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Earthquake Simulation for Buildings

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Any opinions, findings, conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of UK Essays.

Published: Mon, 26 Feb 2018

Abstract

Earthquake is an independent natural phenomenon of vibration of the ground which can become dangerous mainly when it is considered in relation with structures. Earthquakes can be very weak, without even realizing them but (they) can also be strong enough to result serious damages to buildings which can lead to injures or even loss of human lives. In order to avoid any structural damage the legislation sets conditions on the building design. For that purpose, Eurocode 8 is established in European countries and sets up all the appropriate criteria and measures for the design of buildings for earthquake resistance (Eurocode 8 is established in Europe and suggests 4 different methods of analysis.) In this project the response of eight buildings is examined (investigated) under seismic excitation. Firstly, is examined the case of four buildings (1 storey, 2 storey, 3 storey and 4 storey) where all the storeys are facsimile (replica). Afterwards, is examined the case of four buildings (again 1-4 storeys) where while the storeys of each building are increased, the mass, the stiffness and the height of each floor are decreased. Both the lateral method of analysis and the modal response spectrum analysis are used as recommended by EC8 to calculate the inter-storey drifts, the total shear forces and the overturning moments at the base of each building. The results are plotted and compared so that useful outcomes can be obtained.

1. Introduction

One of the most frightening and destructive phenomena of nature is a severe earthquake and its terrible aftereffects especially when they are associated with structures. An earthquake is a sudden movement of the Earth, caused by the abrupt release of strain that has accumulated over a long time. Earthquake intensity and magnitude are the most common used parameters in order to understand and compare different earthquake events.( Ή are the most common parameters used to appreciate and compare.)

In recent years have been giving increasing attention to the design of buildings for earthquake resistance.

Specific (particular) legislation is (have been) established to make structures able to resist at any seismic excitation. In Europe, Eurocode 8 explains how to make buildings able to resist to earthquakes, and recommends the use of linear and non-linear methods for the seismic design of the buildings

Simple structures can be modelled either as equivalent single degree of freedom systems (SDOF) or as a combination of SDOF systems.

In this project 8 different buildings with a variation either on the number of storeys or on their characteristics are simulated as a combination of SDOF systems for which the mode shapes and their corresponding eigenfrequencies and periods are calculated. Afterwards the fundamental frequency is obtained for each case and the elastic design is used in order to obtain the base shear forces and the overturning moments. (INELASTIC DESIGN AND LATERAL FORCE METHOD)

2. Literature review

2.1 Introduction to earthquake engineering

Definition and earthquake derivation or generation or creation or production or formation or genesis

The lithosphere is the solid part of Earth which includes or consists of the crust and the uppermost mantle. The sudden movement of the earth’s lithosphere is called earthquake (technical name seism).

Fractures in Earth’s crust where sections of rock have slipped past each other are called Faults. Most earthquakes occur along Faults. Generally, earthquakes are caused by the sudden release of built-up stress within rocks along geologic faults or by the movement of magma in volcanic areas.

The theory of plate tectonics provides geology with a comprehensive theory that explains “how the Earth works.” The theory states that Earth’s outermost layer, the lithosphere, is broken into 7 large, rigid pieces called plates: the African, North American, South American, Australian- Indian, Eurasian, Antarctic, and Pacific plates. Several subcontinental plates also exist, including the Caribbean, Arabian, Nazca, Philippines and Cocos plates.

Boundaries of tectonic plates are found at the edge of the lithospheric plates and can be of various forms, depending on the nature of relative movements. By their distinct motions, three main types can be characterized. The three types are: subduction zones (or trenches), spreading ridges (or spreading rifts) and transform faults.. convergent, divergent and conservative.

At subduction zone boundaries, plates move towards each other and the one plate subducts underneath the other (ή μπορÏŽ να πω: one plate is overriding another, thereby forcing the other into the mantle beneath it.)

The opposite form of movement takes place at spreading ridge boundaries. At these boundaries, two plates move away from one another. As the two move apart, molten rock is allowed to rise from the mantle to the surface and cool down to form part of the plates. This, in turn, causes the growth of oceanic crust on either side of the vents. As the plates continue to move, and more crust is formed, the ocean basin expands and a ridge system is created. Divergent boundaries are responsible in part for driving the motion of the plates.

At transform fault boundaries, plate material is neither created nor destroyed at these boundaries, but rather plates slide past each other. Transform faults are mainly associated with spreading ridges, as they are usually formed by surface movement due to perpendicular spreading ridges on either side.

Earthquake Location

When an earthquake occurs, one of the first questions is “where was it?”. An earthquake’s location may tell us what fault it was on and where the possible damage most likely occurred. The hypocentre of an earthquake is its location in three dimensions: latitude, longitude, and depth. The hypocentre (literally meaning: ‘below the center’ from the Greek υπÏŒκεντρον), or focus of the earthquake, refers to the point at which the rupture initiates and the first seismic wave is released.

As an earthquake is triggered, the fault is associated with a large area of fault plane.

The point directly above the focus, on the earth’s surface where the origin of an earthquake above ground.

The epicentre is the place on the surface of the earth under which an earthquake rupture originates, often given in degrees of latitude (north-south) and longitude (east-west). The epicentre is vertically above the hypocentre. The distance between the two points is the focal depth. The location of any station or observation can be described relative to the origin of the earthquake in terms of the epicentral or hypocentral distances.

Propagation of seismic waves

Seismic waves are the energy generated by a sudden breaking of rock within the earth or an artificial explosion that travels through the earth and is recorded on seismographs. There are several different kinds of seismic waves, and they all move in different ways. The two most important types of seismic waves are body waves and surface waves. Body waves travel deep within the earth and surface waves travel near the surface of the earth.

Body waves:

There are two types of body waves: P-waves (also pressure waves) and S-waves (also shear waves).

P-waves travel through the Earth as longitudinal waves whose compressions and rarefactions resemble those of a sound wave. The name P-wave comes from the fact that this is the fastest kind of seismic wave and, consequently, it is the first or ‘Primary’ wave to be detected at a seismograph. Speed depends on the kind of rock and its depth; usually they travel at speeds between 1.5 and 8 kilometers per second in the Earth’s crust. P waves are also known as compressional waves, because of the pushing and pulling they do. P waves shake the ground in the direction they are propagating, while S waves shake perpendicularly or transverse to the direction of propagation. The P-wave can move through solids, liquids or gases. Sometimes animals can hear the P-waves of an earthquake

S-waves travel more slowly, usually at 60% to 70% of the speed of P waves. The name S-wave comes from the fact that these slower waves arrive ‘Secondary’ after the P wave at any observation point. S-waves are transverse waves or shear waves, so that particles move in a direction perpendicular to that of wave propagation. Depending in whether this direction is along a vertical or horizontal plane, S-waves are subcategorized into SV and SH-waves, respectively. Because liquids and gases have no resistance to shear and cannot sustain a shear wave, S-waves travel only through solids materials. The Earth’s outer core is believed to be liquid because S-waves disappear at the mantle-core boundary, while P-waves do not.

(3: http://www.globalchange.umich.edu/globalchange1/current/lectures/nat_hazards/nat_hazards.html)

Surface waves:

The surface waves expand, as the name indicates, near the earth’s surface. The amplitudes of surface waves approximately decrease exponentially with depth. Motion in surface waves is usually larger than in body waves therefore surface waves tend to cause more damage. They are the slowest and by far the most destructive of seismic waves, especially at distances far from the epicenter. Surface waves are divided into Rayleigh waves and Love waves.

Rayleigh waves, also known as “ground roll”, are the result of an incident P and SV plane waves interacting at the free surface and traveling parallel to that surface. Rayleigh waves (or R-waves) took their name from (named for) John Strutt, Lord Rayleigh who first described them in 1885 (ή who mathematically predicted the existence of this kind of wave in 1885) and they are an important kind of surface wave. Most of the shaking felt from an earthquake is due to the R-wave, which can be much larger than the other waves. In Rayleigh waves the particles of soil move vertically in circular or elliptical paths, just like a wave rolls across a lake or an ocean. As Rayleigh wave particle motion is only found in the vertical plane, this means that they most commonly found on the vertical component of seismograms.

The Rayleigh equation is:

Love waves (also named Q waves) are surface seismic waves that cause horizontal shifting of the earth during an earthquake. They move the ground from side to side in a horizontal plane but at right angles to the direction of propagation. Love waves took their name from A.E.H. Love, a British mathematician who worked out the mathematical model for this kind of wave in 1911. Love waves are the result from the interaction with SH-waves. They travel with a slower velocity than P- or S- waves, but faster than Rayleigh waves, their speed relate to the frequency of oscillation.

Earthquake size:

Earthquake measurement is not a simple problem and it is hampered by many factors. The size of an earthquake can be quantified in various ways. The intensity and the magnitude of an earthquake are terms that were developed in an attempt to evaluate the earthquake phenomenon and they are the most commonly used terms to express the severity of an earthquake.

Earthquake intensity:

Intensity is based on the observed effects of ground shaking on people, buildings, and natural features. It varies from place to place within the disturbed region depending on the location of the observer with respect to the earthquake epicenter.

Earthquake magnitude:

The magnitude is the most often cited measure of an earthquake’s size.

The most common method of describing the size of an earthquake is the Richter magnitude scale, ML. This scale is based on the observation that, if the logarithm of the maximum displacement amplitudes which were recorded by seismographs located at various distances from the epicenter are put on the same diagram and this is repeated for several earthquakes with the same epicentre, the resulting curves are parallel to each other.

This means that if one of these earthquakes is taken as the basis, the coordinate difference between that earthquake and every other earthquake, measures the magnitude of the earthquake at the epicentre. Richter defined as zero magnitude earthquake one which is recorded with 1μm amplitude at a distance of 100 km. Therefore, the local magnitude ML of an earthquake is based on the maximum trace amplitude A and can be estimated from the relation:

ML= log A – log A’ (3)

Where A’ is the amplitude of the zero magnitude earthquake (ML=0).

The Richter magnitude scale can only be used when seismographs are within 600 km of the earthquake. For greater distances, other magnitude scales have been defined. The most current scale is the moment magnitude scale MW, which can be used for a wide range of magnitudes and distances.

Two main categories of instruments are used for the quantitative evaluation (estimation, assessment) of the earthquake phenomenon: the seismographs which record the displacement of the ground as a function of time, and the accelerographs (or accelerometers) which record the acceleration of the ground as a function of time, producing accelerograms. X the accelerogram of the 1940 El Centro earthquake.

For every earthquake accelerogram, elastic or linear acceleration response spectrum diagrams can be calculated. (obtained, estimated) The response spectrum of an earthquake is a diagram of the peak values of any of the response parameters (displacement, acceleration or velocity) as a function of the natural vibration period T of the SDOF system, subjected to the same seismic input. All these parameters can be plotted together in one diagram which is called the tripartite plot (also known as “four coordinate paper”).

2.2 Earthquake and Structures simulation

2.2.1 Equation of motion of SDOF system

Introduction

Vibration is the periodic motion or the oscillation of an elastic body or a medium, whose state of equilibrium has been disturbed. Η μπορω να πω: whose position of equilibrium has been displaced. There are two types of vibrations, free vibration and forced vibration. Vibration can be classified as either free or forced. A structure is said to be in a state of free vibration when it is disturbed from its static equilibrium by given a small displacement or deformation and then released and allowed to vibrate without any external dynamic excitation.

Number of Degrees of Freedom (DOF) is the number of the displacements that are needed to define the displaced position of the masses relative to their original position. Simple structures can be idealised as a system with a lumped mass m supported by a massless structure with stiffness k. It is assumed that the energy is dissipated through a viscous damper with damping coefficient c. Only one displacement variable is required in order to specify the position of the mass in this system, so it is called Singe Degree of Freedom (SDOF) system.

Undamped Free Vibration of SDOF systems

Furthermore, if there is no damping or resistance in the system, there will be no reduction to the amplitude of the oscillation and theoretically the system will vibrate forever. Such a system is called undamped and is represented in the below:

By taking into consideration the inertia force fin and the elastic spring force fs the equation of the motion is given by:

fin + fs = 0 → m+ ku = 0

Considering the initial conditions u(0) and (0), where u(0) is the displacement and (0) is the velocity at the time zero, the equation (4) has the general solution:

u(t) = u(0) cosωnt + sinωnt

where ωn is the natural frequency of the system and is given by,

ωn = (6)

The natural period and the natural frequency can be defined by the above equations:

Tn = (7) fn = (8)

Viscously damped Free Vibration of SDOF systems

The equation of motion of such a system can be developed from its free body diagram below:

Considering the inertia force fin, the elastic spring force fs and the damping force fD, the equation of the motion is given by:

m+ c+ ku = 0 (9)

Dividing by m the above equation gives:

+ 2ξωn+ ω2u = 0 (10)

where ξ is the critical damping and is given by:

ξ = (11)

and Cc is the critical damping ratio given by:

Cc = 2mωn

* If ξ > 1 or c > Cc the system is overdamped. It returns to its equilibrium position without oscillating.

* If ξ = 1 or c = Cc the system is critically damped. It returns to its equilibrium position without oscillating, but at a slower rate.

* If ξ < 1 or c < Cc the system is underdamped. The system oscillates about its equilibrium position with continuously decreasing amplitude.

Taking into account that all the structures can be considered as underdamped systems, as typically their damping ratio ξ is less than 0.10 the equation (9) for the initial conditions u (0) and (0) gives the solution below:

U (t) = e……………[u(0)cosωn+[….+sinωDt] (13)

where ωD is the natural frequency of damped vibration and is given by:

ωD = ωn (14)

Hence the natural period is:

TD = (15)

Undamped Forced Vibration of SDOF system

The equation of motion of such a system can be developed from its free body diagram below:

Considering the inertia force fin, the elastic spring force fs and the external dynamic load f(t), the equation of the motion is given by:

m+ ku = f(t) (16)

where f(t) = f0 sinωt is the maximum value of the force with frequency ω

By imposing the initial conditions u(0) and (0) the equation (16) has a general solution:

u(t) = u(0)cosωnt + sinωnt + sinωt (17)

Damped Forced Vibration of SDOF system

The equation of motion of such a system can be developed from its free body diagram below:

Considering the inertia force fin, the elastic spring force fs, the damping force fD and the external dynamic load f(t), the equation of the motion is given by:

m+ c+ ku = f(t) (18)

where f(t) = f0 sinωt

The particular solution of equation (18) is:

up = Csinωt + Dcosωt (19)

And the complementary solution of equation (18) is:

 

(20)

uc = e…(AcosωDt + Bsinωnt) (20)

2.2.2 Equation of motion of MDOF system

The equation of motion of a MDOF elastic system is expressed by:

M+ C+ Ku = -MAI(t) (21)

where M is the mass matrix, C is the damping matrix, K is the stiffness matrix, u” is the acceleration vector, u’ is the velocity vector and u is the displacement vector. Finally, AI is a vector with all the elements equal to unity and u”g(t) is the ground acceleration.

2.2 Earthquake and Structures simulation

2.2.1 Equation of motion of SDOF system

Introduction

Vibration is the periodic motion or the oscillation of an elastic body or a medium, whose state of equilibrium has been disturbed. Η μπορω να πω: whose position of equilibrium has been displaced. There are two types of vibrations, free vibration and forced vibration. Vibration can be classified as either free or forced. A structure is said to be in a state of free vibration when it is disturbed from its static equilibrium by given a small displacement or deformation and then released and allowed to vibrate without any external dynamic excitation.

Number of Degrees of Freedom (DOF) is the number of the displacements that are needed to define the displaced position of the masses relative to their original position. Simple structures can be idealised as a system with a lumped mass m supported by a massless structure with stiffness k. It is assumed that the energy is dissipated through a viscous damper with damping coefficient c. Only one displacement variable is required in order to specify the position of the mass in this system, so it is called Singe Degree of Freedom (SDOF) system.

Undamped Free Vibration of SDOF systems

Furthermore, if there is no damping or resistance in the system, there will be no reduction to the amplitude of the oscillation and theoretically the system will vibrate forever. Such a system is called undamped and is represented in the below:

By taking into consideration the inertia force fin and the elastic spring force fs the equation of the motion is given by:

fin + fs = 0 → m+ ku = 0

Considering the initial conditions u(0) and (0), where u(0) is the displacement and (0) is the velocity at the time zero, the equation (4) has the general solution:

u(t) = u(0) cosωnt + sinωnt

where ωn is the natural frequency of the system and is given by,

ωn = (6)

The natural period and the natural frequency can be defined by the above equations:

Tn = (7) fn = (8)

Viscously damped Free Vibration of SDOF systems

The equation of motion of such a system can be developed from its free body diagram below:

Considering the inertia force fin, the elastic spring force fs and the damping force fD, the equation of the motion is given by:

m+ c+ ku = 0 (9)

Dividing by m the above equation gives:

+ 2ξωn+ ω2u = 0 (10)

where ξ is the critical damping and is given by:

ξ = (11)

and Cc is the critical damping ratio given by:

Cc = 2mωn

* If ξ > 1 or c > Cc the system is overdamped. It returns to its equilibrium position without oscillating.

* If ξ = 1 or c = Cc the system is critically damped. It returns to its equilibrium position without oscillating, but at a slower rate.

* If ξ < 1 or c < Cc the system is underdamped. The system oscillates about its equilibrium position with continuously decreasing amplitude.

Taking into account that all the structures can be considered as underdamped systems, as typically their damping ratio ξ is less than 0.10 the equation (9) for the initial conditions u (0) and (0) gives the solution below:

U (t) = e……………[u(0)cosωn+[….+sinωDt] (13)

where ωD is the natural frequency of damped vibration and is given by:

ωD = ωn (14)

Hence the natural period is:

TD = (15)

Undamped Forced Vibration of SDOF system

The equation of motion of such a system can be developed from its free body diagram below:

Considering the inertia force fin, the elastic spring force fs and the external dynamic load f(t), the equation of the motion is given by:

m+ ku = f(t) (16)

where f(t) = f0 sinωt is the maximum value of the force with frequency ω

By imposing the initial conditions u(0) and (0) the equation (16) has a general solution:

u(t) = u(0)cosωnt + sinωnt + sinωt (17)

Damped Forced Vibration of SDOF system

The equation of motion of such a system can be developed from its free body diagram below:

Considering the inertia force fin, the elastic spring force fs, the damping force fD and the external dynamic load f(t), the equation of the motion is given by:

m+ c+ ku = f(t) (18)

where f(t) = f0 sinωt

The particular solution of equation (18) is:

up = Csinωt + Dcosωt (19)

And the complementary solution of equation (18) is:

uc = (AcosωDt + Bsinωnt) (20)

2.2.2 Equation of motion of MDOF system

The equation of motion of a MDOF elastic system is expressed by:

M+ C+ Ku = -MAI(t) (21)

where M is the mass matrix, C is the damping matrix, K is the stiffness matrix, u” is the acceleration vector, u’ is the velocity vector and u is the displacement vector. Finally, AI is a vector with all the elements equal to unity and g(t) is the ground acceleration.

3. Description of the Method

3.1 Simplified Multi-Storey Shear Building Model

It is almost impossible to predict precisely which seismic action a structure will undergo during its life time. Each structure must be designed to resist at any seismic excitation without failing. For this reason each structure is designed to meet the requirements of the design spectrum analysis based in EC8. Also some assumptions are necessary in order to achieve the best and the simplest idealization for each multi store building. Initially it is assumed that the mass of each floor is lumped at the centre of the floor and the columns are massless. The floor beams are completely rigid and incompressible; hence the floor displacement is being transferred equally to all the columns. The columns are flexible in horizontal displacement and rigid in vertical displacement, while they are provided with a fully fixed support from the floors and the ground. The building is assumed to be symmetric about both x and y directions with symmetric column arrangement. The consequence of this is that the centre of the mass of each floor to coincide with the centre of the stiffness of each floor. The position of this centre remains stable up the entire height of the building. Finally, it is assumed that there are no torsional effects for each of the floors.

If all the above assumptions are used the building structure is idealised as a model where the displacement at each floor is described by one degree of freedom. Thus, for a jth storey building, j degrees of freedom required to express the total displacement of the building.

The roof of the building has always to be considered as a floor.

The mass matrix M is a symmetric diagonal nxn matrix for a n-storey building and is given below. Each diagonal value in the matrix represents the total mass of one beam and its two corresponding columns which are assumed to be lumped at each level.

M =

Stiffness method is used to formulate the stiffness matrix. K is the lateral stiffness of each column and is given by the relationship:

K = (22)

where EI is the flexural stiffness of a column.

The lateral stiffness of each column is clamped at the ends and is imposed in a unit sway. The stiffness of each floor is the sum of the lateral force of all columns in the floor. The stiffness matrix is for a n-storey building is:

 

K =

In order to calculate the natural modes of the vibration, the system is assumed that vibrates freely. Thus, g(t)=0, which for systems without damping (c=0) the equation (21) specializes to:

M+ Ku = 0 (23)

The displacement is assumed to be harmonic in time, this is:

= -ω2Ueiωt (24)

Hence equation (23) becomes:

(K – ω2M)U = 0 (25)

The above equation has the trivial solution u=0. For non trivial solutions, u≠0 the determinant for the left hand size must be zero. That is:

|K – ω2 M| = 0 (26)

This condition leads to a polynomial in terms of ω2 with n roots, where n is the size of matrices and vectors as cited above. These roots are called eigenvalues.

By applying the equation (6) & (7), the natural frequency and the natural period of vibration for each mode shape can be determined.

Each eigenvalue has a relative eigenvector which represent the natural ith mode shape. After the estimation of the eigenvector in order to compare the mode shapes, scale factors are applied to natural modes to standarise their elements associated with various degrees of freedom (X). This process is called normalization. Hence, after the estimation of the eigenvectors each mode is normalised so that the biggest value is X: eigenvector notation. unity.

The eigenvectors of a symmetric matrix corresponding to distinct eigenvalues are orthogonal. This aspect is expressed by the following expression:

UiTKUij = UiTMUij (27)

The classical eigenvalue problem has the following form:

(M-1K – λ I) u = 0 (28)

where λ =ω2 and I is the identity matrix.

EC8 suggests that the response in two modes i and j can be assumed independent of each other when

Tj ≤ 0.9 Ti

where Ti and Tj are the periods of the modes i and j respectively (always Ti ≥ Tj). The calculated fundamental period can be checked by the equation that EC8 suggests:

T = Ct*H3/4

where T is the fundamental period of the building, Ct is a coefficient and H is the total height of the building; this expression is valid buildings that their total height is not more than forty metres

3.2 Elastic Analysis

The response method is used to estimate the maximum displacement (uj), pseudo- velocity (j) and acceleration (j) for each calculated natural frequency. It is assumed that the MDOF system oscillates in each of its modes independently and displacements, velocities and accelerations can be obtained for each mode separately considering modal responses as SDOF responses. Each maximum, displacement velocity and acceleration read from the design spectrum is multiplying by the participation factor αi to re-evaluate the maximum values expressed ujmax, jmax, jmax respectively. The participation factor αi is defined by the following equation:

(28)

where UijT is the transpose vector of each of the mode vectors, M is the mass matrix, AI is the unit vector and Uij is the mode shape vector.

The actual maximum displacements of the jth mode are given by:

u = ujmax·Uj

Afterwards, the root-mean-square (RMS) approximation is used in order to calculate the maximum displacement for each floor. In this approach, all the maximum values for each mode, are squared and summed and their square root is derived. If we let Dmax be the maximum displacement then:

Dmax = (29)

A very variable parameter to characterise the seismic behaviour of a building is the Inter-Storey Drift which can be obtained by the following equation:

δi = Di – Di-1/hi (30)

where Di, Di-1, are the horizontal displacements for two contiguous floors and hi is the corresponding height of the floor. The calculated values must be lower than 4% in order to agree with the Eurocode.

Afterwards the horizontal inertia forces Fj’s applied at each floor are obtained by applying the following equation:

Fj = M·Uj·jmax (31)

where M is the mass matrix, Uj is the eigenvector for each mode and jmax is the maximum acceleration.

As it is suggested from the EC8, the root-mean-approximation is used again in order to obtain the total lateral forces. EC8 suggests that the combined lateral force at each floor is given by the square root of the sum of the squares of each lateral force at each floor of all the modes. If we let Ftotal,i the maximum base shear force then:

Ftotal,j = [1] (32)

where Fij is the lateral force at floor i of the mode j.

Once the total lateral forces and the shear forces have been obtained, the maximum overturning moment is calculated.

3.3 Inelastic Analysis

The inelastic response spectra are generally obtained by the scaling of the elastic design spectra via the use of response modification factors. No effect of the energy absorption was assumed in the structure for the calculated values by using the elastic design spectrum. By introducing the ductility factor this parameter is taking into consideration.

Newmark has described the ductility parameter μ as the ratio of maximum displacement to the displacement at yield. Apparently when yielding does not take place the concept of ductility is not relevant and μ is taken equal to unity. Τhe system is described by the damping ratio ς, the natural frequency ωn, and the ductility factor μ.

In order to calculate the new set of values of acceleration, displacement and velocity the design response spectrum has to be constructed. Newmark’s procedure leads to the construction of two modified spectra.

1. For maximum acceleration:

In this case the elastic design spectrum is reduced by the appropriate coefficients. The acceleration region of the graph is multiplie


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