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\title{Stress Increment Solution Charts for Soil Consolidation Analysis}
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             \author[1]{Dr. Muhannad  Ismeik}

             \affil[1]{  University of Jordan}

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\date{\small \em Received: 13 December 2011 Accepted: 3 January 2012 Published: 15 January 2012}

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\begin{abstract}
        


Current practice of estimating average stress increment required for consolidation settlement computations employs mid-depth stress approach or multiple application of sublayer technique, which are tedious and difficult methods to implement for hand calculations. This paper presents simplified charts to estimate such a stress. The influence factor needed to estimate the average stress increment is calculated based on the integration of Boussinesq?s equations for common foundations and various soil configurations. The results are presented in a series of normalized non-dimensional charts, which are independent of structural loads and soil characteristics. The derived charts are useful especially when the compressible layer is not directly located underneath the loaded foundation and they avoid the necessity of dividing the soil into a series of sublayers to obtain a realistic value of average stress increment. They can be readily implemented into design allowing accurate prediction of consolidation settlement or can serve as a powerful tool for optimizing and proportioning the dimensions of footings under certain allowable settlement where otherwise an iterative tedious solution is required. Illustrative examples are presented to demonstrate the applicability and efficiency of the suggested charts for consolidation settlement computations.

\end{abstract}


\keywords{consolidation settlement, stress increment, influence factor, solution charts, shallow foundations.}

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\let\tabcellsep& 	 	 		 
\section[{Introduction}]{Introduction}\par
The variation of the stress produced below the foundation is non-linear in nature as schematically shown in Figure  {\ref 1}. The intensity of the stress decreases from a maximum value just underneath the loaded area to about zero at a very large distance from the Author : Department of Civil Engineering, University of Jordan, Amman 11942, Jordan. E-mail : ismeik@ju.edu.jo foundation. The calculation of the stress increment in a compressible layer is commonly dealt with by the middepth stress approach as suggested in the literature \hyperref[b16]{(Terzaghi 1943;}\hyperref[b8]{Dunn et al. 1980}  \hyperref[b7]{Das 2010)}. Usually, the average stress increment of the entire soil stratum is assumed to be the one calculated at the middle of layer ignoring the non-linear variation of the stress, which may produce a substantial error.\par
Calculations of average stress increment in soil mass are improved by subdividing the soil stratum into a number of horizontal sublayers as illustrated in Figure  {\ref 1}. The technique involves replacing the smoothly varying stress distribution within a soil by a staircase-like distribution. The technique assumes a constant stress over each sublayer and the value at the mid-depth provides an approximation of the stress increment for every sublayer. The stress at the mid-depth of each sublayer is determined and the settlement within every sublayer is separately calculated, and then summed to obtain the total settlement. Although this multiple application of the sublayer technique is recommended in the literature, it is not widely used since it is impractical for manual computations, and the calculations are time-intensive and tedious.\par
The error resulting between the application of mid-depth stress approach for a soil stratum and multiple application of sublayer technique, which might be misleading and unacceptable, depends largely on the size and shape of foundation, thickness of the compressible layer and its location relative to the applied load, and the number of sublayers as rimary consolidation is the time-dependent settlement of soil resulting from squeezing out of water from the voids, due to the dissipation of the excess pore water pressure, following the application of the load increment. The resulting settlement can be particularly large when the drainage is not impeded, but its magnitude is of engineering significance only when reference is made to a tolerable settlement for a given type of structure \hyperref[b0]{(Balasubramaniam and Brenner 1981)}. The magnitude of consolidation settlement depends largely on load and soil characteristics. Thus, a reliable settlement analysis requires accurate determination of the induced stress in the soil layer in addition to reliable consolidation parameters. P II. 
\section[{Global}]{Global} 
\section[{STUDY MOTIVE}]{STUDY MOTIVE}\par
Current practice of estimating average stress increment required for consolidation settlement computations usually employs conventional methods such as mid-depth stress approach or multiple application of sublayer technique, which are both tedious and difficult to implement for hand calculations. In addition, they do not consider the case where the compressible soil layer is not directly located below the loaded foundation.\par
This paper enables the average stress increment beneath the center of a uniformly loaded foundation to be obtained as opposed to the stress increment at a specific depth. A series of normalized non-dimensional charts are developed to estimate the influence factor of a finite soil layer based on size and shape of foundation, thickness of compressible layer, and its location relative to the applied load. Numerical examples are included to illustrate the effectiveness and applicability of the derived charts for settlement computations. A comparison is made between the results obtained by these charts and conventional methods. 
\section[{III.}]{III.} 
\section[{DERIVATION}]{DERIVATION}\par
As proposed by \hyperref[b16]{Terzaghi (1943)}, the magnitude of consolidation settlement of a compressible layer is determined as:dz z z m dS v ) ( ) ( ? ? = (1)\par
in which dS is the differential settlement due to compression of soil thickness dz, m v (z) is the soil coefficient of volume compressibility, and is the vertical stress increment produced below the loaded foundation at a particular depth z. 
\section[{If the coefficient of volume compressibility m v (z)}]{If the coefficient of volume compressibility m v (z)}\par
is taken as a constant, at least at certain depths, the total consolidation settlement S, over the entire thickness of soil stratum H, is the integration of Equation (1) as:? ? = H v dz z m S 0 ) ( ? (2)\par
Based on the theory of elasticity, Boussinesq (1885) provided the equations needed to calculate the stress increment in a soil mass. The equations consider a point load on the surface of a semi-infinite, homogeneous, isotropic, weightless and elastic halfspace. The integration of vertical stress at a depth below a uniformly loaded area was originally described by \hyperref[b13]{Newmark (1935)}. Then the solutions were later improved by \hyperref[b15]{Steinbrenner (1936)} and graphically represented and summarized by \hyperref[b9]{Fadum (1948)}, and \hyperref[b14]{Poulus and Davis (1974)}. Despite of all the unrealistic assumptions used to develop such solutions, they are still traditionally being used in the literature to obtain the stress increment under foundation loads. Using Boussinesq (1885) solutions, the calculation of the stress increment beneath the center of a uniformly loaded foundation is computed as:) ( ) ( z qI z = ?? (3)\par
where is the surface contact stress at the foundation level. is a non-dimensional influence factor defined as:? ? ? ? ? ? + + + + + + + + + = ? 2 2 2 1 2 2 2 2 2 2 2 1 sin ) )( 1 ( 2 1 1 2 n n m m n m n n m n m mn I ?\textbf{(4)}\par
where m and n are dimensionless shape and depth factors, respectively, defined as a function of the rectangular foundation dimensions width B, and length L, as:B L m = and B z n 5 . 0 = (5)\par
If the loaded foundation is circular, of diameter D, the influence factor I is defined as:   {\ref (2012)}. Obviously, as the thickness of sublayer decreases, the precision of the computed settlement becomes greater. However, using a large number of sublayers is not propitious to hand calculations. Thus, practically the soil stratum is usually divided into few sublayers with the intention of providing a reasonable answer with a moderate amount of effort.? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? = 2 / 3 2\par
where I ave is the average influence factor of soil stratum defined as:? = H ave dz z I H I 0 ) ( 1\textbf{(8)}\par
The integration of Equation ( \hyperref[formula_6]{8}) is commonly dealt with numerically since the influence factor I has a ??(z) ??(z) ??(z) q I ??(z), complex non-linear variation, which is a function of shape and size of the foundation, and depth of soil layer as given by Equations ( \hyperref[formula_3]{4}) and \hyperref[b5]{(6)}. Accuracy is improved when the integration is calculated over an infinite number of sublayers each of an infinitesimal uniform thickness dz.\par
IV. 
\section[{RESULTS AND APPLICATION}]{RESULTS AND APPLICATION}\par
Hand calculations of the average influence factor I ave of Equation ( \hyperref[formula_6]{8}), over a series of several sublayers, is impractical and tedious even for a single soil layer. Alternatively, a computer code is developed to evaluate the integral numerically and the results are presented graphically. The solution charts, which are independent of structural loads and soil properties, consider a relative configuration of the compressible layer H 2 /H 1 ranging from 1 to 10, and common foundation types such as square (L/B = 1), rectangular (L/B = 2 and 3), strip (L/B > 10), and circular ones as presented in Figures  {\ref 2, 3}, 4, 5, and 6, respectively.\par
The presented charts are the exact solutions of average influence factor and they can be used confidently in geotechnical design. They enable the average stress increment, beneath the center of a uniformly loaded foundation, to be obtained directly as opposed to the stress value at a specific depth, as provided by Boussinesq's (1885) solutions. The charts, which agree well with the results of Ismeik (2012), have two powerful and practical advantages for preliminary foundation design when hand calculations are carried out, and especially if the compressible layer is not directly located below the loaded foundation. Firstly, the estimation of the average influence factor is far easier when obtained from the charts and thus avoids the use of mid-depth stress approach, which may produce a large error. Secondly, the charts can be used efficiently to optimize the required dimensions of a footing constrained by a tolerable settlement, as an alternative to classical mid-depth stress approach where an iterative method is required to find minimum dimensions. V. 
\section[{EXAMPLES}]{EXAMPLES}\par
The use of the charts in settlement computations is illustrated by considering the 2 m width square footing as shown in Figure 7. The soil profile is The error produced by the use of the proposed charts is about 1.99\% of actual settlement, which is quite acceptable.\par
Had the mid-depth stress approach been used to calculate the influence factor I, for m = 1 and n = 3.5, the computations for settlement predication using Equations ( \hyperref[formula_3]{4}) and (  {\ref 7}) would be: Such a settlement value yields a significant error of about 33.41\%, which is definitely unsatisfactory in geotechnical design. Thus, the direct use of mid-depth stress approach may provide inaccurate results and can be misleading when compared with actual settlement values. As seen, the provided charts simplified the computations and can be used confidently to predict the average stress increment with acceptable accuracy.1371 . 0 5 . 3 1 5 . 3 1 1 sin ) 5 . 3 1 )( 5 . 3 1 ( 5 . 3 2 1 1 5 . 3 1 1 5 . 3 1 2 2 2 2 1 2 2 2\par
Another powerful application of the proposed charts would be to determine the minimum dimensions of a footing required to satisfy an allowable settlement. If design code permits a tolerable settlement of 25.4 mm (1 inch) for the above footing, the average influence factor can be obtained directly from Figure  {\ref 2} for several trials of width B. Then the corresponding settlement is calculated from Equation \hyperref[b6]{(7)}    
\section[{SUMMARY AND CONCLUSIONS}]{SUMMARY AND CONCLUSIONS}\par
Solution charts to predict the average vertical stress increment needed for consolidation settlement analysis are presented based on the numerical integration of Boussinesq's solutions. A software code is developed to provide relationships between the influence factor and shape and size of foundation, thickness of compressible layer, and its depth relative to the location of applied load.\par
The suggested charts provide a refined estimate of the stress increment, which could only be obtained with a large number of sublayers in the routinely used multiple application of the sublayer technique. In addition, if the soil is considered as one layer system, the mid-depth stress approach may provide inaccurate results.\par
The presented charts can be used as an alternative to current conventional methods. They represent an efficient and powerful solution to calculate the average stress increment especially when the compressible layer is not directly located below the loaded foundation, or can serve as a useful tool for optimizing and proportioning the dimensions of footings under an allowable settlement.\par
The most important advantages of these charts, when compared to conventional solutions, are their speed, ease of implementation, and versatility for routine hand settlement calculations required for geotechnical design of shallow foundations. \begin{figure}[htbp]
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\caption{\label{fig_0}11 Fig. 1 :}\end{figure}
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 		 		\backmatter  			  				\begin{bibitemlist}{1}
\bibitem[Steinbrenner ()]{b15}\label{b15} 	 		‘A rational method for the determination of the vertical normal stresses under foundations’.  		 			W Steinbrenner 		.  	 	 		\textit{Proceedings of the International Conference on Soil Mechanics and Foundation Engineering},  				 (the International Conference on Soil Mechanics and Foundation EngineeringCambridge, US, 2)  		1936. p. .  	 
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\bibitem[Boussinesq ()]{b1}\label{b1} 	 		\textit{Applications Des Potentiels a l'etude de l'equilibre et du Mouvement Des Solids Elastiques},  		 			J Boussinesq 		.  		1885. Paris: Gauthier-Villard.  	 
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\end{bibitemlist}
 			 		 	 
\end{document}
