SECTION 13 SUBMARINE PRESSURE HULL - 3D S13-1 NAS120, Section 13, May 2006 Copyright 2006 MSC.Software Corporation

SECTION 13 SUBMARINE PRESSURE HULL - 3D S13-2 NAS120, Section 13, May 2006 Copyright 2006 MSC.Software Corporation

SECTION 13 SUBMARINE PRESSURE HULL - 3D S13-3 NAS120, Section 13, May 2006 Copyright 2006 MSC.Software Corporation n Topics covered in this case study: u Linear buckling analysis

SECTION 13 SUBMARINE PRESSURE HULL - 3D S13-4 NAS120, Section 13, May 2006 Copyright 2006 MSC.Software Corporation n Problem Description u You have been presented with the conceptual design of a next-generation military submarine. The submarine pressure hull is a thick shell structure reinforced with ring frames. The submarine must be capable of operating at depths up to 1000 ft. Will the pressure hull buckle under the pressure loading? n Analysis Objective u Perform a buckling analysis on the submarine pressure hull to determine the buckling load.

SECTION 13 SUBMARINE PRESSURE HULL - 3D S13-5 NAS120, Section 13, May 2006 Copyright 2006 MSC.Software Corporation SAIL FC ER RC FWD MBT AFT MBT Submarine General Spec: Overall Length = 350 ft DIA = 36 ft ER = Engine Room RC = Reactor Compartment FC = Forward Compartment AFT MBT = Aft Main Ballast Tanks FWD MBT = Forward Main Ballast Tanks n Submarine Design Pressure hull

SECTION 13 SUBMARINE PRESSURE HULL - 3D S13-6 NAS120, Section 13, May 2006 Copyright 2006 MSC.Software Corporation n Pressure Hull Details 4

SECTION 13 SUBMARINE PRESSURE HULL - 3D S13-7 NAS120, Section 13, May 2006 Copyright 2006 MSC.Software Corporation n Pressure loading u The cruising depth for the submarine is 1000 ft. The water pressure load at this depth is the main design load. u Compute the pressure: p = gh g = 64 lb/ft 3 h = 1000 ft p = 64 x 1000 = 64,000 lb/ft 2 = 445 psi

SECTION 13 SUBMARINE PRESSURE HULL - 3D S13-8 NAS120, Section 13, May 2006 Copyright 2006 MSC.Software Corporation n Getting started on the case study u The pressure hull shell structure is modeled with plate elements. u The ring frames (typical frames and King frames) are modeled with CBAR elements without offsets. u The two bulkheads are modeled with plate elements. u The model is shown on the next page.

SECTION 13 SUBMARINE PRESSURE HULL - 3D S13-9 NAS120, Section 13, May 2006 Copyright 2006 MSC.Software Corporation n Pressure hull shell structure

SECTION 13 SUBMARINE PRESSURE HULL - 3D S13-10 NAS120, Section 13, May 2006 Copyright 2006 MSC.Software Corporation n Ring frames and bulkheads

SECTION 13 SUBMARINE PRESSURE HULL - 3D S13-11 NAS120, Section 13, May 2006 Copyright 2006 MSC.Software Corporation n Loads and Boundary Conditions u At 1000 ft, the water pressure load is p = 445 psi u The pressure hull is fully fixed at one end point x y

SECTION 13 SUBMARINE PRESSURE HULL - 3D S13-12 NAS120, Section 13, May 2006 Copyright 2006 MSC.Software Corporation n A structure can fail in a number of ways such as u Yield failure (material yield strength exceeded) u Ultimate failure (material ultimate strength exceeded) u Excessive deflection u Buckling n So far in this course we have been designing and analyzing structures to prevent them from material failure and excessive deflection. Lets now examine the concept of buckling.

SECTION 13 SUBMARINE PRESSURE HULL - 3D S13-13 NAS120, Section 13, May 2006 Copyright 2006 MSC.Software Corporation n The Concept of Linear Buckling u A compressive force P is applied to a perfectly straight column. A lateral force is introduced to create a small lateral deflection u If the lateral deflection disappears when the lateral force is removed, the straight form of equilibrium is stable. P P Q

SECTION 13 SUBMARINE PRESSURE HULL - 3D S13-14 NAS120, Section 13, May 2006 Copyright 2006 MSC.Software Corporation n The Concept of Linear Buckling (cont.) u If P is gradually increased, a condition is reached when the straight form of equilibrium becomes unstable and a small lateral force will produce a deflection which does not disappear when the lateral force is removed. u This critical load P cr is defined as the axial force which is sufficient to keep the column in such a slightly bent form. At this load, the bent column is said to be buckled. u The critical loads for the buckling of plates and shells are developed using this same concept.

SECTION 13 SUBMARINE PRESSURE HULL - 3D S13-15 NAS120, Section 13, May 2006 Copyright 2006 MSC.Software Corporation n Thin-walled or slender structures are susceptible to buckling n Examples of such structures are u Columns u Beams u Plates u Cylindrical shell, conical shell, and spherical shell structures

SECTION 13 SUBMARINE PRESSURE HULL - 3D S13-16 NAS120, Section 13, May 2006 Copyright 2006 MSC.Software Corporation n Theory of Linear Buckling u The equilibrium equations for a structure subjected to a constant force system take the following form: u Under loading, the structure deforms and internal loads are developed within the structure. Write the equilibrium equations for this deformed state: [ K ] { u } = { P } ( [ K ] + [ K D ] ) { u } = { P } (1) (2)

SECTION 13 SUBMARINE PRESSURE HULL - 3D S13-17 NAS120, Section 13, May 2006 Copyright 2006 MSC.Software Corporation n Theory of Linear Buckling (cont.) u The matrix [K D ] is the differential stiffness matrix. It is also called the geometric stiffness matrix or the stress stiffness matrix. The differential stiffness is the stiffness that results from including the higher-order terms (non-linear terms) of the strain-displacement relations. u The differential stiffness matrix is proportional to the internal forces in the structure. This allows us to rewrite Equation (2) as where is an arbitrary scalar multiplier for the applied load ( [ K ] + [ K D ] ) { u } = { P } (3)

SECTION 13 SUBMARINE PRESSURE HULL - 3D S13-18 NAS120, Section 13, May 2006 Copyright 2006 MSC.Software Corporation n Theory of Linear Buckling (cont.) u Lets now perturb the structure slightly from its equilibrium position by taking the derivative of both sides of Equation (3): At the critical buckling load, both the reference and the slightly perturbed (buckled) configurations are possible equilibrium positions. Therefore as the displacement { du } takes place, the load does not change. This leads to the eigenvalue problem for buckling: ( [ K ] + [ K D ] ) { du } = { dP } ( [ K ] + [ K D ] ) { du } = (4) (5) ( [ K ] + [ K D ] ) { } = (6)

SECTION 13 SUBMARINE PRESSURE HULL - 3D S13-19 NAS120, Section 13, May 2006 Copyright 2006 MSC.Software Corporation n Solution to the eigenvalue problem u The solution is nontrivial (different from zero) only for specific values of that make the term ([ K ] + [ K D ]) singular. To each eigenvalue i, there is a corresponding distinct eigenvector { i } which represents the buckled shape. u The critical buckling loads for the structure are computed as = i i = 1, 2, …n { P } cr i = i { P }

SECTION 13 SUBMARINE PRESSURE HULL - 3D S13-20 NAS120, Section 13, May 2006 Copyright 2006 MSC.Software Corporation n Solution to the eigenvalue problem (cont.) Usually only the lowest eigenvalue 1 is of interest because it is associated with the lowest buckling load for the structure. The eigenvalue is also called the buckling load factor (BLF). A structure has buckled if the buckling analysis indicates that BLF 1.0

SECTION 13 SUBMARINE PRESSURE HULL - 3D S13-21 NAS120, Section 13, May 2006 Copyright 2006 MSC.Software Corporation n Linear Buckling vs. Nonlinear Buckling u Linear buckling analysis assumes that the structure in the pre-buckled configuration is perfectly straight and elastic. u Nonlinear buckling analysis accounts for the pre- buckled deformations as well as material non- linearity.

SECTION 13 SUBMARINE PRESSURE HULL - 3D S13-22 NAS120, Section 13, May 2006 Copyright 2006 MSC.Software Corporation n Buckling Solutions in MSC.Nastran u SOL Linear Buckling u SOL Nonlinear Buckling u SOL Nonlinear Buckling n SOL 105 may be applicable for column and plate structures with slight manufacturing imperfections or slightly eccentric loadings. Must use engineering judgment. n Some examples of nonlinear buckling problems are shown on the next page.

SECTION 13 SUBMARINE PRESSURE HULL - 3D S13-23 NAS120, Section 13, May 2006 Copyright 2006 MSC.Software Corporation Highly Eccentrically Loaded Column Beam-Column Snap-Through of thin Shell (Large pre-buckled deflection and possible inelastic pre- buckled behavior) n Examples of nonlinear buckling problems

SECTION 13 SUBMARINE PRESSURE HULL - 3D S13-24 NAS120, Section 13, May 2006 Copyright 2006 MSC.Software Corporation n Rules for SOL 105 linear buckling analysis u The Case Control must contain at least two subcases. u The first subcase controls the static analysis run (this step is used to determine the differential stiffness matrix [K D ]). u The second subcase controls the buckling analysis run. The METHOD entry must appear in this subcase to select an EIGRL or EIGB entry from the Bulk Data Section.

SECTION 13 SUBMARINE PRESSURE HULL - 3D S13-25 NAS120, Section 13, May 2006 Copyright 2006 MSC.Software Corporation n Rules for SOL 105 linear buckling analysis (cont.) u For multiple buckling solutions: l All static subcases must appear first l The buckling subcases follow the last static subcase l A METHOD entry must appear in each of the buckling subcases l Each buckling subcase must contain a STATSUB command that references the appropriate subcase ID of the static subcase u The use of offsets in bar, beam, and plate elements in buckling analysis is not recommended.

SECTION 13 SUBMARINE PRESSURE HULL - 3D S13-26 NAS120, Section 13, May 2006 Copyright 2006 MSC.Software Corporation n In order to perform a linear buckling analysis, the following entries are required in the Nastran input data file: u Executive Control Section l SOL 105 u Case Control Section l SUBCASE 1 LOAD = MDefines static loading condition (LOAD, TEMP, DEFORM) l SUBCASE 2 METHOD = NSelects eigenvalue extraction method u Bulk Data Section l EIGRL entry – Lanczos method (recommended) l EIGB entry – Other eigenvalue extraction methods

SECTION 13 SUBMARINE PRESSURE HULL - 3D S13-27 NAS120, Section 13, May 2006 Copyright 2006 MSC.Software Corporation n The EIGRL entry u Defines data needed to perform vibration or buckling analysis with the Lanczos Method EIGRLSIDV1V2NDMSGLVLMAXSETSHFSCL NORM EIGRL FieldContents SIDSet identification number (unique integer > 0) V1, V2Vibration analysis: Frequency range of interest Buckling analysis: Eigenvalue 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 leave it blank).

SECTION 13 SUBMARINE PRESSURE HULL - 3D S13-28 NAS120, Section 13, May 2006 Copyright 2006 MSC.Software Corporation n The EIGRL entry (cont.) FieldContents NDNumber of roots desired (integer > 0 or blank) MSGLVLDiagnostic level (integer 0 through 3 or blank) MAXSETNumber of vectors in block SHFSCLEstimate of the first flexible mode natural frequency (real or blank) NORMMethod for normalizing eigenvectors, either "MASS" or "MAX" MASSNormalize to unit value of the generalized mass (default) MAXNormalize to unit value of the largest displacement in the analysis set

SECTION 13 SUBMARINE PRESSURE HULL - 3D S13-29 NAS120, Section 13, May 2006 Copyright 2006 MSC.Software Corporation n Lets now continue with the case study. n We want to determine the first 5 buckling loads for the submarine structure.

SECTION 13 SUBMARINE PRESSURE HULL - 3D S13-30 NAS120, Section 13, May 2006 Copyright 2006 MSC.Software Corporation Set up the linear buckling analysis

SECTION 13 SUBMARINE PRESSURE HULL - 3D S13-31 NAS120, Section 13, May 2006 Copyright 2006 MSC.Software Corporation Click solution type and select buckling analysis Click solution parameters

SECTION 13 SUBMARINE PRESSURE HULL - 3D S13-32 NAS120, Section 13, May 2006 Copyright 2006 MSC.Software Corporation Next click on Eigenvalue Extraction Select the Lanczos method and enter 5 as the number of desired roots

SECTION 13 SUBMARINE PRESSURE HULL - 3D S13-33 NAS120, Section 13, May 2006 Copyright 2006 MSC.Software Corporation Run the analysis, read the results into Patran, and plot the buckled shapes one at a time

SECTION 13 SUBMARINE PRESSURE HULL - 3D S13-34 NAS120, Section 13, May 2006 Copyright 2006 MSC.Software Corporation n Examine the.f06 file { P } cr = { P } P cr = 2.93 x 445 psi = 1,304 psi

SECTION 13 SUBMARINE PRESSURE HULL - 3D S13-35 NAS120, Section 13, May 2006 Copyright 2006 MSC.Software Corporation n Linear buckling analysis summary: u The critical buckling load factor (BLF) is This is equivalent to a critical buckling load of 1,304 psi. u Mode No.BLFDescription 12.93Pressure hull buckling 22.93Pressure hull buckling 34.42Bulkhead buckling 44.50Bulkhead buckling 55.24Pressure hull buckling u A follow-on nonlinear buckling analysis may be necessary to account for nonlinear effects.

SECTION 13 SUBMARINE PRESSURE HULL - 3D S13-36 NAS120, Section 13, May 2006 Copyright 2006 MSC.Software Corporation EXERCISE Perform Workshop 14 Buckling of a Submarine Pressure Hull in your exercise workbook.