Large rotations: plates with Green-Lagrange membrane strains#
The model 'plate_clpt_gl' is the classical laminated plate theory (CLPT)
of 'plate_clpt_donnell' with the complete Green-Lagrange membrane
strains of the mid-surface, instead of the von Kármán strains. It is meant
for assemblies of plates, e.g. a wing box with MultiDomain, that
undergo large rigid rotations while their strains stay small. The derivation,
with SymPy, is in theory/shells/plate_clpt_gl/plate_clpt_gl.py, with the
note plate_clpt_gl.md, and write_pyx.py generates the kernels
panels.models.plate_clpt_gl and panels.models.plate_clpt_gl_num
from it.
Why the von Kármán strains fail in an assembly#
The von Kármán strains keep the quadratic terms of \(w\) only, \(w_{,x}^2/2\),
\(w_{,y}^2/2\) and \(w_{,x} w_{,y}\). In a box beam bent about a horizontal axis,
the rotation \(\theta\) of a cross-section is an out-of-plane rotation of the
skins, \(w_{,x} = \theta\), and an in-plane rotation of the spars, \(v_{,x} =
\pm \theta\), which enters no strain. A rigid rotation of the box then
stretches the skins by \(\theta^2/2\) relative to the spars, and the box
stiffens spuriously once its tip deflection reaches the order of its height,
instead of its length. For the wing box of the example below, with a height of
12 mm and a span of 1010 mm, the tip deflection at 25 N is 13.9 mm with
'plate_clpt_donnell' against 22.7 mm of the linear solution, and 54.7 mm at
500 N against 454 mm.
Kinematics#
The degrees of freedom of a term are \(u, v, w\), the displacements of the mid-surface along the fixed local axes of the plate, and the strains are
the membrane strains being \(\boldsymbol{E} = (\boldsymbol{F}^\top
\boldsymbol{F} - \boldsymbol{I})/2\) with the deformation gradient of the
mid-surface \(\boldsymbol{F} = [\boldsymbol{e}_x + \boldsymbol{u}_{,x} \;\;
\boldsymbol{e}_y + \boldsymbol{u}_{,y}]\). For a rigid rotation
\(\boldsymbol{u} = (\boldsymbol{R} - \boldsymbol{I}) \boldsymbol{X}\) the
columns of \(\boldsymbol{F}\) are two columns of \(\boldsymbol{R}\), hence
\(\boldsymbol{F}^\top \boldsymbol{F} = \boldsymbol{I}\): the membrane strains
vanish for any rotation, in plane or out of plane, whereas the von Kármán
\(\varepsilon_{xx}\) of an in-plane rotation \(\theta\) is \(\cos\theta - 1\).
Without the quadratic terms of \(u\) and \(v\) the strains are those of
'plate_clpt_donnell'.
Internal forces and tangent stiffness#
At a point, the strains depend on the nine generalized derivatives \(\boldsymbol{d} = (u_{,x}, u_{,y}, v_{,x}, v_{,y}, w_{,x}, w_{,y}, w_{,xx}, w_{,yy}, w_{,xy}) = \boldsymbol{\Phi} \boldsymbol{c}\). With \(\boldsymbol{B} = \partial \boldsymbol{\varepsilon}/\partial \boldsymbol{d}\) (6 x 9, linear in the gradients) and the stress resultants \(\boldsymbol{\sigma} = \boldsymbol{ABD}\, \boldsymbol{\varepsilon}\):
where \(\boldsymbol{H}(\boldsymbol{N})\) has the block \([[N_{xx}, N_{xy}],
[N_{xy}, N_{yy}]]\) on each pair of gradients \((u_{,x}, u_{,y})\), \((v_{,x},
v_{,y})\) and \((w_{,x}, w_{,y})\): the geometric stiffness matrices
Shell.calc_kG() and the analytical one of a constant stress state have
u-u and v-v blocks besides the w-w block of the von Kármán plate.
As for the other models, the stress of the quadratic strains enters
Shell.calc_kC() with NLgeom=True and Shell.calc_kG() stays
homogeneous of degree one in \(\boldsymbol{c}\), see
Tangent stiffness matrix and its product by a vector, whose product calc_kTv is generated with the six
gradients in place of the two rotations of the von Kármán terms. The matrix
\(\boldsymbol{B}^\top \boldsymbol{ABD}\, \boldsymbol{B} + \boldsymbol{H}\) is
formed once per integration point, and each pair of terms costs 27
multiplications.
The integrands are still products of four approximation functions, so the
default numbers of integration points, gauss_points(), are unchanged.
The linear constitutive stiffness, the mass and the aerodynamic matrices are
those of 'plate_clpt_donnell': a linear static analysis and the natural
frequencies are identical, and a linear buckling eigenvalue changes only
through the in-plane blocks of the geometric stiffness, not at all for a
symmetric laminate and at the order of the membrane strains otherwise.
clpt_field.fstrain() gives the Green-Lagrange membrane strains.
Follower pressures use the exact area vector of the plate,
follower='quadratic', see Follower pressure loads: theory; the first-order area
vector of follower='linear' assumes moderate rotations.
Limitations#
The changes of curvature are measured in the fixed local frame of the plate. A plate bent out of plane by a large rotation keeps the bending energy of the linear plate, its membrane strains being kept small by the shortening \(u\): a cantilever strip under a tip force of constant direction bent out of plane follows the linear beam, \(w = \alpha L/3\) at \(\alpha = P L^2/EI = 1\) against \(0.302 L\) of the elastica, whereas bent in its plane, where the bending is carried by the membrane strains, it follows the elastica to 0.15 % at \(\alpha = 2\) (
tests/tests_shell/test_plate_clpt_gl.py). In a box beam the global bending is carried by the membrane strains of the skins and the in-plane bending of the spars, and the out-of-plane bending of the skins contributes little; a local buckle of a skin rotated by \(\theta\) is seen with its bending energy times \(\cos^2\theta\). The model'plate_clpt_tl'measures the curvatures with the deformed normal, see Large rotations: total-Lagrangian Kirchhoff-Love plates.The connections of
MultiDomainare unchanged: they are linear in the displacements in the fixed local frames, exact for the continuity of the displacements at any rotation, and the rotation continuity (\(w_{,y}\) or \(w_{,x}\) across a joint) is that of small rotations, with an error of order \(1 - \cos\theta\) at a joint rotated by \(\theta\). The null-space matrix of the constraints stays constant.The strains are small: the constitutive law is linear in the Green-Lagrange strains.
Example: wing box under a follower tip force#
The High-AR-t5 wing box of Elalfy, De Breuker and Castro
[elalfyWingboxPreprint], half-span 1010.0 mm, chord 50.5 mm, height
12.12 mm, 9 bays, skins of 1.0 mm and spars of 5.0 mm, aluminium, clamped at
the root, is modelled with 45 'plate_clpt_gl' domains (6264 degrees of
freedom) connected with penalties, and loaded at the tip by a follower
pressure on the upper skin (follower='quadratic'), incremented by 25 N
with the Newton-Raphson method of structsolve, without imperfection,
tests/multidomain/test_wingbox_plate_clpt_gl.py. The Abaqus values are
the upper panel of their Fig. 6 (p. 10), read from the vector paths of the
figure (about 1 mm):
tip force (N) |
25 |
100 |
200 |
300 |
400 |
500 |
|---|---|---|---|---|---|---|
Abaqus, Fig. 6 |
23.3 |
93.2 |
189.6 |
298.9 |
410.0 |
515.5 |
|
22.68 |
90.84 |
184.37 |
287.24 |
393.59 |
499.83 |
|
13.9 |
54.7 |
||||
linear |
22.68 |
90.70 |
181.41 |
272.11 |
362.82 |
453.52 |
The Green-Lagrange model is 2.5 to 4.0 % below Abaqus over the whole range, most of which is the 2.7 % of the linear range, where the panels model, whose domains are at the mid-surfaces and connected with penalties, is stiffer than the S4R shell model of Abaqus. Above the first buckling load, 173.11 N in Abaqus (their Table 3, p. 8), the top skin buckles between the ribs, here without imperfection, deflecting in alternate directions from bay to bay (2.4 mm in the second bay at 500 N), and both skins sag towards the mid-plane, the bottom one by 1.4 mm in the root bay at 500 N, 11 % of the height: as in Abaqus, the box is softer at 500 N than the linear beam, and much softer than the elastica of the same bending stiffness, 423.0 mm with the follower force, which only accounts for the large rotations. One Newton-Raphson analysis of the 20 increments takes about 2 min on one core.
Under a dead tip force, follower=False, the same model deflects 22.66,
90.23, 179.69, 270.70, 354.57 and 428.30 mm at 25, 100, 200, 300, 400 and
500 N, against 23.0, 92.4, 184.8, 281.1, 369.0 and 443.8 mm of Abaqus in
their Fig. 5 (p. 10), read in the same way, 1.5 to 3.9 % below.