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A Scrutiny of the Equivalent Static Lateral Load Method of Design for Multistory Masonry Structures

A.R. Touqana Ph.D., S.H. Heloub Ph.D. P.E. a Assistant Professor, Civil Engineering Department, An-Najah National University, Nablus- Palestine, [email protected] b Assistant Professor, Civil Engineering Department, An-Najah National University, Nablus-Palestine, [email protected]

Abstract. Building structures with a soft storey are gaining widespread popularity in urban areas due to the scarcity of land and due to the pressing need for wide open spaces at the entrance level. In earthquake prone zones dynamic analysis based on the Equivalent Static Lateral Load method is attractive to the novice and the design codes leave the choice of the analysis procedure to the discretion of the designer. The following is a comparison of the said method with the more elaborate Response Spectrum Method of analysis as they apply to a repertoire of different structural models. The results clearly show that the former provides similar results of response in structures with gradual change in storey stiffness; while it is over conservative for a bare frame structure, it is however less conservative for structures with a soft storey. Thus the superiority of the Response Spectrum Method becomes evident. Keywords: Multistory Building, Seismic Analysis, Mode Shapes, Shear walls, Soft Storey

INTRODUCTION In urban areas, where land is scarce, the popular practice is to allocate the first storey of public buildings for general services that demand wide open spaces. A reception area or a lobby of a hotel is just one classic example. This prompts architects to remove walls from such buildings and to fully rely on columns as the main supporting structural elements. According to ASCE 7-05 a soft storey is defined as one which enjoys a stiffness of only 70% of the stiffness of the storey directly above it or less than 80% of the average stiffness of the three floors above. When this value drops to about 60% the said storey is labeled extremely soft. This feature renders many high rise public buildings within Palestine and perhaps in the entire Middle East as buildings with a soft storey.

Analysis and design procedures of multi-storey framed structures that have masonry infill walls have not been fully exhausted particularly in regard to resisting the induced seismic actions. Hitherto there are no locally imposed restrictions placed on the design procedures of such structures. However, since the entire Middle East region is known to lie on an active earthquake prone zone, the task of comprehensively understanding the behavior of such structures becomes an indispensable undertaking. Helou and Touqan (Gaza Conference paper) have studied the behavior of shear wall distribution on the behavior of multistory structures using the Response Spectrum Method of analysis. The following is an extension of the same objective except the presentation is limited to the Equivalent Lateral Static Load Method which is a preferred procedure locally. The intension of the present discourse is to present numerical examples that further illustrate that the later method leads to inaccurate and at times erroneous results. Thus this method of analysis may only be used under stringently controlled conditions or during a preliminary design exercise.

Problem formulation In the Equivalent Lateral Static Load Method of analysis the seismic lateral forces are consistently replaced by equivalent lateral static forces. After estimating the fundamental period of vibration of the structure, the maximum acceleration is determined using spectrum curves. The base shear is then calculated using Newton's second law of motion:

V= Mamax (1) In which M represents the seismic mass of the structure including a selected portion of the live load and amax is the maximum acceleration using appropriate spectrum curves. Afterwards the base shear, V, is distributed along the height of the building in accordance with the decomposed modal shapes.

The Equivalent Lateral Static Load analysis continues to be widely used and favored for a number of reasons that include but are not limited to the mathematical simplicity of the method and the rather short execution time and cost that it entails. The popularity of this method was dominant in the era that preceded the advent of personal computers. However, its application is constrained by the following restrictions:

1. General assumptions: these are intended to ensure that the fundamental mode shape governs the behavior of structure. For such structures equivalent two or three dimensional static analysis may be performed. However, the method should not be used for structures with either horizontal iregularity particularly a torsional irregularity, or vertical irregularity such as the presence of a soft-storey. According to Wilson [ ] regular structures are defined as the structures in which a minimum coupling exists between the lateral displacements and the torsional rotations for the mode shapes associated with the lower frequencies of the system. 2. Special assumptions: These are required for the estimation of the fundamental period of the structure. Such assumptions are influenced by the topology of the structure as well as by its material properties and loadings (Touqan 2005 international conference paper TINEE). The above shows that the equations recommended by design codes are valid under specific conditions. If the method is applied to structures that do not meet such conditions erroneous results become inevitable. The following presentation shows a comparison between the three dimensional Equivalent Lateral Static Load Method of analysis and the three dimensional dynamic Response Spectrum Method of analysis as they are both applied to a broad range of structures with different vertical irregularities while keeping a regular plan arrangement.

The Structural Models Studied For the purpose of the present undertaking a symmetrical reinforced concrete structure is randomly selected. The selected building is a typical seven storey residential one with each level having a three meter height. Each floor has two housing units with an access stairway in between. The first level of the selected building is a typical parking area servicing the occupants. The building is comprised of a reinforced concrete structural frame with infill masonry or shear walls. The foundations in all models are assumed fixed for simplicity. Following local considerations the building is envisaged to belong to Seismic Group 2 and Seismic Design category B in accordance with the definitions of the IBC. For the purpose of this presentation the live load is taken to be 3 kN/m2, the floor finish load is taken as 1.5 kN/m2 .Wind loading is not considered. The IBC 2003 response spectrum with 5% damping ratio is adopted in the study. The design spectral response acceleration at short period Sds and at one second period Sd1 are assumed to be 0.333g and 0.133g respectively. The unit weights for concrete and masonry are taken as 25 kN/m3 and 20 kN/m3 respectively. The elastic modulus of concrete is taken as 28,500 MPa and that of masonry is taken as 3,500 MPa. The Poisson's ratio for both concrete and masonry is taken as 0.2. The total height of the building is 21 meters; its length is 21 meters while the width is 12 meters. The numerical models are built using SAP2000 Version 11. The live load contribution to the seismic mass is estimated at 30% in addition to the full dead load and the self weight of the structure. The following five study models are numerically investigated; they vary in the infill walls distribution and in the respective material properties. Both the long and the short directions of the building are investigated. The study models are given the following names and descriptions:

1. BFrame: This is a bare structural frame for all levels. 2. Masonry: This is the bare frame of model 1 but with 15 cm infill un-reinforced masonry walls at all levels but at the soft storey level no walls are included. Window openings are assumed small thus they are totally neglected. 3. Shear wall (soft storey): same as in model 2 but reinforced concrete walls are included to act as shear walls. 4. Stiff column: same as in model 2 but the columns of the first storey are made substantially stiffer (60 cm x 60 cm). 5. Gradual: this is the same as in model 2 but the short direction has one shear wall in the soft storey, two shear walls in the above storey and three shear walls in the third and above levels while in the long direction the soft storey has one shear wall, the second storey has three shear walls and the third and above levels have 5 shear walls. Thus presenting a gradual increase in stiffness with height. A schematic diagram of the five study models is shown in Figure 1.

Models 2, 3 and 4 Model 1 Model 5 Figure 1: A schematic diagram for the five study models. Model Description In constructing the various numerical models except for model 4, all columns are assumed having a square cross section of 40 cm x 40 cm; solid slabs and walls are modeled as shell elements of 20 cm thickness sitting on continuous drop beams of 40 cm x 40 cm section. Beams and columns are modeled as frame elements. Slabs and walls are modeled as shell elements. Window openings, tiny relative to the overall wall area are excluded from the model as they have no appreciable bearing on the general behavior of the structure [Yasir]. Supports at the base are assigned a total fixation. Figure 2 shows the general layout plan of the building used in the study. A set of 12 eigenvectors are requested in the analysis. Masonry walls with no reinforcement bars are modeled as contributing to the mass of the structure but provide no ductility provision.

Figure 2: Floor plan of all structural models. RESULTS OF THE ANALYSIS Comparison of the Natural Periods

0.9 The fundamental period of vibration using the T= 0.0466HN (H in meters) is 0.72 3/4 seconds, and for the rest of the modes T= 0.049HN is 0.48 seconds. Such values are slightly different from the values obtained from the modal decomposition analysis shown in Table 1 which shows also a comparison between the first three modal periods, directions and mass participation ratios. It is clear that the code expression for the period does not make any distinction between the values of the period with respect to the different directions. Model 1, as unrealistic as it is, but most widely adopted by designers has the largest mass to stiffness ratio hence the largest period. It provides almost equal periods in both directions with a mass participation factor of 0.83 for the first three modes. This makes it imperative that additional modes be included in the analysis in order to account for the code desired 90% cumulative mass participation. While reinforced shear walls add substantial stiffness to the structure, pure infill walls add little stiffness if any. Reinforced concrete infill walls in upper floors combined with some side reinforced concrete shear walls at the first storey provide the best alternative from a strict vibration vantage point.

Table 1: Tabular comparison of the fundamental periods for the selected model

Model 1st mode 2nd mode 3rd mode T Direction Mass T Direction Mass T Direction Mass No. (sec) part. partic. partic. ratio ratio ratio 1 0.90 Uy 0.83 0.88 Ux 0.83 0.79 Rz 0.83 2 0.56 Uy 0.94 0.49 Ux 0.98 0.43 Rz 0.99 3 0.47 Uy 0.99 0.43 Ux 0.99 0.39 Rz 0.99 4 0.44 Uy 0.85 0.35 Ux 0.90 0.29 Rz 0.94 5 0.36 Uy 0.79 0.35 Rz 0.98 0.31 Ux 0.85 Comparison of lateral displacements Plots of the storey level displacement in the short and in the long directions versus height are made for all models and superposed on the same graph. Two plots are made for each direction; one using a three dimensional Response Spectrum analysis and the other using an Equivalent Lateral Static Load analysis. These are presented in Figures 3 and 4. It is readily observed that displacements in all cases are consistently lower with increasing number of models 1-5 and in both methods. The decrease in the displacement value is about one order of magnitude. However, although this observation is evident in the short direction; in the long direction the discrepancy is less pronounced. This is attributed to the fact that structures are generally more rigid in the long direction. Furthermore, it is clear that model 3, the shear walls model, behaves almost as a rigid body with a high relative magnitude of displacement at the first storey level.

x-displacement versus height x-displacement versus height

25 25 Bframe 20 Bframe 20 masonry 15 masonry 15 t t h h shear wall shear wall g g

i 10 i 10 e e h stiff coll h stiff coll 5 5 gradual gradual 0 0 Poly. (Bframe) 0 0.02 0.04 0.06 0.08 Poly. (Bframe) 0 0.01 0.02 0.03 0.04 -5 -5 Poly. (masonry) x-displacement 2 per. Mov. Avg. x-displacement (masonry) Poly. (stiff coll) Poly. (stiff coll)

Poly. (gradual) Poly. (gradual)

(a) 2 per. Mov. Avg. (b) 2 per. Mov. Avg. (shear wall) (shear wall) y-displacement versus height y-displacement versus height

25 25 Bframe 20 20 Bframe masonry masonry 15 15 t t h shear wall h shear wall g g i 10 i 10 e e h stiff coll h stiff coll 5 5 gradual gradual 0 0 Poly. (Bframe) 0 0.01 0.02 0.03 0.04 0.05 0 0.02 0.04 0.06 0.08 Poly. (Bframe) -5 -5 Poly. (masonry) Poly. (masonry) y-displacement y-displacement 2 per. Mov. Avg. 2 per. Mov. Avg. (shear wall) (shear wall) Poly. (stiff coll) Poly. (stiff coll)

Poly. (gradual) Poly. (gradual) (c) (d) Figure 3: Displacement in the short and long directions versus height (units in meters): (a, c) using three dimensional Dynamic Response Spectrum Method; (b, d) using the Equivalent Lateral Static Load Method

Comparison of Design Forces

It is prudent to compare moment and shear forces in building columns in all study models. The bar charts in Figures 5, 6 and 7 show the pertinent values. A quick examination of the plots reveals that there is no significant difference in behavior between the short and the long directions in regard to the shear force or bending moment distribution in both methods. Axial forces, however, as evident in Figures 5 and 6 are consistently larger in the short direction than in the long direction.

axial and shear force in corner column in x- axial and shear force in coner column in x- direction direction

1200 1200 1000 1000 N

K 800 N 800 axial force K n i

n axial force e i

shear force

600 c e shear force r 400 c o r 400 f o

f 200 200 0 0 Bframe masonry shear stiff coll gradual wall Bframe masonry shear stiff coll gradual wall

(a) (b) Figure 4: Forces in a typical corner column in the first storey level: (a) using three dimensional dynamic Response Spectrum Method; (b) using Equivalent Lateral Static Load Method

axial force and shear force in corner column in y- axial force and shear force in corner column in y- direction direction

2000 2000 1800 1800 1600 1600

N 1400 N 1400 K K

1200 1200

n axial force n axial force i i

1000 1000 e shear force e shear force c 800 c 800 r r o o

f 600 f 600 400 400 200 200 0 0 Bframe masonry shear stiff coll gradual Bframe masonry shear stiff coll gradual wall wall

(a) (b) Figure 5: Forces in a typical corner column at the first storey level: (a) using three dimensional dynamic Response Spectrum method; (b) using Equivalent Lateral Static Load method bending moment in corner column bending moment in corner column

600 600

m 500 500 m - - N bending moment in x- N bending moment in x-

K 400 K 400

n direction n direction i i

t 300 t 300

n bending moment in y- n bending moment in y- e 200 direction e 200 direction m m o o

M 100 M 100 0 0 Bframe masonry shear stiff coll gradual Bframe masonry shear stiff coll gradual wall wall

(a) (b) Figure 6. Bending moment in a corner column in the short and in the long directions: (a) using three dimensional dynamic Response Spectrum Method; (b) using Equivalent Lateral Static Load Method

Comparison of Lateral Displacement Values and Design Forces

Selecting models 1, 3 and 5 and plotting on the same diagram displacements and forces resulting from both the Equivalent Lateral Static Load Method and from the dynamic Response Spectrum Method versus height yields Figures 9-12. It is obvious from the graphs that the former gives similar response as the later method for gradual distribution of stiffness, however the former is over conservative in the bare frame model and as apparent in Figures 8 and 12 but it is much less conservative for structures with a soft storey.

x-displacement versus height y-displacement versus height

25 25 frame-res frame-res 20 20 frame-eq frame-eq 15 15 shear-res t t

h shear-res h g g

i 10 i 10

shear-eq e e h

h shear-eq 5 grad-res 5 grad-res 0 grad-eq 0 grad-eq 0 0.02 0.04 0.06 0.08 0 0.02 0.04 0.06 0.08 -5 Poly. (frame-res) -5 Poly. (frame- x-displacement Poly. (frame-eq) y-displacement res) Poly. (frame-eq) 2 per. Mov. Avg. (shear-eq) 2 per. Mov. Avg. Poly. (grad-res) (shear-res) 2 per. Mov. Avg. Figure 7 Displacement in the2 per. Mov. short Avg. and the long directions versus(shear-eq) height (units in (shear-res) Poly. (grad-res) meters)

axial and shear force in corner column in x- axial force and shear force in corner column in y- direction direction

1000 1800 900 1600 800 1400

N 700 N 1200 K K

n axial force n 1000 axial force i i

500 e shear force e 800 shear force c 400 c r r 600 o o

f 300 f 200 400 100 200 0 0 frame- frame- shear- shear- grad-res grad-eq frame- frame- shear- shear- grad-res grad-eq res eq res eq res eq res eq

Figure 8: Forces in a typical corner column in the short and in the long directions in the first storey level. bending moment in corner column

N bending moment in x- K 300

n direction i

n 200 bending moment in y- e direction m

o 100 M 0 frame- frame- shear- shear- grad-res grad-eq res eq res eq

Figure 9. Bending moment in a corner column in the short and in the long directions of the first storey level.

Conclusion In earthquake prone zones earthquake analysis is mandatory but the method of analysis is left up to the discretion of the designer. Methods of analysis presently available include the dynamic Response Spectrum Method and the Equivalent Lateral Static Load Method. In the construction industry the later method is preferred over the former. The choice is based on a number of factors such as its apparent mathematical simplicity and its shorter running time. However, when both methods are applied to a range of structural models that have a regular horizontal plan but vertical irregularities it becomes apparent that the two mentioned method give different results especially in buildings with a soft storey. In such cases the dynamic Response Spectrum Method of analysis proves to be the better course for earthquake structural design undertakings. The use of the Equivalent Lateral Static Load method is only applicable under strictly controlled conditions and for a narrow band of structural types.

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