Second-order analysis of a 2D structure

Second-order analysis, also called P-Δ analysis or geometrically nonlinear analysis, is an essential method for studying the behavior of structures where deformations significantly influence the distribution of internal forces. Unlike the classical first-order analysis, which assumes that the structure's geometry remains unchanged under load, second-order analysis accounts for displacements and deformations to determine equilibrium. This approach is crucial for avoiding the risks of instability, such as buckling or lateral-torsional buckling, in slender structures or those subjected to significant compressive loads.

1. What is second-order analysis?

Second-order analysis involves determining the equilibrium of a structure on its deformed geometry, rather than its initial configuration. This means that internal forces (moments, axial forces, shear forces) are recalculated while considering the displacements caused by the applied loads.

This method is particularly important for structures where P-Δ effects (second-order effects) cannot be neglected :

P : Axial load applied.
Δ : Lateral or vertical displacement of the structure under load.

The P-Δ effect generates additional moments that can amplify deformations and, in some cases, lead to global instability ( buckling).

2. Difference between first-order and second-order analysis

Criterion	First-order analysis	Second-order analysis (P-Δ)
Geometry	Equilibrium calculated on the structure undeformed	Equilibrium calculated on the structure deformed
P-Δ Effects	Not taken into account	Taken into account (additional moments)
Accuracy	Sufficient for rigid structures	Necessary for slender structures or those subjected to significant compressive loads
Complexity	Simple, direct calculations	Iterative, requires numerical solvers
Typical applications	Short beams, slabs, walls	Columns, skyscrapers, long beams, cables

3. When is second-order analysis necessary?

Second-order analysis is indispensable in the following cases:

3.1. Slender structures

Columns or columns of great height
Long beams subjected to compressive loads
Skyscrapers or tall structures (wind effects, vertical loads)

3.2. Important compressive loads

Structures where axial forces (compression) are high compared to shear forces
Arches, cables or tensioned structures (like suspension bridges)

3.3. Standards and regulations

Many standards, such as the Eurocodes, require a second-order analysis for structures sensitive to instability. For example :

Eurocode 2 (concrete) : Recommends a second-order verification for compressed elements
Eurocode 3 (steel) : Requires a P-Δ analysis for slender columns

4. MMethodology of second-order analysis

4.1. Iterative process

Second-order analysis is typically iterative:

Initial linear analysis : Calculation of forces and deformations assuming the structure is not deformed
Application of P-Δ effects : Recalculation of forces taking into account the obtained deformations.
Successive iterations : The process is repeated until the deformations and forces **converge** (i.e., they no longer change significantly).

4.2. Convergence criteria

Convergence is achieved when:

Displacements between two iterations are below a defined tolerance (e.g., 1%)
Internal forces no longer change significantly

4.3. Resolution methods

Several numerical methods are used to solve the nonlinear equations:

Newton-Raphson : Fast iterative method, based on the tangent of the force-displacement curves
Picard : More stable but slower method, based on secant approximations.
Dynamic relaxation : Used for complex or unstable problems.

5. Advantages and Limitations of second order analysis

5.1. Advantages

Accuracy : Takes into account the real behavior of deformed structures
Safety : Avoids the risks of instability (buckling, overturning).
Compliance with standards : Meets the requirements of modern regulations.

5.2. Limitations

Complexity : Requires specialized software and expertise in structural analysis.
Calculation time : Iterations can be long for complex structures.
Interpretation : Results can be more difficult to analyze for non-experts.

6. Conclusion

Second order analysis is an indispensable method for analyzing structures where deformations influence the distribution of forces. It helps to prevent the risks of instability and to ensure the safety of slender structures or those subjected to significant compressive loads. Although more complex than first-order analysis, it is now facilitated by BeamZe, which automates the iterations and provides accurate results.