Plate buckling

Eigenvalue methods for computing buckling loads

Plate and shell buckling loads are classically computed by solving the eigenvalue problem $\mathbf {K} -\lambda \mathbf {G} (\mathbf {N} )$ where $\mathbf {K}$ is the shell's/plate's stiffness and $\mathbf {G} (\mathbf {N} )$ is its geometric stiffness, dependent on the in-plane (membrane) stress resultants $\mathbf {N}$, and $\lambda$ is the proportional load factor.

To compute the buckling load in FEAP, one first solves for the membrane stresses and shell stiffness, then one forms the geometric stiffness, and finally one uses an eigensolver to determine the buckling load(s).

The basic MARCO commands are as follows:

TANGent,,1
GEOMetric
SUBSpace,,5

Optionally one can use ARPAck,,5 (if optionally built).

As an example consider a cantilevered plate that is subjected to an in-plane compression, the buckling factor can be computed from the following input file. With this coarse mesh the minimum eigenvalue is found to be $1.30888\times 10^{4}$. Thus the buckling force is found to be ${\bar {p}}\times w\times \lambda _{min}=10^{-3}\times 0.1375\times 1.30888\times 10^{4}=1.8$ in the units of the input file. Note ${\bar {p}}$ is the magnitude of the applied traction (Force per unit length) in the input file and $w$ is the length of the edge of the plate. The validity of the result can be checked using the Euler load $P_{\mathrm {euler} }={\frac {1}{4}}{\frac {\pi ^{2}Eh^{3}}{12(1-\nu ^{2})w}}=1.9$.

feap  * * Buckling of cantilever plate with a in-plane compression * *
ndm = 3  ! 3 spatial dimensions
ndf = 6  ! 6 dofs per node
nen = 4  ! 4 nodes per element


PARAmeters
  h  = 0.001375    ! Thickness
  e  = 436.4*10^6  ! Young's modulus
  nu = 0.3         ! Poisson's ratio
  w  = 0.1375      ! side length

MATErial
  SHELl
    ELAStic isotropic e nu
    THICk plate       h 1   ! Shear factor set to 1
    penalty drill e*1e-6    ! drill stiffness used for flat shells

BLOCk
  CART 8 8
      1  0  0  0
      2  w  0  0
      3  w  w  0
      4  0  w  0

CSURface                    ! Uniform force unit length on edge
 LINEar
 1  w  0 0.0  -1.0d-3
 2  w  w 0.0  -1.0d-3 

EBOUndary                   ! Clamped edge
1  0    1  1  1   1  1  1
  
END


BATCH
 TANGent,,1                 ! Linear problem no need to iterate
 GEOM                       ! Form geomteric tangent once stress-state is known
 SUBSpace,,5                ! Compute eigenvalues
 PLOT PERSpective
 PLOT HIDE
 PLOT MESH                  ! Show the mesh
 PLOT LOAD 1                ! Plot load with arrow head at node
 PLOT DEFOrmed,,,1e-5,1     ! Plot eigvecs with 1e-5 scaling and no re-scale to ref. config
 PLOT EIGVector,1,,3        ! Plot EigVec 1 with contours of 3-displacement
END

INTEractive
STOP