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Linear non linear


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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.

Published in: Design, Technology

Linear non linear

  2. 2. LINEAR ANALYSIS • • • • • • • • 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 • • Solution procedures Introduction to Non-Linear dynamic analysis
  3. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 13. There are four approaches for Linear Dynamic Analysis • Modal Time History Analysis • Harmonic Analysis • Random Vibration Analysis • Response Spectrum Analysis
  14. 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. 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. 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. 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. 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. 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. 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. 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. 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
  23. 23. • • 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
  24. 24. • 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. External Load Calculated Response Error Nonlinear Response Displacement
  25. 25. 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. [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 • • 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. 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. 27. Thank you