Buckling Amplification Factor for Portal Frames Sensitive to 2nd Order Effects

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 α_cr is not applicable.

The axial compression is significant if;

λ ̅  ≥ 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 = (π²EI)/L_cr² = (π² × 210000 × 16000 × 10⁴)/31058² = 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 α_cr,est may be calculated. For frames with pitched rafters;

α_cr,est = min(α_cr,s,est ; α_cr,r,est)

Where;
α_cr,s,est is the estimate of α_cr for the sway buckling mode
α_cr,r,est is the estimate of α_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 α_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)

When the software cannot accommodate a rotational spring, the base fixity may be modelled by a dummy member of equivalent stiffness as shown below;

• 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

In both cases, the length of the dummy member is 0.75L_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 α_cr = h / 200δ_NHF = 7000/(200 × 2.963) = 11.81
α_cr,s,est = 0.8(1 – N_Ed/N_cr)
α_cr = 0.8(1 – 93.9/344)11.81 = 6.869
α_cr,s,est = 6.869 < 10

Therefore, second order effects are significant. Since α_cr,s,est ≥ 3.0, the amplifier is given by;
[1/(1 – 1⁄α_cr,est)] = [1/(1 – 1 ⁄ 6.869)] = 1.17

Note: If α_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;

γ_G = 1.17 × 1.35 = 1.5
γ_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

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