Chapter 1 Introduction to Non-Linear Analysis 비선형유한요소해석 2012 년 2 학기 강의노트

Static or Linear analysis - infinitesimally small displacement : Integration of K and R is based on the volume before deformation. : B matrix of each element, the strain-displacement matrix, is constant and independent of displacement - linearly elastic material : C, the stress-strain matrix is always considered by constant. - Unchanging constraint : The nature of boundary conditions remains unchanged during the application of the loads on the finite element assemblage.

What is non-linear analysis : All case except for linear analysis… ▷ Some important engineering phenomena can only be assessed on the basis of a nonlinear analysis. - Collapse or buckling of structures due to sudden overloads. - Progressive damage behavior due to long lasting severe loads. - For certain structures (e.g. cables), nonlinear phenomena need be included in the analysis even for serviece load calculations. ▷ The need for nonlinear analysis has increased in resent years due to the need for - use of optimized structures. - use of new materials. - addressing safety-related issues of structures more rigorously.

Field of nonlinear analysis ▷ Problems to be addressed by a non-linear finite element analysis are found in almost all branches of engineering, most notably in, - Nuclear Engineering. - Earthquake Engineering. - Automobile Industries. - Defense Industries. - Aeronautical Engineering. - Mining Industries. - Offshore Engineering… and so on.

Classification of NL : Only non-linear material : non-linear curve of stress-strain property

Classification of NL : Large displacement & Small strain : the relationship of stress-strain could be non-linear or not.

Classification of NL : Large displacement & Large strain : the most general case of non-linear analysis. material could be non-linear.

Classification of NL : Contact problem : the case of changing boundary condition during analysis

Example 1.1 A bar rigidly supported at both ends is subjected to an axial loads as show in Fig. E1.1(a). The stress-strain relation and load-versus-time curve relation are given in Figs. E1.1(b) and (c), respectively. Assuming that the displacements and strains are small and that the load is applied slowly, calculate the displacement at the point of load application. (a) Simple bar structure (b) Stress-strain relation (In tension and compression) (c) Load variation Figure E1.1 Analysis of simple bar structure (d) Calculated response

Solution of Example 1.1 Since the load is applied slowly and the displacements and strains are small, we calculate the response of the bar using a static analysis with material nonlinearities only. Then we have for sections a and b, the strain relations (a) The equilibrium relation, (b) And the constitutive relations, under loading conditions, In the elastic region In the plastic region (c)

Solution of Example continue And in unloading In these relations the superscript t denots “at time t”. (i) Both sections a and b are elastic. During the initial phase of load application both section a and b are elastic. Then we have, using (a) & (b) to (c), And substituting the values given in Fig. E1.1, we obtain with (d)

Solution of Example continue (ii) Section a is elastic while section b is plastic. Section b will become plastic at time When, using (d). Afterwards we therefore have (e) Using (e), we therefore have for, And thus

Solution of Example continue we may note that section would become plastic when or. Since the load does not reach this value [see Fig. E1.1(c)], section remains elastic throughout the response history. (iii) In unloading both sections act elastically we have The calculated response is depicted in Fig. E1.1 (d)

Example 1.2 A pretensioned cable is subjected to a transverse load midway between the supports as shown in Fig. E1.2(a). A spring is placed below the load at a distance. Assume that the displacements are small so that the force in the cable remains constant, and that the load is applied slowly. Calculate the displacement under the load as a function of the load intensity (a) Pretensioned cable subjected to transverse load (b) Load (c) Calculated response Figure E1.2 Analysis of pretensioned cable with a spring support

Solution of Example 1.2 As in Example 1.1, we neglect inertia forces and assume small displacements. As long as the displacement under the load is smaller than, vertical equilibrium requires for small, (a) Figure E1.2(c) shows graphically the force displacement relations given in (a) and (b). We should note that in this analysis we neglected the elasticity of the cable; therefore the response is calculated using only the equilibrium equations in (a) and (b), and the only nonlinearity is due to the contact condition established when Once the displacement is larger than, the following equilibrium equation holds : (b)

Classification of NL formulation Type of analysisDescription Typical formulation used stress and strain measures Materially nonlinear only Infinitesimal displacements and strains; the stress-strain relation is nonlinear Materially nonlinear only (MNO) Engineering stress and strain Large displacements, large rotations, but small strains Displacements and rotations of fibers are large, but fiber extensions and angle changes between fibers are small; the stress-strain relation may be linear or nonlinear Total Lagrangian (TL) Updated Lagrangian (UL) Second Piola-Kirchhoff stress vs Green-Lagrange strain Cauchy stress vs Almansi strain Large displacements, large rotations, and large strains Fiber extensions and angle changes between fibers are large, fiber displacements and rotations may also be large; the stress-strain relation may be linear or nonlinear Total Lagrangian (TL) Updated Lagrangian (UL) Second Piola-Kirchhoff stress vs Green-Lagrange strain Cauchy stress vs logarithmic strain

The basic of NL FEM - The basic approach of an incremental solution : We consider a body (a structure or solid) subjected to force and displacement boundary conditions that are changing. : We describe the externally applied forces and the displacement boundary conditions as functions of time. : Find the solution which satisfy equilibrium equation (1.1) : Externally applied nodal load at time t : Nodal force which is calculated from elemental stress

The basic of NL FEM - continue The relation in (1.1) must express the equilibrium of the system in the current deformed geometry taking due account of all nonlinearities. and must be satisfied throughout the complete history of load application; i.e., the time variable may take on any value from zero to the maximum time of interest. - When the applied forces and displacements vary : slowly, meaning that the frequencies of the loads are much smaller than the natural frequencies of the structures, we have a static analysis. frequencies of the structures, we have a dynamic analysis. : fast, meaning that the frequencies of the loads are in the range of the natural

Meaning of Time In a static analysis without time effects other than the definition of the load level (e.g., without creep effects), time is only a convenient variable which denotes different intensities of load applications and correspondingly different configuration) In a dynamic analysis and in static analysis with material time effects, the time variable is an actual variable to be properly included in the modeling of the actual physical situation

Basic requirements which must be achieved ※ At the end of each load (or time) step, we need to satisfy the three basic requirements of mechanics : Equilibrium Compatibility The stress-strain law - This is achieved-in an approximate manner using finite elements-by the application of the principle of virtual work. - In some nonlinear static analysis the equilibrium configurations corresponding to these load levels can be calculated without also solving for other equilibrium configuration. However, when the analysis includes path-dependent nonlinear geometric or material conditions, or time-dependent phenomena, the equilibrium realtions in (1.1) need to be solved for the complete time range of interest.

Incremental Step-by-Step solution is incremental value which is calculated from increments of stress and displacement of element within the range from to. can be evaluated using tangent stiffness matrix which is compatibe with the geometry and material boundary condition Assuming the solution at time is known and the solution at time need to be solved. Assuming is independent of time. (1.2) (1.3) (1.4) is increments of nodal displacement

Incremental Step-by-Step solution - continue (1.5) Inserting (1.4), (1.5) into (1.3) and rearranging. (1.6) (1.7) ① Displacement evaluated from (1.7) is approximate solution, not exact solution by the assumption of (1.4). ② An approximate value of stress can be calculated at the same time using approximate displacement value. ③ Perform calculation of next time step. The accuracy of solution is very dependent on time increment because of the assumption (1.4) ④ The iteration process should be included in solution process for the acquirement of exact solution sufficiently.

Incremental Step-by-Step solution - Iteration For (1.8) In (1.8),,  Nodal displacement evaluated from eq(1.8) in the last place is used in calculation and. Calculation of nodal displacement is repeated until out-of-valance load. and are minimized.

Solution of Non-linear Equations (1.9) : Externally applied load on nodes at time : Nodal force equivalent with the stress of element at time If has dependency non-linearly with nodal point displacement, the solution of eq (1.9) could be resolved using iterative method. The basic equation for non-linear analysis at time

Classfication of Iteration method  Bracketing Methods  Graphical method  Bisection method  False-position method  Modified false-position method  Open Methods  One-point iteration  Newton-Raphson method  Secant method  Newton-like Methods  Quasi-Newton methods  Broyden’s method  Davidon’s method  DFP method  BFGS method  Limited memory quasi-Newton(LMQN) method

Systems of Non-linear Equations f 1 (x 1, x 2, … x n ) = a 1 x 1 + a 2 x 2 +…+ a n x n + c = 0 f 2 (x 1, x 2, … x n ) = b 1 x 1 + b 2 x 2 +…+ b n x n + c = 0  Linear Systems u (x, y) = x 2 + xy - 10 = 0 v (x, y) = y + 3xy = 0  Nonlinear Systems  Most approaches for determining such solutions are extension of the open methods

Newton-Raphson method for single equations  First-order Taylor Series Expansion  Termination criterion x0x0 x1x1 x2x2 y = f (x) y = f (x 0 ) y = f (x 1 ) Merits  fastest method  no require initial interval Demerits  good initial guess needed  require derivative

Newton-Raphson method for 2X2 nonlinear systems  Multivariable Taylor Series for First-order Rearrange,  Matrix Formulation Therefore, N-R method for nonlinear system as

Newton-Raphson method for Non-linear FE system  From Finite Element Formulation

Concept of Full Newton-Raphson method (single equation) Load-displacement relation Function f used : Using re-evaluated tangent stiffness matrix at each every iteration process. → High computational cost

Concept of Modified Newton-Raphson method (single equation) Initial stress method Modified N-R method  Initial stress method : Using initial stiffness matrix of initial step instead of through all step.  Modified Newton-Raphson method : Using initial stiffness matrix of each time step instead of through iteration process.

Comparison Full N-R method with Modified N-R method

Example 1.3 Calcuate the response of the bar assemblage considered in Example 1.1 using the modified Newton-Raphson iteration. Use two equal load steps to reach the maximum load application (a) Simple bar structure (b) Stress-strain relation (In tension and compression) (c) Load variation Figure E1.1 Analysis of simple bar structure (d) Calculated response

Example 1.4 For a single degree of freedom system we have And. Use the Newton-Raphson iteration to calculate.

Concept of Quasi-Newton method f Slope Merits - O(n 2 ) performance - no require derivative Demerits - two initial guess needed - convergence is slower

BFGS Method (Quasi-Newton method)  Step 1 : Evaluate a Displacement Vector Increment  Step 2 : Perform a Line Search in the Direction Until following equation is satisfied: Define displacement increment, out-of-balance loads

BFGS Method (Quasi-Newton method) -continue  Step 3 : Evaluate the Correction to the Coefficient Matrix where the matrix A i+1 is an n x n matrix of the following form

Comparison of Quasi-Newton Method  Broyden’s inverse update:  Davidon’s inverse update:  DFP(Davidon-Fletcher-Powell) inverse update:

Convergence Criteria  Case 1 : using Displacement  Case 2 : using Out-of-valance Load  Case 3 : using Displacement & Out-of-valance Load : allowed tolerance of convergence