neglected). Of course, at cross-sections in some distance from the reference one, the results are inaccurate. This approach was followed in example 2 throughout the whole of the checks.
To check the deviation from the non rotational-symmetric approach, the latter one was used also for the stress calculation due to the wind in example 2, but only for comparison – not in the DBA design checks. The difference in the wind effects is a remarkable 29% - the more accurate one giving the larger values.
3.5 Direct Route (using Elastic Compensation)
3.5.1 Check against Global Plastic Deformation GPD
Elastic compensation[1-6] calculates bounds of the limit load and shakedown load for a - structure for a given load set by using iterative elastic FE-analysis. This method is a generalisation of the technique proposed by Marriot [5] for estimating lower bound limit loads on pressure vessel applications. The procedure involves calculating a series of elastic equilibrium stress fields where the stress is redistributed by altering the elastic modulus of each element based upon the maximum unaveraged nodal stress from the previous iteration, thus
where E is the elastic modulus, i the iteration number, σnom is some nominal value, and σemax the
maximum unaveraged nodal stress in that element from the previous solution. The resulting redistributed stress fields are equilibrium stress fields. By definition, if the equivalent stress anywhere in the equilibrium stress field does not exceed the yield stress of the material then that stress field relates to a lower bound on the limit load. Therefore, scaling the applied loads by the amount given by maximum stress in the redistributed stress field to the yield stress of the material will give the limit load, i.e.
where AL is the limit load for the action(s), Aap is the applied load to the FE-model, σy is the yield
strength of the material and σmax the maximum unaveraged nodal stress in the model. Due to the
simplicity of this method, it lends itself to application in design checks against GPD according to the direct route method in prEN 13445-3 Annex B.
As this method of determining limit loads is wholly elastic it is very simple to apply different yield criteria to the analysis without the convergence difficulties associated with elasto-plastic analysis. In the rules for the check against GPD, the analysis is required to be based on Tresca’s yield condition and associated flow rule, first order theory and an elastic-perfect plastic material model. This can be performed directly using the elastic compensation procedure for solid models.
For analyses utilising shell elements, elastic compensation cannot be applied directly in the same method as described above. As a shell has only one element through thickness, it is not possible to modify the elastic modulus through the thickness. To allow the application of elastic compensation to shell elements a generalised yield model is adopted in the analysis. Ilyushin's[6] generalised yield model for a doubly curved shell is used which is based upon Mises' condition and associated flow
max 1 e nom i i E E σ σ − = max σ σy ap L A A = ⋅
rule. A brief overview of Ilyushin’s generalised yield model is given at the end of this sub-section. As the code specifies the application of Tresca’s, some modification of the results is required when the analysis is based upon Mises' condition. As stated in 3.3.1 the application of a factor of √3/2 to the design stress will result in a conservative result. This method can also be used as a check on the results for solid models utilising Tresca’s condition.
As with the elasto-plastic method above, elastic compensation may be applied in two ways to the check against GPD. First, for specified actions a check on the admissibility of the load set can be made by checking that the equilibrium stress fields satisfy the lower bound limit load theorem. If the maximum equivalent stress anywhere in the equilibrium stress field remains below the design strength of the material then the specified loading is admissible. Second, the limit on the applied action(s) may be found using the above procedure and the maximum admissible action(s) can be determined according to prEN 13445-3 Annex B.
Where there are multiple actions applied, the second case, where the actual limit is calculated, becomes more complex. For example in problems 3.1 and 3.2 there is a constant moment action and an internal pressure action. The limit on the pressure has to be found. In elastic compensation, the applied load set is scaled to give the limit load set. Therefore, in multiple action conditions one analysis is not sufficient to define the limit load, as the ratio of the loads at the limit is not already known. In this situation multiple analyses are made for different ratios of applied load and a limit locus is constructed that describes the limit state for all combinations of load. In the case of problems 3.1 and 3.2, with the constant moment known, the limit pressure can be found directly from the limit locus.
As the code rules in prEN 13445-3 Annex B address elasto-plastic analysis and not any simplified method, some problems arise in applying elastic compensation. The application rule for GPD in prEN 13445-3 Annex B.9.2 states that the maximum absolute value of principal strain should not exceed 5%. As elastic compensation is not a displacement-based approach the values of strain are not accurate in the equilibrium stress fields and cannot be used. Therefore, it is possible for structures that are ‘stiff’ well into the plastic range (where the limit according to the code is defined by the 5% maximum principal strain limit) that elastic compensation non is conservative. In this situation the elastic compensation result would be close to the limit defined by loss of equilibrium in the elasto-plastic analysis (for an elastic-perfect plastic material). This can be noted in problem 3.1 where the elastic compensation result is very much higher than the elasto-plastic result defined by the limit on the principal strain. However, if the elasto-plastic result were defined by the tangent intersection method with no limit on the principal strain, the results would be similar.
3.5.2 Check against Progressive Plastic Deformation (PD)
The principle in the check against PD according to prEN 13445-3 Annex B.9.3 is fulfilled if the structure can be shown to shake down, as progressive plastic deformation and alternating plasticity are the two possible failure modes if the structure fails to shakedown. It is possible using the elastic compensation procedure to calculate lower bounds on the shakedown load using elastic compensation. As used in the elasto-plastic check described in 3.3.2 above Melan’s shakedown theorem is also used in the elastic compensation procedure. The Mises limit stress field for the applied action(s) is calculated using elastic compensation as described above. As the zeroth iteration in elastic compensation is the true elastic stress field (i.e. no modulus modification), the residual stress field can be found by subtracting the elastic stress field from the redistributed limit