5.4 Locking phenomena for NURBS shell elements
5.4.3 Membrane locking
elements with standard difference vector show up in exactly the same way, if, instead of a difference vector formulation, a parametrization with rotation tensors is applied. The same holds true for the Timoshenko beam model, if, instead of rotations, a difference vector is imposed on the undeformed director.
y-z-plane, as shown on the right of Figure 5.12
˜
v =v+ ¯z,yw,
˜
w =w−¯z,yv. (5.21)
By accounting for the approximationsv≈ˆv,w≈wˆ and¯z,yv≪1, the entire displace-ment field, which is composed of mid-surface displacedisplace-ments and curvature contributions, is equal to
˜
u(x,y,z) =u(x,y) + ¯z,xw(x,y),
˜
v(x,y,z) =v(x,y) + ¯z,vw(x,y),
˜
w(x,y,z) =w(x,y),
(5.22)
from which the linearized membrane strain components of the shallow shell can be derived
εxx =∂˜u
∂x = u,x
*+,-lin.
+ ¯z,xw,x
* +, -const.
,
2εxy =∂˜u
∂y+∂˜v
∂x =u,y+v,x
* +,
-lin.
+ ¯z,xw,y+ ¯z,yw,x
* +,
-const.
,
εyy =∂˜v
∂y = v,y
*+,-lin.
+ ¯z,yw,y
* +, -const.
.
(5.23)
Equation (5.23) reveals that nonzero curvature, i.e. ¯z̸= const., leads to contributions of different order in the membrane strain components. An isoparametric interpolation of both the geometry and displacement field therefore leads to unbalanced terms in the kinematics, which are the reason for membrane locking.
The critical parameter for membrane locking is again the thickness of the structure, as for transverse shear locking. The ratio of bending to membrane stiffness is proportional to the square of the thickness. With increasing slenderness of the discrete model, large parts of the internal energy are due to unphysical membrane action, which consequently results in a significant underestimation of the displacements and prevents a uniform convergence to the exact solution.
The application of the constraint count approach of Section 5.4.1 to numerically investi-gate the tendency of displacement-based shell finite elements to membrane locking will be examined in the following.
The number of continuous constraints to be satisfied if shear deformations are accounted for is three, which are the membrane normal forces and shear forces to vanish for the case of pure bending. The number of relevant degrees of freedom per material point consists of three mid-surface displacements and two rotations or difference vector displacements,
respectively. The optimal constraint ratio is therefore equal tocncont.=53≈1.667.
A four node bilinear shell element with standard2×2Gauss integration points gives cn=53··14≈0.417. The biquadratic nine node shell element with standard3×3Gauss points yields a discrete constraint number ofcn=53··49≈0.741. Both values are signifi-cantly smaller than one, which indicates strongly overconstrained formulations that are sensitive to locking.
Analytical investigation of membrane locking – cylindrical shell strip
R t
P x
y z
mˆ·t3 1.0
uxP
Figure 5.13:Cylindrical shell – problem setup.
In this numerical example of a curved cylindrical shell subjected to bending, the behavior of the 3-parameter (3p) and hierarchic 5-parameter (5p-hier.) NURBS shell formula-tions with pure displacement ansatz is analyzed with regard to membrane locking. The 7-parameter shell formulation (7p-hier.) is not considered herein in order to exclude possible curvature-thickness locking effects from the investigations. Transverse shear locking will not show up due to the hierarchic difference vector formulation.
The required material constants for the linear elastic computations are Young’s modulus E= 1000.0and Poisson’s ratioν= 0.0. Thus, material-based locking effects, which are discussed in Section 5.4.5, are excluded a priori.
The only remaining defect in the discrete kinematic equations of the curved shell finite elements may therefore originate from unphysical membrane-bending-coupling.
The shell segment has a radius ofR= 10.0and a width of1.0iny-direction. For the nu-merical simulations, the domain is discretized with a structured mesh of ten biquadratic NURBS shell elements with a maximum continuity ofC1along the circumferential di-rection of the cylindrical strip. Along the edgex= 0the structure is clamped. Clamping
of the rotation-free NURBS shell elements is performed according to Section 5.2.3 by setting all mid-surface displacement degrees of freedomvi atx= 0equal to zero and additionally applying homogeneous displacement boundary conditions inz-direction on the adjacent row of control points in order to fix the horizontal tangent atx= 0.
At the free edge, x=R, a moment loading of mˆ = 1.0·t3 is applied to the discrete model. Due to the lack of rotational degrees of freedom, the moment is modeled with force couples, which are applied to the last two rows of control points. Based on the shape of the cylindrical segment, these rows of control points are both located atx=R and thus the corresponding force couples have an orientation exactly in±x-direction.
An analytical reference solution is based on Bernoulli beam theory and yields the fol-lowing stress resultants for the statically determinate structure
M=−1.0·t3, V = 0.0, N = 0.0. (5.24) The exact radial displacementuxP at the free edge is equal to 1.20, independent of the slenderness Rt.
Beam reference5p-hier.3p
Slenderness Rt DisplacementuxP
10000 1000
100 10
1.25 1 0.75 0.5 0.25 0
Figure 5.14:Cylindrical shell – displacement convergence (membrane locking).
Figure 5.14 clearly demonstrates the sensitivity of both the displacement-based Kirchhoff-Love (3p) and the hierarchic Reissner-Mindlin (5p-hier.) shell elements to membrane locking due to parasitic membrane strains inε11. Whereas for a comparatively thick structure the numerical results conform very well with the analytical reference solution, with increasing slendernessRt, the discrete radial displacement of point P tends to zero.
A moderate slenderness of Rt = 100 already reveals significant unphysical membrane strains, which lead to an underestimation of the reference displacement by almost 30%.
The numerical displacement results are displayed in Table 5.2.
SlendernessRt 10 100 1000 10000 Shell formulation (2nd order NURBS)
3p 1.1909 0.8446 0.0285 0.0003
5p-hier. 1.1948 0.8446 0.0285 0.0003
Analytic result
Beam reference 1.2000 1.2000 1.2000 1.2000
Table 5.2:Cylindrical shell – displacementsuxP(membrane locking).
As mentioned before, the investigated displacement-based NURBS shell formulations may only exhibit membrane locking in theε11strain component. All other locking ef-fects are excluded a priori due to the setup of this model problem. Additionally, no changes in thickness direction will show up during deformation, such that the results for the membrane strain componentε11 can also be carried over to the hierarchic 7-parameter shell element, in principle.
This numerical experiment illustrates that the isogeometric displacement-based NURBS shell element formulations require additional techniques for improving the behavior of the membrane part, which will be dealt with in detail in Chapter 6.
A comprehensive treatment and mathematical analysis of membrane locking for curved beam and shell structures can be found, for instance, inPitkäranta(1992),Arnold and Brezzi(1997), Choi et al.(1998),Bischoff et al. (2004) orWisniewski (2010).