Choice of the Type of Foundation The choice of the appropriate type of foundation is governed by some important factors such as
Therefore to decide about the type of foundation, subsoil exploration must be carried out. Then the soil characteristics within the affected zone below the building should be carefully evaluated. The allowable bearing capacity of the affected soil strata should then be estimated. After this study, one could then decide whether shallow foundations or deep foundations should be used. Shallow foundations, such as footings and rafts, cost less and easier to execute. They could be used if the following two conditions are fulfilled;
This condition is fulfilled when , in Fig.1, is smaller than and smaller than and smaller than and smaller than and so on.
If one or both of these two conditions cannot be fulfilled the use of deep foundations should be considered. Deep foundations are used when top layers of the soil are soft and there exists a good bearing stratum at a reasonable depth. Soil strata lying beneath the bearing stratum should be of ample strength to resist the superimposed stresses (Dp) due to the loads transmitted to the bearing stratum, as shown in Fig.2.
Deep foundations are usually piles or piers which transmits the load of the building to the good bearing stratum. They usually cost more and require well trained engineers to execute. If the explored soil layers are soft for considerable depth and no bearing stratum is found at a reasonable depth, floating foundations could be used. To build a floating foundation, a mass of soil, approximately equal to the weight of the proposed building, is to be removed and replaced by the building. In this case, the bearing stress under the building will be equal to the weight of the removed earth (γD) which is less than (q_{a} = γD +2C) and Dp will be equal to zero. This means that the bearing capacity under the building is less than ( q_{a} ) and the expected settlement equals theoretically to zero. Finally, the engineer should prepare an estimate of the cost of the most promising type of foundation which represents the most acceptable compromise between performance and cost.
Shallow Foundations Shallow foundations are those executed near the ground surface or at shallow depths. As mentioned before in the previous chapter, shallow foundations are used when subsoil exploration proves that all soil strata affected by the building could resist the superimposed stresses (Dp) without causing excessive settlements. Shallow foundations are either footings or rafts. Footings Footing foundation is one of the oldest and most popular type of shallow foundations. A footing is an enlargement of the base of a column or wall for the purpose of distributing the load on the supporting soil at a pressure suiting its properties. Types of Footings There are different types of footings to suit the nature of the structure. Footings could be classified into three main classes Wall or Strip footing It runs underneath the wall by its full length as shown in Fig.3. it is usually used in the bearing wall type structures.
Isolated column footing It acts as a base for a column. It is usually used for reinforced concrete Skelton type of buildings. It can take any shape such as square, rectangular or circular as shown in Fig.4.
Fig.4 Typical spread footings
Combined Column Footing It is a combined base for an exterior and interior columns of a building, Fig.5. It is also used when two adjacent columns of a building are close to each other that their footings overlap
Distribution of stresses under footings The distribution of stresses under footings is considered linear although it is not the case in reality, The error involved in this assumption is small and could be overlooked.
Load Compilations Loads affecting the usual types of buildings are:
Dead Load The full dead load acting on the elements of the structures should be considered in the design.
Live Load It is not probable that the full intensity of the live load will be acting at the same time on all the floors of a multistorey building. Consequently, the codes of practice allow a certain reduction in the intensity of live load. According to the Egyptian Code of practice the following reduction in live load is allowed: No. of floors Reduction in live load % Ground floor zero % 1^{st} floor zero % 2^{nd} floor 10.0 % 3^{rd} floor 20.0 % 4^{th} floor 30.0 % 5^{th} floor and above 40.0 % The live load should not be reduced for ware houses and public buildings such as schools, cinemas, and hospitals.
Wind and Earthquake Loads When the buildings are high and narrow the wind pressure and the earthquake load must be taken into consideration.
Assumption used in the Design of Spread Footings Theory of elasticity analysis indicates that the stress distribution beneath footings, symmetrically loaded, is not uniform. The actual stress distribution depends on the type of material beneath the footing and the rigidity of the footing. For footings on loose cohesionless material, the soil grains tend to displace laterally at the edges from under the load, whereas in the center the soil is relatively confined. This results in a pressure diagram somewhat as indicated in Fig.6. For the general case of rigid footings on cohesive and cohesionless materials, Fig.6 indicates the probable theoretical pressure distribution. The high edge pressure may be explained by considering that edge shear must take place before settlement can take place. Because the pressure intensities beneath the footing depend on the rigidity of the footing, the soil type and the condition of the soil, the problem is generally indeterminate. It is common practice to use a linear pressure distribution beneath the footings and this procedure will be followed in this text. In any case little difference in design results by using a linear pressure distribution
Allowable Bearing Stresses under Footings The factor of safety in the calculating of the allowable bearing capacity under the footing should not be less than 3 if the loads considered in the design is equal to the dead load + the reduced live load. The factor of safety should not be less than 2 when the severest condition of loading is considered, which is, dead load + full live load + wind load or earthquake loads. The loads of the superstructure are usually calculated at ground level. If the net allowable bearing pressure is given, it should be reduced by the volume of concrete below ground surface per unit area of the footing multiplied by the difference between the unit weight of concrete and soil. If we assume equal to the average density of soil and concrete Fig.7, then should be reduced by
Structural design of spread footings For spread footing the following items should be considered 1Shear Shear stresses ate usually control the depth of spread footings. The critical section for wide beam shear is shown in Fig.8a. It is at a distance d from the column or wall face. The values of the shear stresses are given in table 1.The critical section for punching shear (Twoway diagonal shear) is shown in Fig.8b. It is at distance of d/2 from the face of the column. This assumption is according to the American Concrete Institute (A.CI) Code.
Table(1): allowable stresses in concrete and reinforcement:
Punching shear will generally control the depth of the spread footings. From principles of statics Fig.8b , the force on the critical section for shear is equal to the force on the footing beyond the shear section caused by the net soil pressure f_{n}. where q_{p} = allowable punching shear stress = 8 kg/cm^{2} (for cube strength =160) f_{n} = net soil pressure b = Side of column d =depth of punching shear It can be assumed that the critical section for punching shear is at the column face and in this case the allowable punching shear stress can be taken as 10.0 kg/cm^{2} (for cube strength = 160). The footing is usually designed to ensure that the depth is great enough to resist the shear by concrete with out web reinforcement..
2 Bond Bond stress is computed as
where the shear force Q is taken at the same critical section for bending moment, or where changes in concrete cross section or steel reinforcement occur. For footings of constant section, the section for bond is at the face of column or wall. The reinforcing bar should have enough length d_{d} , Fig.9 to avoid the pullout (bond failure) or a splitting of the concrete. The value of d_{d} is computed as follows:
For the first computation take f_{s} equal to allowable working stress. If the calculated d_{d} is larger than the available d_{d } then recalculate d_{d } by taking f_{s } equal to the actual steel stress. The allowable values of bond stress q_{b} are as follows
3 Bending moment The critical sections for bending moment are determined from Fig.10 as follows:
For concrete wall and column, this section is taken at the face of the wall or the column Fig.10a. For masonry wall this section is taken halfway between middle and the edge of the wall Fig.10b. For steel column, this section is located half way between the edge of base plate and the face of the column Fig.(10c). The depth required to resist bending moment is 4 Bearing on top of footing When a reinforced concrete column transmits its load to the footing, the steel of the column, which is carrying a portion of the load, cannot be terminated on top of footing since this may overstress the concrete in column contact area. Therefore it is necessary to transmit the portion of load carried by the column steel by bond stress into the footing by either extending the column steel or by dowels. From Fig.11: where f_{s} is the actual steel stress
5 Plain Concrete Footing Beneath R.C. Footing It is common practice to place a plain concrete layer beneath the reinforced concrete footing. This layer is about 20 cm. to 40 cm. The projection C of the plain concrete layer depends on its thickness t. Referring to Fig.12 , the maximum bending moment per unit length at section aa is given by Where f_{n} = the net soil pressure. The maximum tensile stress at the bottom of section aa is:
DESIGN OF R.C. WALL FOOTING: The wall footing is a strip of reinforced concrete wider than the wall. Fig.13 shows the different types of wall footings. The type shown in Fig.13a is used for footings carry light loads and placed on uniform soil of good bearing capacity. The type shown in Fig.13b is used when the soil under the footing is not uniform and of different bearing capacities. The type shown in Figs.13c and 13d is used for heavy loadings.
Design Procedure: Consider 1.0 meter length of the wall. 1. Find P at ground level. 2. Find if is given then it is reduced by or compute P_{T} . 3. Compute the area of footing
If the bond stress is not safe, we either increase by using steel bars with smaller diameter, or increase ∑ O the depth d. Bending up the steel reinforcement at the edges of the footing helps in resisting bond stresses. The diameter of the main steel reinforcement should not be less than 12 mm. To prevent cracking due to uneven settlement below the wall itself, additional reinforcement is used as shown in Fig.13c and d. It is taken as 1.0% of the concrete cross section under the wall and distributed equally at top and bottom. 19. Check the anchorage bond
Design of Single Column Footing The single column footings are usually square in plan, Rectangular footings are used if there restriction in one direction or If the supported columns are of too elongated .rectangular cross section. In the simplest form, they consist of a single slab FIg.15a. Fig.15b shows a pedestaled column footing, the pedestal provides depth for a more favorable transfer of load and in many cases is required in order to provide the necessary length for dowels. Sloped footings such as those in Fig.15c
Design Procedure for Square Column Footing American Codes of Practice is equal to the moment about the critical section yy of the net stress acting on the hatched .area abcd Fig. 16a. According to the Continental Codes of practiceM_{max}. is equal to either; the moment of the net stresses acting on the hatched area abgh, shown in Fig.16b, about the critical section yy or 0.85 the moment of the net stresses acting on the area abcd in Fig.16a about yy.
8. Determine the depth required to resist punching d_{p}. 9. Calculate d_{m}, the depth to resist
b = B, the side of the footing according to the American Codes of Practice b = (b_{c} + 20) cm where b_{c} is the side of the column according to the Continental Codes of Practice. It should be noted that d_{m} computed by continental method is larger than that compute by American code. The larger depth will decrease the amount of steel reinforcement and usually satisfies the depth required for punching. American code gives smaller d_{m} with higher value of steel reinforcement but by using high tensile, steel, the area of the steel reinforcement can be reduced. In this text, the bending moment will be calculated according to the American Code while b is taken either equal to b_{c} + 20 when the ordinary steel is used or equal to B when high tensile steel is used. The footing depth d may be taken any value between the two values computed by the two above methods. It should be mentioned that, for the same bending moment, a large depth will require smaller area of reinforcement steel which may not satisfy the minimum percentage of steel . Also a small depth will require a large area of steel especially when using ordinary mild steel. 10. Choose the greater of d_{m} or d_{p} 11. Check d_{d}, depth of column dowel.
Design Procedure for Rectangular Footing The procedure is the same as square footing. The depth is usually controlled by punching shear except If the ratio of length to width is large, the wide beam shear may control the depth. The critical sections for shear are at distance d from both sides of the column Fig.17a. The bending moment is calculated for both directions, about 11 axis and about bb axis as shown in Fig.17.b and c.
The reinforcement in the long direction (Side L) is calculated from the bending moment , and is uniformly distributed over the width B. The reinforcement in the short direction (Side B) is calculated from the bending moment M_{11}_{}. In locating the bars in the short direction one has to consider that the support provided to the footing by the column is concentrated near the middle, consequently the area of footing adjacent to the column is more effective in resisting bending. For this reason an adjustment of steel in the short direction is made. This adjustment place a percentage of the steel in a zone centered on the column with a width equal to the length of the short direction of the footing. The remainder of the reinforcement shall be uniformly distributed in two end zones, Fig.18. According to the American Concrete Institute, the percentage of steel in the central zone is given by: where S = ratio of long side to short side, L/B .
SEMELLES Single footings should be tied together by beams known as semelles as shown in Fig.19.a. Their function is to carry the ground floor walls and transfer their loads to the footings. Semelles can prevent relative settlement if they are of very stiff section and heavily reinforced. The semelle is designed as continuous reinforced concrete rectangular beam carrying the weight of the wall. The width of the semelle is equal to the width of the wall plus 5 cm and it should not be less than 25 cm. It should resist the shearing forces and bending moments to which it is subjected, semelles must be reinforced at top and bottom to counteract the differential settlements. by the equal reinforcement A_{s}. The top level of the semelle should be 20 cm below the level of the platform surrounding the building. If the level of the ground floor is higher than that of the platform, the level of the inner semmelle can be taken 20 cm below the ground floor level
Footings subjected to Moment Introduction Many foundations resist, in addition to the concentric vertical load, a moment about one or both axis of the footing. The moment may result from a load applied off the centre of the footing . Examples of foundations that must resist moment are those for retaining walls, abutments , bridge piers, and the columns of the foundations of tall buildings where wind pressure causes appreciable bending moments at the base of columns. The soilpressure resultant under eccentrically loaded footing is assumed to coincide with the axial load P, but not with the centroid of the footing, which results in a linear nonuniform pressure distribution. The maximum pressure must not exceed the maximum allowable pressure on the soil. Tilting of the footing due to the higher intensity of soil pressure at the toe is possible to occur . This can be reduced by using a large safety factor when computing the allowable soil pressure. Chapter 1, Section "Footings with Eccentric or Inclined Loads" provide for a reduction in allowable soil pressure for eccentrically loaded footings.
Footings with Moments or Eccentricity about One Axis
where P = vertical load or resultant force e = eccentricity of vertical load or resultant force q = intensity of soil pressure (+ = compression) and should not be greater than the allowable soil pressure q_{a}
cThe Load P is Outside Middle When the load P is outside the middle third, that is e > L/6, Eq.7 indicates that tension will occur under the footing. However, no tension can develop between the soil and the footing, thus the tension stresses are neglected, and the area of the footing which is in tension is not considered effective in carrying load. Therefore the pressure diagram on the soil should be always in compression as shown in Fig.21.c. For the eccentricity e > L/6 with respect to one axis only, one can drive equations for the maximum soil pressure q_{1} by finding a compressive pressure diagram whose resultant should be equal and on the same line of action of load P. This diagram will take the shape of triangle whose side = q_{1} and base = Footings with Moments or Eccentricity about both Axis For footings with moments or eccentricity about both axis Fig 22, the pressure may be computed by the following equation
a The Neutral Axis Outside the Base : If the neutral axis is outside the base, then all the pressure q is in compression and equation (9) is valid. The location of the maximum and minimum pressures on soil may be determined readly by observing the directions of the moments. The maximum pressure q_{1} is at point (1) Fig.22a and the minimum pressure q_{2} is at point (3). The pressure q_{1} and q_{2} are determined from Eq.(9).
b The Neutral Axis Cuts the Base If the neutral axis cuts the base, then a certain area of the base is subjected to tension Fig.22. As the soil is not likely to grip the footing to hold it in place , therefore the diagram shown in Fig.22b and Eq.(9) cannot be used. The computation of the maximum pressure on soil must be based on the area actually in compression. The compressive diagram is to be found in such a way that its resultant should be equal and on the same line of action of the force P. The simplest way to get this diagram is by trial and error as follows: 1 Find soil pressure at all corners by applying Eq.(9). 2 Determine the position of the neutral axis NA (the line of zero pressure). This is not a straight line, but for the problem it is assumed to be . Therefore it is only necessary to find two points, one on each adjacent side of the footing. 3 Select another neutral axis (N'A') parallel to (NA) but somewhat closer to the location of the resultant load P acting on the footing. 4 Compute the moment of inertia of the area in compression with respect to the N'A' . The simplest procedure is to draw the footing to scale and divide the area into rectangles and triangles
4.4 STRUCTURAL DESIGN OF FOOTINGS SUBJECTED TO MOMENT The principle problem in the design of eccentrically loaded footings is the determination of the pressures distribution under the footings. Once they have been determined, the design procedure will be similar to the concentrically loaded footings, the critical sections are selected and computations of the stresses due to moment and shear are made.
Where the bending moments on a column come from any direction, for example from wind loads, a square footing; is preferable unless the space restrictions dictate the choice of rectangular footing. If the bending moments always act in the same direction, as in columns supporting rigid framed structures, the footing can be lengthed in the direction of the eccentricity The dimensions of the footing B and L are proportioned in such a way that the maximum pressure at the toe does not exceed the allowable soil pressure .
If a column carries a permanent bending moment, for example a bracket carrying a sustained load, it may be advantage to place the column off centre on the footing so that the eccentricity of the resultant loading is zero In this case the distribution of the pressure on the base will be uniform. The long toe section of the footing should be designed as a cantilever about a section through the face of the column, Calculation of the depth to resist punching shear and wide beam shear is the same as in footings support concentric loads Since the bending moment at the base of the column is likely to be large for this type of footing, the column reinforcement should be properly tied into the footing., Reinforcement details for this type of footings are shown in Fig.24. For square footing it is generally most convenient to keep bar diameter and spacing the same in both directions in order to avoid confusion in steel fixing.
Combined Footings
Introduction The preceding section presented elements of the design of spread and wall footings. This section considers some of the more complicated shallowfoundation problems. Among these are footings supporting more than one column in a line (combined footings), which may be rectangular or trapezoidal in shape, or two pads connected by a beam, as for a strap footing. Eccentrically loaded footings and unsymmetrically shaped footings will also be considered.
Rectangular Combined Footings When property lines, equipment locations, column spacing, or other considerations limit the footing clearance at the column locations, a possible solution is the use of a rectangularshaped footing. This type of footing may support two columns, as illustrated in Fig.25 and 26, or more than two columns with only slight modification of the design procedure. These footings are commonly designed by assuming a linear stress distribution on the bottom of the footing, and if the resultant of the soil pressure coincides with the resultant of the loads (and center of gravity of the footing), the soil pressure is assumed to be uniformly distributed, The linear pressure distribution implies a rigid footing on a homogeneous soil. The actual footing is generally not rigid, nor is the pressure uniform beneath it, but it has been found that solutions using this concept are adequate. This concept also results in a rather conservative design.
The design of a rigid rectangular footing consists in determining the location of the center of gravity (cg) of the column loads and using length and width dimensions such that the centroid of the footing and the center of gravity of the column loads coincide. With the dimensions of the footing established, a shear and moment diagram can be prepared, the depth selected for shear (again it is conventional to make the depth adequate for shear without using shear reinforcement to satisfy rigidity requirements implicitly), and reinforcing steel selected for bending requirements. Critical sections for shear, both diagonaltension and widebeam, should be taken as indicated in the previous section. The maximum positive and negative moments are used to design the reinforcing steel and will result in steel in both bottom and top of the beam. In the short direction, obviously, the entire length is not going to be effective in resisting bending. That zone closest to the column will be most effective for bending, and it is recommended that this approach be used. This is basically what the ACI Code specifies in Art. 15.4.4 for rectangular footings If it is accepted that the zone which includes the columns is most effective, what should this zone width be? Certainly, it should be something greater than the width of the column. Probably it should be no greater than the column width plus d to 1.5d, depending on the column location based on the author's analytical work, lack of Code guidance, and recognizing that extra steel will "stiffen" the zone and increase the moments in this zone and reduce the moment out of the zone. An effective width using this method is illustrated in Fig.27 For the remainder of the footing in the short direction, the ACI Code requirement for minimumpercentage steel (Art. 10.5 or 7.13) should be used.
In selecting dimensions for the combined footing, the length dimension is somewhat critical if it is desired to have shear and moment diagrams mathematically close as an error check. This means that unless the length is exactly the computed value from the location of the cg of the columns, an eccentricity will be introduced into the footing, resulting in a nonlinear earthpressure diagram. The actual asbuilt length, however, should be rounded to a practical length, say, to the nearest 0.25 or 0.5 ft (7.5 to 15 cm). The column loads may be taken as concentrated loads for computing shear and moment diagrams. For design the shear and moment .values at the edge (face) of the column should be used. The resulting error, using this approach, is negligible.Fig.(28)
If the footing is loaded by more than two columns, the problem is still statically determinate; the reactions (column loads) are known as well as the distributed loading, i.e. the soil pressure. Design Procedure of rectangular combined footing:
Referring to Fig.29, the steps of the design can be summarized as follows : 1 Find the line of application of the resultant R. This fix L/2 since y is known and limited. It should be indicated that unless the length L is exactly the computed value, an eccentricity will be introduced into the footing, resulting in a nonlinear soil pressure diagram. The actual asbuilt length, however, should be rounded to a practical length, say, to the nearest 5 cm or 10cm. maximum +ve moment at point K where the shearing force = zero 6 Determine the depth for shear. It is conventional to make the depth adequate for shear without using shear reinforcement. The critical section for shear is at distance d from the face of column having maximum shear, Fig.30 7Determine the depth for punching shear for both columns. According to ACI the critical section it at d/2 from column face. Fig.30. 9d is chosen the greatest of t = d + 5 to 8 cm. 11 Check the bond stresses and anchorage length d. 12 Short direction : The column loads are distributed crosswise by transverse beams (hidden), one under each column. The length of the beams are equal to the width of the footing B. The effective width of the transverse beam may be taken the least of the following: a Column width a + 2 d or column width a + d + the projection of the footing beyond the column y, Fig.31. b The width of the footing It should be noted that ACI code consider that the effective width of the transverse beam equal to column width a + d or column width a + d/2 + y The Transverse bending moment M_{T1} at column (1) is equal to
The transverse reinforcement should be distributed over the effective width of the transverse beam. For the remainder of the footing, minimum percentage steel should be used. The bond stresses and anchorage length d_{d}, should be checked.
Combined trapezoidal footing: Combined trapezoidal footing for two columns as used when column carries the largest load is near the property line where the projection is limited or when there is restriction on the total length of the footing. Referring to Fig.32 , The position of the resultant of columns loads R locates the position of the centriod of the trapezoid. The length L is determined and the area A is computed from: The design procedure is the same as rectangular combined footing except that the shear diagram will be a second degree curve and bending moment is a third degree curve.
Design of Strap or Cantilever Footings A strap footing may be used where the distance between columns is so great that a combined or trapezoid footing becomes quite narrow, with resulting high bending moments, or where as in previous section.
A strap footing consists in two column footings connected by a member termed a strap, beam, or cantilever which transmits the moment from the exterior footing. Fig.33 illustrates a strap footing. Since the strap is designed for moment, either it should be formed out of contact with the soil or the soil should be loosened for several inches beneath the strap so that the strap has no soil pressure acting on it. For simplicity of analysis, if the strap is. not very long, the, weight of the strap may be neglected.
In designing a strap footing, it is first necessary to proportion the footings. This is done by assuming a uniform soil pressure beneath the footings; that is, R_{1} and R_{2} (Fig.33) act at the centroid of the footings.
The strap must be a massive member for this solution to be valid. The development of Eq.1 implies rigid body rotation; thus, if the strap is not able to transmit the eccentric moment from column 1 without rotation, the solution is not valid. To avoid exterior footing rotation it is recommended to be . I_{strap}/I_{footing} > 2 It is desirable to proportion both footings so that B and q are as nearly equal as possible to control differential settlements. Design Procedure for Strap Footing reaction under the interior footing will be decreased by the same value, referring to Fig.33 1 The design begins with trial value of e 6 Check that the centroid of the areas of the two footings coincide with the resultant of column loads. 7 Calculate the moments and shear at various parts of the strap footing. 8 Design of the strap The strap represents a singlespan beam loaded upward by the loads transferred to it by the two footings and supported by downward reactions at the centre lines of the two columns. Thus the upward load over the length L is equal to R_{1}/L t/m'. The location of the maximum moment is obtained by equating the shear force to zero. The moment decreases toward the interior column and is zero at the center line of that column. Hence half the strap reinforcement is discontinued where no longer needed and the other half is continued through the interior column. Check shear stresses and use stirrups, and bent bars if necessary. 9 Design of the exterior footing The exterior footing acts exactly like a wall footing of a length equal to L. Even though the column is located at the edge the balancing action of the strap is such as to transmit the reaction R_{1} uniformly over the length L_{1} thus resulting in the desired uniform soil pressure. The design is carried out exactly as for a wall footing. 10 Design of the interior footing The interior footing may be designed as a simple single column footing. The main difference is that the punching shear should be checked along the perimeter fghj, Fig.33.
RAFT FOUNDATIONS Introduction The raft foundation is continuous footing that cover the entire area beneath a structure and supports all the walls and columns. The term mat is also used for foundation of this type. It is used generally on soil of low bearing capacity and where the area covered by spread footings is more than half the area covered by the structure. Raft foundation is also used where the soil mass contains compressible lenses or the soil is sufficiently erratic so that differential settlement would be difficult to control. The raft tends to bridge over erratic deposits and reduces the differential settlement.
Bearing Capacity of Rafts on Sand Beating capacity of foundations on sand increases as the width increases. Owing to the big width of raft compared to the width of an ordinary footing, the allowable bearing capacity under the raft will be much greater than that of the footing.
It was noticed in practice that using an allowable bearing capacity under the raft equal to twice the allowable bearing capacity determined for an ordinary footing. resting on the same sand will give a reasonable and acceptable amount of settlement.
If the water table lies at a depth equal to or greater than B, the width of the raft, the allowable bearing capacity, determined for dry condition, should not be reduced. If there is a possibility that the water table rises up till it floods the site, the allowable bearing capacity should be reduced by 50%. If the water table is at a depth intermediate between B and the base of the raft, an appropriate reduction between zero and 50% should be made.
Bearing Capacity of Rafts on Clay. In clays, the bearing capacity is not affected with the width of the foundation Therefore, the bearing capacity under a raft will be the same as that under an ordinary footing.
If the estimated differential settlement under the raft is more than tolerable or if the weight of the building divided by its area gives a bearing stress greater than the allowable bearing capacity, floating or partially floating foundation should be considered.
To execute a floating foundation, excavation is to be carried out till a depth D is reached where the weight of the excavated soil equals to the weight of the structure, figure 2. In this case, the excess superimposed stress Δp at foundation level equals to zero and consequently, the building will suffer no settlement. If the full weight of the building =Q and the weight of soil removed =W_{s} and the excess load at foundation level =Q_{e} \Q_{e}_{=}QW_{s} In case of floating foundation ; Q = W_{s} and therefore Q_{e} = Zero In case of partially floating foundation, Q_{e}_{ }has a certain value which when divided by foundation area gives the allowable bearing capacity of the soil ;
Design of Raft Foundations ; Rafts may be designed as rigid structures (so called conventional analysis) whereby soil pressure acting against the raft slab is assumed uniformly distributed and equal to the total weight of the building divided by the area of the raft. This is correct if the columns are more or less equally loaded and equally spaced, but it is difficult to fulfill this requirement in practice so it is allowed that the column loads and spacings to be varied within 20%. However if the downward loads on some areas are much heavier than on others, it is advisable to divide the raft into different areas and design each area on its corresponding average pressure. Continuity of the slab between such areas is commonly provided, although for the areas of great differences in pressure it is advisable to construct vertical construction joint through the slab and the superstructure to allow for differential settlement.
In flexible raft foundation the design cannot be based only on the strength requirements but it is necessary to be subjected due to the predicted settlement. The thickness and the amount of reinforcement of the raft should be selected in a such way to prevent development of cracks in the slab. As the differential settlement is not considered in the structural design it is customary to reinforce the raft with twice its theoretical amount of reinforcement. The quantity of the steel may be taken as 1% of the cross sectional area divided at top and bottom. The thickness of the slab should not be greater than 0.01 of the radius of curvature. The thickness may be .increased near columns to^{ }prevent shear failure.
There are two types of raft foundations: 1 Flat slab raft which is an inverted flat slab Fig.34a. If the thickness of the slab is not enough to resist punching shear under columns, pedestals may be .used above the slab Fig.34.b or, below the .slab, by thickening the flat slab under the columns as shown in Fig.34c. 2 Slab and girder raft which, is. an inverted R.C. floor, composed of slabs and beams extending along column, rows in both directions, Fig.34d, it is also, termed ribbed mat. If a continuous floor is desired in the basement, the ribs (beams) may be placed beneath the slab,Fig.34e.
Design of Flat Slab Raft The raft,_{ }which is of uniform thickness, is divided into column strips and middle strips as shown in Fig.35a. The width of column strip is equal to b + 2d, where b = column side. The depth of the raft d may assumed approximately equal to 1/10 the clear span between columns. Also, the width of column strip may be taken equal to 3 b. The column strips are design as continuous beams loaded with triangular loads as shown, in Fig.35b. The net intensity of the uniform upward pressure f_{n} under any area, for example the area DEFG may be assumed equal to one fourth of the total loads on the columns at D,E,F and G divided by the area DEFG. The total loads acting upon the column strip BDEQ, Fig.35a are assumed as triangular loading diagrams shown in Fig.35b. The total load on part DE, P_{DE}, is assumed to be the net pressure acting upon the area DHEJ.
Design of Rigid Raft (Conventional Method) The raft dimension is established and the resultant of all loads is found and the soil pressure is computed at various locations beneath the base from Eq. The raft is subdivided into a series of continuous strips (beams) centered on column rows as shown in Fig.37. The shear and moment diagrams can be established, using either combined footing analysis or beam moment coefficient The beam moment coefficients.The beam moment coefficient PI^{2} / 10 for long directions and PI^{2} / 8 for short directions may be adopted. The negative and positive moments will be taken as equal. The depth is selected to satisfy shear requirements without using stirrups and the tensile reinforcement is selected. The depth will usually be constant but the steel requirements may vary from strip to strip. The perpendicular direction is analyzed similarly. Design of Slab and Girder Raft (Ribbed Mat) If the column loads and spacings are equal or vary within 20%, the net upward pressure f_{n} acting against the raft is assumed to be uniform and equal to Q/A. where Q = weight of building at ground level, and A = area of raft (along the outside of the exterior columns). If this pressure is greater than the net allowable soil pressure, the area of the raft should be increased to an area large enough to reduce the uniform pressure to the net allowable value. This can be achieved by projecting the slab beyond the outside face of exterior columns. Referring to Fig. 38 , the various elements of the raft can be designed as follows:
Design of slab: 1Design of transverse beams B_{1} and B_{2} The uniform distributed load/m' on Let R_{1} and R_{2} be the central reaction of beams B_{1} and B_{2} on the central main beam B_{3} respectively. The end beams B_{1} carries only part of the load carried by the beam B_{2} and hence the central reaction R_{1} is assumed to be equal to KR_{2} where K is a factor based on comparative area, then Also it is assumed that the sum of the central reactions from the transverse beams B_{1} and B_{2} is equals to the total loads from central columns, thus 2R_{1}+8R_{2} =2P_{1} + 2 P_{2} (2) Solving Eqs. (1) and (2), R_{1} and R_{2} can be determined. The bending moment and shearing force diagrams can be drawn as shown in Fig.39. The reactions R_{1} and R_{2} can be determined by equating the sum of vertical forces to zero. The central section of the beams at positive bending moment can be designed as Tbeam as the slab is on the compression side. The sections of the beams below central beam B_{3 }is to be designed as rectangular section.
2 Design of central main beam B_{3} The loading, shearing force, and bending moment diagrams are shown in Fig.40a. The section can be designed as Tbeam.
3 Design of central main beam B_{4} The loading, shearing force, and bending moment diagrams are shown in Fig.40b The section can be designed as Tbeam
