Equivalent calculation model of corrugated web i-beam

Abstract

An equivalent calculation model of corrugated web I-beam, the method for constructing the equivalent calculation model is as follows: assuming the prerequisite of equivalent relationship: same deformation or deflection, and the strain energy of same deformation is equal; according to the shape and size of the corrugated web, the parameters of the cross-sectional moment of inertia and the polar moment of inertia of the corrugated web I-beam are calculated under the premise of meeting the prerequisite, and then the equivalent cross-sectional parameters of the equivalent flat web I-beam are obtained; the equivalent cross-sectional parameters can be directly used to calculate and analyze the corrugated web by using the corresponding formula of the flat web I-beam. The corrugated web I-beam can be calculated by substituting the cross-sectional characteristics calculated according to the method into the corresponding formula of the flat web I-beam, which is simple and convenient, and easy for engineering designers to accept and master.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of steel structure, and particularly relates to an equivalent calculation model of a corrugated steel web I-beam. BACKGROUND

[0002] The corrugated steel web can significantly enhance the local deformation capacity generated by the flat web, thereby adopting a thinner steel plate, and compared with the traditional H-shaped steel, has the advantages of saving steel, energy saving, and environmental protection, so the application range is wide, and the corrugated steel web I-beam is mainly suitable for portal steel frame main beams, high-rise steel structure main and secondary beams, 30-ton crane beams, municipal bridges and truss car main beams, etc. The calculation of the corrugated steel web I-beam generally adopts the following four calculation methods: 1, a fitting formula of the finite element calculation result or the scale test result; 2, direct calculation according to the size of the wave crest; 3, approximate calculation according to the ordinary I-beam; and 4, calculation according to the average value of a long wave band of one period. However, the above existing methods have the following deficiencies: 1, the physical meaning is not clear; and 2, the calculation result of the cross-section characteristics is large, and the structure is unsafe. Therefore, a more effective and accurate method for calculating the corrugated steel web I-beam is needed to ensure the safety of the project. SUMMARY

[0003] The present application provides an equivalent calculation model of a corrugated steel web I-beam, and the equivalent flat steel web I-beam model of the corrugated steel web I-beam is obtained through the principle of the same deformation and the same energy. Based on the mature calculation theory and method of the flat steel web I-beam, the corrugated steel web I-beam can be calculated by substituting the cross-section characteristics calculated according to the method into the corresponding formula of the flat steel web I-beam, and the method is simple and convenient, and easy for engineering designers to accept and master.

[0004] The technical solution for solving the above problems is as follows: an equivalent calculation model of a corrugated steel web I-beam, and the method for constructing the equivalent calculation model is as follows: the premise condition of the equivalent relationship is that the deformation or deflection is the same, and the strain energy of the same deformation is equal; according to the shape and size of the corrugated steel web, the parameters of the cross-section moment of inertia and the polar moment of inertia of the corrugated steel web I-beam are calculated under the premise condition, and then the equivalent cross-section parameters of the equivalent flat steel web I-beam are obtained; the equivalent cross-section parameters can be directly used to calculate and analyze the corrugated steel web by using the corresponding formula of the flat steel web I-beam.

[0005] The further technical solution is that the equivalent cross-section parameters of the equivalent flat steel web I-beam include the equivalent cross-section moment of inertia I y,eff and the equivalent polar moment of inertia I ω,eff .

[0006] The equivalent calculation formula of the equivalent cross-section moment of inertia I y,eff is as follows:

[0007]

[0008] Where: F is the distance from the straight section of corrugated web to the center of the beam; D is the half wavelength of the corrugated web; A f is the flange area of the straight section of corrugated web; A w is the web section area of the straight section of corrugated web; A' w is the web section area of the inclined section of corrugated web; A' is the total web section area of the inclined section of corrugated web; s is the length of the straight section; b is the width of the flange section; d is the horizontal width of the inclined section; t is the thickness of the flange; θ represents the angle between the inclined section and the center line of the beam;

[0009] The equivalent sectorial moment of inertia I ω,eff is calculated as follows:

[0010]

[0011] Where: F is the distance from the straight section of corrugated web to the center of the beam; I f is the moment of inertia of the flange about the x-axis; I w is the moment of inertia of the web of the straight section about the x-axis; I' w is the moment of inertia of the web of the inclined section about the x-axis; I x is the moment of inertia of the section of the straight section about the x-axis; I' x is the moment of inertia of the section of the inclined section about the x-axis; s is the length of the straight section; d is the horizontal width of the inclined section; D is the half wavelength of the corrugated web; θ represents the angle of the inclined section; I ω0 is the sectorial moment of inertia of the section of the flat steel web beam with the same thickness as the corrugated web.

[0012] A further technical solution is a method for calculating the critical load of a simply supported corrugated web I-beam using equivalent section parameters: I y,eff and I ω,eff From the equivalent calculation formula, it can be seen that the equivalent moment of inertia is determined by the section parameters and the size of the corrugated web, therefore, after calculating I y,eff and I ω,eff , the critical load of the corrugated web I-beam can be calculated according to the equivalent flat steel web I-beam model;

[0013] The calculation formula for the critical load M cr of a symmetric section flat steel web I-beam is:

[0014]

[0015] Where: β1 is the correction coefficient of the critical bending moment, which depends on the load acting on the component and the constraint form; β2 is the load point position influence coefficient; a1 is the vertical distance from the load point to the shear center; μ y is the bending calculation length coefficient about the y-axis; μω For torsional calculation, the length factor is denoted by l; l represents the beam length; E is the elastic modulus of steel; G is the shear modulus of steel; I is the length factor. k π is the moment of free torsional inertia; π is pi.

[0016] Will I y,eff with I ω,eff Substituting into the above formula, the critical load of a corrugated steel web I-beam can be easily calculated using the critical load formula for a straight steel web I-beam.

[0017] Due to the adoption of the above technical solution, the equivalent calculation model of the corrugated steel web I-beam of the present invention has the following characteristics and beneficial effects:

[0018] The equivalent calculation model for corrugated steel web I-beams of this invention solves the problem of overly large cross-sectional properties by using this equivalent calculation model method. Furthermore, the physical meaning of the cross-sectional properties calculated by this equivalent calculation model method is clear. At the same time, the cross-sectional properties calculated by this method can be directly applied to the calculation theory and method of straight steel web I-beams to solve the calculation problem of corrugated steel web I-beams. The method is simple and convenient, and easy for engineering designers to accept and master.

[0019] The technical features of the equivalent calculation model of the corrugated steel web I-beam of the present invention will be further described below with reference to the accompanying drawings and embodiments. Attached Figure Description

[0020] Figure 1 This is a schematic diagram of the corrugated steel web of the embodiment on the f(z)-z coordinate axis;

[0021] Figure 2 This is a cross-sectional schematic diagram (on the xy coordinate axis) of the corrugated steel web I-beam structure of the embodiment;

[0022] Figure 3 This is a schematic diagram of a simply supported beam (on the three-dimensional xyz coordinate axis) of a corrugated steel web I-beam according to an embodiment;

[0023] Figure 4 This is a schematic diagram of the strain of the inclined plate segment of the web of the corrugated steel web I-beam in the embodiment;

[0024] Figure 5 This is a cross-sectional static moment distribution diagram of the corrugated steel web I-beam structure in the embodiment.

[0025] Figure 6 This is a calculation and analysis diagram of the sector constant of the corrugated steel web I-beam in the embodiment (on the xy coordinate axis).

[0026] Figure 1 middle:

[0027] F is the distance from the straight plate segment of the corrugated steel web to the center of the beam; D is the half wavelength of the corrugated steel web; s is the length of the straight plate segment; d is the horizontal width of the inclined plate segment; i represents the i-th wave segment, i is a natural number; f(z) represents the wave crest height of the corrugated steel plate.

[0028] Figure 2 In the formula:

[0029] There are two coordinate systems, one is the x-y0 coordinate system with the centroid O' of the ordinary I-shaped steel section corresponding to the corrugated steel web as the origin, and the other is the x-y coordinate system with the centroid O of the I-shaped steel section of the corrugated steel web as the origin; S is the shear center; b is the width of the flange segment; t is the thickness of the flange segment; e x is the distance from O to O'; h w is the web height; t w is the thickness of the corrugated steel web; h is the section height.

[0030] Figure 3 In the formula:

[0031] l represents the length of the beam, which is generally an integer multiple of the half wavelength D of the corrugated steel web, i.e. l = nD, n is an integer;

[0032] Figure 4 In the formula:

[0033] θ represents the included angle of the inclined plate segment; ε w represents the bending strain of the inclined plate segment; ε z represents the normal strain of the inclined plate segment around the y-axis; ε represents the strain of the beam section.

[0034] Figure 5 In the formula:

[0035] f(z) represents the wave crest height of the corrugated steel plate; x0 is the distance from the shear center of the straight plate segment section to the centroid; S is the shear center; O represents the centroid of the inclined plate segment section; e x represents the distance from the section centroid O to the section center O'; t w is the thickness of the corrugated steel web; h is the section height; t is the thickness of the flange segment; b is the width of the flange segment; + represents a positive number, and — represents a negative number.

[0036] Figure 6 In the formula:

[0037] A0 is the starting point of the curve coordinates, and the coordinates of A0 are [-f(z)+e x , 0]; S is the shear center, and the coordinates of S are [-x0, 0]; x0 is the shear center position of the straight plate segment; e x represents the distance from the section centroid O to the section center O'; f(z) represents the wave crest height of the corrugated steel plate. DETAILED DESCRIPTION

[0038] A calculation model for an equivalent straight-web I-beam with corrugated steel web is provided, and the equivalent model adopts the following equivalent relationship:

[0039] (1) Both have the same deformation (deflection);

[0040] (2) The deformation energy is the same.

[0041] Based on the dimensions of the corrugated steel web, and under the two conditions mentioned above, the section moment of inertia and sectoral moment of inertia of the corrugated steel web beam are calculated, thereby obtaining the equivalent section parameters of the equivalent straight steel web I-beam. Using the equivalent section parameters, the relevant theories and methods for straight steel web I-beams can be directly applied to calculate and analyze the corrugated steel web I-beam.

[0042] The following uses a trapezoidal corrugated steel web, which is most widely used in engineering projects, as an example to derive the equivalent moment of inertia I of an equivalent straight steel web I-beam. y,eff and equivalent sectoral moment of inertia I ω,eff The equivalent calculation formula is as follows:

[0043] Step 1: Clarify the physical meaning of the cross-sectional properties. Analyze the shape and dimensions of the corrugated steel web, such as... Figure 1 As shown, f(z) represents the web of the corrugated steel plate in... Figure 1 At a certain position of f(z) on the coordinate axis, due to the moment of inertia I of f(z) about the corrugated steel web about the x-axis... x (or referred to as the moment of inertia I of the straight section of the corrugated steel web about the x-axis) x ) and free torsional moment of inertia I k No effect; the equivalent moment of inertia I needs to be calculated based on the assumption that deformation and strain energy are equal. y,eff I ω,eff .

[0044] like Figure 1 The waveform steel web I-beam shown has a waveform steel web position function, and the i-th half-wave is:

[0045]

[0046] Based on common knowledge, the cross-sectional properties of the straight section are obtained (see...). Figure 2 ):

[0047] A f =2bt, A w =h w t w A = 2bt + h w t w ,

[0048] In the above formula, A fA is the flange area of ​​a corrugated steel web I-beam; w I represents the cross-sectional area of ​​the straight section of the corrugated steel web; A represents the total cross-sectional area of ​​the straight section of the corrugated steel web; I x Let x be the moment of inertia of the straight section of the corrugated steel web about the x-axis.

[0049] Based on common knowledge, the cross-sectional properties of the inclined plate segment are obtained (see...). Figure 2 ):

[0050] A′ w =h w t w cosθ, A′=2bt+h w t w cosθ,

[0051] In the above formula, A' w I represents the web area of ​​the inclined section of the corrugated steel web; A' represents the total web area of ​​the inclined section of the corrugated steel web; I x ' represents the moment of inertia of the inclined section of the corrugated steel web about the x-axis; θ represents the included angle of the inclined section.

[0052] When the web is located on the y0 axis, i.e., f(z) = 0, the following equation is obtained according to the well-known formula for moment of inertia:

[0053] The moment of inertia I of the flange about the x-axis f :

[0054] The formulas for the moments of inertia about the x-axis of the straight and inclined web sections are as follows: I w (Straight section of corrugated steel web), I' w (of the inclined section of the corrugated steel web):

[0055]

[0056] Moments of inertia I about the y0 axis of the straight and inclined sections of the web y0 (Straight section of corrugated steel web), I y0 (Of the inclined section of the corrugated steel web):

[0057]

[0058] The sectoral moment of inertia I of the cross section ω0 :

[0059] Step 2: Strain analysis of the corrugated steel web.

[0060] like Figure 4 As shown, let ρ be the radius of curvature of the beam. Based on common knowledge, we obtain the following formula:

[0061] The beam cross-section strain ε is:

[0062]

[0063] The oblique plate in-plane strain ε w is:

[0064]

[0065] The oblique plate normal strain ε z around the y axis is:

[0066]

[0067] Step 3: Calculate the cross-section moment of inertia parameters of the corrugated steel web steel beam and establish the equivalent cross-section moment of inertia I y,eff equivalent calculation formula.

[0068] 3.1 Establish the calculation formula of the arbitrary cross-section moment of inertia I y (z), I y '(z) (see Figure 2 ):

[0069] When the corrugated steel plate is f(z) away from the y0axis, the distance e x , e x ' from the centroid O of the cross-section of the straight plate segment and the oblique plate segment to the y0axis:

[0070] Straight plate segment cross-section:

[0071] Oblique plate segment cross-section:

[0072] The formulas of the moments of inertia of the straight plate segment and the oblique plate segment around the y axis are as follows:

[0073]

[0074]

[0075] 3.2 Establish the equivalent calculation formula of the equivalent cross-section moment of inertia I y,eff .

[0076] According to the same deformation and deformation energy of the corrugated steel web beam and the equivalent I-beam, for a simply supported beam (see Figure 3 ), the deflection equation can be assumed as:

[0077] In the formula, a is the mid-span deflection;

[0078] According to the reciprocal of the radius of curvature 1 / ρ, the second derivative u'' of the deflection u needs to be substituted into the strain energy U yDefinition: Integrate along the volume V of the I-beam with respect to the web and flanges respectively;

[0079]

[0080] In the formula: σ represents bending stress; E is the elastic modulus of steel.

[0081] Substituting into the deflection equation and integrating, after mathematical calculations and simplification, we obtain:

[0082]

[0083] The trigonometric function summation formula (see Journal of Puyang Teachers College, 2000, 13(4):64-65. Reference: A Brief Discussion on the Summation of Trigonometric Sequences. Authors: Li Zongming, Sun Shaofeng) yields:

[0084]

[0085] but:

[0086]

[0087] In addition, the formula for the equivalent I-beam bending strain energy about the y-axis is as follows (definition):

[0088]

[0089] Based on the equality of strain energy, we have U y =U y,eff Then the equivalent moment of inertia I y,eff for:

[0090]

[0091] In the formula: F is the distance from the straight section of the corrugated steel web to the center of the beam; D is the half wavelength of the corrugated steel web; A f A is the flange area of ​​the straight section of the corrugated steel web; w Let A' be the cross-sectional area of ​​the straight section of the corrugated steel web; w denoted as A', which is the cross-sectional area of ​​the inclined section of the corrugated steel web; denoted as s, which is the length of the straight section; denoted as b, which is the width of the flange section; denoted as d, which is the horizontal width of the inclined section; denoted as t, which is the flange thickness; and θ, which represents the angle between the inclined section and the centerline of the beam.

[0092] Step 4: Calculate the sectoral moment of inertia parameters of the corrugated steel web beam and establish the equivalent sectoral moment of inertia I. ω,eff The equivalent calculation formula.

[0093] 4.1 Calculate the shearing position:

[0094] like Figure 5The static moment distribution diagram of the corrugated web I-beam is shown, and the shear center position x0 of the straight web segment section is calculated according to the diagram: coordinate integration along the web and flange center line;

[0095]

[0096] In the formula, S x represents the area distance, and ρ0 is the distance from the microelement segment to the shear center S; the self-defined C is a constant, is the distance from the corrugated web straight web segment to the section center.

[0097] Similarly, according to the method for determining the shear center position of the straight web segment section, the shear center position x0' of the inclined web segment section can be determined:

[0098]

[0099] In the formula, the self-defined C' is a constant,

[0100] 4.2 Establishing the equivalent sectorial moment of inertia I ω,eff The equivalent calculation formula is:

[0101] As shown in Figure 6 , taking the point A0 as the starting point of the curve coordinate, the section sectorial coordinate is:

[0102] Web: ω s = [-f(z) + e x +x0]y;

[0103] Top flange:

[0104] Bottom flange:

[0105] Due to the symmetry of the section, the average sectorial coordinate is:

[0106]

[0107] The main sectorial coordinate is:

[0108] The warping normal strain of the inclined web segment of the corrugated web I-beam is:

[0109]

[0110] According to the fact that the corrugated web I-beam and the equivalent I-beam have the same deformation and deformation energy, the torsion angle equation of the beam is:

[0111]

[0112] The second derivative of the torsion angle of the beam is: Substitute the warping strain energy U of the corrugated web I-beam into the formula ω Definition: Integrate the web and flange along the volume V of the beam respectively;

[0113]

[0114] In the formula: σ ω represents the warping stress; E is the elastic modulus of steel.

[0115] According to the summation formula of trigonometric functions (see Puyang Education College Journal, 2000, 13 (4): 64-65. Literature: On the Summation of Triangular Series. Author: Li Zongming, Sun Shaofeng.), we can get:

[0116]

[0117] Therefore, the warping strain energy of the corrugated web I-beam is:

[0118]

[0119] The warping moment of inertia of the equivalent straight web I-beam is I ω,eff , and its strain energy is:

[0120]

[0121] Because the strain energy is equal, U ω = U ω,eff Substitute C and C' into the formula, and we get the equivalent warping moment of inertia:

[0122]

[0123] In the formula: F is the distance from the straight web segment to the center segment of the corrugated web; I f is the moment of inertia of the flange about the x-axis; I w is the moment of inertia of the straight web segment about the x-axis; I' w is the moment of inertia of the inclined web segment about the x-axis; I x is the moment of inertia of the straight segment cross-section about the x-axis; I' x is the moment of inertia of the inclined segment cross-section about the x-axis; s is the length of the straight segment; d is the horizontal width of the inclined segment; D is the half wavelength of the corrugated web; θ represents the angle of the inclined segment; I ω0 is the moment of inertia of the straight steel web I-beam cross-section with the same thickness as the corrugated web.

[0124] As I y,eff and I ω,eff , the equivalent moment of inertia is only determined by the cross-sectional parameters and the size of the corrugated steel plate, and has no direct relationship with the span and constraints of the beam.

[0125] Step five: Calculate I y,eff and Iω,eff After that, the corrugated web I-beam can be calculated and analyzed according to the equivalent flat web I-beam model. The unified calculation formula of the critical load M cr of various symmetrical section flat web I-beams is as follows:

[0126]

[0127] In the formula, β1 is a critical moment correction coefficient, which depends on the load acting on the component and the constraint form; β2 is a load action point position influence coefficient; a is the vertical distance from the load action point to the shear center; μ y is the bending calculation length coefficient around the y-axis; μ ω is the torsion calculation length coefficient; l represents the length of the beam; E is the elastic modulus of steel; G is the shear modulus of steel (a material constant, which is the ratio of shear stress and strain); I k is the free torsion moment of inertia; and π is the circular constant.

[0128] I y,eff and I ω,eff are substituted into the above formula, so that the critical load of the corrugated web I-beam can be conveniently calculated by using the critical load formula of the flat web I-beam.

[0129] The parts not involved in the present application are the same as or can be realized by using the prior art. The above content is a further detailed description of the present application in combination with the specific preferred embodiments, and the specific implementation of the present application should not be limited to the above-mentioned embodiments. For ordinary skilled persons in the technical field to which the present application belongs, some simple deductions or substitutions can be made without departing from the concept of the present application, and all of them should be regarded as belonging to the patent protection scope of the present application determined by the submitted claims.

Claims

1. A method of constructing an equivalent computational model of a corrugated steel web I-beam, characterized by: The method for constructing the equivalent calculation model is as follows: the precondition of assuming the equivalent relation is that the corrugated steel web I-beam and the equivalent I-beam have the same deformation and equal strain energy; according to the shape and size of the corrugated steel web, the parameters of the sectional moment of inertia and the polar moment of inertia of the corrugated steel web I-beam are calculated under the precondition, and then the equivalent sectional parameters of the equivalent flat steel web I-beam are obtained; the equivalent sectional parameters can be directly used to calculate and analyze the corrugated steel web by using the corresponding formula of the flat steel web I-beam; The equivalent cross-section parameters of the equivalent flat steel web I-beam include an equivalent cross-section moment of inertia I y,eff and an equivalent cross-section polar moment of inertia I ω,eff ; the equivalent cross-sectional moment of inertia I y,eff The equivalent calculation formula is as follows: ; wherein: F is the distance from the straight web segment of the corrugated steel web to the center of the beam; D is the half wavelength of the corrugated steel web; A f is the flange area of the flat segment of the corrugated steel web; A w is the web cross-sectional area of the flat segment of the corrugated steel web; A is the total web cross-sectional area of the flat segment of the corrugated steel web; A ＇ w is the web cross-sectional area of the inclined web segment of the corrugated steel web; A ＇ is the total web cross-sectional area of the inclined web segment of the corrugated steel web; s is the length of the straight segment; b is the width of the flange segment; d is the horizontal width of the inclined segment; t is the thickness of the flange; θ denotes the included angle of the inclined segment with the centerline of the beam; the equivalent sectorial moment of inertia I ω,eff The equivalent calculation formula is as follows: ; wherein: I f Iy = the moment of inertia of the flange about the x-axis; I w Iy = the moment of inertia of the straight web segment about the x-axis; I ＇ w Iy = the moment of inertia of the inclined web segment about the x-axis; I x Iy = the moment of inertia of the straight web segment about the x-axis; I ＇ x Iy = the moment of inertia of the inclined web segment about the x-axis; I ω0 Iy = the moment of inertia of the straight web segment about the x-axis; The method for calculating the critical load of the I-beam with corrugated steel web by using equivalent cross-section parameters is as follows: I y,eff And I ω,eff As can be seen from the equivalent calculation formula, the equivalent moment of inertia is determined by the cross-section parameters and the size of the corrugated steel plate, therefore, the equivalent moment of inertia is calculated I y,eff And I ω,eff Then, the critical load of the I-beam with corrugated steel web can be calculated according to the model of the I-beam with equivalent flat steel web. Critical load of i-beam with symmetric cross-section and flat steel web M cr The calculation formula is: ； wherein: β 1 is a critical moment modification factor, which depends on the load and the form of restraint acting on the member; β 2 is a load point location factor; a 1 is the vertical distance from the load point to the shear center; μ y is the bending calculation length factor about the y-axis; μ ω is the torsion calculation length factor; l denotes the length of the beam; E is the modulus of elasticity of steel; G is the shear modulus of steel; I k is the free torsional moment of inertia; π is the circle constant; The I y,eff With I ω,eff Substituting the above formula, the critical load of the corrugated steel web I-beam can be easily calculated by using the critical load formula of the flat steel web I-beam.

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