EVALUATION OF PΔ EFFECT FOR SYMMETRICAL BRACED 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 for symmetrical braced frames. A sub-assemblage frame is analyzed by using the buckling slope deflection method, and amplification factors are obtained. An approximate formula for estimating the amplification factors is presented.

 

 2. Analytical work

 The sub-assemblage frame shown in Fig. 2 is analysed, taking the geometric nonlinear effect into consideration. It is assumed that the slope θ_A= θ_A1 = θ_A2 and θ_B= θ_B1 = θ_B2 and the beam moments of node A and B are distributed to upper and lower column according to the column stiffness. Equation (6) is obtained by the fundamental formula of buckling slope deflection method. Moment equation at point A and B becomes Eqs. (10) and (13), respectively, where G_A and G_B are G factors defined as Eqs. (12) and (14). From the story equation (Eq. (15)) together with moment equations, 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. (16), (17) and (18). In absence of axial force P, the rotation angle R and the moment M_AB and M_BA become Eqs. (20), (21) and (22), and then the amplification factors are obtained by Eqs. (23), (24) and (25).

 

 3. Results and discussions

 As the analytical parameters, G factors G_A(=G_B), slenderness ratios γ axial load ratios n_y and non-dimensional horizontal stiffness k^* 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 γ: 20, 40 and 80, axial load ratio n_y: 0.1, 0.3, 0.5, 0.7, non-dimensional horizontal stiffness k^*: 0, 1, 5. Figure 5 shows R/₀R－Z relations and M/₀M－Z relations. The dotted lines and solid lines show the R/₀R－Z relations and M/₀M－Z relations, respectively. A similar tendency is observed between R/₀R and M/₀M. Figure 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. (27), and using this, the amplification factors a_m can be obtained by Eq. (31). In Fig. 5, 6 and 7, white circles indicate the results obtained using Eq. (31). 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 factors is guaranteed below the value of α=1.05.

 

 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. (23) - (25). The governing parameters are G factors, Z and k^* defined in Eqs. (12), (14), (9) and (5).

 2) The amplification factors can be approximately calculated by using Eq. (31).

 3) The limit value of Z to assure the restriction of the amplification factors is proposed by Eq. (35).