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The standard formula for geometric stiffness matrix calculation, which is convenient for most engineering applications, is seen to be unsatisfactory for large strains because of poor accuracy, low convergence rate, and stability. For very large compressions, the tangent stiffness in the direction of the compression can even become negative, which can be regarded as physical nonsense. So in many cases rubber materials exposed to great compression cannot be analyzed, or the analysis could lead to very poor convergence. Problems with the standard geometric stiffness matrix can even occur with a small strain in the case of plastic yielding, which eventuates even greater practical problems. The authors demonstrate that amore precisional approach would not lead to such strange and theoretically unjustified results. An improved formula that would eliminate the disadvantages mentioned above and leads to higher convergence rate and more robust computations is suggested in this paper. The new formula can be derived from the principle of virtual work using a modified Green-Lagrange strain tensor, or from equilibrium conditions where in the choice of a specific strain measure is not needed for the geometric stiffness derivation (which can also be used for derivation of geometric stiffness of a rigid truss member). The new formula has been verified in practice with many calculations and implemented in the RFEM and SCIA Engineer programs. The advantages of the new formula in comparison with the standard formula are shown using several examples.

Stress stiffening is an important source of stiffness and must be taken into account when analyzing structures. The standard formula for geometric stiffness matrices is introduced by a number of authors, such as Zienkiewicz, Bathe, Cook, Belytschko, Simo, Hughes, Bonet, de Souza Neto and others [

Let us show the general calculation algorithm for the geometric stiffness matrix (sometimes also called the stress stiffness matrix or initial stress matrix) of an element in an updated Lagrangian formulation.

Let the following hold for each component

where

Let us define matrix

where

where

Let us define matrix

and matrix

The operator

Further, let us define matrix

If state of the stress is not negligible, the potential energy of the internal forces should be completed by the following term:

Then, the following formula for the geometric matrix of the element can be written:

Integration is carried out on the deformed body

The component of the matrix

or in indicial notation:

Similar formulae also hold for a total Lagrangian formulation, but the second Piola-Kirchhoff stress tensor is then used instead of the Cauchy stress, and integration is carried out on the undeformed body

Let us consider the truss member shown in

where

This formula is independent of any strain measure or pertinent constitutive relations. It can be seen that stiffness

Let us show a derivation of a formula for geometric stiffness matrix of a truss member (see

Let

A geometric (stress) stiffness matrix can be obtained by an equilibrium condition when only the initial stress state and pertinent infinitesimal nodal displacement for each row of the matrix is taken into account. Such a definition of a geometric stiffness matrix is independent of the strain tensor chosen.

To simplify the following derivations let’s introduce both, the coordinates

Let the vector of the nodal displacements of the element be

where

Note that the truss element has no lateral material stiffness.

In general, arbitrary term of a stiffness matrix

The moment equilibrium condition can be written as follows:

For the infinitesimal angle

When introducing a displacement

From equilibrium equations and symmetry of the stiffness matrix it is easy to determine the other coefficients of the geometric stiffness matrix, particularly

The same formula corresponds with Formula (12) and is presented also by Cook in [

The resulting tangent stiffness matrix

When applying the general standard algorithm for geometric stiffness matrices to the truss element in question, we obtain:

where

where

Substituting in the formulae

the formula for the geometric stiffness matrix reads:

This geometric stiffness matrix differs from that in Formula (18) and introduces also an axial stiffening. But no reason was found by the authors for concluding that normal force had led to a change in the axial stiffness of the element. So let us derive the geometric stiffness matrix of a truss element in a more undisputable way based on the principle of virtual work.

With deformation restricted to the

where

For truss the principle of virtual work becomes

where ^{ }Piola-Kirchhoff stress in the

and the linearized equation of the principle of virtual work (virtual displacement) simplifies to:

Assuming we obtain

where

Then the equation of the principle of virtual work can be written as follows:

where

After transformation into global coordinate system

and after elimination of the vector of virtual displacements we get:

The geometric stiffness matrix (45) is the same as that obtained by use the standard Formula (27) and the first row of the matrix does not correspond with Formula (12). Let us try to derive the geometric stiffness matrix of a truss element using a more accurate strain measure.

The approximate nature of the linear relation between the deformation and displacement can be shown on a fibre of initial length

Let us denote by the vector of displacement of the starting point of the fibre. The end-point of the fibre will be displaced by vector

Using the formula for the body-diagonal of a cuboid with dimensions

Introducing stretch

we obtain the following relation for stretch of the fibre:

Let us consider the binomial theorem:

and let us take into account only the first two terms. Then we can write:

and for

If we want to be more accurate and take into account three terms of the binomial expansion, and if we neglect the third and higher powers of the derivatives of the displacement components, we get a more accurate expression for the stretch:

and hence

For a 1D problem, therefore, this more accurate expression would be identical to the formula for

Using the more accurate strain measure we obtain:

where

where is defined a new matrix

The linearized equation of the principle of virtual work (virtual displacement) modifies to:

After transformation into global coordinate system and elimination of the vector of virtual displacements we get different geometric stiffness matrix in the rotated and thus also in global coordinate system:

Resulting stiffness matrix

It can be seen that the standard formula has produced a different geometric matrix for the 2D truss element (27) than Formulae (18), (12) and (61) derived earlier and theoretically unjustified geometric axial stiffness was also produced. This formula would lead to a poor convergence rate, inaccuracy and even, in the case of extreme compression, to singularity. E.g. for^{nd} iteration it would be 1/4, and in the i-th iteration the unbalanced force of

To obtain the same geometric stiffness matrix for the 2D truss element (18) as was derived above from the equilibrium, the influence of the member

Introducing a fibre of constant cross section area A in the direction

where

To evaluate the first part of the expression, a strain measure and pertinent constitutive relation must be chosen. This part represents material stiffness. The second part, which is the matter of our interest, represents geometric stiffness.

The contributions of the two remaining principal stresses to the stiffness could be derived in a similar manner.

Let us introduce the infinitesimal volume element of continuum

It was earlier shown that for a rod (see formula (12)) the first derivative of a displacement component with respect to the same direction does not generate geometric stiffness. For the 2D or 3D continuum a similar formula to (60) can be derived in a similar way as in the case of a rod.

New measure of deformation

where

The linearized equation of the principle of virtual work (virtual displacement) for 2D or 3D continuum is similar to (32)

yielding its following form in terms of finite element matrices:

where

The difference from the standard formula (8) lies in the fact that in the

A particular case where the standard formula was applied to a 2D truss element producing an unintentional change in the axial stiffness was presented earlier. This phenomenon can also be generally observed when the standard formula is used. It is clear that the uniaxial stress state will provide the same result regardless of the way it is modeled, i.e. a truss member modeled as a 3D solid should provide the same result as one modeled by a truss member or by shell elements. To guarantee this and to improve the influence of the stress state on stiffness,

the members

stress component should influence the stiffness in the same direction. To ensure objectivity (independence from any arbitrary coordinate system) of the geometric stiffness matrix, the omission of the above mentioned terms must be evaluated in the principal stress axes

where

The component of the matrix

To obtain the geometric stiffness matrix in the global coordinate system the following transformation must be performed:

The transformation from the global coordinate system

Then, the relations between the first derivatives of the base functions and stresses are the following:

For a quadrilateral plane stress element the following can be obtained:

where C = cos(α); S = sin(α); α is the angle between principal and global directions; t is the element thickness and A is the area of the element.

A similar formula also holds for the total Lagrangian formulation for such an element, but the second Piola- Kirchhoff stress tensor is then used instead of the Cauchy stress, integration is carried out on the undeformed body

An application of the new formula for the geometric stiffness matrix for large strain was demonstrated on the example of a unit cube represented by different computational models (rod, shell, solid elements) with different orientations in space (see

logarithmic strain and Cauchy stress tensors. Let E be the Young modulus and for simplicity let us assume zero Poisson ratio. The cube was exposed to uniform stress of the magnitude E or −E normal on two opposite sides. A logarithmic strain value of 1 or −1 and the prolongation or shortening of the value of

Let the sequence

then p is called the order of convergence of the sequence. The constant

If p is large, the sequence

The numerical solution of the presented example has shown that to reach a sufficiently good result using the standard formula (ANSYS etc.) 15 iterations were needed whereas using the improved approach presented in this paper (RFEM) only 5 iterations were needed to obtain the same precision.

The present formula for a geometric stiffness matrix, which has been published in many books, is widely utilized, objective and simply defined. However, stability and convergence problems occur when analyzing large strains, or, what is more important in practice, in a case of yielding. If the yield criterion is satisfied, then the

material stiffness decreases substantially. The stress state remains high and in case of compression the tangent stiffness in the direction of the compression can become negative even with a small strain.

This is caused by a theoretically unsupported change in pressure stiffness in the direction of compression produced by the standard formula. This results in a correspondingly high nodal force unbalance, poor convergence and possibly also instability. The origin of the problem arises from the approximation of strain, in which only the first two terms of the binomial series are applied.

The presented algorithm is slightly more complicated, but remains objective and provides a solution with increased stability, a higher rate of convergence in the case of a large strain, or plastic yielding, and improved accuracy over the present formula. In case of very large strain, the number of iterations needed could be several times less using the new formula comparing to the standard formula. In many cases the new formula can even provide solutions in cases where the standard formula has failed. This new formula for a geometric matrix has been implemented in the RFEM program and has been demonstrated to be much more stable and faster than the standard formula.

This outcome has been achieved with the financial support of the Czech Science Foundation (GACR) project 14-25320S “Aspects of the use of complex non linear material models”.

I. Němec,M. Trcala,I. Ševčík,H. Štekbauer, (2016) New Formula for Geometric Stiffness Matrix Calculation. Journal of Applied Mathematics and Physics,04,733-748. doi: 10.4236/jamp.2016.44084

_{ }Shape functions

_{ }_{ }Rotation tensor

_{ }_{ }Second Piola-Kirchhoff stress

^{ }Internal and external virtual work

_{ }_{ }Displacement field

_{ }Spatial (Eulerian) coordinates

_{ }_{ }Coordinates in principal aces