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 μπορώ 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 υπόκεντρον), 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 multiplied by the following factor:

(33)

While the displacement region is multiplied by:

(34)

X: Construction of the inelastic maximum acceleration design spectrum.

Where A'B' = [AB]

And C'D' = [CD]

2. For maximum displacement:

In this case the elastic design spectrum is increased by the appropriate coefficients.

The inelastic maximum displacement spectrum is constructed and is presented in X. As it is observed AB is the same as the elastic spectrum, while C'D' and E'F' are each μ times CD and EF on the acceleration scale.

Once the construction of both the above inelastic design spectra is completed, a new set of values of acceleration and displacement can be obtained. Each displacement and acceleration read from the spectrum is multiplying by the participation factor αi, in order to modify the calculated values. After the re-evaluation of the displacement and the accelerations the procedure is the same as in the elastic analysis. The participation factors remain stable for the inelastic analysis as the ductility factor does not affect them. The actual maximum accelerations and displacements of the jth mode can be obtained by applying the equation (X) and then by applying the RMS approximation. Herein, the inter-storey drift and the lateral forces FJ's applied to each floor can be obtained by using the equations (X) and (X) respectively. Once the total lateral forces and the shear forces have been obtained, the maximum overturning moment is calculated.

4 Results

In this project two different cases are examined:

1. Four buildings with a variation in the number of storeys and differentiation in their characteristics.

2. Four buildings from one until four storeys where all the levels are identical between them.

Case 1:

Firstly, a one storey building is examined and the elastic and inelastic responses are analysed. Afterwards one storey is added which means that two degrees of freedom are needed to describe the total displacement of the structure. The second floor of the building has redundant mass, height and stiffness. Afterwards, one more storey is added above the existing two storey building with even less mass, height and stiffness. The elastic and inelastic responses are then analysed for the three storey building. Finally, one more storey is added which is identical as the last one and the four storey building is analysed.

Ø One storey building

The dimensions of both the building and its elements are presented in the below.

X: (a) dimensions of the one storey building, (b) beam cross section (c) column cross section.

By applying the equation (22) stiffness K can be obtained and the calculated value is represented below:

K = 9.6*107

Afterwards, by applying the equation (26) the eigenvalue ω2 is obtained as. The natural frequency, which in this case is the fundamental frequency as well, is obtained by applying the equation (6) and the period by applying the equation (7). The calculated values are represented in the table below:

Eigenvalue ω2 (rad2/s2)

Frequency (Hz)

Period (s)

192

2.205

0.453

Table X: Eigenvalue, Frequency and Period for the one storey building.

Elastic Analysis

The maximum displacement, Pseudo-Velocity and Acceleration are obtained from the elastic design spectrum as:

Pseudo- Velocity (m/s)

Displacement (m)

Acceleration (g)

0.750

0.092

0.636

Table X: Maximum Displacement, Pseudo- Velocity and Acceleration for the one storey building.

The Inter- Storey Drift is obtained in percentages and it is δ=1.840 %. Afterwards, the maximum base shear force and the maximum overturning moment for the elastic analysis are represented in Table X.

Maximum Base Shear Force (N)

Maximum Overturning Moment (Nm)

3.120*106

2.560*107

Table X: The Maximum Base Shear Force and the Maximum Overturning Moment for the one storey building.

Inelastic Analysis

The maximum displacement and Acceleration are obtained from the inelastic design spectrum as:

Displacement (m)

Acceleration (g)

0.193

0.123

Table X: Maximum Displacement, Pseudo- Velocity and Acceleration for the one storey building.

The Inter- Storey Drift is obtained in percentages and it is δ=3.860 %. The maximum base shear force and the maximum overturning moment for the inelastic analysis are calculated and presented below:

Maximum Base Shear Force (N)

Maximum Overturning Moment (Nm)

6.033*106

1.560*107

Table X: The Maximum Base Shear Force and the Maximum Overturning Moment for the one storey building (Inelastic design)

Ø Two storey building

One floor is added above the existing one storey building. The height of the new floor, the mass and the X EI are reduced as it is shown in the below.

The mass matrix M for the above building is a symmetric, diagonal 2x2 matrix and is given below:

The stiffness matrix is derived applying equation (22) to the general form of stiffness matrix ( 15). For a 2 storey building, the stiffness matrix K is a symmetric 2x2 matrix:

By applying the equation x to x the stiffness matrix is obtained. to By applying the stiffness method to a 2 storey building, the stiffness matrix K which is a symmetric 2x2 matrix, becomes:

By using the equation (6) & (7), the natural frequency and the natural period of vibration for each mode shape can be determined. The calculated eigenvalues, and the related natural frequencies and periods are given in the table below:

Mode Shape

Eigenvalue ω2 (rad2/s2)

Frequency (Hz)

Period (s)

1

93.026

1.535

0.651

2

773.974

4.428

0.226

Table X: Eigenvalues, Frequencies and Periods for the 2 storey building.

The modes of the shape and their corresponding periods are shown below:

Using the method of normalisation the eigenvectors become:

The two different mode shapes for the 2 storey building are presented below graphically.

Mode Shape

Frequency (Hz)

Pseudo- Velocity (m/s)

Displacement (m)

Acceleration (g)

1

1.535

0.750

0.113

0.510

2

4.428

0.729

0.045

1.080

Table X: Maximum Displacement, Pseudo- Velocity and Acceleration for the 2 storey building.

The two calculated participation factors αi's are represented in the table below:

α1

α2

1.137

0.145

Table X: Participation Factors.

The maximum displacement, Pseudo-Velocity and Acceleration from Table X are multiplied by the respective participation factors from Table X. The scaled parameters of the motion are given in the table below.

Mode Shape

Frequency (Hz)

Pseudo- Velocity (m/s)

Displacement (m)

Acceleration (g)

1

1.535

0.852

0.128

0.589

2

4.428

0.106

6.541*10-3

1.158

Table X: Scaled parameters of the motion for each mode due to the participation factors.

By applying the root-mean-square (RMS) approximation (equation 29) the maximum displacement can be obtained for both of the floors:

D1 = 0.097 m for the first floor and D2 = 0.129 m for the second floor.

Afterwards the Inter- Storey Drift is obtained in percentages for both of the floors and it is δ1=1.936 % for the first one and δ2 = 0.795 % for the second. The above values are in agreement with the Eurocode as they are lower than 4%.

The Maximum Base Shear Force and the Maximum Overturning Moment are calculated by applying the root-mean-approximation and the results are presented in the above table:

Maximum Base Shear Force (N)

Maximum Overturning Moment (Nm)

4.468*106

3.168*107

Table X: The Maximum Base Shear Force and the Maximum Overturning Moment for the 2 storey building.

Inelastic Analysis

The inelastic maximum acceleration and the inelastic maximum displacement design spectra are constructed by using a ductility factor of 5 (μ=5). Hence the acceleration and the displacement are re-evaluated. The results are given in the table below.

Mode Shape

Frequency (Hz)

Displacement (m)

Acceleration (g)

1

1.535

0.198

0.100

2

4.428

0.100

0.172

Table X: Calculated displacements and accelerations for each mode (Inelastic design).

Afterwards, each displacement and acceleration s multiplied by the respective participation factor from Table (X) and the scaled parameters are given below:

Mode Shape

Frequency (Hz)

Displacement (m)

Acceleration (g)

1

1.535

0.225

0.114

2

4.428

0.015

0.025

Table X: Scaled parameters of the motion for each mode due to the participation factors

(Inelastic design).

Herein, the Inter-Storey Drift is obtained in percentages for both of the floors and it is:

δ1=3.397 % for the first one and δ2 = 1.390 % for the second. It is observed an increment at the above values comparing them with the corresponding values of the elastic analysis but they are still in agreement with the Eurocode as they are lower than 4%.

The Maximum Base Shear Force and the Maximum Overturning Moment are calculated and the results are presented in the table below:

Maximum Base Shear Force (N)

Maximum Overturning Moment (Nm)

8.657*105

6.114*106

Table X: The Maximum Base Shear Force and the Maximum Overturning Moment for the 2 storey building (Inelastic design).

As it is observed, there is a noticeable decrease of the values in Table (X) comparing them with the corresponding values from the elastic design in Table(x). As it is shown, the ductility factor μ reduces the total shear force and the total overturning moment of the building.

3. Three storey building

One floor is added above the existing two storey building. The height of the new floor, the mass and the stiffness are reduced. The below represents the dimensions for the three storey building and its elements.

The mass matrix M for a 3 storey building is a symmetric, diagonal 3x3 matrix and is given below:

The stiffness matrix is derived applying equation (22) to the general form of stiffness matrix ( 15). For a 3 storey building, the stiffness matrix K is a symmetric 3x3 matrix:

By using the equation (6) & (7), the natural frequency and the natural period of vibration for each mode shape can be determined. The calculated eigenvalues, and the corresponding natural frequencies and periods are given in the above table:

Mode Shape

Eigenvalue ω2 (rad2/s2)

Frequency (Hz)

Period (s)

1

70.673

1.338

0.747

2

625.839

3.982

0.251

3

2.17*103

7.414

0.135

Table X: Eigenvalues, Frequencies and Periods for the 3 storey building.

The modes of the shape and their corresponding periods are shown below:

Using the method of normalisation the eigenvectors become:

The different mode shapes for the 3 storey building are presented below graphically.

Mode Shape

Frequency (Hz)

Pseudo- Velocity (m/s)

Displacement (m)

Acceleration (g)

1

1.338

0.750

0.123

0.477

2

3.982

0.750

0.062

1.000

3

7.414

0.345

0.010

1.087

Table X: maximum Displacement, Pseudo- Velocity and Acceleration for the 3 storey building.

The three calculated participation factors αi's are represented in the table below:

α1

α2

α3

1.165

0.213

0.014

Table X: Participation Factors.

The maximum displacement, Pseudo-Velocity and Acceleration from Table X are multiplied by the respective participation factors from Table X. The scaled parameters of the motion are given in the table below.

Mode Shape

Frequency (Hz)

Pseudo- Velocity (m/s)

Displacement (m)

Acceleration (g)

1

1.338

0.874

0.143

0.556

2

3.982

0.160

0.013

0.213

3

7.414

4.705*10-3

1.364*10-4

0.015

Table X: Scaled parameters of the motion for each mode due to the participation factors.

Maximum displacement and the Inter Storey Drift for each floor are given below:

D1 = 0.098m, D2 = 0.136m and D3 = 0.144m

δ1 = 1.951 %, δ2 = 0.958 % and δ3 = 0.263 %

The above values of the Inter Storey Drift are in agreement with the Eurocode as they are lower than 4%.

The Maximum Base Shear Force and the Maximum Overturning Moment are calculating by applying the root-mean-approximation and the results are presented in the table below:

Maximum Base Shear Force (N)

Maximum Overturning Moment (Nm)

5.005*106

4.093*107

Table X: The Maximum Base Shear Force and the Maximum Overturning Moment for the 3 storey building.

Inelastic Analysis

The inelastic maximum acceleration and the inelastic maximum displacement design spectra are constructed by using a ductility factor of 5 (μ=5). Hence the acceleration and the displacement are re-evaluated. The results are given in the table below.

Mode Shape

Frequency (Hz)

Displacement (m)

Acceleration (g)

1

1.338

0.200

0.090

2

3.982

0.131

0.172

3

7.414

0.022

0.172

Table X: Calculated displacements and accelerations for each mode (Inelastic design).

Afterwards, each displacement and acceleration is multiplied by the respective participation factor from Table (X) and the scaled parameters are given below:

Mode Shape

Frequency (Hz)

Displacement (m)

Acceleration (g)

1

1.338

0.233

0.105

2

3.982

0.028

0.037

3

7.414

3*10-4

2.435*10-3

Table X: Scaled parameters of the motion for each mode due to the participation factors (Inelastic design).

Maximum displacement and the Inter Storey Drift for each floor are given below:

D1 = 0.160m, D2 = 0.221m and D3 = 0.234m

δ1 = 3.192 %, δ2 = 1.536 % and δ3 = 0.439 %

It is observed an increment at the values of the Inter-Storey Drift comparing them with the corresponding values of the elastic analysis but they are still in agreement with the Eurocode as they are lower than 4%.

The Maximum Base Shear Force and the Maximum Overturning Moment are calculated and the results are presented in the table below:

Maximum Base Shear Force (N)

Maximum Overturning Moment (Nm)

9.439*105

7.721*106

Table X: The Maximum Base Shear Force and the Maximum Overturning Moment for the 3 storey building (Inelastic design).

Ø Four storey building

One floor is added above the existing three storey building. The new floor has identical characteristics to the third floor. The below presents the dimensions for the four storey building and its elements.

The mass matrix M is given below:

The stiffness matrix K is:

By using the equation (6) & (7), the natural frequency and the natural period of vibration for each mode shapes can be determined. The calculated eigenvalues, and the corresponding natural frequencies and periods are given in the above table:

Mode Shape

Eigenvalue ω2 (rad2/s2)

Frequency (Hz)

Period (s)

1

55.426

1.185

0.844

2

497.489

3.55

0.282

3

1.241*103

5.607

0.178

4

3.739*103

9.732

0.103

Table X: Eigenvalues, Frequencies and Periods for the 4 storey building.

The modes of the shape and their corresponding periods are shown below:

Using the method of normalisation the eigenvectors become:

The different mode shapes for the 3 storey building are presented below graphically

Mode Shape

Frequency (Hz)

Pseudo- Velocity (m/s)

Displacement (m)

Acceleration (g)

1

1.185

0.750

0.123

0.477

2

3.55

0.750

0.069

0.919

3

5.607

0.519

0.026

1.087

4

9.732

0.119

0.002

0.636

Table X: maximum Displacement, Pseudo- Velocity and Acceleration for the 2 storey building.

The four calculated participation factors αi's are represented in the table below:

α1

α2

α3

α4

1.196

0.262

0.047

2.071*10-3

Table X: Participation Factors.

The maximum displacement, Pseudo-Velocity and Acceleration from Table X are multiplied by the respective participation factors from Table X. The scaled parameters of the motion are given in the table below.

Mode Shape

Frequency (Hz)

Pseudo- Velocity (m/s)

Displacement (m)

Acceleration (g)

1

1.185

0.897

0.248

0.102

2

3.55

0.197

0.041

0.045

3

5.607

0.024

2.760*10-3

8.047*10-3

4

9.732

2.464*10-3

1.242*10-5

2.360*10-3

Table X: Scaled parameters of the motion for each mode due to the participation factors.

Maximum displacement and the Inter Storey Drift for each floor are given below:

D1 = 0.09m, D2 = 0.129m, D3 = 0.14m and D4 = 0.148m

δ1 = 1.809 %, δ2 = 0.963 %, δ3 = 0.412 % and δ4 = 0.222 %.

The Maximum Base Shear Force and the Maximum Overturning Moment are calculating by applying the root-mean-approximation and the results are presented in the table below:

Maximum Base Shear Force (N)

Maximum Overturning Moment (Nm)

1.044*106

9.964*106

Table X: The Maximum Base Shear Force and the Maximum Overturning Moment for the 4 storey building.

Inelastic Analysis

The inelastic design spectrum is used with a ductility factor of 5 (μ=5). Hence the acceleration and the displacement are re-evaluated. The results are given in the table below.

Mode Shape

Frequency (Hz)

Displacement (m)

Acceleration (g)

1

1.185

0.207

0.085

2

3.55

0.156

0.172

3

5.607

0.059

0.172

4

9.732

0.006

0.114

Table X: Calculated displacements and accelerations for each mode (Inelastic design).

Afterwards, each displacement and acceleration is multiplied by the respective participation factor from Table (X) and the scaled parameters are given below:

Mode Shape

Frequency (Hz)

Displacement (m)

Acceleration (g)

1

1.185

0.248

0.102

2

3.55

0.041

0.045

3

5.607

2.760*10-3

8.047*10-3

4

9.732

1.242*10-5

2.360*10-4

Table X: Scaled parameters of the motion for each mode due to the participation factors (Inelastic design).

Maximum displacement and the Inter Storey Drift for each floor are given below:

D1 = 0.155m, D2 = 0.217m, D3 = 0.238m and D4= 0.250 m

δ1 = 3.093 %, δ2 = 1.561 %, δ3 = 0.710 % and δ4 = 0.399 %

The values of the Inter-Storey Drift are in agreement with the Eurocode as they are lower than 4%. The Maximum Base Shear Force and the Maximum Overturning Moment are calculated and the results are presented in the table below:

Maximum Base Shear Force (N)

Maximum Overturning Moment (Nm)

9.439*105

7.721*106

Table X: The Maximum Base Shear Force and the Maximum Overturning Moment for the 4 storey building (Inelastic design).

Then one floor is added above the existing building, which is a duplicate of the first one.

Case 2: Four buildings with identical characteristics for all the floors:

In this case the four buildings that will be examined they will have the exact same characteristics at all of the floors. The same procedure as in case one is followed in order to design four different building models able to resist at any seismic excitation. While the procedure is the same, only the final tables and the appropriate s for each case will be presented. The one storey building is the same as the one storey building of case one.

Ø Two storey building

The dimensions of both the building and its elements are presented in the below.

The mass matrix M is given below:

The stiffness matrix is given below:

The calculated eigenvalues, and the corresponding natural frequencies and periods are given in the above table:

Mode Shape

Eigenvalue ω2 (rad2/s2)

Frequency (Hz)

Period (s)

1

73.337

1.363

0.734

2

502.663

3.568

0.280

Table X: Eigenvalues, Frequencies and Periods for the 2 storey building.

The modes of the shape and their corresponding periods are shown below:

Using the method of normalisation the eigenvectors become:

T1= 0.734s T2= 0.280s

The two different mode shapes for the 2 storey building are presented below graphically.

Mode Shape

Frequency (Hz)

Pseudo- Velocity (m/s)

Displacement (m)

Acceleration (g)

1

1.363

0.750

0.118

0.497

2

3.568

0.750

0.067

0.921

Table X: Maximum Displacement, Pseudo- Velocity and Acceleration for the 2 storey building.

The two calculated participation factors αi's are represented in the table below:

α1

α2

1.171

0.276

Table X: Participation Factors.

The scaled parameters of the motion are given in the table below.

Mode Shape

Frequency (Hz)

Pseudo- Velocity (m/s)

Displacement (m)

Acceleration (g)

1

1.363

0.878

0.138

0.582

2

3.568

0.207

0.019

0.255

Table X: Scaled parameters of the motion for each mode due to the participation factors.

Maximum displacement and the Inter Storey Drift for each floor are given below:

D1 = 0.087 m and D2 = 0.139 m.

δ1=1.747 % and δ2 = 1.025 %.

The above values are in agreement with the Eurocode as they are lower than 4%.

The Maximum Base Shear Force and the Maximum Overturning Moment are given in the table below.

Maximum Base Shear Force (N)

Maximum Overturning Moment (Nm)

4.643*106

3.739*107

Table X: The Maximum Base Shear Force and the Maximum Overturning Moment for the 2 storey building.

Inelastic Analysis

The inelastic design spectrum is used with a ductility factor of 5 (μ=5). Hence the acceleration and the displacement are re-evaluated. The results are given in the table below:

Mode Shape

Frequency (Hz)

Displacement (m)

Acceleration (g)

1

1.363

0.199

0.086

2

3.568

0.151

0.172

Table X: Calculated displacements and accelerations for each mode (Inelastic design).

Afterwards, each displacement and acceleration is multiplied by the respective participation factor from Table (X) and the scaled parameters are given below:

Mode Shape

Frequency (Hz)

Displacement (m)

Acceleration (g)

1

1.363

0.233

0.101

2

3.568

0.042

0.048

Table X: Scaled parameters of the motion for each mode due to the participation factors (Inelastic design).

Maximum displacement and the Inter Storey Drift for each floor are given below:

D1 = 0.150m and D2 = 0.234m

δ1 = 2.998 % and δ2 = 1.690 %

The values of the Inter-Storey Drift are in agreement with the Eurocode as they are lower than 4%. The Maximum Base Shear Force and the Maximum Overturning Moment are calculated and the results are presented in the table below:

Maximum Base Shear Force (N)

Maximum Overturning Moment (Nm)

8.041*105

6.471*106

Table X: The Maximum Base Shear Force and the Maximum Overturning Moment for the 2 storey building (Inelastic design).

Ø Three storey building

g

The below represents the dimensions for the two storey building and its elements.

The mass matrix M is given below:

The stiffness matrix K is:

The calculated eigenvalues, and the corresponding natural frequencies and periods are given in the above table:

Mode Shape

Eigenvalue ω2 (rad2/s2)

Frequency (Hz)

Period (s)

1

38.028

0.981

1.019

2

298.552

2.750

0.364

3

623.420

3.974

0.252

Table X: Eigenvalues, Frequencies and Periods for the 3 storey building.

The modes of the shape and their corresponding periods are shown below:

T1= 1.019s T2= 0.364s T3= 0.252s

Using the method of normalisation the eigenvectors become:

T1= 1.019s T2= 0.364s T3= 0.252s

The three different mode shapes for the 3 storey building are presented below graphically

Mode Shape

Frequency (Hz)

Pseudo- Velocity (m/s)

Displacement (m)

Acceleration (g)

1

0.981

0.750

0.133

0.405

2

2.750

0.750

0.082

0.720

3

3.974

0.750

0.063

0.998

Table X: Maximum Displacement, Pseudo- Velocity and Acceleration for the 3 storey building.

The three calculated participation factors αi's are represented in the table below:

α1

α2

α3

1.220

0.349

0.134

Table X: Participation Factors.

The scaled parameters of the motion are given in the table below.

Mode Shape

Frequency (Hz)

Pseudo- Velocity (m/s)

Displacement (m)

Acceleration (g)

1

0.981

0.915

0.162

0.494

2

2.750

0.262

0.029

0.251

3

3.974

0.101

8.451*10-3

0.134

Table X: Scaled parameters of the motion for each mode due to the participation factors

Maxi\mum displacement and the Inter Storey Drift for each floor are given below:

D1 = 0.078 m, D2 = 0.131 m and D3 = 1.164 m.

δ1=1.560 %, δ2 = 1.061 % and δ3=0.658.

The above values are in agreement with the Eurocode as they are lower than 4%.

The Maximum Base Shear Force and the Maximum Overturning Moment are presented in the table below:

Maximum Base Shear Force (N)

Maximum Overturning Moment (Nm)

5.507*106

6.129*107

It is observed an increscent at the above values in table X comparing them with( se sxesi me) the corresponding calculated values in table X for the 2-storey building.

Inelastic Analysis

The inelastic maximum acceleration and the inelastic maximum displacement design spectra are constructed by using a ductility factor of 5 (μ=5). Hence the acceleration and the displacement are re-evaluated. The results are given in the table below.

Mode Shape

Frequency (Hz)

Displacement (m)

Acceleration (g)

1

0.981

0.207

0.072

2

2.750

0.189

0.152

3

3.974

0.134

0.172

Table X: Calculated displacements and accelerations for each mode (Inelastic design).

Afterwards, each displacement and acceleration is multiplied by the respective participation factor from Table (X) and the scaled parameters are given below:

Mode Shape

Frequency (Hz)

Displacement (m)

Acceleration (g)

1

1.338

0.253

0.088

2

3.982

0.066

0.053

3

7.414

0.018

0.023

Table X: Scaled parameters of the motion for each mode due to the participation factors (Inelastic design).

Maximum displacement and the Inter Storey Drift for each floor are given below:

D1 = 0.131m, D2 = 0.205m and D3 = 0.258m

δ1 = 2.623 %, δ2 = 1.486 % and δ3 = 1.055 %

It is observed an increment at the values of the Inter-Storey Drift comparing them with the corresponding values of the elastic analysis but they are still in agreement with the Eurocode as they are lower than 4%.

The Maximum Base Shear Force and the Maximum Overturning Moment are calculated and the results are presented in the table below:

Maximum Base Shear Force (N)

Maximum Overturning Moment (Nm)

9.832*105

1.090*107

Table X: The Maximum Base Shear Force and the Maximum Overturning Moment for the 3 storey building (Inelastic design).

4. Four storey building

The below represents the dimensions for the two storey building and its elements.

The mass matrix M is given below:

The stiffness matrix is:

By using the equation (6) & (7), the natural frequency and the natural period of vibration for each mode shapes can be determined. The calculated eigenvalues, and the corresponding natural frequencies and periods are given in the above table:

Mode Shape

Eigenvalue ω2 (rad2/s2)

Frequency (Hz)

Period (s)

1

23.158

0.766

1.306

2

192

2.205

0.453

3

450.681

3.379

0.296

4

678.161

4.145

0.241

Table X: Eigenvalues, Frequencies and Periods for the 4 storey building.

The modes of the shape and their corresponding periods are shown below:

T1= 1.306s T2= 0.453 T3= 0.296s T4= 0.241s

Using the method of normalisation the eigenvectors become:

T1= 1.306s T2= 0.453 T3= 0.296s T4= 0.241s

Maximum Displacement, Pseudo Velocity and Acceleration for the different frequencies

Mode Shape

Frequency (Hz)

Pseudo- Velocity (m/s)

Displacement (m)

Acceleration (g)

1

0.766

0.750

0.193

0.291

2

2.205

0.750

0.092

0.636

3

3.379

0.750

0.068

0.884

4

4.145

0.750

0.055

1.087

Table X: maximum Displacement, Pseudo- Velocity and Acceleration for the 4 storey building.

The four calculated participation factors αi's are represented in the table below:

α1

α2

α3

α4

1.241

0.333

0.184

0.080

Table X: Participation Factors.

The maximum displacement, Pseudo-Velocity and Acceleration from Table X are multiplied by the respective participation factors from Table X. The scaled parameters of the motion are given in the table below.

Mode Shape

Frequency (Hz)

Pseudo- Velocity (m/s)

Displacement (m)

Acceleration (g)

1

0.766

0.931

0.240

0.361

2

2.205

0.250

0.031

0.212

3

3.379

0.138

0.012

0.162

4

4.145

0.060

4.381*10-3

0.087

Table X: Scaled parameters of the motion for each mode due to the participation factors.

Maximum displacement and the Inter Storey Drift for each floor are given below:

D1 = 0.090 m, D2 = 0.159 m, D3 = 0.211 m and D4 = 0.242 m.

δ1=1.792 % , δ2 = 1.397 %, δ3=1.030 % and δ4=0.613 % for the last one. The above values are in agreement with the Eurocode as they are lower than 4%.

The Maximum Base Shear Force and the Maximum Overturning Moment are presented in the above table:

Maximum Base Shear Force (N)

Maximum Overturning Moment (Nm)

5.218*106

7.363*107

Inelastic Analysis

The inelastic design spectrum is used with a ductility factor of 5 (μ=5). Hence the acceleration and the displacement are re-evaluated. The results are given in the table below.

Mode Shape

Frequency (Hz)

Displacement (m)

Acceleration (g)

1

0.766

0.211

0.040

2

2.205

0.193

0.123

3

3.379

0.187

0.172

4

4.145

0.115

0.172

Table X: Calculated displacements and accelerations for each mode (Inelastic design).

Afterwards, each displacement and acceleration is multiplied by the respective participation factor from Table (X) and the scaled parameters are given below:

Mode Shape

Frequency (Hz)

Displacement (m)

Acceleration (g)

1

0.766

0.262

0.050

2

2.205

0.064

0.041

3

3.379

0.034

0.032

4

4.145

9.160*10-3

0.014

Table X: Scaled parameters of the motion for each mode due to the participation factors (Inelastic design).

Maximum displacement and the Inter Storey Drift for each floor are given below:

D1 = 0.117m, D2 = 0.183m, D3 = 0.232m and D4= 0.271 m

δ1 = 2.335 %, δ2 = 1.331 %, δ3 = 0.983 % and δ4 = 0.764 %

The values of the Inter-Storey Drift are in agreement with the Eurocode as they are lower than 4%. The Maximum Base Shear Force and the Maximum Overturning Moment are calculated and the results are presented in the table below:

Maximum Base Shear Force (N)

Maximum Overturning Moment (Nm)

7.325*105

1.015*107

Table X: The Maximum Base Shear Force and the Maximum Overturning Moment for the 4 storey building (Inelastic design).

5 Discussion of results

Graph 1 and Graph 2 below depict the variation of the fundamental frequency for multi storey buildings, against each multi storey building. In particular, it represents the fundamental frequency of 1, 2, 3, 4 storey building for each storey respectively.

At the above graph it is observed that as the floors are increased the fundamental frequency is decreased. The peak value of the graph is f11= 2.205 Hz, and represents the fundamental frequency in the case of 1 storey building. Afterwards one storey is added to the existing building with different characteristics from the existing one and the two natural frequencies are calculated. The smallest frequency of these two is the fundamental frequency and is f12= 1.535 Hz. The same procedure is applied for the rest two buildings. One storey is added each time to the previous existing building and the fundamental frequency is calculated for each one. This gives the values f13= 1.338 Hz and f14= 1.185 Hz for 3 storey and 4 storey buildings respectively.

By following the same procedure as in Graph 1 the fundamental frequencies are calculated for 1, 2, 3 and 4 storey buildings and plotted the results are represented in Graph 2. However, in this case as the floors are increased the mass, the height of the each floor and the stiffness remain the same as and each floor is a duplicate of the first floor.

The peak value of the graph is f11=2.205 Hz, which represents the fundamental frequency. For 2, 3 and 4 storey buildings the fundamental frequencies are: f12=1.363 Hz, f13=0.982 Hz and f14=0.766 Hz respectively.

At graph 1 and graph 2, is observed that as the floors are increased the fundamental frequencies are decreased. The value of fundamental frequency for one storey building has the same value in both cases, as the two buildings are two exact replicas. After that point, by comparing the two sets of results it is observed a variation of frequencies. There is a greater decrease in the fundamental frequency values for buildings with the same characteristics at each floor.

The four following graphs depict the Inter Storey Drift - Number of storeys relationship. In particular the Inter-storey Drift of the first floor of each building is plotted against the total number of storeys of the corresponding building. The first two graphs represent the results of the Inter-Storey Drift for the elastic analysis.