Example: Design of a tall chimney against wind loads. The primary loads acting on a tall chimney are due to the wind loads. The wind field acting on the chimney ...
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Module 1 !
Lecture 2: Random Vibrations & Failure Analysis Introduction to uncertainty
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Sayan Gupta
Department of Applied Mechanics
Indian Institute of Technology Madras
Design Philosophy: The key steps in design
Estimate forces expected to act on structure during lifetime.
Structure design is conceptualized & dimensions are caluclated.
Select design load (L) = expected maximum load.
Load bearing capacity (C) is calculated.
Condition not satisfied
For optimal design C should be optimally greater than L
Condition satisfied
END
Schematic of steps involved in design
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Design Philosophy (Cont’d) The robustness of the design depend on two key issues:
• What is the chance that the actual loads that the structure will endure in its lifetime is always less that the design load (L) ?
• How can one be sure that the structure when constructed according to the design will indeed exhibit the design capacity (C) during its lifetime?
It is impossible to say with certainty that the loading on the structure will be less than L.
Similarly, the structure capacity may be different from C due to various reasons, such as, construction deviations due to practical problems, inadequate quality control in manufacturing of raw materials and during construction, un-modeled structure features and unforeseen structure behavior
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Design Philosophy (Cont’d) Thus, both loading and the structure resistance are uncertain.
Traditionally the effect if these uncertainties are incorporated in design through safety margins.
For safe designs,
C>L.
The distance between C and L is indicative of the safety margin in the design.
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Example: Design of a tall chimney against wind loads The primary loads acting on a tall chimney are due to the wind loads.! ! ! ! ! ! ! ! ! ! ! ! The wind field acting on the chimney is expressed as u(x, t) and has spatial as well as temporal variations.
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The wind force acting on the chimney is given by
F (x, t) = Cd Au(x, t)2
(1)
where,! ! Cd is the drag coefficient that depends on the structure geometry,! ! ⇢ is the air density,! ! A is the cross-sectional area of the chimney transverse to the air flow! ! x is the height measured along the longitudinal axis, and! ! t is time
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The design wind load, at a specific height, is proportional to the maximum wind force acting on the structure.
! Mathematically, this is expressed as
Fd (x) = Cd A⇢Ud (x)2 , where,
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Fd (x)
is the design wind load at height
x , and
! ! max U (x) = d 0tT {u(x, t)} ! ! is the maximum wind velocity profile along the height of the chimney.
! !
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To compute the expected maximum wind load, one needs to have data in the form of several years of wind speed measurements for the location at which the chimney is to be constructed.
• Wind data usually unavailable at •
•
Figure: ! Schematic diagram; ! blue circles: measurement stations;! red circle: chimney location;! arrows: wind flow directions.
location.
! Data is interpolated from neighboring stations (maybe hundreds of kilometers apart).
! Correction factors introduced to take into account topographical conditions
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• Even if wind field data is available at location, it is not feasible to measure the wind field at all points along the height of the chimney.
! ! • Assumptions are made on principles of aerodynamics and boundary layer theories to model the wind field along the chimney height.
! ! • Usually a power law is assumed for the wind field variation.
Thus,
! 1 U (x, t) = q(x)f (t) = Uz (x/z) ↵ f (t). ! ! ! The parameters are decided empirically from observed data.
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Uncertainties introduced in modeling the wind field because:
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• errors in interpolating the wind data from measuring stations
• statistical errors as continuous measurements are unlikely to be recorded
• topographical effects are empirically modeled
• complicating effects such as temporal variations in the directionality, local turbulence and interaction effects are either not modeled or modeled empirically
This implies that considerable uncertainty exists in estimating the maximum wind velocity, and in turn, the maximum wind load acting on the structure.
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The assumptions that are generally made in calculating the structure capacity, but which need not be true are
! •structure properties are homogeneous and isotropic (usually not true due to variations in quality control)
! • construction is as per design (invariably there are deviations due to practical difficulties during construction)
! • material degrades with time which in turn affects the load bearing capacity. Modeling the degradation is not easy.
The above assumptions imply that significant uncertainties exist in the calculation of the capacity of the structure.
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Factors of Safety The uncertainties in estimating the expected maximum load and the structure capacity have been recognized.
! Thus, traditional designs involve multiplying the expected maximum load by a factor greater than unity to obtain the design load.
! Similarly, the design capacity is decided by multiplying the capacity by a factor less than unity.
Figure: Schematic diagram of design philosophy using the factor of safety approach; f1 is the load factor of safety and f2 is the capacity factor of safety; dotted lines indicate new design load and design capacity.
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Drawbacks in the factor of safety approach •Choice of safety factors depend on ``experience" and expert judgement
and lacks scientific rigor.
! ! •Can be a handicap when designing structural systems where little or no previous experience exists.
! ! •More seriously, using this approach it is difficult to design systems with target safety levels (as is the modern trend in design).
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For vibrating systems, both the loading X(t) and the response Y time varying.
(t) are
The equation of motion for a vibrating system modeled as a single degree of freedom system, is typically represented through a second order differential equation and the response can be expressed as
Y (t) = F [X(t), (t)] Here,
!
(t)
are structure properties which may or may not be time dependent, the function F (·) relates load X(t) with response Y (t) and could be linear or nonlinear. The structure represented by
F (·)
thus behaves like a filter. 14
•For structures subjected to random loadings, such as, earthquakes, wind, sea waves, machine vibrations etc, the loading X(t) is random.
! ! • Consequently, the uncertainties in X(t) gets transmitted through the filter F (·) to the response Y (t) .
! ! • Additionally, there could be uncertainties in the filter itself (due to modeling inaccuracies and limitations).
! •These effects contribute to the uncertainties in the response Y (t).
X(t) loading is uncertain
F (·) (could be uncertain)
Thus, response is uncertain
Y (t) 15
The propagation of these uncertainties can be schematically represented in the following figure.
Figure: Schematic diagram of uncertainty propagation in a vibrating system. 16
Quantification of the uncertainty in the response can be scientifically and rationally addressed by using the mathematical theories of probability, random variables, random processes and statistics.
! ! ! Once suitable probabilistic models are developed for loads X(t) and structure uncertainties F (·) , the uncertainties in the response Y (t) can be quantified in probabilistic terms.
! ! ! Subsequently, the risk of failure can be quantified by estimating the probability of the response exceeding safe levels in the structure lifetime.
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Mathematically, this can be expressed as
Pf = 1
P [Y (t) < Ysafe 8t 2 (0, T )]
Here,
Pf P [·] Y (t) Ysafe t (0, T )
is failure probability,
! is probability measure,
! is structure response,
! safe threshold level indicating that failure does not occur if Y(t) is below this value,
! is instantaneous time,
! indicates the structure lifetime starting from time zero to time 18
“God does not play dice” The above is a quotation that is attributed to Einstein and is a statement that nature is deterministic and is not random, that all events in nature follow certain physical laws which are deterministic in nature.
An apparent contradiction is observed when we experiment with the simple act of tossing a fair coin.
! ! The random outcome of Heads (H) or tails (T) on tossing a coin seems to indicate that the outcome of a coin tossing experiment does not follow deterministic rules.
However, this is not correct.
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If we were able to toss a particular coin exactly in two successive trials
i.e.,
•ensuring that the way the coin is held between the fingers and
the force that is imparted are identical (ensuring identical initial conditions)
! • the ambient air flow affected by the breathing of the person are identical for both the tosses
! •the air speed at the time of the toss are identical, etc. then, the outcome of the successive trials would have been identical.
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If we were able to model all these effects exactly, then the event would no longer have been random but deterministic.
Unfortunately, it is neither feasible nor practical to be able to model all these effects. Thus, we are forced to have un-modeled parameters in our model affecting the outcome, which in turn, implies that the outcome cannot be predicted deterministically.
Thus, uncertainties that enter in the model are due to our ignorance of all parameters that affect the outcome and/or our inability to model these parameters or incorporate the effect of these parameters into our model.
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The Canon Ball Experiment To have a better appreciation of the problem of uncertainty, let us consider the problem of predicting the distance a cannon ball will travel when fired from a cannon. The horizontal distance ud travelled by a cannon ball when fired at an angle of ✓ with the horizontal and with an exit initial velocity v can be derived using Newton's laws and is equal to
ud = v cos ✓t =
v2 g
sin 2✓.
In calculating ud , many parameters, such as, air friction, wind velocity etc., have been neglected and therefore, it is expected that the predictions of ud using the above equation to be approximate. 22
The Canon Ball Experiment Alternatively, the distance ud can be estimated by performing actual experimentation. This involves firing identical cannon balls under as much as possible identical conditions repeated number of times (N ) and measuring the distance travelled by the cannon ball, ui.
! ! It is expected that
ud ⇡
1 N
P
ui
In performing the experiments, the effect of air friction and all parameters that are neglected in deriving the equation in the previous slide exist. However, when the mean effects are considered, we see that these parameters do not seem to have any effect.
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The Canon Ball Experiment This apparent contradiction can be appreciated if one realizes that the Newton's laws, or in fact, most physical laws have been postulated based on physical experimentation.
! ! For parameters which do not have a bias, predictions made by neglecting these parameters are acceptable on an average. For example, wind velocity does not have a bias as its directions and velocity can change from time to time.
! ! On the other hand, air friction always opposes the motion. However, neglecting the effect of air friction simplifies the expressions and the induced errors can be negligible in comparison.
! ! All the unmodeled parameters enter the model as uncertainties.
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Uncertainties Uncertainties can be classified under two broad categories.
Uncertainties
Aleatoric uncertainties
Epistemic uncertainties
We explore in more details these classifications in the following pages.
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Aleatoric Uncertainties Aleatory uncertainties are the inherent uncertainties that are associated with a problem and are not reducible with increased knowledge about the system.
! ! Thus, while designing for the tall chimney, the uncertainty associated with the wind load can be classified as an aleatory uncertainty.
! ! If a sufficiently detailed model is developed that takes into account even the effect of wing flapping of a butterfly on the wind flow, then the wind load could be determined with precision. However, neither such a model that takes into account so much complexities is available, and even it was available, it would be impractical to solve such a complex set of equations to determine the wind flow.
! ! Instead, it is much easier to model the wind field to be random and assume probabilistic properties for the loading.
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Aleatoric Uncertainties Aleatory uncertainties are inherent to the isolated system under study and are irreducible.
We need to develop models for these uncertainties and develop techniques for incorporating their effects into the analysis.
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Aleatoric Uncertainties
Figure:
Apple falling on head; The uncertainty associated with the time at which it falls can be classified as aleatory uncertainty
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Epistemic Uncertainties Epistemic uncertainties arise due to lack of knowledge with respect to a particular system and can be reduced with increased knowledge or through better observations. Examples:
• The use of Timoshenko beam theory over the simpler Euler-Bernoulli
beam theory in analysis of beam vibration problems reduces epistemic uncertainties. This is because the Timoshenko beam theory relaxes some of the assumptions made in the Euler Bernoulli beam theory.
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The Euler-Bernoulli beam theory does not take into account the effects of rotary inertia & shear deformation. These effects are considered in the Timoshenko beam theory.
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Epistemic Uncertainties • If numerical techniques, such as, finite elements (FE) are used in the analysis, the results will depend on the size and type of elements. Considering higher order elements and finer mesh size, improves the results (and hence reduces uncertainties. However, the increased accuracy comes at a cost on increased computational costs).
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The size of the FE problem depends on the number of degrees of freedom (dof). Thus,
! {K}{X} = {F} ! where, the size of the matrices depends on the number of dofs.
For higher order elements, or with finer mesh sizes, the number of dofs increase. Thus, the sizes of the matrices become larger which imply that a larger number of simultaneous equations need to be solved. Hence, the computational costs are higher.
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Epistemic Uncertainties • In numerical integration of the governing equations of a dynamical
system, smaller step size being used implies higher number of modes being considered in the analysis.
If a signal with the highest frequency component is! then the smallest time interval at which the signal needs to be sampled is usually 1/20 times the corresponding time period given by
! T = 2⇡/! ! A continuous vibrating system has infinite number of modes. In numerical integration, if the time integration step is taken to be
dT , then the highest mode that is considered in the analysis is given by
!h = 2⇡/(20dT ) ! ! It is implied that the effect of higher modes is not considered. 31
Epistemic Uncertainties • In experimental analysis, inadequate sampling rate in data acquisition could contribute to uncertainties.
Figure:
Schematic diagram of sampling errors; yellow signal indicates the true waveform; the red signal represents the interpolated waveform obtained by inadequate sampling of the true signal. The sampled points of the true signal are indicated by the intersection points of the true and the interpolated signal.
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Epistemic Uncertainties The examples presented here are merely indicative of the type of epistemic uncertainties.
! ! Techniques or guidelines for minimizing/eliminating many of these uncertainties are available in the literature.
! Thus, for example, the sampling uncertainties can be eliminated by adopting a data acquisition system with higher sampling rate, modeling uncertainties can be reduced by developing more complicated beam vibration theories that relax the assumptions and the truncation errors can be minimized by considering a very large number of terms in the series expansions.
!
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Epistemic Uncertainties The salient point that emerges from this discussion is that these uncertainties arise due to
! •either insufficient knowledge,
! •lack of understanding, or
! •due to mathematical idealizations,
! and can be reduced with increased knowledge.
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Summary It is clear from these discussions that uncertainties exist in any analysis and model and these uncertainties propagate into the structure response.
! Any design procedure which do not take into account these uncertainties are bound to be either inadequate or suboptimal.
! Importantly, for risk assessment, it is imperative that the uncertainties associated with the system are quantified.
! The mathematical theories of probability provide a systematic and elegant tool for their quantification.
! Alternative theories based on possibilistic concepts (as opposed to probabilistic theories) have also been discussed in the literature.
! There is a view that possibilistic methods needs to be treated using possibilistic methods while probabilistic methods are more suited for modeling aleatory uncertainties.
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Summary It is clear from these discussions that uncertainties exist in any analysis and model and these uncertainties propagate into the structure response.
! Any design procedure which do not take into account these uncertainties are bound to be either inadequate or suboptimal.
! Importantly, for risk assessment, it is imperative that the uncertainties associated with the system are quantified.
! The mathematical theories of probability provide a systematic and elegant tool for their quantification.
! Alternative theories based on possibilistic concepts (as opposed to probabilistic theories) have also been discussed in the literature.
! There is a view that possibilistic methods needs to be treated using possibilistic methods while probabilistic methods are more suited for modeling aleatory uncertainties.
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