Mat Foundatins

NPTEL - ADVANCED FOUNDATION ENGINEERING-1

Module 5

(Lecture 19)

MAT FOUNDATIONS

Topics

1.1 STRUCTURAL DESIGN OF MAT FOUNDATIONS Conventional Rigid Method

1.2 Approximate Flexible Method

Foundations on Sandy SoilsFoundations on Clays

1.3 Example

1.4 PROBLEMS

STRUCTURAL DESIGN OF MAT FOUNDATIONS

The structural design of mat foundations can be carried out by two conventional methods: the conventional rigid method and the approximate flexible method. Finite difference and finite element methods can also be used, but this section covers only the basic concepts of the first two design methods.

Conventional Rigid Method

The conventional rigid method of mat foundation design can be explained step by step with reference to figure 5.8.

1. Figure 5.8a shows mat dimensions of and columns loads of iCalculate the total column load as ! i [5.24]


2. Determine the pressure on the soil, q, below the mat at points i, by using the equation! [5.25]

Where !! ! QSQIRXSJMRIVXMEEFSYXXLI E\MW! !QSQIRXSJMRIVXMEEFSYXXLI E\MW! QSQIRXSJXLIGSPYQRPSEHWEFSYXXLI E\MW ! ! QSQIRXSJXLIGSPYQRPSEHWEFSYXXLI E\MW ! The load eccentricities, ERH , in the ERH directions can be determined by using ( coordinates:! [5.26]And ! [5.27]Similarly ! [5.28]And ! [5.29]

3. Compare the values of the soil pressures determined in step 2 with the net allowable soil pressure to determine whether EPP RIX .

4. Divide the mat into several strips in x and y directions (see figure 5.8a). Let the width of any strip be .

5. Draw the shear, V, and the moment, M, diagrams for each individual strip (in the xand y directions). For example, the average soil pressure of the bottom strip in the x direction of figure 5.8a is [5.30]


Where ERH ! WSMPTVIWWYVIWEXTSMRW ERH EWHIXIVQMRIHJVSQWXITThe total soil reaction is equal to . Now obtain the total column load on the strip as . The sum of the column loads on the strip will not equal because the shear between the adjacent strips has not been taken into account. For this reason, the soil reaction and the column loads need to be adjusted, or %ZIVEKIPSEH ! [5.31]Now, the modified average soil reaction becomesQSHMJMIH ! EZIVEKI PSEH [5.32]And the column load modification factor is ! EZIVEKI PSEH [5.33]So, the modified column loads are ERH . This modified loading on the strip under consideration is shown in figure 5.8b. The shear and the moment diagram for this strip can now be drawn. This procedure is repeated for all strips in the x and ydirections.

6. Determine the effective depth of the mat d by checking for diagonal tension shear near various columns. According to ACI Code 318-95. American Concrete Institute, 1995), for the critical section,! ? A [5.34]Where ! JEGXSVIHGSPYQRPSEH12 SVGSPYQRPSEH PSEHJEGXSV! VIHYGXMSRJEGXSV ! ! GSQTVIWWMZIWXVIRKXLSJGSRGVIXIEXHE]W12Q)

The units of ERH in equation (34) are in meters. In English units, equation (34) may be expressed as ! [5.35]


Where MWMRPF ERH EVIMRMR ERH MWMRPFMRThe expression for in terms of which depends on the location of the column with respect to the plan of the mat, can be obtained from figure 5.8c.

7. From the moment diagrams of all strips in one direction (x or y), obtain the maximum positive and negative moments per unit width (that is, ! ).

8. Determine the areas of steep per unit width for positive and negative reinforcement in the x and y directions.! PSEHJEGXSV ! [5.36]And ! [5.37]Where ! EVIESJWXIIPTIVYRMX[MHXL! ]MIPHWXVIWWSJVIMRJSVGIQIRXMRXIRWMSR! JEGXSVIHQSQIRX! ! VIHYGXMSRJEGXSV

Examples 5 and 6 illustrate the use of the conventional rigid method of mat foundation design.

Approximate Flexible Method

In the conventional rigid method of design, the mat is assumed to be infinitely rigid. Also, the soil pressure is distributed in a straight line, and the centroid of the soil pressure is coincidental with the line of action of the resultant column loads (see figure 5.9). In the approximate flexible method of design, the soil is assumed to be equivalent to infinite number of elastic springs, as shown in figure 5.9b. It is sometimes referred to as the Winkler foundation. The elastic constant of these assumed springs is referred to as the coefficient of subgrade reaction k.


Figure 5.9 (a) Principles of design conventional rigid method; (b) principles of approximate flexible method; (c) derivation of equation (42) for beams on elastic foundation

To understand the fundamental concepts behind flexible foundation design, consider a beam of width having infinite length, as shown in figure 5.9c. The beam is subjected to a single concentrated load Q. from the fundamental of mechanics of materials,! [5.38]Where ! QSQIRXEXER]WIGXMSR! QSHYPYWSJIPEWXMGMX]SJJSYRHEXMSRQEXIVMEP! QSQIRXSJMRIVXMESJXLIGVSWWWIGXMSRSJXLIFIEQ ! WIIJMKYVIGHowever ! WLIEVJSVGI !


And ! ! WSMPVIEGXMSRHence ! [5.39]Combining equations (38 and 39) yields ! [5.40]However, the soil reaction is !Where ! HIJPIGXMSR! ! GSIJJMGMIRXSJWYFKVEHIVIEGXMSRO2QSVPFMR)So ! [5.41]Solution of equation (41) yields ! GSW WMR [5.42]Where ERH are constants and ! [5.43]The unit of the term as defined by the preceding equation is PIRKXL . This parameter is very important in determining whether a mat foundation should be designed by conventional rigid method or approximate flexible method. According to the American Concrete Institute Committee 336 (1988), mats should be designed by the conventional rigid method if the spacing of columns in a strip is less than . If the spacing of columns is larger than , the approximate flexible method may be used.


To perform the analysis for the structural design of a flexible mat, you must know the principles of evaluating the coefficient of subgrade reaction, k. before proceeding with the discussion of the approximate flexible design method, let as discuss this coefficient in more detail.

If a foundation of width B (figure 5.10) is subjected to a load per unit area of q, it will undergo a settlement, . The coefficient of subgrade modulus, k, can be defined as ! [5.44]

Figure 5.10 Definition of coefficient of subgrade reaction,

The unit of MWO2Q (or PFMR). The value of the coefficient of subgrade reaction is not a constant for a given soil. T depends on several factors, such as the length, and width, of the foundation and also the depth of embedment of the foundation. Terzaghi (1955) made a comprehensive study of the parameters affecting the coefficient of subgrade reaction. It indicated that the value of the coefficient of subgrade reaction decreases with the width of the foundation. In the field, load tests can be carried out by means of square plate measuring JX JXQ Q and values of k can be calculated. The value of k can be related to large foundations measuring in the following ways.

Foundations on Sandy Soils! [5.45]Where ERH ! GSIJJMGMIRXWSJWYFKVEHIVIEGXMSRSJJSYRHEXMSRQIEWYVMRKQQERH Q Q VIWTIGXMZIP]YRMXWMWO2QIn English units, equation (45) may be expressed as ! [5.46]


Where ERH ! GSIJJMGMIRXSJWYFKVEHIVIEGXMSRSJJSYRHEXMSRQIEWYVMRKJX EJXERH JX VIWTIGXMZIP]YRMXWMWPFMR)Foundations on Clays O2Q ! O2Q QQ [5.47]The definition of k in equation (47) is the same as in equation (45).

In English units,PFMR ! PFMR JXJX [5.48]The definitions of ERH are the same as in equation (46).For rectangular foundations having dimensions of (for similar soil and q),! [5.49]Where ! GSIJJMGMIRXSJWYFKVEHIQSHYPYWSJXLIVIGXERKYPEVJSYRHEXMSR !GSIJJMGMIRXSJWYFKVEHIQSHYPYWSJEWUYEVIJSYRHEXMSRLEZMRKHMQIRWMSRSJ Equation (49) indicates that the value of k of a very long foundation with a width B is approximately 0.67 .The modulus of elasticity of granular soils increases with depth. Because the settlement of a foundation depends on the modulus of elasticity, the value of k increases as the depth of the foundation increases.

Following are some typical ranges of value for the coefficient of subgrade reaction for sandy and clayey soils.

Sand (dry or moist)

Loose: PFMR 12QMedium: PFMR 12QDense: PFMR 12Q


Sand (saturated)

Loose: PFMR 12QMedium: PFMR 12QDense: PFMR 12QClay 7XMJJ PFMR 12Q:IV]WXMJJ PFMR 12Q,EVH" PFMR" QScott (1981) proposed that for sandy soils the value of can be obtained from standard penetration resistance at any given depth, or 12Q ! GSV [5.50]Where ! WXERHEVHTIRIXVEXMSRVIWMWXERGIIn English units,9 7 XSRJX ! 2GSV [5.51]For long beams, Vesic (1961) proposed an equation for estimating subgrade reaction:! [5.52]Where ! QSHYPYWSJIPEWXMGMX]SJWSMP! JSYRHEXMSR[MHXL! QSHYPYWSJIPEWXMGMX]SJJSYRHEXMSRQEXIVMEP! QSQIRXSJMRIVXMESJXLIGVSWWWIGXMSRSJXLIJSYRHEXMSR! 4SMWWSR WVEXMSSJWSMPFor most practical purposes, equation (52) can be approximated as

NPTEL - ADVANCED FOUNDATION ENGINEERING-1! [5.53]The coefficient of subgrade reaction is also very useful parameter in the design of rigid highway and airfield pavements. The pavements with a concrete wearing surface are generally referred to as a rigid pavement, and the pavement with an asphaltic wearing surface is called a flexible pavement. For surface load acting on a rigid pavement, the maximum tensile stress occurs at the base of the slab. For estimating the magnitude of the maximum horizontal tensile stress developed at the base of the rigid pavement, elastic solutions involving slabs on Winkler foundations are extremely useful. Some of the early work in this area was done by Westergaard (1926, 1939, and 1947).

Now that we have discussed the coefficient of subgrade reaction, we will proceed with the discussion of the approximate flexible method of designing mat foundations. This method, as proposed by the American Concrete Institute Committee 336 (1988), is described step by step. The design procedure is based primarily on the theory of plates. Its use allows the effects (that is, moment, shear, and deflection) of a concentrated column load in the area surrounding it to be evaluated. If the zones of influence of two or more columns overlap, superposition can be used to obtain the net moment, shear, anddeflection at any point.

1. Assume a thickness, h, for the mat, according to step 6 as outlined for the conventional rigid method. (Note: h is the total thickness of the mat).

2. Determine the flexural ridigity R of the mat:! [5.54]Where ! QSHYPYWSJIPEWXMGMX]SJJSYRHEXMSRQEXIVMEP! 4SMWWSR WVEXMSSJJSYRHEXMSRQEXIVMEP

3. Determine the radius of effective stiffness:! [5.55]Where ! GSIJJMGMIRXSJWYFKVEHIVIEGXMSRThe zone of influence of any column load will be on the order of 3 to 4 L.

4. Determine the moment (in polar coordinates at a point) caused by a column load (figure 5.11a):

NPTEL - ADVANCED FOUNDATION ENGINEERING-1! VEHMEPQSQIRX ! [5.56]! XERKIRXMEPQSQIRX ! [5.57]Figure 5.11 Approximate flexible method of mat design

Where ! VEHMEPHMWXERGIJVSQXLIGSPYQRPSEH! GSPYQRPSEH ! JYRGXMSRWSJ The variations of ERH with are shown in figure 5.11b (for details, see Hetenyi, 1946).

In the Cartesian coordinates system (figure 5.11a),! WMR GSW [5.58]! GSW WMR [5.59]5. For the unit width of the mat, determine the shear force, V, caused by a column

load:

NPTEL - ADVANCED FOUNDATION ENGINEERING-1! [5.60]The variation of with is shown in figure 5.11b.

6. If the edge of the mat is located in the zone of influence of a column, determine the moment and shear along the wedge (assume that the mat is continuous). Moment and shear opposite in sign to those determined are applied at the edges to satisfy the known conditions.

7. Deflection ( at any point is given by! [5.61]The variation of is given in figure 5.11.

Example 5

The plan of a mat foundation with column loads is shown in figure 5.12. Use equation (25) to calculate the soil pressures at points ERH . The size of the mat is JX JX all columns are MR MR in section, and EPP RIX !OMTJX. Verify that the soil pressures are less than the net allowable bearing capacity.


Figure 5.12 Plan of a mat foundation

Solution

From figure 5.12,'SPYQRHIEHPSEH ! ! OMT'SPYQRPMZIPSEH ! OMTSo 7IVZMGIPSEH ! ! OMT


According to ACI 318-95, factored load, ! (IEHPSEH 0MZIPSEH. So *EGXSVIHPSEH ! ! OMTThe moments of inertia of the foundation are ! ! JX! ! JXAnd ! So ! ! JXAnd ! ! JXSimilarly,! So ! ! JXAnd ! ! JXThe moments caused by eccentricity are ! ! ! OMT JX! ! ! OMT JX


From equation (25)! ! Or ! OMTJXNow the following table can be prepared.

Point OMTJX) JX JX JX JX OMTJX)A 1.2 -38 -0.065 48 -0.053 1.082

B 1.2 -24 -0.041 48 -0.053 1.106

C 1.2 -12 -0.020 48 -0.053 1.127

D 1.2 0 0.0 48 -0.053 1.147

E 1.2 12 0.020 48 -0.053 1.167

F 1.2 24 0.041 48 -0.053 1.188

G 1.2 38 0.065 48 -0.053 1.212

H 1.2 38 0.065 -48 0.053 1.318

I 1.2 24 0.041 -48 0.053 1.294

J 1.2 12 0.020 -48 0.053 1.273

K 1.2 0 0.0 -48 0.053 1.253

L 1.2 -12 -0.020 -48 0.053 1.233

M 1.2 -24 -0.041 -48 0.053 1.212

N 1.2 -38 -0.065 -48 0.053 1.188

The soil pressures at all points are less than the given value of EPP RIX ! OMTJX.


Example 6

Use the results of example 5 and the conventional rigid method.

a. Determine the thickness of the slab.b. Divide the mat into four strips (that is, ERH )

and determine the average soil reaction at the ends of each strips.c. Determine the reinforcement requirements in the y direction for !PFMR and ! PFMR.

Solution

Part a: Determination of Mat Thickness

For the critical perimeter column as shown in figure 5.13 *(ACI 318-95),

Figure 5.13 Critical perimeter column! ! ! OMT! ! MRFrom ACI 318-95

Where ! RSQMREPWLIEVWXVIRKXLSJGSRGVIXI

NPTEL - ADVANCED FOUNDATION ENGINEERING-1! JEGXSVIHWLIEVWXVIRKXL! ! So MRFor the critical internal column shown in figure 5.14,

Figure 5.14 Critical internal column! ! MR ! ! OMTAnd MR


Use ! MRWith a minimum cover of 3 in. over the steel reinforcement and 1-in. diameter steel bars, the total slab thickness is ! ! MRPart b: Average Soil Reaction

Refer to figure 5.12. For strip ABMN ([MHXL ! JX ! EX EX ! ! OMTJX ! EX EX ! ! OMTJXFor strip BCDKLM ([MHXL ! JX ! ! OMTJX ! ! OMTJXFor strip DEFIJK ([MHXL ! JX ! ! OMTJX ! ! OMTJXFor strip FGHI ([MHXL ! JX ! ! OMTJX ! ! OMTJXCheck for ! :7SMPVIEGXMSRJSVWXVMT ! ! OMT7SMPVIEGXMSRJSVWXVMT ! ! OMT7SMPVIEGXMSRJSVWXVMT ! ! OMT7SMPVIEGXMSRJSVWXVMT ! ! OMTOMT 'SPYQRPSEH ! OMT 3/


Part c: Reinforcement Requirements

Refer to figure 5.15 for the design of strip BCDKLM. Figure 5.15 shows the load diagram, in which ! ! OMT ! ! OMT ! ! OMT ! ! OMTThe shear and moment diagrams are shown in figure 5.15b and c, respectively. From figure 5.15c, the maximum positive moment at the bottom of the foundation ! ! OMT JXJX.


Figure 5.15

Figure 5.16 Rectangular section in bending; (a) section, (b) assumed stress distribution across the section

For the design concepts of a rectangular section in bending refer to figure 5.16.'SQTVIWWMZIJSVGI ! 8IRWMPIJSVGI ! !Note that for this case ! JX ! MR ! ! From equation (36),! ! ! MR


Thus ! ! MRMinimum reinforcement, QMR ! ! !Minimum ! ! MRJX. Hence use minimum reinforcement with ! MRJX.Use no. 9 bars at 10 in. center-to-center ( ! ) at the bottom of the foundation.

From figure 5.15c, the maximum negative moment ! OMT JX ! OMTJXJX. by observation, QMR .Use no. 9 bars at 10 in. center-to-center at the top of the foundation.

Example 7

From the plate load test (plate dimensionJX JX) in the field, the coefficient of subgrade reaction of a sandy soil was determined to be PFMR. (a) What will be the value of the coefficient of subgrade reaction on the same soil for a foundation with dimensions of JX JX# (b) if the full-sized foundation has dimension of JX JX, what will be the value of the coefficient of subgrade reaction?Solution

Part a

From equation (46),! Where ! PFMR! JXSo! ! MR


Part b

From equation (49),! JXJX ! PFMRSo! ! PFMRPROBLEMS

1. Determine the net ultimate bearing capacity of mat foundation with the following characteristics:

a. ! O2Q ! ! Q ! Q ! Qb. ! PFJX ! ! JX ! JX ! JX

2. Following are the results of a standard penetration test in the field (sandy soil):

Depth (m) Field value of

1.59

3.012

4.511

6.07

7.513

9.011

10.513

Estimate the net allowable bearing capacity of a mat foundation Q Q in plan. Here, ! Q and allowable settlement mm. assume that the unit weight of soil ! O2Q.

3. A mat foundation on a saturated clay soil has dimensions of Q Q. Given dead and live load ! 12 ! O2Q, GPE] ! O2Q.


a. Find the depth, of the mat for a fully compensated foundation.b. What will be the depth of the mat ( ) for a factor of safety of 2 against

bearing capacity failure?

4. Repeat problem 4 part b for ! O2Q.5. A mat foundation is shown in figure P-1. The design considerations are !Q ! Q ! Q ! 12 ! Q ! Q ! Q, and

preconsolidation pressure ! O2Q. Calculate the consolidation settlement under the center of the mat.

Figure P-16. Refer to figure P-2. For the mat, ! XSRW ! XSRW ! XSRW ERH ! XSRW

all columns are MR MR in cross section. Use the procedure outlined in section 7 to determine the pressure on the soil at A, B, C, D, E, F, G, and H.


Figure P-2

7. The plan of a mat foundation with column loads is shown in figure P-3. Calculate the soil pressure at points A, B, C, D, E, and F. note: all columns are Q Q in plan.


Figure P-3

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