# ANALYSIS OF STRUCTURE SUPPORTED BY ELASTIC FOUNDATION

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ANALYSIS OF STRUCTURE SUPPORTED BY ELASTIC FOUNDATION
A Seminar on ANALYSIS OF STRUCTURE SUPPORTED BY ELASTIC FOUNDATION DEPARTMENT OF STRUCTURAL ENGINEERING MALAVIYA NATIONAL INSTITUTE OF TECHNOLOGY, JAIPUR December 2011 SUBMITTED BY ANKIT SURI (2010PST125) SUPERVISED BY DR.S.K. TIWARI Associate Professor

SOIL STRUCTURE INTERACTION
Soil is a very complex material for the modeling. It is very difficult to model the soil-structure interaction problem. In RCC buildings slab on grade is a very common construction system e.g. mat footing Very heavy slab loads occur in these structures.

For safe and economical design, compute plate displacement and stresses accurately.
Difficult to obtain samples for testing producing results in accordance with ground behavior. Necessary to make simplifying assumptions.

SCOPE OF STUDY To develop a workable approach for analysis of plates on elastic foundations. Structural Engineers go for simplified assumptions of rigid foundation STAAD Pro is used to incorporate the elasticity of soil that will provide approximate solutions as close to the exact solutions.

TYPES OF FOUNDATION MODELS
The plate-foundation system is idealized as a thin elastic plate resting on a linearly elastic foundation. Various foundation models were given by the investigators which are discussed ahead.

WINKLER MODEL Winkler first studied beam on elastic springs
Model based on the pure bending beam theory. p = Kw Here, w = vertical translations of the soil p = contact pressure K = modulus of subgrade reaction

Plates based on Winkler model involve fourth order differential equation:
D ▼4 w+ Kw = q Here D is the plate flexural rigidity, q is the pressure on the plate and▼ is the Laplace operator. The deformations outside the loaded area were neglected and taken as zero.

DEFORMATION OF A UNIFORMLY LOADED PLATE ON TYPICAL WINKLER MODEL
Source : Kerr A. D., "Elastic and visco-elastic foundation models." Journal of Applied Mechanics, ASCE, 31, p

Winkler foundation model has two major limitations:
No interaction between springs is considered. The spring constant may depend on a number of parameters, such as stiffness of beam, geometry of beam, soil profile, and behavior.

FILONENKO BORODICH MODEL
Top ends of springs connected to a elastic membrane stretched to constant tension T. It was done to achieve some degree of interaction between the spring elements, Modulus of subgrade reaction is given by p = Kw – T ▼2 w

FILONENKO-BORODICH FOUNDATION MODEL
Source : Kerr A. D., "Elastic and visco-elastic foundation models." Journal of Applied Mechanics, ASCE, 31, p

HETENYI MODEL Embedded a plate in the three-dimensional case in the material of the Winkler foundation to accomplish interaction among springs. Assumed that the plate deforms in bending only. p = Kw + D▼2 ▼2 w Here, p = load w = vertical translation D = flexural rigidity of plate.

PASTERNAK FOUNDATION MODEL
Pasternak assumed shear interactions between spring elements. Connecting the ends of springs to a beam or plate consisting of incompressible vertical elements, which can deform only by transverse shear. p = Kw - G ▼2 w

TIMOSHENKO MODEL This model is based on Timoshenko beam theory
Plane sections still remain plane after bending but are no longer normal to the longitudinal axis. This model considers both the bending and shear deformations.

Pressure sustained per unit deformation of subgrade at specified deformation or pressure level. Calculated from plate load test from the plot of q versus δ K = q/δ Here , q = mean bearing pressure K = modulus of subgrade reaction δ = mean settlement

LOAD DEFORMATION CURVE FROM PLATE BEARING TEST
Source : Bowles J E., Foundation Analysis and Design, McGraw-Hill, Inc., 1982

DETERMINATION OF MODULUS OF SUBGRADE REACTION
TERZAGHI His work showed that value of k depends upon dimensions of area acted upon by subgrade reaction. He incorporated shape and size effects in his equations

k = desired value of modulus of subgrade reaction
For footings on clay: k = k1 x Bf For footings on sand : k = k1 * For rectangular footing on sand of dimensions b x mb: k = k1* Where, k = desired value of modulus of subgrade reaction k1 = value of k from a plate load test Bf = footing width

VALUES OF K FOR SLAB ON WINKLER FOUNDATION
Boit found that he could obtain a good correlation with the Winkler model for the maximum moment case by setting the value of k as follows: Where, Es = modulus of elasticity of soil vs = Poisson’s ratio of the soil B = modulus of elasticity of the beam I = moment of inertia of the beam

Vesic showed that K depends upon the stiffness of the soil, as well as the stiffness of the structure. Vesic’s work extended Boit’s solution by providing the distribution of deflection, moment, shear and pressure along the beam. He found the continuum solution correlated with the Winkler model by setting

Bowles (1982) suggested an indirect method of approximate estimation of the value of modulus of subgrade reaction. According to him it may be assumed that net ultimate bearing capacity of a footing occurs at a settlement of 25 mm. qnu = cNCSC + γ1DfNqSqrw γ2BN γS γrw’ k = = 40 qnu

Values of modulus of subgrade reaction (suggested by Bowles 1982)
Type of Soil K (KN/m2/m) Loose sand Medium dense sand Dense sand Clayey medium dense sand Silty medium dense sand Clayey soil : qu < 200 Kpa 200<qu<400 Kpa qu> 800 Kpa >48000

PROBLEM DEFINITION AND STRUCTURAL MODELLING
Three-dimensional structure is modeled for the analysis utilizing the STAAD Pro software. The plan dimensions of the building are m x m. The Structure has 10 (G+9) stories with height of 3.66 m each.

The raft is modeled with the structure.
The total area of the raft is divided into finite number of plates. The soil under the raft slab is represented by a set of springs for which the spring constants k, adjusted to reflect the corresponding soil type.

PLAN OF THE STRUCTURE

3 D VIEW OF STRUCTURE

MEMBER AND RAFT SIZES BEAM SIZE - 300mm X 450mm
COLUMN SIZE – 450mm X 600mm RAFT SLAB is divided into finite number of plates Approximately 1.0m x 1.0m plates are used. Thickness is taken as 600mm.

STAAD has a facility for automatic generation of spring supports specified under the SUPPORT command. The modulus of subgrade reaction constant k for each soil type is taken as 10,000 kN/m3, 45,000 kN/m3, and 95,000 kN/m3, representing soft, medium, and stiff soil, respectively

DESIGN LOADS Self weight of floor slabs = 0.15 x 25 = 3.75 kN/m2
DEAD LOAD (IS: 875 PART ) Self weight of floor slabs = 0.15 x 25 = 3.75 kN/m2 Weight of floor finish (4 inches thick) = 0.1 x 20 = 2 KN/m2 Weight of flooring (1 inch thick) = x (marble) = KN/m2 Incidental load due to partition wall = 1.0 KN/m2 (as per clause of IS 875 Part II)

Dead load of wall (230 mm thick) = 19 x 0.23 x 3.66 = 16 kN/m
Dead load of plaster on wall = 2 x x 20 x 3.66 = 1.76 kN/m Dead load of parapet wall = 19x0.23 x x x 20 x 1.0 = 4.85 kN/m

IMPOSED LOAD (IS: 875 - 1987 PART II)
The magnitude of minimum imposed load which has to be considered for the structural safety is provided in IS: (part II). Here imposed load of intensity 3kN/m2 and 4kN/m2 have been taken as per the code and same is applied in all floors. On the roof it is taken as 1.5kN/m2.

SEISMIC LOAD (IS: ) The total design lateral force or design seismic base shear Vb is computed in accordance with the IS 1893 (Part I) -2002 Vb = Ah x w Where

Calculation of base shear is carried out for structure located in seismic zone IV.
I = 1.0 considering the structure is of general category. R = 3 for OMRF

LOAD COMBINATIONS 1.5 (DL + IL) 1.2(DL + IL + ELX) 1.2 (DL + IL - ELX)
1.2 (DL + IL + ELZ) 1.2 (DL + IL - ELZ) 1.5 (DL + ELX) 1.5 (D L - ELX) 1.5 (DL + ELZ) 1.5 (DL - ELZ) 0 .9 DL ELX 0 .9 DL ELX 0 .9 DL ELZ 0 .9 DL ELZ

RESULTS AND CONCLUSIONS
It has been observed that the stiff stratum at the base does not change the design forces significantly. The bending moments at the base of the columns under gravity loadings show a greater increase for soft soils as compared to the medium and soft soil. As the stiffness of the soil strata increased, structure behavior became closer to that observed for rigid supports.

BENDING MOMENT FOR EXTERIOR COLUMNS FOR 1.5(DL+LL) TABLE 7.2
Floor Level MZ (K=10000 KN/m2/m ) MZ (K= KN/m2/m ) MZ (K= KN/m2/m ) MZ Bottom node Top node -1 90.23 -4.288 71.75 3.88 58.22 7.1 -5.78 14.76 -23.32 22.61 -17.78 20.44 -17.38 21.05 -17.6 23.95 1 -23.92 25.44 -24.08 24.61 -25.01 25.34 -27.59 27.83 2 -27.03 28.09 -26.73 27.57 -27.49 28.27 -29.49 30.27 3 -30.1 30.87 -29.44 30.19 -30.06 30.8 -31.86 32.52 4 -32.49 33.19 -31.66 32.35 -32.19 32.87 -33.74 34.36 5 -34.5 35.07 -33.55 34.11 -34.01 34.56 -35.37 35.87 6 -36.27 36.56 -35.19 35.52 -35.59 35.92 -36.79 37.09 7 -38.02 38.25 -36.76 37.06 -37.09 37.4 -38.15 38.46 8 -38.67 44.27 -37.32 41.57 -37.62 41.54 -38.57 42 9 -27.2 35.55 -29.11 36.51 -30.15 37.59 -32.2 39.96

BENDING MOMENT FOR INTERIOR COLUMNS FOR 1.5(DL+LL) TABLE 7.4
Floor Level MZ (K=10000 KN/m2/m ) MZ (K= KN/m2/m ) MZ (K= KN/m2/m ) MZ Bottom node Top node -1 -71.34 34.43 -33.93 18.76 -20.21 12.95 -1.63 1.91 -15.57 20.89 -7.82 11.38 -6.1 8.85 -2.88 4.27 1 -22.85 23.55 -13.44 14.24 -10.63 11.57 -5.38 6.65 2 -24.69 24.53 -15.45 15.36 -12.79 12.74 -7.78 7.84 3 -24.51 24.76 -15.47 15.76 -12.92 13.22 -8.18 8.53 4 -24.68 24.43 -15.78 15.57 -13.3 13.12 -8.72 8.57 5 -23.99 23.64 -15.22 14.92 -12.81 12.52 -8.34 8.08 6 -23.17 22.83 -14.52 14.23 -12.16 11.88 -7.77 7.53 7 -22.52 22.3 -13.95 13.74 -11.62 11.41 -7.29 7.08 8 -22.14 22.26 -13.76 14.16 -11.51 11.99 7.96 9 -22.33 28.4 -13.06 16.72 -10.48 13.48 -5.68 7.42

ABRUPT CHANGE IN BENDING MOMENTS AT THE BASE FOR FOUNDATIONS ON SOFTER SOILS
Generally this portion of the structure is not given consideration in most of the practical designs which are based on the assumption of rigid support system.

DEFLECTION PROFILE FOR CASE OF FIXED SUPPORT FIG 6.12 (a) (EQX)

DEFLECTION PROFILE FOR CASE OF ELASTIC SUPPORT FIG 6.12 (b) (EQX)

For seismic forces, magnitude of bending moments in the columns and beams of the structure increase with the increase in modulus of subgrade reaction. The structure on soft soil deflects as a whole body (Fig 7.12.) The relative displacements between successive floors are less for structure on soft soils.

BENDING MOMENTS AT SUPPORT OF BEAM CONNECTED TO EXTERIOR COLUMN FOR EQX TABLE 7.5
Floor Level MZ (K=10000 KN/m2/m ) MZ (K= KN/m2/m ) (K= KN/m2/m ) -1 -66.13 -82.68 -88.7 -98.66 1 2 3 -137.6 4 -147.7 5 6 -82.38 -93.29 -96.95 7 -52.20 -62.87 -66.44 -73.34 8 -14.78 -25.38 -28.94 -35.87 9 5.16 -2.30 -4.86 -9.89

BENDING MOMENT FOR INTERIOR COLUMNS FOR EQX TABLE 7.1
Floor Level MZ (K=10000 KN/m2/m ) MZ (K= KN/m2/m ) MZ (K= KN/m2/m ) MZ Bottom node Top node -1 43.03 58.72 81.58 50.88 96.99 47.59 132.59 43.88 124.62 -61.32 133.36 -70.07 136.09 -73.02 142.35 -78.43 1 83.7 -76.61 90.08 -83.36 92.4 -85.72 97.32 -90.48 2 75.24 -76.9 82.02 -83.38 84.37 -85.63 89.07 -90.1 3 69.48 -74.61 75.82 -80.76 78 -82.87 82.35 -87.04 4 62.47 -71.04 68.52 -76.92 70.58 -78.91 74.65 -82.83 5 52.8 -65.42 58.61 -71.08 60.58 -72.99 64.43 -76.72 6 40.32 -57.49 45.93 -62.99 47.82 -64.83 51.5 -68.4 7 24.42 -46.36 29.89 -51.74 31.72 -53.54 35.27 -57.03 8 5.45 -33.09 10.79 -38.33 12.57 -40.04 16 -43.31 9 -18.65 5.33 -13.16 -2.17 -11.31 -4.73 -7.65 -9.76

STOREY DRIFT For soft soils very significant increase in displacements of the structure can occur when subjected to lateral forces due to earthquake. For EQX forces deflection at the top floor was 10 to 12% more for structure supported on soft soils than that observed for the case of fixed supports.

Storey drift along exterior column for EQX
Floor Level Fixed K=10000 KN/m2/m K= KN/m2/m K= KN/m2/m 11.512 13.667 12.402 12.111 1 23.868 27.329 25.229 24.752 2 36.398 41.148 38.218 37.56 3 48.657 54.682 50.93 50.09 4 60.334 67.627 63.054 62.037 5 71.903 79.648 74.256 73.059 6 80.541 90.357 84.144 82.77 7 88.23 99.321 92.283 90.726 8 93.7 98.205 96.465 9 96.81 99.861

Storey drift along interior column for EQX
Floor Level Fixed K=10000 KN/m2/m K= KN/m2/m K= KN/m2/m 11.478 13.426 12.281 12.024 1 23.76 26.907 24.997 24.569 2 36.277 40.559 37.883 37.288 3 48.447 53.941 50.509 49.752 4 60.111 66.759 62.579 61.658 5 70.892 78.677 73.754 72.672 6 80.38 89.297 83.635 82.395 7 88.116 98.158 91.759 90.363 8 93.6 97.628 96.078 9 96.633 99.352

MORE BM IN MEMBERS DUE TO DIFFERENTIAL SETTLEMENT IN SOFT SOILS.
The softer the soil, the more the differential settlement. This differential settlement resulted in an increase in bending moments of raft slab.

BENDING MOMENT CONTOURS FOR RAFT UNDER SEISMIC LOADS
EQX and 1.2 (DL+LL+EQX) loading conditions have been studied. The moments in the raft have been affected by the change in the values of the modulus of subgrade reaction K, which is responsible for differential settlement of raft slab.

BM variations in raft slab for K = 10000 kN/m2/m in EQX loading case

BM variations in raft slab for K = 45000 kN/m2/m in EQX loading case

BM variations in raft slab for K = 95000 kN/m2/m in EQX loading case

BM variations in raft slab for K = 10000 kN/m2/m in 1
BM variations in raft slab for K = kN/m2/m in 1.2(DL+LL+EQX) loading case

BM variations in raft slab for K = 45000 kN/m2/m in 1
BM variations in raft slab for K = kN/m2/m in 1.2(DL+LL+EQX) loading case

BM variations in raft slab for K = 95000 in kN/m2/m 1
BM variations in raft slab for K = in kN/m2/m 1.2(DL+LL+EQX) loading case

As the value of modulus of subgrade reaction decreases the differential settlements increase leading to an increase in both the hogging and sagging bending moments. The hogging moments produce tension at the top and can cause the foundation to loose contact with soil. Hence due consideration must be given to the elastic nature of soil in design.

RECOMMENDATIONS The soil structure interaction must be considered in the design of structures. At the design stage, specific effort must be made to find the realistic value of modulus of subgrade reaction depending on the type of soil, so that we can get the exact design forces for optimum design solution.

REFERENCES Bowles J E., Foundation Analysis and Design, McGraw-Hill, Inc., 1982 Kerr A. D., "Elastic and visco-elastic foundation models." Journal of Applied Mechanics, ASCE, 31, p Daloglu A. T. and Vallabhan C. V. G., "Values of K for slab on Winkler foundation" Journal of Geotechnical and Geo-environmental Engineering, Vol. 126, No.5, 2000 p Fwa T.F., Shi X.P. and Tan S.A. , "Use of Pasternak foundation model in concrete pavement analysis" Journal of Transportation Engineering, Vol. 122, No.4, 1996 p

Horvath J. S., "Modulus of subgrade reaction: new perspective," Journal of Geotechnical Engineering, Vol. 109, No. 12, 1983, p Liou G. S. and Lai S.C., "Structural analysis model for mat foundations," Journal of Structural Engineering, Vol. 122, No.9, p Mishra R. C. and Chakrabarti S. K., "Rectangular plates resting on tensionless elastic foundation: some new results", Journal of Engineering Mechanics, Vol. 122, No 4, p Shi X.P., Tan SA and Fwa T.F., "Rectangular thick plate with free with free edges on Pasternak foundation" Journal of Engineering Mechanics, Vol. 120, No.5,

STAAD Pro V8i, Structural Analysis and Design Package, Research Engineers.
Stavridis L. T., "Simplified analysis of layered soil-structure interaction," Journal of Structural Engineering, Vol. 128, No.2, p Wang C. M., Xiang Y. and Wang Q., 2001, "Axisymmetric buckling of reddy circular plates on Pasternak foundation," Journal of Engineering Mechanics, Vol. 127, No 3 Yin J-H., "Comparative modeling study of reinforced beam on elastic foundation" Journal of Geotechnical and Geo-environmental Engineering, ASCE, 126(3), p

IS 875(Part 1): 1987: Indian Standard Code of Practice for Design Loads (Other than earthquake loads) For Buildings and Structures. (Dead Loads) IS 875(Part 2): 1987: Indian Standard Code of Practice for Design Loads (Other than earthquake loads) For Buildings and Structures. (Live Loads) IS 875(Part 5): 1987: Indian Standard Code of Practice for Design Loads (Other than earthquake loads) For Buildings and Structures. (Special Loads and Load Combinations) IS 1893 (Part 1): 2002: Indian Standard Code of Practice for Criteria for Earthquake Resistance Design of Structures. (General Provisions and Buildings) IS 456: 2000: Plain and Reinforced Concrete Code of Practice

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