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HAUSÜBUNG - BAUSTATIK V - DYNAMIK WINTER SEMESTER 14/15 UNIVERSITÄT DUISBURG ESSEN Fakultät für Ingenieurwissenschaften Master of Science in Bauwissenschaften Dynamic Response of a Water Tower to Earthquake Excitation Prof. Dr. - Ing. Jochen Menkenhagen Pascon Tommaso, 0302336000

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  • HAUSBUNG - BAUSTATIK V - DYNAMIK WINTER SEMESTER 14/15

    UNIVERSITT DUISBURG ESSEN Fakultt fr Ingenieurwissenschaften

    Master of Science in Bauwissenschaften

    Dynamic Response of a Water Tower to Earthquake Excitation

    Prof. Dr. - Ing. Jochen Menkenhagen

    Pascon Tommaso, 0302336000

  • CONTENTS

    1. SCRIPT OF THE TASK (AUFGABE) 3 2. INTRODUCTION - STEP OF THE ANALYSIS 4 3. FIRST PART 4

    3.1. Idealization of the Structure 4

    3.2. Determination of the Overall Stiffness of the Frame Structure 7

    3.3. Damped vibration due to Seismic Excitation 10

    4. SECOND PART 13 4.1. The Duhamel - Integral 13

    4.2. The Numeric Integration of the Equation of Motion - Central Difference Method 16

    4.3. Conclusions 22

    2

  • 1. SCRIPT OF THE TASK (AUFGABE)

    The displacement u(t) of the Water Tower structure illustrated in fig. 1 as a result of an earthquake is to be determined. The Earthquake is represented here by the accelerogram shown in fig. 2.

    The displacement must be estimated by numerical integration of the Duhamel - Integral.

    The Water Tower must be idealised as a single degree of freedom (SDOF) system.

    The displacement u(t) must be determined both for the undamped and damped system. The damping ratio D can be calculated with the help of the Chart 1.

    The essential steps of the computation and the results must be shown and explained.

    3

  • 2. INTRODUCTION - STEP OF THE ANALYSIS

    The analysis and the computation steps can be divided overall in two parts:

    First Part: Evaluation of the structures stiness needed for the idealization as one single degree of freedom system. In the end of this part also the Natural circular frequencies () and the corresponding Period and Frequency are calculated.

    Second Part: Introduction of the Seismic Excitation, evaluation of the Duhamel - Integral and its computation with a Numerical Method. In the end of the second part the displacements of the Water Tower subjected to the Earthquake will be determined for both the undamped and damped system.

    3. FIRST PART3.1. Idealization of the Structure

    The main task of the first part of the exercise is to idealize the 3 Dimensional Water Tower into a single degree of freedom system. For example our Water Tower tank can be modeled as a simple oscillator if we make the following simplifying assumptions:

    The mass of the steel column is negligible compared to the mass of the tank;

    The tank is a point mass moving only along the horizontal direction;

    The water is not moving inside the tank, otherwise we should consider also the dynamic behavior of it;

    In the 3 dimensional word we have the x - y - z axis; the ground acceleration due to the Earthquake is given only in the x direction.

    From the 3D model shown in the front page I built the two Elevation Plans with the architectural designing software Archicad; the first is the Elevation in the x - z plane and the second one is in the y - z plane.

    4

  • 5GSEducationalVersion

    0,000 GROUND LEVEL

    +10,001 TANK LEVEL

    WATER TOWERELEVATION PLAN X-Z

    Scale 1:100

    X

    Z

  • Thanks to the repetition of the structure in the y direction and to the fact that the earthquake is characterized by a ground acceleration only in the x direction, we need to focus just on the x - z plane Elevation view. From the 2D model we can gain the static framework of the structure as follow:

    6GSEducationalVersion

    0,000 GROUND LEVEL

    +10,001 TANK LEVEL

    WATER TOWERELEVATION PLAN Y-Z

    Scale 1:100

    Y

    Z

  • 7GSEducationalVersion

    b

    c

    a

    ra ra

    r (t)

    0,5 MN

    WATER TOWERSTATIC FRAMEWORK REPRESENTATION

    Scale 1:100

    X

    Z

    Parameter:a = 2,0 m

    b = 10,0 mc = 5,0 m

    Steel: HE-B 200GSEducationalVersion

    b

    c

    a

    u (t)

    0,5 MN

    g g

    WATER TOWERSTATIC FRAMEWORK REPRESENTATION

    Scale 1:100

    X

    Z

    Parameter:a = 2,0 m

    b = 10,0 mc = 5,0 m

    Steel: HE-B 200

  • 3.2. Determination of the Overall Stiffness of the Frame Structure

    For the computation of the global stiness of the Structure i used a freeware software called Ftool. I modeled the frame structure in the software using the HE-B material propriety (E, A, I). In this simplified model the water tank is replace by the upper crosspiece beam, which can be seen as a rigid body (EA and EI ) due to the stiness of the tank. The global stiness can be obtained applying a unit Force in the x direction at the top of the frame structure, and then divide it by the displacement.

    F = 1 KN

    Steel Columns: HE-B 200

    E = 210000 MPa [N/mm2];

    A = 7808 mm2;

    I = 5,7107 mm4

    Water Tank:

    m = 0,5 MN = 500 KN = 50000 Kg

    8

  • As we can see in the picture above, the displacement Dx is equal to:

    Dx = = 4,99710-2 mm = 4,99710-5 m

    Now that we have obtained the global stiness of the steel structure we are able to idealize the 3d tower in one single degree of freedom cantilever beam with the pointed mass at the top, as follow:

    9

    Kx,eq =1= 14,997 105

    = 20012,007 KNm

    GSEducationalVersion

    x (t)

    u(t)

    u (t)

    0,5 MN

    u (t)

    0,5 MN

    Kx,eq = 20012 KN/m

    g

    xg (t)

    c

    Fs = ku(t)

    FD = cu(t)

    g g g

  • 3.3. Damped vibration due to Seismic Excitation

    When we talk about Earthquake we must consider that there is not an actual load applied to our structures, but more like a ground acceleration; and its from this acceleration that the structures are subjected to Forces, because of the well know relation:

    F = ma;

    The equation of motion for a base point excitation through an acceleration time -history g(t) can be derived from the equilibrium of forces as:

    Where:

    O is the Global Reference System;

    O is the Reference System that is solid to the ground at the structure point;

    u(t) is the displacement of the mass m in respect of O;

    x(t) is the displacement of the mass m in the Global Reference System;

    x(t) = xg(t) + u(t);

    10

    GSEducationalVersion

    x (t)

    u(t)

    u (t)

    0,5 MN

    u (t)

    0,5 MN

    Kx,eq = 20012 KN/m

    g

    xg (t)

    c

    Fs = ku(t)

    FD = cu(t)

    g g g

    O O'

  • The equation of motion in the Global Reference System is:

    There is no forcing function on the right-hand side of the equation, but there is the inertial force, generated by mass and acceleration.

    While in the Relative Reference System we get:

    If then we divide the last equation by m we get:

    with:

    Natural Circular Frequency:

    Damping Ratio:

    The negative sign has no real meaning since the earthquake will move in both directions. The important point is that earthquake forces are generated by the inertial resistance of the structure.

    We can now calculate the Natural Circular Frequency, the Period and the Frequency of our structure:

    11

    GSEducationalVersion

    x (t)

    u(t)

    u (t)

    0,5 MN

    u (t)

    0,5 MN

    Kx,eq = 20012 KN/m

    g

    xg (t)

    c

    Fs = ku(t)

    FD = cu(t)

    g g g

    !!u + 2 D !u + 2 u = !!xg (t )

    = km

    D = c2 m

    m !!x + c !x + k x = 0

    F = m = k u c !u = m !!x

    k u c !u m ( !!xg + !!u) = 0

    m !!u + c !u + k u = m !!xg

    = km =20012 10350000 = 20,006 s

    1

    T = 2

    = 0,314 s

    f = 1T = 3,184 Hz

  • What is more interesting is how the structure might respond to the given earthquake. This is evaluated using a method to determine response of a SDOF system to a general excitation history, such as an earthquake through Duhamel - Integral.

    A full discussion of Duhamels integral is presented in the second part of this paper.

    12

  • 4. SECOND PART4.1. The Duhamel - Integral

    In the second part of this paper the eects of the given earthquake on the Water Tower will be analyzed. The structure will be subjected to the El Centro (1940) earthquake in Mexico. The ground motion is represented here below:

    From the previous figure it can be clearly seen that the time history of an earthquake ground acceleration can not be described by a simple mathematical formula, for instance an external harmonic excitation; as we can see it is more like a random function. So how could we treat the behavior of our single degree of freedom oscillator under this kind of load? The unit impulse response procedure for approximating the response of a structure may be used as the basis for developing a formula for evaluating response to a general dynamic loading.

    We have than our generic external excitation:

    13

    El Centro Earthquake Ground Motion

    g [g

    ]

    -0,400

    -0,320

    -0,240

    -0,160

    -0,080

    0,000

    0,080

    0,160

    0,240

    0,320

    t [s]0,00 1,50 3,00 4,50 6,00 7,50 9,00 10,50 12,00 13,50 15,00

    GSEducationalVersion

    p (t)

    t

    p (t)

    t

    d!

    !

    p (t)

    t

    d!

    !

  • We can treat this external excitation as a summation of momentums. But first we need to know how can we get the solution of the problem for just one momentum.

    A momentum can be expressed as a loading p(t) acting at time t = !. This loading acting during the short interval of time d! produces a short duration impulse on the structure.

    We know from the Balance of Momentum law that the time derivation of the momentum is equal to the loading p(t):

    And we know that the displacement at t = ! is zero:

    After t = ! + d! the structure is not subjected to external loading anymore, so we are

    in the case of the damped free vibrations, that depend only on boundary conditions. In this case the boundary conditions must be picked up from the perturbation eect of the impulse.

    This is the solution of the free damped vibrations:

    Now the following relations must be replaced in the above equation:

    14

    GSEducationalVersion

    p (t)

    t

    p (t)

    t

    d!

    !

    p (t)

    t

    d!

    !

    dd m

    !u( ) = p( )

    d !u( ) = p( )m d

    du( ) = 0

    u(t ) = eDt u0 cost +!u0 +Du0

    sint

  • And we obtain:

    The entire loading history may be considered to consist of a succession of such short impulses, each producing its own dierential response of the form above.

    For this linearly elastic system, then, the total response can be obtained by summing all the dierential responses developed during the loading history, that is, by integrating as follows:

    We than get the integral:

    with:

    Natural Circular Frequency (Undamped):

    Natural Circular Frequency (Damped):

    Damping Ratio:

    15

    u(t ) = du(t )u(0) = 0

    !u(0) = d !u( ) = p( )m d

    du(t ) = eD (t ) p( )m sin( (t ))d

    GSEducationalVersion

    p (t)

    t

    p (t)

    t

    d!

    !

    p (t)

    t

    d!

    !

    u(t ) = 1m p( ) eD (t ) sin( (t ))d

    0

    t

    = km

    = 1D2

    D = c2m

  • The equation found integrating the single impulse response is generally known as the Duhamel - Integral for a damped system. It may be used to evaluate the response of an damped SDOF system to any form of dynamic loading p(t), although in the case of arbitrary loadings (like the ground motion due to Earthquakes) the evaluation will have to be performed numerically.

    4.2. The Numeric Integration of the Equation of Motion - Central Difference Method

    The Duhamel - Integral is surely very ecient when the excitation is defined as a continuous function, but when the external excitation is characterized by values each defined every ti = ti-1 + "t, we are not able to use this method and we have to use some numerical methods. This is exactly the case of accelerations records of the Earthquakes that, even though it is a continuos phenomena, are taken every "t (time step).

    The sample values of the ground acceleration xg(t) are known from beginning to end of the earthquake at each increment of time "t. The solution strategy assumes that the motion quantities of the SDoF system at time t are known, and that those at the time t = t + "t can be computed. Calculations start at the time t = 0 (at which the SDOF system is subjected to known initial conditions) and are carried out time step after time step until the entire time history of the motion quantities is computed.

    In this paper the Central Dierence Method will be discussed and implemented. Lets consider our Seismic Excitation values discretized every time step "t, then we will obtain:

    ui, ui+1 and ui-1: displacements of the mass m at t = i, i+1 and i-1;

    pi: Seismic excitation values at time step i;

    We then gain the equation of motion of the SDOF system at the generic time step i:

    with:

    pi = -mxg(t)

    From the above equation we gain:

    16

    m !!ui + c !ui + k ui = pi

    !!ui =pi c !ui k ui

    m

  • We use for the initial velocity and acceleration the central dierence definition:

    So we get:

    Now we replace these velocity and acceleration in the equation of motion and we obtain:

    Where both ui and ui-1 are known. The above equation can also be manipulated until we get:

    Or in compact form:

    17

    !ui =ui+1 ui12 t

    !!ui =ui+1 2 ui + ui1

    t( )2

    m ui+1 2 ui + ui1t( )2

    + c ui+1 ui12 t + k ui = pi

    mt( )2

    + c2 t

    ui+1 = pi

    mt( )2

    c2 t

    ui1 k

    2mt( )2

    ui

    pi = pi mt( )2

    c2 t

    ui1 k

    2mt( )2

    ui

    k = mt( )2

    + c2 t

    GSEducationalVersion

    ui-1

    !t !tui ui+1

    u

  • We get:

    Where the unknown term is then:

    The solution at time t = ti+1 can be obtained only if we know the solution at time t = ti. To begin the iterative computation and to get u1, u0 and u0-1 must be known; These two first displacements are easy obtained solving the velocity and acceleration defined from the central dierence method for i = 0:

    And substituting the first equation into the second we obtain:

    The initial condition (displacement and velocity at time t = 0) are known, and from the original equation of motion we can get the acceleration 0:

    We are now able to start the recursive computation since the mass, the stiness, the damping of the SDOF oscillator and the values of the earthquake excitation are known at each time step; set the initial displacement and velocity equal to zero we gain 0 = 0 and 0-1 = 0.

    We are able to calculate

    at each time step and until the entire time history of the motion quantities is computed.

    18

    k ui+1 = pi

    ui+1 =pik

    !u0 =u1 u012 t

    !!u0 =u1 2 u0 + u01

    t( )2

    u01 = u0 !u0 t +t( )22

    !!u0

    m !!u0 + c !u0 + k u0 = p0

    !!u0 =p0 c !u0 k u0

    m

    ui+1 =pik

  • In the implementation of the Central Dierence Method in a spreadsheet were introduced also two more constant parameter to get a more compact form:

    for the undamped system for the damped system

    As the Eurocode suggest, the Damping Ratio (for steel structures) D = 2% has been chosen.

    For the time discretization, a time step "t = 0,02 seconds has been selected, in order to have the same discretization both for the Earthquake Excitation and the Displacement.

    Now the two spreadsheet used for the evaluation of the displacements of the Water Tower are presented:

    19

    a = mt2

    + c2 t = 125000,000KNm

    b = k 2mt2

    = 229987,993 KNm

    k = mt2

    + c2 t = 125000,000KNm

    c = 2m D = 40,012 KN sm

    a = mt2

    + c2 t = 123999,700KNm

    b = k 2mt2

    = 229987,993 KNm

    k = mt2

    + c2 t = 126000,300KNm

  • When we plot the results of the displacements of the Water Tower Structure we get:

    The peaks of the displacements for the undamped case are 2,85 cm in one direction and 2,77 cm in the other direction.

    20

    Displacements of the Water Tower due to Earthquake Excitation (D = 0)

    Disp

    lacem

    ent [

    cm]

    -3,000-2,400-1,800-1,200-0,6000,0000,6001,2001,8002,4003,000

    t [s]0,00 1,50 3,00 4,50 6,00 7,50 9,00 10,50 12,00 13,50 15,00

  • The peaks of the displacements for the damped case are 2,09 cm in one direction and 2,39 cm in the other direction.

    21

    Displacements of the Water Tower due to Earthquake Excitation (D = 2%)Di

    splac

    emen

    t [cm

    ]

    -3,000-2,400-1,800-1,200-0,6000,0000,6001,2001,8002,4003,000

    t [s]0,00 1,50 3,00 4,50 6,00 7,50 9,00 10,50 12,00 13,50 15,00

    Comparison of the displacements for D = 0 and D = 2%

    Disp

    lacem

    ent [

    cm]

    -3,000-2,400-1,800-1,200-0,6000,0000,6001,2001,8002,4003,000

    t [s]0,00 1,50 3,00 4,50 6,00 7,50 9,00 10,50 12,00 13,50 15,00

    u (D = 2%) u (D = 0%)

  • 4.3. Conclusions

    As we can see from the above plots, when damping is involved in the system we get a reduction of the displacements and therefore of the structural stresses.

    The damped system get a peak deflection of 2,39 cm instead of 2,85 cm of the undamped system. The main dierence is not only in the maximum values of the displacements, but also in the overall trend of u in the two systems: in fact the undamped model not only has the peak displacement greater than the damped one, but also o course does not tend to decrease with time, instead of the damped system.

    Having a good amount of Damping in a structural system is always desirable , cause dampers dissipate energy within a system by converting it to heat.

    22