Structural Dynamics

In many engineering problems one encounters random loading, e.g. induced by sea-waves, wind, turbulence in flows, roughness of the road, etc. If such a process is inspected at a number of successive time intervals, they all have a similar appearance, although they differ completely in all details. Such type of behaviour is idealized by saying that the process is stationary, i.e. the statistical properties of the process are time invariant. The postulation of stationarity is a very strong assumption which simplifies the mathematical theory considerably. Moreover, the parameters defining the stationary process can be evaluated from a single record of the loading process by assuming the stationary process to be ergodic. The well developed mathematical theory of stationary processes and the ease to identify the parameters of a stationary loading process are utilized in many engineering fields (e.g. wind-, ocean-, noise and vibration engineering) to evaluate the statistical characteristics of the stationary response of structures or mechanical components.

Even in the most seismic areas of the world the occurrence of destructive earthquakes is a rare event. Earthquake loads differ from ordinary design loads in various aspects. Not only the occurrence, but also the loading itself is highly uncertain with respect to magnitude, frequency content and duration of the ground shaking. Naturally, it is not economical to design structures which resists the “strongest possible” earthquake, since it is very unlikely that a structure ever experiences such a strong shaking. It is generally accepted in design, that a structure should resist moderate earthquakes without damage, but to permit yielding and damage for very rare strong earthquake events provided the structure is unlikely to endanger human life by collapse. The required resistance to possible future earthquake loading depends, of course, on the consequences of structural failure. In case of dams, power plants and important life lines, the design will be more conservative.

Methods of structural dynamics find practical applications in the earthquake resistant design of major structures, such as sky-scrapers, nuclear power stations, hospitals and large dams and bridges located in seismically active regions. In the past, earthquake resistant design criteria for such structures usually employed the so-called design response spectrum (see Newmark and Rosenblueth [2.3–1] and Clough and Penzien [2.3–2]). This spectrum is based on estimated values of certain numerical indices of the expected strong ground motion at the structure site. The indices, constructed from empirical relations supplied by the analysis of available ground motion records and other historical data (see, e.g., Hays [2.3–3]), include peak values of the ground displacement, velocity and acceleration as well as their dominant periods. Since the significant amount of reliable data for the most important cases of severe events and small epicentral distances is still not available, the reliability of these empirical relations is doubtful. Consequently, in modern civil engineering practice, the design response spectrum is applicable at preliminary design stages; the ultimate proportioning of a structure requires an explicit description of the expected ground motion at the site (see Clough and Penzien [2.3–2]). (Actually, the design response spectrum is not applicable to structures with nonlinear responses or involving various types of structural interactions. For such structures, an actual time history record of ground motion is, as it was pointed out by Clough and Penzien [2.3–2], indispensable). More reliable criteria for structural design employ more detailed descriptions of the expected ground motion (see Scanlan [2.3–4]). The best information is provided by complete time histories of ground displacement, velocity and acceleration likely to occur at the site.

The velocity field of natural winds in the atmospheric boundary layer is characterized by three-dimensional turbulence. Mathematically, this implies that at a given location the wind velocity cannot be described by one value (i.e. the mean wind speed ū) but moreover by three additional quantities u’, v’, w’ defining fluctuations in three independent coordinates. Identifying the x-axis with the direction of ū and u’ a three-dimensional wind model as shown in Fig 2.4–1 [2.4–1] can be used as a basis for further analysis.

Within the scope of linear random vibration analysis it is frequently assumed that the excitation process possesses Gaussian properties. Although this simplifying assumption may be justified in many cases there are certain — mainly environmental — load processes (e.g. earthquake, wind, water waves) whose time histories (realizations) quite frequently reveal considerably non-normal characteristics. These properties are, of course, reflected in the response of systems to this type of excitation. Consequently, the probabilistic description of the response will, in general, have to be based on Non-Gaussian properties as e.g. reflected in the higher order statistical moments of the response. The non-normality becomes of significant importance when exceedance probabilities are under investigation, i.e. the reliability of a structure is being assessed.

Stability analysis of suspension bridges subjected to wind loading has been a major topic in wind engineering for several decades. However, the question of turbulence effects on stability remained largely unanswered. In many cases, instability of bridge motion appears to be of the flutter type and to be dominated by one torsional mode of vibration. Consequently, the work by Lin and Ariaratnam [3.2–1] investigated the effect of turbulence on the moment stability of one uncoupled torsional mode. It was shown that turbulence has a destabilizing effect on the single-degree-of-freedom model. Experimental evidences, however, indicate that both destabilizing and stabilizing effects of turbulence can occur depending on the shape of the structure (Irwin and Schuyler [3.2–2]; Huston [3.2–3]). An intuitive explanation for the stabilizing effect is that turbulence may help to feed energy from the least stable mode to the more stable modes thus providing stabilization (Lin [3.2–4]). It is the purpose of this chapter to provide a theoretical explanation for the above contradictory phenomena. To allow for energy transfer, coupled modes have to be considered in the analysis. Additionally, unsteady aerodynamics are taken into account to obtain appropriate modeling of fluid-structure interaction and fluid memory. The mathematical formulation of the self-excited forces acting on a moving bridge model is chosen in a way appropriate for stochastic analysis but consistent with experimental data obtained in deterministic (laminar) flow conditions. In the following, the equations of motion are written in the time domain for coupled modes, one being much less stable than the others. The stability of the overall structure-fluid system is investigated in terms of second order stochastic moments of the state variables.

The most powerful tool available for the analysis of the response of non-linear systems to random loading is the Markov vector approach. This, however, requires the excitation to be Gaussian white noise (although non-white processes can be modeled by passing white noise through a shaping filter, e.g. a Kanai-Tajimi filter). In this case the joint probability density of the state vector components is governed by the Fokker-Planck equation. Unfortunately, only very few closed-form solutions are known, most of them for SDOF systems. Even for in the SDOF case only some stationary solutions are available (e.g. Refs[3.3–1, 3.3–2, 3.3–3, 3.3–4]). This means that for more general cases approximate methods such as equivalent linearization have to be utilized. A detailed discussion is given in chapter 3.3.2.

Deterministic and random vibrations of linear elastic plate structures are discussed. At first polygonally shaped thin plates according to Kirchhoff s theory are considered. The undamped frequency response function is calculated by a powerful boundary element method (BEM) with Green’s functions of rectangular domains, which was developed in [3.4–1] for static loading of plates in a first stage. An extension of the method to eigenvalue problems of membranes and plates is given in [3.4–2], forced vibrations are analysed in [3.4–3,4,5,6], and plates with particular orthotropy are treated in [3.4–7,8]. Embedding the actual polygonal domain properly into a rectangular plate, the boundary conditions (b.c.s) are possibly satisfied exactly at the coinciding boundaries. The remaining prescribed b.c.s of the actual problem lead to a pair of coupled integral equations for a density function vector whose components are line loads and moments distributed in the basic domain along the actual boundary. Considering time-harmonic excitation and sweeping the forcing frequency stepwise the undamped frequency response function results, where the roots of the reciprocal yield the eigenfrequencies with high numerical accuracy.

Commonly, macroscopically inelastic behavior is a result of changes of the materials micro-structure. In modern engineering science, frequent use is made of multi-component or multiphase materials, which exhibit a variety of different inelastic mechanisms on the microlevel. It is important to have knowledge of these mechanisms in order to give a thorough description of the macroscopic behavior. In a macroscopic continuum formulation non-uniformities on the microscale can only be considered as averaged quantities referred to a certain reference volume. Consequently, additional variables have to be introduced accounting for the microstructural state. These variables are called internal (sometimes hidden) variables. Since any rearrangements of the material microstructure are connected with energy dissipation, a change of the internal variables indicates a dissipative process. Therefore, variations of the microstate are at least partly irreversible in the thermodynamic sense. This corresponds to the conception of the microstructure being altered by formation and spreading of microdefects. In real materials such microdefects are identified as dislocations, microcracks and -voids, etc. There exists a vast field of literature concerning the formulation of inelastic behavior due to mechanisms on the microscale. In case of plastic flow first attempts to describe this phenomenon based on slip systems of single crystals are due to Bishop and Hill [3.5–1], [3.5–2] later contributions are given by Hill [3.5–3], [3.5–4], Lin [3.5–5], [3.5–6], Budiansky and Wu [3.5–7] and Havner [3.5–8] among others. Dislocation theories of plasticity and viscoplasticity have been introduced by numerous authors [3.5–9] — [3.5–12].

Recently, the analysis of engineering structures, which are driven from the elastic into the physically nonlinear range by severe loadings, has become a main field of research interest. In engineering practice, the statics of inelastic structures is handled by using the plastic hinge approximation, for both steel and R/C structures, [3.6–1], [3.6–2]. However, attempts are made to overcome the drawbacks of localizing the dissipation of plastic work at discrete hinges by considering the effect of finite spread of plastic zones, e.g. [3.6–3], [3.6–4], [3.6–5], [3.6–6]. Furthermore, rate effects are to be included in the quasistatic structural analysis of time-dependent loadings, e.g. [3.6–7], [3.6–8], [3.6–9], [3.6–10].

For computational procedures to estimate the reliability of systems or structures under random dynamic loading it is one of the foremost goals to obtain detailed information on the threshold crossing rate. This is the average number of times a random process crosses a certain, possibly critical, level per unit time. The threshold crossing rate provides most important information on the excursion probability within a given time interval (cf ch. 3.3.1.4). A thorough analysis requires that both loading and system parameters are treated as random quantities reflecting their physical properties.

It is a well known fact that traditional analysis and design procedures of engineering structures neglect the intrinsic, i.e. physical, uncertainties of the parameters and variables, e.g. loading, material, geometrical properties, etc., involved. These procedures are generally labled as deterministic procedures. Stochastic approaches by which, in general, these uncertainties may be treated more realistically are — particularly for dynamic problems — open to criticism with respect to the degree of simplification of the mechanical modeling. In other words, while the load models show a considerable degree of sophistication, in most cases the structural mechanical models are far from reflecting the state of the art. Procedures are generally limited to SDOF or two DOF systems. PΔ effects with large deflections, which are so important, particularly when investigating collapse failure mechanisms, generally cannot be included in the analysis. Moreover, in many cases only second moment properties of the response are provided, which are certainly of limited practical use only for the analyst or designer. It is mainly due to this lack of sophistication of the mechanical modeling, i.e. due to the fact that the mechanical models of the structures for stochastic analysis do not match those which are used in so called deterministic design, that stochastic procedures still lack the recognition they deserve by practicing engineers. Basically stochastic procedures represent a particular way of information processing. While deterministic procedures select “representative” (mean, maximum, minimum, etc.) values to describe the parameters, stochastic methods process the entire information which is available, i.e. instead of arbitrarily selecting one particular value, the entire spectrum of possible values — in terms of probability distributions — is utilized.

In the past major damage to liquid storage tanks has been caused by earthquakes and leakage or spillage combined with fire led to considerable harm in addition to the damage of the tanks. Hence, earthquake resistant design of liquid storage tanks has been an essential topic in the field of earthquake engineering. It is necessary to have proper codes for engineers dealing with the construction of liquid storage tanks. Present codes do not provide fully satisfactory procedures for an earthquake resistant design of anchored and especially unanchored tanks. For example the American Standard API 650 [5.1–1] which is based on the work of Wozniak [5.1–2] does not in general provide a conservative estimate of the dynamically activated loads. The maximum axial compression force in the tank wall which is further increased by uplift of tank bottom edge of unanchored tanks is substantially underestimated in many cases. A design rule is proposed by Priestley et al. [5.1–3] in the New Zealand recommendations for the seismic design of storage tanks. In these recommandations an excellent summary of the design rules of earthquake excited tanks is given. The maximum axial membrane force, increasing due to uplift of unanchored tanks, is estimated by a simple iterative procedure based on results of Clough [5.1–4] and experiments. The modification of the effective flexibility of the interaction vibration mode (common vibration of the flexible tank wall and the liquid), increasing the natural period due to uplift, is, however, not taken into account.

Storage tanks of standard design have natural periods well above one second and, thus, are to be considered likewise to structures with a soft first storey. In case of a site with high seismic risk and soft soil-layers resonance loading at low frequencies will occur. Contrary to ordinary structures the stiffness of the tank cannot be increased considerably. A deep-pile foundation traversing an intermediate layer of low mechanical impedance is analysed as an alternative to the cheaper shallow sand or concrete foundation used commonly under sufficiently good soil conditions. Such dynamic interaction problems of relatively stiff structures on pile foundations are treated by Wolf [5.2–1] and nonlinear constitutive relations of the viscous soil are considered by Penzien [5.2–2]. The behavior of the piles in layered media is discussed by Novak and Aboul-Ella [5.2–3] and, more, recently, by Nogami [5.2–4]. The analysis is kept linear following the latter references and, by considering also the fact that not sufficient data are commonly available to model the soil-layers more sophistically than linear elastic and hysteretically damped. Contrary to the vibrations of a stiff structure on a deep-pile foundation the low-pass filter properties dominate the overall resonance behavior. For a dense population of piles the model of reinforced soil reflects the dynamic properties well in the low frequency range and the foundation comes close to a Thimoshenko beam model. Individual piles contribute to thickness vibrations only at much higher frequencies. The inhomogeneous cross-section of the Timoshenko beam is assumed to exhibit circular symmetry, and its effective rigidities with respect to bending and shear as well as its mass per unit of length vary from layer to layer according to the soil properties. The effective modulus theory is applied in a standard fashion.

Static analysis of tunnels in rock generally neglects the load imposed by blasting during tunnel driving. Today, the influence of time on the redistribution of stresses is largely limited to the rock’s rheology, whereby these are extremely slow load functions. It is precisely the extremely short effect of the blasting load that exerts an additional force on the rock and superposes on the loading from stress redistribution in the destroyed excavation zone. The result is an irreversible change of the rock properties immediately behind the face with a major import on further static analysis. This loosening caused by blasting, that has been a known factor to design engineers for many years, was often used to advocate mechanical tunneling without it being possible to quantify its influence.

Design Codes for R/C industrial chimneys such as CICIND [5.4–1] or DIN 1056 [5.4–2] are based on linear dynamic analysis with uncracked cross-sections. Under extreme loading conditions, however, nonlinear effects may become important, particularly when approaching the ultimate limit state. This fact, of course, directly affects the dynamic amplification of the response, which in general is expressed by the so called gust response factor (GRF). Naturally, the quantification of this factor depends on the respective structural model which is utilized. In addition, it is also affected by taking into account the tensile strength of concrete. In other words, the utilization of most realistic structural models is of great importance particularly when approaching the ultimate limit state.

The reliability of containment structures is within the risk evaluation of nuclear power plants (NPP’s) of paramount interest. Consequently the development of credible methods to quantify the risk of failure has received considerable attention in the past (see e.g. [5.5–1, 5.5–2, 5.5–3]). Containment structures are exposed on one hand to internal hazards, such as loss of coolant accident, H₂ explosion, etc. and on the other hand to external hazards, such as aircraft impact, earthquake loading, etc.. In the following the reliability estimation of a particular containment structure under severe earthquake condition is shown.