NAS101, Page Section 7 Linear Buckling Analysis

NAS101, Page Linear Buckling Analysis PAGE Theory of Buckling 3 Solution of the Eigenvalue Problem 5 Solution Sequences for Buckling and Stability Problems 7 Examples of Nonlinear Buckling 10 Rules for SOL 105 Buckling Analysis 11 Data Entries for Linear Buckling 13 EIGRL Entries 15 Example – Simple Euler Column 17 Example – Simple Euler Column -- Input File 20 Example – Simple Euler Column – Output File 21

NAS101, Page Linear Buckling Analysis (cont.) PAGE References for Buckling and Stability Analysis 22 Workshop 9Buckling Analysis of Plate 24 Boundary Conditions 26 Applied Loads 27 Partial Input File for Workshop 9 28 F06 Output for Workshop 9 30 Lowest Buckling Mode for Workshop 9 31 Solution for Workshop 9 32

NAS101, Page THEORY OF BUCKLING The equilibrium equations for a structure subjected to a constant force system take the following form [ K ] { u } = { P } Include the differential stiffness effects. The differential stiffness is the stiffness [ K d ] that results from including the higher- order terms of the strain-displacement relations. These relations are assumed to be independent of the displacements of the structure associated with an arbitrary intensity of load.

NAS101, Page THEORY OF BUCKLING (cont.) Let be an arbitrary scalar multiplier for another intensity of load. By perturbing the structure slightly at a variety of load intensities, the load intensities can be found that possess unstable equilibrium positions. This leads to the associated eigenvalue problem for buckling.

NAS101, Page SOLUTION OF THE EIGENVALUE PROBLEM [ K – K d ] { } = 0 The solution is nontrivial (different from zero) only for specific values of = i for i = 1, 2, 3,…, n That makes the matrix [ K – K d ] singular

NAS101, Page SOLUTION OF THE EIGENVALUE PROBLEM (cont.) To each eigenvalue i, there is a corresponding distinct eigenvector { i }. { i } can be scaled by any constant multiplier and still be a solution to Equation 1. The components of { i } are real numbers.

NAS101, Page SOLUTION SEQUENCES FOR BUCKLING AND STABILITY PROBLEM SOL 105Linear buckling SOL 106Nonlinear buckling Limitations of SOL 105 In prebuckled configuration: äDeflections must be small. äStresses must be elastic (and linearly related to strain).

NAS101, Page SOLUTION SEQUENCES FOR BUCKLING AND STABILITY PROBLEMS (Cont.) Example:Three classes of columns (loaded at centroid, no material imperfections)

NAS101, Page SOLUTION SEQUENCES FOR BUCKLING AND STABILITY PROBLEMS (Cont.) Note:SOL 105 may be applicable for structures with slight material imperfections or slightly noncentric loadings (i.e., load does not align with centroid producing a small degree of bending). Must use engineering judgment Same arguments hold for plate structures.

NAS101, Page EXAMPLES OF NONLINEAR BUCKLING Highly Eccentrically Loaded Column Snap-Through of Thin Shell (like the Bottom of an Oil Can)

NAS101, Page RULES FOR SOL 105 BUCKLING ANALYSIS (For reference, see section 13 of the MSC/NASTRAN Linear Statics Users Guide) The Case Control must contain at least two subcases. Normally the first subcase is the static solution under loading. METHOD must appear in a separate subcase to select an EIGB or EIGRL entry from the Bulk Data for the buckling solution. If you have multiple static solutions, then use the STATSUB command to select the static subcase for the buckling solution.

NAS101, Page RULES FOR SOL 105 BUCKLING ANALYSIS (Cont.) If desired, different SPC sets may be applied in the static subcase and the buckling subcase Output requests may be placed in any selected subcases. Output requests that apply to both the static solution and the buckling modes may be placed above the subcase level.

NAS101, Page DATA ENTRIES FOR LINEAR BUCKLING Executive Control Section SOL 105 Case Control Section SUBCASE 1 LOAD = M Defines static loading condition (LOAD, TEMP, DEFORM) SUBCASE 2 METHOD = N STATSUB = i Selects eigenvalue extraction method Selects static subcase to use for buckling solution (defaults to first subcase)

NAS101, Page DATA ENTRIES FOR LINEAR BUCKLING (Cont.) The Case Control must contain at least two subcases. Bulk Data Section Static loading condition required EIGBEigenvalue extraction data entry or EIGLREigenvalue extraction data for Lanczos method

NAS101, Page EIGRL ENTRY EIGRL Entry - recommended eigenvalue solution method Defines data needed to perform real eigenvalue or buckling analysis with the Lanczos Method EIGRLSIDV1V2NDMSGLVLMAXSETSHFSCLNORM EIGRL

NAS101, Page EIGRL ENTRY (cont.) FieldContents SIDSet identification number (unique integer > 0) V1, V2Vibration analysis: Frequency range of interest Buckling analysis: l range of interest (V1 < V2, real). If all modes below a frequency are desired, set V2 to the desired frequency and leave V1 blank. It is not recommended to put 0.0 for V1, it is more efficient to use a small negative number or to leave it blank. NDNumber of roots desired (integer > 0 or blank) MSGLVLDiagnostic level (integer 0 through 3 or blank) MAXSETNumber of vectors in block (integer 1 through 15 or blank)

NAS101, Page EXAMPLE - SIMPLE EULER COLUMN Problem Find the critical load and corresponding first mode buckled shape of a solid circular rod.

NAS101, Page EXAMPLE - SIMPLE EULER COLUMN (Cont.) Theoretical Solution where L eff =effective column length =2 x 2" for free-fixed column

NAS101, Page EXAMPLE - SIMPLE EULER COLUMN (Cont.) MSC.Nastran Model 7 7 7

NAS101, Page EXAMPLE - SIMPLE EULER COLUMN – INPUT FILE

NAS101, Page EXAMPLE - SIMPLE EULER COLUMN – OUTPUT FILE First eigenvalue: P cr = 1 x 10 lbs = lbs First eigenvector (buckled shape)

NAS101, Page REFERENCES FOR BUCKLING AND STABILITY ANALYSIS 1.MSC Seminar Notes, MSC/NASTRAN Material and Geometric Nonlinear Analysis: 2.MSC/NASTRAN Linear Static Analysis Users Guide, Section MSC/NASTRAN Verification Problem Manual (Version 64, January 1986 Edition): Problem A, Lateral Buckling of a Cantilever Beam Problem A, Simple Frame Analysis with Buckling Problem S, Euler Buckling of a Simply Supported Beam

NAS101, Page REFERENCES FOR BUCKLING AND STABILITY ANALYSIS (Cont.) 4.MSC/NASTRAN Demonstration Problem Manual (Version 64, March 1985 Edition): Under Elastic Stability Analysis, see Demonstration Problem D0504A, Flexural Buckling of a Beam 5.MSC/NASTRAN Application Notes October 1978Buckling and Real Eigenvalue Analysis of Laminated Plates September 1979Static Stability of Structures with Nonlinear Differential Stiffness February 1982Elastic-Plastic Buckling of a Thin Spherical Shell November 1985Nonlinear Buckling Analysis

NAS101, Page Workshop 9 Buckling Analysis of Plate

NAS101, Page Workshop 9 (cont.) Model description Same plate model as in workshop 5 without the stiffeners The following boundary conditions are applied to the model: Pin at the left end Rollers at the right end Zero vertical deflections at top and bottom edges Apply 100psi compressive loads at the right edge Total loads at right edge = (100) (8) (.01) = 8 –Apply 1 lb each at grid points 11 and 55 –Apply 2 lbs each at grid points 22, 33, and 44

NAS101, Page Workshop 9 -- Boundary Conditions Simply supported Supported on rollers Supported in the Vertical Direction

NAS101, Page Workshop 9 -- Applied Loads

NAS101, Page Partial Input File for Workshop 9

NAS101, Page Partial Input File for Workshop 9 (cont.)

NAS101, Page F06 Output for Workshop 9 Lowest Buckling Load = 1 = x 8 = #

NAS101, Page Lowest Buckling Mode for Workshop 9

NAS101, Page Solution for Workshop 9

NAS101, Page Solution for Workshop 9