CHAPTER 3-DISTRIBUTION OF SOILREACTIONS

3.1- GeneralExcept for unusual conditions, the contact pressures

at the base of a combined footing may be assumed tofollow either a distribution governed by elastic subgradereaction or a straight-line distribution. At no placeshould the calculated contact pressure exceed the max-imum allowable value, qo.

3.2-Straight-line distribution of soil pressureA linear soil pressure distribution may be assumed

for footings which can be considered to be a rigid bodyto the extent that only very small relative deformationsresult from the loading. This rigid body assumptionmay result from the spacing of the columns on thefooting, from the stiffness of the footing itself, or therigidity of the superstructure. Criteria that must be ful-filled to make this assumption valid are discussed in thesections following.

3.2.1 Contact pressure over total base area- If theresultant of all forces is such that all portions of thefoundation contact area are in compression, the maxi-mum and minimum soil pressure may then be calcu-lated from the following formula, which applies only torectangular base areas and only when e is located alongone of the principal axes

q;:g = g 1 6e( >+-Z (3-l)

3.2.2 Contact pressure over part of area-The soilpressure distribution should be assumed to be triangu-lar. The resultant of this distribution has the samemagnitude and colinear, but acts in the opposite direc-tion of the resultant of the acting forces.

The maximum and minimum soil pressure under thiscondition can be calculated from the following expres-sions

At the footing edge

q (3-2)

At distance Z’ from the pressed edge

9 0nl,” = (3-3)

Eq. (3-l) to (3-4) are based on the assumption that notensile (-) stresses exist between footing and soil. Theequations may be applied with the details to be shownin the stability ratio calculation in Chapter 4, Fig. 4.2.Eq. (3-2) through (3-4) apply for cases where the resul-tant force falls out of the middle third of the base.

3.3-Distribution of soil pressure governed bythe modulus of subgrade reaction

The assumption of a linear pressure distribution iscommonly used and is satisfactory in most cases be-cause of conservative load estimates and ample safetyfactors in materials and soil. The acutal contact pres-sure distribution in cohesionless soils is concave; in co-hesive soils, the pressure distribution is convex (Fig.3.1). See Chapter 2 for more discussion of foundationstiff and pressure distributions.

Fig. 3.1-Contact pressure on bases of rigid founda-tions l

The suggested initial design approach is to size thethickness for shear without using reinforcement. Theflexural steel is then obtained by assuming a linear soilpressure distribution and using simplified procedures inwhich the foundation satisfies statics. The flexural steelmay also be obtained by assuming that the foundationis an elastic member interacting with an elastic soil.Simplified methods are found in some textbooks andreferences: Bowles (1974, 1982); Hetenyi (1946); Kram-risch (1984); and Teng (1962).

3.3.1 Beams on elastic foundations - If a combinedfooting is assumed to be a flexible slab, it may be ana-lyzed as a beam on elastic foundation using the meth-ods found in Bowles (1974, 1982); Hetenyi (1946);Kramrisch and Rogers (1961); or Kramrisch (1984). Thediscrete element method has distinct advantages of al-lowing better modeling of boundary conditions of soil,load, and footing geometry than closed-form solutionsof the Hetenyi type. The finite element method usingbeam elements is superior to other discrete elementmethods.

It is common in discrete element analyses of beams touse uncoupled springs. Special attention should begiven to end springs because studies with large-scalemodels have shown that doubling the end springs wasneeded to give good agreement between the analysis


and performance (Bowles 1974). End-spring doublingfor beams will give a minimal spring coupling effect.

3.3.2 Estimating the modulus of subgrade reaction -It is necessary to estimate a value for the modulus ofsubgrade reaction for use in elastic foundation analy-sis.

Several procedures are available for design:a. Estimate a value from published sources (Bowles

1974, 1982, and 1984; Dept. of Navy 1982; Kramrisch1984; Terzaghi 1955).

b. Estimate the value from a plate load test (Ter-zaghi 1955). Since plate load tests are of necessity onsmall plates, great care must be exercised to insure thatresults are properly extrapolated. The procedure (Sow-ers 1977) for converting the ks of a plate kp to that forthe mat ks may be as in the following

Ks = K, 2( >

n(3-8)

m

where n ranges from 0.5 to 0.7 commonly. One mustallow for the depth of compressible strata beneath themat and if it is less than about 4B the designer shoulduse lower values of n.

c. Estimate the value based on laboratory or in situtests to determine the elastic parameters of the foun-dation material (Bowles 1982). This may be done bynumerically integrating the strain over the depth of in-fluence to obtain a settlement _̂ H and back computingks as

Ks = q/AH

Several values of strain should be used in the influencedepth of approximately 4B where B is the largest di-mension of the base. Values of elastic parameters de-termined in the laboratory are heavily dependent onsample disturbance and the quality and type of triaxialtest results.

d. Use one of the preceding methods for estimatingthe modulus of subgrade reaction, but, in addition,consider the time-dependent subgrade response to theloading conditions. This time-dependent soil responsemay be consolidation settlement or partial-elasticmovement. An iterative procedure outlined in Section6.8, and described by Ulrich (l988), Banavalkar andUlrich (1984). and Focht et al. (1978), may be neces-sary to compare the mat deflections with computed soilresponse. The computed soil responses are used in amanner similar to producing the coupling factor toback compute springs at appropriate nodes. Since thesoil response profile is based on contact stresses whichare in turn based on mat loads, flexibility, and modu-lus of subgrade reaction, iterations are necessary untilthe computed mat deflection and soil response con-verge within user-acceptable tolerance.

CHAPTER 4-COMBINED FOOTINGS4.1-Rectangular-shaped footings

The length and width of rectangular-shaped footingsshould be established such that the maximum contact

pressure at no place exceeds the allowable soil pressure.All moments should be calculated about the centroid ofthe footing area and the bottom of the footing. Allfooting dimensions should be computed on the as-sumption that the footing acts as a rigid body. Whenthe resultant of the column loads, including considera-tion of the moments from lateral forces, concides withthe centroid of the footing area, the contact pressuremay be assumed to be uniform over the entire area ofthe footing.

When the resultant is eccentric with respect to thecenter of the footing area, the contact pressure may beassumed to follow a linear distribution based on the as-sumption that the footing acts as a rigid body (see Sec-tion 3.2). The contact pressure varies from a maximumat the pressed edge to a minimum either beneath thefooting or at the opposite edge.

Although the effect of horizontal forces are beyondthe scope of this analysis and design procedure, hori-zontal forces can provide a major component to thevertical resultant. Horizontal forces that can generatevertical components to the foundation may originatefrom (but are not limited to) wind, earth pressure, andunbalanced hydrostatic pressure. A careful examina-tion of the free body must be made with the geotech-nical engineer to fully define the force systems actingon the foundation before the structural analyses are at-tempted.

4.2-Trapezoidal or irregularly shaped footingsTo reduce eccentric loading conditions, a trapezoidal

or irregularly shaped footing may be designed. In thiscase the footing can be considered to act as a rigid bodyand the soil pressure determined in a manner similar tothat for a rectangular footing.

4.3-Overturning calculationsFor calculations that involve overturning, use the

combination of loading that produces the greatest ratioof overturning moment to the corresponding verticalload.

For footings resting on rock or very hard soil, over-turning will occur when the eccentricity e of the loadsP falls outside the footing edge. Where the eccentricityis inside the footing edge, the stability ratio SR againstoverturning can be evaluated from

SR = MR / Mo (4-1)

In Eq. (4-l). Mo is the maximum overturning mo-ment and MR is the resisting moment caused by theminimum dead weight of the structure; both are calcu-lated about the pressed edge of the footing. The stabil-ity ratio should generally not be less than 1.5.

Overturning may occur by yielding of the subsoil in-side and along the pressed edge of the footing. In thiscase, rectangular or triangular distributions of the soilpressure along the pressed edge of the footing as shownin Fig. 4.1 and 4.2, respectively, are indicated. In thiscase the stability ratio SR against overturning is calcu-


I 'v min

Fig. 4.2-Stability ratio calculation (triangular distri-butions of soil pressure along pressed edge of footing)

lated from Eq. (4-l), with

MR = Rv min (c - v) (4-2)

The calculation of the stability ratio is illustrated inFig. 4.1 and 4.2. Since the actual pressure distributionmay fall between triangular and rectangular. the truestability ratio may be less than that indicated by rect-angular distribution. A stability ratio of at least I.5 isrecommended for overturning.

IR v min

SlABIL:TV RAT:O: S.R. * r 2 1 . 50

UHERE bh * R" ml" lc - v)

R ” mm“=z-qt

Fig. 4.1-Stability ratio calculation (rectangular distri-butions of soil pressure along pressed edge of footing)

CHAPTER 5-GRID FOUNDATIONS AND STRIPFOOTINGS

SUPPORTING MORE THAN TWO COLUMNS5.1 -General

Strip footings are used to support two or more col-umns and other loadings in a line. They are commonlyused where it is desirable to assume a constant soilpressure beneath the foundation; where site and build-ing geometries require a lateral load transfer to exteriorcolumns; or where columns in a line are too close to besupported by individual foundations. Grid foundationsshould be analyzed as independent continuous stripsusing column loads proportioned in direct ratio to thestiffness of the strips acting in each direction. The fol-lowing design principles defined for continuous stripfootings will also apply with modifications for gridfoundations.

5.2-Footings supporting rigid structuresContinuous strip footings supporting structures

which, because of their stiffness, will not allow the in-dividual columns to settle differentially, may be de-signed using the rigid body assumption with a lineardistribution of soil pressure. This distribution can bedetermined based on statics.

To determine the approximate stiffness of the struc-ture, an analysis must be made comparing the com-bined stiffness of the footing, superstructure framingmembers, and shearwalls with the stiffness of the soil.This relative stiffness Kr will determine whether thefooting should be considered as flexible or to act as arigid body. The following formulas (Meyerhof 1953)may be used in this analysis

K, = ‘2 (5-l)

An approximate value of E’IB per unit width of build-ing can be determined by summing the flexural stiff-ness of the footing E'IF, the flexural stiffness of eachframed member E'Ib' and the flexural stiffness of anyshearwalls E'twh3

w/12 where tw and hw are the thicknessand height of the walls, respectively

&E’ IB = E' IF + CE’ IB + E’ 12 (5-2)

Computations indicate that as the relative stiffness Krincreases, the differential settlement decreases rapidly.

For Kr = 0, the ratio of differential to total settle-ment is 0.5 for a long footing and 0.35 for a squareone. For Kr = 0.5, the ratio of differential to total set-tlement is about 0.1.

OMBINED FOOTINGS AND MATS 336.2R-9
ANALYSIS AND DESIGN OF C
If the analysis of the relative stiffness of the footingyields a value of 0.5, the footing can be considered rigidand the variation of soil pressure determined on thebasis of simple statics. If the relative stiffness factor isfound to be less than 0.5, the footing should be de-signed as a flexible member using the foundation mod-ulus approach as described under Section 5.4.

5.3-Column spacingThe column spacing on continuous footings is im-

portant in determining the variation in soil pressuredistribution. If the average of two adjacent spans in acontinuous strip having adjacent loads and columnspacings that vary by not more than 20 percent of thegreater value, and is less than 1.75/X. the footing canbe considered rigid and the variation of soil pressuredetermined on the basis of simple statics.

The beam-on-elastic foundation method (see Section2.2.8) should be used if the average of two adjacentspans as limited above is greater than 1.75///\ .

For general cases falling outside the limitations givenabove, the critical spacing at which the subgrade mod-ulus theory becomes effective should be determined in-dividually.

The factor X is

h = (5-3)

5.4-Design procedure for flexible footingsA flexible strip footing (either isolated or taken from

a mat) should be analyzed as a beam-on-elastic foun-dation. Thickness is normally established on the basisof allowable wide beam or punching shear without useof shear reinforcement; however, this does not prohibitthe designer’s use of shear reinforcement in specific sit-uations.

Either closed-form solutions (Hetenyi 1946) or com-puter methods can be used in the analysis.

5.5-Simplified procedure for flexible footingsThe evaluation of moments and shears can be sim-

plified from the procedure involved in the classical the-ory of a beam supported by subgrade reactions, if thefooting meets the following basic requirements (Kram-risch and Rogers 1961 and Kramrisch 1984):

a. The minimum number of bays is three.b. The variation in adjacent column loads is not

greater than 20 percent.c. The variation in adjacent spans is not greater than

20 percent.d. The average of adjacent spans is between the lim-

its 1.75/X and 3.50/X.If these limitations are met, the contact pressures can

be assumed to vary linearly, with the maximum valueunder the columns and a minimum value at the centerof each bay. This simplified procedure is described insome detail by Kramrisch and Rogers (1961) andKramrisch (1984).

CHAPTER 6-MAT FOUNDATIONS6.1-General

Mat foundations are commonly used on erratic orrelatively weak subsurfaces where a large number ofspread footings would be required and a well definedbearing stratum for deep foundations is not near thefoundation base. Often, a mat foundation is used whenspread footings cover more than one-half the founda-tion area. A common mat foundation configuration isshown in Fig. 6.1(a).

The flexural stiffness EI of the mat may be of con-siderable aid in the horizontal transfer of column loadsto the soil (similar to a spread footing) and may aid inlimiting differential settlements between adjacent col-umns. Structure tilt may be more pronounced if themat is very rigid. Load concentrations and weak sub-surface conditions can offset the benefits of mat flex-ural stiffness.

Mats are often placed so that the thickness of the matis fully embedded in the surrounding soil. Mats forbuildings are usually beneath a basement that extendsat least one-half story below the surrounding grade.Additionally, the top mat surface may function as abasement floor. However, experience has shown thatutilities and piping are more easily installed and main-tained if they are placed above the mat concrete. De-pending on the structure geometry and weight, a matfoundation may “float” the structure in the soil so thatsettlement is controlled. In general, the pressure caus-ing settlement in a mat analysis may be computed as

Net pressure = {[Total (including mat) structure weight]

- Weight of excavated soil}/Mat area (6-1)

Part of the total structure weight may be controlledby using cellular mat construction, as illustrated in Fig.6.1(b). Another means of increasing mar stiffness whilelimiting mat weight is to use inverted ribs between col-umns in the basement area as in Fig. 6.1 (c). The cells ina cellular mat may be used for liquid storage or to alterthe weight by filling or pumping with water. This maybe of some use in controlling differential settlement ortilt.

Mats may be designed and analyzed as either rigidbodies or as flexible plates supported by an elasticfoundation (the soil). A combination analysis is com-mon in current practice. An exact theoretical design ofa mat as a plate on an elastic foundation can be made;however, a number of factors rapidly reduce the exact-ness to a combination of approximations. These in-clude:

1. Great difficulty in predicting subgrade responsesand assigning even approximate elastic parameters tothe soil.

2. Finite soil-strata thickness and variations in soilproperties both horizontally and vertically.

3. Mat shape.4. Variety of superstructure loads and assumptions in

their development.


Plan Plan

( a ) S o l i d m a t o f r e i n f o r c e d ( b ) M a t u s i n g c e l l (c) Ribbed mat used toconcrete ; most common c o n s t r u c t i o n . Cel ls control bendingconf igurat ion. D = depth may be f i l led with with minimum con-f o r s h e a r , moment or stab- water or sand to con- c r e t e . Ribs mayi l i t y a n d ranges from about t r o l s e t t l e m e n t s o r f o r be e i ther one or1.5 to 6+ ft (0.5 to 2+ m). s t a b i l i t y . two-way.

Fig. 6.1--Mat configurations for various applications: (a) mat ideally suited for finite element or finite grid method;(b) mat that can be modeled either as two parallel plates with the upper plate supported by cell walls modeled assprings, or as a series of plates supported on all edges; and (c) mat ideally suited for analysis using finite grid method,since ribs make direct formulation of element properties difficult

5. Effect of superstructure stiffness on mat (and viceversa).With these factors in mind, it is necessary to designconservatively to maintain an adequate factor of safety.The designer should work closely with the geotechnicalengineer to form realistic subgrade response predic-tions, and not rely on values from textbooks.

There are a large number of commercially availablecomputer programs that can be used for a mat analy-sis. ACI Committee 336 makes no individual programrecommendation since the program user is responsiblefor the design. A program should be used that the de-signer is most familiar with or has investigated suffi-ciently to be certain that the analyses and output arecorrect.

6.1.1 Excavation heave-Heave or expansion of thebase soil into the excavation often occurs when exca-vating for a mat foundation. The amount depends onseveral factors:

a. Depth of excavation (amount of lost overburdenpressure).

b. Type of soil (sand or clay)-soil heave is less forsand than clay. The principal heave in sand overlyingclay is usually developed in the clay.

c. Previous stress history of the soil.d. Pore pressures developed in the soil during exca-

vation from construction operations.The amount of heave can range from very little-1/2

to 2 in. (12 to 50 mm)-to much larger values. Ulrichand Focht (1982) report values in the Houston, Tex.,area of as much as 4 in. (102 mm). Some heave is al-

most immediately recovered when the mat concrete isplaced, since concrete density is from 1.5 to 2.5 timesthat of soil.

The influence of heave on subgrade response shouldbe determined by the geotechnical engineer workingclosely with the structural designer. Recovery of theheave remaining after placing the mat must be treatedas either a recompression or as an elastic problem. Ifthe problem is analyzed as a recompression problem,the subsurface response related to recompressionshould be obtained from the geotechnical engineer. Thesubsurface response may be in the form of a re-compression index or deflections computed by the geo-technical engineer based on elastic and consolidationsubsurface behavior. If the recovery is treated as anelastic problem, the modulus of subgrade reactionshould be reduced as outlined in Section 6.8, where theconsolidation settlement used in Eq. (6.8) includes theamount of recompression.

6.1.2 Design procedure-A mat may be designed us-ing either the Strength Design Method (SDM) or work-ing stress design according to the Alternate DesignMethod (ADM) of ACI 318-83, Appendix B. The ADMis an earlier method, and most designers prefer to usethe SDM.

The suggested design procedure is to:1. Proportion the mat plan using unfactored loads

and any overturning moments as

(6-2)


The eccentricity ex, ey of the resultant of column loadsC P includes the effect of any column moments and anyoverturning moment due to wind or other effects. Theeccentricities ex, ey are computed using statics, sum-ming moments about two of the adjacent mat edges[say, Lines O-X, O-Y of Fig. 6.1(a)]. The values of de-sign ex, ey will be slightly different if computed usingunfactored or factored design loads.

The actual unfactored loads are used here as thecomparison to the soil pressure furnished by the geo-technical engineer is

q G qil

The allowable soil pressure may be furnished as oneor more values depending on long-term loading or in-cluding transient loads such as wind and snow. The soilpressure furnished by the geotechnical engineer is useddirectly in the ADM procedure. For strength design itis necessary to factor this furnished allowable soil pres-sure to a pseudo “ultimate” value, which may be doneas follows

sum of factored design loads

of unfactored design loads (6-3)

2. Compute the minimum mat thickness based onpunching shear at critical columns (corners, sides, inte-rior, etc.), refer to Fig. 6.1(a), based on column loadand shear perimeter. It is common practice not to useshear reinforcement so that the mat depth is a maxi-mum. This increases the flexural stiffness and increasesthe reliability of using Eq. (6-2).

3. Design the reinforcing steel for bending by treat-ing the mat as a rigid body and considering strips bothways, if the following criteria are met:

a. Column spacing is < 1.75/X, or the mat is verythick.

b. Variation in column loads and spacing is notover 20 percent. For mats not meeting this cri-teria, go immediately to Step 4.

These strips are analyzed as combined footings withmultiple columns loaded with the soil pressure on thestrip, and column reactions equal to the factored (orunfactored) loads obtained from the superstructureanalysis. Since a mat transfers load horizontally, anygiven strip may not satisfy a vertical load summationunless consideration is given to the shear transfer be-tween strips. Bowles (1982) illustrates this problem anda method of analysis so the strips satisfy statics.

4. Perform an approximate analysis (Shukla 1984) or a computer analysis of the mat and revise the rigid body design as necessary. An approximate analysis can be made using the method suggested by ACI 336.2R-66 to calculate moments, shears, and deflections in a mat with the help of charts. Charts for this procedure are given in Shukla (1984) and Hetenyi (1946). A com- pleted example problem is given in the paper to assist the designer. The geotechnical engineer should furnish the designer subgrade response values even when a simplified design method is used.

Computer analysis for mat foundations is usuallybased on an approximation where the mat is dividedinto a number of discrete (finite) elements using gridlines. There are three general discrete element formula-tions which may be used:

a. Finite difference (FD).b. Finite grid method (FGM).c. Finite element method (FEM).

These latter two methods can be used for mats withcurved boundaries or notches with re-entrant corners asin Fig. 6.2. All three of these methods use the modulusof subgrade reaction k, as the soil contribution to thestructural model. This has been considered in Sections2.3, 3.3, and 5.4, and will be considered in Sections 6.7and 6.8. Computers (micro to mainframe) and avail-able software make the use of any of the discrete ele-ment methods economical and rapid. The finite ele-ment model shown in Fig. 6.2(b) indicates a simplegridding that produces 70 elements, 82 nodes, and 246equations with a band width computed as shown.

The finite difference method is a procedure whichprovides quite good results for the approximationsused. This procedure was used extensively in the past,but is sometimes used to verify finite element methods.It is particularly appealing since it does not requiremassive computer resources. Fig. 6.3 shows the finitedifference method with node equation for interior nodegiven.

6.2-Finite difference method

6.3-Finite grid methodThis method discretizes the mat into a number of

beam-column elements with bending and torsional re-sistance (Fig. 6.4). The torsional resistance is used toincorporate the plate twist using the shear modulus G.In finite element terminology, the FGM produces non-conforming elements, i.e., interelement compatibility isinsured only at the nodes. A theoretical development ofthis method specifically for mats is found in Bowles(1974, 1976, and 1982).

6.4-Finite element methodThis method discretizes the mat into a number of

rectangular and/or triangular elements (Fig. 6.5). Ear-lier programs used displacement functions which pur-ported to produce conforming interelement compatibil-ity (i.e., compatibility both at nodes and along elementboundaries between adjacent elements).

For bending, the nodal displacement and slopes inthe X- and Y-direction are required. This results in tak-ing partial derivatives of the displacement function.First, however, one must delete one term for the trian-gle or add two nodes and use the fifteen-term displace-ment function. Similarly, for the rectangle one mustdrop three terms or add a node. There are computerprograms which drop terms, combine terms, and addnodes. Any of the programs will give about the samecomputed output so the preferred program is that onemost familiar to the user.


/rt--*t--c6?,+t-f----+- + --,L\t?3

Nodes 1 2 3 4 5 6 7 8

This grid produces 70 elements, 82 nodes, 246equations

Bandwidth depends on node numberingscheme =3 (largest element node

difference + 1)Here. bandwidth = 3(37 + 25 +1) = 39

a. Mat with wall , notch. and irregular shapeincluding a curved area.

Fig. 6.2-Mat gridding for either finite grid or finite element method

rh h rh

h

“TTh

\

%L

Ib

For r = 1: 20(wo + kshC/D) - 8(-T + wB + wR + wL)

+ 2(WTL + WTR + VBL + vBR)

+ (WTT + WBB + ”LL + wRR) - Ph*/D

Fig. 6.3-Finite difference method with node equationfor interior node given. Separate equations are re-quired for corners, sides, and first nodes in from sides

6.4.1 Iso-parametric elements-An element is of theiso-parametric type if the same function can be used todescribe both shape and displacement. Some computerprograms suitable for mat (or plate) analysis use thismethodology. The method is heavily computation-in-tensive but has a particular advantage over displace-ment functions in that any given element may contain adifferent number of nodes than others as illustrated inFig. 6.5.

Interpolation functions are used together with an ap-proximation for area integration. Good results areclaimed with care in using enough interpolation points.

Some programs using the iso-parametric formulationmay not have the necessary routines to include extranodes on select elements and in these cases there is usu-ally no advantage of the iso-parametric element overthe displacement function approach. Neither of theseapproaches is exact, unless substantial effort is ex-pended.

Cook (1974), Ghali and Neville (1972), and Zien-kiewicz (1977) describe formulation of the finite ele-ment stiffness matrix.

Advantages of the FEM include:a. The solution is particularly mathematically effi-

cient.b. Boundary condition displacements can be mod-

eled.c. The iso-parametric approach can allow use of mid-

side nodes, which produces some mesh refinementwithout the large increase in grid lines of the othermethods.

Disadvantages of the FEM include:a. The formulation of the stiffness matrix is compu-

tationally intensive.b. It is very difficult to include a concentrated mo-

ment at the nodes as the methodology uses moment perunit of width.

ANALYSIS AND DESIGN OF COMBINED FOOTINGS AND MATS
336.2R-13
Pl, P2, P4, P5 = moment (xi = 0,)

P3, P6 = P (Xi = Azi)

PS - x5 // -P

P2 -.x24h;-"

Node K

typical beam element withtorsion included.

p2 PSLp6P4

Start node coding atleft corner of mat

upper

Fig. 6.4-Element coding for nodes and element forces for the finite grid method. Dimension: B and t used bothfor element moment of inertia i and torsion inertia J

The FEM plate elementM

M yxX

Mi + Mx etc in general

p2

P

M’Y t- x2 1--

t

M'3 - x 3 X

PI - Xl M’F

Node P-X coding hassame relationship tonode forces and disp-lacements as in Fig. 6-4.

5.

Additional nodes are allowed with some programswhere isoparametric elements using shape functionsare programmed.

Fig. 6.5-Typical rectangular finite element using the FEM plate formulation.Continuing Pi-Xi coding counterclockwise around the four-node element givestwelve element displacements: four translations and eight rotations


c. Often a statics summation of nodal moments isonly approximated. This is often the case where thecommon elements include triangles or a mix of trian-gles and rectangles.

d. It requires interpretation of the plate twist mo-ment Mxv that is output.

e. It is not directly amenable to increasing the nodaldegree-of-freedom from three to six for pile caps as isthe finite grid method (FGM).

f. The stiffness matrix is the same size as the FGMbut three times the size of the FD method.

The FEM is particularly sensitive to aspect ratios ofrectangular elements and intersection angles of trian-gles. In practical problems, the designer has little con-trol over these factors other than increasing the gridlines (and number of nodes). For example, Elements A,B, and C of Fig. 6.2 may not give reliable computedresults. This deficiency may be overcome for that ex-ample by adding grid lines, but this is at the expense ofincreasing user input and a rapid increase in the size ofthe stiffness matrix since it increases at three times thenumber of nodes.

6.5-Column loadsColumns apply axial loads, shears, and moments to

a mat. Generally the shear is neglected since its effect

would be to slide the foundation laterally. There couldalso be a tendency to compress the mat between twocolumns; however, the AEc/Z term is so large that thismovement is negligible and can be ignored.

There is no practical way to incorporate horizontalloads into the mat, except at the expense of additionalcomputations. For example, the FGM would require sixdegree-of-freedom nodes to allow this. This representsprohibitive effort for the results-except for pile caps,where some (or all) of the horizontal shear force maybe assumed to be carried by the piles.

The mat may be gridded through the column centers(Fig. 6.6) with any of the methods. If column fixity,zero rotation, at the four column nodes of Fig. 6-6(b)is modeled, the grid lines must be taken through thecolumn faces for the FGM or FEM.

a. Gridding through a column centeras commonly done (typically column 1

-I of Fig. 6-2.

b.-P

3

- PO/4

3

C.

K - Po/LlL2

pl- cbK

P2 - dbK

P3 - daK

p4- CaK

Gridding through columnfaces when column is “fixed”with boundary conditionsinput of Qx = Q = 0 asshown. Y

Pro-rating a column axial load (ormoment to adjacent nodes when gridlines are not on column.

Fig. 6.6--Procedures for modeling columns and column loads into one of the dis-crete (finite) element procedures

If the column lies within the grid spacing, the adja-cent nodes are loaded with a portion of the load. Thesimple beam analogy shown in Fig. 6.6(c) is sufficientlyaccurate and may be used for both axial load and col-umn moments.

6.6-SymmetryConsideration should be given to any mat symmetry

to reduce the total number of nodes to be analyzed.This is critical in the FGM and FEM in particular since I

there are three equations per node so that a mat with400 or more nodes will require substantial memory andmay require special matrix reduction routines whichblock the stiffness matrix to and from a disk file.

With symmetry, perhaps only one-quarter or one-alf of the mat will uire modeling. If there is sym-

metry only for selec load cases and not for others,nothing is gained by attempting to analyze some loadcases with a reduced mat plan. Fig. 6.7 illustrates theuse of symmetry to reduce a simple plate problem byapproximately one-quarter. Success in using symmetryto reduce the problem size depends very heavily onbeing able to recognize and input the correct nodalboundary conditions along the axes of symmetry.

supported plate with a (b) concentrated load at the

FEM or FGM model taking advantage of platesymmetry (both load and geometry). For this

giving matrix size shown, model user must recognize the following boun-dary restraints:

For At Nodes =

aBZ 1, 2, 3, 4, 5, 6, 11, 16, 21

@; 1, 2, 3, 4, 5, 10, 15, 20, 25

1, 6, 11, 16, 21, 22, 23, 24, 25

Theereduced model gives: NP = 3 x 25 = 75bandwidth = 3 x 6 = 18matrix size = 75 x 18 = 1350 words

Note that user must recognize the correctboundary restraints as incorrectly identifiedones will give a solution for the model used.

A careful study of whether symmetry applies shouldbe made prior to any modeling, since a substantialamount of engineering and programming effort is oftenrequired in developing the element data.

A major factor where computer memory is adequateis whether there is sufficient advantage in utilization ofany symmetry to reduce the problem size. Some sav-ings of computer resources are offset with engineeringtime of identification and input of the extra boundary

conditions. It is very easy to make an error, and care-ful inspection of the output displacement matrix is nec-essary.

6.7-Node coupling of soil effectsWhen the mat is interfaced with the ground, the soil

effect is concentrated at the grid node for computeranalysis. The most common soil-to-node concentrationeffect produces the so-called Winkler foundation. Thatis, the modulus of subgrade reaction is used and theconcentration is simply

K = Contributory area X ks (6-7)

This is shown for several element shapes in Fig. 6.8.Since the concentration is a product of area and ks, theresult has the units of a spring and is commonly calleda soil spring.

Strictly, nodes might be coupled [Fig. 6.9(b)], butthis is seldom done, If a Winkler foundation is used,the soil spring K direetly adds (superposition) to every

mthird diagonal term of the stiffness matrix,very easy to model excessive displacements into

g itsoil


C G Grid lines dividing

6 2irregular shaped mat

N I

B Fc

e-l

(

Compute spring constants Ki as follows:

Let the area of any element such as A, B, C 1 . . . . = A, B, C....Let the modulus of subgrade reaction within any element be kA, kg, etc.

Then: KT = AkA/3 + BkB/4

K2 = 6kB/4 + CkC/4 (typical side spring)

K3 = CkC/4 (typical corner

K4 = AkA/3 + EkE/4 + DkD/3

K5 = AkA/3 + (BkB + Ek E + FkF)/4

K6 = (BkB + CkC + FkF + Gkti)/4

spring)

(typical interior spring)

A computer routine can develop corner, side and interior springs easily butis substantially more difficult for springs with contributions from triangleelements.

A good computer program should be able to compute the typical corner, side andinterior springs and allow the user to input by hand springs such as for nodes1, 4, and 8 above.

Fig. 6.8-Computation of uncoupled Winkler-type soil node springs

or mat-soil separation at nodes. This allows reuse of thestiffness matrix by removing the spring as necessary onsubsequent cycles. Since the elastic parameters E, and ,4,usually increase with depth from overburden and pre-consolidation it appears this may lessen the effect ofignoring coupling (Christian 1976).

(a) Uncoupled springs

(b) Coupled springs

Fig. 6.9-Soil springs in X-Z plane

If coupling were undertaken, the equations to beprogrammed become much more complicated and, ad-ditionally, fractions of the soil concentration appear inoff-diagonal terms of the stiffness matrix. If nonline-arity is allowed (soil separation or excessive deforma-tion) the stiffness matrix requires substantial modifica-tions and it becomes about as easy to completely re-build it on subsequent cycles.

The most direct and obvious effect of coupling ver-sus noncoupling is that this analysis of a uniformlyloaded mat (an oil tank base) will produce a dishingprofile across the slab with coupling and a constantsettlement profile without coupling. Boussinesq theoryindicates a dishing profile is correct.

It is possible to indirectly allow for coupling as fol-lows (Bowles 1984 and Fig. 6.10):

1. Obtain a vertical pressure profile for the mat us-ing any accepted procedure. The Newmark (1935)method in Bowles (1984) might be programmed or the

pressure bulbs in Bowles (1982) might be used. This isdone at sufficient points so that the mat plan can besubsequently zoned with different values of k,. Use aconvenient influence depth of 38 to 5B, subject to theapproval of the geotechnical engineer. Only slightly

ANALYSIS AND DESIGN 0 FOOTINGS AND MATS 336.2R-17

P

2 has 4 areas with 2 contributions

from

Point 3 has 4 contributing areas

from 3-C-D-E-3

Possible zones are as shown with outer zone using ks furnished

ther zones are bounded by a-b-c-d and e-f-g-h; inner zone is e-f-g-h.

EdgeI . l/4 point Center

are rounded from

Table 6-l

Compute average DQ as:

(1.0 ; 0.028 + 0.40 + 0.24 + 0.146 + 0.095 + 0.066 + 0.048 + 0.036)

= O.l93(qo) Using values directly from Table 6-1

For ks = 500: Obtain edge k = 500; zone 1 = 500(.193/.235) = 410.6 (use 410.)Central values= 500(.193/.250) = 386.

Fig. 6. 10-Method of computing coupled ks

elastic system, therefore, a suggested value of 4B maydifferent results are obtained using 3B versus 10B in an

be sufficiently accurate.Table 6.1 provides values at the 5/s points for several

mat plans which can he used directly or with linear in-terpolation for other mat shapes.

2. Numerically integrate the vertical stress profile to obtain the average pressure DO. Designate the edge

- -value as DO,.3. Compute ks at any point i (refer to Fig. 6.10) as

(6-8)


The furnished value of ks is taken for the mat perime-ter zone.

4. Assign values of ksi in the interior as practical for

the gridding scheme used for the mat. When doing thistake into account how the nodal springs will be com-puted (refer to Fig. 6.8 and 6.10).

Table 6.1-Vertical pressure profiles by Newmark’s 1935 method (inBowles 1985) for selected points beneath a foundation as shown

B/L = 1

PRESSURE PROFILE FOR POINTS:DY 1 2

0 .00 B 1 . 0 0 0 1 . 0 0 00 .50B 0 . 4 0 0 0 . 5 3 01 . 0 0 B 0 . 2 4 0 0 . 2 7 81 . 5 0 B 0 . 1 4 6 0 . 1 6 02 . 0 0 B 0 . 0 9 5 0 . 1 0 02 . 5 0 B 0 . 0 6 6 0 . 0 6 83 . 0 0 B 0 . 0 4 8 0 . 0 4 93 . 5 0 B 0 . 0 3 6 0 . 0 3 74 . 0 0 B 0 . 0 2 8 0 . 0 2 8

AVERAGE PRESSURE INCREASEDQ = 0 . 1 9 3 0 . 2 1 7

B/L = 2PRESSURE PROFILE FOR POINTS:DY 1 2

0 . 0 0 B 1 . 0 0 0 1 . 0 0 00 . 5 0 B 0 . 4 0 8 0 . 6 4 81 . 0 0 B 0 . 2 7 0 0 . 3 6 21 . 5 0 B 0 . 1 8 6 0 . 2 2 92 . 0 0 B 0 . 1 3 5 0 . 1 5 72 . 5 0 B 0 . 1 0 1 0 . 1 1 33 . 0 0 B 0 . 0 7 8 0 . 0 8 53 . 5 0 B 0 . 0 6 1 0 . 0 6 64 . 0 0 B 0 . 0 4 9 0 . 0 5 2


B/L = 3PRESSUREDY

0 . 0 0 B0 . 5 0 B1 . 0 0 B1 . 5 0 B2 . 0 0 B2 . 5 0 B3 . 0 0 B3 . 5 0 B4 . 0 0 B

PROFILE FOR POINTS:1 21 . 0 0 0 1 . 0 0 00 . 4 0 9 0 . 7 1 80 . 2 7 4 0 . 4 0 80 . 1 9 5 0 . 2 6 20 . 1 4 7 0 . l 8 50 . 1 1 5 0 . 1 3 80 .092 0 . 1 0 70 . 0 7 5 0 . 0 8 50 .062 0 . 0 6 9


3 4 51 . 0 0 0 1 . 0 0 0 1 . 0 0 00 . 6 2 8 0 . 6 8 3 0 . 7 0 10 . 3 0 9 0 . 3 2 9 0 . 3 3 60 . 1 7 0 0 . 1 7 7 0 . 1 7 90 . 1 0 5 0 . 1 0 7 0 . 1 0 80 . 0 7 0 0 . 0 7 1 0 . 0 7 20 . 0 5 0 0 . 0 5 0 0 .0510 . 0 3 7 0 . 0 3 8 0 . 0 3 80 . 0 2 9 0 . 0 2 9 0 . 0 2 9

0 . 2 3 5 0 . 2 4 6 0 . 2 5 0

3 4 51 . 0 0 0 1 . 0 0 0 1 . 0 0 00 . 7 5 7 0 . 7 9 2 0 . 8 0 00 .431 0 . 4 6 9 0 .4810 .263 0 . 2 8 5 0 .2930 . 1 7 5 0 . 1 8 6 0 . 1 9 00 .123 0 . 1 2 9 0.1310 .090 0 . 0 9 4 0.0950 .069 0 .071 0 .0720 . 0 5 4 0 . 0 5 6 0 .056

0 . 3 0 4 0 . 3 1 9

41 . 0 0 00 . 8 1 10 . 5 1 70 . 3 4 00 . 2 3 50 . 1 7 00 . 1 2 70 . 0 9 80 . 0 7 8

0 . 3 5 5

0 . 3 2 4

31 . 0 0 00 . 7 9 60 .4860 .3120 .2160 .1570 .1190 .0930 .075

51 . 0 0 00 . 8 1 40 . 5 2 50 . 3 4 80 . 2 4 10 . 1 7 40 . 1 3 00 . 1 0 00 . 0 7 9

0 . 3 4 0 0 . 3 5 9

I

t

l.OB

Y


6.8-Consolidation settlementConsolidation settlement and recompression of heave

can be incorporated into the mat analysis in an approx-imate manner as follows:

1. Compute the estimated consolidation settlementsZJ H,, at the several points as used for the couplinganalysis. This must also be done for the edge.

2. Make a basic mat analysis (including coupling ifthis is going to be the method used). Inspect output foredge pressures and obtain a best estimate of the edge-pressure average. Also obtain node displacements AH,and estimate node consolidation displacements AH,,.The revised k'si can be computed from the basic defini-tions of ks

4, = kst (&Y,j and k'st = q,/(AH, + AH,.,)

to obtain

k'st = ksi (AH,)/(AH, + AH,,) (6-9)

3. Run the mat analysis with these new zoned valuesof k'si Output will include effect of consolidation set-tlement. Edge nodes should compute a displacementapproximately equal to the computed consolidationsettlement from Step 2.

The problem may be cycled several times since theconsolidation settlement computed depends on the matcontact pressure, which may change as computationsprogress.

Since the consolidation settlement is time-dependentit may be appropriate (depending on time for consoli-dation) to use a reduced value of Ec for the concrete toallow for creep effects,

6.9-Edge springs for matsAn alternative to using the Boussinesq coupling pro-

cedure is to double the exterior edge springs (but notany along edges of symmetry where a part of the mat isanalyzed). This doubling will include some coupling ef-fect and account somewhat for the higher theoreticaledge pressures claimed to develop in cohesive soils be-neath rigid plates. Doubling of springs for a plate witha single column (as a spread footing) tends to givecomputed differences of less than 10 percent for mo-ments at the column faces.

Doubling of springs for mats can give differences be-tween computer-generated values as much as 20 per-cent larger with the higher values obtained from dou-bling. A Committee comparison using the FGM and themat given by Shukla (1984) showed that doubling theedge springs produced computed moments about 18 to25 percent higher and that the larger moments gave abetter comparison to the values computed by hand inthat reference.

It should be noted, however, that doubling the edgesprings tends towards the same end result as using theBoussinesq coupling procedure, since the reduction ofthe interior values of ks for a mat is similar to increas-ing the soil resistance at the edge.

In order of increasing difficulty, the suggested pro-cedures are:

1. Using direct area contributions for all springs(common practice but not correct, as stated previ-ously).

2. Doubling the edge springs (not strictly correct, butnot very difficult to program a routine in a mat pro-gram).

3. Using the Boussinesq coupling of Section 6.7. Thisis the most accurate of the approximate procedures, butis also more difficult. Some difficulty can be reducedby programming a routine for most of the mechanicalsteps.

4. Using Methods 2 and 3 in combination.

6.10-Computer outputThe computer program for any of the three methods

(finite difference [FD], finite grid method [FGM], orfinite element method [FEM] should output:

1. X and Y bending moment summations at all nodesso that a statics check can be quickly made at pointswhere this summation should be zero.

2. The displacement matrix. This should be routinelychecked for any symmetry and points of known dis-placements (if any).

3. Nodal soil reactions (forces) computed as

Force = Node spring x Displacement (6-10)

The node forces should be computed, summed, andcompared to the sum of the vertical input forces (col-umn loads and mat weight). Routinely check this towithin computer roundoff.

4. Nodal soil pressures computed as:

4, = k, x Displacement (6-11)

The node pressures q should not exceed the value rec-ommended by the geotechnical engineer. If the pro-gram considers nonlinear effects, both the node forceand node pressure is limited to some maximum value.Verify that the program uses this by inspecting the dis-placements larger than the limiting value (and any thatindicate soil separation).

If a program is reasonably complete in providing theabove output, the designer can make a rapid check ofthe computations. A check of the mat data (node co-ordinates, etc.) will usually take much longer; however,the check may identify that some input is bad if thecomputed output is grossly in error. Where errorsmight occur, a user should modify the program to print(on demand) the full stiffness matrix (in blocks so thepages can be cut and pasted) to display the matrix forinspection of symmetry. Since the stiffness matrix is al-ways symmetrical and has no zeroes on the diagonal,this provides a good check on the internal program-ming used to develop the stiffness matrix. While thischeck produces a large amount of paper output, thepaper cost is negligible compared to a design error.


6.11-Two-dimensional or three-dimensionalanalysis

A mat designed as a plate on an elastic foundationusing soil concentrations at the nodes is considered atwo-dimensional analysis (2-D). If the mat is placed onthe soil and the soil is modeled using three-dimensionalsolids. the analysis is termed three-dimensional (3-D).

A 3-D analysis is extremely costly in modeling timeand use of computer resources. At present, computerprograms are not widely available to model the soil asa 3-D continuum and most use has been for nuclearpower plants.

Since any mat analysis is only as good as the soil pa-rameters, it is usually difficult to justify a 5 to 10 per-cent computational difference when the soil parametersmay only be estimates or a range of values. This is par-ticularly true if:

a. 3-D analysis cost is four or five times that of a 2-Danalysis.

b. Design personnel are not particularly familiar withthe program or methodology.

The major appeal of a 3-D analysis is for soil cou-pling. Since coupling can be handled within the preci-sion of the soil response prediction as outlined inSection 6.7, there is little to justify use of a 3-D matanalysis.

6.12-Mat thicknessThe FD and FEM methods imply use of thin plate

theory. While an investigator may question whether a5-ft (1.8 m) mat is a thin plate, a thin plate analysis isgenerally used and is adequate. The FGM methodmakes no assumption on plate thickness but analyzeswhat is given.

A comparison study by Frederick (1957) showed thata plate would have to be quite thick to invalidate the“thin” plate theory model.

6.13-Parametric studiesIt is very difficult for the geotechnical engineer to

provide accurate elastic design parameters for the soil(Es, p, and ks). Recognizing this, the structural designerand geotechnical engineer may do a parametric study,varying the value of ks over a range of one-half the fur-nished value to five or ten times the furnished value.

The results of the parametric study should be re-viewed by the geotechnical engineer during the courseof the design. If no satisfactory solution is found, thenadjustments in the development concept may be appro-priate. Adjustments may include: reducing the weightof the structure (change from reinforced concrete tostructural steel), enlarging the mat in plan, or deepen-ing the mat base to reduce the net applied pressure.Such adjustments should be made only with the con-currence of the geotechnical engineer.

6.14-Mat foundation detailing/constructionTo simplify steel placement, it is common practice to

provide a layer of top and bottom reinforcement in

both directions. As localized flexure requirements dic-tate, additional reinforcement is placed in the samelayer, or as additional layers to add strength to the sec-tion. It is essential that the engineer prepare thoroughdrawings documenting all phases of the reinforcementplacement. Splice locations must be shown and lengthsnoted. Specification of placement sequence is very im-portant. To insure structural integrity, short staggeredlap splices should be used instead of long splices occur-ring at one location. On large jobs, No. 14 and No. 18bars may be used to minimize the number of pieces onthe job and to reduce the quantity of bar layers.

CHAPTER 7-SUMMARYThe recommendations given in Chapters 1 through 6

may be used as a guide in analysis and design of com-bined footings (two or more columns in a line) or matfoundations. These recommendations represent thecollective opinion of the Committee members based onboth practice and a survey of published literature. TheCommittee recognizes that it is impossible to cover allfoundation cases likely to be encountered by a designerand no attempt has been made to do this.

The Committee recognizes that the state of the art incomputerized analyses is substantially ahead of theability of engineers to determine soil properties accu-rately. The use of computer methods is recommendedsince they allow parametric studies so that a range ofpossible (or probable) soil responses can be investi-gated.

The Committee concedes that the Winkler founda-tion model can be improved, but the increased compu-tational complexity is not warranted when soil proper-ties are taken into account. For this reason the simplecoupling procedure outlined here which relies on thewidely used Boussinesq theory will generally be ade-quate if some additional computational refinement iswanted.

Computer analyses are generally more efficient thanhand computations - particularly of the beam-on-elastic-foundation type since interpolation of table orcurve values is eliminated. This statement also recog-nizes the current widespread availability of desk-top tomain-frame computer systems.

Finally, the Committee notes that, with the same careand attention to detail, the real advantage of the com-puter approach is the ability to better capture the ef-fects of irregular geometry and foundation flexibility.The keywords are “with the same care.” The analysisand design of a mat foundation is a problem in soil-structure interaction requiring a close working relation-ship between the structural and geotechnical engineer.Although published values of soil response parametersare available, geotechnical engineering analysis andjudgement are needed to make the values meaningful tothe design.


CHAPTER 8--REFERENCES8.1 -Specified and/or recommended referencesAmerican Concrete Institute318-83 Building Code Requirements for(Revised 1986) Reinforced Concrete

8.2 -Cited referencesACI Committee 336, 1966, “Suggested Design Procedures for

Combined Footings and Mats,” American Concrete Institute. De-troit, 13 pp.

Ball. Steven C., and Notch, James S., May 1984, “ComputerAnalysis/Design of Large Mar Foundations,” Journal of StructuralEngineering, ASCE, V. 110, No. 5, pp. 1180-1196.

Banavalkar, P. V.. and Ulrich, E. J., Jr., 1984, “Republic BankCenter: Structural and Geotechnical Features,” International Con-ference on Tall Buildings, Singapore, Oct.

Bowles, Joseph E., 1974, Analytical and Computer Methods inFoundation Engineering, McGraw-Hill Book Co., New York, pp.154, 155, and 168-170.

Bowles, Joseph E.. 1975, “Spread Footings,” Foundation Engi-neering Handbook, Van Nostrand Reinhold Co., New York. Chap-ter 15, pp. 490-491.

Bowles. Joseph E., 1976, “Mat Foundations” and “ComputerAnalysis of Mat Foundations,” Proceedings, Short Course-Seminaron Analysis and Design of Building Foundations (Lehigh Univer-sity), Envo Press, Lehigh Valley, pp. 209-232 and 233-256.

Bowles, Joseph E., 1982. Foundation Analysis and Design, 3rdEdition, McGraw-Hill Book Co.. New York, 800 pp.

Bowles. Joseph E.. 1983, “Pile Cap Analysis,” Proceedings, 8thConference on Electronic Computation (Houston), American Societyof Civil Engineers, New York, pp. 102-113.

Bowles, Joseph E.. 1984. Physical and Geotechnical Properties ofSoil, 2nd Edition, McGraw-Hill Book Co., New York, 1984, Chap-ter IO.

Christian, J. T., 1976, “ Soil-Foundation-Structure Interaction.”Proceedings, Short Course-Seminar on Analysis and Design ofBuilding Foundations (Lehigh University), Envo Press, Lehigh Val-ley. pp. 149-179.

Collatz, L., 1960, The Numerical Treatment of Differential Equa-tions, 3rd Edition, Springer-Verlag. Berlin.

Cook, R. D., 1974, Concepts and Applications of Finite ElementAnalysis, John Wiley & Sons, New York, 402 pp.

Department of the Navy, 1982, “Soil Mechanics,” Design ManualNAVFAC DM-7.1. Naval Facilities Engineering Command, Alexan-dria, pp. 7.1-161 - 7.1-241.

Department of the Navy, 1982, “Foundations and Earth Struc-tures,” Design Manual NAVFAC DM-7.2, Naval Facilities Engi-neering Command, Alexandria, pp. 7.2-129 - 7.2-159.

Focht, John A., Jr.; Khan, Fazlur, R.; and Gemeinhardt, J. Peter,May 1978, “Performance of One Shell Plaza Deep Mat Founda-tion,” Proceedings, ASCE, V. 104, GT5, pp. 593-608.

Frederick, Daniel, 1957, “Thick Rectangular Plates on an ElasticFoundation,” Transactions, ASCE, V. 122. pp. 1069-1085. Also, seeDiscussion, p. 1086, which corrects conclusions presented in paper.

Ghali, A., and Neville, A. M.. 1972, Structural Analysis: A Uni-fied Classical and Matrix Approach. Intext Educational Publishers(Now Harper and Row), Chapters 17-20.

Haddadin, M. J.. 1971. “Analysis and Design of Foundation Matsand Combined Footings-Program User’s Manual,” Portland Ce-ment Association, Skokie. 55 pp.

Hetenyi, M., 1946. Beams on Elastic Foundation. University ofMichigan Press. Ann Arbor, pp. 100-106.

Kramrisch, Fritz, and Rogers, Paul, Oct. 1961. “Simplified Designof Combined Footings,” Proceedings. ASCE, V. 88, SM5, pp. 19-44.

Kramrisch, Fritz, 1984, “Footings,” Handbook of Concrete Engi-neering, 2nd Edition. Mark Fintel. Editor, Van Nostrand ReinholdCo., New York, pp. 139-168.

Meyerhof, G. G., June 1953. “Some Recent Foundation Researchand its Application to Design,” The Structural Engineer (London), V.31, No. 6.

Newmark, N. M., 1935. “Simplified Computation of VerticalPressures in Elastic Foundations,” Experiment Station Circular No.24, University of Illinois, Urbana.

Schleicher, Ferdinand, 1926, Kreisplatten auf elastischer Unter-lage, Jules Springer, Berlin.

Scott, R. F., 1981, Foundation Analysis, Prentice-Hall, Inc.. En-glewood Cliffs, pp. 150-197.

Shukla, Shyam N., Sept.-Oct. 1984, “A Simplified Method forDesign of Mats on Elastic Foundations,” ACI JOURNAL, Proceed-ings V. 81. No. 5, pp. 469-475.

Sowers, G. F.. 1977, “Foundation Modulus for a Mat on Sand,”Preprint No. 29371. American Society of Civil Engineers, New York.

Teng. Wayne Chi-Yu, 1962. Foundation Design, Prentice-Hall,Inc., Englewood Cliffs, pp. 174-191.

Terzaghi. K., Dec. 1955, “Evaluation of Coefficients of SubgradeReaction,” Geotechnique (London), V. 5, No. 4, pp. 297-326.

Timoshenko, S., and Woinowsky-Krieger, S.. 1959, Theory ofPlates and Shells, 2nd Edition, McGraw-Hill Book Co., New York,pp. 259-281.

Ulrich, E.J., 1988, “Geotechnical Considerations in Mat Founda-tion Design,” Structural and Geotechnical Lecture Series, Illinoissection, ASCE, Chicago.

Ulrich, E. J., and Focht, J. A., 1982. “First City National Bankto Texas Commerce Tower: Lessons Learned,” Preprint No. 82-532,ASCE National Convention (New Orleans), American Society of CivilEngineers, New York.

U.S. Department of the Interior, 1960. “Moments and Reactionsfor Rectangular Plates,” Engineering Monograph No. 27. Bureau ofReclamation, Denver.

Winterkorn, Hans F., and Fang, Hsai-Yang. 1975, FoundationEngineering Handbook, Van Nostrand Reinhold Co., New York, pp.481-555.

Zienkiewicz. O. C., 1977, The Finite Element Method, 3rd Edi-tion, McGraw-Hill Book Co., New York, 808 pp.

This report was submitted to letter ballot of the committee and was ap-proved according to ACI balloting procedures.

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