Second order analysis - Questions Part 1
Second order analysis takes into account how the structure deforms while loads are being applied on it. Numerically speaking, all load is divided into smaller portions and for each load portion, the stiffness of the structure changes.
Second order effects are also called P-Δ and p-δ effects, and in SCIA Engineer (among other places) these effects are also referred to as Geometrical nonlinearity.
Here are some common questions from users that our Support team often needs to address.
When do I need to perform a second order analysis?
Material-specific Eurocode parts (EC2, EC3, EC4, EC5, etc.) specify in their chapter 5 (Structural analysis) when to perform a second order analysis. For steel structures, low values of the αcr coefficient indicate high flexibility (or slenderness) of the structure, and therefore, high sensitivity to initial imperfections and lateral displacements.
The αcr coefficient is determined by dividing the elastic critical buckling load for a global instability mode, Fcr, by the design loading on the structure, FEd. In order words, we need to rely on the results of stability analysis to find out IF we need to perform second-order analysis or not.
What we can do in SCIA Engineer is create stability combinations from selected design combinations and obtain the critical buckling factors that correspond to these loading scenarios. These factors are simply multipliers of the load that is present in the stability combination. And because our stability combinations are the “design load,” these factors are exactly the αcr coefficients that we are looking for.
It is also important what kind of FEM analysis of the structure we would like to perform to obtain the design internal forces: are we going to stick to elastic analysis, or we would also like to make use of plastic hinges to redistribute moments? According to EC3, when we perform elastic analysis and any of our αcr are lower than 10, then we need to design the structure using second-order analysis results. In the case of plastic (e.g., hinge) analysis, any αcr dropping below 15 indicates the need to go to second-order.
How can I insert global imperfections according to the code?
Geometric imperfections in a FE model make sure that second-order effects are properly triggered during a nonlinear analysis. The ones familiar with Chapter 5 of EC3 often ask how to take global or member imperfections into account in SCIA Engineer.
Imperfections are defined on the level of a nonlinear combination: each nonlinear combination can have its own set of imperfections. This is useful, because different loading scenarios induce different failure modes, which in turn are influenced to a higher or lesser extend by a specific imperfection shape.
To define a global imperfection as a uniform sideways tilt of the structure, use the input type called “Simple inclination” and define the relative inclination along the global X and Y axes, dx and dy. In EC3, a formula is given in Chapter 5.3.2, Figure 5.2 for the angle of the inclination, φ. In the input fields in the nonlinear combination dialog, use dx = 1000*φx (or dx = 1000*tan(φx), depending on how you interpret the figure). The Simple inclination imperfection type is perfect for structures regular in plan and height.
Are there different input possibilities for global imperfections?
Yes, a couple of them are possible in SCIA Engineer. Namely:
- Inclination functions: if you would like to vary the inclination value and even sign along the height (or length) of a structure, you can use imperfection functions. These functions are manually defined multilinear curves, and are inputted and stored via Libraries > Structure, Analysis > Initial imperfections.
- Imperfections based on a load case: SCIA Engineer can calculate the deformation of a structure for a load case you specify, and apply this deformation as an initial imperfection. If you know what imperfection shape you more or less would like to obtain, you can define a load case that would cause such deformation; you may define a load case with a fraction of all design loads; in this way, you are certain that all possible stability effects will be represented in the analysis.
- A stability eigen shape as an imperfection: you can select a stability combination and one of its buckling shapes to apply as an imperfection on the structure.
How can I insert the local imperfections according to the code?
As mentioned above, imperfections are assigned per nonlinear combination. The most practical way to assign the value of imperfections is to refer (via the settings of the combination) to the buckling settings. This lets one specify, via the buckling systems, which members should have imperfections: it is often practical to consider imperfections on specific members that we would like to investigate in more detail, rather than assign imperfections to all members in the structure.
Are there different input possibilities for local imperfections?
Yes. Instead of referring to buckling data, SCIA Engineer lets you define local imperfections directly via the properties of the nonlinear combination. Keep in mind that the specified curvature there will then be applied to all members in the structure.