NON-LINEAR AND DYNAMIC ANALYSIS.pptx

NON-LINEAR AND DYNAMIC ANALYSIS
UNIT V –NON-LINEAR AND DYNAMIC ANALYSIS
COMPUTER AIDED ENGINEERING

By
Prof.(Dr) D Y Dhande
Professor
Department of Mechanical Engineering
AISSMS College Of Engineering, Pune – 411001
Email:dydhande@aissmscoe.com
Prof (Dr) D Y Dhande

COURSE OBJECTIVE (CO5)
• Evaluate and solve nonlinear and dynamic analysis
problems by analysing the results obtained from analytical
and computational method.
Prof (Dr) D Y Dhande

CONTENTS
• Non-linear Analysis- Introduction to nonlinear problems, comparison of
linear and non-linear analysis, types of nonlinearities, Stress-strain measures
for non-linear analysis, Analysis of geometry, Material Nonlinearity, Solution
techniques for non-linear analysis, Newton-Raphson Method, Essential steps
in Nonlinear analysis. (No numerical treatment)
• Dynamic Analysis- Introduction to dynamic analysis, Comparison of static
and dynamic analysis, Time domain and frequency domain, types of loading,
Simple Harmonic motion, Free vibrations, Bounday conditions for free
vibrations, Solution

NON-LINEAR ANALYSIS

INTRODUCTION TO NON-LINEAR ANALYSIS
• When structure response (deformation, stress and strain)is not linearly proportional to
the magnitude of the load, then the analysis is known as non-linear analysis.
• Due to cost and weight advantage of non-metals (polymers, woods, composites) over
metals, non-metals are replacing metals for variety of applications, which have
nonlinear load to response characteristics, even under mild loading conditions.
• Also, the structures are optimized to make most of its strength, pushing the load level
so close to the strength of the material, that it starts behaving nonlinearly.
• In order to accurately predict the strength of the structures in these circumstances, it is
necessary to perform nonlinear analysis.

COMPARISON OF LINEAR & NON-LINEAR ANALYSIS

TYPES OF NON-LINEARITIES
• Geometric Non-linearities:
• Stiffness changes due to geometric deformations are categorized as geometric
nonlinearities. The different kinds of geometric nonlinearities are: (i) Large strain; (ii)
Large deflection (large rotation); (iii) Stress stiffening; (iv) Spin softening
• Large Strain:
• If element’s shape changes (area,thickness, etc), its individual element stiffness will
change.

• Large Rotation:
• If element’s orientation changes (rotation), transformation of its local stiffness to global
components will change which induces large strains.
• Stress Stiffening:
• This is associated with tension bending coupling. The more the tension in the
membrane, more its bending rigidity or stiffness. If element’s strains produce a
significant inplane stress state, the out of plane stiffness can be significantly affected.

• Spin softening:
• This phenomena is related to stiffness change due to rotational speed. Most common
in components such as shrink fit assembly rotation at a high speed and the press fit
interference changing into a clearance fit due to large circumferential deformation.
Another example is change in tension in the cable or rope in brake dynamometer as a
function of rotational speed of drum.Prof (Dr) D Y Dhande

• Material Non-linearities:
• The engineering material behaviour cannot be idealised using a single constitutive law
for the entire range of environmental conditions loading, temp and rate of deformation.
• The linear elastic (also called Hooken) material assumption is simplest of all.
• The martial is nonlinear elastic if the deformation is recoverable and plastic if it is not
recoverable.
• If the temperature effects on material properties are important, then the coupling
between mechanical and thermal behaviour should be properly taken into
consideration through thermo-elasticity and thermo-plasticity.
• If strain rate has significant effects on material, them we have to consider the theories
of visco-elasticity and visco-plasticity. A sample is given in below figures:

• A brief classification can be given as below:
1. Non-linear elastic, 2. Hyper elastic, 3.Elastic-perfectly plastic; 4. Elastic-Time
dependant plastic 5. Time dependant plastic (Creep); 6. Strain rate dependant
elasticity-plasticity, 7. Temperature dependant elasticity and plasticity.
• The material non-linearity can be classified further as: (i) Linear elastic-perfectly
plastic; (ii) Linear Elastic-Plastic.
• The plastic part in stress strain curve is time dependant and can be analysed into two
main types: (i) Elastic-Piecewise linear plastic; (ii) Elastic-actual stress strain curve
• Non-linear elastic model characterizing materials with no fixed definition of yield point
such as say plastic but the strain still limiting well below say 20%
• Hyper elastic materials such as rubber undergoing large displacements (Gaskets).

• Boundary Non-linearities:
• Boundary nonlinearity arise when boundary conditions in a FEA model changes during
the course of analysis. The boundary conditions could be added or removed from the
model due to boundary non-linearity as the analysis progresses.

STRESS STRAIN MEASURES FOR NONLINEAR ANALYSIS
• There are different measures of strain and stress for nonlinear analysis. Some are
explained below:
(i) Engineering strain and engineering stress;
(ii) Logarithmic strain and true stress;
(iii) Green-Lagrange strain and 2nd Piola-Kirchoff stress.
• Let us consider below example to demonstrate these stresses.

• Engineering Strain and Stress:
• Engineering strain measure is a linear measure since it depends upon the original
geometry (length) which is known.
𝜀𝑥 =
∆𝑙
𝐿
• Engineering stress() is the conjugate stress measure to engineering strain (). It
uses the current force F and the original area A0 in its computation.
𝜎 =
𝐹
𝐴0
• Logarithmic Strain and true stress:
• Logarithmic strain/natural strain/true strain is a large strain measure which is
computed as: Prof (Dr) D Y Dhande










  0
0
ln
0
l
l
l
dl
l
l
l

• This measure is a nonlinear strain measure since it is a nonlinear function of the
unknown.
• The 3D equivalent of the log strain in the Hencky strain.
• This strain measure is widely used in nonlinear analysis and is a additive strain
measure as compared to linear strain.
• True stress () is the conjugate 1D stress measure to the log strain (i) which is
computed by dividing the forec F by the current area A. This measure is commonly
referred as Cauchy Stress.
𝜏 =
𝐹
𝐴

• Green-Lagrange Strain and 2nd Piola-Kirchoff Stress:
• This is another large strain measure which is computed in 1D as:
• This measure is non-linear because it depends upon the square of the updated length
l, which is an unknown.
• Its advantage over Hencky strain is that it automatically accommodates arbitrary
large rotations in large strain problems.
• The conjugate stress measure for the Green Lagrange strain is the 2nd Piola-Kirchoff
stress which is computed as:







 
 2
0
2
0
2
2
1
l
l
l
G

0
0
A
F
l
l
S 

SOLUTION TECHNIQUES FOR NONLINEAR ANALYSIS
• The stiffness matrix in a nonlinear static analysis needs to be updated as analysis
progresses.
• When a structure is subjected to external loading, internal loads are generated. These
are caused due to internal stresses in the structure. For the equilibrium of the
structure, each node of the structure must be in equilibrium.
• The condition of equilibrium is checked by determining residual load, which is
difference between internal and external load. For perfect equilibrium, this residual
load must be zero which is unlikely situation in non-linear analysis.
• The FEA codes assume the nodes to be in equilibrium if the residual load is negligibly
small. Prof (Dr) D Y Dhande

• Let us consider point 1 in the load displacement curve for a node in a FE model,
which is in equilibrium (i.e. difference between external load P1 and internal load is
zero).
• When external load is increased by small amount to P2, the incremental displacement
using linear elastic theory is-
d2-d1 = k1 (P2-p1), where K1 is stiffness based on configuration at point.
• The FEA code will now calculate the internal load at point 2 and comapres the
residual load at point 2, which is given by
R2 = P2-l2
• If R2 is more than the acceptable residual load, the FEA code locates the new point 3
using linear relationship- d3-d2 = k2(P2-P1)

• Then new stiffness matrix K3 is formed on the configuration at point 3 and
corresponding internal load l3 is calculated. The residual load at point 3 becomes:
R3 = P2-l3
• If R3 meets the residual load criteria, the solution is said to be converged for that
particular increment. Each attempt of calculating residual load for an increment is
called interation.
• Once the solution is converged for a load increment, FEA code accepts it as a
equilibrium and increases the load by further increment.This process is repeated until
entire load is applied on the structure.
• This procedure of stiffness update at every step is known as Newton-Raphson
method.

• Another common option is to update the stiffness after a number of steps is known as
Modified Newton-Raphson method.
• In FEA analysis, if the load is applied in too many steps or if stress strain curve is
represented in too many segments, the computational cost will be high.
• FEA analyst control the number of steps in a nonlinear static analysis by specifying
number of increments (% of total load).

ESSENTIAL STEPS TO START WITH NONLINEAR ANALYSIS
• Learn first how the software works on a simple model before using a non-linear
feature which you haven’t used.
• Refer and understand software manual and supporting documentation.
• Prepare a list of questions for which you want to carry out analysis. Design the
analysis such as model, material model, boundary conditions in order to answer the
questions.
• Keep the final model as simple as possible. Check all boundary conditions as well as
meshing.
• Verify the results of the nonlinear FEA solution.
• Try to look into the assumptions made with respect to the structural component, its
geometry behaviour and different material models.

DYNAMIC ANALYSIS

• Static analysis does not into account variation of load with respect to time. Output in
the form of stress, displacement etc with respect to time could be predicted by
dynamic analysis.
• In static analysis, velocity and acceleration (due to deformation of component) are
always zero. Dynamic analysis can predict these variable with respect to
time/frequency.
• To determine natural frequency of component which ia also helpful for avoiding
resonance, noise reduction and mesh check.
• When the excitation frequency is close to natural frequency of component, there
would be big difference in static and dynamic results. Static analysis would probably
show stress magnitude within yield stress and safe but in reality it might fail.
NEED FOR DYNAMIC ANALYSIS

STATIC ANALYSIS VS DYNAMIC ANALYSIS

• For a single dof problem, when frequency of excitation is one third of fundamental
frequency, the problem can be treated as static.
• What is frequency:
• Frequency is number of occurrences per unit time. (e.g a doctor prescribe medicines
3 times a day to a patient, the frequency is 3 per day)
• Let us consider a steel ruler and disturb it slightly and observe the vibrations. Ruler
deflects on either sides of the mean position. Time require to complete one cycle is
time period.
• Number of cycles per seconds is known as Hertz (Hz).

• Frequently used formula for natural frequency , n = (k/m) is for circular frequency
and units are rad/sec.
• While, most of the commercial softwares give output in “Hz” i.e. cyclic frequency i.e.
f = (1/2) (k/m)
• How to convert rpm to Hz : Engine speed 6000rpm means vibrations at 6000/60 rps
or 100Hz
• Effect of natural frequency on noise: Noise and vibrations are interrelated.

• When the length of steel ruler is reduced, it changes natural frequency but also
reduces the noise. One important factor in reducing the noise is altering the natural
frequency.
DIFFERENCE BETWEEN FREQUENCY AND TIME DOMAIN

• Imagine you are sitting on a chair and ground starts vibrating first slowly and then
gradually faster and faster. This phenomenon could be presented easily in frequency
domain via a single straight line as shown in the figure.
• Its equivalent representation in time domain is shown in right side figure (for simplicity
constant amplitude of vibration is assumed)
• Frequency and time domain are inter convertible.
f = 1/T

TYPES OF LOADING
Constant
Periodic

• Non-Periodic Transient
response (Random Vibrations) :

SIMPLE HARMONIC MOTION
• Simple harmonic motion, regular vibration in which the acceleration of the
vibrating object is directly proportional to the displacement of the object from
its equilibrium position but oppositely directed. A single object vibrating in this
manner is said to exhibit simple harmonic motion (SHM).

SPRING MASS REPRESENTATION
• In Mechanical Engineering, any system or component could be represented by three
basic elements i.e. mass, stiffness and damping. For example, entire railway bogie,
chassis of automobile or even a human body could be represented mathematically as
a spring , mass and damping of appropriate values.

FREE VIBRATION
• The ordinary homogeneous linear second order differential equation for spring mass
system is given as: .
• Applying Newton's laws of motion, we have
• The solution of above equation is :
• This is Simple Harmonic Motion.

NATURAL FREQUENCY ANALYSIS
• The frequency with which any object will vibrate if disturbed and allowed to vibrate on
its own without any external force is known as natural frequency. It is estimated by
equation, n = (k/m)
• Damping is neglected for natural frequency calculations.
• Any object has first 6 natural frequencies = 0 when run free-free.
• All real life objects have infinite natural frequencies but FEA can compute compute
natural frequencies equal to dof s of the FEA model only.
• Lowest natural frequency is known as fundamental frequency.

NATURAL FREQUENCY ANALYSIS FOR CANTILEVER BEAM
 
  35851
.
449
100
3
10
8
.
7
25
.
2
10
1
.
2
786
.
149
100
3
10
8
.
7
25
.
0
10
1
.
2
inertias,
of
moments
two
to
ing
Correspond
2
:
as
calculated
is
frequency
natural
The
4
9
5
4
2
4
9
5
4
1
4































































AL
EI
AL
EI
AL
EI
n
f i
i




The value of 1st to ‘n’ are 3.51602, 22.0345, 61.6972,
120.902, 199.86, 298.556, 416.991, 555.165, 713.079,
890.732

Exact Solution (Beam Theory) FEA Solution (Quad
4)
1 83.81 83.03
2 251.45 249.01
3 525.28 508.66
4 1470.81 1395.44
5 1575.84 1521.85
6 2882.20 2675
7 4412.43 4158.55
8 4764.39 4294.04
9 7117.32 4513.20
10 8646.61 6060.61
11 9491.35 7498.46

12 13234 7920.43
13 14293.49 12592.44
14 16999.21 12944.66
15 21234.32 13313.31
16 21352.01 17559.74
17 29822.20 21445.83
18 39703 21510.63
19 50996.52 28502.96
20 63703.03 34130.84

NON-LINEAR AND DYNAMIC ANALYSIS.pptx

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