FINITE ELEMENT ANALYSIS OF SHALLOW FOUNDATIONS FOR ECCENTRIC INCLINED LOADS

Abstract

INTRODUCTION Afoundation engineer frequently comes across the problems of foundations subjected to eccentric-inclined loads e.g. foundations of retaining walls, abutments, columns of framed buildings, etc. The criteria for the satisfactory action of such foundations are their ultimate bearing capacity, permissible settlement, tilt and horizontal displacement. Several theories are available for computation of the ultimate bearing capacity of shallow foundations subjected to (i) central vertical loads (Terzaghi, 1943; Meyerhof, 1951; Balla, 1962; Vesic, 1973) (ii) eccentric vertical loads (Parakash and Saran, 1971) (iii) central oblique loads (Meyerhof, 1953; Janbu, 1957; Kezdi, 1961; Sokolovski, 1965) and (iv) eccentric-inclined loads (Agrawal, 1986; Saran and Agrawal, 1991). Further, no work has so far been reported for obtaining the settlement, tilt and horizontal displacement of footings under eccentric inclined loads except of course by Millovic et. al. (1970), who used finite element method and assumed that the soil behaves as an isotropic, homogeneous and linearly elastic material. Few investigators have studied experimentally the behaviour of footings under the action of eccentric-inclined loads (Meyerhof, 1953; Saran andNiyogi, 1970; Agrawal, 1986). Non-dimensional correlations based on experimental data are available for obtaining the settlement, tilt and horizontal displacement ofeccentrically-obliquely loaded footings (Saran, 1969; Agrawal, 1986). PROBLEM IDENTIFICATION Critical review of literature suggests that for foundations under eccentric inclined loads, analytical work by limit equilibrium and limit analysis methods is available for obtaining the ultimate bearing capacity. Further, settlement, tilt and horizontal displacement of the footing can be obtained only by using the empirical relations derived on basis of data obtained through experimental work. However, no attempt has so far been made to predict the pressure-settlement, pressure-horizontal displacement and pressure-tilt characteristics of such foundations taking into account the nonlinear soil behaviour except the work of Agrawal (1986). Keeping the above facts in view, a need was felt to investigate the complete behaviour of shallow foundations under the action of: i) Central vertical loading ii) Central inclined loading iii) Eccentric vertical loading and iv) Eccentric inclined loading The investigation has been carried out primarily theoretically and partly experimentally. ANALYTICAL WORK This includes finite element analysis of footings subjected to above mentioned four loading conditions considering the following constitutive models for underlying soil: i) Nonlinear elastic behaviour using Kondner's(1963) hyperbolic model and ii) Elasto-plastic behaviour of soil considering different yield criteria. The study includes behaviour of footings under two conditions: a) without consideration of the characteristics soil-footing interface and b) with proper representation of interface characteristics The interface element used is a six noded, zero thickness element. Finite element analysis codes written for the purpose of nonlinear elastic and elasto-plastic analyses are general purpose codes. Elasto-plastic analysis can be carried out using any of the following yield criteria: i)Von - Mises Yield Criterion ii)Tresca Yield Criterion iii)Mohr - Coulomb Yield Criterion n iv)Drucker-Prager Yield Criterion v)Compromise Cone Yield Criterion vi)Axial Extension Cone Yield Criterion vii)Extended Von Mises Yield Criterion viii) Naylor-Zienkiewicz model and ix)Critical State model. Incremental-iterative method in conjunction with the residual force approach has been used for the nonlinear analysis. For examining the effectiveness of the analytical results and their verification, experimental data of Agrawal (1986) has been used. Agrawal (1986) studied the behaviour of model rigid strip footings with a rough base. The footing, 100mm x 600mm in size was tested in a steel tank, 1500mm x 1500mm in size and having a height of 1000mm. The tests were conducted on dry sand at a relative density of 84 percent. The footing was tested at two different ratios of depth to width, Df/B = 0.0 (surface footing) and 0.5 (embedded footing), four eccentricity to width ratios, namely e/B = 0.0, 0.1, 0.2 and 0.3 and with five different values of load inclination, namely, i = 0°, 5°, 10°, 15° and 20°. In all, forty model tests were conducted by Agrawal (1986) and the footing was loaded in each case up to failure and its pressure-settlement, pressure-tilt and pressure-horizontal displacement characteristics were obtained. EXPERIMENTAL WORK The experimental work was undertaken in the present study primarily to obtain: i) pressure-settlement and pressure-tilt characteristics of model strip footings in c-c)> soil and ii) the interface characteristics between the soil and the rough footing base. Tests on Footings Four tests were carried out to validate the behaviour of footings resting on c-^> soil and obtained via elasto-plastic analysis. Box type footing, 100mm x 490mm in plan and 100mm in height and made of mild steel plate was used for conducting the tests. The thickness of the rough base of the box was 12.5mm and wall thickness of the box was 6mm. Tests were in conducted in the field where the unit weight of soil was 14.70 kN / m3. The strength parameters of soil were, cohesion, c = 21.75 kPa and friction angle, <j> = 21.8° as obtained from direct shear test. Kondner's hyperbolic stress-strain model was used as the constitutive law of this soil for nonlinear elastic analysis. The footing was tested at four eccentricity to width ratios, namely, e/B = 0.0, 0.1, 0.2 and 0.3 with vertical loading. Each footing was loaded up to failure and its pressure-settlement and pressure-tilt characteristics have been obtained. Sliding Shear Tests The soil-footing interface characteristics were evaluated by conducting sliding shear tests in a shear box, 310mm x 310mm and 150mm high. To represent the behaviour of the interfacial material, half of the shear box was filled with soil and other half included the steel footing with dimensions, 305mm x 305mm and thickness 12.5mm. Two different soil types were used in these tests: a) dry sand at a relative density of 84 %and b) ac- $ soil from field having a unit weight of 14.70 kN /m3. Kondner's hyperbolic model has been used to fit the test data and obtain the necessary constants, namely the normal stiffness and the tangential stiffness between soil and the footing. COMPARISON OF RESULTS i) Comparison with Experimental data of Agrawal (1986) The functional form of soil as defined by Kondner's (1963) hyperbolic stress-strain model and modified by Duncan and Chang (1970) has been used in this study. Mixed incremental-iterative method has been used for nonlinear elastic analysis ofall the forty cases offootings from the work of Agrawal (1986). These include two cases of depth to footing width ratio, (Df / B), four ratios of eccentricity to footing width, (e /B) and five values of load inclination, /°. The pressure-settlement, pressure-tilt and pressure-horizontal displacement characteristics were obtained for the footings, both by considering the interface iv characteristics and by ignoring the interface behaviour and these were then compared with the experimental results of Agrawal (1986). The contact pressure for each case has been obtained and presented graphically. It was found that analytical results predicted by using soil-footing interface characteristics compare very well with the experimental results of Agrawal (1986). Similar attempt was made for analysing the four cases of footings on c-<|> soil by elasto-plastic finite element analysis using Mohr-Coulmob yield criterion and the strain hardening behaviour of soil below the footing. The analysis makes use ofinterface elements between the footing and the soil mass. The numerical results agree well with experimental results ofthe present study only when interface elements are included in the analysis. Thus, the behaviour of shallow foundations subjected to eccentric inclined loads can be predicted reasonably well in cohesionless soils using nonlinear elastic representation ofthe soil when interface characteristics between the soil and the footing are considered. For cohesive-frictional soils (c-<|> soils), elasto-plastic analysis with interface elements predicts pressure-settlement, pressure-horizontal displacement and pressure-tilt characteristics of shallow footings subjected to eccentric vertical load reasonably well. PARAMETRIC STUDY The software for nonlinear elastic finite element analysis of strip footings on cohesionless soil using the interface elements has proven its validity in predicting the behaviour of footings subjected to eccentric-inclined loads. Therefore, a comprehensive parametric study was carried out in the present work for the following parametric values: i) width offooting (B =300mm, 600mm, 1200mm, 1500mm and 1800mm) ii) depth to width ratio ( Df/ B= 0.0, 0.5 and 1.0) iii) factor of safety (1, 2 and 3) iv) relative density of sand ( Dr = 84%, 46%and 9.5%) v) eccentricity to width ratio (e/B = 0.0, 0.1, 0.2 and 0.3) and vi) load inclination with vertical (/ = 0°, 5°, 10°, 15°, and 20°). The methodology for ultimate load criterion suggested by Christiaens (1966) and reported by De Beer (1970) was chosen to calculate the ultimate bearing capacity for sand with a relative density of 84% and depth to width ratios of 0.0 and 0.5. The values of ultimate bearing capacity agree very well with those of Meyerhof (1956). The software for elasto-plastic finite element analysis developed in this study has the capability to find pressure-settlement, pressure-horizontal displacement and pressure-tilt characteristics of shallow foundations subjected to eccentric inclined loads and resting on soil possessing some cohesion. Therefore, the above parametric study has also been carried out by considering the elasto-plastic soil behaviour. Using the data obtained from the parametric study, non-dimensional correlations have been developed for predicting the values of settlement, horizontal displacement and tilt of eccentrically obliquely loaded footings. These correlations have been found to be independent of the size of footing, factor of safety and depth to width ratio, whereas they are dependent on eccentricity to width ratio, inclination of load and type of soil. However, in case of sand, these correlations were found to be independent to the relative density of sand. To use these correlations, the settlement of the footing subjected to central load is required. It may be obtained using the conventional methods. CONCLUSIONS i) The nonlinear elastic finite element analysis using interface elements between the footing base and the soil predicts the complete behaviour of shallow foundations subjected to eccentric inclined loads satisfactorily in cohesionless soil only. ii) The elasto-plastic finite element analysis using interface elements between the footing base and the soil predicts the complete behaviour of shallow foundations vi under eccentric inclined loads satisfactorily inpurely cohesive and c-<J> soils. Such an analysis is not applicable for purely cohesionless soils. iii) The values of ultimate bearing capacity obtained via finite element analysis match well with the equation proposed by Meyerhof (1956). iv) Non-dimensional correlations obtained in this investigation to predict settlement, tilt and horizontal displacement are convenient for use by design engineers. vn