Chapter seven Design Of Mat Foundations 7.1 INTRODUCTION A mat foundation is a large concrete slab used to interface one column, or more than one column in several lines, with the base soil. It may encompass the entire foundation area or only a portion. A mat may be used to support on-grade storage tanks or several pieces of industrial equipment. Mats are commonly used beneath chimneys, and various tower structures. 7.2 Mat foundations are selected : 1. When the area covered by the individual footings exceeds 50% of

the structural plan area (the base soil has a low bearing capacity and/or the column loads are so large)

2. When the building requires a deep basement 3. to provide a water barrier 4. to minimize the differential settlement in variable soils 5. Mat foundations may be supported by piles in situations such as high

groundwater (to control buoyancy) or where the base soil is susceptible to large settlements

7.3 Types of Mat Foundations

Most mat foundations employ a constant thickness ‘T’. This type of constant thickness is called a flat plate mat.

When large column loads must be designed to prevent shear, other thickening designs are common. Below are some typical flat plates with thickening under the columns.

Plate thickened under column

Plate with pedestals

Waffle slab

7.4 Bearing Capacity of Mat Foundations The mat foundation must be designed to limit settlements to a tolerable amount. These settlements may include the following: 1. Consolidation 2. Immediate or elastic 3. A combination of consolidation and immediate amounts A mat must be stable against a deep shear failure, which may result in either a rotational failure or a vertical (or punching) failure. The gross ultimate bearing-capacity of a mat foundation can be determined by the same equation used for shallow:

• The term “B” in the preceding equation is the smallest dimension of the mat .

• The net ultimate capacity can be given as : qu,net = qu –q • A factor of safety of 3 should normally be used under most

working conditions

For saturated clay with Φ = 0 and vertical loading condition, (Nc=5.14; Nq=1.0; Nγ = 0), the general bearing capacity equation takes the form :

From table 3-3 (chapter 3):

• The net ultimate bearing capacity is

• The net allowable soil bearing capacity is :

7.5-Bearing capacity from SPT

Where N : average corrected SPT value B : width (m) Fd : 1+ 0.33(Df/B) ≤ 1.33 S : tolerable settlement (mm) When the width “B” is large, 3.28B +1 ≈ 3.28B

Noting that these equations were originally developed for spread footings and for settlement 25.4 mm (1 in.) with a differential settlement of 19 mm (0.75 in.),however the width of raft foundations are larger than isolated footing, whereas the differential settlement is smaller. Thus it can be assumed that for a maximum raft settlement of 50.8 mm (2 in.) the differential settlement would be 19 mm. We can take conservatively Fd =1.0, we obtain :

7.6 compensated Foundations If zero settlement is desired, the excavated soil weight will be equal to the weight of the building, that is, the Engineer must excavate to a depth Df where:

A: area of raft Q: D.L &L.L of structure

7.6 Mat Settlements Mat foundations are commonly used where settlements may be a problem. The settlement tends to be controlled via the following: 1. Use of a larger foundation to produce lower soil contact pressures 2. flotation effect 3. Allowing somewhat larger settlements, say, 50 instead of 25 mm.

7.7 Design of Mat Foundations The structural design of mat foundations can be carried out by two methods: • The conventional rigid method (CRM) • The approximate flexible method

7.7 Analysis and design procedure for the Rigid Method:

1) Calculate the total column load, Q = Q1 + Q2 + ….. Qn; 2) Determine the soil pressure (q) below the mat at points A,B,C,…J by using the equation:

Where: • A : area of raft = B .L • Ix : moment of inertia about x axis • Iy: moment of inertia about y axis • Mx: moment of the column loads about the x axis= Q.ey • My: moment of the column loads about the x axis= Q.ex • ex,ey: load eccentricities in the x and y direction, they may be

determined from:

3) Compare the resulting soil pressures (q) with the net allowable soil pressure, q≤ qall,net (if not we may increase L or B or both)

4) Divide the mat into several strips in the x and y directions, let the

width be B1(not necessarily equal strips) example: for strip FGHI • qavg ≈ ( qI +qF )/2 • total soil reaction = qavg .B1.L • total column load = Q9 +Q10+Q11+Q12 • Average load =(total soil reaction+ total column load)/2 • The modified average soil reaction is qavg (modified) = qavg.(average load)/ Total soil reaction • The column load modification factor is: F =average load/Q9 +Q10+Q11+Q12 So the modified column loads are F.Q9, F.Q10, F.Q11 and F.Q12

5) Draw the shear V and the moment M diagrams for each strip in both directions;

6) Determine the effective depth d of the mat by checking wide beam or

punching shear. this latter type will be the controlling factor in most cases.

Illustrates the different configurations which may be encountered when considering the punching perimeter around columns 7) From the moment diagrams of all the strips in one direction, choose the maximum positive and negative moments per unit of width, that’s M’=M/B1

8) Determine the areas of steel per unit width for the positive and the negative reinforcement in both directions: Mu = M’.load factor = Φ.As.fy.(d-a/2) and a =As.fy/0.85.f’c.b

7.8 Modulus Of Subgrade Reaction ks For Mats Consider a foundation of width “B” subjected to a load per unit area(q),and undergoing a settlement ∆, K is defined as:

(KN/m3 or lb/in3)

The K value is not constant for a given soil , it depends on several factors ,such as the length L and width B of the foundation and also the embedment depth of the foundation. In the field, load test can be carried out by means of square plates measuring each 0.3mx0.3m(1ft x1ft), and the calculated K value can be related to large foundations (BXB) through:

9)Continue with the routine checks and make a representative sketch Remark: the conventional rigid method supposes that the mat is very rigid and the column pattern is fairly uniform in both spacing and loads. These are large approximations in the analysis of mat foundations and may not be valid except in relatively few real cases.

i-)Foundation on sandy soil:

S.I

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ii-) Foundations on clay:

S.I

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For rectangular foundations BXL

Where: K: coefficient of subgrade modulus of the rectangular foundation (LXB) K(BXB): coefficient of subgrade modulus of square foundation (BXB)

Some typical ranges of value for K0.3 for sandy and clayey soils are given in the following table:

from SPT values for sandy soil, we may use: K 0.3 = 1.8 N (MN/m3)

K0.3 = 6N (U.Ston/ft3) Where N = corrected SPT value

The modulus ks is used to compute node springs based on the contributing plan area of an element to any node For a triangle one should arbitrarily use one-third of the triangle area to any corner node. For these area contributions the fraction of ks node resistance from any element is Ki = ks, (kN/m3) X Area,(m2) = units of kN/m (or kips/ft in Fps)