Numerical analysis of non-linear soil-structure interaction

Transcription

1 Numerical analysis of non-linear soil-structure interaction M. R. Massimino, M. Maugeri Department of Civil and En vironmen tal Engineering, University of Catania, Italy Abstract The present paper deals with a parametric study of soil-foundation-frame interaction in pseudo-static conditions by means of a new finite element code, named SOFIA. The analysis is performed considering strip foundations resting on linear and non-linear soil elements. The seismic actions are simulated by means of a pseudo-static approach. Even if the code is used for soil-foundationframe systems in plane-strain conditions, the 3-D effect is taken into account through an approximate procedure. In particular, in this paper the effects due to the soil-structure relative stiffness and the number of storeys in the superstructure are taken into account. The results, reported in a-dimensional form to be as general as possible. show some interesting differences between the static and pseudo-static behaviour of the soil-structure system and the importance of considering the soil nonlinearity. 1 Introduction The seismic response of structures strongly depends on their interaction with the underlying soil. Then, also for more realistic routine designs, it is necessary to consider the structure (superstructure and foundation) and the soil as parts of a single system. The soil-structure system can show different behaviour in relation to many aspects regarding the geometric and geotechnical characteristics of the soil and the structure. On the one hand. the two classical procedures of frames with fixed-based columns or of frames on a Winkler soil schematisation, used in

2 540 Enrthqt~ake Res~stant Engineering Structwes I11 the structural engineering field, could give very approximated and sometimes not safe results. On the other hand, the geotechnical soil models often do not take into account the important role of the real geometry and stiffness of the structure, already widely analysed for static condition by several geotechnical researches (Poulos [l]). It is, then, important to realise simple and rapid tools to analyse the whole soil-foundation-superstructure behaviour. In this context the most appropriate and the most used approach, either for static or dynamic conditions, is the numerical approach due to its versatility. In the present paper a parametric analysis is performed highlighting, above all, the important role played by the soil-structure stiffness and by the number of storeys in the superstructure. The previous two parameters can change the soilstructure response significantly and sometimes give opposite results from static to pseudo-static conditions. In the last case the effects of the soil-structure stiffness and of the number of storeys on the soil stress and strain level and on the foundation differential settlements are emphasised. Furthermore, the changing of superstructure stresses due to soil movements are considered. The analysis is performed by means of a new finite element code, called SOFIA (Massimino [2]). The SOFIA =oil Frame lnter&t~on) code allows the study of soil-foundation-frame systems in plane-strain conditions, but at the same time it considers the 3-D effect through an approximate procedure. Moreover, for the soil elements a linear and a non-linear stress-strain relationship is used. The seismic actions on the soil-frame schemes are applied by means of a pseudo-static approach according to the Italian Seismic Code [3]. A higher value of acceleration is also considered according to Eurocode 8 [4] to take into account the possible site effects. All the results are reported in a-dimensional form to be as general as possible. 2 Numerical analysis of a parametric soil-structure system The analysis is performed for the seven plane frames shown in Fig. 1 with the new SOFIA code, subdividing the frame elements by means of the classical De Saint Venant monodimensional elements and the interacting subsoil by means of isoparametric quadratic plane elements, as reported in Fig. 2. The frames present three spans of equal size 1 and columns of height h = The storey total number changes from 1 to 7. However, the pseudo-static analysis offers safe results only for buildings of limited height. In all the cases a strip foundation that joins all the base columns is considered. This foundation presents two equal projections of l' = , one on the left of the frame and one on the right of the frame. The width of the foundation in the direction orthogonal to the examined plane is equal to B = I / 2. All the columns and the beams of the superstructure have the same moment of inertia (Ic =IB). A soil mass 14 1 high and 24 1 wide is also considered, so that the boundaries are far enough from the frame and do not significantly disturb the frame behaviour. The soil-element dimensions decrease approaching the frame, where the highest strain and stress variation is expected. The tridimensionality of the real system, i. e. the foundation size in the direction orthogonal to the examined

3 Earthquake Resrstarir Eizgrneerlng Strzlctwes plane, reported in Fig. 2, is also considered, dividing the initial value of each soil-element Young modulus by a corrective coefficient (Ghersi et al. [j]). The generic value of this coefficient represents the ratio between the vertical deformation due to a rectangular load area (B and L plane dimensions) and the vertical deformation due to a strip load area (B and L = m dimensions) computed in a generic point of the soil, following the Boussinesq solution (Massimino [2]). Figure 1 : The analysed frames To analyse the effects of the soil-structure relative stiffness the following soil-superstructure and soil-foundation stiffness parameters are defined (Poulos [l]): E. I, K,, = - E,. being Kss the soil-superstructure stiffness parameter; KSF the soil-foundation stiffness parameter; E, EF and Es the Young modula of the superstructure members, of the foundation and of the soil respectively, IB and IF the moment of inertia of the superstructure beams and of the foundation respectively; l the span: vs the Poisson coefficient of the soil, considered equal to 0.3 in the present work. In particular. in the case of a variation of the soil Young modulus with the depth according to Janbu [6] law, the E, value reported in expression (l) is computed at the medium depth H = All the calculations are developed considering three different phases: I) the excavation to locate the foundation at the depth D = h ) the structure

4 542 Eartlzqztake Resistant Engineering Strzlctures III submitted to the vertical dead and live loads, 3) the structure submitted to the pseudo-static horizontal forces at each storey to simulate the seismic action. The horizontal forces are evaluated following the Italian Seismic Code [3] (Fig. 2), but considering two different values of the seismic coefficient: C = 0.07 for moderate seismicity and C = 0.35 for higher seismicity and also to take into account possible amplification phenomena of the seismic waves moving from the bedrock to the soil surface. Moreover, to approximate as well as possible the modified (Massimino [2]) non-linear soil stress-strain relationship proposed by Duncan and Chang [7], for each of the above three phases, different calculation steps are performed. The non-linear soil analysis is very important to achieve a more realistic evaluation of the soil deformations, as shown by the numerous experimental observations. 3 Soil-foundation behaviour As is well known, the foundation differential settlements can produce different serviceability and structural damage on the structure. So it is important to control the settlements considering the indications reported in the literature (Wahls [g]) and in the Codes (Eurocode 1 [9], Eurocode 7 [10]) and, above all, considering the real effects on the structure by means of a unitary analysis of the soilstructure interaction. Movement amplification factor = 50 1 / 1 1 / I 1 / I I Figure 2: The soil-foundation-superstructure system in pseudo-static conditions

5 Earthqz~ake Resistant Engineering Strzlctwes The entity of the differential settlements depends strongly on the soilstructure stiffness. Taking into account static and pseudo-static conditions and two different values of Kss and KSF, in Figs. 3, 4 and 5 the a-dimensional maximum differential settlement F,,,,, is plotted versus the total storey number of the superstructure for the linear and non-linear analyses, according to the following expression: being Aw,,, the maximum value of the differential settlement along the foundation and W,,the total vertical load for each storey. Comparing the results of Figs. 3 and 4 it is possible to note that the decrease of Kssand KSF gives rise to a decrease of the differential settlements. This trend is opposite that regarding the static condition (Fig. 5). With a greater stiffness of the structure there are less differential settlements in static condition. due to the less deformation of the structure, but there is a higher global rotation of the whole structure during seismic conditions. that can be reduced with a more flexible frame. Also, the total storey number plays an opposite role from static to pseudo-static conditions. In the first case considering the soil non-linearity, the increase of stories gives rise to a decrease of the differential settlements, due to the higher stiffness of the superstructure. + L~near analksis (C=O 07) + Lmear analysis (C=O 35) O0 1 --t Non-imear ana1)sls (C=O 071 &Non-linear anahs~s (C=O 35) Storey number Figure 3: The a-dimensional maximum differential settlement in pseudostatic conditions for KsS = and KSF=

6 544 Enrthyzlnke Resistant Englneerztzg Structures III Linear analysis (C=O 07) * + Lmear analys~s (C=O 35) -t Non-lmear analysis ( C4 07) + Non-llnear analys~s (C435) Stanmg of up-l~ft~ng Storey number Figure 4: The a-dimensional maximum differential settlement in pseudostatic conditions for Kss = and KSF= Storey number Figure 5: The a-dimensional maximum differential settlement in static conditions

7 Earthquake Resistant Engineer-irlg Strzlctzlres For the elastic-linear soil constitutive law the a-dimensional differential settlement given by expression (3) is quite constant in respect to the storey number. While in seismic conditions the increase of the total storey number leads to an increase of the simulated seismic actions and then an increase of the differential settlements. The increase of the seismic coefficient C could significantly increase the differential settlements, so that for a given building with more than 6 storeys submitted to an acceleration of 0.35.g there is a possible foundation up-lifting. This result confirms once more that the geotechnical characterisation of the subsoil is the first important step to a good design. With the SOFIA code it is also possible to have different information as regards the strain and stress on the soil coming from the soil-structure interaction. As an example, in Figs. 6a and 6b the a-dimensional increment of the vertical stresses AoJq are plotted for the pseudo-static condition both for C = 0.07 (Fig. 6a) and for C = 0.35 (Fig. 6b), being q the average contact pressure between the soil and the foundation. Fig. 6 shows the significant increase of the vertical stresses with the increase of the coefficient C. 4 Effects on the superstructure of the soil-structure interaction The previous global schemes (soil, foundation and superstructure) are finally compared with the classical schemes of frames characterised by fixed-based columns. still used in static and dynamic routine designs, taking into account the bending moments, the normal stresses and the shear stresses on the superstructure columns and considering the soil non-linearity. As reported in Ghersi et al. [S] and Massimino [2], it is possible to note that the superstructure most sensitive members to the foundation movements are the columns of the first elevation. For this reason in the present paper only the stresses on these members are reported, through the a-dimensional factors F,v,n,ax, FT.max and Fv The and FT,,,, factors are respectively the ratios between the maximum bending moment and the maximum shear stress of a generic superstructure member computed with the global soil-frame scheme and the maximum bending moment and the maximum shear stress of the same member computed with the fixedbased columns frame-scheme. The Fx factor is the ratio between the normal stress of a generic superstructure member computed with the global soil-frame scheme and the normal stress of the same member computed with the fixedbased columns frame-scheme. Figs. 7, 8 and 9 show respectively the variation of FT,&.; and Fs with the total storey number for the four columns of the first storey (Fig. 7) and considering Kss = KSF= and C = 0.07, as suggested by the Italian Seismic Code [3] for areas with moderate seismicity. In all the three figures 7, 8 and 9 it is possible to see a no negligible divergence of Fr.max and Fy from the unitary value. These results underline how it is necessary, for a more realistic design of the superstructure members, to take into account the soil movements through a unitan view of the soil-structure system.

8 546 Earthquake Resistant Engineering Str-uctwes 111 AoJ ; C = 0.07 P Figure 6: The a-dimensional increment of the vertical stresses in the soil volume

9 Earrhquuke Resfstant Engfneer~ng Strzlcrwes l Store\ number Figure 7: The pseudo-static analysis of F,M,,max for the columns of the first storey 1 2 -l Storey number Figure 8: The pseudo-static analysis of FT,max for the columns of the first storey c condition (C=O.O? ; 7 Store) number Figure 9: The pseudo-static analysis of F,\, for the columns of the first storey

10 548 Earthquake Resistant Engineering Structures Conclusion In the present paper a numerical application, through a new simple finite element code, called SOFIA, is developed to take into consideration, above all, the effects of the soil-structure relative stiffness and the total number of storeys of buildings, underlying the different behaviour under static or pseudo-static conditions. In particular. it is pointed out how the increase of the soil-structure relative stiffness gives rise to less differential settlements in static conditions, while it gives rise to an opposite trend in pseudo-static conditions, especially considering a more realistic non-linear stress-strain relationship for the soil. The elastic-linear approach overvalues the differential settlements in static conditions and undervalues the differential settlements in pseudo-static conditions. In this last case, an increase of the seismic coefficient C can produce such a significant increment of the differential settlements that an up-lifting can occur in high structures. Lastly, the importance of taking into account the soil-foundationsuperstructure interaction to achieve a more appropriate evaluation of the stresses on the superstructure is highlighted. References Poulos H. G. "Settlement analysis of structural foundation systems", Proc. 4th SouthEast Asian Conf. on Soil Engineering Kuala Lumpur, Malaysia, 4: 54-61,1975. Massimino M. R. "Analisi non-lineare dell'interazione terreno-fondazionesovrastruttura mediante un nuovo codice FEM", Ph.D. Thesis, February D. M "Norme tecnice relative alle costruzioni sismiche - Suppi. alla G.U. della Repubblica Italiana N. 29, , Roma, Eurocode 8 "Design provisions for earthquake resistance of structures - Part 5: foundation, retaining structures and geotechnical aspects, European Committee for Standardization, Brussels, Belgium, Ghersi A., Massimino M. R., Maugeri M. "Soil-foundation-superstructure interaction: effects on the superstructure", Proc. ERES99 Conf., June 1999, Catania, Italy, Janbu N. "Soil compressibility as determined by oedometer and triaxial tests", Proc. ECSME, Wiesbaden; 1: 19-24, Duncan J. M,, Chang C. J. "Non-linear analysis of stress and strain in soils", LSMFD, ASCE, 96: , Wahls H. E. "Tolerable deformations". Vert. and Horiz. Deform. of Found. and Embankments, Geotech. Spec. Pubbl. 1994; No. 40, Vol. 2, ASCE, , Eurocode 1 "Basis of design and actions on structures", European Committee for Standardisation, Brussels, Belgium, Eurocode 7 "Geotechnical design - part 1. European Committee for Standardization, Brussels, Belgium, 19941