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.

Comparative Analysis of the Effects of deformed geometry on a short 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

Second-order Effects in RC Structures according to Eurocode 2

Prior to considering a structure for second-order analysis, the standard provides a few criteria to first established that the effect of the deformed geometry on the structure is significant. If not significant then effects of deformed geometry on the structure should be ignored. The below sections discuss the application principles to assess and also allow for sway-sensitivity if significant in a structure.

Assessing the Sway-sensitivity of Building Structures

There are about three guidance in Eurocode 2 for the classification of building as sway (sway-sensitive) or no-sway. The reference to the clauses or annexes associated with this guidance are enumerated below and discussed subsequently in detail.

• Clause 5.8.2 (6)
• Clause 5.8.3.3
• Annex H

Clause 5.8.2 (6)

Clause 5. 8. 2 (6) of EN 1992-1-1 states “Second order effects may be ignored if they are less than 10 % of the corresponding first order effects”.

This clause only mentioned a principle for which when not satisfied, then a structure (or an individual member) should be considered for second-order analysis, application rules to implement this for a structure is given in clause 5.8.3.3 and also Annex H for special cases.

Clause 5.8.3.3

As an application to the general principle given in clause 5.8.2 (6), global second order effects in buildings may be ignored if

$ F_{v,Ed}\,\,\leqslant \,\,K_1.\,\,\frac{n_s}{n_s\,\,+\,\,1.6}\,\,.\,\,\sum{\frac{E_{cd}\,\,I_c}{L^2}} $

Where:

F_v,Ed is the total vertical load (on braced and bracing members)

n_s is the number of storeys

L is the total height of the building above the level of moment restraint

E_cd is the design value of modulus of elasticity of concrete

I_c  is the second moment of area (uncracked concrete section) of bracing member(s)

This expression in clause 5.8.3.3 is only applicable if the below conditions are met:

1. Torsional instability is not governing, i.e., structure is reasonably symmetrical
2. Global shear deformations are negligible (as in a bracing system mainly consisting of shear walls without large openings) –
3. Bracing members are rigidly fixed at the base, i.e. rotations are negligible
4. The stiffness of bracing members is reasonably constant along the height
5. The total vertical load increases by approximately the same amount per storey

Where these conditions are not met the provisions in Annex H should be adopted

Annex H

Annex H gives alternative solutions to structures where the conditions of applicability of clause 5.8.3.3 are not met. It deals with structures with and those without shear deformations.

Bracing system without significant Shear deformation

For a bracing system without significant shear deformations (e.g., shear walls without openings), global second order effects may be ignored if:

F_v,Ed  ≤ 0.1F_v,BB

Where;

F_v,Ed is the total vatical load on braced and bracing members

F_v,BB is the nominal global buckling load for global bending

H.1.2 (2) of EN 1992-1-1 should be consulted for expression for estimating the buckling load ( F_v,BB )

Bracing system with significant Shear deformation

Global second order effects may be ignored if the following condition is fulfilled:

$ F_{v,Ed}\,\,\leqslant \,\,0.1F_{v,B}\,\,=\,\,0.1\frac{F_{v,BB}}{\text{1}+\frac{F_{v,BB}}{F_{v,BS}}} $

where:

F_v,B is the global buckling load taking into account global bending and shear

F_v,BB is the global buckling load for pure bending, 

F_v,BS is the global buckling load for pure shear, F_v,BS = ΣS

ΣS is the total shear stiffness (force per shear angle) of bracing units (see Figure H.1 of the standard)  

 

Allowing for Second-order Effect in Sway sensitive Frames

If the structure is classified as sway sensitive, then there are two options proffered by the standard in accounting for second-order effects:

1. Annex H – Application of increased horizontal forces
2. Perform P-Delta Analysis using a suitable computer programme

Annex H – Application of increased horizontal forces.

This clause is based on linear second order analysis according to 5.8.7. Global second order effects may then be taken into account by analysing the structure for fictitious, magnified horizontal forces

F_H,Ed: $ F_{H,Ed}\,\,=\,\,0.1\frac{F_{H,oEd}}{\text{1}-\frac{F_{v,Ed}}{F_{v,B}}} $

Where;

F_H,oEd  is the first order horizontal force due to wind, imperfections etc.

F_v,Ed  is the total vertical load on bracing and braced members

F_v,B is the nominal global buckling load In cases where the global buckling load

F_v,B is not defined, the following expression may be used instead:

$ F_{H,Ed}\,\,=\,\,0.1\frac{F_{H,oEd}}{\text{1}-\frac{F_{H1,Ed}}{F_{Ho,Ed}}} $

F_H1,Ed is the fictitious horizontal force, giving the same bending moments as vertical load N_Y,Ed acting on the deformed structure, with deformation caused by F_H,oEd (first order deformation), and calculated with nominal stiffness values according to 5.8.7.2  

Perform P-Delta 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.  

Second-order Effects in Slender Members according to EC2

Similar to overall structure, individual members such as walls and columns are also evaluated for second-order effects. This second-order effect in member can be accounted for in either of the two ways enumerated below depending on the behaviour of the overall structure.  

• If the structure is not sway sensitive, then the individual member should be checked for second-order effects, and these effects should be allowed for in non-sway mode if significant.
• When the structure is sway sensitive, the structure should be analysed for the global second-order effects as explained in previous sections. Additionally, the individual members should be checked for second-order effects, and these effects should be allowed for in non-sway mode if significant.

Further details on how to determine second-order effects in sway mode and also allow for them when significant is discussed in the remaining sections.

Assessing the Sway-sensitivity of Individual member.

There are two guidance in Eurocode 2 to assess the classification of isolated members as sway (sway-sensitive) or no-sway. The reference to the clauses or annexes associated with this guidance are enumerated below and treated subsequently in detail

• Clause 5.8.2
• Clause 5.8.3.1

  Clause 5.8.2

Clause 5. 8. 2 of EN 1992-1-1 states “Second order effects may be ignored if they are less than 10 % of the corresponding first order effects”. This clause only mentioned a general principle for which when not satisfied, then a structure or an isolated member should be considered for second-order effects, application rules to implement this for an isolated member is given in clause 5.8.3.1.  

Clause 5.8.3.1

Clause 5.8.3.1 states “A column would be considered for second-order effects if its slenderness exceeds the prescribed limiting slenderness.”  

From the above sentence, the susceptibility of an isolated member (e.g., columns, walls etc.) to second-order effects should be determined in terms of slenderness wherein the slenderness of the column is compared against the limiting slenderness. Hence, we have two important parameters to determine, which are:

a) Slenderness of the column

b) Limiting slenderness  

Slenderness of a Column  

The slenderness of a column is the ratio of its effective length to its radius of gyration. The effective length of the column is dependent on the mode of buckling of the column. The mode of buckling is dependent on whether the whole structure is sway or no-sway. The sway-sensitivity of the structure can only be determined if it can be confirmed whether or not the deflection of the structure increases the first-order moments by more than 10% by iterative analysis or through other alternatives by EN 1992-1-1 discussed previously. This would be a long process. As a walkaround, the standard allows engineers to assume the sway-sensitivity of the structure by virtual inspection by categorizing the structure as braced or unbraced.  

Braced and Unbraced Structure  

Braced: A braced structure resist lateral load and also achieve stability by means of an independent bracing system such as shear wall, core walls etc. Columns in braced structure are generally assume to buckle in “no-sway” mode and their effective length is determined using the expression below. For braced members:

$ _{\,\,l_0=0.5l\sqrt{\left( \text{1}+\,\,\frac{K_1}{\text{0.45}+\,\,K_1} \right) \left( \text{1}+\,\,\frac{K_2}{\text{0.45}+\,\,K_2} \right)}} $

Bracing (Unbraced): A bracing structure. Columns in unbraced structure are generally assume to buckle in “sway” mode and their effective length is determined using either of the expressions below. For Unbraced members, the larger of the two below formulas is used:

$ l_{o\,\,}=\,\,I\sqrt{\text{1}+\,\,10\left( \frac{K_{\text{1}}x\,\,K_2}{K_1+\,\,K_2} \right)} $

or

$ l_{o\,\,}=\,\,I\left( \text{1}+\,\,\frac{K_1}{\text{1}+\,\,K_1} \right) \left( \text{1}+\,\,\frac{K_2}{\text{1}+\,\,K_2} \right) $

Slenderness and Limiting Slenderness

Having determined the effective length of a column in the relevant buckling mode, then the slenderness of the column is determined as the ratio of the effective length and radius of gyration as shown below:

λ = lo/r

The slenderness is then compared against the limiting slenderness. Should the slenderness exceed the limiting slenderness then the column is considered slender and second-order effects is accounted for, if otherwise then second-order effects should be ignored. The slenderness limit is calculated using:

$ \lambda _{\lim}\,\,=\,\,\frac{\text{20}x\,\,A\,\,x\,\,B\,\,x\,\,C}{\sqrt{n}} $

Allowing for Second-order Effect in Slender Members

Once the column has been adjudged slender

1)A general method: This method is based on non-linear analysis of the structure including geometric non-linearity. A sophisticated computer programme is required to implement this method and as such will not be consider further in this article.

(2) Nominal Stiffness Method: In this method, second-order analysis is carried out using the nominal stiffness of the column, taking into account the effects of cracking, material non-linearity and creep on the overall behavior. This also applies to adjacent members involved in the analysis, e.g. beams, slabs or foundations. Where relevant, soil-structure interaction should be taken into account.  

(3) The moment magnification factor method: In this method, the critical moment allowing for second-order effects is obtained by multiplying the first-order moment by a magnification factor.

(4) The nominal curvature method: In this method, the curvature, deflection and finally the critical moment of a column is estimated. The critical moment is always determined by adding an estimated second-order moment to the first-order moment. This method is similar to that used in BS 8110, and is more widely adopted. For practical application of this method read, “Design of slender column using nominal curvature method”.  

 

References

EN 1992-1-1: Design of concrete structures – Part 1-1: General rules and rules for buildings

Designer’s Guide to EN 1992-1-1 and EN 1992-1-2