Second-Order Effects in Reinforced Concrete Buildings According to Eurocode 2 (EC2) 

The design of reinforced concrete buildings, in accordance with Eurocode 2 (EC2), requires a precise evaluation of second-order effects, which include local and global P-Delta effects. This phenomenon occurs due to non-linear interactions between axial forces and the lateral displacements of structures. Therefore, adapting calculation methodologies to these effects is essential to ensure structural safety and avoid over or underestimated design. 

In this article, we will discuss the main practical and theoretical concepts associated with second-order effects, the simplified methods provided in EC2 (nominal stiffness and nominal curvature), and the associated limitations, based on analyses performed by software such as SAP2000, ETABS, CSiBridge, VIS and CSiCol. 

Second-Order Effects (P-Delta Effects) 

Second-order effects in reinforced concrete buildings can be classified into two main levels: 

• Local P-Delta Effects: These relate to the non-linear behavior of isolated elements due to axial forces at their ends and associated local displacements. These effects can be analyzed individually using nominal curvature or nominal stiffness methods, as defined by Eurocode 2. 
• Global P-Delta Effects: These refer to the interaction between the lateral displacements of the structure as a whole and the axial forces in its various elements. These effects significantly depend on the global lateral stiffness of the structure and the interaction between its different components. 

Simplified Methods of Eurocode 2 

Eurocode 2 allows the use of simplified methods for considering second-order effects in individual elements: 

• Nominal Curvature Method: Based on the increase in bending moments due to the introduction of additional deformations along the element. This method is particularly useful when axial forces and buckling effects are low. 
• Nominal Stiffness Method: Uses reduced stiffness values in the calculation of forces and displacements, taking into account local second-order effects. This is a more conservative approach for high axial forces but should only be used to obtain local P-Delta effects, similar to the nominal curvature method. 

It is important to note that simplified methods should not be used in isolation to obtain global P-Delta effects, as they were not designed for this purpose, nor do they consider the global stiffness characteristics of the structure. In structures where these effects are relevant, global non-linear P-Delta analyses should be performed. 

Below, it can be seen that for columns in structures susceptible to significant lateral displacements, the governing moment for design will be the one marked as Msway, which can only be obtained through a P-Delta analysis. In the opposite situation, for structures less susceptible to lateral displacements, the Mnonsway moment will typically be the governing one and can be obtained through simplified methods. 

Limitations of Simplified EC2 Methods 

The simplified methods described have clear limitations for specific situations: 

• Cases where Global P-Delta Effects are Relevant: When the structure exhibits significant lateral displacements in response to horizontal loads (wind, earthquakes, etc.), simplified methods may underestimate global effects. In this case, a non-linear second-order analysis, such as global P-Delta analyses performed by advanced software like SAP2000 and ETABS, is recommended. 
• Potential Over Estimation: The amplification of moments through simplified methods should be performed at the level of equivalent first-order end moments. If, instead, we directly increase the end moments through the nominal curvature or stiffness methods, we may obtain excessively conservative results, especially if global second-order moments are obtained through non-linear analyses. 

For example, in a simple cantilever column, isolated nominal stiffness (or nominal curvature) methods can provide reasonable estimates for end moments. However, if the column is connected to other elements with some lateral stiffness, global P-Delta effects should not be estimated using these simplified methods but rather obtained through non-linear analyses. 

Conclusion 

The treatment of second-order effects in reinforced concrete buildings according to EC2 involves a careful understanding of the available simplified methods (nominal curvature and nominal stiffness) and their limitations. Although useful in local analyses, these methods should not replace global non-linear analyses where global P-Delta effects are relevant. 

With the support of advanced software tools like SAP2000, ETABS, and VIS, engineers can perform more precise and safer analyses, avoiding excessively conservative results or the opposite. These software packages already incorporate calculation algorithms according to EC2, allowing for efficient practical application. 

Thus, for projects where global effects play a significant role, it is recommended to perform global analyses on the complete structural model, ensuring that normative criteria are met rigorously and safely. 

You can see in the article on distributed plasticity how to carry out second order analysis taking into account the material non-linear behavior of reinforced concrete columns and beams modeled through frame elements