15.6. Updated Lagrangian Revisited: A Direct and Consistent Tangent Stiffness Matrix for the Q8 Element

Introduction

StrainSmall

DisplacementLarge

MaterialLinear elastic

Stress statePlane stress

The direct Updated Lagrangian formulation with a consistent tangent stiffness matrix developed in Section 15.5 for Q4 is extended here to the eight-node isoparametric Q8 element. The Q8 interpolation functions are described in Section 9.6.

Section 15.4 extends the approach of Section 15.3 from Q4 to Q8. The present formulation uses the same direct evaluation of total Euler–Almansi strains and consistent linearization as Section 15.5. The derivations are therefore not repeated here.

The spatial kinematics and linearization underlying this formulation are discussed by Bonet and Wood [1].

Q8 element formulation

Extending the element from four to eight nodes gives sixteen displacement degrees of freedom. The strain–displacement matrix $\htmlClass{fea-math-bold-italic}{\boldsymbol{B}}$ has dimensions $3\times16$ , and the element tangent stiffness matrix $\htmlClass{fea-math-bold-italic}{\boldsymbol{k}}_{el}$ has dimensions $16\times16$ . The interpolation-function derivatives and nodal blocks are evaluated for all eight nodes; the material, geometric and thickness contributions retain the definitions given in Section 15.5.

The $2\times2$ Gauss rule used for Q4 is replaced by $3\times3$ integration. The one-dimensional coordinates and weights are:

s_1=-\sqrt{3/5},\quad s_2=0,\quad s_3=+\sqrt{3/5}

w_1=5/9,\quad w_2=8/9,\quad w_3=5/9

The same Gauss points and weights are used in both natural-coordinate directions, $s$ and $t$ . All nine coordinate pairs are used, with two-dimensional weights obtained by multiplication, $w_{ij}=w_i w_j$ . The current geometry is updated at every Newton iteration.

Numerical example

The cantilever beam of Section 15.5 is analyzed using the same geometry, material properties, boundary conditions and applied load. The table compares three mesh refinements for each element type and the ANSYS reference results.

Q4–Q8 comparison. All displacements are in millimeters.
Q4	Q8
MATLAB — first refinement
4×30 120 elements 155 nodes $v_{min}=-276.22$ $u_{max}=2.369$ $u_{min}=-142.65$	2×15 30 elements 125 nodes $v_{min}=-289.69$ $u_{max}=2.450$ $u_{min}=-160.10$
MATLAB — second refinement
8×60 480 elements 549 nodes $v_{min}=-286.60$ $u_{max}=2.431$ $u_{min}=-155.78$	4×30 120 elements 429 nodes $v_{min}=-290.10$ $u_{max}=2.449$ $u_{min}=-160.47$
MATLAB — third refinement
16×120 1920 elements 2057 nodes $v_{min}=-289.29$ $u_{max}=2.449$ $u_{min}=-159.34$	8×60 480 elements 1577 nodes $v_{min}=-290.19$ $u_{max}=2.456$ $u_{min}=-160.54$
ANSYS reference
16×120 1920 elements 2057 nodes $v_{min}=-289.48$ $u_{max}=2.380$ $u_{min}=-159.92$	8×60 480 elements 1577 nodes $v_{min}=-289.92$ $u_{max}=2.369$ $u_{min}=-160.44$

Q8 gives accurate displacement predictions with substantially coarser meshes than Q4. Even the mesh of 30 Q8 elements and 125 nodes gives results close to those of the finer meshes. Q4 requires considerably more refinement, particularly for the vertical displacement and the minimum horizontal displacement.

In the tested examples, the converged Q8 solutions were practically independent of load-step subdivision within the prescribed numerical tolerance. The ANSYS comparison is satisfactory; the remaining differences, including those in the maximum horizontal displacement, are reported without assigning an unverified cause.

MATLAB programs

The MATLAB package implements the consistent Q8 formulation. main.m controls the load steps and Newton iterations; gen.m defines the mesh, boundary conditions and nodal loads. stiff_cons8.m evaluates the strains, stresses, internal forces and consistent tangent using nine Gauss points per element. plot_disp.m and plot_stress.m display the results.

MConsistent UL formulation — Q8MATLAB programs with total Euler–Almansi strains and consistent tangent stiffnessDownload ZIP

References

[1] Bonet, J., and Wood, R. D. (2008). Nonlinear Continuum Mechanics for Finite Element Analysis. 2nd ed. Cambridge University Press. See especially Chapters 4, 8 and 9 for finite-deformation kinematics, linearized equilibrium and finite-element discretization.