If a computer analysis gives a period of 0. SDOF Dynamics In some cases, it is appropriate to remove the conservatism from the empirical period formulas. This is done through use of the C u coefficient. This conservatism arises from two sources: 1.

### Evaluation Copy

The lower bound period was used in the development of the period formula. The empirical formula was developed on the basis of data from California buildings.

Buildings in other parts of the country e. It is important to note that the larger period cannot be used without the benefit of a properly substantiated analysis, which is likely performed on a computer. ASCE will not allow the use of a period larger than C u T a regardless of what the computer analysis says. Note the similarity with the undamped solution. In particular, note the exponential decay term that serves as a multiplier on the whole response. Critical damping c c is defined as the amount of damping that will produce no oscillation.

See next slide. The damped circular frequency is computed as shown. The exception is very highly damped systems. A good example of a critically damped response can be found in heavy doors that are fitted with dampers to keep the door from slamming when closing. However, for low damping values, viscous damping allows for linear equations and vastly simplifies the solution. It is frictional or hysteretic. Viscous damping is used simply because it linearizes the equations of motion.

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Use of viscous damping is acceptable for the modeling of inherent damping but should be used with extreme caution when representing added damping or energy loss associated with yielding in the primary structural system. When the damping is zero, the vibration goes on forever. The values for undamaged steel and concrete upper five lines of table may be considered as working stress values. The structure is subjected to an initial displacement and is suddenly released. Damping is determined from the formulas given.

## Fundamentals of Structural Dynamics

Note that the loading frequency is given by the omega term with the overbar. The loading period is designated in a similar fashion. We have assumed the system is initially at rest. The dynamic magnifier shows how the dynamic effects may increase or decrease the response. This defines the resonant condition. In other words, the response is equal to the static response, times a multiplier, times the sum of two sine waves, one in phase with the load and the other in phase with the structure s undamped natural frequency.

Loading kips Steady state response in. Transient response in. Total response in. The harmonic load amplitude is kips. The static displacement is 5. Note how the steady state response is at the frequency of loading, is in phase with the loading, and has an amplitude greater than the static displacement. The transient response is at the structure s own frequency. In real structures, damping would cause this component to disappear after a few cycles of vibration. The steady state response is still in phase with the loading, but note the huge magnification in response. The transient response is practically equal to and opposite the steady state response.

The total response increases with time. If one looks casually at the steady state and transient response curves, it appears that they should cancel out. Note, however, that the two responses are not exactly in phase due to the slight difference in the loading and natural frequencies. This can be seen most clearly at the time 1. The steady state response crosses the horizontal axis to the right of the vertical 1. In real structures, the observed increased amplitude could occur only to some limit and then yielding would occur.

This yielding would introduce hysteretic energy dissipation apparent damping , causing the transient response to disappear and leading to a constant, damped, steady state response. Note, however, that the steady state response is degrees out of phase with the loading and the transient response is in phase. The resulting total displacement is effectively identical to that shown two slides back. The important point here is that the steady state response amplitude is now less than the static displacement.

At low loading frequencies, the ratio is 1. At very high frequency loading, the structure effectively does not have time to respond to the loading so the displacement is small and approaches zero at very high frequency. The resonance phenomena is very clearly shown. This clearly shows the resonance phenomena. Note the addition of the appropriate term in the equation of motion. The transient response as indicated by the A and B coefficients will damp out and is excluded from further discussion.

Note that there is a component in phase with the loading the sine term and a component out of phase with the loading the cosine term. The actual phase difference between the loading and the response depends on the damping and frequency ratios. Note the exponential decay term causes the transient response to damp out in time. Of significant interest is the resonant response, which is now limited. The undamped response increases indefinitely.

The undamped response has a linear increasing envelope; the damped curve will reach a constant steady state response after a few cycles. For increased damping, the resonant response decreases significantly. Note that for slowly loaded structures, the dynamic amplification is 1.

For high frequency loading, the magnifier is zero. Note also that damping is most effective at or near resonance 0. This is referred to as dynamic amplification. An undamped system, loaded at resonance, will have an unbounded increase in displacement over time. Unlike strain energy, which is recoverable, dissipated energy is not recoverable. Damping is most effective for systems loaded at or near resonance.

Note that some texts use the term absorbed energy in lieu of stored energy. In the first diagram, the system remains elastic and all of the strain energy is stored. If the bar were released, all of the energy would be recovered. In the second diagram, the applied deformation is greater than the elastic deformation and, hence, the system yields. The energy shown in green is stored, but the energy shown in red is dissipated. If the bar is unloaded, the stored energy is recovered, but the dissipated energy is lost. This is shown in Diagrams 3 and 4. By general loading, it is meant that no simple mathematical function defines the entire loading history.

SDOF Dynamics There are a variety of ways to solve the general loading problem and all are carried out numerically on the computer. The Fourier transform and Duhamel integral approaches are not particularly efficient or easy to explain and, hence, these are not covered here. Any text on structural dynamics will provide the required details.

The piecewise exact method is used primarily in the analysis of linear systems. The Newmark method is useful for both linear and nonlinear systems. Only the basic principles underlying of each of these approaches are presented. In a sense, the name of the method is a misnomer because the method is not exact when the actual loading is smooth like a sine wave because the straight line load segments are only an approximation of the actual load.

When the actual load is smooth, the accuracy of the method depends on the level of discretization when defining the loading function. For earthquake loads, the load is almost always represented by a recorded accelerogram, which does consist of straight line segments. There would be little use in trying to interpolate the ground motion with smooth curves. Hence, for the earthquake problem, the piecewise exact method is truly exact.

Given the initial conditions and the load segment, the solution at the end of the load step is determined and this is then used as the initial condition for the next step of the analysis. The analysis then proceeds step by step until all load segments have been processed.

### Practical information

SDOF Dynamics It should be noted that the piecewise exact method may be used for nonlinear analysis in certain circumstances. In FNA, the nonlinearities are right-hand sided, leaving only linear terms in the left-hand side of the equations of motion. It is applicable to both linear and nonlinear systems.

The Newmark method is described in more detail in the topic on inelastic behavior of structures. The principal advantage is that the method may be applied to inelastic systems. The method also may be used without decoupling for multiple-degree-offreedom systems. Instead, the ground moves back and forth and up and down and this movement induces inertial forces that then deform the structure.

It is the displacements in the structure, relative to the moving base, that impose deformations on the structure. Through the elastic properties, these deformations cause elastic forces to develop in the individual members and connections.

SDOF Dynamics Earthquake ground motions usually are imposed through the use of the ground acceleration record or accelerogram. Some programs like Abaqus may require instead that the ground displacement records be used as input. The total acceleration at the center of mass is equal to the ground acceleration plus the acceleration of the center of mass relative to the moving base.

## CUREE : Publications

The inertial force developed at the center of mass is equal to the mass times the total acceleration. The damping force in the system is a function of the velocity of the top of the structure relative to the moving base. Similarly, the spring force is a function of the displacement at the top of the structure relative to the moving base. The equilibrium equation with the zero on the response history spectrum RHS represents the state of the system at any point in time.

The zero on the RHS reflects the fact that there is no applied load. If that part of the total inertial force due to the ground acceleration is moved to the right-hand side the lower equation , all of the forces on the left-hand side are in terms of the relative acceleration, velocity, and displacement.

This equation is essentially the same as that for an applied load see Slide 8 but the effective earthquake force is simply the negative of the mass times the ground acceleration. The equation is then solved for the response history of the relative displacement.

## Earthquake Engineering

When the substitutions are made as indicated, it may be seen that the response is uniquely defined by the damping ratio, the undamped circular frequency of vibration, and the ground acceleration record. A response spectrum is created for a particular ground motion and for a structure with a constant level of damping. The spectrum is obtained by repeatedly solving the equilibrium equations for structures with varying frequencies of vibration and then plotting the peak displacement obtained for that frequency versus the frequency for which the displacement was obtained. The solver indicated in the slide is a routine, such as the Newmark method, that takes the ground motion record, the damping ratio, and the system frequency as input and reports as output only the maximum absolute value of the relative displacement that occurred over the duration of the ground motion.

It is important to note that by taking the absolute value, the sign of the peak response is lost. The time at which the peak response occurred is also lost simply because it is not recorded. Only the frequency of vibration, represented by period T, is changed. The response history from which the peak was obtained is shown at the top of the slide.

This peak occurred at about 5 sec into the response, but this time is not recorded. Note the high frequency content of the response. The first point on the displacement response spectrum is simply the displacement inches plotted against the structural period 0. The computed displacement history is shown at the top of the slide, which shows that the peak displacement was inches.

This peak occurred at about 2. Note that the response history is somewhat smoother than that in the previous slide. The second point on the response spectrum is the peak displacement 0. Again, the response is somewhat smoother than before. Note that only the absolute value of the displacement is recorded.

The complete spectrum is obtained by repeating the process for all remaining periods in the range of 0. A real response spectrum would likely be run at a period resolution of about 0. Perelman, S. Lee, L. Keer, Seismic Response of Structure Foundation Systems. Engineering Structures vol. Computers and Structures vol. Ciampoli, P. Sextos, K. Pitilakis, and A. Part 1: Methodology and Analytical Tools. Mergos, K. Journal of Earthquake Engineering vol. Liu, T. Hutchinson, B. Kutter, M. Hakhamaneshi, M. Aschheim, S. Ghalibafian, R. Foschi, Houman, C. Fourteenth World Conference on Earthquake Engineering.

N Pristley, G. M, Calvi, M. Displacement-Based Seismic Design of Structures. Paolucci, R. Fagini, L. Earthquake Spectra vol. Ni, L. Structure and Infrastructure Engineering vol. Chopra, C. Solomon, and S. Drosos, T. Georgarakos, M. Loli, I. Continuous system. Single Degree of Freedom: If a single coordinate is sufficient to define the position or geometry of the mass of the system at any instant of time is called single or one degree of freedom system.

Multiple degree of freedom MDoF : If more than one independent coordinate is required to completely specify the position or geometry of different masses of the system at any instant of time, is called multiple degrees of freedom system. Continuous system: If the mass of a system may be considered to be distributed over its entire length as shown in figure, in which the mass is considered to have infinite degrees of freedom, it is referred to as a continuous system. It is also known as distributed system.

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Degrees of freedom: —If more than one independent coordinate is required to completely specify the position or geometry of different masses of the system at any instant of time, is called multiple degrees of freedom system. Continuous system: Degrees of freedom: —If the mass of a system may be considered to be distributed over its entire length as shown in figure, in which the mass is considered to have infinite degrees of freedom, it is referred to as a continuous system. Mathematical model - SDOF System Mass element ,m - representing the mass and inertial characteristic of the structure Spring element ,k - representing the elastic restoring force and potential energy capacity of the structure.

Dashpot, c - representing the frictional characteristics and energy losses of the structure Excitation force, P t - represents the external force acting on structure. Simple Harmonic motion 2. Energy methods 4. Rayleights method 5. If the acceleration of a particle in a rectilinear motion is always proportional to the distance of the particle from a fixed point on the path and is directed towards the fixed point, then the particle is said to be in SHM.

Simple Harmonic motion method: SHM is the simplest form of periodic motion. Consider a spring — mass system of figure which is assumed to move only along the vertical direction. It has only one degree of freedom, because its motion is described by a single coordinate x.

ciaglosjobsmehill.gq Energy method: Conservative system: Total sum of energy is constant at all time. Solution of Eq. Such system is said to be over damped or super critically damped. Such system is said to be critically damped. Compute also natural period of vibration. In this case, the restoring force in the form of spring force is provided by AB and CD which are columns. The equivalent stiffness is computed on the basis that the spring actions of the two columns are in parallel.

Determine the logarithmic decrement, ratio of any 2 successful amplitudes. Prasad, Assistant Professor, Civil Engineering Dept, NITS 33 Multiple degree of freedom systems A multi degrees of freedom dof system is one, which requires two or more coordinates to describe its motion. These coordinates are called generalized coordinates when they are independent of each other and equal in number to the degrees of freedom of the system Assume that are of cross section and young's modulus are same for all members.

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