So far our study was confined within static analysis. In this section we shall learn some aspects of dynamic analysis. The theory and field of structural dynamics are very large. Even a decent introduction of this subject would require at least 100 pages. I am just describing some very basic concepts of structural dynamic useful for practicing engineers. If you’ve already studied the theory of structural dynamic then it’s great. If not, I advise that you start reading a textbook of structural dynamic besides this book. It will make you understand the applications of dynamics discussed in this book in an easily understandable manner.
At the beginning, let me explain the difference between static and dynamic analysis. In static analysis, the applied force is constant but in dynamic analysis, the applied force varies with time. It is not necessary that dynamic analysis always involves application force only, it may be due to shaking of ground due to earthquake. In civil engineering applications, dynamic analysis mostly involves determination of maximum response (i.e. displacement etc.) of the structure due to some applied ground acceleration.
To understand dynamic analysis properly, we need understand the concepts of some common terms.
Degrees of Freedom – consider the typical spring mass damper system as shown below.
Mass m Displacement x(t)
Stiffness k
Figure 32-1
It has mass 'm' and stiffness 'k'. It can move in only the direction shown by the arrow. So, this model has one degree of freedom. We call this single degree of freedom (SDF) system. We also assume that the mass 'm' is 'lumped' at the top of model as shown by the filled circle. This means, the mass of the stick, though distributed though out its length, we assume as if it is concentrated at one place as shown by the circle. This model may reflect idealization of a single story
'lumped' at the respective floor levels. In that we shall have multi-degree-of- freedom (MDF) structures. Things may appear little bit terse at the beginning, but gradually everything will seem comprehensive. I assure you!
In practice most structures are of MDF type. However, that does not imply that we need not study theory of SDF structures, because many MDF structures can be 'broken' into separate SDF structures and can be easily analyzed rather than analyzing the whole structures!
Damping – if a structure is displaced from its equilibrium position by a small amount of force, it will vibrate (move from this direction to that direction). Unless there is ‘something’ to prevent vibration, it will go on vibrating forever. But in actual practice, the ‘amplitude’ of vibration will gradually diminish and after some time, the model will come to rest. This process by which free vibration diminishes is known as ‘damping’.
Natural frequency (ωn) – it is the number of cycles per second a structure
vibrates. It’s measured in radian per second. It is related to natural time period Tn = 2π/ωn. Natural frequency is computed using ωn = (stiffness/mass)0.5. For
damped structures, damped natural frequency ωd = ωn(1- ξ ²)0.5.
Damping ratio (ξ) – it is a measurement of how much damping is there. It has values in the range of 0 to 1. It is often expressed in %. For example 100% damping means the structure will not vibrate at all. However, for most practical structures, this value lies in the range of 5% to 20% i.e. ξ = 0.05 to 0.2.
Why bother studying dynamic analysis?
If a dynamic excitation (force or ground acceleration) is applied to a structure, the resulting displacement might be much more than that obtained by simple static analysis. More displacement means higher values of internal forces (bending moment, shear force etc.) and subsequently higher values of stresses in the members. If the stress reaches the yield strength of the material the structure will collapse!
For a time varying applied force, if I apply calculated values of the force at particular instant of time and then perform static analysis with that force, shall I get correct displacement etc.?
No, you won’t (even for a linear structure)! The concept of structural dynamics is different from static analysis theory. You must perform exact dynamic
Is it mandatory to perform dynamic analysis for all structures?
It depends. Dynamic analysis is especially required for multi-story buildings for earthquake analysis. Normally, in low height buildings, dynamic analysis does not produce much different result compared to static analysis. However, for all structures, where vibration is a major factor in design, dynamic analysis must be performed. Designing of machine foundations always requires dynamic analysis. For highly important structures like bridges, dams, nuclear reactors etc. dynamic analyses are very important. Theory of structural dynamics is applicable equally to buildings as well as automobiles!
How the programs calculate dynamic response?
Nearly all analysis programs calculate dynamic response by numerical methods. The main governing equation for dynamic analyses is conventionally written as
Where, [M] = mass matrix (kg or Ns²/m), [C] = damping matrix (Ns/m), [K] = stiffness matrix (N/m), X = displacement (m), X’ = velocity (m/s), X” = acceleration (m/s²) and F(t) = Force (N).
The F(t) in above equation will be replaced by –[M]{u(t)”g} if there is ground
acceleration instead of nodal forces. Here, u(t)”g denotes ground acceleration
(m/s²). If F(t) = 0, then the situation is known as free vibration. If [C] = 0, then we call undamped motion. However, in real life, [C] is never equal to zero. In the subsequent sections, we shall explore various examples of structural dynamic analyses – simple to complex! By this time I expect that you will also study a few pages of dynamics textbook. Suggested chapters for reading in your textbook are – introduction and simple formulation for SDF systems, direct solution of differential equation of motion, damped and undamped motion, response to harmonic and periodic excitation, numerical evaluation of dynamic response, equation of motion for MDF systems, modal and response spectrum analyses. If you are interested, you may read all the chapters of the book, but above topics are enough for understanding the calculations presented in this book. You may read Ref. 5, 16, 17 to start with in dynamics.
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