Structural engineers analyze all types of structures to determine their displacements and internal forces under various load conditions. They choose between linear and non-linear analysis based on the structure’s behavior. When displacements are small and negligible, linear static analysis is sufficient. However, when displacements are large and significantly affect internal forces and bending moment, further analysis is needed to accurately assess the internal forces by considering the deformed geometry of the structure.
To simplify the above analogy – consider the two columns below: Column 1 is short and stiff, while Column 2 is tall and slender. Both columns are subjected to an axial load of wKN and a lateral load PKN. After analysis, the deflection (Δ) of Column 1 is small, making the second-order moment (wΔ) to be negligible. Hence, the column is said not to be sway sensitive (ie: the effect of the deformed geometry is not significant). Such type of columns can also be described as “no-sway”.
Conversely, the deflection (Δ) of Column 2 is large, significantly increasing the first-order moment with an additional second-order moment (wΔ1). Hence, the column is said to be “sway-sensitive” (ie: the effect of the deformed geometry is significant). This type of columns is also be described as “sway”. To accurately capture the internal forces in Column 2, a second-order analysis that will account for the second-order effect (w Δ1) is necessary. This second-order analysis, which accounts for this secondary effect, is also known as P-delta analysis.
column, and a Slender Column
The scenario discussed above is not restricted to individual members but also applicable to frames. As more and more tall, slender, and flexible structures are being built, the need to consider the effect of deformed geometry on internal forces becomes much more necessary than ever. These secondary effects can be captured using rigorous method such as large displacement theory or stress stiffening approach which are only applicable using advanced software. Eurocode 3 and several other international codes provide alternative empirical approaches which are less complex and sometimes amenable to hand calculation for simple structures. Before delving into the Eurocode 3 approach and guidance, it is worthy to note that there are two types of P-delta effects:
- P-Δ effects (pronounced as P “Big delta” effects): This accounts for second-order effect on the whole structure
- P-ծ effects (pronounced as P “small delta” effects): This accounts for second-order effects on individual members
Assessing the Sway-sensitivity of Steel Frame.
Before a structure should be considered for second-order analysis, it has to be first established that the effect of the deformed geometry on the structure is significant. In other words, it has to confirmed that the structure is sway sensitive.
According to clause 5.2.1(3), sensitivity to second-order effect is determined by a parameter αcr. This parameter is the factor by which vertical design loading would have to be increased to cause overall elastic buckling of the frame. The lower the value of αcr the more sensitive the frame to second-order effect and otherwise. The limit at which second-order effects may be neglected when exceeded is given below:
αcr = Fcr/FEd ≥ 10 (for elastic analysis)
αcr = Fcr/FEd ≥ 15 (for plastic analysis)
For multi-storey beams-and-columns type of frames (such as in buildings), and single-storey building such as portal frames without significant compression in its beams, clause 5.2.1(4) gives an approximate method to evaluate the value of αcr which is given below.
$
\alpha _{cr}\,\,=\,\,\frac{H_{Ed}}{V_{Ed}}\,\,.\,\,\frac{h}{\delta _{H,Ed}}
$
Where:
HEd is the horizontal reaction at the bottom of the storey due to horizontal loads such as wind, and equivalent horizontal force (EHF) due to imperfections.
VEd is the total design vertical load on the structure at the level of the bottom of the storey under consideration.
h is the storey height
ծH, Ed is the horizontal displacement of the top of the storey under consideration relative to the bottom of the storey with all horizontal loads.
Allowing for Second-order Effect in Sway sensitive Frames
If a structure is assessed to be sway-sensitive, then this effect has to be allowed for by the designer. The verification of the stability of frames or their parts by considering the significant effect of deformed geometry should be carried out considering both effect of imperfections and second order effects.
Clause 5.2.2(3) of the standard states that the second-order effects and imperfection should be accounted for by one of the following methods:
- both totally by the global analysis
- partially by the global analysis and partially through individual stability checks of members according to 6.3 (of Eurocode 3-1-1),
- for basic cases by individual stability checks of equivalent members according to 6.3 using appropriate buckling lengths according to the global buckling mode of the structure.
These three approaches highlighted by the standard shall be discussed below
-
Allowing for Second-order Effects totally by Global Analysis
The effects of deformed geometry (second-order effect) can be accounted for by performing global analysis of the structure considering both global and local imperfections, and also global second-order effects (P-△ effects) and local second-order effects (P-ʠ effects). This approach requires advanced software and is often beyond the capability of most of the readily available commercial software. We shall not dwell much discussing this approach.
2. Allowing for Second-order Effects by Global Analysis and Individual Stability Checks of Members
This is most practical approach and easily implementable by most commercial software. In this approach, the global second-order effects and initial sway imperfections are allowed for in the frame analysis, while the local second-order effect and local imperfection are accounted for during individual verification of members’ stability. An overview on member stability by verifying local second-order effects and imperfections can be found here for members in pure compression, and here for members subjected to combined compression and moment.
Subsequent paragraphs of this article provide details on global second-order effects, for further details on allowing for structural imperfections read “Analysis of steel frames allowing for imperfections”
The global second-order effects can be accounted for when performing frame analysis by two common methods
i By Increasing Horizontal loads by an amplification factor and performing First-order Analysis
While [performing first-order analysis, the horizontal loads due to wind and equivalent horizontal force (EHF) due to global imperfection are amplified so as to account for the effects of deformed geometry of the structure by the factor below:
$
\frac{1}{\text{1}-\,\,\frac{1}{\alpha _{cr}}}
$
This factor is known as the amplification factor. It should however be noted that this approach is only applicable if the value of αcr is greater than 3.
ii By performing full Second-order (P-delta) Analysis
Performing full scale second-order analysis is indispensable to account for effect of deformed geometry when analysing very flexible structures where the value of αcr is less than 3. This analysis often involves iterations where the deformed shape of the structure from previous analysis is loaded in subsequent analysis. The iteration is halted when it is noticed that the results of two successive analysis have reasonably converged.
3) Allowing for Second-order Effects by performing individual stability checks of equivalent members
In this approach both the global and local second-order effects are accounted for during member design. This can be achieved by using the increased buckling length (effective length) for sway mode during the design of individual members.
Avoiding Second-order Effects by Changing the structural scheme
As an addition, another intuitive solution which is not captured in the standard is to increase the stiffness of the bracing system to change the category of the building to non-sway, with this the building does not become sensitive to deformed geometry.
References
EN 1993-1-1:2005: Eurocode 3, Design of Steel structures – Part 1:1: General rules for buildings.
Handbook of Structural Steelwork: Eurocode Edition
