This article presents guidance on how to allow for imperfections in analysis of steel frame according to EN 1993-1-1 (Eurocode 3). At the end of this article, a practical example of analysis considering imperfection is demonstrated on a simple 2D frame using ETABS software.

♠♠♠♠♠♠

Structural frames in reality are always marred with imperfections which are deviations from idealized state of the structure during simulations. Imperfections could be material imperfections such as when concrete or steel grade are of slight deviation from engineer’s specification, these imperfections at material level are always allowed for by partial factors of safety during designs. Other imperfections that are not catered for by partial safety factors are geometry imperfections during fabrication, frame imperfection during erection, etc. Hence, EN 1993-1-1 stipulates in clause 5.3.1(3) that the following imperfections should be taken into account in steel structures.

1.  global imperfections for frames and bracing systems
2.  local imperfections for individual members

Global Imperfections    

The assumed shape of global imperfections should be derived from the elastic buckling mode of a structure in the plane of buckling considered. Most buildings are prone to buckling in sway mode; hence the effect of imperfections is allowed for in frame analysis by means of an initial sway imperfection.

The initial sway (φ) of the frame due to imperfections is given in EN 1993-1-1 clause 5.3.2(a) as:

φ =   φ₀α_hα_m

where;

φ₀  is the basic value 1/200

α_h is the height reduction factor

α_h = 2/√h (but 2/√3 ≤  α_h ≤ 1.0)

h is the height of the structures in meters

α_m  is the reduction factor for number of columns in a row. α_m  = √(0.5 (1 + 1/m))

m is the number of columns in a row including only those columns which carry a vertical load not less than 50% of the average value of the column in the vertical plane considered.

 

Equivalent Horizontal Force

Ordinarily, it should be required to model the global imperfection in building frames and then analyse the frame as such, having determined the global sway from the expression given above. This approach, however, would be rather tedious and time consuming especially for complex multi-storey frames. To avoid this, rather than modelling these imperfections in a frame it would be better represented with a horizontal force which would have equivalent effects as modelling these imperfections in the structure. This horizontal force that simulates the effects of imperfections is termed equivalent horizontal force (EHF). This horizontal force is obtained by multiplying the sway imperfections (φ) by total vertical load (Ned) as shown in the figure below:

For multi-storey beam-and-column building frames, equivalent horizontal forces should be applied at each floor and roof level as the fraction of the total vertical force in each storey and roof level. It should also be noted that this sway imperfections should apply in all relevant horizontal directions, but need only be considered in one direction at a time. It should also be applied together with all load combinations. The most onerous combinations and direction should be taken as critical. The equivalent horizontal force (EHF) is also referred to as notional loads in other national standards such as ACI.

According to clause 5.3.2 (4(B)) of EN 1993-1-1, the global sway imperfection can be neglected if the expression below is satisfied:

H_Ed ≤ 0.15V_Ed

H_Ed is the design horizontal force due to wind etc.

V_Ed is the design vertical force

Local Imperfections

Local imperfections occur due to several factors such as lack of straightness during erection of members or residual stress in member during fabrication. This local imperfection can be allowed for in frame analysis as initial bow imperfection (e₀/L)

However, local imperfection most times in real project are not considered during analysis as its effects are duly catered for by national standards during member design. Table 6.1 of EN 1993-1-1 provides imperfection factor which is applicable to different buckling modes which is meant to be applied during the verification of individual member for stability. You can click here to read more on stability of axially loaded steel members or click here to read more on the stability of members subjected to combined bending and axial compression.

 

Analysis of Steel Frame in Etabs allowing for Global Imperfections

In this segment, we shall demonstrate how to allow for global imperfections using Etabs software. To aid understanding, we shall use a simple 2d frame model which is as shown below. We shall cater for imperfection in the frame in two ways:

• By manually modelling the EHF in ETABS.
• By using the notional load provision in ETABS.

 

Determine the Initial Sway of the Frame

As earlier stated, global imperfection can be idealized in frames as initial sway (φ).  And the EHF which would cover the effect of imperfection is determined as the product of initial sway (φ), and total vertical loads for each storey.

Let us calculate the initial sway for the frame

φ =   φ₀α_hα_m

φ₀  is the basic value 1/200

α_h is the height reduction factor

α_h = 2/√h (but 2/√3 ≤ α_h ≤ 10) = 2/√6

since 2/√6 < 2/√3, h is taken as 2/√3

α_m  = √(0.5 (1 + 1/m))

m is the number of columns in a row which is 4

α_m  = √(0.5 (1 + 1/4)) = 0.79

φ = 1/200 x 2/√3 x 0.79 = 0.005

Modelling Equivalent Horizontal Force (EHF) Manually in ETABS

We can calculate the EHF and apply it manually in ETABS. To calculate this, we will need the total axial loads per each storey. Hence, we will need to analyse the frame without allowing for imperfection to get the total axial force in each column.

The analysis results of the frame without allowing for imperfection showing the axial forces and bending moments are as shown below:

 

First Story

Total axial load: 58.9578 + 129.2112 + 129.108 + 59.7628 = 377KN

EHF = φNed = 0.005 x 377 = 1.885KN

Second Story

Total axial load: 118.9726 + 256.7445 + 256.2114 + 122.1502 = 754KN

EHF = φNed = 0.005 x 754 = 3.77KN

Now we shall apply the EHFs at appropriate storeys as shown below then we re-analyze the frame.

Bending Moment Diagram of analysis allowing for imperfection

 

Applying EHF in ETABS as Notional loads

The exercise so far allows readers to understand the concept of EHF in its raw form. However, for a complex multi-storey model applying EHF manually can be daunting. ETABS can automatically apply EHF to models using the notional load feature. We shall take us through the step in activating this

Steps in activating Notional Loads in ETABS

Step 1) Go to “Define Load Pattern” (Path: Define > Load patterns)

Step 2) Add a new notional load pattern for each dead and live load patterns for x and y directions (For this specific example, we shall only add notional load in the x direction as the frame is 2d.)

Step 3) Click “modify lateral load” and specify the load ratio for each notional load you have created. (The load ratio is the initial sway (φ) we had calculated earlier. However, for more complex model where it might be difficult to calculate φ, a value of 0.005 may be used as default). The demonstration for dead load case is only shown below.

Step 4) Go to “Define Load Combination” (Path: Define > Load Combinations) and combine all notional load in same direction with all other load patterns with the appropriate load factor. The combination of all loads and notional loads are encircled as shown below.

Step 5) Analyse the Model.

The result of analyzing the model when EHF is applied as Notional load is shown below. A closer look will show that it is similar to the result when we apply EHF manually as point loads

Conclusion

 Applying EHF to allow for imperfections increases the moments in all the columns in the frame except the leftmost columns. This underscores that imperfections should be allowed for in frame analysis so that individual element can be sized as appropriate.

Member Design

The beam in the frame should be designed as an unrestrained beam while the column should be designed as a member subjected to combined bending and compression. Click here and here to study a detailed worked example on design of the beam and the column respectively.