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Finite Element Analysis for Frame
Assistant Professor, Civil Engineering Group A. Amruthavalli and K. Nishamathi Students, Civil Engineering Group |
ABSTRACT
In this paper the frame raft soil interaction has been analysed by finite element method. The frame and raft have been considered as an elastic material while the soil has been modeled as an elastoplastic material by Drucker-Prager Yield criterion. The problem has been analysed by considering it as a plane strain problem. The frame has been considered at each meter location in the transverse direction. The raft and soil have been discretized into four noded isoparametric elements while the frame members have been discretized into two dimensional truss elements in the finite element analysis. Newton Raphson Iterative Procedure has been used to solve the nonlinear finite element equation. The effect of raft flexibility and rigidity on raft and frame, and the effect of frame stiffness on flexibility of raft have been analysed. The effect of flexibility of raft foundation has been found to increase the internal forces in the superstructure. The foundation has been found to undergo differential settlement due to its own flexibility. By increasing the superstructure stiffness the differential settlement of foundation has been found to be almost zero. The differential settlement has also been found to reduce to almost zero by increasing the foundation thickness. The overall settlement of foundation is found more when the stiffness of superstructure is increased keeping the foundation flexible than the case when the foundation stiffness is increased keeping the superstructure flexible.
Keywords: Raft, Frame, Analysis, FEM, Flexibility, Rigidity
INTRODUCTION
The behaviour of superstructure depends on the behaviour of foundation which again depends on the type of soil it interacts and vice versa. This interdependency of soil foundation and structure is defined as Soil Structure Interaction. This soil structure interaction effect is found in all the substructures and hence must be considered while analysing these problems. Figure 1. shows the raft, frame and soil considered in the analysis. The present research paper aims to understand the frame foundation soil interaction by finite element method.
LITERATURE REVIEW
Lot many literatures are available on soil-structure interaction problems as all the substructures and foundations lie under the category of soil structure interaction problems. Here a very brief description of literature review has be cited which is in direct contact with the topic considered in this paper.
Some of the important literatures are Lee and Harrison (1970), Miyahara and Ergatoudis (1976), Desai (1982), Chajes and Churchill (1986) and Yang et al (1996).
Lee and Harrison (1970) analysed combined footings and raft foundations, Miyahara and Ergatoudis (1976) presented in their paper new foundation element fully compatible with the beam element, Desai (1982) presented the finite element procedure for general problem of three dimensional soil-structure interaction. Chajes and Churchill (1986) analysed the frame by nonlinear finite element analysis while Yang et al (1996) demonstrated that the condensation technique well known in structural mechanics can in reality be employed to formulate the soil-structure interaction problems.
Figure 1. The three components of the system:
the Frame, the Raft and the Soil
FINITE ELEMENT FORMULATION
The raft and soil have been discretised into four nodded isoparametric elements. The element stiffness matrix, element load vector, the assembly of element stiffness matrix and load vector, the constitutive model considered, derivation of elastoplastic matrix, the iterative method considered for solving the nonlinear finite element equation and validation of two dimensional finite element model is same as reported by Author, Maharaj (2003-Ppr-0338). The frame has been idealized as two dimensional truss elements.
FINITE ELEMENT ANALYSIS
In this paper the raft has been idealized as a plane strain problem. The frame has been considered to lie at each meter distance in the transverse direction on the raft. This represents closely spaced frames in the transverse direction. Figure 2. shows the finite element discretization for the frame, raft and soil. The soil and raft has been discretized into four noded isoparametric elements while the frame has been idealized by two-dimensional truss elements. The soil has been modeled as an elstoplastic medium by Drucker-Prager Yield Criterion. The Frame has been considered as an elastic member. The nonlinear finite element equation has been solved by Newton Raphson Iterative Procedure. A soil domain equal to 1.5 times the width of raft has been considered on either side from the edge of the raft. A soil domain equal to two times the width of the raft has been considered below the raft. No translation has been allowed at the bottom boundary while vertical translation has been allowed at the two end boundaries. The load on the frame has been applied as concentrated loads.
Parameters Varied and Material Properties:
Width of raft = 10 m
Thickness of raft = 0.20, 1.0 m
Area of cross-section of member of frame = 0.10 m2
Length of each vertical member of frame = 4.0 m
Projection of raft from frame location = 1.0 m
Modulus of Soil = 32000 kN/m2
Modulus of Concrete = 2 x 107 kN/m2
Poisson’s ratio of concrete = 0.30
Poisson’s Ratio of soil = 0.45
Modulus of Steel = 2 x 108 kN/m2
Number of members in frame = 5, 9
Figure 2. Finite element discretization for raft frame soil interaction
RESULTS AND DISCUSSIONS
Figure 3 shows the deflected shape of the raft when a frame with five members on it has been loaded for a load of 100 kN. It can be seen that the raft undergoes maximum deflection at the locations of the truss while minimum at the center of the raft. There is bending in the raft about the center of the raft. This shows the flexible behaviour of the raft.
Figure 4 shows the deflected shape of the raft when a frame with five members on it for a load of 500 kN. It can be seen that the raft undergoes maximum deflection at the end locations of the truss while minimum at the center of the raft. There is bending in the raft about the center of the raft which shows the flexible behaviour of the raft When compared to Figure 3. it is found that the overall settlement in this case is more than the previous case though the trend of deflection is same in both the cases.
Figure 3. Deflected shape of the raft
(Raft thickness = 0.20 m, Load = 100 kN)
Figure 4. Deflected shape of the raft
(Raft thickness = 0.20 m, Load = 500 kN)
ws the deflected shape of the raft when the raft thickness has been increased from 0.2 m to 1.0 meter for a load of 100 kN on the truss. It can be seen that raft undergoes almost uniform deflection throughout. This shows the rigid behaviour of the raft. The rigidity of raft has increased with increase in thickness of raft.
Figure 5. Deflected shape of the raft
(Raft thickness = 1.00 m, Load = 100 kN)
Figure 6 shows the deflected shape for the raft when the raft thickness has been increased from 0.2 m to 1.0 meter for a load of 500 kN on the truss. It looks from the figure that there is bending in the raft. Actually there is no bending in the raft which is clear from the magnitude of deflection at different locations and the maximum differential settlement is 0.2 mm which can be considered negligible. The raft undergoes almost uniform deflection throughout. This shows the rigid behaviour of the raft. This increase in thickness of the raft has been found to reduce the member forces in frame as compared to the case when the raft thickness was 0.2 meter. This shows that the effect of soil structure interaction and the flexibility of raft is to increase the member forces which must be considered in the design.
Figure 6. Deflected shape of the raft
(Raft thickness = 1.00 m, Load = 500 kN)
Figure 7 shows the deflected shape of the raft when the raft thickness is 0.20 meter but the truss members has been increased from 5 to 9. While considering nine members at the middle location a vertical truss member has been considered. On either side of this member two cross members have been considered. It can be seen that by increasing the number of members in the superstructure the raft has undergone almost uniform settlement. This shows that the effect of increase in stiffness of the superstructure is to increase the stiffness of the foundation.
Figure 7. The effect of superstructure stiffness
on the deflected shape of the raft
Figure 8 shows the deflected shape of the raft when the superstructure members have been increased from 5 to 9 keeping raft thickness of 0.20 meter and with the second case when the raft thickness has been increased from 0.20 to 1.0 m keeping five truss member on the raft. It can be seen that the overall settlement of the raft is more when the superstructure stiffness has been increased than the case when the foundation stiffness has been increased. However in both cases uniform settlement has been found.
Figure 8. The effect of foundation and superstructure stiffness
on the overall settlement
CONCLUSIONS
The effect of flexibility of foundation is to increase the internal forces in the superstructure. The foundation undergoes differential settlement due to its own flexibility. By increasing the superstructure stiffness the differential settlement of foundation becomes almost zero. The differential settlement is also reduced to almost zero by increasing the foundation stiffness. The overall settlement of foundation is more when the stiffness of superstructure is increased keeping the foundation flexible than the case when the foundation stiffness is increased keeping the superstructure flexible.
ACKNOWLEDGEMENT
The Authors acknowledge Professor L. K .Maheshwari (Director) Birla Institute of Technology and Science for providing all facilities. The authors also acknowledge Professor A. K. Sarkar (Dean Instruction Division, Dean Faculty Division-I and Unit Chief of Community Welfare Division). The first author acknowledges staffs of all groups specially his Civil Engineering Group. The first author cannot remain without acknowledging his wife and his loving sons Ashish and Manish for their full support in this paper.
REFERENCES
Chajes , A. and Churchill, J.E., Nonlinear frame analysis by finite element method, Journal of Structural Engineering, ASCE, Vol.113, No.6, pp.1221-1235 (1986).
Desai, C.S. Phan, H.V. and Perumpral J.V. Mechanics of three-dimensional soil-structure interaction,. Journal of Engineering Mechanics Division, Proceedings of the American Society of Civil Engineers, ASCE, Vol. 108, No.EM5, pp.731-747 (1982).
Lee, I.K. and Harrison, H.B. Structure and foundation interaction theory, Journal of Structural Engineering Division, Proceedings of Institutions of Civil Engineers, ASCE, Vol.96, No. ST2, pp.177-197(1970).
Maharaj, D.K., Nonlinear finite element analysis of strip footing on reinforced clay, Electronic Journal of Geotechnical Engineering, Vol.8, Bundle(C), Paper 2003-0338 (2003).
Miyahara, F. and Ergatoudis, Matrix analysis of structure-foundation interaction, Journal of Structural Engineering Division, Proceedings of Institutions of Civil Engineers, Vol.102, No.ST1, pp251-265(1976).
Yang, Y.B. , Kuo, S.R. and Liang , M.T., A simplified procedure for formulation of soil-structure interaction problems, Computers and Structures, Vol.60, No.4, pp.513-520 (1996).
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