The Foundation Engineering Handbook Chapter 4

The Foundation Engineering Handbook Chapter 4 Geotechnical earthquake engineering can be defined as that subspecialty within the field of geotechnical engineering that deals with the design and construction of projects in order to resist the effects of earthquakes. Geotechnical earthquake engineering requires an understanding of basic geotechnical principles as well as an understanding of geology, seismology, and earthquake engineering. In a broad sense, seismology can be defined as the study of earthquakes. This would include the internal behavior of the earth and the nature of seismic waves generated by the earthquake.

Geotechnical Design of Combined Spread Footings

Manjriker Gunaratne CONTENTS

4.3 Conventional Design of Rectangular Combined Footings 148

4.6.1 Coefficient of Vertical Subgrade Reaction 1584.6.2 Analysis and Design of Rectangular Combined Footings 1604.6.3 Design of Rectangular Combined Footings Based on Beams on Elastic

Foundations

161

4.6.4 Analysis of Mat Footings Based on Slabs on Elastic Foundations 1664.7 Structural Matrix Analysis Method for Design of Flexible Combined Footings 1694.8 Finite Difference Method of Flexible Mat Footing Design 172

many columns would distribute the load and reduce the bearing stress In addition, thismodification will also reduce the footing settlement.

2 When the exterior columns or walls of a heavy structure are in close proximity to theproperty line or other structures, the designer would not have the

freedom to utilize the area required to design an isolated spread footing In such cases, anyadjoining interior columns can be incorporated to design a combined footing.

3 Isolated spread footings can become unstable in the presence of unexpected lateral forces.The stability of such footings can be increased by tying them to other footings in the

vicinity

4 When adjoining columns are founded on soils with significantly different compressibilityproperties, one would anticipate undesirable differential settlements These settlements can

be minimized by a common combined footing

5 If the superstructure consists of a multitude of column loads, designing a single monolithic

mat or raft footing that supports the entire system of structural columns would be more

economical This is especially the case when the total area required by isolated spreadfootings for the individual columns is greater than 50% of the entire area of the columnplan (blueprint)

4.2 Design Criteria

Two distinct design philosophies are found in the current practice with respect to design ofcombined footings They are: (1) conventional or the rigid method and (2) beams or slabs onelastic foundation or the flexible method Of these, in the conventional design method, oneassumes that the footing is infinitely rigid compared to the foundation soils and that the

contact pressure distribution at the foundation-soil interface is uniform (in the absence of anyeccentricity) or planar (with eccentricity) In other words, the deflection undergone by thefooting is considered to be unrelated to the contact pressure distribution This assumption can

be justified in the case of a spread footing with limited dimensions or a stiff footing founded

on a compressible soil Therefore, the conventional method appears to be inadequate forfootings with larger dimensions relative to their thickness and in the case of footings that areflexible when compared to the foundation medium Although these drawbacks are addressed

in such cases by the flexible footing

TABLE 4.1

From English To SI Multiply by Quantity From SI To English Multiply by

ft-kip kJoule 1.356 kJoule ft-kip 0.7375

Blows/ft Blows/m 3.2808 Blow count blows/m blows/ft 0.3048

design method to some extent, one has to still assume that the soil behaves as an elastic

foundation under the flexible footing

According to ACI (1966), for relatively uniform column loads which do not vary more than20% between adjacent columns and relatively uniform column spacing, mat footings may be

considered as rigid footings if the column spacing is less than 1.75/βor when the mat is supporting a rigid superstructure The characteristic coefficient of the elastic foundation, β , is

defined by Equation (4.14)

4.2.1 Conventional Design Method

Three distinct design criteria are used in this approach

4.2.1.1 Eccentricity Criterion

An effort must be made to prevent the combined footing from having an eccentricity, whichcould cause tilting and the need for a relatively high structural footing rigidity In order toassure this condition, the footing must be dimensioned so that its centroid coincides with theresultant of the structural loads Thus, if the coordinates of the centroid of the footing are

and the locations of the column loads P i are (x i , y i ) (with respect to a local Cartesian

coordinate system) (Figure 4.1), then the following conditions must be ensured:

(4.1)

(4.2)

4.2.1.2 Bearing Capacity Criterion

The allowable stress design (ASD) can be stated as follows:

(4.3)

where qultis the ultimate bearing capacity of the foundation (kN/m2, kPa, or ksf), P is the structural load (kN or kips), A is the footing area (m2or ft2), and F is an appropriate safety

FIGURE 4.1

Combined footing.

factor that accounts for the uncertainties involved in the determination of structural loads (P) and the ultimate bearing capacity (qult).

One can typically use any one of the bearing capacity equations found inSection 3.2toevaluate the bearing capacity of the foundation For conversion of units seeTable 4.1

4.2.1.3 Settlement Criterion

The designer has to also ensure that the combined footing does not undergo either excessivetotal settlement or differential settlement within the footing Excessive settlement of thefoundation generally occurs due to irreversible compressive deformation taking place

immediately or with time Excessive time-dependent settlement occurs in saturated

compressible clays where one will receive advanced warning through cracking, tilting, andother signs of building distress Significant immediate settlement can also occur in loosesands or compressible clays and silts Settlements can be determined based on the methodsdescribed inSection 3.3

4.3 Conventional Design of Rectangular Combined Footings

When it is practical or economical to design a single footing to carry two column loads, arectangular combined footing (Figure 4.2) can be considered

Example 4.1

Use the conventional or the “rigid” method to design a combined footing for the two

columns shown inFigure 4.2, if the average SPT value of the cohesionless foundation soilassumed to be reasonably homogeneous is about 22

Step 1: Compute ultimate loads A Column A

Footing configuration for Example 4.1

(one could follow the same computational procedure even with a nonzero moment on thesecond column)

Step 2: Determine the footing length

The resultant force on the footing=860+1380 kN=2240 kN (at C)

The resultant moment=174 kN m (at A)

Hence the required length of the rigid footing would be 2(2.54+0.5) m or 6.08 m

This requires an additional end-section next to column B of 1.58m (Figure 4.3a)

Option 2

The other option is to curtail the end-section next to column B to about 0.5 m and design aneccentric footing (Figure 4.3b)

This footing would have a base eccentricity of or 0.9 m

Step 3: Determine the footing width

One can employ the bearing capacity criterion (Equation 4.3) and SPT data to determine

the allowable bearing capacity and then footing width as follows It must be noticed that q ais

Alternatively, if the foundation soil is not necessarily cohesionless, one could use the SPTvalue and Equation (4.5) to determine the allowable bearing capacity for an allowable

settlement of s inches.

On the other hand, if soil investigation data is available in terms of CPT (cone penetrationdata), one could use the correlations presented inChapter 2

Using Equation (4.4), q a=30 (0.45)(22)=300 kPa

Then, one can determine the width of the footing (B) using Equation (4.3) as

Shear: clockwise positive Moment: sagging moments positive

FIGURE 4.3

Design configurations for Example 4.1 : (a) design option 1, (b) loading configuration for design

option 1, and (c) design option 2.

The distributed reaction per unit length (1 m) on the footing can be computed as

w=2240/6.08=368.42 kN/m

When x is measured from the left edge of the footing, the shear and the moments of each

segment of the footing can be found as follows:

In the segment to the left of column A (0<x<0.35 m)

S1=368.42x kN

M1=368.42(x2/2) kN m

Within column A (0.35<x<0.65 m)

(with an inflexion point at x=0.63m).

In between the two columns (0.65<x<4.35 m)

Within column B (4.35<x<4.65m)

The distributed column load would be of intensity 1380/0.3 kN/m Then,

S4=368.42(x)−860−(4600)(x−4.35)

M4=368.42(x2/2)−860(x−0.5)+174−(4600)(x−4.35)2/2kN m

(with an inflexion point at x=4.53m).

Within the end-section right of column B (4.605<x<6.08 m)

S5=368.42(x)−860−1380

M4=368.42(x2/2)−860(x−0.5)+174−1380(x−4.5) kN m

The corresponding shear force and bending moment diagrams are plotted inFigure 4.4

4.4 Conventional Design of Mat Footings

4.4.1 Bearing Capacity of a Mat Footing

One can use Equations (3.1)–(3.6) to proportion a mat footing if the strength parameters of theground are known However, since the most easily obtained empirical strength parameter is

the standard penetration blow count, N, an expression is available that uses N to obtain the

bearing capacity of a mat footing on a granular subgrade (Bowles, 2002) This is expressed asfollows: For 0≤Df≤B and B>1.2 m

(4.5)

For B<1.2m

(4.6)

where qn,allis the net allowable bearing capacity in kilopascals, B is the width of the footing, s

is the settlement in millimeters, and Dfis the depth of the footing in meters

Then a modified form of Equation (4.3) has to be used to avoid bearing failure:

Hence the mat can be designed with 0.25 m edge space as shown inFigure 4.5.

For the reinforcement design, one can follow the simple procedure of separating the slabinto a number of strips as shown inFigure 4.5 Then each strip (BCGF inFigure 4.5) can be

considered as an individual beam The uniform soil reaction per unit length (ω) can be

computed as 4000(2.5)/[(5.5)(5.5)]=330.5 kN/m.Figure 4.6indicates the free-body diagram

of the strip BCGF inFigure 4.5

It can be seen from the free-body diagram that the vertical equilibrium of each strip is notsatisfied because the resultant downward load is 2000 kN, as opposed to the resultant upwardload of 1815 kN This discrepancy results from the arbitrary separation of strips at the

midplane between the loads where nonzero shear forces and moments exist In fact,

FIGURE 4.6

Free-body diagram for the strip BCGF in Figure 4.5

FIGURE 4.7

(a) Shear force and (b) bending moment diagrams for Example 4.2

one realizes that the resultant upward shear at the boundaries BF and CG (Figure 4.5)

accounts for the difference, that is, 185 kN However, to obtain shear and moment diagrams

of the strip BCGF, one can simly modify them as indicated in the figure This has been

achieved by reducing the loads by a factor of 0.954 and increasing the reaction by a factor of1.051 The two factors were determined as follows:

For the loads, [(2000+1815)/2]/2000=0.954

For the reaction, 1815/[(2000+1815)/2]=1.051

The resulting shear and moment diagrams are indicated inFigure 4.7

Then, usingFigure 4.7, one can determine the steel reinforcements as well as the matthickness This effort is not repeated here since it is discussed in detail inChapter 5

4.5 Settlement of Mat Footings

The settlement of mat footings can also be estimated using the methods that were outlined inSection 3.3and, assuming that they impart stresses on the ground in a manner similar

Page 155

FIGURE 4.8

Immediate settlement computation for mat footings.

to that of spread footings An example of the estimation of the immediate settlement under amat footing is provided below (Figure 4.8)

4.5.1 Immediate Settlement

The following expression (Timoshenko and Goodier, 1951) based on the theory of elasticity

can be used to estimate the corner settlement of a rectangular footing with dimensions of L' and B',

(4.8)

where q is the contact stress, B' is the least dimension of the footing, v sis the Poisson ratio of

the soil, and Esis the elastic modulus of the soil Factors I1, I2, and IFare obtained from Table4.2andFigure 4.10, respectively, in terms of the ratios N=H/B' (H=layer thickness), M=L'/B' (L'=other dimension of the footing), and D/B.

The same expression (Equation 4.8) can be used to estimate the settlement of the footing atany point other than the corner by approximate partitioning of the footing as illustrated in thisexample It must be noted that even if the footing is considered as a combination of several

partitions (B' and L'), for determining the settlement of an intermediate (noncorner) location, the depth factor, IF, is applied for the entire footing based on the ratio D/B.

FIGURE 4.9

0.041 0.042 0.042 0.042 0.042 0.042 0.043 0.043 0.043 0.043 0.043 0.3 0.019 0.018 0.018 0.017 0.017 0.016 0.016 0.016 0.016 0.016 0.015

0.055 0.056 0.057 0.057 0.058 0.058 0.059 0.059 0.059 0.059 0.059 0.5 0.049 0.047 0.046 0.045 0.044 0.043 0.042 0.041 0.041 0.040 0.040

0.074 0.076 0.077 0.079 0.080 0.081 0.081 0.082 0.083 0.083 0.084 0.7 0.085 0.083 0.081 0.079 0.078 0.076 0.075 0.074 0.073 0.072 0.072

0.082 0.085 0.088 0.090 0.092 0.093 0.095 0.096 0.097 0.098 0.098 0.9 0.123 0.121 0.119 0.117 0.115 0.113 0.112 0.110 0.109 0.108 0.107

value of 10, Esis approximately given by 500(N+15) kPa or 12.5 MPa (Table 1.6) A

Poisson’s ratio of 0.33 can also be assumed in normally consolidated sand (Table 1.4)

Then the uniformly distributed contact stress=4000/(5.5)2=132.23 kPa

D/B for the entire footing=0.5/5.5=0.09

FromFigure 4.10, for L/B=1, IF=0.85

0.043 0.043 0.044 0.044 0.044 0.044 0.044 0.044 0.044 0.044 0.044 0.044 0.3 0.015 0.014 0.014 0.014 0.014 0.014 0.014 0.014 0.014 0.014 0.014 0.014

0.060 0.061 0.061 0.061 0.061 0.061 0.061 0.061 0.061 0.061 0.061 0.061 0.5 0.038 0.037 0.036 0.036 0.036 0.036 0.036 0.036 0.036 0.036 0.036 0.036

0.085 0.087 0.087 0.088 0.088 0.088 0.088 0.088 0.088 0.088 0.088 0.088 0.7 0.069 0.066 0.065 0.065 0.064 0.064 0.064 0.064 0.064 0.064 0.063 0.063

0.101 0.104 0.105 0.106 0.106 0.106 0.106 0.107 0.107 0.107 0.107 0.107 0.9 0.103 0.099 0.097 0.096 0.096 0.095 0.095 0.095 0.095 0.095 0.094 0.094

Plot of the depth influence factor I for Equation (4.8).

Page 158Therefore, the immediate settlement expression (Equation 4.8) can be simplified to:

For the corner settlement

Maximum angular distortion within the footing=(23.596–5.671)/(2)1/2(2.85)(1000) <1/200

It would be safe from any architectural damage

4.6 Design of Flexible Combined Footings

Flexible rectangular combined footings or mat footings are designed based on the principles

of beams and slabs on elastic foundations, respectively In this approach, the foundationmedium is modeled by a series of “elastic” springs characterized by the modulus of vertical

subgrade reaction, ks, and spread in two dimensions First, it is essential to identify this

important empirical soil parameter that is used in a wide variety of designs involving earthenmaterial

4.6.1 Coefficient of Vertical Subgrade Reaction

The coefficient of subgrade reaction is an empirical ratio between the distributed pressure

induced at a point of an elastic medium by a beam or slab and the deflection (w0) undergone

by that point due to the applied pressure, q

k s =q/w0

(4.9)

Egorov (1958) showed that the elastic deformation under a circular area of diameter B

carrying a uniformly distributed load of q is given by

(4.10)

where Es is the elastic modulus of the medium and μsis the Poisson ratio of the medium

(4.11)

foundation soil Typically, the following B values are used in different cases:

(i) Design of a combined footing—the footing width

(ii) Design of a pile—the pile diameter

(iii) Design of a sheet pile or laterally loaded pile—the pile width or pile diameter

Similarly, if one uses plate load test results for evaluating ks, then from a theoretical point of view, it is appropriate to adjust the k sobtained from plate load tests as:

Example 4.4

Estimate the coefficient of vertical subgrade reaction, ks, for a 1 m×1 m footing carrying a

300 kN load, using the data from a plate load test (Figure 4.11) conducted on a sandy soilusing 0.45 m×0.45 m plate

The contact stress on footing=300kN/(1×1)=300 kPa

The modulus of subgrade reaction for the plate=300 kPa/4.3 mm=69,767 kN/m3or 69.77MN/m3

Applying Equation (4.12b)

Alternatively, Equation (3.31) can be used to estimate the settlement of the footing at thesame stress level of 300 kPa

(3.31)

Then, the modulus of subgrade reaction for the footing=300kPa/8.18mm=36.67 MN/m3

4.6.2 Analysis and Design of Rectangular Combined Footings

Based on the definition of k sand the common relationship between the distributed load, shearand moment, the following differential equation that governs the equilibrium of a beam on anelastic foundation can be derived:

(4.13)

In solving the above equation, the most significant parameter associated with the design ofbeams or slabs on elastic foundations turns out to be the characteristic coefficient of the

elastic foundation or the relative stiffness, β , given by the following expression:

For a rectangular beam

(4.14)

where E is the elastic modulus of concrete, ksis the coefficient of subgrade reaction of thefoundation soil usually determined from a plate load test (Section 4.6.1) or Equation (4.10)

(Egorov, 1958), and h is the beam thickness.

One can use the following criteria to determine whether a rectangular combined footingmust be designed based on the rigid method or the flexible beam method: