Analysisof Effective Length Factors of Hinged Pitched-Roof Framed
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E3S Web of Conferences 165, 04069 (2020) https://doi.org/10.1051/e3sconf/202016504069 CAES 2020 Analysis of Effective Length Factors of Hinged Pitched-Roof Framed Column with Sway under Arch Effect PingLi1*, LipingLiu1 and ZunliTeng1 1Department of civil engineering, Long Dong University, Qing Yang, Gan Su, 745000, China Abstract. Aiming at the limitation of the calculated length coefficient of the gable portal rigid frame columns in the standards and regulations, which ignores the influence of the arch effect caused by the axial force of the inclined beams lope, this paper uses the elastic buckling equation of the solid-webbed articulated pitched-roof frame, considered second-order effect and the arch effect of the inclined beam under uniform load together. And the length coefficient calculated is analyzed. It is concluded that the larger the slope of the inclined beam and the ratio− of the rigid frame span are, the more obvious the arching effect is, and the larger the calculated length coefficient of the rigid frame column is. 1 Introduction horizontal load, there will be a second-order effect. This effect that increases the bending moment For the gable portal light rigid frame, reasonable support and displacement is called the span effect[2]. Therefore, − and tie bars can ensure the stability of the rigid frame the total lateral displacement of the column is the sum outside-plane. Therefore, the overall stability of the rigid of the lateral displacement caused by the bending frame is determined by the inside-plane stability. deformation under the vertical load and the 1 Relevant codes and regulations calculate the stability of generated by the horizontal thrust under the arch effect. the rigid frame through calculating the length coefficient. 2 Under the transverse uniform load, there is an axial force caused by the slope at the joint between the inclined beam and the vertical beam. The articulated gabled portal rigid frame exhibits arch-like characteristics in terms of stress and stability, namely the arch effect. At present, considering the stability of the gable portal frame, the related codes and regulations are to simplify the inclined beam as a horizontal beam. Such simplification fails to consider the influence of arch effect on the overall Figure 1. Span variation of pitched-roof frame stability of the rigid frame. This paper will comprehensively consider second-order effect The arch effect causes an axial force at the beam and the arch effect of the inclined beam under the end, as shown in Figure 2, ignoring the small pressure uniform load, and analyze the − relationship between the changes caused by the lateral displacement of the column. calculated length coefficient and the arch effect. According to the deformation characteristics of the rigid frame with lateral displacement, the lateral displacement of the ridge node is equal to that of the beam and column 2 Elastic buckling equation of rigid nodes at both ends. The pedestal is hinged, the bending frame moment is 0, and the ridge point is where the reverse bending point is, so the bending moment is also 0, and The displacement method is used to solve the elastic the vertical displacement is also 0, but there can be side buckling equation of the gable portal frame shown in shifts and corners. The reference [2] derived the elastic Figure 1. Under the vertical uniform load, the gabled buckling equation of the pedestal-shaped rigid frame. rigid frame has a span change effect [1]. As shown in Fig. The equation for the angular displacement of the column 1, the ridge point A moves down, the inclined beam insulator shown in the Fig. 2: pushes the column top outward, and the eaves nodes B and C move simultaneously. Even if there is no * e-mail: [email protected] © The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/). E3S Web of Conferences 165, 04069 (2020) https://doi.org/10.1051/e3sconf/202016504069 CAES 2020 Figure 2. Hinged pitched-roof framed column with sway = [ + ( + ) ] (1) [ ( ) ] = + − + ⁄ (2) 3 Calculation of the length factor Because the lateral shift angle − of the column ⁄ = , In the design of a rigid frame, the traditional method is to = , according to the boundary condition classify the various types of loads acting on the rigid = 0, above equation can be written as: ⁄ frame, determine the internal force according to the ⁄ = ( ) + (1 + ) (3) calculation method of structural mechanics, and combine Substituting equation (3) into equation (2), it can be to obtain the most unfavorable internal force of the obtained: − ⁄ ⁄ component. After the maximum bending moment and = ( )( ) (4) axial force value of the rigid frame column are The equilibrium equation o2f the column insulator is: determined, the in-plane stability of the rigid frame is + = 0, = − ⁄ − calculated according to the bending and bending = ( )( ) (5) members. In this design, the reduction of the bearing Combining with =−2 , Equation (5) can be capacity of the rigid frame by the effect is based rewritten− as: − 2⁄ − on the calculation of the length = to ( ) [ ( ) ] = 0 (6) approximately consider the adverse effects. − Among them, 0 The equation2 of angular2 displacement2 of the beam the horizontal side shift of the top of the insulator − is: ⁄ − − ⁄ − column-shaped portal frame is actually the sum of = (4 + 2 ) = 6 (7) generated by bending deformation under vertical load 1 = 0, + = 0, the buckling equation and generated by horizontal thrust under arch effect. 1 1 of the rigid frame is: − ℎ − ℎ According to the above buckling equation (9) of the ∑[( ) ( ) ] + 6 = gabled2 rigid frame, the precise value of can be 2 2 0 (8) calculate. 1 Substituting − ⁄ =− − into⁄ equation −(8), ℎ the “Steel Structure Design Standard” (GB50017 -2003) buckling equation can be rewritten as: Schedule D-2 gives the inside-plane simplified method ( )tan( ⁄) 6 = 0 (9) (approximate method)[4] of the pedestal-type portal frame. It has been proved in practice that the Through the above derivation, it can be seen that the ⁄ ⁄ − − ℎ1 approximate calculation method of flattening the inclined arch effect exists, which reduces the bending moment beam in the standard is unsafe[5]. acting on the inclined beam and effectively reduces the cross-sectional size of the beam. LinfengLu, QiangGu and QiXie[3] who are using a finite element method 4 Calculation results and analysis analyzed the of the portal frame, and concluded that the rectangular portal frame cannot be simply This paper selects the H-shaped equal-section gable replaced by a rectangular− portal frame. From the above portal frame as model. By changing the height and span process of calculating the elastic buckling equation, it of the gable portal frame, the slope of the inclined beam, can be seen that the effect of the effect and arch and the beam-column stiffness ratio, the axial force effect (axial force N) on the gable portal frame is obvious. generated by the arch effect is considered to stabilize the Given the section size of beams and columns,− we plug in gable portal frame as a whole. The calculated length C and S, And through repeated trial calculation of coefficient of the column under the arch effect in the formula (9), the μ value can be calculated to range from table is calculated by the finite element method. The 2 to ∞. whole is divided into multiple units of equal cross-section. Three groups of models with different high-span ratios are selected, and the calculated results are compared through looking up table of “Steel 2 E3S Web of Conferences 165, 04069 (2020) https://doi.org/10.1051/e3sconf/202016504069 CAES 2020 Structural Design Standard” (GB50017-2017). Select the first group of models: The portal frame has a span of 24 meters, a column height of 6 meters, and a beam column using H400 × 200 × 6 × 12, as shown in Figure. Change the slope of the inclined beam, the length of the inclined beam is , represents the calculated length coefficient of the rigid frame column considering the influence of the arch effect, represents the calculated length coefficient of the rigid frame column determined according to the standard[5]0. Figure 3. Calculation model Figure 2. Hinged pitched-roof framed column with sway Table 1. Effective length factors of hinged pitched-roof framed column [ ( ) ] - 0 = + + (1) 100% even load slope Lb i 0 0 = [ + ( + ) ] (2) − ⁄ 3 Calculation of the length factor 10° 12185 0.492 3.559 6.23 Because the lateral shift angle − of the column ⁄ = , In the design of a rigid frame, the traditional method is to 15° 12423 0.483 3.666 9.45 = , according to the boundary condition classify the various types of loads acting on the rigid 20° 12770 0.470 3.781 12.87 = 0, above equation can be written as: ⁄ frame, determine the internal force according to the 25° 13241 0.453 3.902 16.49 ⁄ = ( ) + (1 + ) (3) 6000N/m2 3.35 calculation method of structural mechanics, and combine 30° 13856 0.433 4.034 20.41 Substituting equation (3) into equation (2), it can be to obtain the most unfavorable internal force of the 35° 14649 0.410 4.179 24.76 obtained: − ⁄ ⁄ component. After the maximum bending moment and = ( )( ) (4) axial force value of the rigid frame column are 40° 15665 0.383 4.344 29.67 The equilibrium equation o2f the column insulator is: 45° 16971 0.356 4.534 35.35 determined, the in-plane stability of the rigid frame is + = 0, = − ⁄ − calculated according to the bending and bending = ( )( ) (5) members. In this design, the reduction of the bearing Select the second group of models: column is increased to 8m.