About Limit Analysis

29 July 2026 10 August 2026 Administrator

Limit analysis is one of the analytical methods permitted by nuclear codes such as ASME and RCC-MR.

Compared to conventional methods that linearise stress (by drawing evaluation lines) and evaluate it, this method offers the advantage of more easily evaluating plastic collapse. In linearisation evaluation, there is a challenge that the evaluation lines are not always explicitly defined, nor is the criterion for judging whether the calculated stress is peak stress or local stress, and the evaluation method can vary slightly depending on the analyst.

Limit analysis has been used in the structural analysis of ITER's vacuum vessel, and is a method employed particularly for evaluating areas prone to stress concentration, such as near the base of ports.

In elastic analysis, once the generated stress exceeds the yield stress, the strain amount cannot be appropriately evaluated thereafter, making it impossible to evaluate the collapse load itself. On the other hand, while elasto-plastic analysis (analysis that considers material non-linearity) allows for evaluation closer to actual behaviour, it presents challenges such as setting up preconditions for handling non-linearity and increased computational cost. Limit analysis is an analytical method positioned between these two, and by combining the two concepts of the lower bound theorem and a perfectly plastic body, it is possible to evaluate the collapse load on the safe side through static structural analysis.

Lower Bound Theorem

This is a theorem stating that if a stress distribution can be found that is in equilibrium and does not exceed the yield stress anywhere, then the load is guaranteed to be less than or equal to the true collapse load. This theorem mathematically guarantees that an evaluation on the safe side is achieved, even if the analyst does not strictly satisfy the compatibility conditions of deformation when determining the stress distribution, as long as equilibrium and yield conditions are met.

Perfectly Plastic Body

This refers to a model that approximates the stress-strain diagram with two straight lines (bilinear). It is a simplified model where stress and strain increase proportionally and monotonically until the yield stress is reached, after which strain increases while stress remains constant.

Actual Judgement Method

There are broadly two approaches to the practical determination of collapse load. One is the "twice elastic gradient method", where the load is increased while plotting the displacement (or angle of rotation) at a representative point, and the intersection of the collapse load with a straight line that is half the gradient in the elastic region is considered the collapse load. The other is a method where the load at the point when local strain reaches a certain reference value (which varies depending on the object and applicable standards) is considered the collapse load. In either method, by comparing the calculated collapse load with the actual design load, the design margin can be calculated.

Verification with ANSYS: Collapse Load Evaluation in a Cantilever Beam Model

To deepen the understanding of the theory, a numerical analysis was performed using ANSYS with a simple cantilever beam model, and its consistency with the theoretical value was confirmed.

Model Conditions

Cross-section: 10 mm × 10 mm (rectangular cross-section)
Length: 100 mm
Material Model: Perfectly plastic body (bilinear model with yield stress of 200 MPa and zero hardening coefficient)
Boundary Conditions: Root completely fixed (Fixed Support), concentrated load gradually applied at the tip

Stress-strain curve for a perfectly elastic-plastic body (yield stress 200 MPa)

[Figure 1: Material Model (Stress-Strain Diagram)]

As for the material properties, a perfectly plastic body was set up, which behaves elastically until the yield stress of 200 MPa is reached, and after yielding, the strain increases while the stress remains constant. It can be confirmed that it is input as a bilinear model, which is a two-straight-line model as per the theory.

Boundary conditions for the cantilever beam model (fixed support and tip load)

[Figure 2: Boundary Conditions]

Comparison with Theoretical Value

The plastic moment Mp for a rectangular cross-section cantilever beam is given by

$M_p = \sigma_y \cdot \frac{bh^2}{4}$

Substituting the conditions for this case ( σy = 200 MPa, b = h = 10 mm ), we get

$M_p = 200 \times \frac{10 \times 10^2}{4} = 50{,}000 \, \text{N・mm}$

From the relationship between the tip load P and the root moment Mp (Mp = P・L), the theoretical collapse load is calculated as

$P_{collapse} = \frac{M_p}{L} = \frac{50{,}000}{100} = 500 \, \text{N}$

which is obtained.

Analysis Results: Load-Strain Curve and Twice Elastic Gradient Method

Collapse load evaluation of cantilever beam model (graph by twice-elastic-gradient method)

[Figure 3: Collapse Load Evaluation Graph (Twice Elastic Gradient Method)]

The load-strain curve obtained while gradually increasing the tip load is shown in the figure above. By drawing a straight line with half the gradient of the elastic region (reference line) and reading the intersection point with the load-strain curve, the collapse load is approximately 500 N, which is almost perfectly consistent with the theoretical value (500 N).

In the load-displacement curve using tip displacement, the influence of local yielding is relatively small compared to the deformation of the entire beam, so nonlinearities are less likely to appear as clearly as when viewed by strain. The reason for using strain to determine the collapse load in this case is to more directly capture the behaviour at the fixed end where the plastic hinge forms.

Verification by Stress and Strain Distribution

Equivalent stress (von Mises stress) distribution diagram for cantilever beam

[Figure 4: Stress Contour (Elemental Mean Display)]

contour.png Equivalent strain distribution diagram for cantilever beam

[Figure 5: Strain Contour]

Checking the stress distribution under load conditions exceeding the collapse load (700 N), the maximum stress near the root is 199.96 MPa, confirming that it does not exceed the set yield stress of 200 MPa. Concurrently, checking the strain distribution under the same conditions reveals that significant plastic strain (maximum approximately 0.004 mm/mm) occurs near the root where the stress has reached the yield stress. The fact that both stress and strain distributions correspond near the root supports that this behaviour is a result of plasticisation actually progressing, rather than numerical error.

Note that depending on how the stress contour is displayed (nodal average or elemental average), values that apparently slightly exceed the yield stress may be displayed due to extrapolation and averaging between adjacent elements. In this case, by using the Elemental Mean display, we have confirmed results consistent with theory, which do not exceed the yield stress set in the material model.

Summary

Although it is a simple cantilever beam model, we have confirmed that the results of theoretical calculations (collapse load from fully plastic moment) and numerical analysis by ANSYS (determination by the twice-elastic-gradient method) are almost perfectly consistent. In addition, from both stress and strain contour diagrams, we were able to visually confirm that plasticisation is actually progressing in the region exceeding the collapse load.

Evaluation method for RCC-MRx (added 2026/08/09)

In the previous chapter, the collapse load of a cantilever beam model was evaluated using the "Twice Elastic Slope (TES)" method, which is widely used in ASME etc. On the other hand, this Twice Elastic Slope method is not adopted in the nuclear code RCC-MRx. Here, the evaluation method for Limit analysis in RCC-MRx will be organised in comparison with the model in the previous chapter.

Provisions of RCC-MRx (RB 3251.113)

RCC-MRx RB 3251.113 stipulates the following equation as the criterion for judgement by Limit analysis.

$S_o = \left(\frac{C}{C_L}\right) \cdot R_L \leq S_m(\theta_{max})$

C: Load under evaluation (including load combinations)
C_L: Collapse load obtained when load C is proportionally increased
R_L: Yield stress in the perfectly elastic-perfectly plastic material model (stress value at the knee point of the bilinear curve) used to calculate the collapse load C_L
S_m(θ_max): S_m value (design stress intensity) of the material at the maximum temperature θ_max occurring within the structure under evaluation

S_o is characterised by being an index that expresses the margin against the collapse load, C/C_L, in terms of stress, independent of how R_L is taken. Furthermore, the criterion S_m is gradually relaxed according to the severity of the load conditions in Level A/B (normal operation), Level C (RB 3251.12), and Level D (RB 3251.13).

Proportional increase method for load combinations

Limit analysis can be applied not only to a single load but also to conditions where multiple loads are combined (e.g., internal pressure + earthquake + self-weight). However, in this case, instead of increasing each load component independently, it is necessary to proportionally increase the entire combination of loads at once, maintaining their ratios.

For example, if the load combination is

$C = P_{\text{internal pressure}} + F_{\text{earthquake}} + W_{\text{self-weight}}$

then, by multiplying by a common factor λ and

$[\lambda \cdot C = \lambda P_{\text{internal pressure}} + \lambda F_{\text{earthquake}} + \lambda W_{\text{self-weight}}]$

while increasing λ, and from λ_collapse at the point when the structure forms a collapse mechanism,

$[C_L = \lambda_{\text{collapse}} \cdot C]$

is determined. This is a condition clearly stated in RB 3251.113 as "loading C_L must be proportional to loading C", and it originates from the fact that both the lower and upper bound theorems of Limit analysis are theories that assume "proportional loading with a fixed load pattern". If each load component is individually increased or decreased at arbitrary ratios, there can be an infinite number of combinations of loads that cause collapse, making it impossible to define a single value C_L; hence, proportional loading is a theoretical requirement.

How to determine the collapse load C_L: Twice Elastic Slope method is not used

The method for calculating C_L in RCC-MRx is limited to the following two:

Analysis based on the lower bound theorem (lower bound method): An analytical approach that mathematically guarantees that a load is less than or equal to the true collapse load by constructing a stress distribution that satisfies equilibrium conditions and does not exceed the yield stress R_L anywhere. This approach determines the lower bound of the collapse load by actually constructing such a stress distribution.
Elasto-plastic analysis using a perfectly elasto-plastic material (FEM): An approach that sets the material model as bilinear (perfectly elasto-plastic) with a yield stress R_L, and numerically determines the collapse load by performing FEM analysis while proportionally increasing the load C.

The "double elastic slope method" used in the ANSYS verification in the previous chapter is merely an empirical graphical method for defining the "plastic load" in the elasto-plastic analysis of real materials considering strain hardening. It is not included within the framework of the perfectly elasto-plastic material model assumed by RCC-MRx. In RCC-MRx, the formation of plastic hinges (i.e., the completion of the collapse mechanism for the entire structure) can be numerically captured by using a perfectly elasto-plastic material. Therefore, there is a difference in positioning, as there is no need to redefine the "plastic load" with a convenient graphical method like TES.

Determination of collapse load: Saturation of the load-displacement curve

When numerically determining C_L within the framework of RCC-MRx, the response of the **load-displacement curve (or load-rotation curve) for the entire structure** should be observed. As the load is progressively increased, a behaviour where the gradient of the curve approaches zero (saturates) near the collapse load appears, which represents the numerical manifestation of plastic hinge formation, i.e., the completion of the collapse mechanism.

In the cantilever beam model from the previous chapter, results that agree well with the theoretical value (500 N) were obtained using the double elastic slope method with local strain. This is because, in simple structures like cantilever beams, the local strain at the fixed end is directly linked to the degree of plastic hinge formation, and in such cases, determination using local strain is less likely to cause problems.

On the other hand, in complex structures (shapes including stress concentration areas or discontinuities), local strain is strongly influenced by stress concentration and can increase divergently, unrelated to the collapse mechanism. Therefore, in RCC-MRx evaluations, it is a safer and more general approach to determine the collapse load based on the saturation of the load-displacement curve, by selecting a displacement point representative of the anticipated collapse mode (rotational behaviour of plastic hinges), rather than using local strain.

It should also be noted that all of these analyses assume the theory of small deformations (without considering geometric nonlinearity). In ANSYS, this is equivalent to not enabling the Large Deflection option (i.e., performing the analysis using the theory of small deformations). If the theory of large deformations is used, there is a risk that the collapse load will be overestimated (i.e., become non-conservative) due to the apparent increase in stiffness caused by the change in shape.

Evaluation of the previous chapter's model in the style of RCC-MRx

Applying the RCC-MRx framework to the cantilever beam model from the previous chapter (cross-section 10 mm × 10 mm, length 100 mm, yield stress 200 MPa) can be organised as follows.

Material model: Perfectly elastoplastic body (bilinear model, R_L = 200 MPa, same as the previous chapter)
Method for determining collapse load C_L: Proportionally increase the load, track the load response against tip displacement (load-displacement curve), and read the load at which saturation occurs.
Theoretical value: From the fully plastic moment M_p = σ_y・bh²/4 = 50,000 N・mm, P_collapse = M_p/L = 500 N (theoretical value same as the previous chapter)
Judgment: For the actual design load C, calculate S_o = (C/C_L)・R_L and evaluate the margin by checking if S_o ≤ S_m(θ_max) is satisfied.

Although the theoretical collapse load itself is consistent with the evaluation by TES in the previous chapter, the differences in the evaluation process are that RCC-MRx uses the "load-displacement curve" rather than the "load-local strain curve" for judgment, and after determining the collapse load, it is converted into a stress-dimensional index called S_o, rather than a simple margin (C_L/C), and compared with the temperature-dependent S_m(θ_max).

If you have any concerns about structural analysis or strength evaluation

We propose analysis methods tailored to the target structure and evaluation objectives, including plastic collapse evaluation using Limit Analysis for stress concentration areas, which tend to be overly conservative in elastic analysis, as well as elastic and elastoplastic analysis. We can also support strength evaluations based on nuclear codes such as RCC-MR and ASME, so please feel free to contact us.

Category: Analysis