This is an Introductory material for those who want to understand the basic difference between linear and nonlinear analysis in the context of civil and structural engineering.

1. OVERVIEW OF LINEAR AND NONLINEAR ANALYSIS FOR
PRACTICING STRUCTURAL
ENGINEERS
BY
VARANASI .V.S.H.RAMA RAO
DISCIPLINE PROJECT MANAGER ( CIVIL AND STRUTURAL)

2. LINEAR ANALYSIS
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CONTENTS
Linear Static Analysis
Frequency Analysis
Linear Dynamic Analysis
Modal Time History Analysis
Harmonic Analysis
Random Vibration Analysis
Response Spectrum Analysis
Linearized Buckling Analysis
NON LINEAR ANALYSIS
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Solution procedures
Introduction to Non-Linear dynamic analysis

3. LINEAR STATIC ANALYSIS
When loads are applied to a body the body deforms and the effects of loads are
transmitted throughout the body
The external forces induce internal forces and reactions to render the body into a state of
equilibrium
What are the assumptions for Linear Static Analysis?
•All loads are applied gradually and slowly until they reach their full magnitude
•After reaching full magnitude the loads remain constant
•Inertial and damping forces to small velocities and accelerations are neglected

4. You can make linearity assumption if:
All material in the model comply with Hooke’s Law
The induced displacements are so small that they cause negligible change in the
geometric and material properties and hence the stiffness
The structure subjected to loading has negligibly small Accelerations and
Velocities
The boundary conditions doesn’t change during loading.
Time variant loads that induce considerable inertial and damping forces may
warrant Dynamic Analysis

5. What does linear static analysis do?
It calculates the displacements, stresses, strains and reaction forces under the affect of
applied loads.
General Equation of motion
[M] x’’(t) + [C] x‘ ( t) + [K] x(t) = F
Since static analysis ignores time dependent effects i.e acceleration
and velocities due to relatively small magnitude, the above
equation shrinks to
[K] x = F
In the above equation x is independent of time.

6. External Load
Linearity
Non Linearity
Displacement
Linear Elastic : The curve is the linear and holds the same equation for both loading
and un loading
Non Linear –Elastic: The curve is non linear and holds the same equation for both
loading and unloading ( not true for structural steels but can be true for materials
like rubber)

7. Before entering into the subject of linear dynamic analysis we
will learn the following :
FREQUENCY ANALYSIS
•Every structure has a tendency to vibrate at a certain frequency
known as natural frequency or resonant frequency
•Each natural frequency is associated with a particular deflection
pattern of the structure and this pattern is known as mode
shape
Mode shapes
•When a structure is properly excited by dynamic load with
frequency that coincides with natural frequency of the structure,
it undergoes large displacements and stresses. In such cases
Static Analysis cannot be used.

8. If your design is subjected to dynamic environments of considerably severe nature,
Static studies cannot be used to evaluate the response
Frequency studies:
•can help us to design a structure which has natural frequencies considerably away
from the frequency for the loading.
•help us to design vibration isolation systems
•Form the basis for evaluating the response of linear dynamic systems where the
response of a system to dynamic environment is assumed to be summation of the
responses of various modes considered in the analysis.

9. LINEAR DYNAMIC ANALYSIS:
Static analysis assumes that the loads are constant or applied very slowly until
they reach their full values. Because of this assumption, the velocity and
acceleration of each particle of the model is assumed to be zero. As a result, static
studies neglect inertial and damping forces.
For many practical cases, loads are not applied slowly or they change with time or
frequency. For such cases, use a dynamic analysis. Generally if the frequency of a
load is larger than 1/3 of the lowest (fundamental) frequency, a dynamic study
should be used
Objectives of a dynamic analysis include:
•Design structural and mechanical systems to perform without failure in dynamic
environments.
•Modify system's characteristics (i.e., geometry, damping mechanisms, material
properties, etc.) to reduce vibration effects.

10. Linear Static vs Dynamic analysis
[M] x’’(t) + [C] x‘ ( t) + [K] x(t) = F(t)
In linear static analysis the Mass, Acceleration, Damping velocity are
neglected.
Where as , In dynamic analysis the above are considered and also force
is time dependent.
In dynamic analysis the response is give in terms of time history (
response vs time or in terms of peak response vs frequency)
In Linear Dynamic Analysis the basic assumption is Mass, Damping and
stiffness matrices in the above equation remain unchanged during the
duration of loading and un loading.

11. Dynamic loads
Dynamic loads are two types deterministic and non deterministic
Deterministic loads are well defined functions of time and can be predicted precisely.
They can be harmonic, periodic or non periodic- Example :centrifugal machine loading
Non deterministic loads cannot be defined explicitly as functions of time and they are
best described by statistical parameters- Example :earthquake loading
Typical dynamic loadings

12. Damping effects
•If you apply some force and leave a system to vibrate, it will come to rest after
some time. This phenomenon is called damping
•Damping is a physical phenomenon that dissipates energy by various
mechanisms like internal and external friction, air resistance etc
•It is difficult to represent damping mathematically as it happens through
several mechanisms
•For many cases damping effects are represented by equivalent viscous dampers
•A viscous damper generates a force that is proportional to velocity .

13. There are four approaches for Linear Dynamic Analysis
• Modal Time History Analysis
• Harmonic Analysis
• Random Vibration Analysis
• Response Spectrum Analysis

14. MODAL TIME HISTORY ANALYSIS
Use modal time history analysis when the variation of each load with time is
known explicitly, and you are interested in the response as a function of time.
Typical loads include:
•shock (or pulse) loads
•general time-varying loads (periodic or non-periodic)
•uniform base motion (displacement, velocity, or acceleration applied to all
supports)
•support motions (displacement, velocity, or acceleration applied to selected
supports non-uniformly)
•initial conditions (a finite displacement, velocity, or acceleration applied to a
part or the whole model at time t =0)

15. Modal Analysis Procedure
[M] x’’(t) + [C] x‘ ( t) + [K] x = F(t)
The above differential equation is a system of n simultaneous ordinary differential
equations with constant coefficients.
The objective of the modal analysis is to transform the coupled system into a set
of independent equations by using modal matrix as transformation matrix
The above is modal matrix
The normal modes and eigenvalues of the system are derived from the solution of
the eigenvalue problem:

16. For linear systems, the system of n equations of motion can be de-coupled into
n single-degree-of-freedom equations in terms of the modal displacement
vector {x}:
[x]= {Φ}u
Substituting this in the main equation of motion and pre multiplying {Φ} T with
we get
{Φ} T[M] {Φ} u’’(t) + {Φ} T [C] {Φ} u‘ ( t) + {Φ} T [K] {Φ} u = {Φ} T F(t)
The normal modes satisfy the orthogonality property, and the modal matrix is
normalized to satisfy the following equations:
{Φ} T[M] {Φ} =1
{Φ} T [C] {Φ} = 2 [ζ] [ω]
{Φ} T [K] {Φ}= [ω2]

17. The resultant equation after substituting the above is
u’’ + u‘2 [ζ] [ω] + [ω2] u = {Φ} T F(t)
The above is system of n –independent second order differential equations which is
solved by step by step integration methods like wilson theta
HARMONIC ANALYSIS
This analysis is used to calculate steady state peak response due to harmonic
loading or base excitations.
Although you can create a modal time history study and define loads as functions of
time, you may not be interested in the transient variation of the response with
time. In such cases, you save time and resources by solving for the steady-state
peak response at the desired operational frequency range using harmonic analysis.

18. RANDOM VIBRATION ANALYSIS
Use a random vibration study to calculate the response due to non-deterministic
loads.
Examples of non-deterministic loads include:
•loads generated on a wheel of a car traveling on a rough road
•base accelerations generated by earthquakes
•pressure generated by air turbulence
•pressure from sea waves or strong wind
In a random vibration study, loads are described statistically by power spectral density
(psd) functions. The units of psd are the units of the load squared over frequency as a
function of frequency.
The solution of random vibration problems is formulated in the frequency domain.
After running the analysis, you can plot root-mean-square (RMS) values, or psd results
of stresses, displacements, velocities, etc.

19. RESPONSE SPECTRUM ANALYSIS
What is response spectrum?
Response spectrum is a plot of peak response vs modal frequency ( for a given
damping)of various single degree freedom systems ( representing various modes of
vibration of the structure) subjected to same dynamic loading.
The normal modes are calculated first to decouple the equations of motion with the use of
generalized modal coordinates. The maximum modal responses are determined from the base
excitation response spectrum. With the use of modal combination techniques, the maximum
structural response is calculated by summing the contributions from each mode

20. LINEARIZED BUCKLING ANALYSIS
Slender structural members tend to buckle under axial loading.
Buckling is a sudden deformation which occurs when stored axial energy is converted
in to bending energy without change in the externally applied load
Mathematically when buckling occurs the stiffness matrix becomes singular
The linearized buckling model solves an eigen value problem to determine the critical
buckling factors and the associated mode shapes
A model can buckle in different shapes under different levels of loading. The shape the
model takes while buckling is called buckling mode shape and the corresponding
loading is called the critical buckling load
Engineers are interested in the lowest buckling mode because it is associated with
the lowest critical buckling load

21. NON LINEAR ANALYSIS
All structures behave non linearly in one way or other beyond a particular level
of loading.
In some cases linear analysis may be adequate but in many cases the linear
analysis may produce an erroneous results as the assumptions on which linear
analysis is done may be violated in real time structure.
Non linear analysis is the most generalized form of analysis and linear analysis is
a sub-set of it.
Non linear analysis is needed if the loading produces a significant changes in the
stiffness

22. Major sources of structural non-linearities:
Geometrical Non Linearity
Large displacements change geometry
Material Non linearity
Non linear relationship between stress and strain
E.g. Yielding of beam column connections during earthquake
Contact Non linearity
E.g. gear-tooth contacts, fitting problems, threaded connections, and impact
bodies

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When a load causes significant changes in stiffness, the load-deflection curve
becomes nonlinear.
The challenge is to calculate the nonlinear displacement response using a linear
set of equations.
External Load
Linear Response
Nonlinear Response
Displacement

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One approach is to apply the load gradually by dividing it into a series of
increments and adjusting the stiffness matrix at the end of each increment.
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The problem with this approach is that errors accumulate with each load
increment, causing the final results to be out of equilibrium.
External Load
Calculated
Response
Error
Nonlinear Response
Displacement

25. Other Approach : Newton-Raphson algorithm:
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Applies the load gradually, in increments.
Also performs equilibrium iterations at each load increment to drive the incremental
solution to equilibrium.
[KT]
Solves the equation [KT]{Du} = {F} - {Fnr}
[KT] = tangent stiffness matrix
F
{Du} = displacement increment
Fnr
{F}
= external load vector
{Fnr} = internal force vector
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Iterations continue until {F} - {Fnr}
(difference between external and internal
loads) is within a tolerance.
2
3
4 equilibrium
iterations
1
Du
Displacement
Some nonlinear analyses have trouble converging. Advanced analysis techniques are
available in such cases.

26. NON LINEAR DYNAMIC ANALYSIS
In this analysis ,unlike linear dynamic analysis the mass , damping and stiffness matrix
are varying and get updated during each iteration.
In nonlinear dynamic analysis, the equilibrium equations of the dynamic system at
time step, t+δt, are:
[M] t+ δ t {U '' } (i) + [C] t+ δ t {U ' } (i) + t+ δ t [K] (i) t+ δ t [ D U] (i) = t+ δ t {R} - t+ δ t {F} (i-1)

27. Thank you