Raft Foundation Analysis and Design Example

Chapter 6: Design of Foundations 79

6.1 INTRODUCTION

The substructure or foundation is the part of a structure that is usually placed below the surface of the ground.

Footings and other foundation units transfer the loads from the structure to the soil or rock supporting the structure.

Because the soil is generally much weaker than the concrete columns & walls that must be supported, the contact area between the soil & the footing is much larger than that between the supported member & the footing.

The more common types of footings are illustrated in figure (6.1). Strip footings or wall footings display essentially one-dimensional action, cantilevering out on each side of the wall. Spread Footings are pads that distribute the column load to an area of soil around the column. These distribute the load in two directions. Sometimes spread footing have pedestals, are stepped, or are tapered to save materials. A pile cap transmits the column load to a series of piles, which in turn, transmit the load to a strong layer at some depth below the surface “hard strata”. Combined footings transmit the loads from two or more columns to the soil. Such a footing is often used when one column is close to a property line. A mat or raft foundation transfers the loads from all the columns in a building to the underlying soil. Mat foundations are used when very weak soils are encountered.

The choice of foundation type is selected in consultation with the geotechnical engineer. Factors to be considered are:

• The soil strength, • The soil type, • The variability of the soil type over the area and with increasing depth, and • The susceptibility of the soil and the building to deflections.

The most basic and most common types are strip, spread, combined footings.

Eng Ahmad Al Omari Eng Essam Ghaith Eng Qutaiba Hameedi


Figure 6.1: (Types of Footings)

The two essential requirements in the design of foundation are that the total settlement of the structure be limited to a tolerably small amount and that differential settlement of the various parts of the structure be eliminated as nearly as possible. With respect to possible structural damage, the elimination of differential settlement, i.e., different amounts of settlement within the same structure, is even more important than limitations on uniform overall settlement.



To limit settlements as indicated, it is necessary to:

• Transmit the load of the structure to a soil stratum of sufficient strength. • Spread the load over a sufficiently large area of that stratum to minimize bearing

pressure.

A shallow single Foundation unit that supports all columns & walls of a structure or parts of a structure may be called a raft foundation. A raft foundation is also called as mat foundation. They are usually provided for multi-story buildings, overhead water tanks, chimneys, etc. A raft foundation becomes unavoidable in submerged structure, in some multi-story structures with basement and in retaining walls, etc. The raft foundation is usually designed as a flat slab.

Foundation engineering often consider mats when dealing with any of the following conditions:

The structural loads are so high or the soil conditions so poor that spread footings would be exceptionally large. As a general rule of thumb, if spread footings would cover more than about one-third of the building footprint area a mat or some type of deep foundation will probability be more economical,

The soil is very erratic & prone to excessive differential settlements. The structural continuity & flexural strength of a mat will bridge over these irregularities. The same is true of mats on highly expansive soils to prone to differential heaves,

The structural loads are erratic, and thus increase the likelihood of excessive differential settlement. Again, the structural continuity and flexural strength of the mat will absorb these irregularities,

Lateral loads are not uniformly distributed through the structure and thus may cause differential horizontal movement in spread footing or pile caps. The continuity of a mat will resist such movements,

The uplift loads are larger than spread footings can accommodate. The greater weight and continuity of a mat may provide sufficient resistance, and

The bottom of the structure is located below the ground table, so waterproofing is an important concern. Because mats are monolithic, they are much easier to waterproof. The weight of the mat also helps resist hydrostatic uplift forces from the groundwater.

In this project; due to the heavy load, earthquake design, and some of the previous provisions the mat foundation might be used.



Many buildings are supported on mat foundations, as are soils, chimneys, and other types of tower structures. Mats are also to support storage tanks and large machines. Typically, the thickness, (T), is 1-2 m (3-6 ft), so mats are massive structural elements. Although most mat foundation are directly supported on soil, sometimes engineers use pile –or shaft- supported mats, these foundation are often called piled rafts, and they are hybrid foundations that combine features of both mat and deep foundations.

Figure 6.2: (a Mat Foundation Supported Directly on Soil)

Figure 6.3: (A Pile or Shaft – Supported Mat Foundation)



6.2 RIGID VS. NON-RIGID

There are various methods have been used to mat foundations. They can be divided into two categories: RIGID METHOD & NON-RIGID METHODS. 6.2.1 Rigid method:

The simplest approach to structural design of mats is the rigid method (also known as the conventional method or the conventional method of static equilibrium). This method assumes the mat is much more rigid than the underlying soils, which means any distortion in the mat are too small to significantly impact the distribution of bearing pressure depends only on the applied loads and the weight of mat, and either uniform across the bottom of the mat (if the normal acts through the centroid and no moment load is present) or varies linearly a cross the mat (if eccentric or moment loads are present) as shown in figure (6.4), this is the same simplifying assumption used in the analysis of spread footings.

Figure 6.4: (Bearing Pressure Distribution for Rigid Method)

This simple distribution makes it easy to compute the flexural stresses and deflections (differential settlements) in the mat. For analysis purposes, the mat becomes an inverted and simply loaded two-way slab, which means the shears, moments, and deflection may be easily computed using the principles of the structural mechanics. The engineer can then select the appropriate mat thickness & reinforcement. Although this type of analysis is appropriate for spread footings, it doesn't accurately model mat foundations becomes the width-to-thickness ratio is much greater in mats and the assumption of rigidity is no longer valid.



Portions of a mat beneath columns and bearing walls settle more than the portions with loss load, which means the bearing pressure will be greater beneath the heavily-loaded zones, as shown in figure (6.5).

Figure 6.5: (The distribution of Soil Bearing Pressure)

This redistribution of bearing pressure is most pronounced when the ground is stiff compared to the mat as shown in figure (6.6), but is present to some degree in all soils.

Figure 6.6: (The distribution of Bearing Pressure under a Mat Foundation)



Because the rigid method does not consider this redistribution of bearing pressure, it doesn't produce reliable estimates of the shear, moments, and deformations in the mat. In addition, even if the mat was perfectly rigid, the simplified bearing pressure distribution in figure (6.6) are not correct-in reality; the bearing pressure is greater on the edges and smaller in the center than shown in this figure.

6.2.2 Non-Rigid methods:

To become the in accuracies of the rigid method by using analyses that consider deformations in the mat and their influence on the bearing pressure distribution. These are called non-rigid methods, and produce more accurate values of mat deformations and stresses, unfortunately non-rigid analyses also are more difficult to implement because they require consideration of soil-structure interaction and because the bearing pressure distribution is not as simple.

Coefficient of subgrade reaction:

Because non-rigid method consider the effects of local mat deformations on the distribution of bearing pressure, it is necessary to define the relation slip between settlement & bearing pressure. This is usually done using the coefficient of subgrade reaction, Ks (also known as the modulus of subgrade reaction, or the subgrade modulus).

Ks =

Where:

Ks = coefficient of subgrade reaction.

q = Bearing pressure.

δ = Settlement.

The coefficient Ks has units of force length cubed. Although we use the same units to wt., Ks is not the same as the same as the unit wt. and they are not numerically equal.

The interaction between the mat and the underlying soil may there be represented as a "bed of springs" each with a stiffness Ks per unit area, as shown in fig (6.7). Portions of the mat that experience more settlement produce more compression in the "springs," which represents the higher bearing pressure, whereas portions that settle less don't compress the springs as for and thus have less bearing pressure. The sum of these spring forces must equal the applied structural loads plus the wt. of the mat:

∑ + - uD = =



Where:

∑ = sum of structural loads acting on the mat.

Wf = Pore of the mat.

uD = Bearing pressure between mat & soil.

A = mat-soil contact Area.

δ = settlement at a point on the mat.

Figure 6.7: (The Coefficient of Subgrade Reaction – bed of springs)

This method of describing bearing pressure is called a soil-structure interaction analysis because the bearing pressure depends on the mat deformations, and the mat deformations depends on the bearing pressure.

Methods in non-rigid:

1. Winkler method. 2. Coupled method. 3. Pseudo-coupled method. 4. Multiple-parameter method. 5. Finite element method.



Determination of the coefficient of subgrade reaction:

Most mat foundation designs are currently developed using either the Winkler method or the pseudo-coupled method, both of which depend on our ability to define the coefficient of subgrade reaction, Ks. Unfortunately, this task is not as simple as it might first appear because Ks is not a fundamental soil property. Its magnitude also depends on many other factors, including the following:

1. The width of the loaded area: A wide mat will settlement more than a narrow one with the same q because it mobilizes the soil to a greater depth, therefore, each has a different (ks).

2. The shape of the loaded area: The stresses below long narrow loaded areas are different from those below square loaded areas therefore, ks will differ.

3. The depth of the loaded area below the ground surface – At greater depths, the change in stress in the soil due to q is a smaller percentage of the initial stress, so the settlement is also smaller and ks is greater.

4. The position on the mat – To model the soil accurately, ks need to be larger near the edges of the mat and smaller near the center.

5. Time - Much of the settlement of mats on deep compressible soils will be due

to consolidation and thus may occur over a period of several years. Therefore, it may be necessary to consider both short-term and long-term cases.

Actually, there is no single ks value, even if we could define these factors because the q-δ relationship is nonlinear and because neither method accounts for interaction between the springs.

Engineers have tried various techniques of measuring or computing ks. Some rely on plate load test to measure ks in situ. However, the test results must be adjusted to compensate for the differences in width, shape, and depth of the plate and the mat. Plate load tests include dubious assumption that the soils within the shallow zone of influence below the plate are comparable to those in the much deeper zone below the mat. Therefore, plate load test generally do not provide good estimates of ks for mat foundation design.

Others have derived relationships between ks and the soils modulus of elasticity, E (Vesic & Saxena, 1970). Although these relationships provide some insight, they too are limited.

Another method consists of computing the average mat settlement using the techniques of settlement and expressing the results in the form of ks using equation:

ks =



6.3 STRUCTURAL DESIGN:

6.3.1 General Methodology:

The structural design of mat foundations must satisfy both strength and serviceability requirements. This requires two separate analyses, as follows: Step (1):

Evaluate the strength requirements result from the load combinations and LRFD design methods (which ACI calls ultimate strength design). The mat must have a sufficient thickness, T, and reinforcement to satisfy resists these loads. As with spread footings, T should be large enough that no shear reinforcement is needed. Step (2): Evaluating mat deformations (which is the primary serviceability requirement) using the unfactored loads. These deformations are the result of concentrated loading at the column locations, possible non-uniformities in the mat, and variations in the soil stiffness. In effect, these deformations are the equivalent of differential settlement. If they are excessive, then the mat must be made stiffer by increasing its thickness.

6.3.2 Closed-Form solutions:

When the Winkler method is used (i.e., when all “springs” have the same Ks) and the geometry of the problem can be represented in two-dimensions, it is possible to develop closed-form solutions using the principles of structural mechanics. These solutions produce values of shear, moment, and deflection at all points in the idealized foundation. When the loading is complex, the principle of superposition may be used to divide the problem into multiple simpler problems. These closed-form solutions were once very popular, because they were the only practical means of solving this problem. However, the advent and widespread availability of powerful computers and the associated software now allows us to use other methods that are more precise and more flexible.



6.3.3 Finite Element Method:

Today, most mat foundations are designed with the aid of a computer using the finite element method (FEM). This method divides the mat into hundreds or perhaps thousands of elements. Each element has certain defined dimensions, a specified stiffness and strength (which may be defined in terms of concrete and steel properties) and is connected to the adjacent elements in a specified way. The mat elements are connected to the ground through a series of “springs,” which are defined using the coefficient of subgrade reaction. Typically, one spring is located at each corner of each element. The loads on the mat include the externally applied column loads, applied line loads, applied area loads, and the weight of the mat itself. These loads press the mat downward, and this downward movement is resisted by the soil “springs.” These opposing forces along with the stiffness of the mat can be evaluated simultaneously using matrix algebra which allows us to compute the stresses, strains, and distortions in the mat. If the results of the analysis are not acceptable, the design is modified accordingly and reanalyzed. This type of finite element analysis does not consider the stiffness of the superstructure. In other words, it assumes the superstructure is perfectly flexible and offers no resistance to deformations in the mat. This is conservative. The finite element analysis can be extended to include the superstructure, the mat, and the underlying soil in a single three-dimensional finite element method. This method would, in principle, be a more accurate model of the soil structure system, and thus may produce a more economical design. However, such analyses are substantially more complex and time-consuming, and it is very difficult to develop accurate soil properties for such models. Therefore, these extended finite element analyses are rarely performed in practice.



6.4 DESIGN PROCESS:

6.4.1 General Description:

After performing analysis on 3-D ETABs software Model, all results were obtained; the base plan then imported with all loads and load combinations to SAFE software in order to analyze the base as raft foundation.

Figure 6.8: (Undeformed Shape for Raft Foundation)

Raft dimensions were selected in order to primarily achieve no bearing capacity and punching shear problems then serviceability criteria to be checked based on the selected dimensions.

The raft covers the base plan with the following offsets relative to the above raft sketch:

• 2.4 m to the right, • 1.0 m to the left, • 1.3 m up, and • 1.5 m down.

Starting with depth, T = 500 mm and by trial and error procedure a depth of T

= 700 mm was selected.

Due to large variation in loads on columns; for economic wise a suitable drop of 600 mm was made under heavy load columns.



6.4.2 SAFE Outputs:

Figure 6.9: (Deformed Shape of Raft Foundation for Ultimate Combination)

Figure 6.10: (Bearing Pressure for Soil beneath the Raft Foundation for Service Combination)



Figure 6.11: (Punching Shear Ratios under Interior Columns for Ultimate Combination)

Figure 6.12: (Punching Shear Ratios under all Columns for Ultimate Combination)



Figure 6.13: (Bending Moment Diagrams for Given X‐Strips)

Figure 6.14: (Bending Moment Diagrams for Given Y‐Strips)



Figure 6.15: (Shear Diagrams for Given X‐Strips)

Figure 6.16: (Shear Diagrams for Given Y‐Strips)



6.4.3 Sample Calculation:

• General Raft (depth = 700 mm): As, min = 0.0018 b h = (0.0018) (1000) (700) = 1260 mm2 / m

252 mm2/ 200 mm Use 1Φ18 / 200 mm … Top & Bottom

• Drop (depth = 600 mm) : As, min = 0.0018 b h = (0.0018) (1000) (1300) = 2340 mm2 / m

268 mm2/ 200 mm

Use 1Φ25/ 200 mm … Top & Bottom

• Wherever minimum reinforcement exceeded; an additional reinforcement must be added. This depends on the values of moment in each strip in both X & Y directions. The following tables can ease this mission.

Table 6.1: (Reinforcement Guideline for General Raft)

Reinforcement Bars Area of Steel (mm2 per meter)

5Φ18 1272 5Φ18 + 5Φ12 1838 5Φ18 + 5Φ14 2042 5Φ18 + 5Φ16 2277 5Φ18 + 5Φ18 2544 5Φ18 + 5Φ20 2843 5Φ18 + 5Φ25 3173 5Φ18 + 5Φ32 3726

Table 6.2: (Reinforcement Guideline for Drops)

Reinforcement Bars Area of Steel (mm2 per meter)

5Φ25 2454 5Φ25 + 5Φ12 3020 5Φ25 + 5Φ14 3224 5Φ25 + 5Φ16 3459 5Φ25 + 5Φ18 3726 5Φ25 + 5Φ20 4025 5Φ25 + 5Φ25 4908 5Φ25 + 5Φ32 6475



• For X-strip:

Mu, bottom = 2150 kN.m / m

h = 1300mm

d = 1150mm

Ru = M .

= .

= 1.806

ρ = . ′ ( 1 – 1 R

. ′ )

= . (1 – 1 ..

)

= 0.0045

As, required = ρ * b * d = 0.0045 * 1000 * 1150 = 5175 mm2

Use 5Φ25 + 5Φ32

Mu,Top = - 250 kN.m / m h= 700mm

d= 575mm

Ru = M

. =

. = 0.840

ρ = . ′ ( 1 – 1 R

. ′ )

= . ( 1 – 1 . .

)

= 0.0020

As req = ρ * b * d = 0.0020 * 1000 * 575 = 1150 mm2

As req < As min ……… Use As,min = 1272 mm2

Use 5Φ18



• For Y-strip:

Mu,bottom = 1686 kN.m / m h= 1300mm

d= 1150mm

Ru = M

. =

. = 1.416

ρ = . ′ ( 1 – 1 R

. ′ )

= . ( 1 – 1 ..

)

= 0.0035

As req = ρ * b * d = 0.00348 * 1000 * 1150 = 4025 mm2

Use 5Φ25+ 5Φ25

Mu,Top = - 490 kN.m / m h = 700mm

d = 575mm

Ru = M

. =

. = 1.647

ρ = . ′ ( 1 – 1 R

. ′ )

= . ( 1 – 1 . .

)

= 0.00406

As req = ρ * b * d = 0.00406 * 1000 * 575 = 2338 mm2

Use 5Φ18 + 5Φ18



• Check Punching shear:

For the purpose of checking punching shear; an Equivalent 800 mm side Square column corresponding to the circular column of diameter of D = 900 mm was taken;

C = 800 mm.

d = 1150 mm. “ assumed”

b0 = c +d = 800 + 1150 =1950 mm

Mu= 3110 kN.m "From SAFE Model"

Pu= 8791 kN “ from ETABS Model”

Vuc = )

= . .

. . = 3085.2 kN

ΦVc = Φ ′ = (0.75) 1950 1150 √28 = 3955.4 kN

Punching Shear Ratio = ΦV

= 0.78 "OK"



Figure 6.17: (Raft Foundation Details)



Figure 6.18: (Sections in Raft)


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Raft Foundation Analysis and Design Example