linear-nonlinear for structures

7/24/2019 linear-nonlinear for structures

1/27

BYVARANASI .V.S.H.RAMA RAO

DISCIPLINE PROJECT MANAGER ( CIVIL AND STRUTURAL)

OVERVIEW OF LINEAR AND NON

LINEAR ANALYSIS FOR

PRACTICING STRUCTURAL

ENGINEERS


2/27

CONTENTSLINEAR ANALYSIS

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

Solution procedures

Introduction to Non-Linear dynamic analysis


3/27

LINEAR STATIC ANALYSIS

When loads are applied to a body the body deforms and the effects of loads aretransmitted 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


4/27

You can make linearity assumption if:

All material in the model comply with Hookes 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 doesnt change during loading.

Time variant loads that induce considerable inertial and damping forces may

warrant Dynamic Analysis


5/27

What does linear static analysis do?

It calculates the displacements, stresses, strains and reaction forcesunder 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.


6/27

Displacement

Extern

alLoad

Linearity

Non Linearity

Linear Elastic : The curve is the linear and holds the same equation for both loading

and un loading

Non LinearElastic: 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 materialslike rubber)


7/27

Before entering into the subject of linear dynamic analysis wewill learn the following :

FREQUENCY ANALYSIS

Every structure has a tendency to vibrate at a certain frequencyknown as natural frequency or resonant frequency

Each natural frequency is associated with a particular deflectionpattern of the structure and this pattern is known as modeshape

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 casesStatic Analysis cannot be used.

Mode shapes


8/27

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.


9/27

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, staticstudies 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.


10/27

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 andstiffness matrices in the above equation remain unchanged during the

duration of loading and un loading.


11/27

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


12/27

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 .


13/27

There are four approaches for Linear Dynamic Analysis

Modal Time History Analysis

Harmonic Analysis

Random Vibration Analysis

Response Spectrum Analysis


14/27

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)


15/27

Modal Analysis Procedure

[M] x(t) + [C] x ( t) + [K] x = F(t)

The above differential equation is a system of nsimultaneous ordinary differential

equations with constant coefficients.

The objective of the modal analysis is to transform the coupled system into a setof 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 ofthe eigenvalue problem:


16/27

For linear systems, the system of nequations of motion can be de-coupled into

n single-degree-of-freedom equations in terms of the modal displacement

vector {x}:

[x]= {}u

Substituting this in the main equation of motion and pre multiplying {} T with

we get

{} T[M]{}u(t) + {} T [C]{}u ( t) + {} T [K]{}u = {} T F(t)

The normal modes satisfy the orthogonality property, and the modal matrix is

normalized to satisfy the following equations:

{} T[M]{} =1

{} T [C]{} = 2 [] []

{}T

[K]{}= [2

]


17/27

The resultant equation after substituting the above is

u+ u2 [] [] + [2] u = {} T F(t)

The above is system of nindependent 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 oftime, 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.


18/27

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.


19/27

RESPONSE SPECTRUM ANALYSIS

What is response spectrum?

Response spectrum is a plot of peak response vs modal frequency ( for a givendamping)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 ofgeneralized 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


20/27

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


21/27

NON LINEAR ANALYSIS

All structures behave non linearly in one way or other beyond a particular levelof 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 isa sub-set of it.

Non linear analysis is needed if the loading produces a significant changes in the

stiffness


22/27

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 impactbodies


23/27


24/27

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.

The problem with this approach is that errors accumulate with each load

increment, causing the final results to be out of equilibrium.

Nonlinear Response

Displacement

External Load Error

Calculated

Response


25/27

Other Approach : Newton-Raphson algorithm:

Applies the load gradually, in increments.

Also performs equilibrium iterations at each load increment to drive the incremental

solution to equilibrium.

Solves the equation [KT]{Du} = {F} - {Fnr}

[KT] = tangent stiffness matrix

{Du} = displacement increment

{F} = external load vector

{Fnr} = internal force vector

Iterations continue until{F} - {Fnr}

(difference between external and internal

loads) is within a tolerance.

Some nonlinear analyses have trouble converging. Advanced analysis techniques are

available in such cases.

Displacement

F

[KT]

1

23

4 equilibrium

iterationsFnr

Du


26/27

NON LINEAR DYNAMIC ANALYSIS

In this analysis ,unlike linear dynamic analysis the mass , damping and stiffness matrixare varying and get updated during each iteration.

In nonlinear dynamic analysis, the equilibrium equations of the dynamic system at

time step, t+t, are:

[M] t+t{U ''} (i)+ [C] t+t{U '} (i)+ t+t[K] (i) t+t[ D U] (i)= t+t{R} - t+t{F} (i-1)


27/27

Thank you

linear-nonlinear for structures

Documents

Transcript of linear-nonlinear for structures