design of mat foundations - geoestahban.ir

Design of Mat Foundations

Mahdi Mokhberi

Islamic Azad University

Foundation Analysis and Design

Type of Mat Foundation

Hemsley (2000)

Rigid vs. Non-rigid Mat

Coduto (2001)

the mat and their influence of bearing pressuredistribution.

mat deformations and stresses

than rigid methods, because of soil-structure

Nonrigid methods consider the deformation of

These methods produce more accurate values of

These methods are more difficult to implement

interaction

Non-rigid Methods

Non-rigid Methods

●

●

●

Pseudo-Coupled Method

Coefficient of Subgrade ReactionWinkler MethodsCoupled Method

●

●

●

Multiple-Parameter MethodFiniteelement Method

Coefficient of Subgrade Reaction

●

both the soil andthe foundation have deformationcharacteristics.

●or non-linear (especially in the case of the soils)

●

quantifier in the coefficient of subgrade reaction,

Nonrigid methods must take into account that

These deformation characteristics can be either linear

The deformation characteristics of the soil are

or subgrade modulus, which is similar to themodulus of elasticity for unidirectionaldeformation

the mat and their influence of bearing pressuredistribution.

mat deformations and stresses

than rigid methods because of soil-structure

Nonrigid methods consider the deformation of

These methods produce more accurate values of

These methods are more difficult to implement

interaction

Non-rigid Methods

Modulus of Subgrade Reaction,k

Modulus of Subgrade Reactionp

δ

1k

k1

●Nonlinear●Dependent on size of plate●Typicalk values based ona plate size of 300mm●Reduction needed fora largely loaded area

Modulus of Subgarde Reaction

Dry or moist sandLooseMediumDense

Submerged sandLooseMediumDense

Soillb/in kN/m

ks3 3

6.5- 8.020- 2555- 65

3.5- 5.012- 1832- 45

1800- 22005500- 700015,000- 18,000

1000- 14003500- 45009000- 12,000


reactionPlate load test for coefficient of subgrade

Coefficient of subgrade Reaction

larger mats

Coefficient ofSubgradeReaction

●

settlement produce more compression in thesprings

●

structural loads plus the weight of the mat

Portions of the mat that experience more

Sum of these springs must equal the applied

●the interaction between soil and foundation wasThe earliest use of these "springs" to represent

done by Winkler in 1867; the model is thusreferred to as the Winkler method

●"beam on elastic foundation," thus sometimes itis called the "beam on elastic foundation" method

●

The one-dimensional representation of this is a

Mat foundations representa two-dimensionalapplication of the Winkler method

Winkler Methods

Beams on Elastic Foundations

Equations of the beam on elastic foundation

For the vertical condition of equilibrium of a differential beam element:


For its moment condition of equilibrium, we get:


By combining results (1) and (2), we obtain the

equilibrium equation of a beam element on an

elastic foundation:

Qʹ= - q+ r (1)

Q= Mʹ (2)

(1) and (2) ⇒ M′′ = −q + r


Using the elementary theory of bending and linearelastic beams. The most essential other equations arethen the relationship between rotation and deflection:

ϕ = v′ (4)the relationship between curvature and deflection:

κ = −v′′ (5 )the relationship between bending moment and curvature:

M = EIκ (6)and the relationship between shear force and bending moment

Q = M′ (7)


The (already known) relationship between bending moment anddeflection follows from expressions (5) and (6).

(11)By inserting this relationship and the relationship (8) between thedeflection and the foundation pressure into equilibrium equationwe get

(12)This is the differential equation of the beam on elastic foundation. In the case of a uniform (EI = constant) beam, it gets the form

(13)This can be expressed as

(14)Where

(15)

Beam on Springs


Deflection,δ

Moment,M

Shear force,T

P

Application to Spread Footings

haviour

Non-Linear Characteristics ofSoil Deformation

●

must make a linear approximation to use theLoad-settlement vurves are not really linear; we

Winkler model●

mat underlain by a perfectly uniform soil willuniformly settle into the soil.

●

Winkler model assumes that a uniformly loaded

Actual data show that such a mat-soil interaction willdeflect in the centre more than the edges

●Schmertmann's) to determine settlementthis is one reason why we uhe other methods (such as

Limitations of Winkler Method

pressure on one part of the mat influences bothSoil springs do not act independently. Bearing

the "spring" under it and those surrounding it(due to leteral earth pressure)

interaction between the soil and the mat

largest problem with the Winkler model

No single value of k truly represents the

The independent spring problem is in reality the

s


Coupled Method

●

additional springs as shown below, is moreIdeally the coupled method, which uses

accurate then the Winkler method●

selecting the values of k for the coupling springsThe problem with the coupled method comes in

s

in the Winkler method and the difficulties of theAn attempt to overcome both the lack of coupling

coupling springs

(like Winkler springs), but have different kvalues depending upon their location on the mat

Does so by using sprinns that act independentlys

Most commertial mat design software uses theWinkler method; thus, pseudo-coupled methodscan be used with these packages for moreconservative and accurate results


ImplementationDivide the mat into two or more concentric zones

The innermost zone should be about half as wide and half aslong as the mat

These should progressively increase from the centre

●

Assign a kvalue to each zone●

The outermost zone k should be about twice as large as theinnermost zone

the Winklermethod

s

●

Evaluate the shears, moments and deformations using

Adjust mat thickness and reinforcement to satisfystrength and serviceability requirements

s


linear springs of the Winkler method with sprinnsThis method replaces the independently-acting

and other mechanical elements

●

distributing the k values in the pseudo-coupledmethod; should be more accurate

The additionalelements define the coupling effectsMethod bypasses the guesswork involved in

s

packages and thus is not routinely used on designprojects

Method has not been implemented into software

Multiple-Parameter Method

dimensional wayModels the entire soil-mat systemin a three-

●

In theory, should be the most accurate methodeMethed is not yet practical because

Requires largeamount of computing power to performDifficult to determine soil properties in such a way asto justify the ptecision of the analysis, espetially whensoil parameters are highly variable

addressedwill become more in use as these problems are

Finite Element Method

analysisFinite element method is used for structural

structures with springs connected at the nodes ofthe elements

Mat is modelled in a similar way to other plate

Mat is loaded with column loads, apllied lineloads, applied area loads, and mat weight

(conservative)●

Usually superstructure stiffness is not considered

Can be done but is rarely performed in practice

Finite Element Method

Finite Element method

Determining the Coefficient of Subgrade Reaction

Not a straightforward process due to:●

wide mat

Width of the loaded area; wide mat will settle more

● Not astraightforwardprocess due to:

●Loaded area:stresses beneathlong, narrowloaded area is

Shape of the

different fromthose belowsquare loadedareas


● Not aStraight forwardprocess due to:

●loaded areabelow theground surface

Change in stress

Depth of the

●in the soil due toq in a smallerpercentage of theinitial stress atgreater depths


Not a straightforward process due to:

●To model the soil accurately, k needs to be larger near theedges ofthe mat and smaller near the centre

●

The position of the mat

●

Time

With compressible (and especially cohesive compressiblesoils) mat seetlement in a process which mat take severalyearsMay be necessary to consider both shortand long term cases

s

●

●●

non-existentNon-linear nature of soil deformation makes uniquevalue of k

s


● Methods used to determine coefficient●

Test results must be adjusted between the shape of theloading plate and the actual shape of the foundation

Adjestment must also be made for the size of the plate vs.the side of the foundation, and the influence of size on the

Plate load tests●

●

depth of soil stress

Attempts to make accurate adjustments have not been verysuccessful to dane

●

Relationships developed are too limited in their applicationpossibilities

●

Derivedrelationships between k and E●

s s


Methods used to determine coeffitient

●consolidation theory, Schmertmann's method, etc., andexpress the results in a k value

If using a pseudo-coupled value, use valees of k in the centreof the mat which are half those along the perimeter

Use settlement techniques such as Terzaghies

s

●

This methodology has the potential of eliminating theproblems described earlier while at the same time yieldingvalues of k which then can be used in astructural analysis ofthe mat with some degree of confidence

s

●

s


●long mat foundation

●

●settlement analysis method

GivenStructure to be supported on a 30 m wide by 50 m

Average bearing pressure is 120 kPaAverage settlement determined δ= 30 mm using

●

FindDesign values of ks used in a pseduo-coupled analysis

Example of Determining the Coefficient of Subgrade Reaction

Solution


● Solution


●Evaluate these requirements using factored loads and LRFDdesign methods

Mat must have sufficient thickness T and reinforcement to

Structural design requires two analysesStrength

●

●safety resist these loads

T should be large enough so that no shear reinforcement isrequired

●

●

●

ServicabilityEvaluate using unfactored loads for excessive deformation asplaces of concentrated loads, such as columns, soil non-uniformities, mat non-uniformities, etc.

This is the equivalent of a differential settlement analysis

mat must be made thicker if this is a problem●

●

Structural Design of Mats

●Have fallen out of favour

closed form solutionsOnce popular; however, with the advent of computers,

Finite difference methodsFinite element methods

● Springvalues as computed in the example can then beused in finite element analysis

●foundation to sag in the centre, which is what weactually see in foundations

the stiffer springs at the edges will encourage the


Total settlement●

compute total settlement; this should be done usingothermethods

"Bed of springs" solution should not be used to

Bearing capacity●

capacity probleme●

to be watched●

foundations, including presumptive bearing capacities

Mat foundations generally do not have bearing

With undrained silts and clays, bearing capacity needs

Methods for spread footings can be used with mat

Other Considerations in Mat Foundations

design of mat foundations - geoestahban.ir

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