In order to evaluate the sensitivity of a portal frame to 2nd order effects, the buckling amplification factor Î±

_{cr}has to be calculated. This calculation requires the deflections of the frame to be known under a given load combination. Check §5.2.1 of EN 1993-1-1.

In this post, elastic first order analysis is performed on a single bay portal frame using

*Staad.Pro*in order to calculate the reactions under vertical loads at ULS (see figure below). The actions on the portal frames are as given below;

g

_{k}= 2.31 kN/m

q

_{k}= 3 kN/m

Design load = (1.35 g

_{k}) + (1.5 × q

_{k})

Roof load = (1.35 × 2.31) + (1.5 × 3.0) = 3.1185 + 4.5 = 7.6185 kN/m

When analysed;

Vertical base reaction V

_{Ed}= 118.263 kN

Horizontal base reaction H

_{Ed}= 65.651 kN

Maximum Axial Load on rafter N

_{R,Ed}= 93.9 kN

**Axial Compression in the rafter**

According to clause 5.2.1(4), if the axial compression in the rafter is significant, then the Î±

_{cr}is not applicable.

The axial compression is significant if;

Î» ̅ ≥ 0.3√((Af

_{y})/N

_{Ed}) and this can be rearranged to show that the compression is significant if

N

_{Ed}≥ 0.09N

_{cr}

N

_{Ed}is the design axial load in the rafter

*L*

_{cr}is the developed length of the rafter pair from column to column;

*L*

_{cr}= 30/cos 15° = 31.058m

N

_{cr}= (Ï€

^{2}EI)/

*L*

_{cr}

^{2}= (Ï€

^{2}× 210000 × 16000 × 10

^{4})/31058

^{2}= 343789.059 N = 344 kN

0.09N

_{cr}= 0.09 × 344 = 30.94 kN

N

_{Ed}= 93.9 kN > 30.94 kN, therefore axial load is significant.

When the axial force in the rafter is significant, a conservative measure of frame stability defined as Î±

_{cr,est}may be calculated. For frames with pitched rafters;

Î±

_{cr,est}= min(Î±

_{cr,s,est }; Î±

_{cr,r,est})

Where;

Î±

_{cr,s,est}is the estimate of Î±

_{cr}for the sway buckling mode

Î±

_{cr,r,est}is the estimate of Î±

_{cr}for the rafter snap-through buckling mode. This is only relevant when the frame has more than two bays or if the rafter is horizontal.

To calculate Î±

_{cr}, a notional horizontal force (NHF) is applied to the frame, and the horizontal deflection of the top of the column is determined under this load.

H

_{NHF}= V

_{Ed}/200 = 118.263/200 = 0.591 kN

For the assessment of frame stability and for the assessment of deflections at SLS, the base may be modelled with a stiffness assumed to be a proportion of the column stiffness as follows;

- 10% when assessing frame stability (10% of the column stiffness may be modelled by using a spring stiffness equal to 0.4EI
_{column}/*L*_{column}) - 20% when calculating deflections at SLS (20% of the column stiffness may be modelled by using a spring stiffness equal to 0.8EI
_{column}/*L*_{column})

- For assessing frame stability, the second moment of area of the dummy member should be taken as I
_{xx}= 0.1I_{xx,column} - For calculating deflection at SLS, the second moment of area of the dummy member should be taken as I
_{xx}= 0.2I_{xx,column}

*L*

_{column}and pinned at the far end.

On modelling the nominally pinned base using dummy members, the deflection below was obtained for the notional horizontal forces.

Therefore Î±

_{cr}= h / 200Î´

_{NHF}= 7000/(200 × 2.963) = 11.81

Î±

_{cr,s,est}= 0.8(1 - N

_{Ed}/N

_{cr})

Î±

_{cr}= 0.8(1 - 93.9/344)11.81 = 6.869

Î±

_{cr,s,est}= 6.869 < 10

Therefore, second order effects are significant. Since Î±

_{cr,s,est}≥ 3.0, the amplifier is given by;

[1/(1 - 1⁄Î±

_{cr,est})] = [1/(1 - 1 ⁄ 6.869)] = 1.17

Note: If Î±

_{cr,s,est}is less than 3.0, second order analysis must be used. The simple amplification is not sufficiently accurate.

Therefore the modified partial factor of safety to account for second order effects are as follows;

Î³

_{G}= 1.17 × 1.35 = 1.5

Î³

_{Q}= 1.17 × 1.5 = 1.75

You can now use these modified partial factors to multiply the characteristic permanent and variable actions. The ultimate vertical action on the rafter (taking into account second order effects)is;

Roof load = (1.5 × 2.31) + (1.75 × 3.0) = 8.715 kN/m

Thank you for visiting Structville today and God bless you.

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