A method for calculating the ultimate strength of thin-walled beams under combined bending, shear and torsion loads

Abstract

A method for calculating the ultimate strength of thin-walled beams under combined bending, shear and torsion loads, according to the different loads and the positions of stiffeners and longitudinal members in the cross section, discretizes the thin-walled beam section into a series of independent non-interactive units, establishes a single-span calculation model of the thin-walled beam section, and inputs basic material properties; applies a bending moment and shear force to the thin-walled beam, and constructs a relationship curve between shear stress and shear strain during torsion; applies a torsion angle to the thin-walled beam, calculates the shear strain of each plate grid, calculates the shear stress of the plate grid according to the shear stress-strain relationship, calculates the torque by summing up the contribution of each plate grid to the torque, increases the torsion angle for iterative calculation, and takes the maximum value of the torque in the iterative process as the ultimate torque after considering the shear force. The present invention is based on the distribution law of stress under the ultimate transition state of thin-walled beams under the combined bending, shear and torsion loads in finite elements combined with the ultimate transition state failure equation of the plate grid under the combined stress, and the calculation is simple, accurate and efficient.

Description

Technical Field

[0001] The present invention relates to the field of thin-walled beam structure design. For beam sections with small double moments, a method for calculating the ultimate strength of thin-walled beams under combined bending, shear and torsion loads is proposed. The ultimate strength of thin-walled beams under different combined bending, shear and torsion loads can be quickly calculated, and the structure can be designed and optimized according to the calculation results. Background Art

[0002] At present, the ultimate strength research of thin-walled beams mainly focuses on single loads, and the research under combined loads is very scarce. However, in actual working conditions, thin-walled beams are easily subjected to multiple effects such as wind, waves and loads, and are not subjected to single loads. Therefore, it is very important to analyze its ultimate strength under combined bending, shear and torsion loads.

[0003] At present, the calculation method for the ultimate bearing capacity of thin-walled beams under the combined loads of bending, shear and torsion mainly adopts the finite element method, which requires a lot of time to establish a sophisticated model and nonlinear solution, is time-consuming and labor-intensive, and has high requirements for calculation personnel. Therefore, for beam sections with smaller double moments, the industry needs to seek a simple, accurate and efficient method to calculate the ultimate bearing capacity of thin-walled beams under the combined loads of bending, shear and torsion. Summary of the invention

[0004] In order to overcome the deficiencies of the prior art, the present invention provides a method for calculating the ultimate strength of a thin-walled beam under a combined bending, shear and torsion load that is simple in calculation, accurate and efficient.

[0005] The technical solution adopted by the present invention to solve its technical problem is:

[0006] A method for calculating the ultimate strength of a thin-walled beam under a combined bending, shear and torsion load comprises the following steps:

[0007] Step S1, according to different loads and the positions of stiffeners and longitudinal members in the cross section, the thin-walled beam cross section is discretized into a series of independent non-interacting units, a single-span calculation model of the thin-walled beam cross section is established, and basic material properties are input, wherein the basic material properties include the yield strength, elastic modulus and Poisson's ratio of the cross-section material;

[0008] Step S2, applying a bending moment and shear force to the thin-walled beam, calculating the section inertia moment, calculating the normal stress distribution of each unit in the section according to the plane section assumption, summing up the areas of all vertical units, calculating the shear stress of the vertical units, and calculating the shear stress of the horizontal units according to the limit state stress distribution law under pure shear action;

[0009] Step S3, judging the normal stress of each unit, if the normal stress of the unit in the compression zone exceeds the buckling stress or the stress of the unit in the tension zone exceeds the yield stress, the unit fails and does not participate in the calculation of the torque, and the remaining units participate in the torsion calculation;

[0010] Step S4, bringing the calculated normal stress and shear stress of each unit into the limit state equation of the plate panel to calculate the torsional limit shear stress after considering bending and shear force;

[0011] Step S5, using the calculated torsional limit shear stress, calculating the width-to-thickness ratio of each panel, and constructing a relationship curve between shear stress and shear strain during torsion;

[0012] Step S6, applying a torsion angle to the thin-walled beam, calculating the shear strain of each panel, calculating the shear stress of the panel according to the shear stress-strain relationship, summing up the contribution of each panel to the torque to calculate the torque, increasing the torsion angle for iterative calculation, and taking the maximum value of the torque during the iteration process as the limit torque after considering the shear force.

[0013] Further, in step S2, a bending moment M is applied to the thin-walled beam, and the normal stress distribution of each unit of the thin-walled beam under the action of M is calculated. The normal stress calculation is expressed as:

[0014]

[0015] Among them, σ x represents the normal stress of the unit, y represents the vertical distance from the unit to the neutral axis, and I is the moment of inertia of the cross section;

[0016] The shear stress calculation of each unit is obtained based on the shear flow distribution under the ultimate strength of the thin-walled beam. The vertical panel shear stress is:

[0017]

[0018] Among them, F is the applied shear force, and A is the total area of ​​the vertical components of the thin-walled beam section. In addition, due to the circulation effect of the shear flow, there is a certain shear stress in the horizontal plate grid of the thin-walled beam;

[0019] According to the shear flow distribution law of thin-walled beams under pure shear action and combined with the shear stress of the vertical plate grid, the shear stress of the horizontal plate grid is calculated.

[0020] Furthermore, in step S3, the yield stress of the unit is taken as the yield strength of the material, and the calculation formula of the buckling limit strength under the bending action of the unit is:

[0021]

[0022] Where E is the material elastic model, N / mm 2 , L is the calculated length of the stiffener, i.e. the length of one span, mm, A E is the cross-sectional area of ​​the stiffener, mm 2 , I E is the moment of inertia of the stiffener, mm 4 .

[0023] In step S4, the calculated normal stress σ of each unit is x and shear stress τ s Substitute it into the limit state equation of the plate panel and calculate the torsional limit shear stress τ after considering bending and shear force cr , the limit state equation of the plate panel under the combined stress is:

[0024]

[0025]

[0026] Among them, σ x ≥0 means the unit is under pressure, σ x <0 means the unit is in tension, and the β width-to-thickness ratio calculation formula is as follows:

[0027]

[0028] Where β is the width-to-thickness ratio of the panel; R eH-P is the material yield strength of the panel, N / mm 2 ; E is elastic modulus, N / mm 2 ; t is the thickness of the panel, mm; b is the length of the short side of the panel, mm;

[0029] The ultimate shear stress τ of the plate panel with four simply supported edges C , the calculation formula is as follows:

[0030]

[0031] Among them, τ C is the ultimate shear strength of the panel; C τ is the buckling reduction factor, which is:

[0032]

[0033] λ is the reference slenderness ratio of the plate panel, which is:

[0034]

[0035] K is the buckling factor, which is:

[0036]

[0037] σ E is the reference stress of the panel, N / mm 2 ;

[0038]

[0039] Where E is the elastic modulus of the material, N / mm2 ; t is the net thickness of the panel, mm; a and b are the length and width of the panel, mm;

[0040] σ cx The calculation formula for the ultimate normal stress of the plate is:

[0041] σ cx =C x R eH

[0042] Among them, R eH is the yield strength of the material, N / mm 2 ; C x is the reduction factor, and the calculation formula is

[0043]

[0044] ψ is the stress ratio, which is taken as 1, and λ is the reference slenderness ratio of the plate panel. c The calculation formula is:

[0045]

[0046] In step S5, according to the property of the plate width-to-thickness ratio, a curve relationship between shear stress and shear strain during the plate panel torsion process is constructed as follows:

[0047]

[0048] In the formula, γ E is the unit strain, γ y is the strain when the element reaches yield, which is determined by the following formula:

[0049]

[0050] Among them, R eH-P is the material yield strength of the panel, N / mm 2 , G is the shear modulus.

[0051] In the step S6, an initial torsion angle is given to the thin-walled beam, and the strain of each panel under the initial torsion angle is calculated according to the distance from the panel to the centroid of the cross section. The shear stress of each panel is obtained according to the relationship between shear stress and shear strain, and the shear force of each panel is obtained by multiplying it by the area of ​​the panel, and then the moment is obtained by multiplying it by the distance of the panel to the centroid of the cross section. The moments of all the panel are accumulated to obtain the torque of the thin-walled beam cross section constrained torsion under the initial torsion angle; the initial torsion angle is increased, and iterative calculation is performed, and the maximum value of the torque in the iterative process is taken as the limit torque after considering the shear force.

[0052] The beneficial effects of the present invention are mainly manifested in: the present invention is applied to calculate the ultimate bearing capacity of thin-walled beams under the combined bending, shear and torsion loads. Two actual container ships are verified, and the formula calculation values ​​and finite element calculation values ​​are analyzed and compared. It is found that the formula calculation values ​​are slightly different from the finite element simulation results, and the formula calculation has a higher accuracy.

[0053] The invention can calculate the ultimate bearing capacity of thin-walled beams under the combined bending, shearing and torsion loads relatively quickly and accurately. BRIEF DESCRIPTION OF THE DRAWINGS

[0054] Figure 1 It is a schematic diagram of unit division, where (a) represents the unit division under bending moment, and (b) represents the unit division under shear force and torque.

[0055] Figure 2 It is the shear flow distribution diagram of thin-walled beam under pure shear action, where (a) represents the hypothetical shear flow distribution diagram at the side wall, and (b) represents the hypothetical shear flow distribution diagram at the double bottom.

[0056] Figure 3 It is an iterative flow chart.

[0057] Figure 4 It is a flow chart of the method for calculating the ultimate strength of thin-walled beams under combined bending, shear and torsion loads. DETAILED DESCRIPTION

[0058] The present invention will be further described below in conjunction with the accompanying drawings.

[0059] Reference Figure 1 to Figure 4 , a method for calculating the ultimate strength of a thin-walled beam under a combined bending, shear and torsion load, comprising the following steps:

[0060] Step S1, according to different loads and the positions of stiffeners and longitudinal members in the cross section, the thin-walled beam cross section is discretized into a series of independent non-interacting units, a single-span calculation model of the thin-walled beam cross section is established, and basic material properties are input, wherein the basic material properties include the yield strength, elastic modulus and Poisson's ratio of the cross-section material;

[0061] Step S2, applying a bending moment and shear force to the thin-walled beam, calculating the section inertia moment, calculating the normal stress distribution of each unit in the section according to the plane section assumption, summing up the areas of all vertical units, calculating the shear stress of the vertical units, and calculating the shear stress of the horizontal units according to the limit state stress distribution law under pure shear action;

[0062] Step S3, judging the normal stress of each unit, if the normal stress of the unit in the compression zone exceeds the buckling stress or the stress of the unit in the tension zone exceeds the yield stress, the unit fails and does not participate in the calculation of the torque, and the remaining units participate in the torsion calculation;

[0063] Step S4, bringing the calculated normal stress and shear stress of each unit into the limit state equation of the plate panel to calculate the torsional limit shear stress after considering bending and shear force;

[0064] Step S5, using the calculated torsional limit shear stress, calculating the width-to-thickness ratio of each panel, and constructing a relationship curve between shear stress and shear strain during torsion;

[0065] Step S6, applying a torsion angle to the thin-walled beam, calculating the shear strain of each panel, calculating the shear stress of the panel according to the shear stress-strain relationship, summing up the contribution of each panel to the torque to calculate the torque, increasing the torsion angle for iterative calculation, and taking the maximum value of the torque during the iteration process as the limit torque after considering the shear force.

[0066] In step S1, the thin-walled beam section is discretized into a series of independent non-interactive units according to the different loads and the positions of the stiffeners, longitudinals and other components in the section. A single-span calculation model for the thin-walled beam section is established, and basic material properties such as the yield strength, elastic modulus, and Poisson's ratio of the section material are input. See the schematic diagram of unit division for details. Figure 1 .

[0067] In step S2, a bending moment and a shear force are applied to the thin-walled beam, the section inertia moment is calculated, and the normal stress distribution of each unit in the section is calculated based on the flat section assumption. The areas of all vertical units are summed up, the shear stress of the vertical unit is calculated, and the shear stress of the horizontal unit is calculated based on the limit state stress distribution law under pure shear action.

[0068] Apply a bending moment M to the thin-walled beam, and calculate the normal stress distribution of each unit of the thin-walled beam under the action of M. The normal stress calculation is expressed as:

[0069]

[0070] Among them, σ x represents the normal stress of the unit, y represents the vertical distance from the unit to the neutral axis, and I is the moment of inertia of the section.

[0071] The shear stress calculation of each unit can be obtained based on the shear flow distribution under the ultimate strength of the thin-walled beam. The vertical plate grid shear stress is:

[0072]

[0073] Among them, F is the applied shear force, and A is the total area of ​​the vertical components of the thin-walled beam section. In addition, due to the circulation effect of the shear flow, there is a certain amount of shear stress in the horizontal plate grid of the thin-walled beam. The distribution law is as follows: Figure 2 shown.

[0074] According to the shear flow distribution law of thin-walled beams under pure shear action and combined with the shear stress of the vertical plate grid, the shear stress of the horizontal plate grid is calculated.

[0075] In step S3, the normal stress of each unit is judged. If the normal stress of the unit in the compression zone exceeds the buckling stress and the stress of the unit in the tension zone exceeds the yield stress, the unit fails and does not participate in the calculation of the torque, and the remaining units participate in the torsion calculation. The yield stress of the unit is the yield strength of the material, and the calculation formula for the buckling limit strength under the bending action of the unit is:

[0076]

[0077] Where E is the material elastic model, N / mm 2 , L is the calculated length of the stiffener, i.e. the length of one span, mm, A E is the cross-sectional area of ​​the stiffener, mm 2 , I E is the moment of inertia of the stiffener, mm 4 .

[0078] In step S4, the calculated normal stress σ of each unit is x and shear stress τ s Substitute it into the limit state equation of the plate panel and calculate the torsional limit shear stress τ after considering bending and shear force cr , the limit state equation of the plate panel under the combined stress is:

[0079]

[0080]

[0081] Among them, σ x ≥0 means the unit is under pressure, σ x <0 means the unit is in tension, and the β width-to-thickness ratio calculation formula is as follows:

[0082]

[0083] Where β is the width-to-thickness ratio of the panel; R eH-P is the material yield strength of the panel, N / mm 2 ; E is elastic modulus, N / mm 2 ; t is the panel thickness, mm; b is the short side length of the panel, mm.

[0084] The ultimate shear stress τ of the plate panel with four simply supported edges C The "Rules for Classification of Sea-going Steel Ships" have detailed descriptions, and the specific calculation formula is as follows:

[0085]

[0086] Among them, τ C is the ultimate shear strength of the panel; C τis the buckling reduction factor, which is:

[0087]

[0088] λ is the reference slenderness ratio of the plate panel, which is:

[0089]

[0090] K is the buckling factor, which is:

[0091]

[0092] σ E is the reference stress of the panel, N / mm 2 ;

[0093]

[0094] Where E is the elastic modulus of the material, N / mm 2 ; t is the net thickness of the panel, mm; a and b are the length and width of the panel, mm;

[0095] σ cx The calculation formula for the ultimate normal stress of the plate is:

[0096] σ cx =C x R eH

[0097] Among them, R eH is the material yield strength, N / mm 2 ; C x is the reduction factor, and the calculation formula is

[0098]

[0099] ψ is the stress ratio, which is taken as 1, and λ is the reference slenderness ratio of the plate panel. c The calculation formula is:

[0100]

[0101] In step S5, the width-to-thickness ratio of each panel is calculated based on the calculated torsional limit shear stress, and a relationship curve between shear stress and shear strain during torsion is constructed;

[0102] According to the property of the plate width-to-thickness ratio, the relationship between shear stress and shear strain during plate grid torsion is constructed as follows:

[0103]

[0104] Among them, γ E is the unit strain, γ yis the strain when the element reaches yield, which is determined by the following formula:

[0105]

[0106] Among them, R eH-P is the material yield strength of the panel, N / mm 2 , G is the shear modulus.

[0107] In the step S6, an initial torsion angle is given to the thin-walled beam, and the strain of each panel under the initial torsion angle is calculated according to the distance from the panel to the centroid of the cross section. The shear stress of each panel is obtained according to the relationship between shear stress and shear strain, and the shear force of each panel is obtained by multiplying it by the area of ​​the panel, and then the moment is obtained by multiplying it by the distance of the panel to the centroid of the cross section. The moments of all the panel are accumulated to obtain the torque of the thin-walled beam cross section constrained torsion under the initial torsion angle; the initial torsion angle is increased, and iterative calculation is performed, and the maximum value of the torque in the iterative process is taken as the limit torque after considering the shear force.

[0108] The shear force of the panel under the initial torsion angle is known, and the moment of the panel on the cross-section centroid is calculated. The moments of all the panel are accumulated to obtain the torque of the thin-walled beam section to constrain torsion under the initial torsion angle. The initial torsion angle is increased and iterative calculation is performed. The specific iterative flow chart is as follows: Figure 3 shown.

[0109] To verify the accuracy of the proposed method, two container ships and an offshore platform were calculated and compared with the finite element method.

[0110] Table 1 is a comparison of the ultimate strength results of the bending shear-torsion combination of container ships (shear 10 8 N, torque and bending moment 10 13 N·mm);

[0111]

[0112]

[0113] Table 1

[0114] Table 2 is a comparison of the ultimate strength results of the bending shear-torsion combination of container ships (shear 10 8 N, torque and bending moment 10 13 N·mm);

[0115]

[0116] Table 2

[0117] Table 3 is a comparison of the ultimate strength results of the two-bend shear-torsion combination of container ships (shear 10 8 N, torque and bending moment 10 13 N·mm);

[0118]

[0119] Table 3

[0120] Table 4 is a comparison of the ultimate strength results of the two-bend shear-torsion combination of container ships (shear 10 8 N, torque and bending moment 10 13 N·mm);

[0121]

[0122]

[0123] Table 4

[0124] The contents described in the embodiments of this specification are merely enumerations of implementation forms of the inventive concept and are for illustrative purposes only. The protection scope of the present invention should not be considered to be limited to the specific forms described in this embodiment, and the protection scope of the present invention also extends to equivalent technical means that can be thought of by ordinary technicians in this field based on the inventive concept.

Claims

1. A method for calculating the ultimate strength of thin-walled beams under combined bending, shear and torsion loads, characterized in that: The method comprises the following steps: Step S1, according to different loads and the positions of stiffeners and longitudinal members in the cross section, the thin-walled beam cross section is discretized into a series of independent non-interacting units, a single-span calculation model of the thin-walled beam cross section is established, and basic material properties are input, wherein the basic material properties include the yield strength, elastic modulus and Poisson's ratio of the cross-section material; Step S2, applying a bending moment and shear force to the thin-walled beam, calculating the section inertia moment, calculating the normal stress distribution of each unit in the section according to the plane section assumption, summing up the areas of all vertical units, calculating the shear stress of the vertical units, and calculating the shear stress of the horizontal units according to the limit state stress distribution law under pure shear action; Step S3, judging the normal stress of each unit, if the normal stress of the unit in the compression zone exceeds the buckling stress or the stress of the unit in the tension zone exceeds the yield stress, the unit fails and does not participate in the calculation of the torque, and the remaining units participate in the torsion calculation; Step S4, bringing the calculated normal stress and shear stress of each unit into the limit state equation of the plate panel to calculate the torsional limit shear stress after considering bending and shear force; Step S5, using the calculated torsional limit shear stress, calculating the width-to-thickness ratio of each panel, and constructing a relationship curve between shear stress and shear strain during torsion; Step S6, applying a torsion angle to the thin-walled beam, calculating the shear strain of each panel, calculating the shear stress of the panel according to the shear stress-strain relationship, and calculating the torque by summing up the contribution of each panel to the torque; increasing the torsion angle for iterative calculation, and taking the maximum value of the torque in the iterative process as the limit torque after considering the shear force; In step S2, a bending moment M is applied to the thin-walled beam, and the normal stress distribution of each unit of the thin-walled beam under the action of M is calculated. The normal stress calculation is expressed as: Among them, σ x represents the normal stress of the unit, y represents the vertical distance from the unit to the neutral axis, and I is the moment of inertia of the cross section; The shear stress calculation of each unit is obtained based on the shear flow distribution under the ultimate strength of the thin-walled beam. The vertical panel shear stress is: Among them, F is the applied shear force, A is the total area of ​​the vertical components of the thin-walled beam section, and due to the circulation effect of the shear flow, there is a certain shear stress in the horizontal plate grid of the thin-walled beam; According to the shear flow distribution law of thin-walled beam under pure shear action and combined with the shear stress of vertical panel, the shear stress of horizontal panel is calculated. In step S3, the yield stress of the unit is the yield strength of the material, and the calculation formula for the buckling limit strength under the bending action of the unit is: Where E is the material elastic model, N / mm 2 , L is the calculated length of the stiffener, i.e. the length of one span, mm, A E is the cross-sectional area of ​​the stiffener, mm 2 , I E is the moment of inertia of the stiffener, mm 4 ; In step S4, the calculated normal stress σ of each unit is x and shear stress τ s Substitute it into the limit state equation of the plate panel and calculate the torsional limit shear stress τ after considering bending and shear force cr , the limit state equation of the plate panel under the combined stress is: Among them, σ x ≥0 means the unit is under pressure, σ x <0 means the unit is in tension, and the β width-to-thickness ratio calculation formula is as follows: Where β is the width-to-thickness ratio of the panel; R eH-P is the material yield strength of the panel, N / mm 2 ; E is elastic modulus, N / mm 2 ; t is the thickness of the panel, mm; b is the length of the short side of the panel, mm; The ultimate shear stress τ of the plate panel with four simply supported edges C , the calculation formula is as follows: In the formula, τ C is the ultimate shear strength of the panel; C τ is the buckling reduction factor, which is: λ is the reference slenderness ratio of the plate panel, which is: K is the buckling factor, which is: σ E is the reference stress of the panel, N / mm 2 ; Where E is the elastic modulus of the material, N / mm 2 ; t is the net thickness of the panel, mm; a and b are the length and width of the panel, mm; σ cx The calculation formula for the ultimate normal stress of the plate is: s cx =C x R eH Among them, R eH is the material yield strength, N / mm 2 ; C x is the reduction factor, and the calculation formula is ψ is the stress ratio, which is taken as 1, and λ is the reference slenderness ratio of the plate panel. c The calculation formula is: In step S5, according to the property of the plate width-to-thickness ratio, a curve relationship between shear stress and shear strain during the plate panel torsion process is constructed as follows: Among them, γ E is the unit strain, γ y is the strain at which the element reaches yield, determined by the following formula: In the formula, R eH-P is the material yield strength of the panel, N / mm 2 , G is the shear modulus; In the step S6, an initial torsion angle is given to the thin-walled beam, and the strain of each panel under the initial torsion angle is calculated according to the distance from the panel to the centroid of the cross section. The shear stress of each panel is obtained according to the relationship between shear stress and shear strain, and the shear force of each panel is obtained by multiplying it by the area of ​​the panel, and then the moment is obtained by multiplying it by the distance of the panel to the centroid of the cross section. The moments of all the panel are accumulated to obtain the torque of the thin-walled beam cross section constrained torsion under the initial torsion angle; the initial torsion angle is increased, and iterative calculation is performed, and the maximum value of the torque in the iterative process is taken as the limit torque after considering the shear force.

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