Incremental formulation up to macro-cracking .1 Incremental Newton–Raphson method

The essence of the non-linear FE procedure is the incremental Newton–Raphson method (INRM) described previously. Irrespective of the updating strategy adopted, its basic formu-lation may be summarised in flowchart form as depicted in Figure 4.7.

The external-load vector is applied in load steps Δfe (typically, Δfe = 5%–10% of the esti-mated failure load), to which the unbalanced nodal forces (i.e., the vector of residual forces, Δfr) of the previous iteration must be added. Then, a decision on whether or not to update the various D-matrices – and, hence, the incremental stiffness matrices [k] – is made. If the current iteration is an updating iteration, the result is an update of the incremental stiff-ness matrix of the structure (usually known as the tangent k-matrix – see Section 4.2.2). It should be noted that, although the incremental k-matrix may be the tangent k-matrix, it is not necessary to use fully tangent properties; in fact, any k-matrix derived from initial secant or tangent properties can be used. In the present work, whenever updating of the k-matrix is required, the matrix is only quasi-tangent because it is more convenient to use simply the material tangent moduli which, it will be recalled (see Section 3.1.2.3), expressions (3.41) through (3.43), neglect σid, so that the true constitutive law is not actually implemented. If the system of equations can be solved, the increments of the nodal-displacement vector Δd are obtained, from which the new increments in strains (Δε) and stresses (Δσ) at all Gauss points are calculated through the matrices [B] and [D], respectively; thus, the total (cumula-tive) strains (ε) and stresses (σ) may be ascertained. The new total stresses are now balanced, that is, they satisfy, at this stage, equilibrium (namely, ∫[B]^Tσ′dV = fi ≠ fe, where f_i is the set of (nodal) internal forces), but, in general, they are not compatible with the actual material

stress–strain relationships (unless, of course, convergence has been obtained). Therefore, these equilibrated stresses are corrected so as to satisfy the constitutive equations (at this point, obviously, the coupling effect σid must formally be taken into account), and this requires the additional stress increments Δσr, which lead to the total stresses σ′ that are now unbalanced (as equilibrium is no longer satisfied since ∫[B]^Tσ′dV = fi ≠ fe). These corrective stresses Δσr create new residual or unbalanced forces Δfr which are applied to the structure in the next iteration in order to re-establish equilibrium conditions. If unbalanced forces do

Add external load, ∆f_e

Add residual forces

∆f = ∆f_e + ∆f_r

∆d = [k]^–1 ∆f

Update D-matrices [k] = ∫_v [B]^T [D][B]dV

STOP. unable to solve

STOP. divergence Decide on updating

Not convergence and not divergence

Convergence No

No Yes

Yes

Updating? Yes

No

Balanced stresses

∆ε = [B] ∆d

∆σ = [D] ∆ε σ = σ′_p + ∆σ Unbalanced stresses

σ′ = σ + ∆σ_r

∆f_r = –∫_v [B]^T ∆σ_r dV

∆f_e = 0

Figure 4.7 Basic flowchart for the non-linear analysis based on the incremental NR method.

not satisfy convergence criteria, the external load is kept constant and further iterations are carried out; otherwise a new external-load increment is applied and the whole procedure repeated. There are only two possible reasons for stopping the analysis: unrealistic solutions to the set of equations (e.g., owing to ill conditioning), or divergence of residual forces. It should be stressed at this point that the preceding outline, as well as much of the subsequent material in the present section devoted to the micro-cracking regime is, clearly, also relevant in the presence of macro-cracking, although, as stated earlier, the specific features of the latter regime will be covered in detail in following section.

An essential requirement of any non-linear package is the ability to store intermediate results which are then used to obtain more accurate values as the iterative solution proceeds and con-vergence criteria are eventually met. The need for such a storage facility is even more apparent when an incremental procedure is being implemented. Such a procedure imposes on the struc-ture under consideration additional external loading when the iterative solution has converged at a given load level, thus enabling the program to follow automatically a monotonic loading path up to overall failure. It is clear that further storage is necessary here in order to follow the structural response at each external-load level, which implies storing strains, stresses, displace-ments and cracking/yielding information. The storage of such vast quantities of data might, at first, appear somewhat wasteful, in the sense that, once a given external-load increment has achieved convergence, information on previous load steps could be dispensed with; however, the retention of the data pertaining to the whole of the analysis is useful for post-processing purposes such as, for example, plotting and, more generally, the capacity for studying in detail and at leisure the various parameters at given solution stages. While the necessary storage additions and accompanying programming strategies are beyond the scope of the present dis-cussion, the relevant background may be found in Bedard 1983; Gonzalez Vidosa 1989.

So far, no specific choice of updating strategy has been mentioned, all INRMs sharing the common layout of Figure 4.7, as mentioned previously. However, this question of updating technique must be addressed when certain aspects associated with the convergence and effi-ciency of the adopted algorithm are being considered. Although the rates of convergence of the modified and mixed INRMs are slower than that corresponding to the pure INRMs (see Figure 4.6), the former methods usually economise on computer time since they cut down on the high cost of the numerical integration of stiffness matrices. Furthermore, if instead of the set of equations being solved by iterative methods, a single reduction or decomposi-tion of the stiffness matrix is carried out, then the former methods (i.e., modified and mixed INRMs) also save in the number of factorisations of the k-matrix of the structure. Now, it must be stressed at this point that the analysis of concrete structures has two sources of high localised non-linearities which make the above reasoning about efficiency of second-ary importance: cracking of concrete and yielding of steel at given Gauss points. These, however, will be discussed later (for concrete, see Section 4.3.3) and, before their occur-rence, the solution searches corresponding to the micro-cracking regime associated with a much milder form of non-linearity may follow safely the mixed INRM in which the stiff-ness matrix is reformulated or updated only periodically. This combination of the pure and modified INRMs represents a sensible compromise between the high convergence of the former and the low cost of the latter (Philips and Zienkiewicz 1976; Bedard 1983).

To summarise, therefore, the incorporation, in the solution strategy, of the constitutive relationships throughout the micro-cracking regime consists, very broadly, of the following three specific steps, which are carried out at each integration point:

• At each iteration, a check is made that the state of stress lies within the failure envelope.

• At each iteration, the state of stress corresponding to the state of strain generated by the FE solution is corrected so that the constitutive laws are satisfied.

• When the updating of the stiffness matrix is required, the concrete material properties K_t, G_t and hence E_t, νt which correspond to the actual state of stress, are obtained.

In order to outline more fully the overall FE procedure described in Figure 4.7, the for-mulation of the B-, D- and k-matrices needs to be discussed in some detail, as well as the residual-force implementation and the criteria adopted for convergence and divergence.

Such aspects are dealt with in the subsequent five sections and, whenever relevant, this will be done, in generic form, by reference to the 3-D brick element.

4.3.2.2 Incremental strain–displacement relationships

The incremental version of Equation 4.1 that link the 3-D displacement field and the nodal displacements is given by

Δu = [ ]N dΔ (4.70)

where Δu = (Δux, Δuy, Δuz) are increments of displacements, Δd = (…, ΔdxI, ΔdyI, ΔdzI,…) are increments of nodal displacements and [N] is the matrix of shape functions. By reference to brick elements, (4.70) may be written as

Δ

in which N_I is the shape function of the Ith node.

The vector of strains ε (εx, εy, εz, γxy, γxz, γyz) is defined on the assumptions of the linear the-ory of elasticity (Timoshenko and Goodier 1970). Accordingly, the relevant expressions are

εx = ϑux/ϑx (4.72)

where the (engineering) strain definitions adopted differ from those corresponding to the components of the strain tensor εij(= (1/2)(ϑui/ϑxj +ϑuj/ϑxi)). By reference to Equation 4.8, the incremental strain-nodal displacement relations are

Δε = [ ]B d Δ (4.78)

which, in 3-D problems, is obtained by combining Equations 4.71 through 4.77, the result pletely define the 6 × 3 block of the B-matrix corresponding to the contribution of the dis-placements of node I to the strain increments at a given point within the element. Since, as pointed out earlier, the analysis does not include geometrical non-linearities, such deriva-tives remain constant throughout the analysis. Therefore, they are calculated for all Gauss points only once (at the first iteration of the analysis). Their calculation, in terms of the local coordinates and the ensuing Jacobian matrix, has already been explained in Section 4.1.3.

4.3.2.3 Incremental stress–strain relationships for uncracked concrete

The increments of stresses and strains are related by the D-matrix adopted. By reference to Equation 4.12, and neglecting initial strains and/or stresses, its incremental counterpart is simply

Δσ = [ ]DΔε (4.80)

For uncracked-concrete Gauss points, the D-matrix may be calculated by reference to a linearly-elastic isotropic material which is usually described in the following concise form

σij = 2Gεij + 3με δo ij (4.81)

where G and μ are the shear and Lame’s moduli, the former being given by (3.2) while the latter is also related to E and ν by the expression

μ = νE/[(1 + ν)(1 − 2ν)] (4.82) On the basis of (4.78), therefore, the incremental constitutive relations (4.72) through (4.77) for uncracked concrete may be written as

Δσ coordinates, while G and μ are derived from the tangent shear and bulk moduli described

in section 3.1.2.3 (i.e., expressions [3.41] through [3.43]). Clearly, the coefficients of the D-matrix are functions of the state of stress (i.e., G(τo), μ(σo,τo)), but, at the same time, it is worth noting that the constitutive matrix is isotropic throughout the micro-cracking regime and, hence, invariant with respect to any set of orthogonal axes.

Although the D-matrices for cracked Gauss points will be described in detail in Section 4.3.3, it is convenient to note at this stage that they are anisotropic and that they will be defined with respect to cracked directions. Thus, all their coefficients in global coordinates will, in general, be non-zero, since cracked non-isotropic D-matrices require a transforma-tion from local to global directransforma-tions (see Appendix B for such a transformatransforma-tion). However, it is evident that axes transformations do not affect the D-matrix in the present case of isotro-pic behaviour before macro-cracking.

4.3.2.4 Incremental force–displacement relationships

The incremental stiffness matrix of an element connects the increments of nodal forces and nodal displacements. The relevant expression may readily be written down by reference to either Equation 4.21 or 4.22 which relates total displacements and forces up to a given stage of loading, the result being

Δf = [ ]k dΔ (4.84)

with the expression for [k] given by Equation 4.20. As explained previously, this k-matrix is calculated in global directions by numerical integration and, hence, may be written as

[ ]k = ([ ] [ ] [ ] )B D B Ji i i i

∑

_i=ⁿ₁ ^{T wi (4.85)}

where J_i and w_i are the Jacobian and the weight of the ith Gauss point, respectively, and n is the total number of Gauss points in the element. As was the case with the coefficients of the B-matrix, Jacobians do not change throughout the analysis and, hence, they are calculated only at the first iteration. Furthermore, as [k] is symmetric, it is necessary to calculate only the coefficients of its upper (or lower) triangle.

While [k] can be calculated directly from expression (4.85) without reference to the actual B- and D-matrices, it is worth noting that the expressions for its coefficients can be com-puted more efficiently by taking into account any special features (e.g., sparsity) of the relevant B- and D-matrices. This will be discussed briefly below, as it leads to efficient pro-cedures for the numerical calculation of [k], and also prepares the ground for the discussion (in Section 4.3.3) of the effect of the smeared representation of cracking on the conditioning of stiffness matrices.

Expression (4.85) can split into blocks of 3 × 3 coefficients relating to the DOF of pairs of nodes. Let [k]^IJ be one such 3 × 3 block relating to the DOF of nodes I and J

This [k]^IJ block is equal to ∑i I=[ ] ,k where [ ]i k_i is the contribution of the ith Gauss point to such a block. For the brick elements adopted herein, and for the general case of cracked Gauss points, this contribution is given by the following expression:

[ ]k_i^IJ i i

The sparsity of the B-matrix is evident: half of its coefficients are zero and placed at known positions. Thus, the number of computations may be reduced quite significantly once all the multiplications involving zero terms are identified and left out of subsequent numerical opera-tions. The D-matrix in Equation 4.87 is not sparse, as it refers to the general case of a cracked Gauss point expressed in terms of global coordinates. On the other hand, uncracked Gauss points are described by Equation 4.83 and, for such isotropic conditions, further reduction of computing effort is clearly possible. Various ways of achieving such computational sav-ings for both isotropic and anisotropic material descriptions have been explored in Gonzalez Vidosa (1989), with subsequent implementation in the computer program as appropriate.

4.3.2.5 Residual forces

In accordance with the present FE model, the non-linear force–displacement relationships at the structural level arise exclusively as a result of the non-linearities in the stress–strain expressions. The iterative procedure required to follow these non-linear σ − ε laws relies on the residual-forces method, by means of which stress corrections (at the material level) of balanced stresses cause the appearance of equivalent unbalanced nodal forces that must be applied in the next iteration in order to re-establish the equilibrium conditions for the overall structure. This may be summarised through the following expression, in which the equivalence between external and internal forces is implicit:

fe =

∫

^{[ ]B}T^σʹdV +

∫

^{[ ] (B}T ^{σ − σ}ʹ⁾dV ^(4.88) where σ and σ′ are balanced and unbalanced stresses respectively; and the first term denotes unbalanced (internal) forces (but satisfying the constitutive relations) while the second term re-establishes overall equilibrium conditions (but causing, in turn, lack of compatibility between stresses and strains). Therefore, the residual forces are given by

Δfr = −

∫

^{[ ]}B^{T(σ −}ʹ ^σ)dV = fe −

∫

^{[ ]}B^Tσʹ^{dV (4.89)} where, it should be recalled, the last term represents nodal internal forces.

When the constitutive laws are expressed in the form σ = σ(ε), as is the case for cracked Gauss points (to be discussed in Section 4.3.3) and, also, for steel Gauss points both before and after yielding, unbalanced stresses are worked out in accordance with such laws in the standard manner shown in Figure 4.8. The residual stresses are given by the components of Δσr = σ′ − σ and must be checked (together with residual forces and, possibly, other criteria) for convergence.

The stress correction for uncracked Gauss points, on the other hand, follows an initial-strain technique, since the constitutive law (for uncracked concrete) is given in the form ε = ε(σ) (see expressions (3.44) in Section 3.1.2.3). Figure 4.9 summarises schematically the implemented initial-strain technique for uncracked concrete points. First, increments of strains (and, then, the total strains) are computed from increments of nodal displacements (subscripts other than r indicate the iteration number)

σ

σ

σ′_p

σ′

∆σ = [D] ∆ε ∆σ_r

∆ε

ε

Figure 4.8 Stress correction, by the standard initial-stress technique that achieves satisfaction of the consti-tutive law but disturbs the equilibrium of the previously balanced stresses σ.

σ

σ_n

∆ε_n (∆ε_r)_n

(∆σ_r)_n

1 2

3

0

σ′_n–1

ε_n–1 ε_n ε(σ_n) σ′_n

∆σn = [D] ∆εn

ε

Figure 4.9 Stress correction, by the initial-strain technique, used for uncracked concrete points (for cracked concrete points, and for all steel Gauss points, refer to Figure 4.8).

Δεn = [ ]B dΔ n (4.90)

εn = εn−1 +Δ εn (4.91)

Balanced stresses are next computed using the D-matrix incorporated into the set of equations

σn = σn−1 +[ ]D Δ εn (4.92)

These balanced stresses are corrected by the following expression (see Figure 4.9)

ʹ = − −

σn σn [ ( )]{ ( )Dσn ε σn n ε n} (4.93)

where [D(σn)] and εn(σn) are, respectively, the tangent D-matrix given by Equation 4.83 and strains, both of these being in accordance with the constitutive law corresponding to the balanced stress level. The components of (Δεr)_n = εn(σn) – εn are residual strains that have to be checked for convergence. It is worth noting that this initial-strain technique converges very quickly in practice owing to the mild nature of the uncracked material non-linearities.

(Clearly, while the standard initial-stress method of Figure 4.8 involves the satisfaction of the material law at the end of every iteration, the initial-strain technique satisfies neither equilibrium nor the material law at the end of an iteration unless, of course, convergence has taken place.)

Once stresses are corrected at a given Gauss point, the numerical integration of Equation 4.89 leads to the following contribution to the residual-forces vector

( )

4.3.2.6 Convergence and divergence criteria

The convergence of solutions obtained by iterative procedures can be checked in terms of one or more vector increments of various parameters, the norms or elements of which must all be smaller than certain prescribed values. The parameters in question include quantities such as displacements, residual forces, residual stresses and residual strains (Bergan and Clough 1972). In practice, only one of these vector increments is checked, since all of them are interrelated. Displacement and force criteria are usually preferred to stress and strain criteria. With regard to non-linear analyses of structural concrete, it would appear that force criteria have mostly been used (Suidan and Schnobrich 1973; Lin and Scordelis 1975, Philips and Zienkiewicz 1976, Cedolin and Dei Poli 1977, Cristfield 1982).

In the earlier part of the work (Bedard 1983) the two residual-stress criteria

i ri

max|Δσ ( )|< 0 1 MPa . (4.95)

i ri ri

max|Δσ ( )|< 0 01. |σ( )| (4.96)

were adopted in recognition of the fact that a single criterion might prove unrealistic for the whole of the loading path. Thus, for example, while Equation 4.95 can ensure a rea-sonable level of accuracy in the early load steps, at more advanced stages of the loading such a residual-stress value might become impractical to achieve, especially as failure is approached, when the numerical solution tends to become unstable. This is why the second convergence criterion – Equation 4.96 – based on residual stresses given as percentages of total-stress values, is more attractive as one nears ultimate-load conditions, and is in keeping with the notion of accepting a larger force imbalance as the total load increases (Cedolin and Nilson 1978). Clearly, the satisfaction, at each Gauss point, of either of the criteria defined by Equations 4.95 and 4.96 is sufficient for convergence.

In subsequent work (Gonzalez Vidosa 1989), several additional convergence criteria were studied, of which the following were implemented in the non-linear procedure. First, a max-imum residual strain of 2.5 mm/m, or less than 0.5 of the total strain, was adopted for uncracked concrete, which was found to be slightly more restrictive than the stress criteria adopted earlier (i.e., Equations 4.95 and 4.96). In addition, once cracking was implemented, it was prescribed that no concrete Gauss points should exceed the failure envelope, with a maximum residual stress of 0.1 MPa set as the limit at cracked Gauss points. The latter stress criterion was also adopted for steel Gauss points, and for these any change from one linear branch to another was treated as a lack of convergence requiring further iterations.

Now, it was found that these various convergence criteria are usually met on specification of the following (additional) residual-force criterion:

i f ir F

max|Δ ( )|< 0 001. |Δ | (4.97)

where |ΔF| is the applied load step (between 5% and 10% of the ultimate experimental load). This last condition is a very restrictive requirement indeed when compared with other reported criteria (see e.g., Suidan and Schnobrich 1973, Lin and Scordelis 1975, Philips and Zienkiewicz 1976, Cedolin and Dei Poli 1977, Cristfield 1982 mentioned previously).

However, all the above criteria were kept (including Equation 4.97) in order to avoid conver-gence in certain situations, such as concrete Gauss points being outside the failure envelope (‘cracking’ criterion) or new yielding of the steel (‘yield’ criterion), as may occur with less restrictive convergence criteria. Nevertheless, it should be said that residual forces become negligible as soon as material properties are updated, and no new cracking and/or yield occur.

The incremental process stops either because of divergence of residual forces, which is taken to have occurred when

The incremental process stops either because of divergence of residual forces, which is taken to have occurred when

In document Finite Element Analysis (Page 149-159)