Assessment of Response Modification Factor of Reinforced Concrete Table Top Frames Structures Subjected to Seismic Loads
Abstract
In this study, the seismic performance of tabletop reinforced concrete frame structure which is commonly found in large oil refineries. Nonlinear static pushover analysis is conducted to study the inelastic behaviour of these frames. The analysis accounts for their unique detailing especially to elements that contribute to energy dissipation during major seismic events. The results show that the reserve strength of structure is greater than that prescribed by the ASCE710 for Special reinforced concert moment frame.
1 Introduction
Tabletop structures are unique type of structures and commonly found in large refineries where they are used to process heavy crude oil. The unit is a complex structure that consists of massive reinforced ordinary reinforced concrete frame which supports pressure vessels and steel towers carrying maintenance platforms. The pressure vessels are connected to the frames using a circular pattern of relatively deep anchor bolts. To provide sufficient embedment of the anchor bolts and the required strength for the anchor bolts to support uplift force due to wind loads, the beams of the frames are very deep and wide in cross section. For Frame action computability, the columns cross sections are seized based on the sizes of the beams rather than the size that are required to resist gravity and lateral loads. In other words, the sizes and layout of the pressure vessels is what determine the cross section of the frame beams and columns.
The current practice is to estimate the seismic loads for these structures using parameters of similar buildingtype structures. The structures are substantially different from typical building structures where there are no diaphragms with lumped mass. Therefore, building code design equations are not necessarily suitable to predict their performance during earthquakes. Many of these structures are constructed or planned to be constructed in an area of high seismic activities and a safe and economic design of these units is of a great value to the society. The dilemma exists in active seismic sites which dictate all reinforced concrete moment frames to be designed as a Special Moment Frame (SMF) with its restricted requirements in ties spacing, lap splice, support of vertical bars, and the continuation of ties into the beamcolumn connection. This requirement complicates the construction of these frames in active seismic zones. Due the massive nature of these frames, it is believed that these structures will most likely remain elastic during major earthquake.
This main purpose of this research is to assess the vulnerability of these unique structure during major seismic event by evaluating the Response Modification Factor (R factor) for these structures through numerical simulations in order to determine the proper classification of theses frames from the building code stand point.
2 Table Top Structure
3 Response Modification Factor
Conventional seismic design in most modal codes is forcebased, with a final check on structural displacements. The forcebased design is suited to design for actions that are permanently applied. The seismic design follows the same procedure, except for the fact that inelastic deformations may be utilized to absorb certain levels of energy leading to a reduction in the forces for which structures are designed. This leads to the creation of the Response Modification Factor (R factor); the allimportant parameter that accounts for overstrength, energy absorption, and dissipation as well as the structural capacity to redistribute forces from inelastic highly stressed regions to other less stressed locations in the structure. This factor is unique and different for different type of structures and materials used. Hence, classification of Response Modification Factor for various structural systems is extremely important in order to do an evaluation based on demand and capacity of the structure.
An elastic design would assume the structure to have a linear forcedeformation relationship, and thus be able to deform less and dissipate less energy. On the other hand, inelastic design acknowledges the ability to continuously deform and keep on dissipating energy. The use of right ductility and Response Modification Factors can result in a safer and more cost effective structure than one designed using the elastic design force, which does not acknowledge the inelastic energy dissipating ability.
 \( R_{0} \)

= Over strength factor
 \( R_{\mu } \)

= Ductility factor
 \( R_{r}\)

= Redundancy
The response modification factor is determined as the product of the overstrength factor and the ductility factor and redundancy factor, these factors can be idealized by Base shear verses Displacement, it can be seen in Fig. 3, which can be developed by a nonlinear static pushover analysis.
Overstrength Factor (\( R_{0} \))
Where:
\( V_{y} \) = available yielding strength.
\( V_{d} \) = design base shear determined from ASCE 710.
Ductility Factor (\( R_{\mu } \))
Where:
\( V_{e} \) = Linear elastic force.
\( V_{y} \) = available yielding strength.
Redundancy Factor (\( R_{r} \))

Seismic Design Category B or C.

There are four or more columns and three or more bays at each level.
Otherwise redundancy factor shall equal to 1.3.
For this study, the redundancy factor is equal to 1.0 due to many moment frames.
4 Numerical Simulations
4.1 Design of Model Structure
Table top frames usually consists of two parts; first the bottom part, the structure is usually composed of reinforce concrete frame i.e. it is usually a Moment frame therefore in this thesis the bottom part of the frame is special reinforced concrete moment frame (SMRF’s). Which is used to support maintenance platforms and secondly, it is consisting of steel cage system at the top of the reinforced concrete frame to support the crude oil distillation unit. The ordinary braced frames used to resist lateral force. It is a conventionally reinforced concrete tabletop vessel support structure, massive concrete frame that is pile supported.
Structure configuration
Group  Structure model  Compressive strength of concrete (fc’)  Size of column (ft)  Size of beam (ft)  

Group I  Group IA  G13ksiD  3ksi  8’ × 8’  6’ × 9’ 
G14ksiD  4ksi  8’ × 8’  6’ × 9’  
G16ksiD  6ksi  8’ × 8’  6’ × 9’  
Group IB  G13ksiND  3ksi  8’ × 8’  6’ × 9’  
G14ksiND  4ksi  8’ × 8’  6’ × 9’  
G16ksiND  6ksi  8’ × 8’  6’ × 9’  
Group II  Group IIA  G23ksiD  3ksi  6’ × 6’  4’ × 7’ 
G24ksiD  4ksi  6’ × 6’  4’ × 7’  
G26ksiD  6ksi  6’ × 6’  4’ × 7’  
Group IIB  G23ksiND  3ksi  6’ × 6’  4’ × 7’  
G24ksiND  4ksi  6’ × 6’  4’ × 7’  
G26ksiND  6ksi  6’ × 6’  4’ × 7’ 
Detailing of cross sections
The base of the structure is fixed, and the loading applied into the structure is; Dead load 100 k/ft at the top and 10 k/ft at the bottom stories and Live load 50psf.
4.2 Material Properties
The mixed design of concrete used for this thesis is aimed at design cylinder strength is 3ksi, 4ksi, 6ksi. Typically design for this strength has a slump test is about 1–2 in., the maximum size of course aggregate is ¾ in. The mix proportion is about 1:3:5 as refer to cement, sand and course aggregate.
The steel reinforcement used in reinforced concrete frames is grade 60 (yielding strength is fy = 60 ksi).
4.3 Nonlinear Static Analysis (Push over Analysis)
Nonlinear static pushover analysis is conducted to determine the ultimate lateral load resistance as well as the sequence of yielding/buckling events. Eigen value analysis was conducted first to determine the elastic natural periods and mode shapes of the structure. Then pushover analysis were carried out to evaluate the global yield limit state and the structural capacity by progressively increasing the lateral story forces proportional to the fundamental mode shape.
Nonlinear pushover analysis serves the basis for determining the capacity of the structure in terms of base shear and roof displacement (Δ), is a method for determining the ultimate load and deflection capability of a structure. In SAP2000, moment curvature curves with post yield behavior are predetermined and can be used to determine hinge properties. The hinges are placed to predict possible hinge formation locations, which are usually near the joints between members. Incremental lateral load applied, the model is then run to view the conceptual force capacity. Local nonlinear effect, such as flexural hinges at the member joints, are modelled and the structure is deformed until it reaches to enough hinges form to develop a collapse mechanism or until it reaches to the plastic deformation limit of a hinges is reached (Fig. 5).
4.4 Nonlinear Dynamic Time–History Analysis
The basic Newmark Constant acceleration method can be extended to nonlinear dynamic analysis. This requires that iteration must be performed at each time step in order to satisfy equilibrium. Also, the incremental stiffness matrix must be formed and triangularized at each iteration or at selective points in time. Many different numerical tricks, including element by element methods, have been developed in order to minimize the computational requirements. Also, the triangularization of the effective incremental stiffness matrix may be avoided by the introduction of iterative solution methods.
5 Results and Discussions
For this study, the result shown below the demand capacity curve.
 If compressive strength (f’c) increases the capacity of the structure also increases and it is also reaches to the demand of the structure (ATC40). From Fig. 10, the design base shear, yielding shear and the maximum seismic demand for the elastic response could be calculated for each case; resulting in the calculated R factors for this study.

As the size of the cross section increase the capacity of the structure also increases.

The maximum base shear does not reach to the maximum base shear from pushover curve,

It is roughly coincide to the yielding range of the pushover, which indicate that the structure itself a huge capacity to reach the failure mechanism of the structure during earthquake,

These 22 earthquakes does not reach to the pushover and these structure remain elastic and did not reach to plastic b/c of the massive size of beams and columns.

And it is also indicating that SMRF’s behaved elastically when subjected to the earthquake obtained from the dynamic analysis.
There is no added value for the special ductiling requirement and ductility that usually exhibit in the special reinforced concrete moment frames. Therefore, based on this study the ordinary moment frame was a nonductile crosssection are suitable for this types of structure in high seismic zone thus the structure does not require the additional ductility. Furthermore, the Rfactor equal to 9, for these types of frames.
The above histograms discuss a detailed comparison of the Overstrength, Ductility, and Response Modification factor of Group I and Group II.
In this case study, the over strength factor for the Tabletop Reinforced Concrete Moment Frames had calculated value is 4, which is above the current code value of 3 in ASCE 710.
The ductility factor for tabletop reinforced concrete moment frames was almost equal in all cases of Group I, while it is increased as the cross section sizes decreased of Group II.
In typical tabletop reinforced concrete frames, in group II the Response Modification Factor was very close to the code value of 7, while the value prescribed in ASCE 710 is 8 seem to greatly underestimate it. Figure 9 is an illustration of the different R factors due to the various parameters applied in the different cases. The ASCE 710 value show with Black Doted line.
As you can see that in Group IB and IIB (6ksi) the response modification factor is decreases as compared to the 3ksi and 4ksi. It is because of the less ductility into the system as you can see the Pushover curve the strength is increases as compressive strength increases but the displacement also decreases that is why the response modification factor in Group IB and IIB (6ksi) decreases. So, I can say that the increase in compressive strength is not beneficial and it seem that the structure is fail in concrete.
6 Conclusions
As the SMF’s were designed based on preliminary response modification factor and their tentative values were evaluated. According to the mentioned procedure all models were analyzed and their final seismic response modification factors were calculated.
Response modification calculated values
Group  Structure model  Ductility  Over strength factor  Response modification coefficient R  

Group I  Group IA  G13ksiD  2  5  10 
G14ksiD  2  5  11  
G16ksiD  2  5  11  
Group IB  G13ksiND  2  5  11  
G14ksiND  2  5  11  
G16ksiND  2  5  10  
Group II  Group IIA  G23ksiD  3  3  8 
G24ksiD  3  3  8  
G26ksiD  3  3  9  
Group IIB  G23ksiND  3  3  7  
G24ksiND  3  3  7  
G26ksiND  2  2  6  
Average  3  4  9 

In (Group I) the R factor is greater than Group II, which means that larger the cross section higher the R factor.

The system has a considerably high overstrength value which is around (3–5),

Response modification factor is between (710), see Table 3, below for response modification factor calculated for prototypes frames compared to that value which is prescribed in ASCE 7–10 values i.e. 8,

And ductility factor comes out to be (2–3), for these types of frames.

As we noticed that the increase in the cross section increased the overstrength and response modification factors.
We can see, there is no added value imposes this ductile requirement. to the structures and it is not beneficial the structure remains elastic so, easy reinforced configuration will probably suffice the requirement and make these structure easy to build and more cost effective.
Notes
Acknowledgments
The authors would like to acknowledge the support provided by the department of Civil Engineering at Cal Poly Pomona in offering the computer recourses that was necessary to conduct this study.
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