Design of steel CBF structures through Linear Time-History Analysis

Scuola di Ingegneria

Corso di Laurea Magistrale in Ingegneria Edile e delle

Costruzioni Civili

Tesi di Laurea Magistrale

Design of steel CBF structures through

Linear Time-History Analysis

Relatori:

Prof. Ing. Walter Salvatore

Dr. Ing. Flavia De Luca Candidato:

Dr. Ing. Silvia Caprili Alessia Di Cuia

Tables of contents

CHAPTER 1

INTRODUCTION ... 5

CHAPTER 2 DYNAMIC OF STRUCTURES ... 8

2.1 BASIC PRINCIPLES OF SEISMIC ANALYSES ... 8

2.1.1. Resonance ... 8

2.1.2. Damping ... 10

2.1.3. Structural periods of buildings... 11

2.1.4. SDOF... 12

2.1.5. MDOF ... 15

2.2 EARTHQUAKE RESPONSE SPECTRA ... 17

2.2.1. Equation of motion ... 17

2.2.2. Response spectra ... 19

2.2.3. Elastic design spectrum ... 24

2.3 STRUCTURAL ANALYSIS METHODS ... 25

2.3.1. Linear static analysis or lateral force method ... 26

2.3.2. Linear dynamic analysis ... 27

2.3.3. Nonlinear static analysis or pushover method ... 28

2.3.4. Non-linear dynamic analysis ... 29

2.4 RESPONSE SPECTRUM ANALYSIS AND LINEAR TIME-HISTORY ANALYSIS .... 30

2.4.1. Response spectrum analysis ... 30

2.4.2. LTHA analysis ... 35

2.4.3. Difference between RSA and LTHA ... 37

CHAPTER 3 SEISMIC DESIGN OF CBF STRUCTURES ... 39

3.1 SEISMIC DESIGN CRITERIA ... 39

3.1.1. Capacity design ... 39

3.1.3. CBFs ... 44

3.1.4. EBFs ... 46

3.2 CONCENTRICALLY BRACED FRAMES ... 48

3.2.1. Brace Inelastic Cyclic Behaviour ... 50

3.2.2. Brace Slenderness and local buckling ... 51

3.2.3. Design according to Eurocode8... 53

3.3 ARCHETYPE STRUCTURES ... 56

CHAPTER 4 EC-8 ARCHETYPE DESIGN THROUGH RSA AND LTHA ... 65

4.1 DESIGN OF STEEL FIVE-STOREYS CBF ... 65

4.1.1. Load analyses ... 69

4.1.2. Preliminary design ... 75

4.1.3. Capacity design ... 90

4.1.4. Model in Midas Gen ... 96

4.2 RESPONSE SPECTRUM ANALYSIS ... 100

4.2.1. Dynamic properties of the structure ... 100

4.2.2. Second-order effects (P-Δ effects) ... 102

4.2.3. Design with Elastic Response Spectrum ... 103

4.2.4. Martinelli et al. (2000) design approach with RSA ... 105

4.2.5. Optimized design of CBF through RSA ... 115

4.3 LINEAR TIME HISTORY ANALYSIS ... 121

4.3.1. EC8 compliant record selection ... 121

4.3.2. Optimized design of CBF through LTHA ... 129

4.4 COMPARISON OF DESIGN APPROACHES ... 143

4.5 RSA VS LTHA ... 150

CHAPTER 5 CONCLUSION AND FURTHER RESEARCH ... 155

REFERENCES ... 158

INTRODUCTION

Design of structures in seismic countries is made through linear response spectrum analysis (RSA). According to the technical jargon of many countries, such as Italy, RSA is also defined as "linear dynamic analysis" or "modal analysis". Strictly speaking, RSA is an approximate approach for the evaluation of linear dynamic response of structures. It accounts for the effects of higher modes through the approximate combination of results of static analyses defined on the basis of modal properties of structures. The Complete Quadratic Combination is the combination rule for RSA since 1980s, and it is suggested as the main combination approach by many codes, e.g., Eurocode 8 (CEN, 2004), Italian Seismic code (NTC, 2008). The main reason for RSA in lieu of proper linear Time-History analysis (THA) is that linear THA requires selection and availability of accelerograms, and it can be time-consuming. On the other hand, the combination in RSA has additional critical aspects such as the choice of the proper sign when the strength in of the element in compression and tension is different. A relevant example is the case of braces in steel frames.

Eurocode 8 conforming design of a five storeys Steel Concentrically Braced Frame building is carried out through both Linear Time History Analysis and the conventionally employed Response Spectrum Analysis.

The aspect investigated in this research is whether the Eurocode 8 design process is affected by the change in the linear analysis methodology, i.e. linear time history versus response spectrum analysis. The investigation includes the design of a five storeys steel CBF archetype building in High Ductility Class to check the resulting design configuration according to EC8 in the case of the two linear analyses approaches.

This thesis deals, at the same time, with design related aspect and a methodological-related one. On one side, in fact, there are all the critical aspects methodological-related to the seismic design of CBFs according to Eurocode 8, on the other side there is the investigation of

the possibility to introduce a new, less approximated analysis methodology as design method.

EC8 defines several limitations for design of CBFs. Some of these conditions are very difficult to be verified (Brandonisio et al. 2012; Faggiano et al. 2014). The problem is also related to the fact that, as predicted by American (AISC 341-05) and European seismic codes, in the design of CBFs compressed braces must be neglected, thus leading to a tension-only approach. As long as the structure is quite simple and regular, the identification of the compressed brace is straightforward and design can be performed with linear static analysis. The real critical design aspect arises when the structure is not symmetric or it has irregularity in plan and elevation preventing the design through linear static analysis.

The methodological issue is due to the results obtained through RSA and LTHA in terms of axial loads in bracing elements. This research proposes a new methodological approach, which takes into account the contribution not only of the brace in tension, but also of the compressed one, as long as it does not attains its buckling strength, in the 3D model under consideration. This new approach is the basis of the optimized design through Linear Time History Analysis and Response Spectrum Analysis of the examined structure. The new approach investigated, even if applicable also for the case of RSA, is specifically suitable for the LTHA situation in which the knowledge of the compression or tension phase of the bracing allows an accurate application of the methodology.

The above design and methodological discussion on the design of steel structures according to EC8 is organized as follows.

Chapter 2 describes the basic principles of seismic analysis, helpful to understand the behaviour of structures under seismic actions. Then the single and the multi degrees of freedom systems will be described, followed by the description of the equation of motion and the response spectrum concept is finally introduced. Chapter 2 focuses on the description of structural analysis methods, with particular attention to RSA and LTHA.

In Chapter 3 seismic design criteria for steel structures are discussed in details. The behaviour of concentric braces and the design limitations imposed by Eurocode 8 are discussed. A literature review on steel archetype structures is provided, in order to define the structural shape and dimensions of the five storeys building integrating

available studies and reflecting the most common building practice in different countries.

Chapter 4 is the core and original part of this research work. After the description of the pre-design step through Lateral Force Method, three linear dynamic analysis methods are explained: (i) Martinelli et al. RSA approach, (ii) RSA and (iii) LTHA with a new bracing optimized design. From the analyses, in chapter 4, more general conclusions and comparison are drawn in the light of further development for a full validation of the proposed design approach.

The study of CBF suggests that Linear Time History Analysis can be systematically included in future versions of Eurocode 8, and it can be employed for a more accurate performance-based design. On the other hand, some methodological aspects still need to be addressed and a more extensive test of the methodology has to be made for different structural typologies, materials and including different cases for accelerograms selection.

DYNAMIC OF STRUCTURES

“A violent and essentially unpredictable dynamic ground motion imposes extreme cyclic loads on engineering materials whose response under such conditions is complex

and incompletely understood” (Booth et al. 2006). A basic understanding of analytical

principles is essential as basis for seismic analysis and design.

2.1 Basic principles of seismic analyses

Seismic forces in a structure do not arise from applied loads, but response is the result of cyclic motions at the base of the structure causing accelerations. This response is dynamic so it is determined by dynamics properties of the structure, especially natural period and damping.

2.1.1. Resonance

Resonance, in physics, describes when a vibrating system or external force drives another system to oscillate with greater amplitude at a specific preferential frequency.

All buildings have a natural period, which is the number of seconds it takes for the building naturally vibrating back and forth. In addition, the ground has a specific resonant frequency. If the period of ground motion matches the natural period of a building, it will undergo the largest oscillations possible and suffer the greatest damage.

Figure 2.1 shows resonance effect for various input frequencies and damping coefficients. The response is shown in terms of peak acceleration of the system, divided by the peak ground acceleration to give a normalized response. It shows a peak when the system period matches that of the input motion, causing resonance.

The normalised response tends to unity when the system period tends to zero or when the ground motion, in comparison to the period of the structure, becomes very long. Moreover, the normalised response tends to zero when the system period is too long compared to that of the ground motion, or when the ground motion becomes very short.

Frequencies at which the response amplitude is a relative maximum are known as the system’s resonance frequencies. At resonance frequencies, small periodic driving forces can produce large amplitude oscillations.

Figure 2.1: Resonance effect shown for various input frequencies and damping coefficients; [from Katsuhiko Ogata, 2005]

Figure 2.1 represents the response to constant-amplitude single period motions but, otherwise, earthquakes are temporary phenomena and the associated ground motions contain a range of periods, many of which predominate depending on the magnitude earthquake and the soil conditions at the site. Pseudo-velocity response spectra for a typical earthquake are shown in Figure 2.2. It is possible to note that, increasing the structural period, the response increases up to a maximum, and then it decrease at zero. There are as many curves as considered damping.

Figure 2.2: Pseudo-velocity response spectra, El Centro, California earthquake, 1940; [from Chopra, 2007]

2.1.2. Damping

Damping is an influence on an oscillatory system that has the effect of reducing, restricting or preventing its oscillations. Damping has a very important influence on the response of the structure. A system may be undamped, when it oscillates at its natural resonant frequency, or underdamped, when it oscillates with the amplitude gradually decreasing to zero. Besides, there are overdamped systems, which return to equilibrium without oscillating.

The damping is assumed viscous, that is the damping force varies with the velocity of the system relative to the ground. Viscous damping is expressed in terms of

percentage of critical damping  , where  100% is representative of a system which

converges to zero as fast as possible without oscillating (Figure 2.3).

Figure 2.3: Time dependence of the system behaviour on the value of the damping ratio ζ; [from Weisstein et al., 2007]

For a sinusoidal excitation,  is related to the ratio of energy dissipated by damping cycle (area of ellipse ED) and to the strain energy stored (area of triangle ES), as Figure 2.4 shows, with the equation (2.1):

4 D S E E





 (2.1) (a) (b)

Figure 2.4: Energy dissipated (a) and stored (b) in a damped system; [from Clough and Penzien, 1997]

2.1.3. Structural periods of buildings

The crucial parameters in determining the structural response to an earthquake ground motion are damping and natural period. The period of an undamped mass supported on

a spring is equal to 2 M

k

 .

For most of the buildings natural period can be determined from mass and stiffness: for mass doubling there are period increases, and the same for stiffness halving. While the mass of a building structure may be easy to determine, its stiffness is usually much more uncertain. Non-structural elements, for example partitions, tend to add stiffness and thus to decrease natural period (Booth et al. 2006).

To determine structural period, Eurocode 8 provide empirical formulae based on building height. For buildings with heights of up to 40m, the value of T (s) may be ₁

approximated by the following expression (2.2):

3/4 1 1

T C H (2.2)

where C₁ is 0.085 for moment resisting steel frames, 0.075 for moment resisting concrete frames and for eccentrically braced steel frames and 0.050 for all other structures; H is the height of the building in m, calculated from the foundation.

Alternatively, T (s) may be determined by using the expression (2.3): ₁

1 2

T   d (2.3)

where d is the lateral elastic displacement of the top of the building in m, due to the gravity loads applied in the horizontal direction.

2.1.4. SDOF

A simple structure can be idealised as a system with a lumped mass m supported by a massless structure with stiffness k in the lateral direction, and a viscous damper that dissipates vibrational energy of the system (Figure 2.5). Roof is very stiff, its mass is equal to the lumped mass m. Columns support the roof and they provide entirely the flexibility of the structure in lateral motion. Lateral stiffness k of the structure is equal to the sum of the stiffnesses of the single columns. Columns and beams are assumed inextensible axially.

Figure 2.5: Single degree of freedom system; [from Dindar et al., 2015]

The number of degrees of freedom (DOF) is the number of the independent displacements required to define the displacement of all the masses. The structure represented in Figure 2.5 has only one DOF, i.e. the roof level displacement.

If the system is subjected externally to an applied forcef , there is an internal force _S

resisting the displacement u and equal but opposite in direction to the force f . The _S

relation between displacement and f is linear: _S S

f ku (2.4)

where k is the lateral stiffness of the system, evaluated as force/length.

As previously said, there is a viscous damper for the dissipation of the energy. If a linear viscous damper is subjected to a force f , there is an internal force in the damper _D

equal but opposite in direction to the force f_D. The relation between the damping force

D

f and the velocity u is linear:

D

f cu (2.5)

c is the viscous damping coefficient, evaluated as force x time/length.

Figure 2.6 shows a single-degree of freedom system, subjected to a dynamic force ( )

p t in the direction of the displacement u. As well as the force p varies with time, the

resulting displacement of the mass u varies with time.

Figure 2.6: SDOF subjected to a dynamic force p(t); [from Chopra, 2007]

Figure 2.6b schematizes the forces acting on the mass: the external force ( )p t , the

elastic resisting force f and the damping force_S f . All values are assumed positive in _D

the direction of the x-axis. Newton’s second law of motion gives:

S D

p



f



f



mu

(2.6)

After substituting (2.4) and (2.5), (2.6) becomes:

( )

mu cu ku

 



p t

(2.7)

This is the equation of motion for a SDOF, if we consider the spring and damper to be massless, the mass to be rigid and all motion to be in the x-axis direction.

The most important problem for structural dynamics is the behaviour of structures subjected to earthquake motion. The total displacement of the mass

u

t is equal to the sum of the displacement of the ground

u

_g and the relative displacement between the

mass and the groundu. The relation varies with time:

( )

( )

( )

t

g

u t



u t



u t

(2.8)

The equation of motion for a SDOF derives from (2.7), with the difference that the external force ( )p t is equal to zero.

Figure 2.7: SDOF subjected to earthquake motion; [from Chopra, 2007]

The equilibrium, in Figure 2.7, is represented by the equation

0

I D S

f



f



f



(2.9)

Equations (2.4) and (2.5) are still valid; instead, the inertial force

f

_I is relatedto the

acceleration

u

tof the mass. Relation (2.7) becomes: ( )

g

mucuku mu t (2.10)

The equation of motion for a SDOF system subjected to external force or to earthquake ground motion is a second order differential equation, that can be solved considering the initial displacement

u

(0)

and initial velocity

u

(0)

equal to zero, because the structure is at rest because excitation.

The differential equation governing the lateral displacement

u t

( )

of a SDOF without external forces and damping (

c



0

) is:

0

mu ku

 

(2.11)

The solution of the homogeneous differential equation is simple to obtain by standard methods: (0) ( ) (0) cos _n sin _n n u u t u t  t    _(2.12)

where



_n is the natural circular frequency of vibration, equal to:

n k M

  (2.13)

If an initial displacement

u

(0)

is given to the structure, it will start to oscillate back and forth without stopping. The curve in Figure 2.8 describes one cycle of free vibration of the system.

Figure 2.8: Free vibration of a system without damping; [www.efunda.com/formulae/vibrations/sdof_free_undamped.cfm]

The time required for the undamped system to complete one cycle of free vibration is the natural period of vibration

T

_n (sec.):

2 n n T





 (2.14)

The natural frequency of vibration

f

_n (Hertz) is: 1 2 n n n f T     _(2.15)

2.1.5. MDOF

The equations of motion can be written also for multi-degree of freedom systems (MDOF). A general formulation is reported for a two-storey shear frame, subjected to external forces

p t

₁

( )

and

p t

₂

( )

, as represented in Figure 2.9.

It is supposed that beams and floors are rigid in flexure, axial deformation of the beams and columns is neglected and axial force has no effect on the stiffness of the columns. It is possible to idealize the masses concentrated at the floor levels. A two-storey shear frame has two degrees of freedom, i.e. the lateral displacements of the two floors in the direction of x-axis,

u

₁ and

u

₂. Newton’s second law of motion (2.6) becomes:

j s j D j j

p  f  f mu (2.16)

where j refers to each considered level. In this case there are two equations, one for j=1 and one for j=2. These relations can be written in matrix form:

1 1 1 1 1 2 2 2 2 2 0 ( ) 0 ( ) S D S D f m u f p t f m f p t u   _{ }  _{  } _{ }              _{ }       (2.17)

Written in a compact way, the equation (2.17) becomes:

( )

mu f  _D f_s p t (2.18)

where m is the mass matrix of the system. The storey shear V_j is related to the storey

drift  _j u_i u_j1as follows:

j j j

V  k (2.19)

with k the story stiffness equal to the sum of the lateral stiffnesses of all columns in the _j

storey.

For a MDOF, the equations (2.4) and (2.5) assume the same form:

f_s ku (2.20)

f_D cu (2.21)

considering that k is the stiffness matrix, u the displacement vector, c the damping

matrix and u the velocity vector. Definitely, for a two-storey shear frame subjected to

external dynamic forces

p t

₁

( )

and

p t

₂

( )

, the relation (2.18) can be written in matrix form as follows:

( )

mu cu ku=p t

 

(2.22)

As previously said for SDOFs, also for MDOF systems it is important to understand

of the mass m_j is equal to the sum of the displacement of the ground u_g and the relative displacement between the mass m_j and the ground u . The relation varies with time:

( )

( )

( )

t

j j g

u t



u t



u t

(2.23)

By considering all the N masses, the (2.23) is in vector form:

( )

( )

( )

u

t

t



u

t



u t

_g

1

(2.24)

with 1 a vector of order N with each element equal to unity (Chopra 2007). The dynamic equilibrium for a MDOF, for external force equal to zero, is given by:

0

fI   fD fs (2.25)

Figure 2.10: MDOF subjected to earthquake motion

The inertial force f_I is related to the acceleration

u

tof the masses. Relation (2.7), for a multi degrees of freedom system, becomes:

( )

mu cu ku   m1u_g t (2.26) which contains “N differential equations governing the relative displacements

u

t_j

( )

t

of a linearly elastic MDOF system subjected to ground acceleration u t_g( )” (Chopra 2007).

2.2 Earthquake response spectra

2.2.1. Equation of motion

Because earthquakes can cause damage to many structures, structural dynamic analyses the response of structures to ground motion. The most useful way of defining the

ground acceleration u t during an earthquake is using the strong-motion _g( ) accelerograph (Figure 2.11), the basic instrument to record shaking components. It does not work continuously, but it starts recording by the first waves of the earthquake.

Figure 2.11: Strong-motion accelerograph; [from http://www.trimble.com]

Accelerograph records numerical values of ground acceleration at discrete time instants, so they can describe the high the high variability of the acceleration with time.

“For a given ground motion u t , the deformation response _g( )

u t

( )

of an SDOF system depends only on the natural vibration period of the system and its damping ratio” (Chopra 2007). Figure 2.12 shows the deformation response of three systems, different for their natural periods but with the same value of damping ratio,





2%

. To complete a cycle of vibration, a SDOF needs a time very close to the natural period of the system. It is important to notice that, the longer is the vibration period, the greater is the peak deformation u.

Once known the deformation response history

u t

( )

through dynamic analysis, a static analysis of the structure can determine the internal forces in the structure. The more used method is based on the equivalent static force f_S concept. As previously defined in (2.4), the variation of static force f in the time is due to: _S

( ) ( )

S

f t ku t (2.27)

k

is the lateral stiffness of the frame that, expressed in terms of the mass m, changes the (2.27) in:

2

( ) ( ) ( )

S n

f t m u t mA t (2.28)

where



_n, as reported in (2.13), is the natural frequency of vibration. The

pseudo-acceleration

A t

( )

is so defined:

2

( ) _n ( )

A t  u t (2.29)

The pseudo-acceleration is obtained by multiplying each deformation response

u t

( )

with the corresponding natural frequency of vibration

2 2 ₂ n n T    _ _ , since T_n is a known value of the structure.

After all, for the one-storey frame, it is possible to calculate the base shear V tb( ) and

the base overturning moment M tb( ):

( ) ( ) ( ) ( )

b s b s

V t  f t M t hf t (2.30)

where h is the height of the mass above the base. By considering the (2.28):

( ) ( ) ( ) ( )

b b b

V t mA t M t hV t (2.31)

2.2.2. Response spectra

“A plot of the peak value of a response quantity as a function of the natural vibration period T_n of the system, or a related parameter such as circular frequency



_n or cyclic frequencyf_n, is called the response spectrum for that quantity” (Chopra 2007).

Spectra can be plotted for spectral acceleration, velocity and displacement. Acceleration spectra are used for determining maximum inertia forces that develop during an earthquake and so the strength that a structure need to resist. Displacement spectra are convenient in methods for evaluating non-linear response in earthquakes.

Velocity spectra are helpful to derive power spectral densities, which are used in probabilistic analysis.

As mentioned above, it is important to know only the deformation ( )u t to compute

internal forces. Figure 2.13 represents a complete displacement spectrum, for several values of damping:

Figure 2.13: Displacement response spectrum of registered at AQV station during the 2009 L'Aquila earthquake [Chioccarelli et al., 2009]

The pseudo-velocity V for a SDOF system with natural frequency



n is related to

its peak displacement, indicated as D, with the relation: 2 n n V D D T     _(2.32)

“The pseudo-velocity response spectrum is a plot of V as a function of the natural vibration periodTn, or natural vibration frequencyfn, of the system” (Chopra 2007).

The pseudo-velocity response spectrum, shown in Figure 2.14, is obtained from the displacement response spectrum, by the relation:

2 V d n S S T   _(2.33)

Figure 2.14: Velocity response spectrum of registered at AQV station during the 2009 L'Aquila earthquake [Chioccarelli et al., 2009]

The pseudo-acceleration A for a SDOF system with natural frequency



_n is linked

to its peak displacement, indicated as D, with the relation:

2 2 2 n n A D D T      _{  }   (2.34)

“The pseudo-acceleration response spectrum is a plot of A as a function of the natural vibration periodT_n, or natural vibration frequencyf_n, of the system” (Chopra 2007). The pseudo-acceleration response spectrum, shown in Figure 2.15, is obtained from the displacement response spectrum by the relation:

2 2 a d n S S T     _{ }   (2.35)

Figure 2.15: Acceleration response spectrum of registered at AQV station during the 2009 L'Aquila earthquake [Chioccarelli et al., 2009]

Each of these three spectra (deformation, pseudo-velocity and pseudo-acceleration) contains the same information for a given ground motion. They are simple different ways to show the same information on structural response. Figure 2.16 shows the response spectrum for El Centro ground motion together with u , _g0 u_g0 and u , using _g0

normalized scales: D u , / _g0 V u/ _g0 and A u . In the graphic, there are four curves, / _g0 each for a different value of damping ratio, equal to 0, 2, 5 and 10%.

Figure 2.16: Response spectrum for El Centro ground motion plotted with normalized scales; [form Chopra 2007]

By observing response spectra, it is possible to notice that, for very short period, the

peak pseudo-acceleration A is equal to u_g and the displacement D approaches to zero.

Indeed for a fixed mass, a very short period is obtained by a very high stiffness k, as reported by (2.13) and (2.14). The system can be considered rigid, so its mass would start to move rigidly with the ground.

For very long period, the peak pseudo-acceleration A is very small and the

displacement D, for all damping values, approaches to u . Indeed for a fixed mass, a _g0

very long period is obtained by a very small stiffness k, as reported by (2.13) and (2.14). The system can be considered flexible, so its mass would remain stationary while the ground moves.

For intermediate values of period T , the pseudo-velocity _n V may be considered constant and equal toug0.

ranges” (Chopra 2007), as shown in Figure 2.17. The region to the left of point c,

n C

T T , is called the acceleration-sensitive region because the structural response is related to the ground motion. The region between points c and d, T_C T_n T_D, is called the velocity-sensitive region because the structural response is related to the ground velocity. At the end, the region to the right of point d, T_n T_D, is called the

displacement-sensitive region because the structural response is related to ground

displacement.

Figure 2.17: Spectral regions of response spectrum for El Centro ground motion; [from Chopra 2007]

For a particular ground motion, in the idealised spectrum, just T and _C T_D vary with damping; the other periods are independent of it. The idealization of the response spectrum allows creating a design spectrum representative of many ground motions. The period separating the spectral regions depends on the earthquake magnitude, fault-to site distance and soil conditions at the site (Chopra 2007).

Regarding the damping, it reduces the response of the structure only in the velocity-sensitive region. Indeed, in the acceleration-velocity-sensitive region the structure moves rigidly with the ground and in the displacement-sensitive one the mass stays while the ground moves. The forces associated to the motion could be reduced by using the effective damping of the structure.

2.2.3. Elastic design spectrum

The design spectrum is useful for the design of new structures or for the evaluation of seismic safety of existing structures. “The design spectrum should, in a general sense, be representative of ground motions recorded at the site during past earthquakes” (Chopra 2007). The necessary factors for a design spectrum are the magnitude of the earthquake, the distance of the site from the fault, the geology of the ground path of the seismic waves and the soil conditions of the site.

The design spectrum is obtained from the analysis of a several number of ground motions. For each ground motions considered, there is a value of peak displacement

0

i g

u , a value of peak velocity u_g0 and a value of peak accelerationui_g0. Each ground motion, denoted by u ti_g( ), is normalised, so all ground motions have the same peak ground accelerationu_g0. In correspondence of the period T , there are as many spectral _n

values as the number of ground motions considered

D

i, Vi and

A

i. Values represented in a normalised scale are the average values of the peak displacement, velocity and acceleration of the i-th ground motions considered.

Definitely, a design spectrum (Figure 2.18) is different from a response spectrum. The response spectrum is a plot of the peak response for a particular ground motion, valid for all SDOF systems. The design spectrum, indeed, is a plot of seismic deformation, function of the natural period of the structure and of the damping ratio.

Figure 2.18: Design response spectrum

Elastic design response spectra characterize ground motions and assess demands on various types of simple structures. They are the basis for computing design

displacements and forces in SDOF and MDOF systems, expected to remain elastic. The design spectrum is not intended to display to response spectrum for any particular ground motion. It is plotted to represent the average characteristics of many ground motions. (Chopra 2007)

Once design response spectra are established, it is simple to evaluate seismic forces for a building. Estimated the period of vibration of the building, it is possible to find the corresponding S and so to determine the lateral forces on the building. _a

2.3 Structural analysis methods

Analysis methodologies can be divided into two different families, linear and nonlinear, according to the actions. We can also divide each family in two categories, static and

dynamic, according to the structural response. Definitely, there are four different

analysis methods, provided by most of the recent seismic codes: linear static, linear

dynamic, nonlinear static and nonlinear dynamic analysis. Structural engineers must

choose the best seismic analysis method in order to obtain the best estimate of the seismic displacement demands.

These analyses show an increasing level of application difficulty, which corresponds to a higher degree of reliability and accuracy in evaluating the seismic response of the structure.

The linear analysis can be used to determine the seismic action effects both on dissipative that not dissipative systems. For dissipative systems, effects of earthquake should be evaluated by referring to the spectrum obtained with a reduction factor said

behaviour factor q. For non-dissipative systems, effects are determined by referring to

the design spectrum, considering a unit behaviour factor q. This factor depends on the structural type, the degree of hyperstaticity and the adopted design criteria. It takes into account the non-linearity of the material.

Behaviour factor is obtained with the relation:

0 R

qq K (2.36)

where q0 is the maximum value of the structure factor, which depends on the ductility

level, on the structure type and on the relationship  _u / ₁. The values of this factor are tabulated in seismic codes. K is a reduction factor, which depends on the regularity in _R

height of the construction. It takes value 1 for buildings regular in height, 0.8 for non-regular structures.

The non-linear analysis may be used for not dissipative systems and it takes into account the non-linearity of material and of geometry.

2.3.1. Linear static analysis or lateral force method

This structural analysis method is the basic approach to earthquake engineering. In this analysis, the first mode of the structure is considered representative of the whole structural behaviour. The effects of the earthquake on the building are shown schematically as a system of horizontal forces applied statically to the structure and distributed to the various floors. The distribution of forces has an inverted triangle shape, as shown in Figure 2.19, which recalls the shape of the first mode of regular buildings.

This schematisation can be adopted on condition that the period of the main vibration mode in the direction in examination T does not exceed 2.5₁ T or _C T , and _D

that the construction is regular in height. For civil or industrial buildings that do not exceed 40m in height and having a mass approximately evenly distributed along the height, T can be estimated, in the absence of more detailed methods, using the relation ₁

(2.2).

This method is force-based, as it allows estimating seismic response in term of forces. The forces applied to the masses of the various plans are determined with the following relationship: i i i h j j j z W F F z W      (2.37)

in which W and _i W are the weights of mass i_j th and jth, z and _i z are the levels of i_j th and jth mass and F is the seismic base shear, determined using the equation: _h

1 ( ) i d F S T W g     (2.38)

where S_d ( )T is the acceleration due to the mass of the building to be designed, W is ₁

the whole structure mass,  is a coefficient equal to 0.85 or 1:00 depending on the own

The equivalent static analysis can work well for building without significant lateral-torsional modes. This method is not suitable for tall buildings, where second and higher modes can be important.

Figure 2.19: Distribution of forces

2.3.2. Linear dynamic analysis

The analysis of the vibration modes, or modal analysis, is a method that allows identifying the modes of vibration of a structure, each of them defined by frequency, damping and mode shape. It is considered as a simple method for the design stresses definition. This method takes into account the dynamic characteristics of the structure using own ways of vibrate, considering all those with participating significant mass. Linear dynamic analysis can be applied to a linear structural model and the input given to the structure is characterized by accelerometric waveforms. As the linear static analysis, also the linear dynamic one can be considered force-based, with the difference that this method considers the interaction of the vibration modes of the structure, with the typical frequencies.

When seismic codes describe linear dynamic analysis, they generally refer to a modal response spectrum analysis. The seismic action, indeed, is represented by the design response spectrum, for each of the identified vibration modes. The response spectrum gives the peak response of the building, helpful for the structural design.

Definitely, the modal response analysis could be considered as a combination of different linear static analyses, in which the shapes of the force distributions are proportional to the modal shape of the ith mode. For each static analysis, modal response analysis considers the ith peak response referred to the ith mode and evaluates the base shear force as the product of the elastic spectral acceleration at the period of the ith mode and the mass excited by the ith mode.

necessary to consider for the structural response. Seismic codes require considering all the modes with participant mass ratios exceeding a specific value; moreover, it is necessary to consider a number of total modes so that the sum of participant mass ratios is equal to a percentage of the total mass of the structure.

Another aspect of this analysis approach, as well as the most critical, is the combination of all peak modal responses. Seismic codes describe mainly two combinations of modal responses, the square root of the sum of the squares (SRSS) or the complete quadratic combination (CQC).

Afterwards, the two linear dynamic analyses, RSA (response spectrum analysis) and LTHA (linear time history analysis) will be described in detail.

2.3.3. Nonlinear static analysis or pushover method

To obtain an accurate and realistic prediction of the seismic response of a structure, that allows taking the nonlinear behaviour and its evolution in time, is possible to use procedures of nonlinear static analysis. These procedures, while retaining remarkable simplicity of use and interpretation of results typical of the linear static analysis, allow more estimates realistic and reliable structural response even in nonlinear field. Non-linear static analysis gives direct information on the magnitude and the distribution of plastic strains in a structure.

Following this structural method, we can define a set of lateral forces, gradually increased by the same proportion, which are applied as a static load case to a non-linear model (Booth et al.2006). Then it is possible to register the response in a graph, considering on the vertical axis the base shear, and on the horizontal one the displacement of a control point, generally on the top of the structure. The result of this analysis is a plot (Figure 2.20), "capacity curve" or "curve pushover ", representative of the true monotonic behaviour of the structure. With this analysis, we can also get the evaluation of the maximum displacement reached by structure before the collapse.

Figure 2.20: Capacity curve or curve-pushover; [from thesis of Roberta Belfiore 2010]

The main idea, on which the analysis method is based, is to approximate the response of a MDOF (multi degree of freedom) system with the response of an equivalent SDOF (single degree of freedom) structure. The peak displacement of this SDOF structure is determined directly from the design response spectrum.

To obtain a curve pushover, at first it is necessary to define a capacity curve of an equivalent single degree of freedom and then to transform this curve in a bilinear function. After this, with an R-μ-T (reduction factor-ductility-period) relationship for the linear function, we can obtain SDOF seismic demand for a given spectrum. The SDOF response is translated to the MDOF displacement of the roof level and, with the pushover curve, the whole MDOF structure response is extracted. The analysis ends with a comparison between demand and capacity.

2.3.4. Non-linear dynamic analysis

Non-linear analysis allows the evaluation of the seismic response of a structure through direct integration of equations of motion, applying the appropriately selected accelerograms to the structure nodes, considering related to the ground. Among the various methodologies of analyses, the non-linear dynamic one is the most complex and consequently, the most complete. It allows knowing and checking for brittle behaviour of the elements that shape the structure; and it lets you estimate non-linear behaviour of the entire structure.

The nonlinear dynamic analysis is the most comprehensive type of analysis, but it is also the most complex. Particularly sensitive issues are the identification of a model that is able to describe the post-elastic behaviour under load cycles and unloading of the

elements and the consequent dissipation of energy, and the choice of ground motion, representative of expected events in the area where the building is located.

This analysis requires the use of calculation programs able to describe the non-linearity of the material. The structural models currently available are two: focused

plasticity models and diffused plasticity models. In focused plasticity models, all the

elements of the structure (usually the beam type elements) remain elastic and, where it is expected the formation of a plastic hinge, the hinge elements are introduced with inelastic behaviour. Therefore, the non-linearity of the structure remains concentrated in a few elements. In diffused plasticity models, in elements with inelastic behaviour, the elasticity is widespread around the element, both longitudinally and transversely and the section is divided into a set of "fibers" uniaxial.

This approach is the most rigorous, and is required by some building codes for buildings of unusual configuration or of special importance

2.4 Response Spectrum Analysis and Linear Time-History

Analysis

2.4.1. Response spectrum analysis

The earthquake response is difficult to calculate, even for a simple structure idealized as spring-mass-dashpot system shown in Figure 2.21.

Figure 2.21: Spring-mass-dashpot system; [from: Dr.Drang, 2014]

Response-spectrum analysis (RSA) is a linear-dynamic statistical analysis method, which measures the contribution from each natural mode of vibration to indicate the maximum seismic response of an elastic structure. Response-spectrum analysis provides insight into dynamic behaviour by measuring the maximum response of a series of simple systems with a range of periods from short to long and with various level of damping. Spectra values, that are the maxima, are then plotted against the natural

periods of the structure to produce the response spectrum shown in Figure 2.22.

Figure 2.22: Response spectrum

Since the structure can be idealised as a simple linear spring-mass-dashpot system, knowing mass, damping and structural period and given an acceleration response spectrum, it is possible to derive the following quantities:

a

FM S (2.39)

where F is the peak spring force, M is the mass and Sa is the pseudo spectral

acceleration; d

F

S

k



(2.40)

where S_d is the peak deflection, F is the peak spring force and

k

is the spring

constant. Because

T



2



M k

, combining equations (2.39) and (2.40), it is obtained:

2 2 4 d a T S S



 (2.41)

With an earthquake response spectrum, it is possible to determine two of the most quantities of most use to earthquake engineers, that is peak force and peak deflection in a given earthquake (Booth et al. 2006).

As previously said, it is possible to idealise simple structure as spring-mass-dashpot system shown in Figure 2.21, but almost all structures are more complex than these “single degree of freedom” systems (SDOF).

Generally, a SDOF model cannot describe the dynamic response of a structure because complex structures need to be studied by considering not only the fundamental

mode but also the higher natural modes of vibration. These modes are characteristics of the stiffness and the mass distribution of the structure (Booth et al. 2006). Natural mode shapes of a three-storeys building are represented in Figure 2.23.

Figure 2.23: Mode shapes of a three-storeys building; [from Lorant, FAIA 2012]

The combination of the different modal responses allows the calculation of a linear structure response, because each mode of vibration has a unique period and a unique mode shape. It is possible, for each mode, to define the entire deformation of the structure by using just one parameter, primarily top deflection.

The basic form of equations (2.39) and (2.40) must be modified for MDOF. The total mass in (2.39), for the base shear in each mode, must be replaced by the “effective” mass. Base shear factor in mode i is equal to (2.42):

2 i ai i L S M (2.42) where 2 i i L

M has the dimensions of mass. 2

i i

L M

 is equal to the total mass of the system. Codes allow considering a number of modes for which the 90% of the mass is captured. It is possible to apply similar factors to acceleration (2.43) and displacement (2.44) equations, which vary with height (Figure 2.24) (Booth et al. 2006).

Acceleration at level x in mode i i ( )

ai i i L S  _M  x   _{  }     (2.43)

Displacement at level x in mode i ( )



2 2



4 i ai i i L _T S x M       _  _     (2.44) ai

S is the spectral acceleration for ith mode period, _i( )x is the modal deflection at height x, L and _i M , for distributed two-dimensional, are the properties defined in _i





0 ( ) ( ) H i i L 



m x  x dx (2.45)





2 0 ( ) ( ) H i i M 



m x  x dx (2.46)

where m(x) is the mass per unit length at the height x and _i( )x is the modal deflection at height x in mode i.

Figure 2.24: Mode shape of a distributed system

The results of these equations give the maximum response of the structure for each mode of vibration. The total response, i.e. the maximum modal response, is got by adding the response in each mode at any time. A simple addiction, however, gives an overestimate value of the maximum. A good estimation of the real maximum is given by SRSS (square root of the sum of the squares) or CQC (complete quadratic combination) combination of modal responses. The main factor that contributes to the base shear, for these combination methods, is the fundamental mode. For this reason, “a building can be treated as an SDOF system corresponding to the fundamental mode. However, where the first mode is well out of resonance with the earthquake motion but the second and the third mode periods are close to resonance, shears and deflection at

higher levels are likely to be strongly influenced by higher modes”(Booth et al. 2006),

Figure 2.25: Modal contributions to shear force in a typical frame building

Response spectrum analysis considers also the torsional response (Figure 2.26) that occurs when the centre of mass and stiffness of the structure do not coincide.

Figure 2.26: Torsional response of a structure; [from Tabatabaei, 2011]

Structures with significant torsional eccentricity cannot be analysed with two-dimensional models because their behaviour is three-two-dimensional. Therefore, response can be underestimated with static analysis, by using forces applied at the centre of mass. At the same time, response can be underestimated also with linear dynamic analysis, because the less stiff side becomes flexible and so eccentricity increases. Seismic codes address this problem in different ways. Some codes require more sophisticate analyses if eccentricity exceeds prescribed limits, as Japanese code that require non-linear dynamic analysis, and EC8 and US codes, which require a three-dimensional modal response spectrum analysis (Booth et al. 2006). Generally, seismic codes allow the use of an “accidental eccentricity”, considered by moving the point of application of lateral loads of 5-10% of the size of the building from the centre of mass.

2.4.2. LTHA analysis

It is likely that response spectrum represents the basis of the design of earthquake-resistant buildings. The problem is that response spectrum does not give any information about the duration of the motion. This problem could be relevant when soil or structural properties changes appear with the time, under repeated cyclic loading. Time history analysis represents the solution to this issue (Booth et al. 2006).

Linear Time History Analysis (LTHA) is an alternative linear methodology for seismic design according to Eurocode8. This methodology allows the comparison of linear and nonlinear response with the same set of accelerograms. This analysis, applied to a linear model of the structures, is based on accelerograms input selection.

“Time-history analysis is a step-by-step procedure where the loading and the response histories are evaluated at successive time increments, t steps” (Costa 2003). The response depends, in each time step, from the loading history during the step and the conditions at the beginning of the step, which are displacements and velocities. Time step should be sufficiently short to allow the extrapolation of results from one calculation time to the next. In case of linear time history analysis, “the time step should not exceed a quarter of the period of the highest structural mode of interest” (Booth et al. 2006).

LTHA considers the interaction of the modes of vibration with the typical frequencies of an earthquake defined by accelerometric waveforms. The following differential equation represents the seismic response of a linear n-degree of freedom structure, characterized by mass

 

M

, damping

 

C

and stiffness

 

K

matrices.

           

M u t( )  C u t( )  K u t( )   M 

 

u t_g( ) (2.47) where

 

u t

( )

is the relative acceleration vector,

 

u t

( )

is the relative velocity vector,

 

u t

( )

is the relative displacement vector,

 



is the influence coefficient vector and

 

u t_g( ) is the earthquake-induced ground motion acceleration. In this case, the structural response is a function of the time, accounting for the whole duration of the earthquake.

In order to derive design time histories for a specific site, it is necessary to select the records of artificial, simulated or real seismic motion. Seismic codes specify that at least three records are necessary for the LTHA (Booth at al. 2006).

time histories for which the average spectrum envelopes a specified design spectrum” (Booth at al. 2006). There is a package of software, which allows producing a set of three or four independent records, as required by seismic codes. The problem is that these records are different from those of real earthquakes; in particular, they have many damaging cycles. Therefore, artificial records are conservative but this conservatism may be excessive and so the response may be not realistic.

For low-seismicity countries, where there is a very small number of ground motion records, it is useful to use simulated seismic motions. Thanks to sensitive instruments, it is possible to record the accelerations of many small earthquakes in these sites. “Since the number of earthquakes occurring increases exponentially with decreasing magnitude, a sufficient number of records can be acquired over a period of a few years” (Booth at al. 2006). The combination of a large number of these records, using a “Green’s function” technique, simulates much larger earthquakes at the chosen sites.

The most direct and useful way is to select real seismic motions from most common ground motion databases. These records should have a magnitude and a distance appropriate to the seismic hazard of the place. Real accelerograms are preferable compared to others, thanks to the real frequency content and the correct time correlation between the components and the realistic energetic content referred to seismological parameters. In order to consider the mean results of the analysis, each set of accelerograms should contain at least seven couples of records.

Indeed, it is not possible to use the same accelerograms for both of horizontal directions. For this reason, it is necessary to swap the pairs of horizontal records in the two horizontal directions of the structure: the real number of analyses is, therefore, fourteen. Analyses do not consider the vertical component of the seismic action.

“Time histories are measured by strong motion accelerographs set into action by the earthquake itself when the ground acceleration exceeds a preset threshold” (Booth et al.2006).

Figure 2.27: Time-history plots for a record from the Nepal earthquake on 25/04/2015: the upper plot shows acceleration, middle shows velocity and bottom shows displacement of the ground;[from PEER,

2015]

Figure 2.27 shows time-history plots from the Nepal earthquake on April 25, 2015, recorded by PEER (Pacific Earthquake Engineering Research Center). The first plot shows horizontal acceleration, the middle plot shows velocity, and the bottom one shows displacement of the ground. The first graph is a record from a strong-motion accelerograph, while the others are simply integrations of the first one. Time-history plots provide specific information about one precise earthquake. “Important parameters associated with time-based records are the peak values and the duration of strong motion” (Booth et al. 2006).

2.4.3. Difference between RSA and LTHA

Response spectrum analysis (RSA) has many advantages compared with the more complex time-history analysis, but also some limitations.

Response spectrum analysis is a linear-dynamic analysis method, which measures the contribution from each natural mode of vibration to indicate the likely maximum seismic response of an essentially elastic structure. This analysis provides insight into dynamic behavior by measuring pseudo-spectral acceleration, velocity or displacement as a function of structural period for a given time history and level of damping. Response-spectrum analysis is useful for design decision making because it relates structural type-selection to dynamic performance. Response spectra are simply plots of

the peak response (displacement, velocity or acceleration) of a series of oscillators of varying natural frequency that are forced into motion by the same base vibration or shock. The science of strong ground motion may use some values from the ground response spectrum (calculated from recordings of surface ground motion from seismographs) for correlation with seismic damage. Modal analysis is performed to identify the modes, and the response in that mode can be picked from the response spectrum. These peak responses are then combined to estimate a total response. The result is typically different from that which would be calculated directly from an input, since phase information is lost in the process of generating the response spectrum.

A full time history will give the response of a structure over time during and after the application of a load. To find the full time history of a structure's response, you must solve the structure's equation of motion. In time history analyses the structural response is computed at a number of subsequent time instants. In other words, time histories of the structural response to a given input are obtained as a result. In response spectrum analyses, the time evolution of response cannot be computed. Only the maximum response is estimated. No information is available also about the time when the maximum response occurs. Time history analysis is detailed analysis in which response is calculated for each time step. It requires more time but gives good results.

Definitely, RSA reduces the size of the problem “to finding only the maximum response of a limited number of modes of the structure, rather than calculating the entire time history of responses during the earthquake. This makes the problem much more tractable in terms both of processing time and size of computer output” (Booth et al. 2006). Response spectrum analysis is, in this way, a very useful method to obtain a preliminary analysis of the structural behavior.

SEISMIC DESIGN OF CBF STRUCTURES

3.1 Seismic Design Criteria

3.1.1. Capacity design

It is economically impractical to design structures to resist such severe but rare earthquakes without damage. Therefore, building codes have adopted a design philosophy intended to provide safety by avoiding earthquake-induced collapse in severe events, while permitting extensive structural and non-structural damage. Seismic design codes allow the realization of ductile structures able to dissipate the seismic energy stored during the earthquake through cyclic plastic deformations located in dissipative zones. The design of these dissipative zones or plastic hinges, which are located in selected points of the elements, is possible with the use of the methodology called capacity design.

A structure has a ductile behaviour when the yield capacity is reached first in ductile response modes rather than brittle modes. A ductile structure is able to preserve its stability after repeated cyclical deflections that are greater than its yield deflection. The extreme earthquake resistance of ductile structures is due to the reached plastic deformations, which absorb kinetic energy induced by the ground motion. Capacity design aim is “to provide sufficient strength to minimise damage in an earthquake with a high probability of occurrence, but to accept that the structure may yield in a low-probability event with the accompanying risk of damage, while ensuring that the post-yield response is ductile rather than brittle” (Booth et al.2006). When an extreme earthquake, the structure is subjected to considerable structural damage and it may not be repaired. “Given the huge uncertainties both in predicting earthquake motions and calculating response, the provision of ductility is the best insurance policy against

destruction of human lives” (Booth et al. 2006). The basis of the capacity design is the overstrength concept: the brittle elements must withstand the forces induced by yielding of ductile elements, so that the brittle members may not reach their failure loads. Therefore, strength required in brittle elements is based on the strength demand in the ductile elements. This strength value may exceed the minimum one described in seismic code. Seismic analyses allow checking if structure follows hierarchy of strength.

The capacity of a structure to dissipate energy is taken into account through a reduction factor, known as behaviour factor q. The capacity of structures to resist seismic forces in the non-linear range permits their design by using seismic forces smaller than those corresponding to a linear elastic response. Therefore, the behaviour factor reduces the response spectrum in an elastic one, called design spectrum. The behaviour factor q approximates the ratio of the seismic forces that the structure would experience if its response were completely elastic with 5% viscous damping, to the seismic forces that may be used in the design, with a conventional elastic analysis model, still ensuring a satisfactory response of the structure (EC8). The values of q, given in Eurocode 8, may be different in the two different horizontal directions of the structure. For regular structures, the behaviour factor should be taken with upper limits to the reference values given in Table 6.2 of EC8 and reported in Table 3.1.

STRUCTURAL TYPE Ductility Class DCM DCH

a) Moment resisting frames 4 5αu/α1

b) Frame with concentric bracings Diagonal bracings

V-bracings

4 4

2 2.5

c) Frame with eccentric bracings 4 5αu/α1

d) Inverted pendulum 2 2αu/α1

e) Structures with concrete cores or concrete walls See section 5 f) Moment resisting frame with concentric bracing 4 4αu/α1

g) Moment resisting frames with infills

Unconnected concrete or masonry infills, in contact with the frame

Connected reinforced concrete infills

Infills isolated from moment frame (see moment frames)

2 2

See section 7 4 5αu/α1

Table 3.1: Upper limit of reference values of behaviour factors for systems regular in elevation

For each structural type, there are two values, one for each ductility class DCM (medium ductility) and DCH (high ductility) of the structures. The two classes are different according to their hysteretic dissipation capacity. If the building is non-regular