EVALUATION OF PΔ MOMENT AND THE EFFECT ON DEFORMATION OF COLUMNS IN SYMMETRICAL FRAMES

Abstract

 1. Introduction

 The objective of this study is to examine the amplification factor of moment and rotation angle due to the PΔ effect. A sub-assemblage frame is analyzed by using the buckling slope deflection method, and amplification factors are obtained. Numerical analysis is performed taking the G factors, slenderness ratio λ and axial load ratio n_y as the analytical parameters. An approximate formula for estimating the amplification factors is presented on the basis of the buckling load proposed by the second author.

 

 2. Analytical work

 The sub-assemblage frame shown in Fig. 2 is analysed, taking the geometric nonlinear effect into consideration. Equation (3) is obtained by the fundamental formula of buckling slope deflection method. Moment equation at point A and B becomes Eqs. (7) and (10), respectively, where G_A and G_B are G factors defined as Eqs. (9) and (11). From the story equation (Eq. (12)) together with moment equation, the rotation angle R of the member AB and the moment M_AB and M_BA of node A and B are obtained as Eqs. (13) ~ (15). In absence of axial force P, the rotation angle R and the moment M_AB and M_BA become Eqs. (18) ~ (20), and then the amplification factors are obtained by Eqs. (21) ~ (23).

 

 3. Results and discussions

 As the analytical parameters, G factors G_A(=G_B), slenderness ratios λ and axial load ratios n_y are selected, and they vary as follows: G factors G_A(=G_B) : 0 (rigid beams), 0.5, 1, 2.5 and 5, slenderness ratio λ: 20, 40 and 80, axial load ratio n_y: 0.1, 0.3, 0.5, 0.7. Figure 4 shows R/₀R－Z relations and M/₀M－Z relations. A similar tendency is observed between R/₀R and M/₀M. Figures 6 and 7 show the effect of axial load ratio n_y and G factors. From Fig. 6, it is shown that the ratios R/₀R and M/₀M become large as the axial load ratio increases. Approximate effective length factor K_dsn has been proposed by Eq. (29), and using this, the amplification factors a_m can be obtained by Eq. (33). The values of a_m agree well with the R/₀R and M/₀M, and the relative error is shown in Fig. 8. While the error increases as the value Z becomes large, the a_m fits with an error of 2% or less when Z is smaller than 1. Figure 9 shows the limit Z_α where the amplification factor is guaranteed below the value of α. Moreover, an example for calculating the amplification factors of six-story and five-span steel frame shown in Fig. 10 is performed.

 

 4. Conclusions

 The conclusions derived from this study are as follows:

 1) The amplification factors of the moments and rotation angle are presented by Eqs. (21) ~ (23). The governing parameters are G factors and Z defined in Eqs. (9), (11) and (6).

 2) The amplification factors can be approximately calculated by using Eq. (33).

 3) The limit value of Z to assure the restriction of the amplification factors is proposed by Eq. (38).