ACCIDENTAL TORSION IN THE MOROCCANSEISMIC CODE: PARAMETRIC STUDY
Nabil MEKAOUIMEE17714

Supervisor:

Taiki SAITOABSTRACT
Many researchers have studied the natural torsion of buildings subjected to earthquake groundmotions, but only a few of them studied the accidental torsion. The increase in response in the flexible
and the stiff edges of the building due to accidental torsion is investigated in this study, using both
static and time history analyses. The accidental torsion is introduced by the accidental eccentricity
provided in the Moroccan seismic code: RPS2000 version 2011. Based on the results of the time
history analyses, a new practical method is proposed to assess the maximum increase in response due
to accidental torsion. This method is generalized to a certain category of multi-storey buildings andpresents the main advantage to avoid performing many extra seismic analyses as required in thecurrent method of the Moroccan seismic code. The new proposed method is limited to the elastic
domain to be compliant with the design procedure of the Moroccan seismic code. Further studies haveto be performed to assess the efficiency of the elastic seismic design procedure that guaranties that thebuilding will reach the required seismic level of performance by inelastic deformations.
Keywords: Accidental torsion, eccentricity, frequency ratio, earthquake, Morocco.
1. INTRODUCTION
During a ground shaking, the forces applied to the lateral resisting elements of a building are inertia
forces. Their resultant passes through the Center of Mass (CM) of the structure. These forces areresisted by the lateral resisting elements via their stiffness. In the elastic range, the resultant of all theresisting forces passes through the Center of Stiffness (CS) or Center of Rigidity (CR), because theforce in a resisting plane is proportional to its stiffness. When the CM and the CR are coincident, thestructure is referred to as nominally a symmetric-plan building. Otherwise, the distance separatingthem is defined as static eccentricity (es) and the structure is referred to as an asymmetric-planbuilding. Subjected to a pure translational earthquake ground motion, an asymmetric-plan building issubjected to story torques in addition to shears and overturning moments. Hence, it undergoes arotational motion in addition to the lateral one. This rotational motion is referred to as natural torsion.
It has been proved from real earthquake records of existing nominally symmetric-plan buildings, thatthe latter category of buildings can also undergo a rotational motion in addition to the lateral one (Dela Llera and Chopra, 1994a). The main origins of these rotational vibrations are the discrepancies thatexist between the mass, stiffness and strength plan-distributions used in the analysis and the true
distributions at the time of the earthquake; and the base rotational motion due to a non-uniform
distribution of the ground motion in the space (De la Llera and Chopra, 1994b). To account for thetorsional vibrations caused by these uncertainties, an additional eccentricity is introduced in many
seismic codes which is referred to as accidental eccentricity (eacc). The rotational motion caused by theaccidental eccentricity is referred to as accidental torsion.

Mohammadia School of Engineers (EMI), Mohammed V University of Rabat, Morocco.
Professor, Toyohashi University of Technology (TUT), Japan.

12. DATAThis study is carried out on single storey buildings using the software STructural Earthquake Response
Analysis 3D (STERA 3D), elaborated by Dr. Taiki SAITO. Then the results are generalized to acertain category of multistorey buildings. The plan view from one of the studied single stories ispresented in Figure 1.

Figure 1. Plan view and perspective of the studied building.

The system is areinforced concrete structure
formed by a grid of identicalcolumns and two different typesof shear walls. During all thestudy, the structure remains
nominally symmetric in the X
direction, and the location ofthe shear walls in the Y
direction is controlled toproduce the target staticeccentricity. The floor isconsidered rigid with a uniform
distribution of the mass.

Because of the unavailability of the earthquake records in Morocco, artificial earthquake
ground motions were generated with compatibility with the design spectrum proposed in the Moroccan
Seismic Code. Table 1 summarizes the five seismic waveforms used in this study and Figure 2 shows
an example of the generated ground motions and its compatibility with the design spectrum of theMoroccan Seismic Code.
Table 1. Generated artificial earthquake ground motions.
Artificial Earthquake Ground Motion (GM) Nᵒ GM
Al Hoceima (random phase)
1
Al Hoceima (El Centro phase)
2
Al Hoceima (Kobe phase)
3Agadir (random phase)
4Rabat (random phase)5

Three target spectra wereconsidered: Al Hoceima city, Agadircity and Rabat the capital. Two phases
are used: The Kobe phase and the El
Centro phase. A third case is added byconsidering a random phase andmultiplying the seismic wave by
Jennings Envelope.

Figure 2. Artificial earthquake ground motion GM1 (Al Hoceima city).

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3. THEORY AND METHODOLOGY
3.1. Theory
The ratio of the uncoupled circular frequencies of the model building is referred to as the parameter Ω,and it’s given by equation (1). It’s the main parameter that controls the increase in response due totorsion. The response of the building subjected to an earthquake ground motion can be expressed byequation (2):(1)(2)
where ωy and ωθ are the uncoupled circular frequencies of the building, ky is the lateral stiffness in theY direction, kθ is the rotational stiffness around the CM, es is the static eccentricity in the Y direction,uy and uθ are respectively the translation in Y direction and the rotation around the vertical axis Z
through the CM, r is the mass-radius of gyration, ed is the design eccentricity, and ugy is the grounddisplacement in Y direction.
3.2. Methodology
Three different static analyses are needed to calculate the increase in displacement in the flexible edge
(If,st), respectively in the stiff edge (Is,st), for a specific model building. Figure 3 highlights the threeanalyses with three different colors: black color for the 1st seismic analysis using only the staticeccentricity as a design eccentricity (the effect of the accidental torsion is not included), red color forthe 2nd seismic analysis with the CM shifted by 0.5es + 0.05b to the flexible side, and green color forthe 3rd seismic analysis with the CM shifted by 0.05b to the stiff side. By the 1st analysis (black color),
the displacements in Y direction Uy,f and Uy,s respectively in the flexible and the stiff edge aredetermined. These latter displacements do not include the effect of the accidental torsion. The 2ndanalysis (red color) produces the greatest displacement in the flexible edge U *y,f including the effect of
the accidental torsion. The 3rd analysis (green color) produces the greatest displacement in the stiffedge U*y,s including the effect of the accidental torsion. Finally, the ratios of these 4 determined
displacements are calculated to get the investigated parameters If,st and Is,st.

Figure 3. Calculation of the investigated parameters by the static analysis.
Three different time history analyses are needed to calculate the increase in displacementin the flexible edge (If,dy), respectively in the stiff edge (Is,dy) for a specific model building. Figure 43highlights the three analyses with three different colors: black color for the 1st seismic analysis using
only the static eccentricity as a design eccentricity (the effect of the accidental torsion is not included),
red color for the 2nd seismic analysis with the CM shifted by 0.05b to the flexible side, and green color
for the 3rd seismic analysis with the CM shifted by 0.05b to the stiff side. By the 1 st analysis (blackcolor), the maximum displacements in Y direction Uy,f and Uy,s respectively in the flexible and the stiffedges are determined. These latter displacements do not include the effect of the accidental torsion. By
the 2nd analysis (red color), the maximum displacements in Y direction U*y,f and U*y,s respectively inthe flexible and the stiff edges are determined, whose displacements include the effect of theaccidental torsion (eacc=-0.05b). By the 3rd analysis (green color), the maximum displacements in Y
direction U*y,f and U*y,s respectively in the flexible and the stiff edges are determined, whosedisplacements include the effect of the accidental torsion (eacc=+0.05b). Finally, the maximum ratios ofthese 6 determined displacements are calculated to get the investigated parameters If,dy and Is,dy.

Figure 4. Calculation of the investigated parameters by time history analysis.
4. RESULTS AND DISCUSSION
4.1. Design envelopes
The new proposed method to evaluate the effect of the accidental torsion is based only on the resultsof the time history analysis, which is considered more reliable than the static analysis because it
represents the true behavior of the building under earthquake ground motions. It has been proved thatthe greater the aspect ratio is, the greater is the increase in edge displacements. Hence, the aspect ratio
is fixed to 3.5, which is the highest value permitted for regular buildings in the Moroccan SeismicCode. The target spectrum does not affect on the investigated parameters, which are normalized
displacements, as long as the analysis is limited to the elastic range. More than 2000 time history
analyses have been performed by changing the parameter Ω (from 0.7 to 1.6), the uncoupled lateral
period Ty (from 0.2s to 2.0s), the phase (El Centro, Kobe, Random) and the static eccentricity e s (0.00b,0.05b, and 0.09b). Envelopes of the final results are proposed for each value of static eccentricity andboth edges. Figures 5 and 6 summarize the proposed design envelopes.

Figure 5. Design envelopes for the stiff edge

Figure 6. Design envelopes for the flexible edge

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4.2. New Proposed Method
To improve the torsional provisions on the Moroccan seismic code RPS2000 Version 2011, a newpractical method is proposed to evaluate the effect of the accidental torsion. The main advantage is to
avoid performing many extra seismic analyses as required in the current version of the code, which istime-consuming and cumbersome. The new method is so far applied only to regular multi-storey
buildings, and it’s summarized in the following steps:
 Do the linear seismic analysis without considering the accidental eccentricity.
 Determine the equivalent single degree of freedom (Ωequi, es,equi).
 Determine Ifd and Isd, respectively for the increase in the flexible and stiff edges due toaccidental torsion using the design envelopes (Figures 5 and 6).
 Calculate the increase in displacement of any resisting element due to accidental torsion, using
only its distance from the center of rigidity (Figure 7).

Figure 7. The Distance of resisting elements from the CR.
 Amplify the element forces obtained in the first step by the calculated increase indisplacement.
An example of three storey building has been studied to show how the new proposed
method to account for the influence of the accidental torsion can be used by practitioner engineers, andto highlight its advantages with respect to the one proposed in the Moroccan seismic code. Threeearthquake ground motions are considered. The increase in edge displacement of each storey
calculated by the current method of the Moroccan Seismic Code (27 dynamic analyses for each groundmotion) and the maximum increase in edge displacement calculated by the proposed method (only one
seismic analysis for each ground motion) are summarized in Figure 8. The increase in edge
displacement, calculated by the proposed method is equal to 1.5, whereas the maximum value from thecurrent method of the Moroccan Seismic Code is estimated at 1.4816. This example shows how
practical and how accurate is the proposed procedure.

Figure 8. Increase in edge displacements of a regular three storey building55. CONCLUSIONS
The investigation of the increase in edge displacements due to torsion has led to the following
conclusions: Many parameters influence the increase in edge displacements due to the accidental torsion.
The circular frequency ratio Ω is the most significant. For values of Ω greater than 1.1, thebuilding is considered torsionally stiff, and the total response is mainly due to the lateral
movement rather than the torsional one. For small values of Ω, let’s say less than 1.1, thebuilding is torsionally flexible and the influence of the torsional motion becomes significant.
 The increases calculated by the equivalent static method and the time history analysis aredifferent, which means that the torsional provisions of the Moroccan and other worldwide
seismic codes have to be revised.
 The increases in the stiff and the flexible edges are different because the torsional movementin the first one is opposite to the lateral one, whereas they are complementary in the flexibleedge.
 Nominally symmetric-plan buildings are more sensitive to the introduction of the accidental
eccentricity, and the resulting increase in edge displacements can reach 1.6 in some cases.
 The new proposed method to improve the torsional provisions of the Moroccan seismic codeis based on design envelopes. This method can quickly and easily evaluate the maximum
increase in edge displacements due to accidental torsion.
6. RECOMMENDATION
The proposed design envelopes are obtained from results of buildings with an aspect ratio of 3.5 andthree different static eccentricities (0.00b, 0.05b, and 0.09b). Hence, these results cover all the regularbuildings as defined in the Moroccan seismic code, but not always accurate (especially if the aspectratio is small). A Similar procedure of determining the design envelopes has to be done for buildings
with other values of the aspect ratio and the static eccentricity, to increase the accuracy of the resultsand to expand its use to all categories of buildings. An analytical study is to be performed to generalize
the results of the one-story building to all types of multi-storey buildings. A stiffness proportional
damping matrix is considered in this study. The contribution of the higher modes of vibration, thus theresults, can change if other type of damping matrix is to be considered.
This study is performed in the elastic domain to be compliant with the Moroccan seismiccode. The efficiency of this design procedure is to be assessed by studying the effect on the ductilityon the torsional response and by performing nonlinear time history analyses.
ACKNOWLEDGEMENTS
I would like to express my sincere gratitude to my supervisor Dr. Taiki SAITO for his advice andguidance which allowed me to master the subject of this study.
REFERENCES
Chopra, A.K., and De La Llera, J.C., 1996, Proc. 11th World Conf. Earthquake Eng., Paper No 2006.
De La Llera, J.C., and Chopra, A.K., 1994, Earthquake Eng. and Str. Dynamics, 23, 947-967.
Kashima, T., 2018, IISEE Lecture Note 2017-2018, IISEE, BRI.
Ministry of housing and urban policy of Morocco, Moroccan Seismic Code RPS2000 Version 2011.
Paulay, T., 2000, Bull. of the New Zealand society for earthquake engineering, 33, No. 4, 403-420.
Saito, T., 2018, IISEE Lecture Note 2017-2018, IISEE, BRI.
Saito, T., 2017, STERA 3D User Manual Version 9.6, Toyohashi University of Technology (TUT).
Shibata, A., 2010, Dynamic analysis of earthquake resistant structures, Tohoku University Press.

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