Arab J Sci Eng (2012) 37:277–290 DOI 10.1007/s13369-012-0178-5
R E S E A R C H A RT I C L E - C I V I L E N G I N E E R I N G
N. Al-Shayea · H. Zeedan
A New Approach for Estimating Thickness of Mat Foundations Under Certain Conditions
Received: 13 June 2010 / Accepted: 21 June 2011 / Published online: 25 January 2012 © King Fahd University of Petroleum and Minerals 2012
Abstract This paper presents work based on a modern approach that offers the potential for modeling both the whole structure (superstructure and mat) and the subgrade (soil) component quite accurately, to overcome the shortcoming of the separate modeling of each part. In addition, this approach considers the rigidity of superstructure (flexural stiffness of each floor) and rigidity of mat within one 3-D soil structure interaction model. To implement the model, a complete 3-D model was used for the superstructure, the mat foundation and the soil. The soil was modeled as a 3-D solid finite element elastic material connected to the mat foundation. The mat foundation was modeled as a 3-D finite plate element. Both soil and concrete material of mat were taken as an elastic material, by modulus of elasticity and Poisson’s ratio. The superstructure was modeled as a multistory building consisting of 2–15 stories with different column spacing varying from 3 to 7 m. STAADPRO software was used for the analysis of this model. The numerical results are studied, and summarized in the form of design charts to show the relationship between thickness of mat foundation and each of the following: number of stories, column spacing, subgrade material (type of soil), total settlement, differential settlement and soil pressure. Thickness of mat foundation can be estimated from the developed curves and charts, for various design parameters, including: soil type, maximum allowable settlement, maximum allowable differential settlement, and maximum allowable bearing capacity. Keywords 3-D model · Design charts · High-rise buildings · Mat foundation · Mat thickness · Settlement · Soil structure interaction · Subgrade reaction
N. Al-Shayea (B) · H. Zeedan Civil Engineering Department, King Fahd University of Petroleum and Minerals, Box 368, Dhahran 31261, Saudi Arabia E-mail:
[email protected]
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1 Introduction A mat or raft foundation is usually a large concrete slab supporting a number of columns or an entire structure. Mat foundations are used where soil has low bearing capacity and when the column loads are so high (in multistory buildings) that spread footings will cover more than 50% of the building area [1]. Combining individual footings in one mat will not only reduce the contact pressure, but also will increase bearing capacity. Other advantages of mat foundations are to reduce the differential settlements depending on the rigidity, or to bridge over subsurface cavities. Mat foundations were used to support important buildings, such as historic monuments [2], nuclear facilities, [3], and multistory buildings. The Washington Monument is a classical example for mat foundation, which was construction in 1848; as a 24.38 m2 [2]. Mat foundations are also used with pile (piled-raft foundations) to support the load of high-rise buildings or towers, when the raft has adequate load-carrying capacities, but the settlement or differential settlement exceed allowable values [4]. In practice, most of the analysis and design of mat foundation focuses on separate solutions for geotechnical or structural aspects. Thus, the interaction between the soil and the overall structure (mat + superstructure) are ignored, and the contact soil pressure and/or the coefficient of subgrade reaction are assumed as input data. However, the contact soil pressure and the coefficient of subgrade reaction are supposed to be output results. This paper attempts to obtain an estimation of mat foundation thickness (flat plate) for multistory buildings by considering the ideal soil structure interaction, modeling the superstructure (the building), substructure (the mat), and the subgrade (the soil) in one structural analysis model. The objective of this paper is to develop design charts, for helping practicing engineers to estimate the thickness of mat foundations. To achieve this objective, for different type of soil and column spacing, investigation was made on the relationship between mat thickness and rigidity factor (relative stiffness), and also on the effect of the acting loads (number of stories) on the mat thickness, total settlement, differential settlement, maximum contact soil pressure, and modulus of subgrade reaction. 2 Background Mat foundations should be safe against structural failure, bearing capacity failure, and settlement failure. Bearing capacity failure could be due to either one of the following three modes: general shear failure, local shear failure, or punching shear failure. On the other hand, settlement could be due to elastic/immediate settlement, consolidation settlement, and creep settlement, depending on the soil type. In many instances, the allowable bearing capacity of mat foundation is limited by the tolerable/allowable settlement [5]. Total settlement of mat foundations can range from 75 to 125 mm for clays and 50–75 mm for sands [1]. Settlement of mat foundations is also investigated in the literature [6]. A problem of more considerable concern is the differential settlement, which induces moments and shear forces in the superstructure depending on the relative (differential) movement. Differential settlement can be computed as the difference in settlement between two adjacent points [1]. Mat tends to reduce the value of differential settlement, depending on its own rigidity. Differential settlement can be estimated as a 3/4 of the computed maximum total settlement. ACI Committee 336 suggested a method for calculating differential settlement of mat foundations, based on the relative stiffness between the structure and the soil (rigidity factor, K r ) presented by [7], as: K r = E Ib /E s B 3
(1)
where: E = modulus of elasticity of the structure, E s = modulus of elasticity of the soil, B = foundation width, and Ib = moment of inertia of the structure per unit length perpendicular to B. Based on the value of K r , the ratio of the differential settlement to total settlement can be estimated to be 0.0 for K r > 0.5 (rigid mat), 0.1 for K r = 0.5, and 0.35 (for square mats) 0.5 (for long mats) for K r = 0. Thus, K r also determines whether the mat is flexible or rigid. Modulus of subgrade reaction is a conceptual relationship between soil pressure and deflection, which is used in the analysis of various foundations. The coefficient of subgrade re-action (ks ) is a ratio between the subgrade reaction/pressure (q), and the corresponding settlement (δ), as follows: ks = q/δ
(2)
Stress distribution under a symmetrical loaded footing is not uniform. The actual stress distribution depends on both rigidity of the footing and type of the soil. For cohesionless soils, sand, the pressure distribution
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depends on the depth of embedment of the foundation, but it is usually higher in the middle. For clays, similar to elastic materials, the pressure near the edges is greater, which is similar to the pressure distribution for rigid foundation on any soil material. The pressure distribution for very stiff or dense soils is high under the columns. For settlement, flexible uniformly loaded base settle more at the center than at the edges. The settlement of a relatively flexible footing supported on a clay soil (elastic soil) will be greatest at the center of the footing although the contact soil pressure is uniform. If the footing is considered rigid, the settlement is uniform and the unit soil pressures are greater at the edge of the footing. So, modulus of subgrade reaction is not constant under the foundation. The finite deference method and the finite element method utilize thin plate theory. The finite grid method does not account for the plate thickness. Whether a 2-m-thick mat is a thin plate or not is a valid question. But literature has shown that a plate would have to be quite thick to invalidate the “thin” plate theory model. So, a plate around 2 m thick can be used in a plate model without disturbing the plate theory. A thin plate analysis is generally used and is adequate, but a thick plate modeling is more accurate and closer to the actual situation. Estimation of mat thickness is obtained by comparing the mat thickness requirements from structural and geotechnical aspects. Structural aspects include moment, shear, punching shear, and ACI minimum requirements. The largest value is selected to be the governed design value. However this value needs to be checked against geotechnical parameters. While shear failure in soil below mat foundations is very rare, settlement criteria usually governs the allowable bearing capacity. The total vertical settlement does not generally cause structural or architectural distress, but the differential settlement is the most important geotechnical parameter that controls the mat foundation design. So, checking against allowable differential settlement should be performed to get the final design of the mat thickness. There are different methods for the analysis and design of mat foundations; i.e., the conventional rigid method, the approximate flexible method, the numerical methods (finite difference, finite element, and finite grid methods), and the soil–structure interaction (SSI) approach. The conventional rigid method is an approximate method where the mat is assumed to be infinitely rigid. The mat is divided into several strips in x and y directions loaded by a line of columns and resisted by soil pressure. The soil pressure is linearly distributed and the center of the soil pressure coincides with the line of action of the resultant column loads. These strips are then analyzed and designed as combined footings. This method can be used when the mat is very rigid, the column spacing pattern is fairly uniform in both directions, and column loads do not vary much over 20% [7]. This method is not recommended at present because of the substantial amount of approximations and the wide availability of computer programs using the finite element method. The approximate flexible method assumes the soil to be equivalent to an infinite number of elastic springs (Winkler foundation), with elastic constant as the coefficient of subgrade reaction (ks ). The moments, the shear, and deflection can be computed using the various equations. Kashikar et al. [3] suggested the use of more rational iterative methods like variable ‘k’ approach. The finite element method considers the mat foundation as a plate on elastic foundation, and transforms it into a computer-oriented procedure of matrix structural analysis. The plate is idealized as a mesh of finite elements interconnected only at the nodes (corners), and the soil may be modeled as a set of isolated springs (Winkler foundation). A computer program was developed in which the plate is divided into a mesh of rectangular or triangular elements as well [8]. The same mesh applies to the soil, and the stiffness (represented by elastic springs) is concentrated at the nodal points. Finite element analysis is also used to study the behavior of pile-raft foundations [9]. Mat foundation is one of the most common SSI applications encountered in practice [10]. The actual mat foundation problem is highly complex and statically indeterminate. Prior to the availability of digital computers, the problem was traditionally broken into its three basic components (superstructure, mat and subgrade “soil”). In the traditional solution, structural analysis and design of the mat is performed using the loads applied from the analysis of the superstructure, as a known input. The key aspect of this step is that the subgrade reaction (soil pressure), must be assumed beforehand, magnitude and distribution. It is noted that soil pressure is not necessarily uniform although it is generally continuous. A geotechnical settlement analysis is performed to calculate the pattern of mat settlements. There are three modern SSI solution/alternatives. A “structural” alternative is the approach in which the superstructure and mat are combined into one structural model (mega-structure). Analysis of the mega-structure can be accomplished easily, even in three dimensions if desired, with the commercially available computer software for structural analysis. The primary shortcoming of this alternative is that the subgrade reaction must either be assumed beforehand or modeled mathematically.
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The “geotechnical” alternative is to model both the mat and subgrade in one analysis. This allows very sophisticated constitutive models to be used for the subgrade material(s). The geotechnical alternative has at least two significant shortcomings, namely the lack of ability to modeling superstructure interaction effects directly and the difficulty in performing a three-dimensional geotechnical analysis. The ideal solution combines mat and superstructure together as a single structural system (mega-structure) that is in contact with the ground. As a result of some system of loads applied to the mega-structure, there will be displacements, including vertical downward displacement (into the ground) at foundation level, i.e. settlement. The ability to analyze both the mega-structure and subgrade requires using numerical tools, such as the finite-element method. A 3-D model would be required for such problems. The modeled portion of the subgrade would have to continue for some distances in both horizontal directions beyond the limits of the foundation, as well as extending to some depth below foundation level. Computer software would be required in such a case. Both geotechnical and structural engineers would be required to prepare the input as well as to interpret the results. The parameters considered necessary for any mat foundation problem (bending moments and total settlements) would be produced as part of solving the overall system. Since there are no assumptions or approximations involved in obtaining the necessary results, the ideal solution was considered and adopted in this paper. 3 Model, Methodology, and Assumptions The model is a 3-D model developed for superstructure, mat foundation and soil, as shows in Fig. 1 from STAAD-PRO software. Soil was modeled as a 3-D solid finite element connected to the mat foundation. The mat foundation was modeled as a 3-D finite plate element. The superstructure was modeled as a multistory building consisting of 2, 4, 7, 10, 12, or 15 stories, and with column spacing of 3, 4, 5, 6, or 7 m. 3.1 Superstructure The superstructure was modeled as a multi-story concrete building with five grid column lines, equally spaced in both directions, to form a square shape. The distances between column grid lines are 3, 4, 5, 6, or 7 m. The numbers of stories are 2, 4, 7, 10, 12, or 15. The height of every story (column height) is taken equal to 3 m, as a normal height for residential buildings. The column size was taken 600 mm × 600 mm. Beams are assumed on all grid lines, with a cross sections taken equal to 200 mm × 500 mm, for spans up to 5 m; 200 mm × 600 mm for spans of 6 and 7 m. The total number of member/beam elements in the model varies from 130 to 975 for 2–15 stories, respectively. The material for the superstructure and the mat foundation were taken as a reinforced concrete with constant properties of modulus of elasticity E = 25, 000 MPa, Poisson ratio μ = 0.17, density of 24kN/m3 , fc = 28 MPa (4 ksi), f y = 415MPa (60 ksi), shear strength = 652 kPa, punching shear strength = 1.75 MPa, flexural resistance factor ( R) = 1.4 MPa (204 psi), and the minimum steel reinforcement ratio (ρ) = 0.0035. Slab thickness is 100 mm for S = 3 and 4 m, 150 mm for S = 5 m, and 200 mm for S = 6 and 7 m). Thickness of internal and external walls is 150 and 200 mm, respectively. Height of parapet wall is 1 m. 3.2 Mat Foundation The mat foundation was modeled as a concrete solid flat slab on grade with constant thickness. It is square mat with a width equalling four times the column spacing (B = 4*S) with 1 m projection all around, i.e., B = 14, 18, 22, 26 and 30 m for S = 3, 4, 5, 6 and 7 m, respectively. An assumed thickness was set as an initial design thickness, and then the design thicknesses were computed from the analysis. Mat foundation was divided into square plate elements 1 m × 1 m, and the numbers of these plate elements were 196, 324, 484, 676 and 900 for S = 3, 4, 5, 6 and 7, respectively. Mat concrete sections are design as per ACI code. 3.3 Soil The soil was assumed to be elastic material with different modulus of elasticity (E s ) of 10, 50, 100, and 250 MPa, and a constant Poisson ratio μ = 0.35. The soil under the mat foundations was modeled as 3-D
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Fig. 1 Model for superstructure, mat foundation and soil
solid element. The influenced soil body was taken to extend to a depth equal to 2*B beneath the mat, and to a distance equal to B from all sides beyond mat foundation limit, where B is the width of the mat. The dimensions of solid element were set as 1 m × 1 m × H ; where, H is the depth of the solid element, which was taken as 4 m under the mat and increased by increasing the soil depth. Every one solid element row represents one of the soil layers. Effect of water table was not considered in this work. The depth of the upper soil layer was kept constant equal to 4 m for all cases of column spacing. Depths of other layers were increased gradually to achieve the required soil depth without increasing the number of solid elements to avoid huge stiffness matrix and to optimize numerical computations. The self weight of the soil was not considered in the model, so as to get the additional soil pressure under the mat foundation due to the external loads only. Maximum allowable total settlement (δ) = 65 mm, and maximum allowable differential settlement () = 48 mm.
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Table 1 Allowable bearing capacity (kPa) E s (MPa)
Mat width, B (m) 12
16
20
24
28
10 50 100 250
85 440 > 500 > 500
65 325 > 500 > 500
≤ 50 265 > 500 > 500
≤ 50 220 435 > 500
≤ 50 185 375 > 500
Under conditions of no ground water
Fig. 2 Soil pressure under a rigid mat
3.4 Supports The supports of the structural model were selected to be at the bottom soil layer (at distance equal to 2B from the mat foundation level). All supports are modeled as pinned supports, where rotation and displacements are equal to zero in three directions (x, y and z), which represent the most actual case at depth of 2B.
3.5 Loads All applicable loads (such as dead loads, live loads, wind loads, and seismic loads) were considered as per the guidelines of ACI, UBC, and ASCE codes, and Saudi Aramco standards, as listed below. The model was subjected to various load combinations as per ASCE and/or ACI. 3.5.1 Live Loads The uniformly distributed floor live loads used in the analysis was considered to act vertically on the floor, and was set equal to 2 kN/m2 . Roof live load acting vertically on the roof was equal to 1 kN/m2 .
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Fig. 3 Settlement under a flexible mat
3.5.2 Dead Loads Dead loads include weight of all permanent constructions, such as walls, floor, roof framing and covering materials (floor finish). Collateral loads are all specified as additional dead loads other than the material or building framing, such as mechanical and electrical system and ceiling. Dead loads were taken as follows: wall loads hollow block unit weight = 20kN/m3 , concrete unit weight = 24kN/m3 , floor finish = 1.5kN/m2 , and collateral loads = 0.5kN/m2 .
3.5.3 Wind Loads The following information related to wind loads is used regardless of whether wind load govern the design of the lateral force resisting system of the building: wind speed was considered for Dhahran area (Saudi Arabia) as per Saudi Aramco standard SAES-A-112 [11], meteorological seismic design data, basic wind speed 78 mph, wind importance factor, Iw = 1, wind exposure = C, the applicable pressure qs = 0.79 kN/m2 for wind speed =78 mph, and wind loads are applied from one direction because of symmetry.
3.5.4 Seismic Loads The following information related to seismic loads is used regardless of whether seismic load govern the design of the lateral force resisting system of the building: for Dhahran area seismic zone is not applicable where the zone is ZERO as per Aramco standard SAES-A-112 [11], meteorological seismic design data. However, seismic loads are not applicable as per local code of Saudi Aramco, seismic zone area was considered as zone ONE as per international code (UBC [12]) for Dhahran area, seismic importance factor, I E = 1, numerical coefficient R=3.5 for resisting moment concrete frame as per UBC.
3.5.5 Load Combinations For factored/ultimate load combinations, ACI-factored loads were applied in the model for concrete section design, and for soil pressure for punching calculations. For serviceability (settlements and soil pressure), unfactored loads combinations of ASCE were used. Factored-load combinations were considered of soil pressure for punching calculations, as per ACI.
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Fig. 4 Settlement (a) and soil pressure (b) under flexible mat on very stiff soil
4 Results and Discussions Based on theory of elasticity, settlement and bearing pressure charts for different soil types (different E s ) are obtained for different widths of mat foundation (different column spacing). The settlement criteria, rather than shear failure criteria, usually governs the allowable bearing pressure. Considering the allowable settlement for mat foundation equals to 65 mm, the value of allowable bearing pressure corresponding to the allowable total settlement is given in Table 1.
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Fig. 5 The total settlement (a) and the soil pressure distribution (b) for a rigid mat under lateral loads
Under gravity loads, for the case of soil with E s = 10 MPa, S = 3 m, and number of stories= 15 floors, the following are found. The design thickness of the mat foundation = 1,100 mm, K r = 0.754 (> 0.5), the total settlement = 203 mm at the center, 201 mm at the edge, and 198 mm at the corner. This means that the mat is acting as rigid, according to ACI-336 definition. Also, the nodal soil pressure (i.e. soil pressure at nodes of the solid element) under the mat is greater at the edge (336 kPa) than at the center (129 kPa). This criterion matches ACI-336 for rigid mat on elastic soil. Figure 2 shows the soil pressure distribution below this mat. For soil with E s = 50 MPa, S = 3 m, four floors, the followings were obtained. The design mat thickness = 400 mm, K r = 0.033 (< 0.5). The total settlement = to 14 mm at the center, 11.0 mm at the edge, and 8.0 mm at the corner of the mat (the ratio = 0.42). This means that the mat flexible according to ACI-336.
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Fig. 6 Distribution of Mx for rigid mat—vertical loads
Fig. 7 Distribution of Mx for flexible mat—vertical loads
The dish shape of the total settlement under the mat is clearly shown in Fig. 3. The nodal soil pressure under the mat is almost uniform (at the center = 29.0 kPa, at the edge = 28.0 kPa, and at the corner = 26.0 kPa). This matches ACI-336 for flexible mat of elastic soil where the contact soil pressure is uniform although the settlement is greatest at the center.
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Fig. 8 Distribution of Mx for flexible mat on stiff soil—vertical loads
Es = 50 Mpa 0.55 0.5
Rigidity Factor (Kr)
0.45
S=3, (B=12m) S=4, (B=16m) S=5, (B=20m) S=6, (B=24m) S=7, (B=28m)
0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0
0
500
1000
1500
2000
Mat Thickness (mm)
Fig. 9 Rigidity factor Kr versus mat thickness
For very stiff or dense soil (E s = 250 MPa,S = 7 m, two floors, the followings are obtained. The mat thickness = 450 mm and governed by shear requirements, K r = 0.0002 (< 0.5), the total settlement was found equal to 4.134 mm at the center, 2.609 mm at the edge, and 1.701 mm at the corner, and the ratio of differential settlement to total settlement = 0.58. This means that the mat is flexible, according to ACI-336. The shape of the total settlement below the mat is shown in Fig. 4a. When a mat is on very stiff soil, the column loads are transmitted to a relatively small area of the soil directly below the columns. The maximum soil pressure is found at the center to be 99 kPa. The pressure distribution is shown in Fig. 4b which is in accordance with the one in the literature. 4.1 Lateral Loads The effect of horizontal forces was considered in this study, and they include wind and earthquake (seismic) loads. Results indicated that the lateral loads have a major impact on the distribution and the magnitude of both the contact pressure and the settlements. This effect depends on the soil type and the height of the building (number of floors). The effect increases with decreasing E s , and with increasing the building height. For weak
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Es = 50 Mpa S=3, (B=12m) S=4, (B=16m) S=5, (B=20m) S=6, (B=24m) S=7, (B=28m)
90
Max. Settlement (mm)
Max. soil pressure - Nodal (KN/m2 )
100
80 70 60 50 40 30 20 10 0
0
1
2
3
4
5
6
8
7
Es = 50 Mpa
500.0 S=3, (B=12m) S=4 (B=16m) S=5 (B=20m) S=6 (B=24m) S=7 (B=28m)
450.0 400.0 350.0 300.0 250.0 200.0 150.0 100.0 50.0 0.0
9 10 11 12 13 14 15 16
0
1
2
3
4
NO. OF FLOORS
5
6
7
(a)
Es = 50 Mpa 9000.0 8000.0
50
7000.0 S=3 (B=12m) S=4 (B=16m) S=5 (B=20m) S=6 (B=24m) S=7 (B=28m)
40 30
Ks (Kn/m 2 )
Max. Diff Settlement (mm)
9 10 11 12 13 14 15 16
(c)
Es = 50 Mpa
60
8
LOADS - NO. OF FLOORS
20
6000.0 5000.0 4000.0 3000.0
S=3, (B=12m) S=4, (B=16m) S=5, (B=20m) S=6, (B=24m) S=7, (B=28m)
2000.0 10
1000.0 0.0
0 0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16
NO. OF FLOORS
(b)
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16
NO. OF FLOORS
(d)
Fig. 10 a Maximum settlement versus number of floors. b Maximum differential settlement versus number of floors. c Maximum soil pressure versus number of floors. d Maximum soil Ks versus number of floors
soils (E s = 10 to 50 MPa), load combination with lateral loads always governs. The only exception is for E s = 50 MPa with two floors, which is governed by gravity loads. For E s = 100 MPa, the cases governed by gravity loads are up to four floors. For E s = 250 MPa, the cases governed by gravity loads are up to seven floors. For a rigid mat under load combinations including lateral loads, the total settlement and the soil pressure distribution under the mat are shown in Fig. 5. No tension or soil separation was encountered.
4.2 Bending Moment and Shear Stress Under vertical loads, for rigid mat (E s = 10 MPa, S = 3 m, and 15 floors), the analysis of bending moment and shear stress for shows that all bending moment are positive. The maximum bending moment (1,212 kN m/m) occurred at the center of the mat. Figure 6 shows the distribution of Mx . Maximum shear stress (537 KPa) is at the edge of column. All values of shear stresses are less than allowable as per ACI criteria. For flexible mat soil (E s = 50 MPa, S = 3 m, and 4 floors), the distribution of Mx is shown in Fig. 7, with the maximum moment (70 kN m/m) occurred at the column locations. In between columns, the bending moment is minimum. Shear stress values are less than allowable as per ACI. However, the moment is more critical than shear; and the mat thickness is governed by ACI minimum requirements. For flexible mat on stiff soil, the distribution of Mx is shown in Fig. 8, with the bending moment has positive and negative signs. The maximum values (88 kN m/m) at column locations and center of the mat. In between columns the bending moment is minimum. All values of shear stresses are less than allowable as per ACI criteria. Shear stresses are more critical than moment and govern the design mat thickness. The relationship between the mat thickness and the rigidity factor/relative stiffness (K r ) for soil with E s = 50 MPa, are presented in Fig. 9.
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Es = 10 Mpa
2500
Es = 100 Mpa 2500
S=3, (B=12m) S=4, (B=16m) S=5, (B=20m) S=6, (B=24m) S=7, (B=28m) >δ (S7,B28) >δ (S4,B16) >δ (S3,B12) >δ (S5,B20) >δ (S6,B24)
2000 1500
MAT THICKNESS (MM)
MAT THICKNESS (MM)
289
1000 500
2000 1500 1000 500 0
0
S=3,( B=12m) S=4, (B=16m) S=5, (B=20m) S=6, (B=24m) S=7, (B=28m)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
0 1 2 3 4 5 6 7
NO. OF FLOORS
Es = 50 Mpa
2000 1500 1000 500 0
2000
S=3,( B=12m) S=4, (B=16m) S=5, (B=20m) S=6, (B=24m) S=7, (B=28m)
1500 1000 500 0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Es = 250 Mpa
2500
MAT THICKNESS (MM)
MAT THICKNESS (MM)
S=3,( B=12m) S=4, (B=16m) S=5, (B=20m) S=6, (B=24m) S=7, (B=28m) >δ (S7,B28) >δ (S6,B24)
9 10 11 12 13 14 15 16
(c)
(a) 2500
8
LOADS - NO. OF FLOORS
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
LOADS - NO. OF FLOORS
(b)
LOADS - NO. OF FLOORS
(d)
Fig. 11 a–d Mat thickness versus number of floors
The results obtained from the analysis of the model were compiled into sets of curves to simplify presentations for all soil types. For different column spacing, the effect of acting loads (number of stories) on the same parameters, for soil with E s = 50 MPa, are presented in Fig. 10. These parameters are maximum total settlement, differential settlement, maximum contact soil pressure, and modulus of subgrade reaction (ks ). The relationship between the thickness of mat and the acting loads (number of stories) for different column spacing and different soil types E s = 10, 50, 100, and 250 MPa, are shown in Fig. 11a, b, c, and d, respectively. These plots These design charts were developed for square mat foundation, supporting columns that are equally spaced in both directions, with five columns along each line. These charts can be used as follows: 1. Select the appropriate chart, base on the soil type (E s ). 2. Determine the thickness of the mat, based on the number of floors. 3. Interpolation can be used for other values of E s . 5 Conclusions The following conclusions are drawn from the results obtained, and are limited to the assumptions indicated before in Sect. 3. The developed charts and curves can be used as a design aids to estimate the thickness for solid mat foundation type. The charts can be used within the allowable values of differential settlement (48 mm), and for cases subject to wind loads up to 78 mile/h, for zone-1 seismic load in addition to live load. The water table was not encountered in this analysis. The mat thickness increases with the increase of the number of floors and/or spacing between columns. Usually the mat thickness is governed by ACI minimum requirements only for low-rise buildings (up to 3 floors). The mat thickness is governed by bending moment and/or shear for almost all cases for medium-stiff
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soils (E s ≥ 50 MPa). For weak soils (E s < 50 MPa), differential settlement governs the number of floors and design mat thickness. The relative stiffness K r decreases, as the column spacing increases and/or the mat thickness decreases. The modulus of subgrade reaction (K s ) increases as the umber of floors and/or the modulus of elasticity of the soil (E s ) increases. For elastic soil and rigid mat under gravity loads, the contact soil pressure is greater at the edge and the settlement is uniform. The settlement of a flexible mat on elastic soil under gravity loads is maximum at the center of the mat, although the contact soil pressure is uniform. When a flexible mat on a very dense soil under gravity loads, the column loads are transmitted to the soil on smaller areas directly below the column. The total and differential settlement increases as the number of floors and/or spacing between columns increases. When lateral loads are applied, both maximum contact pressure and maximum settlements are located on the edges of a rigid mat on weak soil, regardless the height of the building. Lateral loads have no significant effect on flexible mat for relatively low-rise buildings (up to 7 floors). However, the effect becomes clear on contact pressure and settlements for high-rise buildings (more than 7 floors). Acknowledgments The authors acknowledge the support of King Fahd University of Petroleum and Minerals.
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