1 The Islamic university - Gaza Faculty of Engineering Civil Engineering Department CHAPTER (6) Mat Foundation Instructor : Dr. Jehad Hamad
2 Introduction Under normal conditions, square andrectangularfootings such as those described in Chapters 3 and 4 are economical for supporting columns and walls. However, under certain circumstances, it may be desirable to construct a footing that supports a line of two or more columns. These footings are referred to as combined footings. When more than one line of columns is supported by a concrete slab, it is called amat foundation. Combined footings can be classified generallyunderthe following categories:::: orectangular Footings. otrapezoidal Footings. ocantilever or Strap Footings
3 A combined footing is usually used to support two columns of unequal loads. In such acase, the resultant of the applied loads would not coincide with the centroid of the footing, and the consequent the soil pressure would not be uniform. Another case where acombined footingis an efficient foundation solution is when there are two interior columns which are so close to each other that the twoisolated footings stress zonesin the soilareas would overlap. The area of the combined footing may be proportioned for a uniform settlement by making its centroid coincide with the resultant of the column loads supportedbythe footing.
4 There are many instances when the load to be carried by acolumn and the soil bearing capacity are such that the standard spread footing design will require an extension of the column foundation beyond the property line. In such acase, two or more columns can be supported on asingle rectangular foundation. If the net allowable soil pressure is known,the size of the foundation BxLcan bedetermined.
5 This photo shows an example of combined footings used in a heavy industrial plant, where the machinery loads place very large loads upon relatively confinedspace. The use of combined footings helps spread out the loads out to the adjacent footings in order to minimize stresses in the footings and reduce the differential settlement between them.
6 A third case:: of a useful application of acombined footing is if one (or several) columns are placedright at thepropertyline. The footings for those columns can not be centered around the columns,the consequent eccentric load would generate a large momentinthe footing. By tying the exterior footing to an interior footing through a continuous footing, the moment can be substantially reduced,andamoreefficient design is attained.
7 Rectangular Combined Footing:::::
8 Geometric design of rectangular combined footing:
9 Rectangular Combined Footing.
10 Rectangular Combined Footings. Step #1. The required design area A of a footing can be found from, where Q1, Q2 are the loads in columns #1 and #2, and q all (net) is the net allowable soil bearing capacity. Step #2. Determine the location of the resultant of the column loads. Step #3. For a uniform distribution of soil pressure under the footing, the resultant of the column loads should pass through the centroidof the foundation. Thus, where L = length of the foundation
11 Step #4.Once the length L is determined from above, the value of L1 can be obtainedfrom, The magnitude of L2 will be known and depends on the location of the property line. The width B is then found from,
12 Trapezoidal Combined Footing
13 Trapezoidal Combined Footing: This type of combined footing, is sometimes used as an isolated spread foundationfor acolumnthatis requiredtocarryalarge load inatight space. The size of the trapezoidal footing that will generate auniform pressure on the soilcanbefound throughthe followingprocedure. Step #1. If the net allowable soil pressure is known, determine the area of the footing, Step #2. Determine the location of the resultant for the column loads,
14 From the property of a trapezoid, With known values of A, L, X, and L2,solve Eqs. to obtain B1 and B2 Note that, for a trapezoid,
16 Cantilever footing- Strap beam
17 A strap footing is used to connect an eccentrically loaded column footing to an interiorcolumn. The strap is used to transmit the moment caused from an eccentricity to the interior column footing so that auniform soil pressure is generated beneath bothfootings. The strap footing may be used instead of arectangular or trapezoidal combined footing if the distance between columns is large and /or the allowable soil pressureis relatively large so thatthe additionalfootingareais notneeded.
20 Grade Beams and Strap Footings: Their purpose is to redistribute Excesses stresses, and possible differential settlements between adjacent spreadfootings.
21 Example1) Find the Dimensions of the combined footing for the columns A and B that spaced 6.0m center to center, column A is 40cm x 40cm carrying dead loads of 50tons and 30tons live load and column B is 40cm x 40cm carrying 70tons dead load and 50 tons live loads.
22 Solution 1-Find the required area: 2-Find the resultant force location (Xr): 3-To ensure uniform soil pressure, the resultant force (R) should be in the center of rectangular footing:
24 Example2) Find the Dimensions of the trapezoidal combined footing for the columns Aand B that spaced 4.0m centerto center,columnais 40cm x40cm carryingdead loads of 80tonsand 40tonslive load and column Bis 30cm x30cm carrying 50tonsdead load and 25 tonslive loads.
25 Solution 1-Find the required area: 2-Determine the resultant force
26 3-Put the resultant force location at the centroidof trapezoid to achieve uniform soil pressure. The censored equation is:
27 o For uniform soil pressure:
28 Example 3) Design astrap footing to support two columns, that spaced 4.0m center to center exteriorcolumnis 80cm x80cm carrying 1500 KN and interiorcolumnis 80cmx80cm carrying 2500KN.
29 1-Find the resultant force location: 2-Assume the length of any foot, let we assume L1=2m.
30 3-Find the distance a:
31 4-Find the resultant of each soil pressure: 5-Find the required area for each foot:
32 Example (4): Design a rectangular combined footing, given that f c = 3.5 ksi, fy = 50 ksi, qall = 5 ksf with a SF = 3, Df = 5 feet, the edge of column #1 is at the property line, and the spacing between columns is 18 feet center-to-center (c.c.).
33 Solution: Step 1: Determine the ultimate column loads and the soil stress at ultimate loads qult. Step 2: Determine the footing dimensions L and B.
34 Step 3: Draw the shear (V) and moment (M) diagrams. The column loads are treated as concentrated loads acting at the centers of the columns.
35 Step 4: Design the strap.
36 Example (5): Design a strap-footing for the following conditions, f c = 3.5 ksi, fy = 60 ksi, and qa = 2.5 ksf for both the footing and the strap, with a FS=4. The edge of column 1 is placed at the property line, and the center of the columns are 25 feet center-to-center (c.c.).
37 Types of Shallow Foundations 1. Spread Footings 2. Combined Footings 3. Continuous Footing 4. Mat Foundations
38 Mat Foundations:
39 Geometric and structural design of Mat foundation: & Geometric design (Working loads):
41 Common Types of Mat Foundations.
42 Bearing capacity of Mat Foundations:
43 The net allowable bearing capacity for mats constructed over granular soil deposits can be adequately determined from the standard penetration resistance numbers. From Eq. (5.64), for shallow foundations When the width B is large, the preceding equation can be approximated as
44 The following Assuming Fd = 1.0, we can approximate eq 6.12 and 6.13 as The net allowable pressure applied on a foundation
45 Compensated Foundation The net average applied pressure on soil is For no increase in the net pressure on soil below a mat foundation, q should be zero. Thus, This relation for Df is usually referred to as the depth of afully compensated foundation. The factor of safety against bearing capacity failure for partially compensated foundations (Df,Q>Ag)
46 For saturated clays, the factor of safety against bearing capacity failure
47 Conventional Rigid Method The conventional rigid method of mat foundation design can be explained step by step::: Step 1. Figure 6.10a shows mat dimensions B,L of and column loads of Q Calculate the total column load as Step 2. Determine the pressure on the soil, q, below the mat at points A,B,C,..by using the equation
48 The load eccentricities, ex and ey in the x and y directions can be determined by using (x,y ) coordinates:
49 Step 3. Compare the values of the soil pressures determined in Step 2 with the net allowable soil pressure to determine whether q < qall(net) Step 4. Divide the mat into several strips in the x and y directions. (See Figure 6.10). Let the width of any strip be B1 Step 5. Draw the shear,v,andthe moment,m,diagrams for each individual strip (inthexandydirections). For example,theaverage soilpressure of the bottom stripinthexdirectionoffigure 6.10a is
50 Now, the modified average soil reaction becomes and the column load modification factor is Step 6. Determine the effective depth d of the mat by checking for diagonal tension shear near various columns.
51 Example (6): For the shown mat foundation: * Check the adequacy of the foundation dimensions. * Calculate the modified soil pressure under the strip ABCD which is 2m width. * Draw SFD and BMD for the strip.
53 Checkthe adequacyof thefoundationdimensions. 1-Find the centerof gravity of mat footing: The distances are taken from (x-y) axes shown in the figure. 2-Find the resultant force R: 3-Find the location of the resultant force:
54 4-Find the eccentricities: 5-Find M Y and M X : 6-Find the stresses:
55 X,Y: Distances from the point to the center of gravity Calculate the modified soil pressure under the strip ABCD which is 2m width. * Locate the points E and F at the middle of strip edges. * Find the stresses at E and F and be careful that we use ultimate loads:
56 * Find the average stress:
57 We have to make adjustment for the loads as follow: Find the modified column loads: Multiply each column load by Find the modified soil pressure: Draw SFD and BMD.
58 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 astraight line, and the centroid of the soil pressure is coincident with the line of action of the resultant column loads. (See Figure 6.11a.) In the approximate flexible method ofdesign, the soil is assumed to be equivalent to an infinite number of elastic springs, as shown in Figure 6.11b. This assumption 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. To understand the fundamental concepts behind flexible foundation design, consider abeam ofwidthhavinginfinite length, asshown in Figure 6.11c. The beam is subjected to asingle concentrated load Q. From the fundamentals of mechanicsof materials,
61 where and are constants and
62 If a foundation of width B (see Figure) is subjected to a load per unit area of q, it will undergo a settlement,the coefficient of subgrade modulus can be defined as
63 Foundations on Sandy Soils: For foundations on sandy soils, Foundations on Clays For foundations on clays, For rectangular foundations having dimensions of B,L (for similar soil and q),
64 For long beams,vesic (1961) proposed an equation for estimating subgrade reaction,namely, For most practical purposes, Eq. (6.46) can be approximated as