Method for calculating equivalent section modulus of cold-formed thin-walled steel beam with hole under pure bending

Abstract

The invention discloses a method for calculating the equivalent section modulus of a cold-formed thin-walled steel beam with a hole under a pure bend, the cold-formed thin-walled steel beam has a C-shaped section, and long round holes are continuously formed along a web plate; comprising the following steps that the width b of a web straight section of the cold-formed thin-walled steel beam, the height h of a flange straight section, the length d of a curled edge straight section, the plate thickness t, the total length e of an oblong hole, the appearance period S of the oblong hole, the semi-circle radius r of the two ends of the oblong hole and the steel yield strength fy are determined; based on an effective width method, considering local buckling and an oblong hole weakening effect, respectively calculating effective widths of a web, a flange and a curled edge, combining the effective widths to form an equivalent effective section, and calculating an effective area Aeff of the equivalent effective section and a distance eeff from an effective centroid to a center line of the web; and calculating an effective inertia moment Ieff and an equivalent section modulus Weff. According to the method, the section modulus after reduction can be quickly obtained by substituting the geometric dimension once, the method is simple and quick, and high-precision and zero-iteration preposed input can be provided for subsequent deformation checking calculation and flexural capacity design.

Description

Technical Field

[0001] This invention relates to cold-formed thin-walled steel component technology, specifically to a method for calculating the equivalent section modulus of a cold-formed thin-walled steel beam with holes under pure bending. Background Technology

[0002] Continuously perforated cold-formed thin-walled steel beams are widely used in seismic bracing, utility tunnel crossbeams, and industrial plant purlins due to their lightweight, high strength, and quick assembly. To meet functional requirements such as pipeline crossings, weight reduction, or reduced wind resistance, the web is often densely perforated with elongated circular holes (hole length 20–60 mm, hole spacing 50–150 mm) in a 200–600 mm module, forming beam segments with "periodic stiffness degradation." However, the current standard GB 50018-2013 only provides reduction coefficients for perforation ratios <15% for circular holes, and lacks design provisions for densely perforated arrays of elongated circular holes. Directly applying the complete section formula can overestimate stiffness by 10%–30% and bearing capacity by up to 25%, becoming a hidden risk point in seismic deformation verification. The existing technical approaches and their shortcomings are as follows: ① Experience deduction method The "area moment of inertia reduction method" derived from Timoshenko beam theory deducts the opening section as the net cross-section and then multiplies it by an empirical coefficient of 0.8–0.9, completely ignoring the coupling effect of hole shape, hole spacing and local buckling. For long circular holes, the error increases exponentially with the hole length / hole spacing ratio. The circular hole reduction formula provided by AISI S100-2016 has an error of >40% under some conditions and does not cover the cold-bent section with rolled edge stiffening.

[0003] ② Simplification of the effective width method Treating the perforated area as "zero stiffness" and proportionally reducing the web height before substituting it into the standard effective width formula. This method assumes that the cross-sectional strain is constant along the beam length, which fails to reflect the additional shear hysteresis and distortion buckling caused by periodic openings. This leads to an overestimation of the flange-flanged stiffening contribution, resulting in generally unsafe calculation results.

[0004] ③ Finite element refined modeling Using SOLID186 or SHELL181 to create millimeter-level meshes for single holes / single spans, combined with sub-models or periodic boundaries, can yield high-precision results, but the mesh is significantly sensitive: in a case of a pipe gallery crossarm, when the mesh at the hole edge is refined from 2 mm to 1 mm, the equivalent cross-sectional stiffness fluctuates by 7%, making it difficult to promote in the batch design of supports and hangers.

[0005] ④ Experimental regression formula The weighted moment of inertia method or piecewise regression method is only applicable to specific cross sections and specific load combinations. It lacks an explanation of the mechanism of the triple coupling of "local-distortion-aperture". After changing the aperture type or load mode, it needs to be refitted. It has poor versatility and design units dare not use it directly.

[0006] In summary, existing technologies suffer from several shortcomings: empirical subtraction ignores hole shape and buckling coupling, often resulting in errors exceeding 25%; the effective width method treats the opening as having "zero stiffness," failing to reflect periodic shear hysteresis and leading to unsafe results; while the finite element method is accurate, it is mesh-sensitive, time-consuming (taking several hours) and dependent on experienced engineers, making it difficult to use in large-scale applications; experimental formulas fail when changing hole shape or load, exhibiting poor versatility. These shortcomings have resulted in the industry facing a long-standing dilemma of "theoretical formulas being inapplicable, finite element method being infeasible, and empirical formulas being unreliable," necessitating an analytical-level calculation method that balances accuracy and efficiency. Summary of the Invention

[0007] The technical problem to be solved by this invention is to overcome the shortcomings of the prior art and provide a method for calculating the equivalent section modulus of cold-formed thin-walled steel beams with holes under pure bending. Based on the effective width method and the explicit parameterized calculation method of the influence of holes, it can quickly obtain the reduced section modulus by substituting the geometric dimensions once, avoiding the traditional trial-and-error effective width iteration and expensive shell element finite element analysis, and providing high-precision, zero-iteration pre-input for subsequent deformation verification and bending bearing capacity design.

[0008] To solve the above-mentioned technical problems, the technical solution provided by the present invention is as follows: A method for calculating the equivalent section modulus of a cold-formed thin-walled steel beam with holes under pure bending, wherein the cold-formed thin-walled steel beam has a C-shaped section and elongated holes are continuously formed along the web. The calculation method includes the following steps: Step 1. Determine the geometric and material parameters of the cold-formed thin-walled steel beam, including the width of the straight section of the web. b Height of straight section of wing flange h Length of the straight section with rolled edge d , plate thickness t Total length of oblong hole e The periodicity of oblong holes S The radius of the semicircles at both ends of the oblong hole r The material parameters include the yield strength of steel. f y ; Step 2. Based on the effective width method, considering the local buckling effect and the weakening effect of the elongated hole, calculate the effective width of the web, the effective width of the flange, and the effective width of the rolled edge, respectively. Step 3. Based on the effective flange width, effective web width, and effective flange width obtained in Step 2, combine them to form an equivalent effective section, and calculate the effective area of ​​the equivalent effective section. A eff and the distance from the effective centroid to the center line of the web y eff ; (1) (2) (3) In the above formula, r w The flexibility index of the web. r f The flexibility index of the flange. r d This is the flexibility index of the curled edge. , where 1.27 is the equivalent effective width coefficient of the web, and 1.27 is the shape equivalence coefficient; Step 4. Based on the effective area A eff Distance from the effective centroid to the center line of the web y eff Calculate the circumference x Effective moment of inertia of the axis I eff ; I eff = (4) Step 5. According to the formula W eff = I eff / y max Calculate the equivalent section modulus W eff ,in y max The distance from the outermost fiber of the compression flange to the centroidal axis in the equivalent effective cross section. y max = ; The final result is: (5) As a further improvement to the above technical solution: Preferably, the web is an unstiffened plate, and its effective width calculation needs to consider both the local buckling effect and the weakening effect of the elongated hole. The flexibility index of the web... r w It is obtained through the following calculation formula: (6) (7) in, l w The dimensionless width-to-thickness ratio of the web is given. k w denoted as the local buckling coefficient of the web.

[0009] Preferably, the flange is a plate that is supported on one side and free on the other, and its effective width is calculated considering only the local buckling effect. The flexibility index of the flange is... r f It is obtained through the following calculation formula: (8) (9) in, l f The dimensionless width-to-thickness ratio of the flange. k f denoted as the local buckling coefficient of the flange.

[0010] Preferably, the rolled edge is a stiffening rib, and its effective width is calculated considering only the local buckling effect. The flexibility index of the rolled edge... r d It is obtained through the following calculation formula; (10) (11) in, l d The dimensionless width-to-thickness ratio of the rolled edge. k d is the local buckling coefficient of the rolled edge.

[0011] Preferably, the effective moment of inertia I eff The specific calculation formula is as follows: (12) (13) (14) (15) in, I web The effective moment of inertia of the web is... I flange The effective moment of inertia of the flange, I lip The effective moment of inertia of the rolled edge.

[0012] Preferably, the total length of the elongated hole e The diameter is 20-60mm, and the oblong holes appear periodically. S It is 50-150mm.

[0013] Preferably, the calculation method is applicable to perforated cold-formed thin-walled steel beams in building electromechanical seismic bracing systems, prefabricated integrated pipe gallery bracing systems, or high-temperature and high-pressure pipeline seismic supports in industrial plants.

[0014] Compared with the prior art, the beneficial effects of the present invention are: (1) It can solve the problem of lacking an analytical formula for the equivalent section modulus of cold-formed steel beams with openings. Current domestic and international standards only provide algorithms for the effective width of sections without holes. There are no explicit formulas for the simultaneous weakening of the flange and web caused by continuous elongated circular holes. Designers can only use empirical reductions of 0.8-0.9 times, often resulting in errors exceeding 15%. This invention is the first to derive the section modulus of sections with holes. W eff The closed parameter formula includes hole spacing S, hole length e, and plate thickness. t By directly substituting the values, an accurate value with a deviation of less than 5% from the shell element result can be obtained, allowing for a single manual calculation that can be used for stiffness reduction and bending strength verification.

[0015] (2) It can solve the bottleneck of local-distortion-aperture triple coupling that cannot be calculated manually. Traditional effective width methods only consider local buckling and lack analytical representation of the interaction between edge stiffening and stress concentration at the hole edge. This invention introduces a buckling reduction coefficient and an equivalent effective width coefficient for the web, embedding them into the same... W eff The formula allows the triple coupling effect to be completed within an Excel spreadsheet, eliminating tedious iterations and finite element modeling, and shortening the design cycle from hours to minutes, meeting the needs of batch design.

[0016] (3) It can solve the pain point of "excessive cost of finite element iteration". During the scheme comparison phase, each adjustment to the hole spacing or plate thickness requires rebuilding the shell element model and solving it, consuming a large amount of computational resources. This invention compresses the calculation time to the second level, and can quickly scan thousands of hole type combinations in the optimization of support and hanger layout, pipe gallery crossarms or industrial plant beam openings, directly outputting the optimal hole spacing and plate thickness, saving more than 90% of the iteration cost, and meeting the requirements of simultaneous verification of deformation and strength in both GB 50018 and AISI S100 standards.

[0017] In summary, this invention achieves [the following] by embedding the local buckling reduction coefficient and the equivalent effective width coefficient of the web into the effective section modulus formula in one step. W eff Achieving shell element precision in a single manual calculation provides a universal, iterative, programmable, and standardized solution for subsequent seismic deformation verification and bearing capacity design. Compared to shell element finite element analysis, this invention maintains... W effWith an error of ≤5%, the calculation time for a single operation is reduced from hours to minutes, without the need for high-performance hardware or professional software such as ABAQUS. Compared with the effective width empirical reduction method, the accuracy is improved by more than 10%, and the applicable range is expanded to any combination of long oval hole spacing, hole diameter and pure bend. It can overcome the shortcomings of the traditional "area subtraction method" for insufficient accuracy of long oval holes and achieve the best technical effect. Attached Figure Description

[0018] Figure 1 This is a schematic diagram of the cross-sectional structure of the cold-formed thin-walled steel in this invention.

[0019] Figure 2 This is a schematic diagram of the layout structure of the openings in the web of the cold-formed thin-walled steel in this invention. Detailed Implementation

[0020] The invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0021] like Figure 1 and Figure 2 As shown in the embodiment of the present invention, a method for calculating the equivalent section modulus of a cold-formed thin-walled steel beam with holes under pure bending is provided. The cold-formed thin-walled steel beam has a C-shaped section and elongated holes are continuously formed along the web. The calculation method of the present invention includes the following steps: Step 1. Determine the geometric and material parameters of the cold-formed thin-walled steel beam. The geometric parameters include the width of the straight section of the web. b Height of straight section of wing flange h Length of the straight section with rolled edge d , plate thickness t Total length of oblong hole e The periodicity of oblong holes S The radius of the semicircles at both ends of the oblong hole r Material parameters include the yield strength of steel. f y ; Step 2. Based on the effective width method, considering the local buckling effect and the weakening effect of the elongated hole, calculate the effective width of the web, the effective width of the flange, and the effective width of the rolled edge, respectively. Step 3. Based on the effective flange width, effective web width, and effective flange width obtained in Step 2, combine them to form an equivalent effective section. Calculate the effective area of ​​the equivalent effective section using the explicit parameterization formula. A eff and the distance from the effective centroid to the center line of the web y eff (In this embodiment, the upper side is under pressure), thus: (1) (2) (3) In the above formula, r w The flexibility index of the web. r f The flexibility index of the flange. r d This is the flexibility index of the curled edge. 1.27 is the equivalent effective width coefficient of the web, and 1.27 in the formula is the shape equivalence coefficient (the semicircular area at both ends of the elongated hole is equivalent to the shape coefficient of a rectangle). Step 4. Based on effective area A eff Distance from the effective centroid to the center line of the web y eff Calculate the circumference x Effective moment of inertia of the axis I eff ; I eff = (4) Step 5. According to the formula W eff = I eff / y max Calculate the equivalent section modulus W eff ,in y max The distance from the outermost fiber of the compression flange to the centroidal axis in the equivalent effective cross section. y max = ; The final result is: (5) Compared with shell element finite element analysis, this invention maintains... W eff With an error of ≤5%, the calculation time for a single operation is reduced from hours to minutes, without the need for high-performance hardware or professional software such as ABAQUS. Compared with the empirical reduction method for effective width, the accuracy is improved by more than 10%, and the applicable range is expanded to any combination of long circle hole spacing, hole diameter and pure bend, achieving the best technical effect.

[0022] In this embodiment, the web is an unstiffened plate, and the calculation of its effective width needs to consider both the local buckling effect and the weakening effect of the elongated hole. The flexibility index of the web... r w It is obtained through the following calculation formula: (6) (7) in, l w The dimensionless width-to-thickness ratio of the web is given. k w denoted as the local buckling coefficient of the web.

[0023] In this embodiment, the flange is a plate that is supported on one side and free on the other. The calculation of its effective width only considers the local buckling effect, and the flange's flexibility index... r f It is obtained through the following calculation formula: (8) (9) in, l f The dimensionless width-to-thickness ratio of the flange. k f denoted as the local buckling coefficient of the flange.

[0024] In this embodiment, the rolled edge serves as a stiffening rib, and its effective width is calculated considering only the local buckling effect. The flexibility index of the rolled edge... r d It is obtained through the following calculation formula; (10) (11) in, l d The dimensionless width-to-thickness ratio of the rolled edge. k d is the local buckling coefficient of the rolled edge.

[0025] In this embodiment, the effective moment of inertia I eff The specific calculation formula is as follows: (12) (13) (14) (15) in, I web The effective moment of inertia of the web is... I flange The effective moment of inertia of the flange, I lip The effective moment of inertia of the rolled edge.

[0026] In this embodiment, the total length of the oblong hole eThe diameter is 20-60mm, and the oblong holes appear periodically. S It is 50-150mm.

[0027] The calculation method of this invention is applicable to perforated cold-formed thin-walled steel beams in building electromechanical seismic support systems, prefabricated integrated pipe gallery supports, or high-temperature and high-pressure pipeline seismic supports in industrial plants.

[0028] The method for calculating the equivalent section modulus of a perforated cold-formed thin-walled steel beam under pure bending, as described in this invention, has been proven feasible through experiments, simulations, and practical application. The verification steps are as follows: S1. Fabricate a cold-formed steel simply supported beam with a continuous elongated circular hole of the same dimensions as the theoretical model, and accurately measure the total length of the elongated circular hole. e The appearance of oblong holes is periodic. S and plate thickness t Four-point bending loading is adopted, at mid-span and... L / 4、3 L / 4 LVDTs (a type of sensor that accurately converts "linear displacement" into electrical signals) are placed at the locations to synchronously record the load-displacement curves until the elastic limit, ensuring that the specimen is in a pure bending state. S2. For continuously perforated cold-formed thin-walled beams, a four-point bending method is used. Load-displacement curves are recorded in the elastic segment, and the data is then analyzed. IE eq = Yes (3 L ²-4 a ²) / (24 d Inverse calculation of equivalent bending stiffness (where P For load, a The distance from the loading point to the nearest support. L For the span of the support, d (For the actual deflection of the pure bending segment), then press W. eff,e = IE eq / y max The equivalent section modulus is thus obtained; S3. Comparison and verification with the formula of this invention: Input the same geometric parameters into the W of this invention. eff The analytical expression yields the theoretical equivalent section modulus. S4. Compare the difference between the two (i.e., the equivalent section modulus in S2 and the theoretical equivalent section modulus in S3). If the error is ≤5%, the formula of the present invention is deemed reliable.

[0029] During the experiment, strain contour maps of the borehole edge were simultaneously captured to verify whether the plane section assumption and local buckling occurred synchronously, thus confirming the validity of the theoretical premise. The above process was repeated using six beam segments of different lengths, covering commonly used parameter ranges, to experimentally verify the method for calculating the equivalent section modulus of a cold-formed thin-walled steel beam with a bore under pure bending. The following calculation results were obtained: The parameters selected in the experimental process are: b = h = 41.3mm, t = 2.0mm, r =6.5mm, e =28mm, S =50mm.

[0030] To facilitate comparison, relative error is introduced. E The calculation formula is: (16) Table 1. Comparison results of four-point bending

[0031] As shown in Table 1, the theoretical calculation method of this invention fits the experimental results well, with the error basically within 5%.

[0032] This invention has a wide range of applications, mainly involving the following areas: 1. Building electromechanical seismic bracing system In seismic bracing systems for electromechanical pipelines in high-rise and super high-rise buildings, cold-formed thin-walled channel steel typically has elongated holes continuously drilled along its flanges for inserting hangers or cable trays. Using the method of this invention, the equivalent section modulus after the hole reduction can be obtained in a single step. W eff By directly substituting the values ​​from GB 50981-2014, the inter-story drift angle and flexural bearing capacity can be verified, avoiding the error of more than 15% caused by the traditional "experience reduction factor". This reduces the amount of steel used in the supports and hangers by about 10%, while meeting the requirements for seismic toughness.

[0033] 2. Prefabricated integrated utility tunnel support system In urban underground utility tunnels, columns and crossarms typically utilize cold-formed thin-walled steel with a wall thickness of approximately 2 mm, and are continuously perforated with elongated holes in a 50 mm module to facilitate rapid assembly and subsequent expansion. Using the parametric formulas of this invention, designers can quickly calculate the equivalent stiffness of large-span crossarms without repeatedly building finite element models. This provides real-time data support for rapid tunnel assembly, clearance optimization, and adjustment of reserved hole positions, significantly shortening the design cycle and reducing construction change costs.

[0034] 3. Seismic supports for high-temperature and high-pressure pipelines in industrial plants In industrial plants such as petrochemical, pharmaceutical, and data center facilities, high-temperature and high-pressure pipeline supports often utilize double-rolled C-shaped steel or rectangular steel pipes as load-bearing beams, balancing weight reduction with adjustable suspension rod functionality. The calculations based on this invention... W effIt can be directly used for buckling verification in GB50018-2013 and AISI S100-16 to ensure that the strength and deformation requirements are still met after the continuous elongated hole is weakened, avoid repeated shell element modeling, and realize an efficient design process of "one calculation, two standards passed".

[0035] The above are all preferred embodiments of this application, and are not intended to limit the scope of protection of this application. Therefore, all embodiments based on this application are subject to change. Equivalent changes to the structure, shape, and principle of the object shall be covered within the scope of protection of this application.

Claims

1. A method for calculating the equivalent section modulus of a cold-formed thin-walled steel beam with holes under pure bending, characterized in that, The cold-formed thin-walled steel beam has a C-shaped cross-section and continuous elongated holes along the web. The calculation method includes the following steps: Step 1. Determine the geometric and material parameters of the cold-formed thin-walled steel beam. The geometric parameters include the width of the straight section of the web. b Height of straight section of wing flange h Length of the straight section with rolled edge d , plate thickness t Total length of oblong hole e The periodicity of oblong holes S The radius of the semicircles at both ends of the oblong hole r The material parameters include the yield strength of steel. f y ; Step 2. Based on the effective width method, considering the local buckling effect and the weakening effect of the elongated hole, calculate the effective width of the web, the effective width of the flange, and the effective width of the rolled edge, respectively. Step 3. Based on the effective flange width, effective web width, and effective flange width obtained in Step 2, combine them to form an equivalent effective section, and calculate the effective area of ​​the equivalent effective section. A eff and the distance from the effective centroid to the center line of the web y eff ; ；(1) ；(2) ；(3) In the above formula, ρ w The flexibility index of the web. ρ f The flexibility index of the flange. ρ d This refers to the flexibility index of the curled edge. , where 1.27 is the equivalent effective width coefficient of the web, and 1.27 is the shape equivalence coefficient; Step 4. Based on the effective area A eff Distance from the effective centroid to the center line of the web y eff Calculate the circumference x Effective moment of inertia of the axis I eff ; I eff = ；(4) Step 5. According to the formula W eff = I eff / y max Calculate the equivalent section modulus W eff ,in y max The distance from the outermost fiber of the compression flange to the centroidal axis in the equivalent effective cross section. y max = ; The final result is: (5)。 2. The calculation method according to claim 1, characterized in that, The web is an unstiffened plate, and its effective width calculation needs to consider both local buckling effects and the weakening effect of elongated holes. The flexibility index of the web... ρ w It is obtained through the following calculation formula: ；(6) ；(7) in, λ w The dimensionless width-to-thickness ratio of the web is given. k w denoted as the local buckling coefficient of the web.

3. The calculation method according to claim 1, characterized in that, The flange is a plate with one side supported and the other side free. Its effective width is calculated considering only local buckling effects. The flange's flexibility index... ρ f It is obtained through the following calculation formula: ；(8) ； (9) in, λ f The dimensionless width-to-thickness ratio of the flange. k f denoted as the local buckling coefficient of the flange.

4. The calculation method according to claim 1, characterized in that, The rolled edge serves as a stiffening rib, and its effective width is calculated considering only local buckling effects. The flexibility index of the rolled edge... ρ d It is obtained through the following calculation formula; ；(10) ； (11) in, λ d The dimensionless width-to-thickness ratio of the rolled edge. k d is the local buckling coefficient of the rolled edge.

5. The calculation method according to claim 1, characterized in that, The effective moment of inertia I eff The specific calculation formula is as follows: ；(12) ；(13) ；(14) ；(15) in, I web The effective moment of inertia of the web is... I flange The effective moment of inertia of the flange, I lip The effective moment of inertia of the rolled edge.

6. The calculation method according to claim 1, characterized in that, The total length of the oblong hole e The diameter is 20-60mm, and the oblong holes appear periodically. S It is 50-150mm.

7. The calculation method according to any one of claims 1-6, characterized in that, The calculation method is applicable to perforated cold-formed thin-walled steel beams in building electromechanical seismic support systems, prefabricated integrated pipe gallery supports, or high-temperature and high-pressure pipeline seismic supports in industrial plants.

Images