S. Bhargavi1 and Ramancharla Pradeep Kumar2
1MSby Research Student, 2Professor
Earthquake Engineering Research Centre, International Institute of Information Technology, Hyderabad Email: 1[email protected], 2[email protected]
ABSTRACT
Response of a structure subjected to gravity and lateral loads depends on the boundary conditions assigned at the base of structure in numerical modeling. Most of the structures are analyzed considering fixed base, but in reality, foundation is not fixed. The fixity depends on the interaction between the soil and foundation. In most of the cases base of a structure undergoes small amount of rotation because of flexibility induced by soil especially at the time of earthquake. Difference in boundary conditions used in analysis and in actual conditions will lead to improper estimation of the design forces, to reduce such effects in numerical analysis fixed base of structure can be replaced by springs or structure base resting on soil to obtain results closer to that of actual base conditions. Stiffness of the spring depends on geotechnical parameters as well as on the dimensions of the foundation. In the present paper an attempt is made to understand the difference between the Linear and Non Linear response of a frame having fixed base, spring base and with soil. Stiffness of springs for spring base structure is calculated as per ATC-40 [1] and JICA [7]. Analysis of structure with soil modeling is complex, specifically for Non Linear behavior for this Applied Element Method is used. Linear and Non Linear response is compared in terms of stress resultants and capacity of the frame respectively, for non linear response static pushover analysis is done. From the study, it is observed that the actual capacity of structure is overestimated assuming fixed base. The fact that the initial stiffness of the structure is governed by the soil stiffness can be clearly stated from the obtained pushover curves. It is also recommended that analysis of structure should be done considering spring base in place of fixed support.
Keywords— Continuum Approach, Structure Approach, Static Pushover Analysis, Applied Element Method.
INTRODUCTION
In general practice analysis of superstructure with fixed base is done separately and reactions are used in designing substructure, in this type of analysis interaction between foundation and soil is neglected by assuming fixity at the base; soil is assumed to be rigid as a result actual base condition effect on the structure response is neglected. The base condition of the structure depends upon the geotechnical parameters of soil media on which the structure is standing. Former can be cohesive or cohesionless based on site conditions. The fixity of the structure solely does not depend on those characteristics of the soil. Structure is considered to have fixed base due to the difficulties faced in considering base as flexible.
Generally base is considered as fixed in design codes to simplify the design procedure. In real world, upliftment of the foundation is affected both by vertical loads and lateral loads, which is not in case of theoretical fixed base structure. To take into account the effect on structure due to soil structure interaction, it is very important to consider the actual base condition in performing seismic analysis using design packages.Response of the structure can be known from linear and nonlinear analysis. The
strength capacity of frame can be analyzed by non linear static pushover analysis.
NUMERICAL MODELING
Effect of soil in analysis can be considered by two approaches; Structural (Substructure) and Continuum (Direct) approach. In structural approach soil is represented by structural elements like spring, in continuum approach soil and structure is modeled together [3]. In substructure method in place of fixity, resistance offered by the foundation from soil is represented by springs having stiffness equivalent to that of soil. When foundation is considered as rigid in numerical modeling, then at base 3 springs are provided for 2D structure and 6 springs are provided for 3D structure. Figure: 1 shows arrangement of springs at the base in three directions corresponding to the stiffness direction offered by foundation and soil.
For the current study a 2D frame of span 3 m and height 3 m with column and beam dimensions 0.3 x 0.3 m and 0.25 x 0.3 m respectively is considered with following boundary conditions:
Seismic Behaviour of Fixed and Flexible 2D RC Frame: A Case Study
1. Fixed Base 2. Spring Base 3. Frame with Soil
Fig. 1: Spring Model
2D frame considered is designed for gravity loads as per IS: 456 (2000)
Fixed Base Model
Two techniques were adopted to analyze 2d RC frame with fixed base;
Finite Element Method using SAP2000 package.
Applied element method.
The reason behind analyzing fixed frame in two techniques is to make the comparison simple.
(a)
(b)
Fig. 1: 2D RC frame model with fixed base (a) SAP2000 (b) AEM
Spring Base Model
To model RC 2D frame with spring base, geotechnical parameters are converted into stiffness of spring as per ATC-40 and JICA. Stiffness of spring is calculated along three directions i.e. vertical, horizontal and rotation about Y-axis. Three mechanisms, related to axial, shear or rotation can lead to soil failure. In order to accommodate these three mechanisms, springs at the base of the structure are provided with calculated stiffness (Eq. 1-10).
The first mechanism, associated with axial, is the punching of the soil. The second mechanism, associated with shear, is a translation mechanism (sliding of the foundation) with the activation of a passive zone in front of the foundation and an active zone behind the foundation. The last mechanism, associated with rotation, is a global rotation with active and passive zones around the foundation [5].
Stiffness calculations: As per Japanese practice [PWRI, 1998; JICA, 1992]
Where and are moduli of sub grade reaction in horizontal and vertical directions respectively and is modulus of sub grade reaction corresponding to in-plane shear at the base of the foundation.
= width of foundation perpendicular in the considered direction of shaking (m).
= Equivalent base dimension of the foundation (m) = Ratio of horizontal shear sub grade reaction to that of vertical modulus of sub grade reaction = 0.5
√
Where and are basic moduli of sub grade reaction in horizontal and vertical directions.
{
Where is shear wave velocity in the soil layer calculated from the uncorrected N values.
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{
Where the design shear wave velocity is calculated using .
is the dynamic shear modulus of the soil layer calculated from mass density of soil layer ( ) and .
Where is dynamic modulus of elasticity calculated from Poisson’s ratio of 0.25 for soil and .
Similarly stiffness of soil is calculated as per ATC-40 chapter 10 section 10.4.1.
Stiffness values calculated as per JICA and ATC-40 code, mentioned in Table1 are used to model springs as shown in Figure: 4, to carry out linear and non-linear analysis for 2D RC frame.
Table 1: Stiffness Values Stiffness Stiffness as per
JICA (kN/m ) Stiffness as per ATC-40 (p/ft)
70625.69 1234908
100439.2 2551503
1506.587 1345455
Fig. 2: (a) 2D frame Model (b) Spring arrangement at base Frame Modeled along with Soil
Generally, springs behave independently, whereas in reality soil is continuous medium; the soil property will
change depending upon the loading conditions but whereas in spring system soil properties remain constant in terms of stiffness. Thus to achieve better results 2D frame is modeled with base resting on soil. This model is analyzed using AEM. Cohesionless Soil parameters are considered in soil block modeling. Dimensions of soil block are 9.7 m x 3 m x 3 m, boundary conditions of soil block are considered to be fixed as shown in Figure 4.
Fig. 3: AEM model of frame with soil ANALYSIS TECHNIQUES
Finite Element Method
FEM is numerical technique used to analyze different boundary condition systems to obtain approximate solutions. In this study FEM is used to perform both linear and nonlinear analysis for 2D frame with fixed base and 2D frame with springs at the base using SAP2000 package. To carry out nonlinear pushover analysis in SAP2000 it is important to model hinges in the frame, this hinges gives nonlinear behavior of frame. Hinges are assigned at the locations where frame is expected to enter into inelastic deformation. Hinge length and location are calculated as per ATC-40.
Applied Element Method
AEM is used for numerical analysis to know continuum and discrete behavior of structures. In this study AEM is used to perform both linear analysis and nonlinear analysis, to know the response of 2D frame with discrete soil system and 2d frame with fixed base. In nonlinear static pushover analysis displacement control method is adopted as a result plot between force and displacement is obtained as shown in Figure: 10.
PUSHOVER ANALYSIS
Pushover analysis is a procedure which involves applying monotonically increased lateral loads in a predefined load pattern along the height of the frame. The corresponding stress resultants are found in each step individually and the global capacity of the structure is estimated. The global capacity is represented as a plot between the base shear and corresponding lateral load displacement. This procedure can be carried out in two ways; load controlled and displacement controlled. Load controlled procedure involves applying lateral loads in constant proportions whereas displacement controlled procedure involves applying lateral load corresponding to displacement,
Seismic Behaviour of Fixed and Flexible 2D RC Frame: A Case Study which is increased constantly. Lateral load profile will
affect the response of the structure in static nonlinear pushover analysis. There are three types of lateral load patterns of which one load pattern is considered; load is applied in first mode.
Plastic Hinge Length Calculations
=plastic length
Section depth in the direction of loading
= Plastic hinge length at the starting of column
Fig. 4: Hinge Locations
Plastic hinge length at the end of the column
Plastic hinge length at the starting of beam
Non-linear analysis is carried out through pushover technique considering displacement control method as a result plot between base shear and top displacements are obtained as shown in Figure: 9.
Table 2: Hinge Locations of beam and column Element Starting
(Relative Length)
Ending (Relative Length)
Beam 0.025 0.925
Column 0.025 0.975
Hinge Formation Sequence
Figure 6 shows hinge status at the final loading step of pushover analysis; it is observed that hinge status for fixed base and spring base (JICA) is same whereas differing for spring base (ATC40).
Fig. 5: Final hinge status of (a) Fixed base (SAP) (b) JICA spring SAP (c) ATC-40 spring SAP
Fig. 6: Base Shear vs. Roof Displacement curve for fixed base
Fig. 7: Base Shear vs. Roof Displacement curve for fixed and spring base (FEM)
0 10 20 30 40 50 60
0 0.05 0.1 0.15 0.2
Base Shear (kN)
Roof Displacement (m)
Fixed Base (FEM)
0 10 20 30 40 50 60
0 0.05 0.1 0.15 0.2
Base Shear (kN)
Roof Displacement (m)
Fixed Base
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Fig. 8: Base Shear vs. Roof Displacement curve for fixed and soil base (AEM)
RESULTS AND INTERPRETATIONS Linear Analysis
Linear analysis results are shown in Table 3. Moment and Shear force results of 2D fixed frame are more compared to the results of 2D frame with spring base as tabulated due to difference in the base conditions of frames, but vertical deflection in beam is very large compared to fixed base. Compared design forces are from gravity load only the effect will differ in case of lateral load. From this observation it can be stated that base condition of frame effect the design forces.
Initial Stiffness of Frame
Due to change in the base condition, 67.2% is the variation in initial stiffness of 2D frame soil model
compared to2D frame fixed, 67.1% is the variation of initial stiffness of 2D frame JICA spring model compared to 2D frame fixed model and 70.3% is the variation of initial of 2D frame ATC spring model compared to2D frame fixed model in Table: 6.
Table 4: Initial stiffness results
AEM results FEM results
Model
Type Stiffness
(kN/m) Model Type Initial Stiffness (kN/m)
Fixed 9367.38 Fixed 5902.36
Soil 3080.68 JICA Spring 1946.36
- - ATC-40 1756.73
Nonlinear Analysis
Figure (6-9) shows that capacity and strength of structure is more in case of fixed base, thus if performance of structure is estimated on the basis of fixed base condition the structure will be safe but in actual condition its performance is very low comparatively.
CONCLUSIONS
Study done clearly shows that base support condition has an impact on the behavior of structure in Linear and Non linear analysis. Soil structure interaction should be considered in analysis, it is difficult to go for continuum modeling but structure approach can be used in analysis.
Capacity and performance of the structure is to be
Table 3: Moments and Shear forces comparison for spring and fixed base
Element
Table 5: Comparison of stiffness and strength
Analysis Type Model-1 Model-2 Comparison
Parameter
% Variation from Model-2
to Model-1
Linear(AEM) 2D Soil frame 2D Fixed frame Initial Stiffness 67.2
Linear(FEM) 2D spring Frame JICA 2D Fixed frame Initial Stiffness 67.1 Linear(FEM) 2D ATC-40 spring Frame 2D Fixed frame Initial Stiffness 70.3
Non-linear (AEM) 2D JICA spring Frame 2D Fixed frame Strength 25.2
Non-linear(AEM) 2D ATC-40 spring Frame 2D Fixed frame Strength 66.4
Non-linear (AEM) 2D Soil frame 2D Fixed frame Strength 2.65
NON-linear 2DFixed frame(AEM) 2DFixed frame(FEM) Strength 26
Seismic Behaviour of Fixed and Flexible 2D RC Frame: A Case Study
REFERENCES
[1] Applied Technology Council ATC-40. Seismic Evaluation
& Retrofit of concrete Buildings.Volume.1 1996
[2] Applied Technology Council ATC-40. Seismic Evaluation
& Retrofit of concrete Buildings.Volume.2 1996
[3] Caselunghe Aron & Eriksson Jonas. Structural Element Approaches for Soil-Structure Interaction. MS report Chalmers University of Technology 2012
[4] Jonathan P. Stewart University of California, Los Angeles.
Overview of Soil-Structure Interaction Principles. Stewart University of California, Los Angeles.
[5] Damien Dreier. Influence of soil-structure interaction on structural behavior of integral bridge piers. PhD Symposium in Stuttgart, Germany 2008.
[6] Tagel-Din Hatem and Kimiro Meguro. Applied Element Method for Simulation of Nonlinear Materials: Theory and Application for RC Structures. Structural Eng.
/Earthquake Eng. JSCE, Vol 1, No.2, 2000.
[7] JICA, (1992), Specifications for highway bridges, part IV:
Substructures, Earthquake Disaster Prevention Research Centre, Public Works Research Institute, Tsukuba-Shi, Japan, 445p.
Proceedings of the National Conference on Advances in Civil Engineering and Infrastructure Development (ACEID-2014), Vasavi College of Engineering, Hyderabad, A.P. 6 - 7 February, 2014. pp.130-133.