evaluation of turkish seismic code for mass irregular...

Indian Journal of Engineering & Materials Sciences Vol. 14, June 2007, pp. 220-234

Evaluation of Turkish seismic code for mass irregular buildings

Kamil Aydin Department of Civil Engineering, Engineering Faculty, Erciyes University, 38039 Kayseri, Turkey

Received 19 October 2005; accepted 11 April 2007

Several earthquake-prone countries including Turkey do not consider vertical mass discontinuity as a type of structural irregularity in their seismic codes. However, other earthquake prone countries clearly explain the mass irregularity and have limitations on the use of approximate methods to determine the earthquake forces developed in an irregular structure. The applicability of equivalent lateral force procedure of the Turkish earthquake code for mass irregular buildings is examined in this study. This is achieved by comparing the approximate results to those obtained by linear and non-linear time history analyses. 75 real and 100 simulated earthquake records are used in time history analyses. Two-dimensional 5, 10, and 20-story structures idealized as shear and frame buildings are studied. Mass irregularity is resulted from varying the mass of one floor and keeping the other story masses constant. Effect of altering the mass of different floors is also investigated. The comparison of the analysis results shows that the approximate method always overestimates the linear behaviour regardless of structure height, building rigidity and degree of mass irregularity. The method, however, underestimates the non-linear response of story columns. In light of the results obtained in this study, the inclusion of mass irregularity to the code appears appropriate.

IPC Code: G01V1/00

Most of the buildings codes such as the uniform buildings code (UBC)1 have the same basic philosophy to provide an estimate of the forces developed in a structure during an earthquake: the earthquake force is treated as an inertial problem. This philosophy is applied in keeping with Newton’s Second Law of Motion (F=ma, where F is the inertia force, m is the mass, and a is the acceleration of the mass). Stated differently, the dynamic forces generated in a structure during an earthquake are approximated to be proportional to the maximum ground acceleration. These forces are then formed as a set of equivalent lateral forces which are distributed in proportion to the mass of the element or component over the height of the structure. This simplified method is referred to as the equivalent lateral force (ELF) procedure. The method is based on a number of assumptions such as that the structure uniform. Uniform structure refers to one with a uniform mass, stiffness and strength distribution in vertical plane and with symmetry with respect to orthogonal axes in horizontal plane. No real-life structure, however, falls within the definition of a uniform structure. For this reason, the building codes have set the provisions regarding the application of the ELF procedure for

seismic analysis. In order to categorize a structure as regular or irregular, the codes have also defined certain limits in the values of mass, stiffness, strength, setbacks and offsets of adjacent stories. The definitions of regular and irregular structures and the magnitudes of the ratio of mass, stiffness, strength, setbacks and offsets of one floor to that of the corresponding value of an adjacent floor slightly change from one building code to another. Not all types of irregularities, however, are included in every building code. For example, the seismic codes of such earthquake-prone countries as Italy, Romania and Turkey do not consider the mass irregularity as a type of vertical irregularity whilst several other countries include this kind of discontinuity in their codes. The latter codes state that the ELF procedure can be resorted if the structure, regardless of its configuration, resides in a region of low seismicity or the structure of irregular configuration is not higher than a specified number of stories. In the cases where the ELF is not applicable, these codes either require a dynamic analysis method or do not permit the irregularity. Since the former codes do not count for mass irregularity, no limitations on the use of approximate methods exist in these codes for structures with this type of irregularity. This study investigates the applicability of the ELF procedure outlined in specification for structures to be

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AYDIN: TURKISH SEISMIC CODE FOR MASS IRREGULAR BUILDINGS

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]

built in disaster areas2, which is termed shortly ‘Turkish seismic code (TSC)’, for structures with vertical mass irregularity. The failure of Olive Oil Medical Center under the San Fernando earthquake in 1971 and the collapse of the two-story ‘computer’ building in Bucarest, Romania during the Bucarest earthquake of 1977 are two examples commonly known to earthquake engineering community. Both of the failures were caused or at least aggravated by the presence of heavy mass in one floor. Among the researchers who studied the vertical irregularities, Fernandes3, Hidalgo et al.4, Valmunddson and Nau5, Ozmen et al.6 and Das and Nau7 also included the mass irregularity in their work. This study differs from the previous research due to the approximate method employed to determine the earthquake design forces and number of input earthquakes. Analytical Model Structural modeling Two-dimensional 5, 10, and 20-story structures represented as shear buildings and frame buildings are considered. The structures are single bay buildings with a constant story height of 3.65 m and beam width of 7.30 m. Story masses are assumed to be lumped at each floor level. An initial value of 3.6 tons for the story mass is selected for uniform buildings. Various fractions of this value are applied in order to establish irregular buildings. For each structure height, six different fundamental periods of vibration are considered. The periods are obtained from the measured accelerograph records in the 1971 San Fernando earthquake. Accordingly, the values of 0.5, 0.6, 0.7, 0.8, 0.9, 1.0 seconds for 5-story structures; 0.8, 1.0, 1.2, 1.4, 1.6, 1.8 seconds for 10-story structures; and 1.4, 1.7, 2.0, 2.3, 2.6, 2.9 seconds for 20-story structures are calculated. A stiffness ratio ρ is used for frame buildings to define relative beam-to-column stiffness values. This value is the ratio of beam properties to column properties formulated as

1 1

[ ( / ) ] /[ ( / )n n

b ci i

EI L EI Lρ= =

= ∑ ∑ … (1)

where b stands for beam, c stands for column, E is the modulus of elasticity, I is the area moment of inertia, and L is the length of the member. The summations in the equation include all the beams and columns in the

story closest to the mid-height of the frame. A ρ value of 0 assumes very flexible girders relative to the columns and a ρ value of ∞ presumes infinitely rigid girders, i.e., shear building case. In the study herein these two values are approximated by ρ = 0.0625 and ρ = 50 respectively. Five intermediate ρ values, 0.125, 0.25, 0.5, 1.0 and 2.0, are utilized such that a large number of frame responses can be evaluated. Mass irregularity in a structure is created by varying the mass of one floor while keeping the other story masses constant. The mass ratios of 0.1, 0.5, 1.5, 2.0, and 5.0 times the story mass for the uniform structure are considered. An additional mass ratio of 1.0, which corresponds to the regular building, is studied. The effect of varying the mass of different floors is also investigated. For 5-story structures the masses of third and fifth floors are varied for elastic response, and the first floor mass is also altered for inelastic response. For 10-story structures, the masses of fifth and tenth floors for elastic response and also the mass of the first floor for inelastic response are varied. For 20-story structures, the fourteenth and twentieth floor masses are varied to compute elastic response. For inelastic response, the masses at the first, tenth and twentieth floors are changed. Analysis methods Lateral force procedure of TSC The fundamental expression for the inertia force induced in a rigid body by an acceleration is given as the product of mass of the structure and the acceleration caused by input earthquake forces, i.e., F = ma. Most seismic codes of present time approximate the lateral seismic forces generated in a structural system from the same philosophy. However, the code-form of this equation is to some extent different as an earthquake ground motion is an extremely complex phenomenon and several parameters that affect inertial forces induced in a structure are present. The TSC specifies the seismic force in terms of base shear by

1 1 0= ( ) / ( ) 0.10t aV W A T R T A IW≥ … (2) where Vt is the base shear induced in the structure, W is the total seismic dead load, A(T1) is the spectral acceleration coefficient, Ra(T1) is the earthquake load reduction factor, T1 is the fundamental period of oscillation of the building in the direction of consideration, A0 is the effective ground acceleration

INDIAN J. ENG. MATER. SCI., JUNE 2007

222

⎟

tV

coefficient, and I is the importance factor. Assuming that all the structures analyzed in the study are located in a region of high seismicity, i.e., earthquake zone 1 according to TSC and earthquake zone 4 according to UBC, the value of A0 is taken 0.4. In the region, the soil is assumed to be of stiff and rock type and importance factor is taken 1.0. The earthquake load reduction factor Ra(T1) is 1.0 for elastic design and analysis, and a value corresponding to respective ductility level is accepted for non-elastic design. The distribution of lateral forces over the height of the building is based on the value determined from Eq. (2). This distribution is carried out according to the following formula for the lateral force at the ith floor:

1 ( ) /

N

i t N i i j jj

F V F W H W H=

⎛ ⎞= − Δ ⎜

⎝ ⎠∑ … (3)

with the exception that the force at the top floor computed from this equation is increased by an additional force due to the whipping effect, if the total height of the building HN is greater than 25 m. In the above equation, Wi and Wj are the portions of seismic load at levels i and j, Hi and Hj are the heights in m above the level of application of the seismic action to levels i and j, respectively. The seismic design values of element forces, story shears, base overturning moments etc. are determined by static analysis of the building under the equivalent lateral forces. For example, the story shear in any story x is calculated as the sum of the lateral forces acting above that story and is given by

3/ 40.005 0.2N N tF H VΔ = ≤

Nx ii x

V=

=∑ F , where N refers to the top level. Time history analysis Elastic analysis — The time history analysis is utilized to obtain actual linear elastic response of structures under earthquake excitation. The procedure to compute the response of a building is summarized in a step-by-step form:

(i) Select the values of fundamental natural period and floor masses leading to a mass regular or irregular configuration as required.

(ii) Determine the stiffness of structural elements. This is an iterative process of updating the rigidities of the elements at each iteration step. The iteration continues until the required fundamental period is calculated. Note that, during the iteration process, the pre-selected floor masses are unchanged.

(iii) Having determined the structural properties, compute the exact response of the system to earthquake forces based on linear differential equations of motion ( )u t

%. The well-known

differential equations governing the response of a multi-degree-of-freedom system to an earthquake excitation is given by

( )eff% % % % % % %mu cu ku p t&& & + = , where ( ) - ( )eff gp t m u t&&

% %%+ ι=

and is called the effective earthquake force vector. In the equation ,m

% ,c%

and k%

are the mass, damping, and stiffness matrices respectively. ( )gu t&& is the earthquake-induced ground motion and

%ι is the influence vector.

The displacement vector u%

of an N-degree-of-freedom system can be expressed as the superposition of the modal contributions:

… (4) 1

( ) ( )N

n nn

u t q tφ=

=∑%

where nφ is the modal shape and the generalized coordinates corresponding to the nth mode. The contribution of the nth mode to the displacements is given by

( )nq t

( )u t%

( ) ( ) ( )n n n n n nu t q t D tφ φ= = Γ … (5)

where Гn is referred to as modal participation factor and Dn(t) is the solution to the equation:

22 (n n n n n n g )D D D u&& & &&ξ ω ω+ + = − t … (6) A program engaging piece-wise linear

interpolation used to integrate Eq. (6) is prepared. The equations are solved at a variable time step in order to eliminate the errors resulted from the discrezation of earthquake accelerograms. The displacements can then be computed from Eq. (4).

( )u t%

(iv) Determine the element forces. Solving the equations of motion yields the displacement vector consisting of lateral displacements (and rotational displacements in the case of frame buildings). The individual member forces, base shear and moments etc. are then computed using the Direct Stiffness Method.

Non-elastic analysis — Determination of non-elastic response of buildings comprises of two stages.


223

In the first stage, the structural properties and story yield strengths are computed. The steps involved in this stage can be enumerated as follows: (i) Select a value of design ductility. This value is

defined as the ductility level in which the structure is allowed to respond non-linearly to a given earthquake time history.

(ii) Compute the story yield displacements. The study uses an iterative solution of the differential equations of motion to determine the yield response. The iteration carried out for each earthquake acceleration record and design ductility value continues until the computed ductility and desired (design) ductility levels are matched.

(iii) Determine the floor level yield strengths. Having computed the yield displacements in step (ii), the floor yield strengths can be statically determined in two ways: (a) use of stiffness coefficients of structural elements to find element forces at each time instant, or (b) use of equivalent static forces approach. At any instant of time, these static forces are the external forces that will produce the displacements

%

associated with the nth mode in the stiffness component of the structure, i.e.,

( )nu t

( ) ( )n nf t ku t% %%

= .

The latter approach is preferred in this study as it facilitates a comparison of dynamic analysis procedure with the earthquake forces determined from the procedures specified in the building codes8. The equivalent static forces ( )nf t

% may

be expanded into the vector of forces fjn at various floor levels:

, =1,2,...,Njn n j jn nf m A jφ= Γ … (7)

in which An(t) is the pseudo-acceleration response of the nth mode single-degree-of-freedom system to

. The yield pseudo-acceleration response is readily determined from the yield displacements previously calculated. The floor level yield strengths can then be computed. Finally, the total story forces (story yield strengths) need to be determined. This is done by combining the response contributions of all the modes. The peak modal values are combined according to the square-root-of-sum-of-squares (SRSS) rule. The buildings analyzed in this study have well-separated natural frequencies and it is

known that the SRSS rule provides excellent response estimates for such structure8. It is noted that the contributions of all the modes are taken into account in the analysis procedures.

( )gu t&&

The second stage consists of determining the non-linear response of the structure to earthquake excitation designed in accordance with the required ductility. The differential equation to be solved is given by

( , ) - ( )s gmu cu f u u mıu t&& & & &&% % % % % % % %%

+ + = … (8) in which ( , )sf u u&

% %% is the inelastic force vector. At each

time instant, ( , )sf u u&% %%

is related through statics to story shears. These story forces are in turn functions of the respective story displacements and velocities. The function between story shears and displacements is assumed to be an elastoplastic relationship. This is the simplest load-deformation relationship that can be employed to capture the non-linear behaviour of real structures. The non-linear relationship is given by (1- ) s yf r k u r f z= + , where k is the elastic spring constant, fy is the yield force, r is the specified ratio of post-yield stiffness to elastic stiffness (k), and z is an internal hysteretic variable. The value of r is taken 0.0 in the study. z has a range of |z|≤1 and evolves with time according to differential equation9. The differential equations of motion are solved iteratively in each time step performing modal analysis. The iterations are carried out until solution converges. Design ductility levels of 1, 2, 4, 6, and 10 are considered for 5-story buildings and 2, 6, and 10 are used for 10 and 20-story buildings. Design ductility level of 1 corresponds to elastic analysis. A design ductility level larger than 10 is only applicable in special cases, hence, design ductilities greater than this value are not considered in the study. Earthquake excitations Two types of earthquake forces are considered in the study: real and artificially generated earthquake time histories. Real earthquakes 75 real earthquake motions exhibiting different characteristics are used for response computations. In all the earthquake records, most of the strong ground motion occurred within the first twenty seconds of excitation. However, the time history analyses are


224

performed using the full durations of the excitations. A prefixed pulse of 2.0 s is added to each ground motion in order to account for the low frequency distortions. Artificial earthquakes The procedure to generate earthquake time histories is based on the fact that a function can be represented by the sum of a series of sinusoids. The amplitude of the series is a power spectral density function to be determined.

It has been common practice to represent the design ground acceleration by the design spectrum. This design spectrum is called the target spectrum. The design spectrum of TSC (1997) which has a damping ratio of 5% is used in this study. It is assumed that the ground accelerations to be generated from this spectrum will occur in a rock and stiff soil type. The spectrum characteristic periods defined in TSC are determined accordingly. In order to generate ground accelerations that are compatible with this spectrum, it is first needed to covert the specified spectrum to an equivalent power spectral density (PSD) function. The manner in which such a conversion can be made has been well studied. A methodology based on the work of Park10 is accepted in this study. Having determined the amplitude of the series from the target spectrum, the series needs to be multiplied by a deterministic function referred to as intensity function to reflect the transient character of real earthquakes. A variety of intensity functions have been used in the literature of earthquake generation history. The compound intensity function is considered in this study.

Using the above-mentioned technique, one can generate an ensemble of earthquake time histories. The size of the ensemble should be chosen such that statistically meaningful results may be obtained.

The intensity of the ensemble is controlled by the peak acceleration of earthquake time histories denoted by Agxg, g being the gravitational acceleration. Varying this Agx value over a reasonable range, the intensity of real earthquake motions can be reflected. It should be noted here that when dealing with the ensemble of earthquakes, Agx refers to the average of peak acceleration of generated earthquakes, not the peak acceleration of an individual earthquake. Therefore, the peak acceleration of the ensemble will

most likely be different from that of a single earthquake time history11. Results and Discussion

Frame action Frame action is investigated through the plots of linear story shear ratios, which are computed by dividing the story shear responses of frame building by those of shear building model. The story shears determined from the time history analysis are the responses of the buildings to real earthquake accelerograms. The story shears are averaged over the ensemble of real earthquake excitations used in the study. Based on the previous studies12,13, a ρ value of 2 is selected as an approximation of shear building behaviour. Figures 1 and 2 show the diagrams for 5 and 10-story building frames, respectively. The diagrams are given for extreme and middle values of fundamental periods and mass ratios, those corresponding to each period and mass ratio. The figures show that the shear building model acceptably estimates the frame responses for a ρ value of 1.0 or greater. For smaller values of ρ, the model under or

Fig. 1 — Ratios of story shears determined from the time history analysis for 5-story frame buildings with (a) 3rd floor, (b) 5th floor mass variations subjected to real earthquakes


225

overestimates the behaviour depending on the story level at which the mass irregularity occurs.

Fig. 2 — Ratios of story shears determined from the time history analysis for 10-story frame buildings with (a) 5th floor, (b) 10th floor mass variations subjected to real earthquakes Mass irregularity If the linear story shear responses of 20-story shear buildings to real earthquakes are plotted against the corresponding story level, Fig. 3 is obtained. The story shears are again the averaged values and determined from the time history analysis. The first three plots in the figure are for the case in which the fourteenth floor mass is varied and the remaining plots for the case where the twentieth floor mass is altered. The figure displays that, for the uniform structure (m14=1.0), the curves of shear responses are almost smooth. As the mass ratio is changed, for example m14 becomes 0.1 or 5.0, however, the curves get jagged at the floor level where a severe mass irregularity is present. Such an observation is also made for all the 5 and 10-story shear and frame buildings analyzed with either time history analysis or the ELF procedure. Figure 4 shows the effect of mass irregularity on the linear story shear response. The figure is acquired

for 10-story frame buildings subjected to real earthquake time histories. The shear responses are averaged over the 75 real earthquakes. The first three plots are for frame buildings whose fifth floor masses are varied and the last three are for the buildings whose tenth floor masses are changed. The values of fundamental period of vibration (T1) and the stiffness ratio (ρ) of the frames are indicated on the plots. The responses corresponding to each and every value of T1, ρ and structural height could not be presented here. Figure 4 suggests that the shear responses of the stories below the level at which the mass variation takes place increase linearly with the increasing mass ratios. The shears of the other stories are not affected by mass variations. Similar results are obtained for 5 and 20-story structures and, hence, the respective figures are not shown for brevity.

Fig. 3 — Story shears determined from the time history analysis for 20-story shear buildings with (a) 14th floor, (b) 20th floor mass variations subjected to real earthquakes

Comparison of analysis methods The ratio of story shear response obtained by linear time history analysis to that obtained by the ELF procedure of 1997 TSC is determined. This ratio gives


226

some measure of accuracy of the ELF procedure compared to linear time history analysis. Table 1 shows the computed ratios of shear responses of the shear building model. For the time history analysis, the story shears are responses to the real earthquake ground motions. The results are presented for a selection of 5, 10 and 20-story buildings, each with six different fundamental periods of vibration. The average values of the ratios over the periods are also given as indicated in the table. In order to be able to present the ratios for each mass irregularity case, the results are further condensed by averaging them over the periods of vibration. Table 2 shows these values. It is noted from the tables that all the ratios are less than unity, indicating the overestimates of the ELF procedure for the linear story shear responses. The amount of overestimates varies from as low as 7% to as high as 95%. The tabulated values also show that, for 5-story shear buildings, as the fundamental period increases, the story shear ratios also increase, i.e., the amount of overestimates decreases. On the other hand, as the period increases the magnitude of

overestimates also increases for 10 and 20-story shear buildings.

Fig. 4 — Story shears determined from the time history analysis for 10-story frame buildings with (a) 5th floor, (b) 10th floor mass variations subjected to real earthquakes

Another comparison between the linear time history analysis and the ELF procedure can be made for the responses of frame buildings. This comparison reveals the applicability of the ELF approach for flexible frame buildings. Figure 5 shows the quantities of story shear ratios for various fundamental periods, stiffness ratios and floor mass ratios of 5-story frame buildings. From this figure, it is seen that the ELF procedure overestimates all the story shears except those of the fourth and fifth stories of the frame buildings with ρ values smaller than 0.5, fundamental periods larger than 0.8 s, and floor mass ratios greater than 1.0. It can be stated that for ρ values greater than 0.5, the overestimates of the ELF routine is invariant with the fundamental periods and stiffness ratios of a frame buildings. This observation is also valid for 5-story buildings whose fifth story masses are varied. However, the underestimates of the ELF routine in this case occurs for low values of the fifth floor mass ratios, namely, mass ratios less than 1.0. Figure 6 displays the story shear ratios for 10-story frame buildings (those for 20-story buildings are not shown here due to lack of space). The ELF procedure overestimates all the story shears regardless of 10 or 20-story frame buildings being relatively flexible or rigid, or the buildings with low or high values of ρ. A trend is observed for 10 and 20-story frames: the ELF process is more conservative for shear responses of the stories above the floor in which the mass irregularity occurs. As in the case of 5-story buildings the overestimates of the ELF process do not change as the ρ values change from 0.5 to 50. The applicability of the ELF procedure for structures with mass irregularity is next examined by comparing the non-linear responses of the buildings designed according to the provisions of 1997 TSC with those of the buildings designed by using the non-linear dynamic analysis. For each method, a total of 4950 buildings are designed and their response to real accelerogram records is then computed. Considering the number of buildings and earthquakes, it would be best to condense the analysis results by averaging the imposed ductility demands over, first, the fundamental periods of vibration and, second, over the ensemble of earthquake records. Figure 7 shows the responses of 5-story buildings designed in accordance with the inelastic time history analysis. The figure displays the ductility demand


227

ratios for the mass irregularities on the first, third and fifth floors. Ductility demand ratio is computed by dividing the design ductility level by the ductility demand imposed on the story columns by the earthquake accelerations. The plots in each row correspond to design ductility levels μ = 1, 4, 6 and 10 (those for μ = 2 are not shown due to space problem) and show the demand ratios for every story. The plots of design ductility value of 1.0 indicate that story ductility demands ratios are approximately 1.0, i.e.,

the story columns behave elastically with a few exceptions. It is observed from the other plots of μ = 4, 6 and 10 corresponding to the buildings whose first floor masses are varied that as the level of design ductility level increases, the ductility demands imposed on the story columns also increase. It is noted that all the stories yield and ductility demand ratios almost linearly decrease with the increasing mass ratios. Only the ductility demand in the first story exceeds the allowable ductility level. A

Table 1 — Ratios of story shear responses determined from linear time history analysis to those determined from the ELF approach for 5, 10 and 20-story shear buildings

Story Period (seconds) No 0.5 0.6 0.7 0.8 0.9 1.0

Aver.

1 0.74 0.80 0.83 0.88 0.87 0.88 0.83 2 0.72 0.75 0.77 0.83 0.82 0.81 0.78 3 0.71 0.77 0.78 0.84 0.85 0.84 0.80 4 0.71 0.77 0.78 0.84 0.86 0.84 0.80

m3=0.1

5 0.69 0.77 0.80 0.85 0.89 0.89 0.82 1 0.82 0.86 0.87 0.93 0.91 0.90 0.88 2 0.81 0.85 0.86 0.93 0.92 0.90 0.88 3 0.81 0.84 0.86 0.93 0.92 0.90 0.88 4 0.68 0.72 0.75 0.80 0.80 0.80 0.76

5-st

ory

build

ing

Mas

s var

iatio

ns o

n th

ird fl

oor

m3=5.0

5 0.63 0.68 0.72 0.76 0.77 0.77 0.72

Period (seconds)

Story No 0.8 1.0 1.2 1.4 1.6 1.8

Aver.

1 0.86 0.85 0.85 0.79 0.76 0.72 0.81 3 0.82 0.80 0.80 0.75 0.72 0.69 0.76 5 0.79 0.78 0.81 0.75 0.72 0.68 0.76 6 0.79 0.78 0.82 0.75 0.72 0.68 0.76 8 0.78 0.81 0.84 0.80 0.79 0.73 0.79

m5=0.1

10 0.68 0.71 0.73 0.69 0.69 0.65 0.69 1 0.90 0.88 0.88 0.82 0.80 0.74 0.84 3 0.89 0.86 0.86 0.80 0.77 0.72 0.82 5 0.88 0.85 0.87 0.81 0.80 0.74 0.83 6 0.77 0.77 0.79 0.73 0.71 0.65 0.74 8 0.72 0.73 0.75 0.70 0.67 0.63 0.70

10-s

tory

bui

ldin

g M

ass v

aria

tions

on

fifth

floo

r

m5=5.0

10 0.55 0.55 0.56 0.50 0.48 0.45

0.52

Period (seconds)

Story No 1.4 1.7 2.0 2.3 2.6 2.9

Aver.

1 0.77 0.73 0.69 0.67 0.64 0.60 0.68 4 0.75 0.71 0.66 0.63 0.61 0.57 0.66 7 0.74 0.69 0.65 0.62 0.58 0.56 0.64 10 0.74 0.70 0.66 0.64 0.60 0.57 0.65 13 0.77 0.72 0.69 0.68 0.65 0.61 0.69 14 0.78 0.73 0.70 0.69 0.67 0.63 0.70 17 0.76 0.74 0.69 0.68 0.67 0.63 0.70

m20=0.5

20 0.33 0.31 0.27 0.26 0.25 0.23 0.28 1 0.77 0.73 0.69 0.67 0.64 0.60 0.68 4 0.75 0.71 0.66 0.64 0.61 0.58 0.66 7 0.74 0.69 0.65 0.62 0.58 0.56 0.64 10 0.74 0.69 0.66 0.63 0.59 0.56 0.65 13 0.77 0.71 0.68 0.67 0.64 0.60 0.68 14 0.77 0.72 0.69 0.68 0.66 0.62 0.69 17 0.78 0.75 0.71 0.70 0.68 0.65 0.71

20-s

tory

bui

ldin

g M

ass v

aria

tions

on

twen

tieth

floo

r

m20=2.0

20 0.66 0.64 0.59 0.58 0.57 0.54 0.60


228

maximum demand ratio of 1.81 is determined for the first story columns when the mass ratio on the first

Table 2 — Averaged ratios of story shear responses determined from linear time history analysis to those determined from the ELF approach for 5, 10 and 20-story shear buildings

Mass ratios

Story No 0.1 0.5 1.0 1.5 2.0 5.0

Mass variations on third floor 1 0.83 0.84 0.84 0.85 0.86 0.88 2 0.78 0.8 0.82 0.83 0.84 0.88 3 0.8 0.8 0.81 0.82 0.83 0.88 4 0.8 0.81 0.82 0.82 0.81 0.76 5 0.82 0.83 0.84 0.83 0.82 0.72

Mass variations on fifth floor 1 0.85 0.85 0.84 0.84 0.85 0.86 2 0.82 0.82 0.82 0.82 0.82 0.84 3 0.82 0.81 0.81 0.81 0.81 0.84 4 0.84 0.82 0.82 0.82 0.82 0.84

5-st

ory

shea

r bui

ldin

g

5 0.71 0.8 0.84 0.84 0.84 0.86

Mass ratios

Story No 0.1 0.5 1.0 1.5 2.0 5.0

Mass variations on fifth floor 1 0.81 0.81 0.8 0.81 0.81 0.84 3 0.76 0.77 0.78 0.78 0.79 0.82 5 0.76 0.76 0.77 0.77 0.78 0.83 6 0.76 0.76 0.77 0.77 0.77 0.74 8 0.79 0.79 0.79 0.79 0.78 0.7

10 0.69 0.69 0.68 0.66 0.64 0.52

Mass variations on tenth floor 1 0.81 0.81 0.8 0.81 0.81 0.82 3 0.78 0.78 0.78 0.78 0.78 0.79 5 0.77 0.77 0.77 0.76 0.76 0.78 6 0.78 0.78 0.77 0.77 0.77 0.78 8 0.79 0.79 0.79 0.79 0.79 0.8

10-s

tory

shea

r bui

ldin

g

10 0.18 0.52 0.68 0.75 0.78 0.83

Mass ratios

Story No 0.1 0.5 1.0 1.5 2.0 5.0

Mass variations on fourteenth floor 1 0.68 0.68 0.68 0.68 0.69 0.7 4 0.65 0.66 0.66 0.66 0.66 0.67 7 0.64 0.64 0.64 0.64 0.64 0.66

10 0.65 0.65 0.65 0.65 0.65 0.66 13 0.68 0.68 0.69 0.69 0.69 0.71 14 0.7 0.69 0.7 0.7 0.7 0.71 17 0.7 0.7 0.71 0.71 0.71 0.7 20 0.45 0.44 0.44 0.43 0.43

0.38

Mass variations on twentieth floor 1 0.68 0.68 0.68 0.68 0.68 0.69 4 0.66 0.66 0.66 0.66 0.66 0.67 7 0.64 0.64 0.64 0.64 0.64 0.64

10 0.66 0.65 0.65 0.65 0.65 0.65 13 0.69 0.69 0.69 0.68 0.68 0.68

20-s

tory

shea

r bui

ldin

g

14 0.7 0.7 0.7 0.69 0.69 0.69


229

17 0.69 0.7 0.71 0.71 0.71 0.72 20 0.07 0.28 0.44 0.54 0.6 0.71

Fig. 5 — Ratios of story shears determined from the time history analysis to those from ELF approach for 5-story frame buildings with (a) 3rd floor, (b) 5th floor mass variations subjected to real earthquakes floor is 0.5. When the third floor masses are varied, the ductility demand ratios remain approximately unchanged. Again, the ductility demands for all the stories except for the first story are less than the design ductility levels. When the fifth floor mass ratios are varied from 0.1 to 5.0, the ductility demands imposed on the first story columns increase

whereas the demands on the other stories remain unaltered. The manner in which the 5-story buildings respond to earthquake excitations is also observed for 10 and 20-story buildings: the first story columns are always imposed on the greatest ductility demands and the other stories behave within their allowable design limits. The maximum ductility demands on the first story columns are determined to be 2.74 and 3.85 for 10 and 20-story structures respectively. Both maxima occur when the top floor mass ratio is 5.0. The non-linear responses of the buildings designed in accordance with the ELF procedure are not given directly instead they are presented in terms of ductility demand ratios of the two analysis methods. The values given in Tables 3 and 4 are the ratios of ductility demands imposed on the buildings proportioned in conformance with the provisions of 1997 TSC to those of the buildings designed according to the inelastic time history analysis. The tables are for 10 and 20-story structures respectively (the one for 5-story buildings is not given for brevity). Each datum in the tables refers to the computed ductility demand ratio averaged over the six fundamental periods and real earthquake time histories. The estimates of the ELF procedure for the first story columns of 10-story buildings are about two times higher than those of the dynamic analysis method. The ductility demands for the other stories determined by the static analysis procedure appear to be safer. A similar trend is observed for 20-story buildings but in this case the differences between the first story ductility demands of the two procedures are not as great.


230

Type of input earthquake It is previously stated that the ELF procedure overestimates all the linear shear responses as compared with the time history analysis using real earthquakes as input forces. In the case of simulated earthquakes utilized for the time history analysis, the comparison depends on the ensemble peak average Agx of the simulated records. As the Agx values vary from, say, 0 to 1.0, the story shear responses systematically increase such that for a specific value of Agx the time history results match with those of the ELF. A match can also be achieved between the shear responses to real and simulated earthquakes. The following briefly discusses the Agx values for which the two analysis results are equal. Note that the average peak acceleration value of the 75 real earthquakes is 0.32 g. Table 5 shows, for 5-story shear buildings, the Agx values determined such that the responses obtained by the ELF or time history analysis employing real earthquakes are equal to those obtained by time history analysis using simulated earthquakes. Both the third and fifth floor mass ratios are considered. It is seen from the table that an Agx value of 0.382 is

computed for a shear building with third floor mass ratio of 0.1. As the mass ratio changes, the computed Agx values remain almost unchanged. On the average Agx=0.374 is calculated for the shear buildings with the third floor mass variations. From the same table, it is seen that Agx = 0.307 is obtained for 5-story shear buildings subjected to real and simulated earthquakes. It is noted again that this value does not change as the floor mass ratio changes.

Fig. 6 — Ratios of story shears determined from the time history analysis to those from ELF approach for 10-story frame buildings with (a) 5th floor, (b) 10th floor mass variations subjected to real earthquakes

Fig. 7 — Maximum ductility demands determined from the non-linear time history analysis for 5-story buildings with (a) 1st floor, (b) 3rd floor, and (c) 5th floor mass variations subjected to real earthquakes

For the fifth floor mass ratios, Agx values of 0.372 and 307 are determined from the comparison of simulated earthquake responses with the ELF and real earthquake responses respectively. It is interesting to notice that an Agx value of 0.307 is close to 0.32, the average peak acceleration of the ensemble of real earthquakes. In order to see the effect of frame flexibility on this comparison, Table 6 is formed for 5-story frame buildings with the third or fifth floor mass ratio of 1.0. The table suggests that the ρ values have no considerable effect on the computed Agx quantities.


231

A similar comparison is made for 10 and 20-story buildings. It is found that the average Agx values of

0.408 and 0.311 are determined for the ELF and time

Table 3— Ratios of ductility demands of 10-story buildings designed as per the ELF approach of 1997 TSC to those of the buildings designed according to the time history analysis

Mass ratio at first floor

Story No 0.1 0.5 1.0 1.5 2.0 5.0

1 2.45 2.54 2.66 2.79 2.92 2.84 3 1.12 1.00 0.95 0.89 0.85 0.65 5 0.86 0.88 0.89 0.90 0.85 0.59 8 1.22 1.22 1.30 1.29 1.24 1.35

μ=2

10 0.89 0.98 1.10 1.12 1.18 0.75 1 2.20 2.20 2.39 2.39 2.46 2.92 3 1.89 1.64 1.52 1.32 1.15 0.69 5 1.05 1.00 0.92 0.91 0.86 0.64 8 1.19 1.25 1.28 1.31 1.38 1.31

μ=6

10 0.89 1.01 1.37 1.20 1.06 0.47 1 1.70 1.63 1.77 1.74 1.81 2.20 3 1.64 1.64 1.30 1.20 1.13 0.61 5 1.05 0.99 0.96 0.91 0.93 0.59 8 1.09 1.21 1.20 1.13 1.19 1.19

Allo

wab

le d

uctil

ity

μ=10

10 0.84 0.98 1.20 1.13 0.97 0.46

Mass ratio at fifth floor

Story No 0.1 0.5 1.0 1.5 2.0 5.0

1 2.77 2.64 2.66 2.59 2.54 2.52 3 0.94 0.91 0.95 0.96 1.01 1.12 5 0.93 0.83 0.89 0.99 1.07 1.39 8 1.29 1.25 1.30 1.25 1.20 0.88

μ=2

10 1.05 1.02 1.10 1.01 0.96 0.61 1 2.45 2.37 2.39 2.26 2.25 2.29 3 1.34 1.48 1.52 1.47 1.47 1.48 5 0.97 0.92 0.92 1.07 1.26 1.70 8 1.29 1.31 1.28 1.23 1.17 0.94

μ=6

10 1.20 1.26 1.37 1.12 0.95 0.42 1 1.81 1.76 1.77 1.73 1.67 1.68 3 1.25 1.32 1.30 1.32 1.40 1.35 5 0.92 0.88 0.96 1.18 1.30 1.63 8 1.15 1.22 1.20 1.11 1.12 0.94

Allo

wab

le d

uctil

ity

μ=10

10 1.18 1.24 1.20 0.92 0.94 0.42

Mass ratio at tenth floor

Story No 0.1 0.5 1.0 1.5 2.0 5.0

1 2.63 2.67 2.66 2.69 2.82 2.82 3 0.93 0.93 0.95 0.95 0.97 1.04 5 0.96 0.98 0.89 0.86 0.93 0.75 8 1.42 1.39 1.30 1.27 1.24 1.12

μ=2

10 0.16 0.67 1.10 1.49 1.61 1.98 1 2.32 2.33 2.39 2.30 2.30 2.38 3 1.40 1.44 1.52 1.54 1.57 1.77 5 0.99 0.92 0.92 0.97 1.06 1.17 8 1.43 1.38 1.28 1.26 1.37 1.21

μ=6

10 0.19 0.68 1.37 1.51 1.79 1.99 1 1.73 1.72 1.77 1.71 1.69 1.74 3 1.24 1.27 1.30 1.32 1.36 1.51

Allo

wab

le d

uctil

ity

μ=10

5 0.96 0.94 0.96 0.96 0.99 1.03


232

8 1.33 1.27 1.20 1.18 1.24 1.35 10 0.17 0.65 1.20 1.47 1.66 1.89

Table 4 — Ratios of ductility demands of 20-story buildings designed as per the ELF approach of 1997 TSC to those of the buildings designed according to the time history analysis

Mass ratio at first floor

Story No 0.1 0.5 1.0 1.5 2.0 5.0

1 1.78 1.85 1.96 2.05 2.12 2.07 5 0.74 0.74 0.71 0.69 0.68 0.56 10 0.76 0.79 0.72 0.74 0.77 0.72 15 1.17 1.18 1.18 1.17 1.17 0.95

μ=2

20 0.49 0.54 0.70 0.65 0.80 0.53 1 1.27 1.24 1.09 1.32 1.36 1.56 5 0.70 0.69 0.66 0.62 0.60 0.50 10 0.88 0.89 0.89 0.86 0.85 0.73 15 1.25 1.25 1.26 1.28 1.31 1.22

μ=6

20 0.27 0.45 0.81 0.60 0.60 0.31 1 0.72 0.72 0.70 0.79 0.80 1.01 5 0.45 0.43 0.42 0.41 0.41 0.37 10 0.65 0.66 0.67 0.66 0.65 0.68 15 1.18 1.17 1.15 1.05 1.11 1.01

Allo

wab

le d

uctil

ity

μ=10

20 0.35 0.42 0.51 0.56 0.53 0.37

Mass ratio at tenth floor

Story No 0.1 0.5 1.0 1.5 2.0 5.0

1 1.99 1.97 1.96 1.94 1.93 1.92 5 0.75 0.73 0.71 0.69 0.68 0.76 10 0.83 0.79 0.72 0.76 0.78 0.98 15 1.15 1.17 1.18 1.17 1.32 1.02

μ=2

20 0.59 0.74 0.70 0.59 0.56 0.38 1 1.31 1.14 1.09 1.24 1.23 1.23 5 0.64 0.65 0.66 0.68 0.70 0.74 10 0.97 0.85 0.89 0.96 1.01 1.33 15 1.27 1.21 1.26 1.24 1.23 1.14

μ=6

20 0.53 0.79 0.81 0.55 0.49 0.28 1 0.76 0.77 0.70 0.72 0.71 0.71 5 0.45 0.43 0.42 0.43 0.47 0.50 10 0.75 0.69 0.67 0.76 0.86 1.19 15 1.17 1.17 1.15 1.14 1.09 1.06

Allo

wab

le d

uctil

ity

μ=10

20 0.50 0.63 0.51 0.50 0.44 0.28

Mass ratio at twentieth floor

Story No 0.1 0.5 1.0 1.5 2.0 5.0

1 1.95 1.96 1.96 1.95 1.98 2.01 5 0.71 0.71 0.71 0.71 0.69 0.73 10 0.82 0.80 0.72 0.74 0.70 0.66 15 1.25 1.20 1.18 1.14 1.09 0.85

μ=2

20 0.08 0.36 0.70 0.81 1.06 1.55 1 1.22 1.27 1.09 1.19 1.27 1.28 5 0.64 0.65 0.66 0.67 0.68 0.72 10 0.94 0.91 0.89 0.85 0.82 0.79 15 1.39 1.32 1.26 1.17 1.11 1.08

μ=6

20 0.06 0.30 0.81 0.78 0.96 1.48 1 0.74 0.75 0.70 0.70 0.73 0.75 5 0.42 0.42 0.42 0.43 0.44 0.44

Allo

wab

le d

uctil

ity

μ=10

10 0.69 0.68 0.67 0.65 0.65 0.57


233

15 1.17 1.19 1.15 1.09 1.05 0.94 20 0.05 0.28 0.51 0.87 0.86 1.36

Table 5 — The Agx values based on the comparison of shear responses of the ELF procedure to those of time history analysis for 5-story shear buildings

Agx Average

0.1 0.382 0.5 0.377 1.0 0.373 1.5 0.371 2.0 0.369

From the Comparison of ELF procedure to Time History Analysis with Simulated Earthquakes

5.0 0.370 0.374 0.1 0.307 0.5 0.307 1.0 0.307 1.5 0.307 2.0 0.307

From the Comparison of Time History Analysis with Real Earthquakes to that with Simulated Earthquakes

Third

Flo

or M

ass R

atio

5.0 0.305 0.307 0.1 0.380 0.5 0.374 1.0 0.373 1.5 0.373 2.0 0.371


5.0 0.361 0.372 0.1 0.307 0.5 0.307 1.0 0.307 1.5 0.307 2.0 0.307


Fifth

Flo

or M

ass R

atio

5.0 0.306 0.307 Table 6 — The Agx values based on the comparison of shear responses of the ELF procedure to those of time history analysis for 5-story frame buildings

Agx Average

0.0625 0.375 0.125 0.375 0.25 0.375 0.5 0.374 1.0 0.374 2.0 0.374


50 0.373 0.374 0.0625 0.304 0.125 0.304 0.25 0.305 0.5 0.306 1.0 0.306 2.0 0.307


Stiff

ness

ratio

ρ

50 0.307 0.306

history results for 10-story shear buildings under real earthquakes regardless of the location of mass irregularity. The respective values for 20-story

structure responses remain almost unchanged, although those from the comparison of the ELF and simulated earthquake responses slightly increase with the increasing number of stories. Conclusions A large number of shear and frame buildings and earthquake records are exploited to study the effect of (i) frame action, (ii) type of earthquake excitation, (iii) mass variation of different floors and (iv) the use of the ELF procedure of 1997 TSC on the response of structures with mass irregularity. The results of the ELF procedure are compared to ‘true’ results determined from linear and non-linear time history analyses. Based on the comparisons, the following conclusions can be drawn: (i) A stiffness ratio of 1.0 or greater captures the

shear building behaviour. As the stiffness ratio gets smaller, the shear building and frame building responses diverge. The extent and direction of the deviation depend on which story the mass variation appears and the amount of mass ratio.

(ii) Varying the mass ratio of a floor affects the linear shear responses of the stories below the location of mass irregularity. This happens in a way that as the mass ratio increases, the mentioned story shears also increase linearly. For those stories above the floor in which the mass irregularity occurs, however, the shear responses are not influenced by the changes in floor mass ratio values

(iii) A specific Agx value is determined by equating the linear shear responses of the ELF procedure and those of the time history analysis with simulated earthquakes. Using the time history analysis only, similar matching can be made for the responses of the buildings to real and simulated earthquakes. Evaluation of such Agx values indicates that the ELF approach of TSC estimates the linear shear responses well with a safety (overestimation) factor.

(iv) An additional direct comparison of the results of the ELF procedure and time history analysis shows that the ELF approach of TSC always overestimates the story shears regardless of (i) the number of stories in a building, (ii) building rigidity, and (iii) degree of mass irregularity.


234

(v) The amount of overestimates, however, depends on the height of the building and floor mass ratio. Noticing that the shear responses are not significantly affected by the fundamental periods of vibration of a building, the responses may be averaged over the periods. Comparing the averaged shears from the time history analysis to those from the ELF procedure yields that as the number of stories increases, the amount of the ELF overestimates also increases. Accordingly, a maximum overestimate of 93% is obtained for a 20-story building with a twentieth floor mass ratio of 0.1 and a minimum overestimate of 12% is obtained for a 5-story structure with a fifth floor mass ratio of 5.0.

(vi) The story ductility demands for all the multistory buildings considered in the study vary over the height and differ from the allowable ductility used in computing the story yield strengths. Only the ductility demands in the first story exceed the design ductility. The ductility demands in the other stories are less than the allowable values and approximately independent of the changes in the floor mass ratios. However, the ductility demands in the first story linearly changes with the mass ratios such that the demands decrease with an increase in the mass ratios when the first floor mass ratios are varied, and the demands increase with the increasing mass ratios when the fifth floor masses are varied. This observation holds for all 5, 10 and 20-story buildings.

(vii) The deviation of story ductility demands from the design ductility increases for taller buildings. Maximum ductility demand ratios of 1.94, 2.73, and 3.85 are obtained for 5, 10, and 20-story buildings respectively. In all the cases of building height and mass irregularity, the ductility demand in the first story is the largest among all the stories.

(viii) The relative story yield story strengths, which were selected in accordance with the height-wise distribution of the earthquake forces specified in the 1997 TSC do not result in equal ductility demands in all stories. Moreover, they lead to

the ductility demand ratios in the first story of all the buildings higher than those predicted by the time history analysis. A previous study7 noted that the ELFR of 1997 UBC procedure estimated the design forces within reasonable approximations. The ELFR procedure considers the actual first mode shape, actual fundamental period and the corresponding effective mass. None of these considerations, however, are accounted for in the ELF procedure of 1997 TSC.

(ix) It is found that the code-based procedure overestimates the story shear responses in each and every case regardless of the structure being regular or irregular and the extent of irregularity introduced to the structure.

(x) The ductility demand ratios examination indicated that design strengths determined from the ELF procedure yield unsafe non-linear response values of the buildings with mass discontinuity.

References

1 International Conference of Building Officials, Uniform Building Code (UBC) Vol 2, (Whittier, California, USA), 1997.

2 Ministry of Public Works and Settlement, Turkish Earthquake Resistant Design Code, (Ankara, Turkey), 1997.

3 Fernandez J, Bull Int Inst Seismol Earthqu Eng, 19 (1983) 203-215.

4 Hidalgo P, Arias A & Cruz E, Influence of vertical structural irregularity on the selection of the method of seismic analysis, Proc Fifth US National Conf Earthquake Eng, (Chicago, Illinois, USA), 1994.

5 Valdmundsson E V & Nau J, ASCE J Struct Eng, 123 (1997) 30-41.

6 Ozmen G, Pala S, Gulay G & Orakdogen E, Effect of structural irregularities to earthquake analysis of multistory structures TDV/TR017-28, (Istanbul, Turkey), 1998.

7 Das S & Nau J, Earthqu Spect, 19 (2003) 455-477. 8 Chopra A K, Dynamics of Structures: Theory and

Applications to Earthquake Engineering, (Prentice Hall, Upper Saddle River, NJ, USA), 1996.

9 Wen Y K, ASCE J Eng Mech Div, 102 (1976) 249-263. 10 Park Y J, ASCE J Eng Mech, 121 (1995) 1391-1392. 11 Aydin K & Tung C C, Earthqu Spect, 17 (2001) 209-220. 12 Blume J A, ASCE J Struct Eng, 94 (1968) 337-402. 13 Cruz E F & Chopra A K, ASCE J Struct Eng, 112 (1986)

443-459.

evaluation of turkish seismic code for mass irregular...

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