UNIVERSITY DUKE Department of Civil & Environmental Engineering CE 130L. Uncertainty, Design, and Optimization Spring 2009 Henri P. Gavin Design of a Collapse-Sensitive Structure Subjected to Earthquake Loads Due Wednesday, April 22, 2009
1 Objective
The objective of this assignment is to use numerical simulation to design and observe the performance of a single degree of freedom structure (SDOF) subjected to earthquake loads. You will begin by running animated simulations of linear and nonlinear structures subjected to sinusoidal and earthquake ground accelerations. The purpose of these simulations is to observe the eects of the excitation frequency, damping ratio, yield force, ductility, and lateral buckling on the dynamic response of structures subjected to idealized and realistic loading. In the earthquake response simulations you will determine the yield force and yield displacement of an inelastic SDOF structure such that both the accelerations and the deections are kept within acceptable limits. The acceleration limit will be satised if a slender block sitting on top of the mass does not topple over during the earthquake. In your report you will describe the simulations and your design method, and will assess the the importance of uncertainty in earthquake loads.
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The response of elastic SDOF structures to sinusoidal loads
An linear SDOF structure consists of an elastic spring, k, a viscous dashpot, c, and a rigid mass, m, as shown in Figure 1. The base of the structure moves with displacement z(t) with respect to a xed point of reference. The mass of the structure moves with displacement x(t) with respect to the base. The displacement across the spring is x(t) and the velocity across the dashpot is x(t). The total acceleration of the mass with respect to a xed point is x(t) + z (t). A detailed analysis of the transient and steady-state response of this kind of system is in the course notes. Figure 2 illustrates the magnitude and phase of
z(t)
x(t)
111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000
. k x(t) + c x(t)
k c
m k x(t)
x(t)
Figure 1. A linear single degree of freedom system. the steady-state response of SDOF structures to sinusoidal loadings. The plots on the left illustrate the response to sinusoidal force loading. The two plots on the right are of particular interest to us as they illustrate the response of structures to support motion. The plot labeled frequency response Z to X illustrates the dynamic amplication of the displacement response (X) of structures subjected to ground displacements (Z). We see that when the frequency ratio (/n ) equals one, the excitation frequency () equals the resonant frequency (n = k/m) and the displacements of the structural response can be 1
frequency response: F to X 10 =0.05 8 =0.05 10 10
frequency response: Z to X
transimissibility: Z to (X+Z) =0.05 8
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|X/X |
0.1 4 0.2 2 0.5 0 0 1 2 3 0 1.0 0 1 2 0.5 1.0 2 0.2 4
|X/Z|
st
6 0.1
6
6 0.1 4 0.2 2 0.5 0 3
1.0 0 1
2
3
Figure 2. Magnitude and phase of the steady-state response of SDOF structures subjected to sinusoidal loadings.
| (X+Z) / Z |
0 =0.05
0 =0.05
0
45 phase, degrees
45
45
=1.0
90 =1.0 135
90 =1.0 135
90
135 =0.05
180
0
1 2 frequency ratio, = /
3
n
180
0
1 2 frequency ratio, = /
3
n
180
0
1 2 frequency ratio, = /
3
n
2
many times greater than the ground displacement (X/Z 1). As the damping () increases the amount of amplication decreases. The displacement amplication at resonance ( = 1) is equal to 1/(2). For example, if the damping ratio, , equals ve percent, 1/(2 0.05) = 10 and the amplication at resonance is 10. Also note that at resonance, the displacement response lags the base displacement by ninety degrees no matter what the damping is. When the frequency of the ground displacement is zero, then the ground simply has a constant static displacement, (Z = const.) and the relative displacement of the structure with respect to the ground is zero (X/Z = 0). When the frequency of the ground displacement becomes high ( > 2) then the structure remains relatively stationary while the ground moves under the structure. In this case the structural displacement is equal and opposite to the ground displacement, (X/Z = 1). The absolute value |X/Z| = 1 and the displacement response lags the ground displacement by 180 degrees. Turning our attention to the plots labeled transmissibility: Z to (X+Z) we see the relationship between the ground displacement, Z, and the total (or absolute) motion of the structure, which is the sum of the ground displacement and the relative displacement of the ground with respect to the structure. For vibration isolation applications, we want the total motion of the structure to be small, and the transmissibility is the item we would like to minimize. As in the previous case, adding damping reduces the dynamic amplication near resonance. However, at higher frequencies ( > 2), increasing damping increases the dynamic amplication. The phase relationship for transmissibility is more complicated than the phase relationship for the frequency response function from Z to X. The phase lag at resonance ( = 1) depends on the value of damping and does not necessarily equals ninety degrees only if the damping is zero. To develop your understanding of seismically-excited vibrations download and save the Matlab/Simulink simulations found in: http://www.duke.edu/hpgavin/ce130/Dynamics.zip 1. Unzip the le ... unzip Dynamics.zip 2. Change directories ... cd Dynamics 3. Start Matlab ... matlab 4. Load the SDOF Simulink demonstration ... sdof . . . various windows will appear 5. Start the Simulink demonstration ... select Start from the Simulation menu of the Simulink block You will observe the response of a SDOF elastic structure to sinusoidal base motion. Double click on the Simulink block labeled FREQUENCY OF BASE MOTION and set the FREQUENCY equal to 1 cycle/second. Similarly set the NATURAL FREQUENCY to 1 cycle/second and the DAMPING RATIO to 0.10. Start the simulation and observe the dynamic amplication and phase of the response. Next increase the FREQUENCY OF BASE MOTION to 3 and start the simulation again. Notice that the structure remains relatively stationary while the ground moves beneath it. This is considered good isolation performance. The absolute acceleration response is much less than the ground acceleration, and the relative displacement response is equal and opposite to the ground displacement. Now, increase the damping ratio from 10 percent to 50 percent and repeat the simulation. Notice that the absolute acceleration response is larger than when the damping was 10 percent. Also note that the absolute acceleration response is more in phase with the ground acceleration. Finally increase the damping to 100 percent. Notice how the absolute accelerations increase further as the high damping makes the structure more tightly coupled to the ground motion. The Restoring Force Hysteresis is a plot of the force in the spring and damper (f (t) = k x(t)+c x(t)) versus the deformation of the spring, x(t). Note that when the damping ratio is low, then the restoring force is basically just k x(t) and the hysteresis plot shows the force vs displacement relationship for the spring. As the damping increases, the hysteresis plot becomes more like an ellipse since the restoring
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force approaches c x(t). The area of the ellipse has units of force displacement and is equal to the energy dissipated per cycle in the damper. The bigger the loops, the more energy is dissipated into heat. Explore the eect of other values of the FREQUENCY OF BASE MOTION and the DAMPING RATIO and notice the link between the animation, the time history display, and the frequency response (X/Z vs. ), transmissibility ((X+Z)/Z vs. ), and hysteresis plots (kx + cx vs. x). Make notes on how the amplitude ratio and the phase of the frequency response and the transmissibility are aected by the FREQUENCY OF BASE MOTION and the DAMPING RATIO.
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The response of inelastic SDOF structures to earthquake loads
Actual earthquakes are not sinusoidal and the response of structures to strong earthquakes is inelastic. Figure 3 shows an inelastic base-excited building structure. The forces, R(x) between the mass m and the base are inelastic and hysteretic with a yield force of Fy , as shown.
x(t) m H R(x) d 111 000 z(t) 111 000 111 000 111 000 11111111111 00000000000 111 000 11111111111 00000000000 111 000
Figure 3. An inelastic base-excited structure.
F p
R(x)
Dp
x(t)
The next Simulink animation illustrates the inelastic and dynamic behavior of SDOF structures subjected to strong ground motion. In this demonstration energy is dissipated through the yielding and inelastic deformation of the structure. 1. Start Matlab ( if you havent already ) ... matlab 2. Load the EQUAKE Simulink demonstration ... equake 3. Start the Simulink demonstration ... select Start from the Simulation menu of the Simulink block diagram. You will observe the response of an inelastic SDOF structure to earthquake-like ground motions. The restoring force now consists of the inelastic structural force plus a small amount of viscous damping force. The restoring force is limited by the yield force, Fy , and the yielding behavior creates loops in the hysteresis plot, which indicates the dissipation of energy. Also, if the restoring forces are limited by yielding, then the total accelerations of the structure ((t) + x(t)) are also limited. The equation of z motion, in the simplest terms is (Mass)(total acceleration) + Restoring Force = 0 , or, m ((t) + z (t)) + R(x(t)) = 0 . x (1) Since the mass is constant, limiting the restoring force by limiting the yield force is equivalent to limiting the total accelerations of the structure. Furthermore, energy is dissipated by inelastic deformation. For 4
these reasons, inelastic behavior is generally desirable for good response to earthquakes. A by-product of inelastic behavior is a certain amount of permanent deformation, as seen at the end of the displacement time history record. In the animation, the structural columns turn red when the structure yields. Set the YIELD FORCE to only 10 Nt and the YIELD DISPLACEMENT to 1 cm and run a simulation. Notice that the acceleration response is limited by the level of yielding, ... that is, until the structure collapses due to the overturning moment, i.e., (mg x(t)) exceeds the plastic moments which form at the ends of the columns. This illustrates that the appropriate level of yield force has a lower and an upper bound. To prevent collapse the YIELD FORCE can be made larger, but doing so increases the absolute acceleration response, which can cause damage to the contents of the building.
4 Acceptable oor acceleration levels
Acceleration limits for building oors should ensure that slender objects, like bookshelves or statues, do not topple. The toppling of rigid objects due to motion of their bases is a very complex and subtle problem in dynamics and has been an active area of research for decades. One of the rst proposed criteria for this problem (Housner, 1963), gives the total acceleration required to topple the block | + z |max in x terms of the period of the motion and slenderness of the block. A block of width w and height h will be likely to topple if the total acceleration at the base of the block is greater than Acrit , where Acrit Io = g 1 + mgR 2 T
2 1/2
(2) (3) (4) (5)
= arctan(w/h) , R = (h + w )/4 , Io = m R2 ,
2 2 2
and T is the period of motion at the base of the block, in seconds. Note that for blocks of uniform density, this relationship is independent the of mass of the block. A plot of this relationship between the critical acceleration level Acrit and the period of motion T , as given by equation (2), is shown in gure 4 for blocks with aspect ratios of 2:1 and 4:1. Notice that if the period of the structural response is long ( greater than 1 second ) then the acceleration required to topple the block is close to g times the arc tangent of the slenderness ratio. For short period structural responses, much greater accelerations are required to topple the block. The analysis behind equation (2) depends on many assumptions, including that the motion at the base of the block is sinusoidal and not transient. The toppling of slender objects subjected to accelerations at their base is dicult to predict when the base accelerations are irregular, as they would be during an earthquake.
5 Modeling
The primary structure is modeled as of a rigid steel mass M of 10 tons supported by four exible and ductile steel columns made of grade A-366 ductile steel plate with a thickness, D, width B, and height H. The motion of the primary system is described by X(t). The statue is modeled as a secondary structure with a mass of 1 ton. The motion of the secondary system with respect to the mass of the primary system is described by x(t) X(t). The modulus of elasticity of the steel is 200,000,000 kN/m2 (29,000 ksi) and the yield stress is approximately 350,000 kN/m2 (50 ksi).
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Total Acceleration (peak) required to topple a rectangular block Total Acceleration, x"(t)+z"(t), (cm/sec/sec) 1400 1200 1000 800 600 400 200 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Period of Total Acceleration, Tg, (seconds) 0.9 1 g arctan(1/2) = 455 cm/s/s g arctan(1/4) = 240 cm/s/s Aspect Ratio = 2:1 Aspect Ratio = 4:1
Figure 4. Acceleration required to topple a slender block (Housner 1963). If the top of the statue displaces by an amount x X greater than about 0.15 m the statue will topple over. This is modeled by a force f (x X) on the secondary mass that has a stable equilibria at x X = 0 and at x X = and unstable equilibria at x X = , as shown in Figure 6. f (u) = k (u )(u )u(u + )(u + ) , 2 2 u=xX (6)
The coupled nonlinear ordinary dierential equations describing this system are: m(t) + f (x(t) X(t)) x M X(t) f (x(t) X(t)) + R(X(t)) mgX(t)/H = m(t) z = M z (t) (7) (8)
where z (t) is the earthquake ground acceleration. The earthquake ground acceleration is modeled as a sum of sinusoids with dierent amplitudes, a and frequencies, fk .
N
z (t) = e(t)
k=1
a(fk ) sin(2fk k ) ,
(9)
where the amplitudes a(fk ) depend upon the frequencies as follows a(fk ) = (2g fk /fg )2 , (1 (fk /fg )2 )2 + (2g fk /fg )2 (10)
fg = 1.1 Hz, g =1.4, k is a random phase angle between 0 and 2, the earthquake ground motion is enveloped by the function e(t) e(t) = exp (2t/T )2 exp (5t/T )2 , (11)
the frequencies fk are evenly spaced within a range of frequencies, and the maximum ground acceleration is Amax .
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x(t) m X(t) M H R(X)
11 00 11 00 11111111111 00000000000 11 00 11111111111 00000000000 11 00 11111111111 00000000000 11 00
Figure 5. Inelastic single degree of freedom structure supporting a secondary mass, m.
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F p
R(X)
Dp
X(t)
z(t)
d
restoring force, f(x-X)
5
-
0
- 0
-5
-10 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2
potential energy function, U(x-X)
0.4 0.3 0.2 0.1 0
stable unstable unstable
-0.1 -0.2 -0.3 -0.2
stable stable
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
lateral displacement (x-X)
Figure 6. Restoring force and the associated potential energy function showing stable equilibria (points where
f = 0 and U > 0) and unstable equilibria, (points where f = 0 and U < 0).
6 Design objective
The objective of this project is to determine the smallest size steel columns of cross section b d and height H of to hold up a ten-ton mass such that: the structure does not undergo excessive displacements during an earthquake; a priceless slender statue sitting on the structure does not topple during an earthquake; H < 3 meters; and b > 4d meters. Design-oriented analysis will be accomplished with the Matlab function earthquake analysis.m which is part of the Dynamics.zip archive. This function approximately models the physics of yielding in the columns, the potential for collapse of the structure, and the toppling of the statue, as described above.
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The dierential equations (7) and (8) are solved numerically in earthquake analysis.m using a 4th order Runge-Kutta method implemented in the function ode4u.m. Figures 7 and 8 show sample results. The usage of earthquake analysis.m is described below.
earthquake ground acceleration envelope
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quake accel, m/s2
1 0 -1 -2 0 0.5 5 10 15 20 structure, X statue, (x-X)
response displacements, m
0.4 0.3 0.2 0.1 0 -0.1 0 5 10 15
20
Figure 7. Example of earthquake ground motion and response simulation. Amax = 3.5 m/s2 . The statue topples
at around t = 6 seconds.
function [ cost, constraints, random_variables ] = earthquake_analysis( param ) % [cost,constraints,random_variables] = earthquake_analysis( param ) % simulate the response of an inelastic SDOF system to an earthquake % the SDOF system supports an object that can topple % % Input: % param(1) D depth of column, m % param(2) B width of column, m % param(3) H height of column, m % % Output: % cost volume of steel utilized in the columns, m^3 % constraints(1) maximum response displacement of the primary system, m % constraints(2) maximum response displacement of the secondary system, m % random_variables(1) peak earthquake ground acceleration, Amax, m/s/s % random_variables(2) duration of the earthquake, T, s randomize = 0; % 1: model random variability in earthquakes % 0: no random variation in earthquakes my_favorite_number = 7; Plots = 1; % used to seed the random number generators
% 1: draw plots, 0: dont draw plots
You will set randomize = 1 for the safety analysis. You may set Plots = 0 to suppress the plotting of results. You may change the value assigned to my favorite number to change the earthquake waveform used in the analysis. 8
30
20
structural force, R(X)-MgX/H, kN
10
0
-10
-20
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
structural displacement, X, m
Figure 8. Example yielding response simulation. The yield force is about 40 kN.
7 Project tasks
1. Get the les: Download http://www.duke.edu/~pgavin/ce130/Dynamics.zip h Uncompress or unzip Dynamics.zip , cd Dynamics, start Matlab, 2. Using basic tools of intuition, try to nd a feasible design (for which the constraints are satised). Run some analyses with . . . randomize = 0 . . . >> [J,g,r] = earthquake analysis( [ d b H ] ) to nd a set of values of d, b, and H that result in acceptable responses. 3. Write a Matlab script earthquake opt.m which makes use of the initial guess you found in the previous step along with optimization routines old sqp.m or ossrs.m to converge upon a solution that minimizes the steel in the columns without resulting in excessive displacements or toppling the statue. 4. Repeat the design optimization step three or seven times, for three or seven dierent values of my favorite number. If you run three design optimizations, the largest of the column sizes will be your design. If you run seven design optimizations, the average of the seven optimal column sizes will be your design. This is the infamous worst of three, average of seven rule in the building design code, Minimum Design Loads for Buildings and Other Structures, ASCE/SEI 7-05. 5. Once you have determined an optimal design, carry out a safety analysis. Set randomize = 1 in earthquake analysis.m Put your optimal design into the rst line of earthquake safety.m Set Ns = 1000. in earthquake safety.m and Run earthquake safety.m. The probability of structural collapse and the probability of toppling statues are computed. Histograms of g1 , g2 , Amax , and T are plotted in Figure 1. Scatter plots of g1 , g2 , Amax , and T are plotted in Figure 2. Observe the correlation between the input random variables, the peak ground acceleration Amax , and the earthquake duration T , to the constraints. If the failure probabilities are not acceptable, how could the design be improved?
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To hand in: 1. Background Write the dierential equation of motion for linear SDOF structures subjected to ground accelerations. Write the dierential equation of motion for nonlinear SDOF structures subjected to ground accelerations. Dene all the the terms in the equations. What are the major assumptions in these equations? What are the key dierences between linear elastic and inelastic behavior of earthquake excited structures? For vibration isolation should a structure be sti or exible? Why? How should the stiness of the structure be determined? For vibration isolation in linear elastic structural systems, is damping good or bad? What does the right amount of damping depend on? For vibration isolation in inelastic structures is yielding good or bad? How much is the right amount of yielding? What would be an approach to nd the right amount of yielding? Can we know this before an earthquake? Can the frequency response plots and transmissibility plots for linear elastic structures give us insight into how inelastic structures will behave? How could these gures be at all useful in the design of an inelastic structural system, in determining Fp and Dp ? 2. Design and Optimization How did you go about determining b, d and H for your design? What values of b, d, and H did you settle on in task 4? What are the values of Fp , Dp for your values of b, d, and H? For a 10-ton mass, what would the elastic natural frequency be . . . n = Fp /(M Dp ). Include plots of the displacement time histories and the restoring force hysteresis produced by earthquake analysis.m. For these values of Fp and Dp was there signicant yielding (hysteresis) in the response of the primary system? 3. Safety Analysis Include the histogram plots and the scatter plots generated by earthquake safety.m What is the probability of structural collapse? What is the probability of statue toppling? Which of these probabilities do you think would be more dicult to reduce using the methods available to you in this project? Is the structural response correlated signicantly to the earthquake amplitude Amax ? Is the structural response correlated signicantly to the earthquake duration T ? Is the statue toppling response correlated signicantly to the earthquake amplitude Amax ? Is the statue toppling response correlated signicantly to the earthquake duration T ? In your opinion, what special challenges does this situation present?
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References
1. Chopra , Anil K., (2000) Dynamics of Structures: Theory and Applications to Earthquake Engineering, Prentice-Hall. 2. Clough, Raymond W., and Penzien, Joseph, (1975) Dynamics of Structures, 2nd ed. McGraw-Hill. 3. Fielder W.T., Virgin L.N., Plaut R.H. (1997) Experiments and simulation of overturning of an asymmetric rocking block on an oscillating foundation, European Journal of Mechanics A - Solids, 16 (5): 905-923. 4. Housner, George W., (1963), The Behavior of Inverted Pendulum Structures During Earthquakes, Bulletin of the Seismological Society of America, 53 (2): 403-417. 5. Tedesco, Joseph W., McDougal, William G., and Ross, C. Allen, (1998) Structural Dynamics : Theory and Applications, Prentice-Hall. 6. Yim, C.S., Chopra, A.K., and Penzien, J., (1980) Rocking Response of Rigid Blocks to Earthquakes, Earthquake Engineering & Structural Dynamics, 8 (6): 565-587. 7. Zhang J., and Makris N., (2001) Rocking response of free-standing blocks under cycloidal pulses, Journal of Engineering Mechanics, 127 (5): 473-483.
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