ENCE 4610 Foundation Analysis and Design

Mat Foundations

Nonrigid Methods
Nonrigid methods consider the deformation of the mat and their influence of bearing pressure distribution. These methods produce more accurate values of mat deformations and stresses These methods are more difficult to implement than rigid methods because of soil-structure interaction

Coefficient of Subgrade Reaction Winkler Methods Coupled Method Pseudo-Coupled Method Multiple-Parameter Method Finite Element Method

Coefficient of Subgrade Reaction

Nonrigid methods must take into account that both the soil and the foundation have deformation characteristics.
These deformation characteristics can be either linear or non-linear (especially in the case of the soils)

The deformation characteristics of the soil are quantified in the coefficient of subgrade reaction, or subgrade modulus, which is similar to the modulus of elasticity for unidirectional deformation

Coefficient of Subgrade Reaction

Definition of Coefficient of Subgrade Reaction

ks = coefficient of subgrade reaction, units of force/length3 (the units are the same as the unit weight, but not the significance!) q = bearing pressure = settlement

q ks =

Coefficient of Subgrade Reaction

Plate load test for coefficient of subgrade reaction

Coefficient of Subgrade Reaction

Application of coefficient of subgrade reaction to larger mats

Coefficient of Subgrade Reaction

Portions of the mat that experience more settlement produce more compression in the springs Sum of these springs must equal the applied structural loads plus the weight of the mat

P + W f u D = qdA = k s dA

Winkler Methods
The earliest use of these "springs" to represent the interaction between soil and foundation was done by Winkler in 1867; the model is thus referred to as the Winkler method The one-dimensional representation of this is a "beam on elastic foundation," thus sometimes it is called the "beam on elastic foundation" method Mat foundations represent a two-dimensional application of the Winkler method

Beams on Elastic Foundations

Application to Spread Footings

Note non-linear behaviour

Limitations of Winkler Method

Load-settlement curves are not really linear; we must make a linear approximation to use the Winkler model Winkler model assumes that a uniformly loaded mat underlain by a perfectly uniform soil will uniformly settle into the soil.
Actual data show that such a mat-soil interaction will deflect in the centre more than the edges This is one reason why we use other methods (such as Schmertmann's or Houghs) to determine settlement

Soil springs do not act independently. Bearing pressure on one part of the mat influences both the "spring" under it and those surrounding it (due to lateral earth pressure) No single value of ks truly represents the interaction between the soil and the mat The independent spring problem is in reality the largest problem with the Winkler model

Coupled Method
Ideally the coupled method, which uses additional springs as shown below, is more accurate than the Winkler method The problem with the coupled method comes in selecting the values of ks for the coupling springs

Multiple-Parameter Method
This method replaces the independently-acting linear springs of the Winkler method with springs and other mechanical elements
The additional elements define the coupling effects

Method bypasses the guesswork involved in distributing the ks values in the pseudo-coupled method; should be more accurate Method has not been implemented into software packages and thus is not routinely used on design projects

Finite Element Method

Models the entire soil-mat system in a threedimensional way In theory, should be the most accurate method Method is not yet practical because
Requires large amount of computing power to perform Difficult to determine soil properties in such a way as to justify the precision of the analysis, especially when soil parameters are highly variable
Finite element method is used for structural analysis Mat is modelled in a similar way to other plate structures with springs connected at the nodes of the elements Mat is loaded with column loads, applied line loads, applied area loads, and mat weight Usually superstructure stiffness is not considered (conservative)
Can be done but is rarely performed in practice

Will become more in use as these problems are addressed

Finite Element Method

Pseudo-Coupled Method
An attempt to overcome both the lack of coupling in the Winkler method and the difficulties of the coupling springs Does so by using springs that act independently (like Winkler springs), but have different ks values depending upon their location on the mat Most commercial mat design software uses the Winkler method; thus, pseudocoupled methods can be used with these packages for more conservative and accurate results Implementation
Divide the mat into two or more concentric zones
The innermost zone should be about half as wide and half as long as the mat

Assign a ks value to each zone

These should progressively increase from the centre The outermost zone ks should be about twice as large as the innermost zone

Evaluate the shears, moments and deformations using the Winkler method Adjust mat thickness and reinforcement to satisfy strength and serviceability requirements

Pseudo-Coupled Method

Difficulties in Determining the Coefficient of Subgrade Reaction

Width of the loaded area; wide mat will settle more than a narrow one because more soil is mobilised by a wide mat Depth of the loaded area below the ground surface
Change in stress in the soil due to q is a smaller percentage of the initial stress at greater depths Shape of the loaded area: stresses beneath long, narrow loaded area is different from those below square loaded areas

The position of the mat

To model the soil accurately, ks needs to be larger near the edges of the mat and smaller near the centre

Time
With compressible (and especially cohesive compressible soils) mat settlement is a process which may take several years May be necessary to consider both short and long term cases

Non-linear nature of soil deformation makes unique value of ks non-existent

Determining the Coefficient of Subgrade Reaction

Methods used to determine coefficient
Plate load tests
Test results must be adjusted between the shape of the loading plate and the actual shape of the foundation Adjustment must also be made for the size of the plate vs. the size of the foundation, and the influence of size on the depth of soil stress Attempts to make accurate adjustments have not been very successful to date

Methods used to determine coefficient

Use settlement techniques such as Terzaghi's consolidation theory, Schmertmann's or Houghs method, etc., and express the results in a ks value

Derived relationships between ks and Es

Relationships developed are too limited in their application possibilities

If using a pseudo-coupled value, use values of ks in the centre of the mat which are half those along the perimeter This methodology has the potential of eliminating the problems described earlier while at the same time yielding values of ks which then can be used in a structural analysis of the mat with some degree of confidence

Example of Determining Coefficient of Subgrade Reaction

Given
Structure to be supported on a 30 m wide by 50 m long mat foundation Average bearing pressure is 120 kPa Average settlement determined = 30 mm using a settlement analysis method

Solution
Compute average ks for entire mat

ks =

120 kPa ks = = 4000 kN / m 3 0.030 m

Find
Design values of ks used in a pseudo-coupled analysis

Example of Determining Coefficient of Subgrade Reaction

Solution
Divide mat into three zones as shown

W L (ks)A (ks)B = 1.5 (ks)A (ks)C = 2 (ks)A

Example of Determining Coefficient of Subgrade Reaction

Solution
AB = (37.5)(22.5) 375 = 469 m AC = (50)(30) - 469 - 375 = 656 m2

Compute the area of each zone 2

AA = (25)(15) = 375 m2

Example of Determining Coefficient of Subgrade Reaction

Solution
Write the averaging equation A (k ) + A (k ) + A (k ) = ( A + A + A )(k ) A s A B s B C s C A B C s avg for the ks values Substitute into (ks)b and (ks)c 375(k s )A + (469)(1.5)(k s )A + (656)(2)(k s )A = 1500(k s )avg the equivalent value of (ks)a 2390(k ) = 1500(k )

(k s )A = 0.627(k s )avg
Compute the design ks values

s A

s avg

(k s )A = (0.627 )(4000 ) = 2510 kN / m 3 (k s )B = (0.627 )(1.5 )(4000 ) = 3765 kN / m 3 (k s )C = (0.627 )(2 )(4000 ) = 5020 kN / m 3

ACI suggests varying ks from its computed value to 5 or 10 times the computed value, then base the structural design on the worst condition

Structural Design of Mats

Structural design requires two analyses
Strength
Evaluate these requirements using factored loads and LRFD design methods Mat must have sufficient thickness T and reinforcement to safety resist these loads T should be large enough so that no shear reinforcement is required Evaluate using unfactored loads for excessive deformation at places 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

Serviceability

Structural Design of Mats

Structural Design of Mats

Structural Design of Mats

Closed form solutions
Once popular; however, with the advent of computers, have fallen out of favour

Finite difference methods Finite element methods

Spring values as computed in the example can then be used in finite element analysis The stiffer springs at the edges will encourage the foundation to sag in the centre, which is what we actually see in foundations

Other Considerations in Mat Foundations

Total settlement
"Bed of springs" solution should not be used to compute total settlement; this should be done using other methods

Bearing capacity
Mat foundations generally do not have bearing capacity problems With undrained silts and clays, bearing capacity needs to be watched Methods for spread footings can be used with mat foundations, including presumptive bearing capacities

Questions