APCBEE Procedia 9 ( 2014 ) 354 – 359

2212-6708 © 2014 Farid Chalah. Published by Elsevier B.V.Selection and peer review under responsibility of Asia-Pacifi c Chemical, Biological & Environmental Engineering Societydoi: 10.1016/j.apcbee.2014.01.062

ScienceDirectAvailable online at www.sciencedirect.com

ICCEN 2013: December 13-14, Stockholm, Sweden

Fundamental Vibration Period of SW Buildings

Farid Chalaha, , Lila Chalah-Rezguia, Kamel Faleka, Salah Eddine Djellaba and Abderrahim Balib

aUsthb, Faculty of Civil Engineering, Algiers, Algeria. bURIE, Ecole Nationale Polytechnique, Algiers, Algeria.

Abstract

This paper focuses on the fundamental period vibration evaluation of civil engineering constructions achieved in reinforced concrete which use the shear walls bracing system. As known, in the vibration modes assessment phase of a given structure, much of the seismic energy is absorbed by the fundamental mode. For this reason, many authors have provided empirical formulas using many approaches. Based on an elastic behavior and the Dunkerley formula, a simplified expression is proposed, which takes into account the geometrical and the mechanical characteristics of the structure. Following this development, the period value is a function of the number of floors of the considered structure. It is in good agreement with that obtained by the FEM that uses numerical methods requiring a sophisticated and more complex process. It may constitute a practical tool for practicing engineers in the preliminary design phase of any SW building. © 2013 Published by Elsevier B.V. Selection and/or peer review under responsibility of Asia-Pacific Chemical, Biological & Environmental Engineering Society Keywords: Fundamental period, SW buildings, earthquake code, mode vibration

1. Introduction

The objective of this work is to propose a formula for the determination of the fundamental period of shear wall buildings. In this field, experimental works [1] have been conducted and often followed by statistical treatments of results. These are generally obtained from ambient vibrations measurements or from occurred

Corresponding author. Tel.: +213-21-24-72-24; Fax: +213-21-24-79-14. E-mail address: f_chalah@yahoo.fr.

© 2014 Farid Chalah. Published by Elsevier B.V.Selection and peer review under responsibility of Asia-Pacifi c Chemical, Biological & Environmental Engineering Society

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earthquakes (recorded data). They are used to suggest simplified formulas. Also, theoretical developments have been already done for this purpose by solving differential equations with assumptions on the vibration modes.

Some methods consisted on modifying existing formulas in the current codes which allow the evaluation of the lateral shear force applied at the base of a projected structure.

In its preliminary design phase, the building structure is not yet completely and entirely defined. Thus, only some parameters are known, namely; steel or concrete material and bracing system (shear walls or other…). Using seismic records, many research investigations led to proposal formulas which are more and more refined that take into account more effectively the bracing system, construction dimensions and mechanical characteristics of the used material. Thus, a number of empirical formulas [2-5] were developed for a better estimation of the fundamental vibration period of SW buildings and this was at the base of the earthquake codes evolution.

Improved formulas are continuously suggested. They generally include the construction dimensions such as the area, the height, the width and the length easily measurable and intuitively deductible. Many of them, are based on the characteristics of the building which are ignored and replaced by none really representative external dimensions in terms of in plane area’s length in the direction of the movement and the total height of the studied structure.

The finding of this research work is to propose a simplified formula for assessing the fundamental vibration period of SW buildings. It allows the evaluation of the seismic acting shear force at the base of the studied structure.

2. Modal Analysis of a Structure

For carrying out the free vibration analysis of an undamped multiple degrees of freedom system, the homogeneous system of differential equations given in 1., is considered.

0xKxM (1)

where: [M] : mass matrix [K] : stiffness matrix {x}, x : displacement and acceleration vectors with the assumption on the displacement:

)sin( tXx (2)

Substituting equation 2 in equation 1, without considering the nontrivial solution implies that its determinant must satisfy, the following condition:

| [K]-ω2[M] |=0 (3)

Many solving numerical methods exist depending on the dimension of the two matrices [6]. However, a formulation of a simple and practical expression of the fundamental period is needed. Its value for a single degree of freedom is given by the relation:

T1 =2 (m/k)1/2 (4)

with; k : stiffness m : floor mass.

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3. Evaluation of the Fundamental Period in Compliance with Existing Codes

In the following presentations, the formulas to evaluate the fundamental periods in function of the bracing systems are given for three earthquake codes; RPA99 version 2003[7], UBC-97[8] and Eurocodes[9].

3.1. Evaluation of the vibration period

In order to evaluate the base shear force acting on a civil engineering structure, different formulas for determining the fundamental vibration period which consist on an intermediate parameter are given by empirical methods (generally based on the Rayleigh quotient method) as expressed below:

T1=2. .[1/g( mi. xi2)/ ( fi.xi)]1/2 (5)

where: mi : mass of the ith story. xi : lateral displacement of the ith story under the application of a vector force.

3.2. Proposed formula for the fundamental vibration as given by actual earthquake codes

In the other hand, a version of the Rayleigh formula can be expressed as indicated in earthquake codes:

T1=2. .[1/g( Wi. i2)/ ( fi. i)]1/2 (6)

where: fi : ith component of a horizontal vector force system distributed along the structure height. δi : lateral displacements of the ith floor.

3.3. Assessment of the fundamental period in compliance with earthquake codes:

The T fundamental period of a given structure is assessed by applying formulas largely admitted and used.

3.3.1. Algerian seismic standard Depending on the bracing system, the empirical formula to be used is:

3/4T NT C h (7)

where : hN : Total height of the structure in meters. CT : coefficient which depends on the bracing system and the in filling nature.

The CT value to be considered for a bracing system partially or completely made by concrete SW is 0.05. Adding to this, another expression is given, under the form:

DhT N /09.0 (8)

where: D : building dimension, measured at its base, in the vibration direction. Nota: The appropriate value to be retained is the smaller of the values given by the two formulas and this

concerns each direction of vibration.

3.3.2. Expression of the period by UBC-97 The 1997 Uniform Building Code, expresses the period T1 by the relationship :

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T= CT.hn3/4 (9)

where: hn : is the building height in ft.

The following Table 1, shows the value CT to be considered:

Table 1. CT value to be considered according to the bracing system, as given in the UBC97.

Lateral force resisting system CT Steel moment frames 0.035 Steel moment frames eccentrically braced steel frames

0.030

All others buildings 0.020

3.3.3. Evaluation of the fundamental period by the Eurocodes For buildings up to 40 m of heights, the value of T1 (in s) may be approximated by the following

expression:

T1=Ct.H3/4 (10)

where: Ct :is 0.085 for moment resistant space steel frames, 0.075 for moment resistant space concrete frames

and for eccentrically braced steel frames and 0.050 for all other structures. H : height of the building, in m, from the foundation or from the top of a rigid basement.

Alternatively, the estimation of T1 (in s) may be made by using the following expression: T1=2 d1/2 where d :lateral elastic displacement of the top of the building, in m, due to the gravity loads applied horizontally.

4. Proposed Formula for the Fundamental Vibration Period Assessment for SW buildings

The response of a given structure to a seism depends only on its eigenvalues and eigenvectors. These parameters defined as vibration modes (periods, frequencies or pulsations and vectors) are well recognized when the structure is in free oscillations. After having highlighted these parameters by a stiffness formulation in statics and concentrating the mass of each level at its floor, a process is then usually used. At this step, an eigenvalue problem is formulated were various methods of resolution exist, from handle to numerical ones. Initially developed around the analysis field of rotating shafts by Dunkerley[10] and used in [11,12], its extension was widely used to the analysis of many vibrating systems. Also, this use concerned the determination of the first eigenvalue of various dynamic system representing constructions differently braced. The equation 11., expresses the circular frequency:

1/ 1D2=m1. 11+m2. 22 +m3. 33 +…+ mk. kk +…+ mn. nn (11)

where : 1D : Assessed fundamental vibration period by the Dunkerley-Southwell method

mk : kth floor mass kk : lateral displacement of the kth floor.

The expression of the kk displacement at a level k is given by the relationship 12.:

kk=1/3.hk3/(E.I) (12)

where:

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E : Young modulus I : level inertia in m4 hk : kth floor position in meters

Assuming a constant level height, the hk value is given by the following relationship: hk=k.h where: h : floor height

The previous equation is rewritten under the form of equation 13:

1/ 12=m1.131/3. h3/(E.I)+m2. 23.1/3. h3/(E.I)+ …+ mk. k3. 1/3.h3/(E.I)+…+mn. n3. 1/3.h3/(E.I) (13)

By using a suplementary assumption of a constant mass for all the levels of m value, a new expression is obtained, see equation 14 :

1/ 12= [13+23+33+…+k3+…+n3]. 1/3. m. h3/(E.I) (14)

Then, a final expression for 12 results as given in equation 15:

12= 12.[(E.I)/( m. h3)]/(n2.(n+1)2) (15)

which leads to the period expression of the equation 16:

T1= 1.814 n(n+1)[ m h3/(EI)]1/2 (16)

5. Results

The Table 2 below summarizes the results on the period values obtained by the proposed formula in equation 16 and those issued from the Jacobi method with an error calculus limited to 1.50% largely accepted.

Table 2. Some values of the T10 fundamental period of SW building as a function of the floors number.

N Reference value Assessed value Error N Reference value Assessed value Error 2 10,762 10,883 0,0112 7 100,11 101,573 0,0146 3 21,482 21,766 0,0132 8 128,702 130,593 0,0147 4 35,778 36,276 0,0139 10 196,609 199,518 0,0148 5 53,648 54,414 0,0143 13 325,274 330,111 0,0149 6 75,092 76,179 0,0145 20 750,584 761,795 0,0149

with: T1= T10 m h3/(EI)]1/2

A simple correction is proposed to improve the above formula, as written in 17:

T1= 1.8 n(n+1)[ m h3/(EI)]1/2 (17)

6. Conclusion

Various formulas have been proposed to determine empirically or theoretically the fundamental vibration period of shear wall buildings. These are obtained by mathematical formulations and experimental measurements of ambient vibrations with a post processing statistical treatment. For this, only the geometrical characteristics of the buildings are taken into account with additional coefficients of correction. Formulas based only on the dimensions, will give the same period value, whatever, the SW density repartition.

In this present work, starting from the Dunkerley formula, using hypothesis on constant height and mass of the floor, a realistic formula is found. It allows the period assessment by a direct and simplified relationship expressed in terms of mechanical and geometrical characteristics and the total floors number of the SW

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building.

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