1 Course Introduction
2 Overview
•
• Fundamental Concepts
• Ground Motions and Their Effects
• Structural Dynamics of Linear SDOF Systems
• Response Spectra
• Structural Dynamics of Simple MDOF Systems
• Inelastic Behavior
• Structural Design
3 Fundamental Concepts (1)
• Ordinarily, a large earthquake produces the most severe loading that a building is expected to survive. The probability that failure will occur is very real and is greater than for other loading phenomena. Also, in the case of earthquakes, the definition of failure is altered to permit certain types of behavior and damage that are considered unacceptable in relation to the effects of other phenomena.
• The levels of uncertainty are much greater than those encountered in the design of structures to resist other phenomena. The high uncertainty applies both to knowledge of the loading function and to the resistance properties of the materials, members, and systems.
• The details of construction are very important because flaws of no apparent consequence often will cause systematic and unacceptable damage simply because the earthquake loading is so severe and an extended range of behavior is permitted.
•
•
4 Fundamental Concepts (2)
• During an earthquake the ground shakes violently in all directions. Buildings respond to the shaking by vibration, and the movements caused by the vibration and the ground motion induce inertial forces throughout the structure.
•
• In most parts of the country the inertial forces are so large that it is not economical to design a building to resist the forces elastically. Thus inelastic behavior is necessary, and structures must be detailed to survive several cycles of inelastic behavior during an earthquake.
• The structural analysis that is required to exactly account for the dynamic loading and the inelastic response is quite complex, and is too cumbersome for most projects. The NEHRP Provisions and ASCE 7 provide simplified approximate analysis approaches that overcome these difficulties.
• Rules for detailing structures for seismic resistance are provided by standards such as ACI
318 and the AISC Specification and the AISC Seismic Provisions
5 Overview
•
• Fundamental Concepts
• Ground Motions and Their Effects
• Structural Dynamics of Linear SDOF Systems
• Response Spectra
• Structural Dynamics of Simple MDOF Systems
• Inelastic Behavior
• Structural Design
6 Seismic Activity on Earth
7 Tectonic Plates
8 Section of Earth Crust at Ocean Rift Valley
9 Section of Earth Crust at Plate Boundary (Subduction Zone)
10 Fault Features
11 Faults and Fault Rupture
12 Types of Faults
13 Seismic Wave Forms (Body Waves)
14 Seismic Wave Forms (Surface Waves)
15 Arrival of Seismic Waves
16 Effects of Earthquakes
• Ground Failure
• Rupture
• Landslide
• Liquefaction
• Lateral Spreading
• Tsunami
• Seiche
• Ground Shaking
•
17 Recorded Ground Motions
18 Shaking at the Holiday Inn During the 1971 San Fernando Valley EQ
19 Overview
•
• Fundamental Concepts
• Ground Motions and Their Effects
• Structural Dynamics of Linear SDOF Systems
• Response Spectra
• Structural Dynamics of Simple MDOF Systems
• Inelastic Behavior
• Structural Design
20 NEHRP Seismic Hazard Maps
•Probabilistic / Deterministic (Separate Maps)
•Uniform Risk (Separate Maps)
•Spectral Contours (PGA, 0.1, 0.2 sec)
•5 % Damping
•Site Class B/C Boundary
•Maximum Direction Values
21 Structural Dynamics of SDOF Systems
22 Mass
• Includes all dead weight of structure
• May include some live load
• Has units of force/acceleration
23 Linear Viscous Damping
• In absence of dampers, is called inherent damping
• Usually represented by linear viscous dashpot
• Has units of force/velocity
24 Damping and Energy Dissipation
25 Elastic Stiffness
• Includes all structural members
• May include some “seismically nonstructural” members
• Requires careful mathematical modelling
• Has units of force/displacement
26 Inelastic Behavior
• Is almost always nonlinear in real seismic response
• Nonlinearity is implicitly handled by codes
• Explicit modelling of nonlinear effects is possible
(but very difficult)
27 Undamped Free Vibration
28 Undamped Free Vibration (2)
29 Periods of Vibration of Common Structures
20-story moment resisting frame T = 2.4 sec
10-story moment resisting frame T = 1.3 sec
1-story moment resisting frame T = 0.2 sec
•
20-story steel braced frame T = 1.6 sec
10-story steel braced frame T = 0.9 sec
1-story steel braced frame T = 0.1 sec
•
Gravity dam T = 0.2 sec
Suspension bridge T = 20 sec
30 Damped Free Vibration
31 Damped Free Vibration (2)
32 Damped Free Vibration (3)
True damping in structures is NOT viscous. However, for low damping values, viscous damping allows for linear equations and vastly simplifies the solution.
33 Damping in Structures
Welded steel frame x = 0.010
Bolted steel frame x = 0.020
•
Uncracked prestressed concrete x = 0.015
Uncracked reinforced concrete x = 0.020
Cracked reinforced concrete x = 0.035
•
Glued plywood shear wall x = 0.100
Nailed plywood shear wall x = 0.150
•
Damaged steel structure x = 0.050
Damaged concrete structure x = 0.075
•
Structure with added damping x = 0.250
34 Undamped Harmonic Loading and Resonance
35 Damped Harmonic Loading and Resonance
36 Resonant Response Curve
37 General Dynamic Loading
1 • Fourier transform
• Duhamel integration
• Piecewise exact
• Newmark techniques
•
2 All techniques are carried out numerically.
38 Effective Earthquake Force
39 Simplified SDOF Equation of Motion
40 Use of Simplified Equation of Motion
For a given ground motion, the response history ur(t) is function of the structure’s frequency
w and damping ratio x.
41 Use of Simplified Equation
Change in ground motion or structural parameters x and w requires re-calculation of structural response
42 Creating an Elastic Response Spectrum
An elastic displacement response spectrum is a plot of the peak computed relative displacement, ur, for an elastic structure with a constant damping x, a varying fundamental frequency w (or period T = 2p/ w), responding to a given ground motion.
44 Pseudoacceleration is Total Acceleration
The pseudoacceleration response spectrum represents the total acceleration of the system, not the relative acceleration. It is nearly identical to the true total acceleration response spectrum for lightly damped structures.
45 Using Pseudoacceleration to Compute Seismic Force
46 Response Spectra for 1971 San Fernando Valley EQ (Holiday Inn)
47 Averaged Spectrum and Code Spectrum
48 NEHRP/ASCE 7 Design Spectrum
49 Overview
•
• Fundamental Concepts
• Ground Motions and Their Effects
• Structural Dynamics of Linear SDOF Systems
• Response Spectra
• Structural Dynamics of Simple MDOF Systems
• Inelastic Behavior
• Structural Design
50 MDOF Systems
51 Analysis of Linear MDOF Systems
• MDOF Systems may either be solved step by step through time by using the full set of equations in the original coordinate system, or by transforming to the “Modal” coordinate system, analyzing all modes as SDOF systems, and then converting back to the original system. In such a case the solutions obtained are mathematically exact, and identical. This analysis is referred to as either Direct (no transformation) or Modal (with transformation) Linear Response History Analysis. This procedure is covered in Chapter 16 of ASCE 7.
• Alternately, the system may be transformed to modal coordinates, and only a subset (first several modes) of equations be solved step by step through time before transforming back to the original coordinates. Such a solution is approximate. This analysis is referred to as Modal Linear Response History Analysis. This procedure is not directly addressed in ASCE 7 (although in principle, Ch. 16 could be used)
52 Analysis of Linear MDOF Systems
•Another alternate is to convert to the modal coordinates, and instead of solving step-by-step, solve a subset (the first several modes) of SDOF systems system using a response spectrum. Such a solution is an approximation of an approximation. This analysis is referred to as
Modal Response Spectrum Analysis. This procedure is described in Chapter 12 of ASCE 7.
•
•Finally, the equivalent lateral force method may be used, which in essence, is a one-mode (with higher mode correction) Modal Response Spectrum Analysis. This is an approximation of an approximation of an approximation (but is generally considered to be “good enough for design”.) The Provisions and ASCE 7 do
place some restrictions on the use of this method.
53 Overview
•
• Fundamental Concepts
• Ground Motions and Their Effects
• Structural Dynamics of Linear SDOF Systems
• Response Spectra
• Structural Dynamics of Simple MDOF Systems
• Inelastic Behavior
• Structural Design
54 Basic Base Shear Equations in
NEHRP and ASCE 7
SDS and SD1 are short and one second (T=0.2 s and 1.0 s)Design Basis Spectral Accelerations, including Site Effects
•
Ie is the Importance Factor
•
R is a Response Modification Factor, representing Inelastic Behavior (Ductility, Over-strength, and a few other minor ingredients).
55 Building Designed for Wind or Seismic Load
56 Comparison of EQ vs Wind
• ELASTIC earthquake forces 6 to 9 times wind!
• Virtually impossible to obtain economical design
57 How to Deal with Huge EQ Force?
1 • Pay the premium for remaining elastic
• Isolate structure from ground (seismic isolation)
• Increase damping (passive energy dissipation)
• Allow controlled inelastic response
2 Historically, building codes use inelastic response procedure. Inelastic response occurs through structural damage (yielding). We must control the damage for the method to be successful.
58 Nonlinear Static Pushover Analysis
59 Mathematical Model and Ground Motion
60 Results of Nonlinear Analysis
61 Response Computed by Nonlin
62 Interim Conclusion (the Good News)
1 The frame, designed for a wind force which is 15% of the ELASTIC earthquake force, can survive the earthquake if:
2 • It has the capability to undergo numerous cycles of INELASTIC deformation
• It has the capability to deform at least 5 to 6 times the yield deformation
• It suffers no appreciable loss of strength
REQUIRES ADEQUATE DETAILING
63 Interim Conclusion (The Bad News)
1 As a result of the large displacements associated with the inelastic deformations, the structure will suffer considerable structural and nonstructural damage.
2 • This damage must be controlled by adequate detailing and by limiting structural deformations (drift).
64 Development of the Equal Displacement Concept
65 The Equal Displacement Concept
“The displacement of an inelastic system, with stiffness K and strength Fy, subjected to a particular ground motion, is approximately equal to the displacement of the same system responding elastically.”
•
(The displacement of a system is independent of the yield strength of the system.)
66 Repeated Analysis for Various Yield Strengths (and constant stiffness)
67 Constant Displacement Idealization of Inelastic Response
68 Equal Displacement Idealization of Inelastic Response
• For design purposes, it may be assumed that inelastic displacements are equal to the displacements that would occur during an elastic response.
•
• The required force levels under inelastic response are much less than the force levels required for elastic response.
•
69 Equal Displacement Concept of Inelastic Design
70 Key Ingredient: Ductility
71 Application in Principle
Using response spectra, estimate elastic force demand FE
Estimate ductility supply, m, and determine inelastic force demand FI = FE /m. Design structure for FI.
Compute reduced displacement. dR, and multiply by m to obtain true inelastic displacement,
dI. Check Drift using dI.
72 Application in Practice
(NEHRP and ASCE 7)
Use basic elastic spectrum but, for strength, divide all pseudoacceleration values by R, a response modification factor that accounts for:
•
• Anticipated ductility supply
• Overstrength
• Damping (if different than 5% of critical)
• Past performance of similar systems
• Redundancy
73 Ductility/Overstrength
First Significant Yield
74 First Significant Yield and Design Strength
First Significant Yield is the level of force that causes complete plastification of at least the most critical region of the structure (e.g., formation of the first plastic hinge).
•
The design strength of a structure is equal to the resistance at first significant yield.
75 Overstrength
76 Sources of Overstrength
• Sequential yielding of critical regions
• Material overstrength (actual vs specified yield)
• Strain hardening
• Capacity reduction (f ) factors
• Member selection
• Structures where the proportioning is controlled by the seismic drift limits
•
77 Definition of Overstrength Factor W
78 Definition of Ductility Reduction Factor Rd
79 Definition of Response Modification Coefficient R
80 Definition of Response Modification Coefficient R
81 Definition of Deflection Amplification Factor Cd
82 Example of Design Factors for Reinforced Concrete Structures
R Wo Cd
Special Moment Frame 8 3 5.5
Intermediate Moment Frame 5 3 4.5
Ordinary Moment Frame 3 3 2.5
•
Special Reinforced Shear Wall 5 2.5 5.0
Ordinary Reinforced Shear Wall 4 2.5 4.0
Detailed Plain Concrete Wall 2 2.5 2.0
Ordinary Plain Concrete Wall 1.5 2.5 1.5
83 Design Spectra as Adjusted for Inelastic Behavior
84 Using Inelastic Spectrum to Determine Inelastic Force Demand
85 Using the Inelastic Spectrum and Cd to Determine the Inelastic Displacement
Demand
86 Overview
•
• Fundamental Concepts
• Ground Motions and Their Effects
• Structural Dynamics of Linear SDOF Systems
• Response Spectra
• Structural Dynamics of Simple MDOF Systems
• Inelastic Behavior
• Structural Design
87 Design and Detailing Requirements
88 Questions
Title slide.
Fundamentals -1
This slide provides an overview of the topics presented in this slide set.
Fundamentals -2
These bullet items are taken directly from the text of Chapter 2 of FEMA P-751.
Fundamentals -3
Fundamental concepts, continued.
Fundamentals -4
This slide provides an overview of the topics presented in this slide set.
Fundamentals -5
This slide shows a series of dots where earthquakes have occurred historically in
the world. As may be seen, the dots are not randomly oriented, but instead for
distinct patterns. It has been determined that the dot patterns outline the
boundaries of tectonic “plates”
that form the crust of the earth. The Sand Andreas
fault in western California lies along one of the plate boundaries. Note also the
heavy concentration of dots along the west coast of South America, and along the
north-western Pacific coast, and the Aleutian Island Chain. Most of the historic
“great earthquakes”
have occurred in these locations.
Note the scarcity of dots in the central and eastern U.S. Earthquakes are rare here,
but large earthquakes have occurred in central Missouri and in South Carolina.
Fundamentals -6
This slide shows the tectonic plates and the plate names. Heavy black lines mark
the main plate boundaries.
Fundamentals -7
This slide, from Bolt, shows a section of the Earth’s crust, and the driving
mechanism for plate movement in ocean rift valleys.
Fundamentals -8
If the plates continue to diverge at the ridges, the surface of the earth would have to
grow (or buckle) unless there were some mechanism to return some cool rock into
the asthenosphere. This slide, also from Bolt, shows how the plate submerges (or
subducts) under the continental shelf at the plate boundary. The sudden release of
frictional forces that develop at this interface are a major source of earthquakes.
Volcanic activity is also a source of earthquakes but the resulting ground motions
are usually not as severe. A note about the buckling: this does occur where two
continental plates converge, such as the boundary between India and Asia; the
result is the highest mountain chain on earth.
Fundamentals -9
Zones of relative weakness in the Earth’s crust are called faults. After the stresses
build up in the rock, one particular area will rupture with relative movement. Define
a few key terms for description of faults.
Fundamentals -10
When an earthquake occurs the rupture spreads over a portion of the fault. The
point where the rupture begins is called the focus or hypocenter. The rupture will
then propagate at a very high speed forming a fault plane. The vertical projection of
the focus to the surface is the epicenter. For shallow earthquakes, the fault plate
may intersect the surface, causing a visible fault rupture and possible escarpment.
For some deeper earthquakes, the fault may not be seen at the surface. These are
called blind faults.
Fundamentals -11
These are the various types of faults representing either lateral or vertical
movement. The fault plane may be vertical or skewed as shown. For strike slip
faults, the type designation comes from the movement of one block relative to an
observer. If the observer is standing on one of the blocks looking across the fault
and the far block moves to the observer’s right, it is a right lateral fault. For a
normal fault, the two blocks move away from each other (extensional) . For a thrust
fault, the blocks are moving towards each other (compressional). The visible wall
formed from the movement is called an escarpment. Of course, the fault may be a
combined strike-slip or normal/reverse fault.
Fundamentals -12
The energy released during an earthquake propagates in waves. The two types of
waves are body waves and surface waves. The principal body waves are the
Compression (P) wave and the Shear (S) wave. Compression and shear waves
move on a spherical front. Sometimes the compression waves are called “pushpull”
as they work like an accordion. Compression waves travel the fastest of all
waves (4.8 km/second in granite), and they travel through both solids and liquids.
Shear waves move from side to side. Because fluids (e.g. water and magma) have
no shear stiffness, shear waves do not pass through. Shear waves are the second
wave type to arrive, moving at about 3.0 km/second.
Fundamentals -13
The next waves to arrive are the surface waves. The two main types are Love
waves and Rayleigh waves. These waves have a somewhat longer period than P
or S waves.
Fundamentals -14
This recording shows the sequential arrival of P, S, and Love waves. With travel
speeds of the various waves known, this type of diagram can be used to estimate
the distance to the wave source.
Fundamentals -15
There are numerous hazards related to earthquakes. While ground shaking is
emphasized in this topic, it is not necessarily the greatest hazard. Tsunamis in
December 2004 and March 2011 were incredibly destructive. Ground shaking is
responsible for several of the other effects: landslide, liquefaction, lateral spreading
and seiche, which is oscillation of a body of water with effects similar to tsunami,
but a completely different cause.
Fundamentals -16
In most seismically active areas of the world seismologists have laid out vast
instrument arrays that can capture the ground motion in terms of a recording of
ground acceleration vs time. Usually each instrument can record two horizontal
components and one vertical component at the same station.
This slide shows some of the horizontal component recordings for a variety of
earthquakes world wide. All of the recordings are of the same horizontal (time) and
vertical (acceleration) scale, with the maximum acceleration approximately 1.0 g.
The character of the ground motion recording depends on many factors, including
the soil type, the distance from the epicenter, and the direction of travel of seismic
waves. The 1984 Mexico City Earthquake, shown at the bottom, is noted for its
long duration and low frequency. This is characteristic of recordings on very soft
soil, taken at some distance from the epicenter.
Fundamentals -17
Holiday Inn ground and building roof motion during the M6.4 1971 San Fernando
earthquake: (a) north-south ground acceleration, velocity, and displacement and (b)
north-south roof acceleration, velocity, and displacement (Housner and Jennings,
1982). The Holiday Inn, a 7-story, reinforced concrete frame building, was
approximately 5 miles from the closest portion of the causative fault. The recorded
building motions enabled an analysis to be made of the stresses and strains in the
structure during the earthquake.
.
Fundamentals -18
This slide provides an overview of the topics presented in this slide set.
Fundamentals -19
In the 2009 NEHRP Provisions and in ASCE 7-05 and -10, ground motion is
represented by a quantity called Spectral Acceleration, which represents the total
expected acceleration that a mass of a Single Degree of Freedom Structure
(SDOF) would feel at a given location in the country. Chapter 3 of P-751 and Topic
3 of this slide series discusses the maps in some detail. The point made here is
that the spectral accelerations are Response Spectrum ordinates, and that some
grasp of structural dynamics is needed to understand what a response spectrum is.
Fundamentals -20
This slide shows a simple, highly idealized structure. In this structure the columns
are flexible, and the beam is rigid. All of the mass of the structure is assumed to
reside in the beam. A (fictitious) dashpot is shown for the purpose of providing
some damping in the system. The properties of the system are the stiffness, k, the
damping constant, c, and the mass, m. A time-varying horizontal load F(t) is
applied to the mass, and the displacement history u(t) is to be obtained. The
solution is obtained by solving the equation of dynamic equilibrium shown at the
bottom of the slide, which represents a time-wise balance of inertial, damping,
elastic, and applied forces.
Fundamentals -21
The mass, m, of the system represents all of the weight of the structure and its fixed
attachments. It may include some live load if it can be assumed that the live load
(e.g. storage loads) will move in phase with the structure when it vibrates. Note that
the units of force/acceleration.
Fundamentals -22
Experience shows that a system set in motion and allowed to vibrate freely will
eventually come to rest. This is due to damping in the system, which is a means of
converting energy into heat, which is then irrecoverably dissipated. IN real
structure damping occurs due to a variety of reasons, ranging from material
damping, to friction in connections, and friction in nonstructural components and
contents. In structural dynamics it is convenient to represent the damping as a
linear viscous “dashpot”
for which the resistance is proportional to the
deformational velocity in the dashpot (which is the same as the velocity of the mass
relative to the base of the structure. The damping constant, c, is impossible to
specify directly, and instead, a damping ratio is used in computations (as shown
later).
Fundamentals -23
When the damping force is plotted vs displacement, an elliptical hysteresis occurs
(when the system is under steady state vibration). The area within the hysteresis
curve represent the energy dissipated per cycle.
Fundamentals -24
The stiffness, k, of the structure is described in this slide. For real structures it is
difficulty to obtain an exact value of k because of a variety of uncertainties. For
example, in concrete structures, cracking has a large influence on stiffness, but the
amount of cracking due to environmental and service loads can not be predicted
with any precision. Similarly, in steel structures, it is difficult to quantify the effect of
connection stiffness, or the influence of partially composite slabs.
Fundamentals -25
A mentioned earlier, we can generally not afford to design structures to remain
elastic during major earthquakes. Thus, we must allow the structure to deform
inelastically in a controlled manner. Like damping, inelastic behavior produces an
irrecoverable energy dissipation, shown here as the area enclosed within the cyclic
force-deformation plot.
While it is possible to perform a nonlinear analysis, this is quite difficult, and is done
only in special circumstances. The vast majority of analysis is performed using
linear procedures which implicitly account for the inelastic behavior in the structure.
Fundamentals -26
It is beyond the scope of this topic to present the details of computing the dynamic
response. However, the computed responses of a few simple loadings will be
presented to provide some needed nomenclature.
The first such loading is undamped free vibration, where the damping is assumed to
be absent and the structure is set in motion by an initial displacement and/or
velocity. The response is in the form of a sine wave. The frequency of vibration,
(omega) is called the circular or angular frequency, and has units of
radians/second. As can be seen, this frequency is in effect a structural property as
it involves only mass and stiffness.
Fundamentals -27
This slide shows the dynamic response of a system in undamped free vibration. In
structural engineering, it is common to use the period of vibration, T, instead of
circular frequency (omega) or cyclic frequency (Hz).
The proper units of period of vibration is seconds/cycle, although the word “cycle”
is almost universally omitted.
Fundamentals -28
This slide shows some periods of vibration of common structures. The Provisions
provides approximate formulas for computing period, based on material, structural
system, and height.
Fundamentals -29
In this slide the equation for motion is shown for the system under damped free
vibration is shown. Two important quantities result from the solution to the
equation, the damping ratio (Greek letter xi), and damped frequency of vibration
omega sub D. In earthquake engineering, xi is almost universally taken as 0.05,
and c is then back-calculated using this value, m, and omega. For xi=0.05, there is
practically no difference between the damped and undamped frequency, so the
undamped value is used for convenience.
Fundamentals -30
The important point on this slide is the use of xi as a ratio versus a percentage.
Some computer programs require this number as input and it is important to get the
value entered correctly.
Fundamentals -31
In free vibration, damping reduces the response over time, and the greater the
damping, the more rapid the decay in vibration amplitude.
Fundamentals -32
This slide lists some reasonable damping ratios in structures, and can be seen the
values can vary considerably. In the Provisions, values of damping other than 5%
critical are (very approximately) accounted for in the response modification
coefficient, R. Note that the seismic hazard maps are based on 5% critical
damping.
Fundamentals -33
When an undamped system is loaded harmonically, the response will grow without
bound when the loading frequency (omega bar) is equal to the structure’s own
natural frequency. Such a phenomena is known as “resonance”. An unbounded
response is, of course, only mathematical because any system will become
damaged and fail at some point of deformation if the system is brittle, or will yield
(changing the structure’s frequency and adding damping in the form of hysteretic
energy dissipation) and fall out of resonance. Nevertheless, resonance is
undesirable. Unfortunately, all ground motions will contain at least one frequency
which is in phase with the structure’s own natural frequency, so all earthquake
response is resonant. In most cases the structure will not fall out of resonance
when the system yields because there is some other frequency component of the
ground motion which will be in resonance with the new structural frequency.
Fundamentals -34
When the damped system is loaded at resonance the response will initially build up
rapidly, but will eventually reach a steady state displacement as shown in the plot.
The maximum displacement achieved (assuming no yielding) is 1/(2 times xi) times
the static displacement (e.g. the displacement under p sub o applied as a static
load).
Fundamentals -35
This plot is a set of resonance response curves, with one curve plotted for each of
four damping values. The vertical axis (amplifier) is the ratio of maximum dynamic
response to static response, and the horizontal axis is the ratio of loading frequency
to structural frequency. As may be seen, when beta=1 the system is in resonance,
and damping is extremely effective in reducing response. Since earthquake
loading is a resonant phenomena, damping is always very important, and the more
the better.
Fundamentals -36
For most types of dynamic loading there is not a closed-form solution. In such
cases it is necessary to solve for the response using numerical procedures. This
slide lists some of the more common methods. The last two methods are most
commonly used. The piecewise exact method is restricted to linear systems, and
produces an mathematically exact response of the loading consists of linear load
between discreet time increments. The Newmark method is close to exact for liner
systems, and may be used for nonlinear systems as well.
Fundamentals -37
Seismic load in the form of a ground acceleration history is converted to an
equivalent seismic load as shown in this slide. Note that the response quantities
(displacement, velocity, acceleration) are relative to the base of the structure,
hence the subscript r. Given the digital nature of the loading, the piecewise linear
method is most suitable for solution when the response is linear.
Fundamentals -38
In this slide a revised equation of motion is shown where all terms have been
divided through by mass (never zero). Now the response to a given ground motion
can be computed “generically”
for a system with a given damping ratio and
frequency of vibration (actual mass and stiffness need not be specified). This form
of the equation of motion is useful for computing response spectra.
Fundamentals -39
This slide repeats some of the points made on the previous slide, but in a more
clear fashion.
Fundamentals -40
This slide illustrates the use of a “Solver”, such as the Piecewise Exact Method, to
compute the response history. For the purpose of computing response spectra, the
key parameter of interest in the solution is the absolute value of maximum or
“Peak”
displacement, regardless of the sign or time of occurrence.
Fundamentals -41
An elastic displacement response spectrum is computed by repeatedly solving a
system with a given ground motion, given damping, and varying period of vibration.
Note the jagged appearance of the curve, and mention that no two earthquakes will
produce the same response spectrum. Note also that the displacement quantity
being computed is the relative displacement.
Fundamentals -42
A very important curve, derived from the displacement spectrum, is the
pseudoacceleration spectrum. It is obtained by dividing each displacement
ordinate by the square of the circular frequency at given period. The mapped
values of acceleration given by the Provisions are, essentially, pseudoacceleration
ordinates at periods of 0.2 and 1.0 seconds.
Fundamentals -43
It is very important to recognize that pseudoacceleration is total acceleration,
even though it is derived from relative displacement. Thus, for T=zero,
pseudoaccelertion is equal to the peak ground acceleration. At very large period
(greater than about 10 seconds) pseudoacceleration will approach zero.
Fundamentals -44
Given the pseudoacceleration spectrum, it is easy to compute the peak base shear
developed in a SDOF system. Assuming 5% damping and the earthquake used to
generate the spectrum (El Centro), the shear can be computed as shown. Once
the shear is known the relative displacement can be found by dividing by k (thus
the displacement spectrum is not needed)
Fundamentals -45
This slide shows several pseudoacceleration spectra for the 1971 San Fernando
Valley, 1971 EQ. Here, the different spectra are for different damping values,
ranging from zero to 20% critical. The bold red line is for 5% damping. Note the
tremendous influence of damping on amplitude and shape, particularly at the lower
periods.
Fundamentals -46
This chart shows the mean spectrum among seven ground motion spectra, the
mean plus one standard deviation, and the “smoothed”
code version of the
spectrum (red line). Note the close fit between the average spectrum and the code
spectrum (such fits are not always possible). The Provisions, as well as ASCE 7
use the smoothed spectrum in lieu of true ground motion spectra because of the
ease of use and uniformity.
Fundamentals -47
This slide shows the basic design spectrum, which is Figure 11.4-1 from ASCE 7
10. This spectrum would be used in Modal Response Spectrum Analysis as
described in Section 12.9 of ASCE 7-10. The shape of this spectrum can be traced
back to work done by Newmark in the 1960’s. The Constant Velocity label
indicates that if the spectrum were to be converted to a velocity spectrum, this
segment of the spectrum would be constant. Similar for the Constant Displacement
branch. Note the unit inconsistency of the values; for example, the division by T in
the constant velocity region produces units inconsistency if the units of seconds are
attached to T.
See the slide set on “Ground Motions”
for more detail.
Fundamentals -48
This slide provides an overview of the topics presented in this slide set.
Fundamentals -49
Slide shows a Multiple Degree of Freedom System
Fundamentals -50
The points in this slide are self-explanatory.
Fundamentals -51
The points in this slide are self-explanatory.
Fundamentals -52
This slide provides an overview of the topics presented in this slide set.
Fundamentals -53
These formulas from Chapter 12 of ASCE 7 have the value R in the denominator.
The next several slides provide the background on R.
Fundamentals -54
This slides shows (without detailed computations) the total factored wind forces
acting on a building at some location, and the ELASTIC seismic forces for the same
location. This could represent, for example, a building bear Charleston, S.C.
Fundamentals -55
The computation shows that the seismic loads are 6.4 to 8.5 times the wind forces.
When designing for wind it is assumed that the structure remains elastic up to the
factored wind loads. It would seem economically prohibitive to design the building
to remain elastic for the earthquake loads.
Fundamentals -56
There are a variety of strategies for dealing with the large earthquake forces. The
first choice should be avoided except in very special circumstances, such as the
design of nuclear power plants. The second two approaches are viable (and
supported by the Provisions and ASCE 7 with separate chapters), but are not
commonly used due to extra cost and required expertise. The vast majority of
buildings are designed using the inelastic response method. This is true even in
the highest seismic risk areas.
Fundamentals -57
In this slide we will assume the building with a 2592 k EQ load is designed for a
strength of about 500 kips, well less that the seismic demand. Assuming the
structure has sufficient ductility, it is mathematically “Pushed Over”
to form the
solid blue line force-displacement shown. Note that the building has considerable
reserve strength (called over-strength) beyond first yield. We will make a rather
large simplification and assume that the entire building can then be represented
dynamically as a SDOF system with the bilinear force-deformation curve shown in
red, perform a nonlinear response history analysis to assess the expected behavior.
Note that ductility is the ability to deform beyond first yield without excessive loss of
strength.
Note that actual ductility above 10 (as shown in the slide) is not realistic for real
buildings, and even when R=8 a significant portion of that value is related to entities
other than ductility (as explained later in this set).
Fundamentals -58
This slide shows the simplified mathematical model and the ground motion used for
response history analysis.
Fundamentals -59
The analysis results, produced by the NONLIN Program, is show on this slide. The
maximum displacement achieved is 4.79 inches and the maximum shear is 542
kips. There are 15 yield events, seven in one direction and eight in the other.
Fundamentals -60
This slide gives the basic definition for ductility demand. If this much ductility is not
actually supplied by the structure, collapse may occur.
Note the difference in the second and third force-displacement panels of the slide.
The second panel is member shear only and the third panel is member shear plus
damping, or total base shear.
Write basic design equation:
Ductility Demand < Ductility Supply
Fundamentals -61
This slide is the “Good News”. It appears that the building can survive the
earthquake, but ONLY if the above conditions are met.
Fundamentals -62
The bad news is that he building will probably suffer considerable damage to the
structural and nonstructural system. It may be reparable after the earthquake, but
there is no guarantee. Also, the Provisions and ASCE 7 are based on the
recognition that there is a small (about 1% in 50 years) but real probability of
collapse. Neither the Provision or ASCE 7 provides any explicit protection from
damage during moderate earthquakes, and thus they are both considered as “Life
Safety”
provisions. New concepts in earthquake engineering, called Performance
Based Design are being developed to alleviate this issue, but it will be several
years before these concepts are brought into the code.
Fundamentals -63
The equal displacement concept is the basis for dividing the “Elastic”
force
demands by the factor R. This is one of the most important concepts in earthquake
engineering. The basis for the equal displacement concept is illustrated in the
following slides.
Fundamentals -64
The Equal Displacement Concept in words.
Fundamentals -65
This slide is based on a series of NONLIN analyses wherein all parameters were
kept the same as in the original model except for the yield strength, which was
systematically increased in 500 kip increments to a maximum of 3,500 kips. The
structure with a 3,500 kip strength remains elastic during the earthquake.
Note that the displacement appears to be somewhat independent of yield strength,
but the ductility demand is much higher for relatively lower strengths.
Fundamentals -66
This slide shows simplified force-displacement envelopes from the different
analyses. An apparently conservative assumption (with regard to displacements) is
shown on the right. The basic assumption is that the displacement demand is
relatively insensitive to system yield strength. This is often referred to as the “equal
displacement”
concept of seismic-resistant design.
Fundamentals -67
This slide summarizes the previous points.
Fundamentals -68
The equal displacement concept allows us to use elastic analysis to predict
inelastic displacements. For the example system, the predicted elastic
displacement (red line) is 5.77 inches, and it is assumed that the inelastic response
(blue line) displacement is the same.
Fundamentals -69
For this simple bilinear system, the ductility demand can now be computed. The
system must be detailed to have this level of ductility.
For design purposes, we typically reverse the process. We assume some ductility
supply (based on the level of detailing provided) and, using this, we can estimate
the strength requirements.
Fundamentals -70
The procedure mentioned in the previous slide is explained in more detail here.
Building codes allow for an elastic structural analysis based on applied forces
reduced to account for the presumed ductility supplied by the structure. For elastic
analysis, use of the reduced forces will result in a significant underestimate of
displacement demands. Therefore, the displacements from the reduced-force
elastic analysis must be multiplied by the ductility factor to produce the true
“inelastic”
displacements.
Fundamentals -71
The approach to using the equal displacement concept is discussed in the next
several slides. One of the key aspects of the method is the use of the response
modification factor, R. This term includes a variety of “ingredients,”
the most
important of which are ductility and overstrength.
Note that overstrength did not enter into the previous discussion because we were
working with idealized systems. Real structures are usually much stronger than
required by design. This extra strength, when recognized, can be used to reduce
the ductility demands. (If the overstrength was so large that the response was
elastic, the ductility demand would be less than 1.0.)
Fundamentals -72
In this slide and several that follow as structure is being subjected to a pushover
analysis. The structure remains essentially elastic until the first full plastic hinge
forms. The formation of this “first significant yield”
occurs at a level of load referred
to as the design strength of the system.
If the hinging region has adequate ductility, it can sustain increased plastic rotations
without loss of strength. At the same time, the other potential hinging regions of
the structure will attract additional moment until they begin to yield.
Fundamentals -73
These definitions come from the commentary to the NEHRP Provisions and the
commentary to ASCE 7.
Fundamentals -74
This slide show the sequential formation of plastic hinges in the structure. With
sufficient ductility, the apparent strength can be considerably greater than the
design strength. The reserve capacity is called Overstrength.
Fundamentals -75
This slide lists most of the sources of overstrength. It is not uncommon for the true
strength of a structure, including overstrength, to be two to three times the design
strength.
Fundamentals -76
The apparent strength divided by the design strength is called the “overstrength
factor.”
Note that the symbol .
used for the overstrength factor is similar to the
term O0 in ASCE 7.
As implemented in ASCE 7, O0 is intended to be a high estimate of true
overstrength (although not an upper bound).
Fundamentals -77
This is the definition of that part of R due to ductility.
Fundamentals -78
The NEHRP/ASCE 7 response modification factor, R, is equal to the ductility
reduction factor times the overstrength factor.
Caution the student that the required Rd is not equal to R divided by the tabulated
O0, because the tabulated values are definitely not lower bound values.
Fundamentals -79
This figure is a graphical version of the information presented in the previous slide.
The response modification factor, R, is used to reduce the expected elastic strength
demand to the DESIGN level strength demand.
On the basis of the equal displacement theory the inelastic displacement demand is
the same as the elastic displacement demand. For design purposes, however, the
reduced design strength is applied to the structure to determine the member forces.
The analysis domain represents the response of the linear elastic system as
analyzed with the reduced forces. Clearly the displacement predicted by this
analysis is too low. The Provisions (ASCE 7) compensates through the use of the
Cd factor.
Fundamentals -80
To correct for the too-low displacement predicted by the reduced force elastic
analysis, the “computed design displacement”
is multiplied by the factor C sub d.
This factor is always less than the R factor because R contains ingredients other
than pure ductility. In theory the primary factor reducing displacement demand is
damping that is higher than the standard 5% damping used to develop the design
spectra in the Provisions, but some of the C sub d factors are not consistent with
this explanation. Overstrength do not contribute to reduction of displacement
demand.
Fundamentals -81
These are the design coefficients for a few selected concrete systems. Note the
very low R values for the plain walls. These plain wall systems are allowed only in
SDC A and B buildings. ACI 318 only permits plain concrete where it is
continuously supported.
Note that the values Omega sub 0 are not exactly the same as the overstrength
factor Omega, and might be considered as reasonable upper bounds on Omega.
They are used in certain load combinations that require extra capacity in key
elements and components of a structure.
Fundamentals -82
In the previous slides, the concept of the reduction factor, R, was presented, and
several values were illustrated. Here, the effect of the R value of the design
response spectrum is illustrated for R = 1 (elastic) through 6. The value for R = 1
has been normalized to produce a peak short period acceleration of 1.0g.
Fundamentals -83
This slide simply shows how the design base shear is determined for a system with
T = 0.8 seconds and R = 4. The pseudoacceleration spectrum is used.
Fundamentals -84
At the period of 0.8 the displacement is read off the red line, which includes Cd.
In practice, one would not generally use a displacement spectrum Instead, the
displacements would be determined from the elastic model with the reduced (1/R)
loads, and these would be multiplied by Cd.
Fundamentals -85
This slide provides an overview of the topics presented in this slide set.
Fundamentals -86
While the Provisions and ASCE 7 provide the basic configuration and loading
requirements, the Detailing that is necessary to support a given R value is provided
in materials standards such as ACI 318 and AISC 341. In some cases the
International Building Code or local jurisdictions will have additional requirements.
Fundamentals -87
Slide to initiate questions from participants.
Fundamentals -88