This document summarizes key concepts from the textbook on structural dynamics. It discusses moment of inertia, single and multi-degree of freedom systems, equations of motion, eigen values, forced and free vibrations, damping, modal response, and applications to tall buildings and bridges. Health monitoring of structures is also addressed, covering instrumentation used and typical causes of structural issues.

1. PART - 4
IN ORDER TO EMPHASIZE
the conceptual contents, the mathematical equations are rarely used in
the slides, particularly if it means repetition with the contents of the book
The aspects covered are:
1. Moment of Inertia, an Important property
2. Single degree Mechanism, and Response of Structures of Single
Degree Freedom
3. Stable structures with Multi Degree Freedom System
4. Equation of Motion, and Eigen value Problem
5. Application –Tall Buildings
6. Health Monitoring of Structures

2. Structural Dynamics
(Displacement Method)
1. Force method is ideally suited for simple skeletal structures. The practical cases
have been elaborated in the book. Flexibility approach has been success fully
used for the following two cases (2 and 3)
2. Even for the plates, flexibility approach for member flexibilities has been
successful used. The member flexibility matrix, however, is inverted to obtain the
member-finite element stiffness matrix . Subsequent steps of analysis follows the
Displacement method
3. It is, sometimes more expedient to invert the global stiffness matrix and use F to
obtain the Eigen values (natural frequencies) and mode shapes
4. Otherwise, the stiffness approach is synonymous with the Dynamic analysis
5. Nodal displacements are also the degrees of freedom for Dynamic analysis
6. Several numerical techniques including Runge-Kutta technique use modal
superposition valid for elastic structures
7. New mark β-method has been used for the dynamic Response of the structures
using inelastic materials.
8. FFT analyzer has enabled extensive use of Frequency domain approach for
elastic systems and random loading. A revolution in dynamic analysis.

3. Moment of Inertia
an understanding (through interaction)
QUIZ
Mass Moment of Inertia and Area Moment of Inertia are
Mass and Area based respectively,
Why do we call the quantity ‘I’, moment of Inertia?
Scan the number of locations EI has appeared in the text. Its
companion is EA.

4. Cases of Single Degree Freedom
1. Single Degree Freedom Mechanism – Rotation to Translation
2. Single Degree Freedom structure – No damping
3. Typical Single Degree Freedom structure
Portal with rigid girder Water tank

5. EQUATION OF MOTION WITHOUT DAMPING
FREE VIBRATION
INEXTENSIBILITY IS NOT ASSUMED, Δ 9X1
M Δ + K Δ = 0
NINE FREQUENCIES AND NINE MODE SHAPES
Multi Degree Freedom System
Uniform masses of girders
converted to consistent masses

6. Eigen Values λ
Natural Frequencies and Mode shapes
are Eigen values and Eigen vectors
respectively
(Number of Mode shapes is Degrees of Freedom)
A similar engineering values are Principal stresses
and associated direction cosines.
A X = λ X

7. WITH DAMPING THE FREQUENCIES DO NOT CHANGE SIGNIFICANTLY
Equation of Motion with Damping, Free and Forced
M δ + CΔ + K Δ = 0 Free Vibrations
Nine frequencies for the structure slide 5, wherein inextensibility is not
assumed. However, vertical deflections at the nodes are not significant.
M δ + CΔ + K Δ = P(t) Forced Vibrations

8. Earthquakes and Inertial Loading
Mass Loading
• Many believe that there is only one who is static. In the Universe every thing else is
Dynamic. Einstein** and similar other philosophers have been exploring this
interesting aspect of the creation
• The Structures are in motion and the motion is the same as the earth’s crust at the
location of supports of our structure. The crust also at some locations is weak and
stressed up.
• When earth’s crust is cracking, it gives rise to Ground Motion (a, 𝑎, 𝑎) which we call as
earthquakes at macro scale. Our structure (relatively a micro) develops the inertial
forces, and because of its Mass, it is subjected to Mass Loading. Large dams may not be
treated as micro.
• The mass loading = mass x 𝑎. In the limit, if the mass is zero, there is no earthquake
loading.
• The equation of motion for the earthquake loading is on pp 515*
• Our interest is more on the lateral loading
• FFT Analyzer helps the use of Frequency Domain Approach. It is preferred for random
loading on elastic systems. The examples of random loading are:
Wind, EQ, Ocean waves, Vehicle loading on Bridges.
** The pages refer to the book.

9. Modal Response and Participation
Factor
• Modal Equation of motion and Participation
Factor. pp 555 and 559**
• The coefficients of a generalized Fourier series
are the Participation Factors.
• Because of good convergence, the response is
generally limited to the first few mode shapes.

10. Relevance of the Approximate
methods
1. Approximate methods are made possible by the
robust understanding of the structural behavior
2. A student of structural dynamics should not entirely
rely on software, associated mathematics and
MATLAB. After all, the numerical techniques used in
the Engineering software are ALSO approximate.
3. The following are some approximate methods, using
approximation like Rigid slab-girders as horizontal
diaphragm and masses discretized at the floor levels:
Stodola method, and Holzer’s method.

11. Dynamic Response
Time Domain
• Elastic and Inelastic
response
Frequency domain
• Spectral graph of the
Ground Motion and our
Structure’s ‘response
spectral graph’.
• The analysis is characterized
by Linear system and
Irregular excitation
• Crucial examination of
Response spectra for:-
Maximum amplitudes and
corresponding Frequencies

12. Response based on
Frequency Domain
• Transform Pair, p(t) and c(ω ) and
• Response X(t) also in Integral form
• Relevance of DFT and FFT *
• The Frequency Domain is used for
1. Linear systems, and
2. Irregular excitations.
* Read Clough and Penzien

13. Application
TALL BUILDINGS
Loadings:
Gravity: Dead and Live
Lateral Loading: (a) Wind critical for 20 storeys and above
Return period- almost every year
(b) Earth quake is equally critical for low rise buildings and Tall buildings
Return period is ?
In the case of Bridges Vehicular Loading is the random loading
also need to be considered

14. Bridges and Inextensibility
1. Axial effect is delinked from rotations, and therefore
2. The degrees of freedom in the case of Grid Decks,
there are three degrees of freedom per node
3. They are two transverse rotations and the third is
transverse deflection.

15. Assumption of Inextensibility and
Degrees of Freedom
• QUIZ:
• 13 storeys and 2 bays
• Cases:
1. All members are inextensible,
2. Girders only are inextensible,
and
3. All girders are flexurally rigid,
and all members are
inextensible.

16. HEALTH MONITORING OF STRUCTURES
(A) Typical following Causes
• Lack of durability of the constituent Materials
• Seepage arising due to various factors
including Faulty Plumbing
• Concealed Electrical wiring and the faulty
sheathing
• Trees close to the building
• Fatigue in Tall buildings, Steel Bridges, and
Machine loading on the Industrial Structures
• Neighboring construction activity

17. (B) INSTRUMENTATION USED FOR
REMOTE MONITORING
Careful planning of Transducer points
(i) Displacements are critical at top, and
(ii) Stress resultants are critical close to bottom.

18. Please read the Book
If you liked reading and found it useful
Send your review to McGraw-Hill and my
blog, kavetiseetharamulu.blogspot.com
Thank you
Kaveti Seetharamulu
Faculty of the IITB from 1958 to Nov. 30, 1966
Faculty of the IITD from Dec 1, 1966 to Nov. 30, 1993