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(2003)
A Method for Nonlinear Analysis of Soil in Loading Test on Square Plate
(2000)
Axis Symmetric Static Analysis of Surface Displacements of a Three-dimensional Elastic Layer on a Rigid Base Using Pastternak Model, Journal of Structural and Construction
(2000)
Two-dimensional Analysis of Ground Surface Displacements Using Pasternak Model, Journal of Structural and Construction Engineering, AIJ, No
Axis Symmetric Static Analysis of Surface Displacements of a Three-dimensional Elastic Layer on a Rigid Base Using Pastternak Model
M. Georgiadis, R. Butterfield (1988)
DISPLACEMENTS OF FOOTINGS ON SAND UNDER ECCENTRIC AND INCLINED LOADSCanadian Geotechnical Journal, 25
R. Kondner (1963)
Hyperbolic Stress-Strain Response: Cohesive SoilsJournal of the Soil Mechanics and Foundations Division, 89
K. Terzaghi, R. Peck (1967)
Soil mechanics in engineering practice; second edition
H. Tanahashi (2000)
TWO-DIMENSIONAL ANALYSIS OF GROUND SURFACE DISPLACEMENTS USING PASTERNAK MODELJournal of Structural and Construction Engineering (transactions of Aij), 65
(1963)
A Hyperbolic Stress-Strain Formulation for Sands
(1981)
The Base Resistance of a Non-displacement Pile in Sands
Qunli Chen, Ting Xu, K. Tominaga (2002)
Analytical Method for Nonlinear Behavior of Soil under Vertically Loaded Circular PlateJournal of Asian Architecture and Building Engineering, 1
(1963)
A Hyperbolic Stress-Strain Formulation for Sands”, Proceedings, 2 Pan-American
J. Duncan, C. Chang (1970)
Nonlinear Analysis of Stress and Strain in SoilsJournal of the Soil Mechanics and Foundations Division, 96
In previous studies, an analytical method has been developed for design of spread foundations. A hybrid analysis model with nonlinear discrete springs was proposed (2002) and predictions were made of load-settlement behavior of circular plates in loading tests. The analysis results were in good agreement with test results for small settlement but departed gradually from the test results as the settlements increased, because soil shear failure was neglected. Then, a method was developed for taking into account local soil shear failure. A model was built up and applied in loading tests on square plates for clay and sandy soils (2003). The purpose of current study is to expand the use of this hybrid model considering local shear failure for analysis of circular and square plates for both clayey and sandy soils. Vertical loads are predicted for given settlements. Comparisons are made between test results and analyses with and without consideration of local shear failure. And the results are shown to be greatly improved after consideration of local shear failure. It shows that the proposed method can be very validly used for nonlinear behavior analysis of steel plate even considering local shear failure. Keywords: nonlinear behavior; local shear failure; loading test Introduction comprising a rigid plate, elastic soil and nonlinear In structural design, loading tests are sometimes discrete springs connecting one side to the plate and the performed on steel plate to evaluate bearing capacity of other to soil has been proposed. As shown in Fig.1, the spread foundations. Test results often show a nonlinear soil’s non-linearity is considered in the spring while the relationship between plate settlement and external load. soil in this model is assumed as a linearly elastic body This indicates that the soil under the plate probably expressed by Boussinesq solution. Plate behaviors are behaves nonlinearly under working loads. It is now predicted for circular plates without consideration of soil 7) necessary for designers to consider this soil nonlinearity shear failure and it is found that for large settlements in design of building foundations. the predictions were not in good agreement with test results. Further study was made for square plates 8) Analytical methods for predicting the behavior of a considering local shear failure and it showed that vertically loaded plate have been presented by some analysis considering soil shear failure greatly improved investigators, but most of them deal with ultimate the results. resistance only [for example, K. Terzaghi & R. Peck 1) (1967) ], or elastic behavior only [for example, H. This study expands the use of the proposed model for 2, 3) Takahashi (2000) ]. Only a few of the studies analysis of both circular and square plates taking into investigated the vertical behavior of a plate to the ultimate account local soil shear failure. Spring stress is limited 4) state [H. Yamaguchi (1977) , M. Georgiadis et al. (1988) to the ultimate stress that the actual soil can bear. The 5) 6) , Architectural Institute of Japan (2001) and so on]. elastic modulus, ultimate soil stress and nonlinear coefficient of spring are back-figured from test data of The authors are developing an analytical method to study soil from its elasticity up to general failure through analysis of a vertically loaded steel plate. A hybrid model *Contact Author: Qunli Chen, graduate student, IDEC, Hiroshima University, Kagamiyama 1-5-1, Higashi-Hiroshima, Japan Tel: +81 824 24 6925 Fax: +81 824 24 6925 e-mail: qlch@hiroshima-u.ac.jp (Received November 8, 2003 ; accepted April 6, 2004 ) Fig.1. Spring-Soil Model Journal of Asian Architecture and Building Engineering/May 2004/31 25 loads and settlements. Using these parameters, external loads are estimated for given settlements. Analyses with and without consideration of local shear failure are compared with test results for clay, loam, sand and gravel. Analysis Model and Governing Equations As proposed in previous studies, the steel plate on the tested soil in loading tests as shown in Fig.1a is analytically modeled by a hybrid spring-soil system as Fig.1b. The system consists of a rigid plate, linearly elastic soil and nonlinear springs connected one side to the soil the other to the plate. Fig.2. Division Model for Analysis 1. Analysis model 2. Equations for analysis As specified in previous studies, the spring-soil system If the displacement of the rigid plate under external 1/2 can be summarized as follows: force F is S, letting y =x , (i=1,…,n), the governing i spi equations for this spring-soil system can be expressed 1) The response of soil in this model can be expressed as: by Boussinesq solution as: (1) where δ represents the deformation of soil at the evaluation point; E is the soil’s elastic modulus and (3) ν is the Poisson’s ratio; P is the external vertical load; and ς is the distance from external load’s location to the evaluation point. 2) Non-linearity of actual soil is represented by where i (i=1,…,n) is the number of the evaluation section. discrete springs, where the force-displacement F =F/m for circular plate and F =F for square plate. 1 1 relation can be expressed as: The partial matrix of Eq.(3) is called linear characteristics (2) matrix of soil and can be expressed by Boussinesq solution as: where N is the internal spring force; k is the spring’s nonlinear coefficient (taken as constant everywhere in the soil); and x is the spring deformation from sp its natural state. In this study, the spring stress is limited to ultimate stress of tested soil, which is back-figured from the load-settlement relationship of loading test data. 3) The elastic soil under rigid plate in this model is discretized accordingly with the nonlinear springs, as shown in Fig.2. Reactions of soil and spring are For circular plate, (4-1) evenly distributed on each section. For circular plate, the under soil is divided into m sectors and each sector is divided into n sections, For square plate, (4-2) each of equal area. In view of axial symmetry of load and plate, the reaction along the circumference (a : i = 1,…, n; j = 1,…, n) is the same and difference appears along the radial ij In Eq.(4-1), λ(λ=1,…, m) denotes the number of direction only. circumference vector; ζ is the distance from the ij(λ) external load on section j of sector λ to the evaluation For square plate, the soil is divided into n sections point i. with equal grid span in x- and y-directions along The internal spring force is equivalent to the soil its two perpendicular edges. reaction of corresponding section, and the soil reaction of any section is limited because of soil shear failure. 26 JAABE vol.3 no.1 May. 2004 Qunli Chen Therefore, it is considered in this study that the spring At the same time, the F-S relationships from the force should not exceed the ultimate soil resistance. For vertical loading plate test can also be simulated by two instance, if the spring force N (j=1, ···, n) at section j intersecting lines on logarithmic coordinates. The solid based on elastic calculation reaches the ultimate soil curve shown in the small window in Fig.3 indicates the resistance N (j), it is assumed that shear failure occurred hyperbolic relationship of data included in the first line, in soil of this section. The internal spring force remains while the dotted one indicates that of all the test data. constant and equal to the ultimate value, and Eq.(1) is This indicates that the E -value back-figured using the defined by: first section of the logarithmic F-S relationship is closer N =N (j) (5) to the real elastic modulus of the tested site soil. This is j 0 And the governing equation should be changed used to evaluate the elastic modulus of soils in this study. accordingly with Eq.(5) substituted into Eq.(3). Ultimate resistance and stress distribution 11) Analysis Procedure According to J.M. Duncan et al. who have presented In using the above hybrid model for estimation of plate similar research to Konder et al., the ultimate resistance behavior in loading tests, the elastic modulus, ultimate of soil is expected to be about 0.90 times the asymptotic resistance of soil and nonlinear coefficient of spring value of the hyperbolic relation curve. In this study, the should be determined first. ultimate resistance F of soil is calculated by: F =0.90*F (8) b ult 1. Determination of parameters in which F is the asymptotic load value, that is 1/a. ult Elastic modulus 9, 10) Konder et al. have proposed that the nonlinear On the basis of the data of vertically loaded model 12) stress-strain curve of soil may be approximated by pile tests, Takano proposed the parabolic shape as the hyperbolic equation with a high degree of accuracy. This distribution of soil stresses acting on the pile tip at the is used in this study to simulate the load-settlement curve general failure state in sand. From this result and the of loading test: knowledge of soil mechanics, the distribution of soil stresses under a rigid plate at ultimate state is assumed to be in the shape as shown in Fig.4, that is, uniform for (6) cohesion part and parabolic for friction part. in which F is the working load on the plate; S is the For both circular and square plates, the ultimate stress plate settlement; and a and b are constants and, as shown Q can be expressed in two parts as in Fig.3, a is the reciprocal of the asymptotic load of the Q =Q +Q (9) u c hyperbolic function and b is the reciprocal of the slope where Q =cN (N =5.70) denotes the cohesion part and c c c of the initial tangential line of the hyperbola. The soil’s Q =Q [1-(r/r ) ] denotes the friction part. max 0 elastic modulus is approximated from the initial tangential line together with the Boussinesq equation as For circular plate, in view of axial symmetry of load follows: and plate, the stresses are evenly distributed along each (7) in which B is the width of plate; I denotes the settlement coefficient. I =1.00 for circular plate and I =0.88 for s s square plate. Fig.4. Side View of Ultimate Stress Distribution Fig.3. Hyperbolic Load-Settlement Relationship Fig.5. Upward View of Ultimate Stress Distribution JAABE vol.3 no.1 May. 2004 Qunli Chen 27 circle as indicated in Fig.5a. If r is the radius of a circle on which the evaluation point located, the ultimate stress of friction part can be expressed as: (10-1) For square plate, we assume that the stresses are evenly distributed along each concentric square, as indicated in Fig.5b. If x and y are the distances from the evaluation point to the loaded point in the x- and y- directions, respectively, the ultimate stress of friction part can be expressed by: (10-2) Q for both square plate and circular plate is γmax determined by equating the ultimate resistance to the integration of ultimate stress. The equilibrium can be expressed as: (11) where A is the area of plate, A=πB /4 for circular plate and A=B for square plate. It can be derived that Q =2F /A (12) rmax b At any loading stage, the spring force of a certain section should not exceed the ultimate resistance defined by soil shear failure of that section. The spring force N of an arbitrary failed section j can be expressed as N =Q dA, in which dA indicates the area of soil section j j u and Q is expressed by Eqs.(9) and (10) according to x Fig. 6 Main iterative process u j and y values or radius r. Nonlinear coefficient of spring 2. Estimation of external loads After the elastic modulus and ultimate stress are After all the parameters of the analysis model are determined, the proposed model is used to back-figure determined, it can be used to estimate external loads nonlinear coefficient of spring from load-settlement test under measured plate settlements. Program is developed data. A program using Newton-Raphson’s iterative to solve Eq.(3) with S and k known while y , y , …, y 1 2 n method is built up to solve Eq.(3) with F and S known and F are to be solved. The flow chat is same as in Fig.6, and k, y , y , …, y to be solved. Iterative calculation but initial assumptions are made for F and {x (i)}, (i=1, 1 2 n sp steps are made to ensure N≤N (j) while considering local …, n). j 0 soil shear failure. The program flow chart is as shown in Fig.6, with initial assumption made for k and {x (i)}, Analysis of Test Results sp (i=1, …, n). Test results of 6 sites from Obayashigumi, Ltd. and Tokyo Soil Research, Ltd., performed on clayey soil As in previous studies, k-values solved from Eq.(3) including loam and sandy soil including gravel are are different for each pair of F-S test data even for the analyzed. There are 3 for circular plate and 3 for square same test. And the average of these k-values for different plate. In this section, estimation of elastic modulus E , loading steps is taken as the nonlinear coefficient of ultimate soil resistance F and nonlinear coefficient of spring for the tested soil. spring k are presented first. Then, predictions of vertical 28 JAABE vol.3 no.1 May. 2004 Qunli Chen load from plastic analyses considering local shear failure without consideration of local shear failure and “shear” are compared with elastic analyses without considering means plastic analysis considering local shear failure. local shear failure and test results. From these figures, it is found out that the tendencies of the calculated behavior are similar to the observed ones. 1. Parameters for analysis model Tests were performed on circular steel plates of Fig.7, for test T-43 performed on a square plate for Ø30cm×2.5cm and square steel plates of clay, shows that “shear” has the same results as “elastic”, 30cm*30cm*2.5cm. Outlines of loading tests are shown indicating that there was no shear failure during the test. in Table 1, in which q is unconfined compressive strength of soil. The Poisson’s ratio is taken to be 0.50 Figs.8 - 10 are for tests 48, 104 and 183, performed for clayey soil and 0.30 for sandy soil in this study. Soil’s on circular plates on loam. From these figures, it is clear elastic modulus, ultimate resistance and non-linear that the analysis results are in very good agreement with coefficient of spring back-figured from test result data test results when settlement is small. When settlement using the method mentioned above are also listed. becomes larger, the “Elastic” analysis gives results much larger than the test results, while “Shear” gives results From the data of maximum settlements, some in much closer agreement with test results, as expected. judgment can be made as in practical engineering. For test T-43, the settlement is about 1% of the plate width, Figs.11 and 12 are for test T-6 and 118 performed on from which it can be concluded that the soil is still in square plates on sand and gravel, respectively. As shown, elastic state and there is no shear failure. For test 48 and the analyzed results are in very good agreement with 104, the settlements are about 3% and 5% of the plate test results when settlement is small. When the settlement width, from which it can be assumed that local shear becomes larger, the elastic analysis gives results 1.5 times failure has happened in the soil but the soil has not of test results for test T-6 and almost 2 times of test results reached the ultimate state of general failure. For test 183, for test 118. However, the plastic analysis greatly the settlement is already about 15% percent of the plate improves the results, showing predictions nearly the same width, from which it is considered that the soil had as the test results. undergone general failure and no further resistance could be developed. Conclusions and Discussions This paper has presented an analytical method for It is considered that cohesion of sand and gravel is predicting nonlinear behavior of vertically loaded zero and neither friction nor cohesion of loam or sandy circular and square steel plates in loading tests up to the sandy clay is zero. Thus, the ultimate stress of sand and ultimate state of general soil failure. gravel has a friction part only and is distributed in the form of a parabolic curve with zero at the edges and The plastic analysis method takes into consideration maximum at the center of the plate. However, the soil shear failure with ultimate stress distribution along ultimate stress of loam and sandy clay has both a the plate. cohesion part and a friction part, with an even distribution plus a parabolic distribution as shown in Results show that, for relatively small settlements, both Fig.4. analyses considering and not considering local shear failure in soil show good agreement with test results; 2. Comparisons between measured and analyzed for much larger settlements, elastic analysis without load-settlement behavior consideration of shear failure shows a disagreement with Figs.7 to 12 compare the predicted overall load- the test results; while plastic analysis considering local settlement curves with and without consideration of local shear failure shows results in good agreement with test shear failure in soil and the tested curves . In the figures, data. “Test” means test results, “Elastic” means elastic analysis Table 1. Outline of Tests Applied for Analysis JAABE vol.3 no.1 May. 2004 Qunli Chen 29 Fig.7. Comparison for Test T-43 (Clay) Fig.8. Comparison for Test 48 (Loam) Fig.9. Comparison for Test 104 (Loam) Fig.10. Comparison for Test 183 (Loam) Fig.11. Comparison for Test T-6 (Sand) Fig.12. Comparison for Test 118 (Gravel) This is expected from general knowledge of as a homogeneous half-space. The same nonlinear- geotechnics. It means that the proposed analysis method spring-linear-soil system can be applied for each layer can be flexibly used for analysis of nonlinear behavior in multi-layer soil case. It is expected that both nonlinear of vertically loaded circular and square steel plates for a coefficient and steel plate behavior prediction will be wide range of soil types from clay and loam to sand and improved if a multi-layer model is applied. This is a gravel. subject for further study in the future. However, the results of this study show that predictions made on In this study, the nonlinear coefficient of spring is assumption of half-space are so little different from test back-figured from load-settlement results of loading test. results that the method is already acceptable in If empirical relationships are built up between this engineering practice. coefficient and unconfined compressive strength or SPT N-values, this analysis method can be applied in practice Acknowledgments even if loading tests are not performed. Thanks are given to Obayashigumi, Ltd. and Tokyo Soil Research, Ltd. for supplying test data. This paper has presented the analysis for soil assumed 30 JAABE vol.3 no.1 May. 2004 Qunli Chen Behavior of Soil under Vertically Loaded Circular Plate, Journal References of Asian Architectural and Building Engineering, Vol.1, No.2, pp.1- 1) K. Terzaghi and R.B.Peck: Soil Mechanics in Engineering Practice, 7, 2002. Second edition, Willy International Edition, 1967. 8) Q.L.Chen, K.Tominaga & T.Xu: A Method for Nonlinear Analysis 2) H. Takahashi: Two-dimensional Analysis of Ground Surface of Soil in Loading Test on Square Plate, Proceedings, China-Japan Displacements Using Pasternak Model, Journal of Structural and Geotechnical Syposium, Beijing, pp.408-413, 2003. Construction Engineering, AIJ, No. 530, 85-91, Apr., 2000. 9) Konder, R.L., “Hyperbolic Stress-Strain Response: Cohesive soils”, 3) H. Takahashi: Axis Symmetric Static Analysis of Surface Journal of the Soil Mechanics and Foundations Division, ASCE, Displacements of a Three-dimensional Elastic Layer on a Rigid pp.115-143, 1963. Base Using Pastternak Model, Journal of Structural and 10) Konder, R.L., “A Hyperbolic Stress-Strain Formulation for Sands”, Construction Engineering, AIJ, No. 531, 101-107, May, 2000. nd Proceedings, 2 Pan-American Conference on Soil Mechanics and 4) H.Yamaguchi: Panel Discussion, 9th ICSMFE Vol.3, 1977. Foundations Engineering, Brazil, Vol.1, pp.289-324, 1963. 5) M.Georgiadis and R.Butterfield: Displacement of Footing on Sand 11) James M. Duncan, et al., “Nonlinear Analysis of Stress and Strain under Eccentric and Inclined Loads, Canadian Geotechnical in Soils”, Journal of the Soil Mechanics and Foundations Division, Journal, Vol.25, No.2, pp.199-212, 1988. ASCE, pp.1631-1653, 1970. 6) Architectural Institute of Japan: Recommendations for Design of 12) N. Takano: The Base Resistance of a Non-displacement Pile in Building Foundations, pp.276-278, 2001. Sands, Doctoral thesis of Tokyo Institute of Technology, 1981. 7) Q.L.Chen, T.Xu & K.Tominaga: Analysis Method for Nonlinear JAABE vol.3 no.1 May. 2004 Qunli Chen 31
Journal of Asian Architecture and Building Engineering – Taylor & Francis
Published: May 1, 2004
Keywords: Keywords; nonlinear behavior; local shear failure; loading test
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