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In this study dynamic analysis of Soil Structure Interaction (SSI) effect on multi story reinforced concrete (RC) frame founded on soft soil (flexible base) is made and compared with fixed base. Two model 2D RC frames with 7 and 12 story are selected for analysis. Winkler Spring and half space direct method models are used for flexible base for the frames founded on two types of soft soils with shear velocity Vs < 150 m/s Asper Seismic Codes of Chinese GB50011-2010 Soil IV and Ethiopian ES8-2015 soil D. The frames are subjected to strong ground motion matched to response spectrums of soft soil of Chinese GB50011-2010 and Ethiopian ES8-2015 for linear time history analysis. The dynamic analysis result shows Spring and Fixed base mass participation 90% reaches in 2 or 3 modes but in direct method 11 to 30 modes for story 12 and 7 respectively. However, both flexible base models have bigger fundamental period of vibration and inter story drift but smaller base shear than fixed base. In addition, within the flexible base models the inter-story drift, second order effect (P-Δ) and Story shear distribution are different along the height of frames. The spring model shows larger Story drift and second order effect (P-Δ) at the bottom of Story for both soft soils types. On the other hand, half space direct method model indicates value reverse to spring model; it gives bigger Story drift and P-Δ effect in the top stories than fixed base. Finally, this study concludes that base shear reduction due to SSI may not be always beneficial. Because the gravity load is constant in both fixed and flexible bases that cause bigger P-Δ effect at the bottom stories due to increase, inter story drift and decrease story shear in flexible base.

In dynamic analysis of a building structure, the base support condition is very essential for calculating its dynamic behavior useful in estimating structural responses and distribution within structural members. The building base condition will be different depending on the type of supporting ground. Fixed base foundation could be assumed on stiff soil and flexible base foundation on soft soil. Flexibility of base causes decrease in structural stiffness and increase period of vibration during earthquake ground motion. Consequently, the building structural responses such as displacement drift, Story shear, and P-∆ effects will be different from fixed base that could beneficial or detrimental. As a result, in the past the dynamic analysis building on soft soil has gained serious attention in seismic active areas.

Wolf 1985 [

_{s}) supported on soft soil foundation lateral stiffness (K_{y}), and rotational stiffness (K_{θ}) foundation base radius (r) subjected to ground motion acceleration ü_{g}. This causes the base deformation in rotation (θ) and larger translation of at the top of structure that results base flexibility (SSI). Consequently, the dynamic behavior of the structure changes affected structural responses like period of vibration, displacement, base shear, and secondary moment effects P-Δ

(mainly axial load carrying vertical members such as column in multi-story building).

In the past, many studies showed that soil structure interaction (SSI) has both beneficial and detrimental effect on structure. Because SSI increases flexibility of structure, lengthening of structural vibration period and damping. As a result, in building structures, the base shear decreases; however at the same time displacement increases. The decrease in base shear may be advantages, but the increment in displacement induces secondary moments P-∆ effect due to high inter-story drift. Moreover, excessive deflection of building could lead to collision of nearby structures. In addition, P-∆ is highly emphasized structural members supporting big axial load such as tall building, and consequence can be catastrophic which leads to instability of the whole structure. Moreover, there are researches those stating that for some special cases fixed base models can lead to an underestimation of seismic response [

Additionally, the SSI effect has been included in some seismic codes. European regulation for seismic design Eurocode 8 (EC8-2004) [^{nd} order) effects play a significant role; structures with massive or; slender tall structures, such as towers and chimneys.

On the other hand, the American standard ASCE Standard ASCE/SEI 7 - 10 [

The reduction ΔV, shall be computed as follows and shall not exceed 0.3 V

where C_{s} = the seismic design coefficient computed from Equations 12.8-2, 12.8-3 calculated using fixed base fundamental period of vibration (T) and C ˜ s = the value of C_{s} computed from Equations 12.8-2, 12.8-3 calculated using flexibly supported structure ( T ˜ ) in ASCE/SEI 7-10 Section 12.8.

β ˜ = the fraction of critical damping for structure-foundation system.

W ¯ = the effective seismic weight of structure which shall be taken as 0.7 W, except for structures where effective weight is concentrated at a single level, taken as equal to W.

The effective period T ˜ shall be determined as follows in Equation (3)

where T = the fundamental period of the structure as determined in 12.8.2 of ASCE7-10.

k ¯ = the stiffness of the structure where the fixed base, defined by Equation (4)

where, h ¯ = the effective height of the structure, which shall be taken as 0.7 times the structural height (h_{n}) except for structures where gravity load is concentrated at a single level, equal to that level.

K_{y} = the lateral stiffness of the foundation as the horizontal force at level of the foundation necessary to produce a unit deflection at that level, the force and deflection being measured in the direction in which the structure is analyzed.

K_{θ} = the rocking stiffness of the foundation defined as the moment necessary to produce a unit average rotation of the foundation, the moment and the rotation being measured in the direction in which the structure is analyzed.

g = acceleration due to gravity.

Effective damping factor for structures foundation system β ˜ shall be computed by Equation (5)

β_{0} = the foundation damping factor as specified in

Similar to EC8-2004 Part 5 and ASCE/SEI 7-10,Chinese seismic code GB 50011-2010 [

So far, several SSI study on building report mainly focuses only on period of vibration, displacement, story drift, story shear and geotechnical parameters that affects SSI biased to geotechnical engineering. However, little attention is given to structural responses such as P-∆ effect due to gravity load of building itself affecting vertical structural members for example column, even if many building collapsed as a result of P-∆ secondary moments and instability. In this paper SSI effect (flexible base) on P-∆ effect is additionally studied using two methods: half space direct method of soil structure interaction, and Winkler Spring and then compared with fixed base using SAP2000 structural analysis software [

For comparative analysis of fixed base and flexible base structures, two residential buildings frame regular in plan and elevation are considered to avoid secondary

effects due to irregularity.

for S7 and CL70 cm × 70 cm for S12. Furthermore, the beam and column stiffness in the frame are proportioned to behave in shear mode according to [

In seismic weight calculation, given imposed load on residential building, according to ES1-2015 [^{2}, Roof live load LL = 1.0 kN/m^{2} and dead load floor finish and other walls including self-weight is 8.30 kN/m^{2}, and wall load on beam including self-weight of beam is 15.90 kN/m. For seismic weight calculation, GB50011-2010 is used 50% of live load and 100% dead load for dynamic analysis. Total dead load transferred on floor beam using tributary area including self-weight is 55.00 kN/m and roof floor beam is 37.28 kN/m for analysis. Live load transferred on floor beam 8.00 kN/m and 4.00 kN/m on roof beam.

Structural members material strength specifications from [_{c} = 31 × 10^{6} kN/m^{2}, Concrete Unit Weight = 24 kN/m^{3} and Steel S-400/HRB400, E_{s} = 200 × 10^{6} kN/m^{2}, Steel Unit Weight = 78.5 kN/m^{3}.

Soil properties supporting the structure: Assumed soft soil category according to GB50011-2010 and ES8-2015 for depth greater than 30 m below the structure Shear Velocity V_{s} = 150 m/s corresponds to soil D in ES8-2015 [^{3}, Poisson Ratio, ν = 0.40, Shear modulus, G = 33.75 Mpa, and Elastic modulus, E = 94.50 Mpa.

To date, many general-purpose structural analysis softwares are available for modelling structural members with well-defined member properties and boundaries of structures in either 3D or 2D analysis. However, Soil Structure Interaction (SSI) analysis model involves both structural member and foundation soil properties, which does not have well defined engineering material properties and boundaries. Because of this many simulation software’s may be suggested for soil structure interaction analysis but may not be suitable in design office for practice [

So far, for SSI analysis for both 2D and 3D model has been used using different methods. Using direct half space method, the base soil modelled together structure using 3D solid element [

Most structural analysis computer programs including SAP2000, chosen for this study, automatically apply the seismic loading to all mass degrees of freedom within the computer model and cannot solve the SSI problem. However, this can be solved using the most common soil-structure interaction (SSI) approach, used three dimensional soil-structure systems, is based on the ‘‘added motion’’ formulation [

For modelling of infinite soil surrounding structure to consider the effect of wave propagation the assumption adopted from [

V p = M ρ , P-Wave Velocity (6c)

V s = G ρ , Shear wave Velocity (S-Wave) (6d)

G = E 2 ( 1 + ν ) , Shear Modulus (6e)

M = E ( 1 − ν ) ( 1 + ν ) ( 1 − 2 ν ) , P wave Modulus (6f)

where K_{BN} and K_{BT} are the normal and tangential stiffness coefficients, respectively. C_{BN} and C_{BT} are the normal and tangential damping coefficients, respectively. A is the total area of all elements around the node at the boundary. r_{b} is the distance from the scattering wave source to the artificial boundary point. V_{s} and V_{p} are the wave velocities of the S wave and P wave, respectively. G is the medium’s shear modulus, E―Elastic modulus, ν―Poisson ratio and ρ is the medium’s mass density.

Given Soft Soil Properties above;

ρ = 1500.00 kg/m^{3}, ν = 0.40, G = 33.75 Mpa, E = 94.50 Mpa, V_{s} = 150 m/s, then, V_{p} = 367.42 m/s, K_{BN} = 2250.0 kN/m, K_{BT} = 1125.0 kN/m, C_{BN} = 1653.0 kN∙s/m and C_{BT} = 675.0 kN∙s/m. This values are used for modelling viscous spring dashpot in SAP2000 [

In addition to the direct method of modelling, for evaluating the seismic response of reinforced concrete multi-story buildings with raft foundation on soft soil, the underneath soil is modeled by Winkler spring approach with equivalent static stiffness based on soil modulus of elasticity [_{s} = 150 m/s., equivalent static stiffness for different direction vibration mode, the soil spring stiffness can be calculated using Gazetas 1991 [

Vertical direction (Z) K z = G L ( 1 − ν ) [ 0.73 + 1.54 ( B L ) 0.75 ] (7a)

Horizontal (lateral direction y) K y = G L ( 2 − ν ) [ 2.0 + 2.50 ( B L ) 0.85 ] (7b)

Horizontal (Longitudinal direction x) K x = K y − 0.2 G L ( 0.75 − ν ) [ 1 − B L ] (7c)

where G is shear modulus of soil defined in Equation 6(e), E is the modulus of elasticity of soil; ν is the Poisson’s ratio of soil. L and B are the length and width of Raft foundation of the whole building, respectively. For Raft foundation plan Length L = 32 m and Width B = 15 m, Soft soil with shear velocity V_{s} = 150 m/s, Poisson ratio ν = 0.4, Shear Modulus G = 33.75 Mpa, Modulus of Elasticity E = 94.50 Mpa, Dynamic stiffness in the vertical direction, K_{z} = 6.00 × 10^{3} (kN/m^{2})/m, Horizontal in lateral direction K_{y} = 4.66 × 10^{3} (kN/m^{2})/m, Horizontal Longitudinal direction K_{x} = 3.98 × 10^{6} (kN/m^{2}) /m. For 2D frame analysis, the total stiffness of raft is distributed to each column support according to its tributary area.

In this study strong ground motion Loma Prieta 1989 from PEER [

Design response spectrum with 10% exceedance in 50 years’ PGA = 0.30 g, M = 8 measured with reference to Chinese Code GB50011-2010 for Soil II is used. This is equivalent to 0.25 g of soil B of Ethiopia ES8-2015 based on Chinese and European seismic code seismic soil equivalency study [

For dynamic analysis, strong ground motion of Loma Prieta earthquake of USA in 1989 ground motions, PGA = 0.367 g is used [_{1} and 2T_{1} minimum and maximum for specific building structure, where T_{1} fundamental period of vibration, using modal analysis in SAP2000 with fixed base model the fundamental period of vibration. In this study 12 Story frame with fundamental period of vibration T_{1} = 2.12 s is used. Accordingly, the matching period for Loma Prieta earthquake is T_{min} = 0.424 s and T_{max} = 4.0 s.

Using the matched strong ground motion linear time history analysis is made for flexible and fixed base support conditions of the frames Story 7 (S7) and Story 12 (S12). For all support conditions, the seismic responses are calculated using SAP2000 structural analysis software [

This section presents the dynamic analysis results. This includes dynamic properties

Earthquake Ground Motions | PGA (g) | PGV (cm/s) | PGD (cm) |
---|---|---|---|

Original Loma Prieta | 0.367 | 44.69 | 19.61 |

Matched to Soft Soil D ES82015 | 0.384 | 50.11 | 19.63 |

Matched to Soft Soil IV GB50011-2010 | 0.396 | 46.84 | 17.35 |

and other response of the model structures S7 and S12 shown in table and graph form. All response comparisons are made with reference to fixed base for both types of flexible base models presented in subsequent sections.

Modal analysis of the model frames S7 and S12, both fixed and flexible base (SSI), is made using finite element method software, SAP2000 [

Story displacement is very essential parameters for nearby building collision effect in seismic event for making enough separation between nearby structures. The deflection profile is different based on fixity of the base.

Story drift is one of the important parameter for lateral load effect on vertical members in stability analysis.

Model Building | FB (Fixed Base) | SSI-WS (Spring) | SSI DM (Direct) | FB (Fixed Base) |
---|---|---|---|---|

S7 | 1.20 | 1.89 | 2.04 | 1.20 |

S12 | 2.12 | 3.23 | 3.53 | 2.12 |

179%, decreasing trend. Similarly,

bottom and top stories respectively. Similarly soil IV, 92% and 120% of fixed base in bottom and top stories. On the other hand, spring model for soil D varies 67% to 60% bottom to top and the same trend for soil IV 77% to 70% bottom to top stories of fixed base model decreasing trend.

Furthermore, (

Spring model for soil D varies 53% to 76% bottom to top and the same trend for soil IV 60% to 75% bottom to top stories of fixed base model increasing trend compared to S7. This shows, the decreasing effect is not always true for multi-Story building.

One of the detrimental effect of flexible base is P-Delta effect due to excessive deflection. Because in both fixed base and flexible base the vertical gravity axial load in columns remain constant, but Story shear changes depending on ground motion magnitude, ES8-2015 [_{tot}/h remains the same. The effect of change in base fixity is noted in the ratio of story drift to storey shear (Δ/V_{tot}). This effect can be shown in

θ = P t o t ⋅ Δ V t o t ⋅ h ≤ 0.10 (8)

where

θ is the inter Story drift sensitivity coefficient;

P_{tot} is the total gravity load at and above the Story considered in the seismic design situation;

Δ is the design inter story drift, evaluated as the difference of the average lateral displacements d_{s} at the top and bottom of the Story under consideration and calculated in accordance with 4.3.4 of ES8-2015 [

V_{tot} is the total seismic Story shear; and

h is the inter story height.

In addition,

This study shows both flexible base models give bigger fundamental period of vibration and drift with reduced base shear than fixed base, good agreement with seismic codes provisions [

In addition, closer examination of spring model of ground motion effects of D and IV on model frames S7 and S12 is different. The smaller ground motion with soil D gives bigger story drift, P-delta effect and Story shear on S7 than S12 while larger ground motion with Soil IV shows the opposite. This indicates bigger earthquake effect more on taller than short building. Because soil IV ground motion (Chinese GB50011-2010) is greater than soil D (Ethiopian ES8-2015), which is related to the design response spectrum of the corresponding seismic codes magnitude.

To sum up, SSI effect may not be always beneficial in multi-story RC frame compared to fixed base. Because the beneficial effect reduction in base shear may be smaller than detrimental effect of P-delta increment on vertical load carrying members. The results obtained in this study is limited to linear time history analysis regular 2D RC frame; however it is good indicator of SSI effect; further study can be made in future to take non linearity effect both in structure and soil.

The first Author thanks China Scholarship Council (CSC) for sponsoring this study in China Three Gorges University. In addition, all references and softwares used from different sources are duly acknowledged.

The authors declare no conflicts of interest regarding the publication of this paper.

Kabtamu, H.G., Peng, G. and Chen, D.H. (2018) Dynamic Analysis of Soil Structure Interaction Effect on Multi Story RC Frame. Open Journal of Civil Engineering, 8, 426-446. https://doi.org/10.4236/ojce.2018.84030