Stress Increment Solution Charts for Soil Consolidation Analysis

Transcription

1 Global Journal of Researches in Engineering General Engineering Volue 1 Issue 4 Version Year 01 Type: Double Blind Peer Reviewed International Research Journal Publisher: Global Journals Inc. (USA) Online ISSN: & Print ISSN: Stress Increent Solution Charts for Soil Consolidation Analysis By Muhannad Iseik University of Jordan Abstract - Current practice of estiating average stress increent required for consolidation settleent coputations eploys id-depth stress approach or ultiple application of sublayer technique, which are tedious and difficult ethods to ipleent for hand calculations. This paper presents siplified charts to estiate such a stress. The influence factor needed to estiate the average stress increent is calculated based on the integration of Boussinesq s equations for coon foundations and various soil configurations. The results are presented in a series of noralized nondiensional charts, which are independent of structural loads and soil characteristics. The derived charts are useful especially when the copressible 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 increent. They can be readily ipleented into design allowing accurate prediction of consolidation settleent or can serve as a powerful tool for optiizing and proportioning the diensions of footings under certain allowable settleent where otherwise an iterative tedious solution is required. Illustrative exaples are presented to deonstrate the applicability and efficiency of the suggested charts for consolidation settleent coputations. Keywords : foundations. Consolidation settleent, stress increent, influence factor, solution charts, shallow GJRE-J Classification : FOR Code : , StressIncreentSolutionChartsforSoilConsolidationAnalysis Strictly as per the copliance and regulations of : 01 Muhannad Iseik. This is a research/review paper, distributed under the ters of the Creative Coons Attribution- Noncoercial 3.0 Unported License peritting all non coercial use, distribution, and reproduction in any ediu, provided the original work is properly cited.

2 Stress Increent Solution Charts for Soil Consolidation Analysis Muhannad Iseik Abstract - Current practice of estiating average stress increent required for consolidation settleent coputations eploys id-depth stress approach or ultiple application of sublayer technique, which are tedious and difficult ethods to ipleent for hand calculations. This paper presents siplified charts to estiate such a stress. The influence factor needed to estiate the average stress increent is calculated based on the integration of Boussinesq s equations for coon foundations and various soil configurations. The results are presented in a series of noralized nondiensional charts, which are independent of structural loads and soil characteristics. The derived charts are useful especially when the copressible 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 increent. They can be readily ipleented into design allowing accurate prediction of consolidation settleent or can serve as a powerful tool for optiizing and proportioning the diensions of footings under certain allowable settleent where otherwise an iterative tedious solution is required. Illustrative exaples are presented to deonstrate the applicability and efficiency of the suggested charts for consolidation settleent coputations. Keywords : Consolidation settleent, stress increent, influence factor, solution charts, shallow foundations. I. Introduction Priary consolidation is the tie-dependent settleent of soil resulting fro squeezing out of water fro the voids, due to the dissipation of the excess pore water pressure, following the application of the load increent. The resulting settleent can be particularly large when the drainage is not ipeded, but its agnitude is of engineering significance only when reference is ade to a tolerable settleent for a given type of structure (Balasubraania and Brenner 1981). The agnitude of consolidation settleent depends largely on load and soil characteristics. Thus, a reliable settleent analysis requires accurate deterination of the induced stress in the soil layer in addition to reliable consolidation paraeters. The variation of the stress produced below the foundation is non-linear in nature as scheatically shown in Figure 1. The intensity of the stress decreases fro a axiu value just underneath the loaded area to about zero at a very large distance fro the foundation. The calculation of the stress increent in a Author : Departent of Civil Engineering, University of Jordan, Aan 1194, Jordan. E-ail : iseik@ju.edu.jo 1 Copressible layer P z Fig. 1 : Stress increent distribution below the center of a uniforly loaded footing copressible layer is coonly dealt with by the iddepth stress approach as suggested in the literature (Terzaghi 1943; Dunn et al. 1980; oltz and Kovacs 1981; Cernica 1994; Bowles 1995; Budhu 000; Craig 004; Coduto et al. 010; Das 010). Usually, the average stress increent of the entire soil stratu is assued to be the one calculated at the iddle of layer ignoring the non-linear variation of the stress, which ay produce a substantial error. Calculations of average stress increent in soil ass are iproved by subdividing the soil stratu into a nuber of horizontal sublayers as illustrated in Figure 1. The technique involves replacing the soothly varying stress distribution within a soil by a staircase-like distribution. The technique assues a constant stress over each sublayer and the value at the id-depth provides an approxiation of the stress increent for every sublayer. The stress at the id-depth of each sublayer is deterined and the settleent within every sublayer is separately calculated, and then sued to obtain the total settleent. Although this ultiple application of the sublayer technique is recoended in the literature, it is not widely used since it is ipractical for anual coputations, and the calculations are tie-intensive and tedious. The error resulting between the application of id-depth stress approach for a soil stratu and ultiple application of sublayer technique, which ight be isleading and unacceptable, depends largely on the size and shape of foundation, thickness of the copressible layer and its location relative to the applied load, and the nuber of sublayers as σ Year Global Journal of Researches in Engineering ( D ) Volue XII Issue viv Version I DJ 01 Global Journals Inc. (US)

3 Stress Increent Solution Charts for Soil Consolidation Analysis Year 01 1 Global Journal of Researches in Engineering ( D J ) Volue XII Issue viv v Version I deonstrated by McPhail et al. (000) and Iseik (01). Obviously, as the thickness of sublayer decreases, the precision of the coputed settleent becoes greater. owever, using a large nuber of sublayers is not propitious to hand calculations. Thus, practically the soil stratu is usually divided into few sublayers with the intention of providing a reasonable answer with a oderate aount of effort. II. STUDY MOTIVE Current practice of estiating average stress increent required for consolidation settleent coputations usually eploys conventional ethods such as id-depth stress approach or ultiple application of sublayer technique, which are both tedious and difficult to ipleent for hand calculations. In addition, they do not consider the case where the copressible soil layer is not directly located below the loaded foundation. This paper enables the average stress increent beneath the center of a uniforly loaded foundation to be obtained as opposed to the stress increent at a specific depth. A series of noralized non-diensional charts are developed to estiate the influence factor of a finite soil layer based on size and shape of foundation, thickness of copressible layer, and its location relative to the applied load. Nuerical exaples are included to illustrate the effectiveness and applicability of the derived charts for settleent coputations. A coparison is ade between the results obtained by these charts and conventional ethods. III. DERIVATION As proposed by Terzaghi (1943), the agnitude of consolidation settleent of a copressible layer is deterined as: ds v ( z) σ ( z) dz = (1) in which ds is the differential settleent due to copression of soil thickness dz, v (z) is the soil coefficient of volue copressibility, and σ(z) is the vertical stress increent produced below the loaded foundation at a particular depth z. If the coefficient of volue copressibility v (z) is taken as a constant, at least at certain depths, the total consolidation settleent S, over the entire thickness of soil stratu, is the integration of Equation (1) as: S = v 0 σ ( z) dz () Based on the theory of elasticity, Boussinesq (1885) provided the equations needed to calculate the stress increent σ(z) in a soil ass. The equations consider a point load on the surface of a sei-infinite, hoogeneous, isotropic, weightless and elastic halfspace. The integration of vertical stress at a depth below a uniforly loaded area was originally described by Newark (1935). Then the solutions were later iproved by Steinbrenner (1936) and graphically represented and suarized by Fadu (1948), and Poulus and Davis (1974). Despite of all the unrealistic assuptions used to develop such solutions, they are still traditionally being used in the literature to obtain the stress increent σ(z) under foundation loads. Using Boussinesq (1885) solutions, the calculation of the stress increent beneath the center of a uniforly loaded foundation is coputed as: σ ( z ) = qi( z) (3) where q is the surface contact stress at the foundation level. I is a non-diensional influence factor defined as: I = π n 1+ + n 1+ + n + sin (1 + n )( + n ) 1 + n 1+ n where and n are diensionless shape and depth factors, respectively, defined as a function of the rectangular foundation diensions width B, and length L, as: = L B (4) z and n = (5) 0.5B If the loaded foundation is circular, of diaeter D, the influence factor I is defined as: 1 I = 1 (6) 3 / D 1 + z Substituting the value of stress increent σ(z), as defined by Equation (3), into Equation (), the settleent of the copressible soil stratu is thus coputed as: S = qi (7) v ave where I ave is the average influence factor of soil stratu defined as: I ave 1 = 0 I( z) dz The integration of Equation (8) is coonly dealt with nuerically since the influence factor I has a (8) 01 Global Journals Inc. (US)

4 Stress Increent Solution Charts for Soil Consolidation Analysis coplex non-linear variation, which is a function of shape and size of the foundation, and depth of soil layer as given by Equations (4) and (6). Accuracy is iproved when the integration is calculated over an infinite nuber of sublayers each of an infinitesial unifor thickness dz. / 1 = Rectangular Footing (L/B=) IV. RESULTS AND APPLICATION and calculations of the average influence factor I ave of Equation (8), over a series of several sublayers, is ipractical and tedious even for a single soil layer. Alternatively, a coputer code is developed to evaluate the integral nuerically and the results are presented graphically. The solution charts, which are independent of structural loads and soil properties, consider a relative configuration of the copressible layer / 1 ranging fro 1 to 10, and coon foundation types such as square (L/B = 1), rectangular (L/B = and 3), strip (L/B > 10), and circular ones as presented in Figures, 3, 4, 5, and 6, respectively. 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 increent, beneath the center of a uniforly 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 Iseik (01), have two powerful and practical advantages for preliinary foundation design when hand calculations are carried out, and especially if the copressible layer is not directly located below the loaded foundation. Firstly, the estiation of the average influence factor is far easier when obtained fro the charts and thus avoids the use of id-depth stress approach, which ay produce a large error. Secondly, the charts can be used efficiently to optiize the required diensions of a footing constrained by a tolerable settleent, as an alternative to classical id-depth stress approach where an iterative ethod is required to find iniu diensions. 0. / 1 = Square Footing (L/B=1) /B Fig. : Influence factor under the center of a uniforly loaded square footing (L/B = 1) /B Fig. 3 : Influence factor under the center of a uniforly loaded rectangular footing (L/B = ) 0. / 1 = Rectangular Footing (L/B=3) /B Fig. 4 : Influence factor under the center of a uniforly loaded rectangular footing (L/B = 3) 0. / 1 = Strip Footing (L/B>10) /B Fig. 5 : Influence factor under the center of a uniforly loaded strip footing (L/B > 10) 0. / 1 = Circular Footing /D Fig. 6 : Influence factor under the center of a uniforly loaded circular footing Year Global Journal of Researches in Engineering ( D ) Volue XII Issue viv Version I DJ 01 Global Journals Inc. (US)

5 Stress Increent Solution Charts for Soil Consolidation Analysis Year Global Journal of Researches in Engineering ( D J ) Volue XII Issue viv v Version I V. EXAMPLES The use of the charts in settleent coputations is illustrated by considering the width square footing as shown in Figure 7. The soil profile is 1 5 γ s = 17 kn/ 3 γ c = 19 kn/ 3 v = 00 /kn P = 600 kn x q = 150 kn/ Fig. 7 : Loading and soil profile data of the exaples coposed of a 5 clay layer overtopped by a 3 sand layer and underlain by an ipereable hard base. Soil characteristics are γ s = 17 kn/ 3, γ c = 19 kn/ 3, v = 00 /kn, and 1 = 1, = 6, and = 5. The structural loading values are P = 600 kn and q = 150 kn/. Exact values of average influence factor and settleent are 59 and 385, respectively. With / 1 = 6 and 1 /B = 0.5, the average influence factor I ave is picked up fro Figure as about 0.1. Thus, the consolidation settleent is coputed by Equation (7) as: S = Sand Clay Rock = (9) The error produced by the use of the proposed charts is about 1.99% of actual settleent, which is quite acceptable. ad the id-depth stress approach been used to calculate the influence factor I, for = 1 and n = 3.5, the coputations for settleent predication using Equations (4) and (7) would be: I = + sin = (10) π (1 3.5 )(1 3.5 ) S = = (11) Such a settleent value yields a significant error of about 33.41%, which is definitely unsatisfactory in geotechnical design. Thus, the direct use of id-depth stress approach ay provide inaccurate results and can be isleading when copared with actual settleent values. As seen, the provided charts siplified the coputations and can be used confidently to predict the average stress increent with acceptable accuracy. Another powerful application of the proposed charts would be to deterine the iniu diensions of a footing required to satisfy an allowable settleent. If design code perits a tolerable settleent of 5.4 (1 inch) for the above footing, the average influence factor I ave can be obtained directly fro Figure for several trials of width B. Then the corresponding settleent is calculated fro Equation (7) as shown below. B () 1 /B q (kn/ ) I ave S () As seen, a foundation width B of about.6 would satisfy the settleent requireent. Exact value of width is.64 obtained by an iterative tedious solution of Equations (7) and (8) siultaneously. The resulting error is 1.59%, which is reasonably acceptable. Such an application of the charts saves effort and tie when reliable fast values of diensions are required by anual hand calculations. VI. SUMMARY AND CONCLUSIONS Solution charts to predict the average vertical stress increent needed for consolidation settleent analysis are presented based on the nuerical 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 copressible layer, and its depth relative to the location of applied load. The suggested charts provide a refined estiate of the stress increent, which could only be obtained with a large nuber of sublayers in the routinely used ultiple application of the sublayer technique. In addition, if the soil is considered as one layer syste, the id-depth stress approach ay provide inaccurate results. The presented charts can be used as an alternative to current conventional ethods. They represent an efficient and powerful solution to calculate the average stress increent especially when the copressible layer is not directly located below the loaded foundation, or can serve as a useful tool for optiizing and proportioning the diensions of footings under an allowable settleent. The ost iportant advantages of these charts, when copared to conventional solutions, are their speed, ease of ipleentation, and versatility for routine hand settleent calculations required for geotechnical design of shallow foundations. 01 Global Journals Inc. (US)

6 Stress Increent Solution Charts for Soil Consolidation Analysis Notation Sybol Unit Definition B Width of foundation D Diaeter of foundation 1 I - Influence factor Thickness of copressible stratu Depth to upper boundary of copressible layer Depth to lower boundary of copressible layer I ave - Average influence factor L Length of foundation - Shape factor v /kn n - Depth factor P kn Coefficient of volue copressibility Vertical load at the center of foundation q kn/ Surface contact stress S Consolidation settleent z Depth below the loaded area σ kn/ Vertical stress increent γ kn/ 3 Unit weight of soil π - PI constant References Références Referencias 1. Balasubraania, A. S., and Brenner, R. P Consolidation and settleent of soft clay. In: Brand, E. W., Brenner, R. P. (Eds.). The Soft Clay Engineering, Chapter 7. Elsevier publishers, Boussinesq, J Applications Des Potentiels a l'etude de l'equilibre et du Mouveent Des Solids Elastiques. Gauthier-Villard, Paris. 3. Bowles, J. E Foundation analysis and design. McGraw-ill publishers, New York. 4. Budhu, M Soil echanics and foundations. Wiley publishers, New York. 5. Cernica, J. N Geotechnical engineering: soil echanics. Wiley publishers, New York. 6. Coduto, D. P., Yeung, M. R., and Kitch, W. A Geotechnical engineering: principles and practices. Prentice-all publishers, New Jersey. 7. Craig, R. F Soil echanics. Taylor and Francis publishers, London. 8. Das, B. M Principles of foundation engineering. Cengage publishers, London. 9. Dunn, I. S., Anderson, L. R., and Kiefer, F. W Fundaentals of geotechnical analysis. Wiley publishers, New York. 10. Fadu, R. E Influence values for estiating stresses in elastic foundations. Proceedings of the Second International Conference on Soil Mechanics and Foundation Engineering, Rotterda, Netherlands, 3: oltz, R. D. and Kovacs, W. D An introduction to geotechnical engineering. Prentice- all publishers, New Jersey. 1. Iseik, M. 01. Practical evaluation of induced stress for calculation of consolidation settleent of soil. Soil Mechanics and Foundation Engineering. 49(3): McPhail, J., ellen, P., Britton, S., Colvin, C., Silvey, T., and Jones, J Evaluation of consolidation settleent using the sublayer ethod. The Electronic Journal of Geotechnical Engineering, 5. Available fro Abs000.ht [accessed 30 Septeber 011]. 14. Newark, N. M Siplified coputation of vertical stress in elastic foundations. University of Illinois, Engineering Experient Station, Circular No. 4, Illinois, US. 15. Poulus,. G. and Davis, E Elastic solutions for soil and rock echanics. Wiley publishers, New York. 16. Steinbrenner, W A rational ethod for the deterination of the vertical noral stresses under foundations. Proceedings of the International Conference on Soil Mechanics and Foundation Engineering, Cabridge, US, : Terzaghi, K Theoretical soil echanics. Wiley publishers, New York. Year Global Journal of Researches in Engineering ( D ) Volue XII Issue viv Version I DJ 01 Global Journals Inc. (US)

7 Stress Increent Solution Charts for Soil Consolidation Analysis Year Global Journal of Researches in Engineering ( D J ) Volue XII Issue viv v Version I This page is intentionally left blank 01 Global Journals Inc. (US)