Method for determining critical load of instability of locally reinforced U-shaped steel sheet piles

Abstract

The invention discloses a method for determining the critical load of instability of a partially reinforced U-shaped steel sheet pile. Firstly, the corresponding critical load values ​​under two limit states, that is, the original U-shaped steel sheet pile without a reinforcement plate and the U-shaped steel sheet pile axial compression component with a whole reinforcement plate fully arranged longitudinally along its length direction, are calculated by a standardized stability calculation method. Then, the reduction effect of partial reinforcement on the stiffness of the fully reinforced U-shaped steel sheet pile axial compression component is simplified into a linear function. Finally, the reinforcement effect of the reinforcement plate is converted into a position function, and the conversion coefficient, that is, the dynamic coordination coefficient, is obtained, and then the stable critical load value of the U-shaped steel sheet pile after partial reinforcement is obtained. The invention effectively and quickly obtains the stable critical load value of the U-shaped steel sheet pile after partial reinforcement, and can be used for optimizing the design of the longitudinal reinforcement arrangement of the U-shaped steel sheet pile.

Description

Technical Field

[0001] The present invention relates to calculation of critical load of steel sheet pile instability, and in particular to a method for determining critical load of steel sheet pile instability with local reinforcement Background Art

[0002] For open uniaxially symmetrical steel members, if subjected to axial loads, instability is likely to occur. U-shaped steel sheet piles can be divided into hot-rolled U-shaped steel sheet piles and cold-bent thin-walled U-shaped steel sheet piles according to their forming process. Among them, the buckling modes of the axial members of cold-bent thin-walled U-shaped steel sheet piles are local buckling, distortional buckling and overall buckling. For hot-rolled U-shaped steel sheet piles, overall buckling mainly occurs due to their process limitations. When the slenderness of the member is relatively small and the width and thickness of the plate that constitutes the member are relatively large, the member is prone to local buckling; when the slenderness of the member is relatively large, overall buckling may occur.

[0003] At present, most studies focus on the stability performance of cold-formed thin-walled U-shaped steel sheet piles, explore the unique distortion buckling in thin-walled components, and propose a web-reinforced U-shaped steel sheet pile column for the distortion buckling of cold-formed thin-walled rolled-edge U-shaped steel sheet pile columns, that is, a stiffening plate is longitudinally arranged at the opening of the U-shaped steel sheet pile axial compression component. However, there is currently no calculation method for the critical load of stability of U-shaped steel sheet pile axial compression components. Summary of the invention

[0004] Purpose of the invention: The purpose of the present invention is to provide a method for determining the critical load of instability of locally reinforced U-shaped steel sheet piles, so as to solve the problem that there is currently no method for calculating the critical load of instability of locally reinforced U-shaped steel sheet piles.

[0005] Technical solution: The method for determining the critical load of instability of a locally reinforced U-shaped steel sheet pile according to the present invention is characterized in that it comprises the following steps:

[0006] (1) Calculation of the critical load N of the axial compression member without U-shaped steel sheet piles 1max ;

[0007] (2) Calculate the critical load N of the U-shaped steel sheet pile axial compression member with full longitudinal stiffening plates 2max ;

[0008] (3) Based on the spatial coordination method, the critical load N of the locally reinforced U-shaped steel sheet pile axial compression member is obtained. max , N max The calculation formula is as follows:

[0009] N max =δ(N 2max -N 1max )+N 1max

[0010] =(1-δ)·N 1max+δ·N 2max

[0011] Wherein, δ is the conversion factor.

[0012] Wherein, in step (1), N 1max The calculation formula is as follows:

[0013]

[0014] In the above formula, A is the total cross-sectional area of ​​the component; f is the design value of the compressive strength of the steel, and the stability coefficient The solution is as follows:

[0015] when hour:

[0016]

[0017] when hour:

[0018]

[0019] In the above formula To calculate the relative slenderness ratio, λ is the slenderness ratio, and its calculation formula is Where i is the radius of gyration, I is the moment of inertia of the component, μ is the calculated length coefficient, l is the length of the component; f y is the yield strength of steel; E is the elastic modulus of steel; coefficients α1, α2, α3 are 0.65, 0.965 and 0.3 respectively.

[0020] In step (2), N 2max The calculation formula is as follows:

[0021]

[0022] In the above formula, A=S1+S2, S1 is the cross-sectional area of ​​the original U-shaped steel sheet pile, S2 is the cross-sectional area of ​​the stiffening plate, f is the design value of the compressive strength of the steel, and the stability coefficient is The solution is as follows:

[0023] when

[0024] when

[0025] In the formula To calculate the relative slenderness ratio, The formula for calculating λ is in, I y =I y1 +S1·h12 +I y2 +S2·h2 2 , I y1 is the inertia moment of the original U-shaped steel sheet pile component, I y2 is the moment of inertia of the stiffener.

[0026] The calculation process of the conversion coefficient δ in step (3) is as follows:

[0027] When a U-shaped steel sheet pile member with a length of l is fully reinforced, the entire stiffening plate is regarded as n stiffening plates with a plate width of b0 (n·b0=l). The stiffening effect of each stiffening plate only affects the area between the two adjacent plate edges. Therefore, the three stiffening plates are taken as one unit for the calculation of the conversion coefficient. The calculated length is the distance between the edges of the two outer stiffening plates.

[0028] The center axis of the length direction of the U-shaped steel sheet pile axial compression member is taken as the y-axis of the position function, and the length direction of the U-shaped steel sheet pile axial compression member is taken as the x-axis to establish a coordinate system. The horizontal coordinates of the ends of the U-shaped steel sheet pile axial compression member and the axes of several stiffening plates are set from left to right as l 01 , l 02 …l 0i-1 , l 0i , l 0i+1 …l 0n , the horizontal coordinate of the axis of the stiffening plate to be tested is l 0i , (1≤i≤n), then the position function corresponding to the stiffening plate to be tested is:

[0029]

[0030] The position function of all stiffening plates is a piecewise function, and each segment point can be obtained by combining two straight line equations;

[0031] The conversion factor is calculated as follows:

[0032] The conversion factor for arranging a stiffening plate is:

[0033]

[0034] Conversion factor for arrangement of n stiffeners and n ≥ 2:

[0035]

[0036] Among them, S 阴影 is the area of ​​the graph formed by the position function and the x-axis within the width b0 of the stiffening plate, S0 is the total area of ​​the graph formed by the position function and the x-axis when one stiffening plate is arranged, ΣS 阴影,i is the sum of the shadow areas of the stiffening plates when n and n ≥ 2 stiffening plates are arranged, maxΣS 阴影,iIt is the maximum value of the sum of the shadow areas corresponding to each stiffening plate when n stiffening plates are arranged and n≥2.

[0037] Beneficial effect: The present invention simplifies the reduction effect of partial reinforcement on the stiffness of the fully reinforced U-shaped steel sheet pile axial compression component into a linear function, and then converts the reinforcement effect of the stiffening plate into a position function to obtain the conversion coefficient, thereby effectively and quickly obtaining the reduced critical load, that is, the stable critical load value of the U-shaped steel sheet pile after partial reinforcement; it can be used to optimize the design of the longitudinal reinforcement arrangement of the U-shaped steel sheet pile. BRIEF DESCRIPTION OF THE DRAWINGS

[0038] Figure 1 Schematic diagram of two extreme working conditions of U-shaped steel sheet pile reinforcement in an embodiment of the present invention;

[0039] Figure 2 This is a schematic diagram of the partial reinforcement of the U-shaped steel sheet pile;

[0040] Figure 3 It is a schematic diagram of the position function of U-shaped steel sheet piles when arranging a stiffening plate;

[0041] Figure 4 Schematic diagram of the position function of U-shaped steel sheet piles when arranging n (n≥2) stiffening plates;

[0042] Figure 5 S is a reinforcement condition 阴影 Schematic diagram;

[0043] Figure 6 This is a schematic diagram of S0 under a stiffening condition;

[0044] Figure 7 ΣS under n (n≥2) stiffening conditions 阴影 Schematic diagram;

[0045] Figure 8 ΣS for the case where two reinforcements are arranged at equal intervals 阴影 Schematic diagram;

[0046] Fig. 9 ∑S is the condition of n (n≥2) equally spaced reinforcements 阴影 Schematic diagram;

[0047] Fig.10 This is the relationship between the number of stiffeners and the critical load value. DETAILED DESCRIPTION

[0048] The present invention will be further described below in conjunction with the accompanying drawings.

[0049] The calculation method of the axial compression bearing capacity of non-standard reinforced U-shaped steel sheet piles based on the spatial coordination method is to determine the critical load value of the U-shaped steel sheet pile axial compression member. First, the calculation formula for the overall stability of the axial compression member is obtained as follows:

[0050] Right now

[0051] Where A is the total cross-sectional area of ​​the component; f is the design value of the compressive strength of the steel (i.e., the yield strength divided by the partial coefficient); the stability coefficient The solution is as follows:

[0052] when hour:

[0053]

[0054] when hour:

[0055]

[0056] In the formula, To calculate the relative slenderness ratio, λ is the slenderness ratio, and its calculation formula is Where i is the radius of gyration, I is the moment of inertia of the component, μ is the calculated length coefficient, l is the length of the component; f y is the yield strength of steel; E is the elastic modulus of steel; coefficients α1, α2, α3 are 0.65, 0.965 and 0.3 respectively.

[0057] Then, there are two extreme conditions when the U-shaped steel sheet pile axial compression member is longitudinally reinforced, such as Figure 1 As shown in the figure, one is that no stiffening plate is arranged, that is, the original U-shaped steel sheet pile, and the other is that the stiffening plate is arranged longitudinally along the length direction of the U-shaped steel sheet pile axial compression member. The above two limit states can be solved by the standard stability calculation method to obtain their corresponding critical loads.

[0058] The critical load N of the U-shaped steel sheet pile axial compression member is 1max The calculation process is as follows:

[0059] A is the cross-sectional area of ​​the component, which is equal to the cross-sectional area S1 of the U-shaped steel sheet pile. y is the moment of inertia of the component, which is equal to the moment of inertia I of the original U-shaped steel sheet pile component y1 ;

[0060]

[0061]

[0062]

[0063] when

[0064] when

[0065]

[0066] when

[0067] when

[0068] The critical load N of the U-shaped steel sheet pile axial compression member in working condition 2 is 2max The calculation process is as follows:

[0069] A is equal to the sum of the cross-sectional area S1 of the original U-shaped steel sheet pile and the cross-sectional area S2 of the stiffening plate. y is the moment of inertia of the fully stiffened member, I y1 is the moment of inertia of the original U-shaped steel sheet pile, I y2 is the moment of inertia of the stiffened plate, h1 is the distance between the centroid of the original U-shaped steel sheet pile and the centroid of the stiffened member, and h2 is the distance between the centroid of the stiffened plate and the centroid of the stiffened member;

[0070] A=S1+S2

[0071] I y =I y1 +S1·h1 2 +I y2 +S2·h2 2

[0072]

[0073]

[0074] when

[0075] when

[0076]

[0077] when

[0078] when

[0079] However, in actual projects, stiffening plates with different spacing and thickness are often used for reinforcement, such as Figure 2 As shown, the U-shaped steel sheet pile axial compression member in the present invention is partially stiffened.

[0080] In the present invention, the stiffening plates have the same shape and size, and are symmetrically arranged along the axis of the U-shaped steel sheet pile axial compression component. The position of the single plate arrangement has a position function, and the stiffness conversion coefficient δ is related to the stiffening plate width b0 and the spacing l0 between the stiffening plates. The reduction effect of partial stiffening on the stiffness of the fully stiffened U-shaped steel sheet pile axial compression component can be simplified to a linear function. The stiffening effect range of a single stiffening plate is only between the edge lines of two adjacent plates, and the stiffening effect of each stiffening plate is superimposed to form the total stiffening effect.

[0081] Taking the arrangement of a plate in the center as an example, the position function at this time is expressed as Figure 3 As shown in Figure 2, the position function has a maximum reduction value a at the midline of the reinforcement range and at the center of the length direction of the member.

[0082] Through working conditions 1 and 2, the critical load N of the unstiffened U-shaped steel sheet pile axial compression member can be obtained respectively: 1max and the critical load N under full stiffening 2max Based on the idea and method of spatial coordination method and considering the conversion factor δ, the critical load N of the U-shaped steel sheet pile axial compression member under partial stiffening can be obtained. max The calculation formula is as follows:

[0083] N max =δ(N 2max -N 1max )+N 1max

[0084] =(1-δ)·N 1max +δ·N 2max

[0085] The process of solving the conversion coefficient in the present invention is:

[0086] When a U-shaped steel sheet pile member with a length of l is fully reinforced, the entire stiffening plate can be regarded as n stiffening plates with a plate width of b0 (n·b0=l). The stiffening effect of each stiffening plate only affects the area between the edges of two adjacent plates. Therefore, the conversion coefficient can be calculated with three stiffening plates as one unit, and the calculated length is the distance between the edges of the two outer stiffening plates. The center axis of the length direction of the U-shaped steel sheet pile axial compression member is used as the y-axis of the position function, and the length direction of the U-shaped steel sheet pile axial compression member is used as the x-axis to establish a coordinate system, such as Figure 4 As shown; the horizontal coordinates of the ends of the U-shaped steel sheet pile axial compression member and the axes of several stiffening plates are l from left to right. 01 , l 02 …l 0i-1 , l 0i , l 0i+1 …l 0n , the horizontal coordinate of the axis of the stiffening plate to be tested (the i-th stiffening plate) is l 0i, (1≤i≤n), then the position function corresponding to the stiffening plate to be tested is:

[0087]

[0088] The position function of all stiffening plates is a piecewise function, and each segment point can be obtained by combining two straight line equations.

[0089] The conversion factor is calculated as follows:

[0090] (1) Conversion factor for arranging one stiffening plate:

[0091]

[0092] (2) Conversion factor for arranging n (n ≥ 2) stiffeners:

[0093]

[0094] S 阴影 is the shaded area corresponding to a single stiffening plate on the partially stiffened U-shaped steel sheet pile, that is, the area of ​​the graph enclosed by the position function and the x-axis within the width b0 of the stiffening plate, and S0 is the total area of ​​the graph enclosed by the position function and the x-axis when one stiffening plate is arranged; ∑S 阴影,i is the sum of the shadow areas of the stiffening plates when n (n≥2) stiffening plates are arranged; max∑S 阴影,i It is the maximum value of the sum of the shadow areas corresponding to each stiffening plate when n (n≥2) stiffening plates are arranged.

[0095] The method of the present invention is used to calculate the actual project, as follows:

[0096] The stability of the U-shaped steel sheet pile axial compression member with width h and length l is calculated by symmetrically arranging n stiffening plates with the same plate width b0.

[0097] The conversion coefficients of multiple working conditions were calculated using the method of this embodiment to verify the accuracy. The specific illustrations and calculation results are as follows.

[0098] (1) Calculation of conversion factor under one stiffening condition, such as Figure 5 Figure 6 As shown;

[0099] By calculation:

[0100]

[0101]

[0102]

[0103] Therefore

[0104] (2) Calculation of conversion coefficients for two symmetrical stiffeners, such as Figure 8 As shown;

[0105] By calculation:

[0106]

[0107]

[0108]

[0109]

[0110] Therefore

[0111] (3) Calculation of conversion factor for n (n ≥ 2) symmetrical stiffeners, such as Fig. 9 Shown

[0112]

[0113]

[0114]

[0115]

[0116] Therefore

[0117] The specific working condition is: for a 30m-class U-shaped steel sheet pile partially reinforced component, when the cumulative length of the arranged stiffening plates (∑b) reaches 7m, the critical load value Nmax of the partially reinforced U-shaped steel sheet pile component is calculated according to the above formula, and the relationship between the number of stiffening plates and the critical load value is plotted as shown in the figure below: Fig.10 shown.

[0118] like Fig.10 As shown in the figure, under this condition, when two stiffening plates are arranged at equal intervals, the critical load value of the component reaches a maximum value, and the stiffening effect of the U-shaped steel sheet pile is the best.

[0119] Substituting the conversion coefficient calculated under the above working conditions into the calculation formula of the critical load can obtain the critical load of the U-shaped steel sheet pile under the above partial reinforcement working condition, and the calculated critical load value is close to the critical load value of the actual U-shaped steel sheet pile after partial reinforcement, indicating that the method of this embodiment has a certain accuracy.

Claims

1. A method for determining the critical load of instability of a locally reinforced U-shaped steel sheet pile, characterized in that: The following steps are involved: (1) Calculation of the critical load N of the axial compression member without U-shaped steel sheet piles 1max ; In step (1), N 1max The calculation formula is as follows: In the above formula, A is the total cross-sectional area of ​​the component; f is the design value of the compressive strength of the steel, and the stability coefficient The solution is as follows: when hour: when hour: In the above formula To calculate the relative slenderness ratio, λ is the slenderness ratio, and its calculation formula is Where i is the radius of gyration, I is the moment of inertia of the component, μ is the calculated length coefficient, l is the length of the component; f y is the yield strength of steel; E is the elastic modulus of steel; coefficients α1, α2, and α3 are 0.65, 0.965, and 0.3, respectively; (2) Calculate the critical load N of the U-shaped steel sheet pile axial compression member with full longitudinal stiffening plates 2max ; In step (2), N 2max The calculation formula is as follows: In the above formula, A=S1+S2, S1 is the cross-sectional area of ​​the original U-shaped steel sheet pile, S2 is the cross-sectional area of ​​the stiffening plate, f is the design value of the compressive strength of the steel, and the stability coefficient is The solution is as follows: when when In the formula To calculate the relative slenderness ratio, The formula for calculating λ is: in, I y =I y1 +S1·h1 2 +I y2 +S2·h2 2 , I y1 is the inertia moment of the original U-shaped steel sheet pile component, I y2 is the moment of inertia of the stiffening plate, h1 is the distance between the centroid of the original U-shaped steel sheet pile and the centroid of the stiffened component, and h2 is the distance between the centroid of the stiffening plate and the centroid of the stiffened component; (3) Based on the spatial coordination method, the critical load N of the locally reinforced U-shaped steel sheet pile axial compression member is obtained. max , N max The calculation formula is as follows: N max =δ(N 2max -N 1max )+N 1max =(1-δ)·N 1max +δ·N 2max Where δ is the conversion factor; The calculation process of the conversion coefficient δ in step (3) is as follows: When a U-shaped steel sheet pile member with a length of l is fully reinforced, the entire stiffening plate is regarded as n stiffening plates with a plate width of b0, n·b0=l. The stiffening effect of each stiffening plate only affects the area between the two adjacent plate edges. Therefore, the three stiffening plates are taken as one unit for the calculation of the conversion coefficient. The calculated length is the distance between the edges of the two outer stiffening plates. The center axis of the length direction of the U-shaped steel sheet pile axial compression member is taken as the y-axis of the position function, and the length direction of the U-shaped steel sheet pile axial compression member is taken as the x-axis to establish a coordinate system. The horizontal coordinates of the axes of several stiffening plates of the U-shaped steel sheet pile axial compression member are set to be l from left to right. 01 , l 02 …l 0i-1 , l 0i , l 0i+1 …l 0n , the horizontal coordinate of the axis of the stiffening plate to be tested is l 0i , 1≤i≤n, then the position function corresponding to the stiffening plate to be tested is: The position function of all stiffening plates is a piecewise function, and each segment point is obtained by combining two straight line equations; The conversion factor is calculated as follows: The conversion factor for arranging a stiffening plate is: Conversion factor for arrangement of n stiffeners and n ≥ 2: Among them, S 阴影 is the area of ​​the graph formed by the position function and the x-axis within the width b0 of the stiffening plate, S0 is the total area of ​​the graph formed by the position function and the x-axis when one stiffening plate is arranged, ΣS 阴影,i is the sum of the shadow areas of the stiffening plates when n and n ≥ 2 stiffening plates are arranged, maxΣS 阴影,i The shaded area and maximum value of each stiffening plate when n stiffening plates are arranged and n ≥ 2.

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