A zero state form finding method for a catenary beam

Abstract

The application discloses a zero-state form-finding method for a cantilever beam, and belongs to the technical field of long-span spatial steel structure construction. The method comprises the following steps: obtaining a design target line shape, constructing a reference chord line and arranging control points; defining a shape coefficient, establishing a parameterized geometric mapping model between the reference chord line, the design target line shape and a zero-state installation line shape; taking a root mean square error of the control point positions as an objective function, and iteratively optimizing the shape coefficient by using an intelligent optimization algorithm; substituting the current shape coefficient into the mapping model to generate the zero-state installation line shape, establishing a finite element model, applying a dead load simulation, extracting the coordinates of the deformed control points, calculating the objective function and feeding back to the optimization algorithm; outputting the optimal shape coefficient after iterative convergence, reconstructing the zero-state installation line shape and outputting construction data. The application replaces discrete node local correction with overall curve parameter shaping, ensures that the cantilever beam has continuous and smooth curvature throughout the whole process, has high form-finding accuracy, and is suitable for intelligent construction of long-span cantilever curve beam structures.

Description

Technical Field

[0001] This invention relates to the fields of construction form control, parametric geometric modeling, and intelligent iterative optimization technology for large-span spatial steel structures and large-span suspended curved beam structures. Specifically, it relates to a reverse parametric form-finding method for zero-state installation alignment of a suspended beam under the condition that the geometric positions of both ends are constant. Background Technology

[0002] Large-span suspended curved components include structural forms such as stiffened cables, catenary beams, and large-span irregular curved beams. With their smooth catenary shape and excellent overall mechanical properties, they are widely used in engineering fields such as stadiums, exhibition buildings, and large-span landscape steel structures.

[0003] The design control line of the suspended beam is based on the stress-formed state under dead load, while the factory bending and on-site assembly must be carried out based on the zero-state installation line. If the designed formed line is directly used as the on-site installation benchmark, the structure will experience significant vertical deflection and horizontal displacement under its own weight and dead load. The final formed line will deviate from the preset architectural shape and structural internal force design values, which will not only damage the smoothness of the appearance, but also easily cause local stress concentration and stress exceeding the limit.

[0004] Currently, traditional zero-state form-finding methods in engineering often employ a local adjustment mode of pre-cambering at key points and reverse superposition of discrete node coordinates. This mode only performs independent displacement correction on local nodes and does not consider the continuous change in the curvature of the entire catenary of the suspended beam. This easily leads to a broken line shape and abrupt curvature changes, making it difficult to balance the smooth architectural form with the overall structural stress requirements. Existing conventional curve component form-finding methods are mostly suitable for simple curve shape control in segmented and discrete structures. They are difficult to adapt to the continuously changing catenary forming characteristics of the entire suspended beam, and directly applying them will not yield a smooth and even overall form-finding result.

[0005] Meanwhile, traditional form-finding processes rely heavily on engineering experience for manual calculations and repeated adjustments. The form evaluation dimensions are singular, lacking unified and quantifiable iterative convergence standards. This results in poor controllability of form-finding accuracy and low operational efficiency, making it difficult to meet the high-precision construction requirements of large-span cantilever beams. While existing intelligent optimization algorithms have been applied in structural form-finding, they have not yet been specifically adapted to the forming characteristics of continuous catenary curves in cantilever beams, and therefore cannot be directly applied to zero-state precise form-finding of such components. Summary of the Invention

[0006] This invention provides a zero-state form-finding method for suspended beams, which replaces the traditional local adjustment method of node coordinates with an overall curve parameter adjustment strategy. Through the technical route of benchmark chord layout, shape coefficient parameterized mapping, quantitative objective function evaluation, and finite element simulation closed-loop iteration, it solves the problems of curve discontinuity and curvature change caused by traditional local adjustment methods, thereby improving the zero-state form-finding accuracy and engineering adaptability of suspended beams.

[0007] To achieve the above objectives, the present invention adopts the following technical solution: a zero-state form-finding method for a suspended beam, comprising the following steps:

[0008] (1) Baseline construction and control point layout

[0009] Obtain the design target line shape of the suspended beam under constant load, and connect the two ends of the suspended beam to construct the reference chord line; evenly distribute several control points along the design target line shape for curve morphology tracking and quantitative evaluation of deviation.

[0010] (2) Establish a parametric mapping model for the shape coefficient of the suspended beam.

[0011] Define a dimensionless shape factor, and under the premise of keeping the geometric positions of the two ends of the suspension beam constant, establish a geometric mapping relationship between the reference chord point, the design target line point, and the zero-state installation line control point for the same curve parameter position; by continuously adjusting the shape factor value, achieve smooth and continuous adjustment of the entire suspension beam curve.

[0012] The shape factor ranges from 0 to 1; when the shape factor is 0, the zero-state installation line degenerates into the reference chord; when the shape factor is 1, the zero-state installation line coincides with the design target line.

[0013] (3) Construct the optimization objective function

[0014] Using the root mean square error of control point positions as the optimization objective, the deviation between the formed profile and the design target profile is quantitatively measured, with the minimum error value serving as the criterion for iterative optimization. The control point deviation is quantified using the root mean square method, and the relevant calculation expression is as follows:

[0015]

[0016] in, Objv To optimize the objective function, its value is equal to the root mean square error of the control point positions. RMSE η is the root mean square error of the spatial distance between each control point and the corresponding point of the design target alignment; N is the shape factor; P is the number of control points; η is the root mean square error of the spatial distance between each control point and the corresponding point of the design target alignment. z,i Let P be the coordinates of the i-th control point on the zero-state installation line; d,i Let be the coordinates of the i-th control point on the target design alignment. This objective function represents the average deviation of the corresponding alignment from the design alignment under load, given the shape factor η.

[0017] (4) Set the optimization iteration control parameters and convergence termination conditions

[0018] An initial value for the shape coefficient is set, and a preset convergence error limit is used as the quantization criterion for stopping the iteration; an intelligent optimization algorithm is used to automatically optimize the shape coefficient.

[0019] (5) Finite element simulation closed-loop iterative shape finding

[0020] The intelligent optimization algorithm generates the shape coefficient corresponding to the current iteration, substitutes it into the parametric mapping model for interpolation fitting, and generates the initial installation line of the cantilever beam with continuous and smooth curvature in the zero state. Based on the initial installation line, a finite element model is established, the structural self-weight and the design uniformly distributed dead load are applied, and the entire construction process is simulated to extract the position coordinates of each control point after deformation under the load. The position parameters of the control points extracted in the simulation are substituted into the objective function to calculate the root mean square error of the control point position, and the deviation result is fed back to the intelligent optimization algorithm. If the preset convergence error limit is not met, the algorithm automatically updates the shape coefficient and enters the next iteration until the convergence condition is met, and outputs the optimal shape coefficient.

[0021] (6) Generate zero-state installation line and output engineering application data.

[0022] By substituting the optimal shape coefficient into the parametric mapping model, the optimal zero-state installation line of the suspended beam with reasonable shape and continuous curvature is reconstructed; the zero-state curve morphological features, component processing arc parameters, and three-dimensional installation coordinates of control points are extracted to generate construction guidance data for component bending and cutting in the factory and on-site assembly and positioning.

[0023] (7) Optimization steps for predicting the optimal interval

[0024] Based on the ratio of the maximum vertical deformation of the suspended beam under dead load to the calculated span of the structure, and combined with engineering experience, the optimal search range of the shape coefficient is defined, thereby narrowing the optimization range of the algorithm, reducing the number of invalid iterations, and improving the efficiency of shape-finding convergence.

[0025] Compared with the prior art, the present invention has the following beneficial effects:

[0026] First, this invention constructs a parameterized mapping model of the shape coefficient of the cantilever beam, and replaces the traditional discrete node local correction method with overall curve parameter adjustment. Under the premise of keeping the geometric positions of the two ends constant, it ensures that the curvature of the cantilever beam is continuous and smooth throughout the entire process, without broken lines or abrupt changes in cusps, effectively avoiding the hidden danger of local stress concentration in the structure.

[0027] Second, this invention uses standardized root mean square error of position as a quantitative evaluation index to replace the traditional empirical judgment method, resulting in higher accuracy of shape finding results and unified evaluation standards; at the same time, relying on the overall shape adjustment method, it ensures that the curve shape is natural and smooth.

[0028] Third, this invention adopts a quantitative closed-loop iterative mechanism that couples finite element simulation with intelligent algorithms, which makes the convergence standard quantization controllable, the form finding efficiency high, and the results reproducible.

[0029] Fourth, the present invention adds a fast prediction optimization interval strategy based on the deformation span ratio, which can significantly reduce the search range of the algorithm, reduce invalid iterations, and significantly improve the shape-finding convergence efficiency.

[0030] Fifth, the scope of application of this invention covers various types of beam components with constant end-point geometric positions and smooth downward deflection curves under self-weight, such as stiffened cables, catenary beams, and curved suspended grid beams, making it highly versatile. The output zero-state control point coordinates and curve shape data can be directly used for bending and arc processing, pre-assembly, and on-site positioning and installation of steel structures in factories. Based on this data, construction guidance information such as component processing arc and curvature parameters can be further generated, making it highly practical for engineering and adaptable to the needs of intelligent construction and prefabricated construction of large-span irregular steel structures. Attached Figure Description

[0031] Figure 1 This is a flowchart illustrating the overall process of zero-state form finding for a suspended beam according to an embodiment of the present invention.

[0032] Figure 2 This is a schematic diagram illustrating the geometric relationship between the reference chord, the design target line, and the zero-state installation line in an embodiment of the present invention.

[0033] Figure 3 This is a schematic diagram illustrating the principle of shape coefficient parameterization mapping in this invention;

[0034] Figure 4 This is a schematic diagram comparing the changes in line shape and curvature between the present invention and traditional shape-finding methods;

[0035] Figure 5 This is a schematic diagram of the overall layout of the suspended grid roof structure according to an embodiment of the present invention.

[0036] Figure 1 Meaning of the symbol C: d To design the target linear shape, C b As the reference chord, C z The zero-state installation line shape is given by η, which is the shape factor, and RMSE is the root mean square error of the control point position. P z Reconstruct the coordinates of control points on the linear axis for the zero state. P z,i For the first i The coordinates of the control points P d The coordinates of the control points on the target alignment are determined.

[0037] Explanation of reference numerals in the attached drawings: 1. Suspended grid roof structure; 2. Suspended beam; 3. Bottom support; 4. Top support; 5. Purlin. Detailed Implementation

[0038] The following is in conjunction with the appendix Figures 1 to 5 The specific implementation of the zero-state form-finding method for a suspended beam of the present invention will be further described in detail.

[0039] This invention uses stiffening cables as a typical embodiment of a suspension beam, such as... Figure 1 As shown, the overall form-finding process of this invention follows the main line of "benchmark definition - parameter mapping - closed-loop iteration - result output", and specifically includes the following steps:

[0040] (1) Baseline construction and control point layout

[0041] Obtain the design target linearity C under constant load. d The reference chord C is constructed by connecting the endpoints of the two fixed supports. b Control points are evenly distributed along the design target line to serve as the basis for quantitative evaluation of line deviation.

[0042] (2) Establish a parameterized mapping model for shape coefficients

[0043] Define a dimensionless shape factor η and establish a reference chord C. b Design target linearity C d With zero-state installation of linear C z The parametric geometric mapping model among the three achieves continuous and smooth adjustment of the overall curve of the suspension beam by interpolating the coordinates of the same curve parameter positions.

[0044] (3) Construct the optimization objective function

[0045] The position of each control point after deformation under constant load is compared with the design target line C. d The root mean square value (RMSE) of the deviation between the two is used as the optimization objective function, and minimizing the RMSE is used as the criterion for terminating the iteration.

[0046] (4) Set the optimization iteration control parameters and convergence conditions

[0047] An initial value for the shape coefficient is set, and a preset convergence error limit is used as the quantitative criterion for terminating the iteration. An intelligent optimization algorithm is used to automatically optimize the shape coefficient.

[0048] (5) Finite element simulation closed-loop iterative shape finding

[0049] During the iteration process, the current η value is substituted into the mapping model to generate the zero-state installation linear C. zBased on this, a finite element model is established and the self-weight of the structure and uniformly distributed dead load are applied to carry out simulation of the entire construction process; the coordinates of each control point after deformation are extracted, the RMSE value is calculated and fed back to the optimization algorithm, forming a closed-loop iterative mechanism of "parameter generation → simulation verification → deviation feedback → parameter update"; if the preset convergence error limit is not met, the algorithm automatically updates the shape coefficient and enters the next round of iteration until the convergence condition is met and the optimal shape coefficient is output.

[0050] (6) Generate zero-state installation line and output engineering application data.

[0051] After iterative convergence, the optimal shape coefficient η is output, the optimal zero-state installation line of the cantilever beam is reconstructed, and the curvature features of the line and the three-dimensional coordinates of the control points are extracted to form engineering application data that can directly guide component processing and on-site installation.

[0052] (7) Optimization interval prediction auxiliary steps

[0053] Before iterative optimization, an auxiliary step can be added to estimate the optimization interval based on the ratio of the maximum vertical deformation under constant load to the calculated span of the structure. This will narrow the search range of the shape coefficient to the effective range in advance, thereby reducing the global optimization space of the intelligent algorithm, reducing the number of invalid iterations, and further improving the overall convergence efficiency.

[0054] This embodiment uses a stiffened cable structure with a calculated span of 59m, which is... Figure 5 The suspended beam shown in section number 2 has a cross-sectional specification of B660×325×15×15 mm. The elastic modulus of the steel is taken from the conventional value for building steel structures. The structure bears the self-weight of the component and a uniformly distributed dead load of 5 kN / m beam unit. 127 control points are evenly distributed along the line of the suspended beam to meet the requirements of accurate tracking of curve shape and curvature change and quantitative evaluation of deviation.

[0055] First, obtain the design target alignment of the stiffening cable under constant load, and connect the two endpoints to generate the reference chord. The geometric relationship between the three is as follows: Figure 2 As shown; establish a shape coefficient parameterized geometric mapping model, the mapping principle is as follows: Figure 3 As shown. An initial value for the shape factor is set, and a preset convergence error limit is established. In this embodiment, the convergence error limit is preferably set to 1×10⁻⁶. -2 In actual projects, adjustments can be made flexibly according to the required construction precision.

[0056] This embodiment uses the root mean square error of the control point positions as the objective function for iterative optimization, and the minimum error value is used as the criterion for terminating the iteration. The control point deviation is quantified and calculated using the root mean square method, and the relevant calculation expression is as follows:

[0057]

[0058] In the formula, Objv To optimize the objective function, its value is equal to the root mean square error of the control point locations. RMSE η is the root mean square error of the spatial distance between each control point and the corresponding point of the design target alignment; N is the shape factor; P is the number of critical nodes; η is the root mean square error of the spatial distance between each control point and the corresponding point of the design target alignment. z,i Let P be the coordinates of the i-th node on the linear axis in the zero state. d,i Let be the coordinates of the i-th node on the design alignment. This objective function represents the average deviation of the corresponding alignment from the design target alignment under load, given the shape factor η.

[0059] In this embodiment, the improved PGSA algorithm is preferred as the specific implementation of the intelligent optimization algorithm for iterative optimization of the shape coefficient. Alternatively, other global intelligent optimization algorithms such as particle swarm optimization or simulated annealing can be used as equivalent replacements.

[0060] After multiple rounds of closed-loop iterative simulation calculations, the algorithm reached the preset convergence criterion after 229 iterations, and the optimal shape coefficient was found to be 0.9893. Substituting the optimal shape coefficient into the mapping model, the zero-state installation line of the suspended beam was reconstructed, and the line exhibited a reasonable anti-arch shape, which can effectively offset the structural deflection deformation under constant load.

[0061] Simulation verification was performed by applying a constant load to the optimal zero-state installation alignment. The final alignment closely matched the design target alignment, with continuous and smooth curvature throughout the curve, without any broken lines or abrupt curvature changes. A comparison of the alignment smoothness and curvature differences between this invention and traditional form-finding methods is provided. Figure 4 As shown, the present invention can effectively avoid the problems of linear zigzag and curvature abrupt change caused by traditional local adjustments.

[0062] Deviation statistics were performed on 127 control points. After the structure was formed, the maximum geometric deviation of the control points was 10.933 mm, and the average deviation was 6.687 mm. The error level was much smaller than the structural span, which met the construction accuracy requirements of large-span cantilever beam projects.

[0063] Based on this, the ratio of the maximum vertical deformation under structural dead load to the calculated span of the structure can be used as a priori criterion. Combined with engineering experience, a reasonable search range for the optimal shape coefficient can be flexibly defined. By quantitatively predicting the overall deflection of the structure through the deformation span ratio, the upper and lower limits of the shape coefficient optimization can be constrained in advance, and the search range of the algorithm can be narrowed to the effective range. This can significantly reduce large-scale invalid iterations, reduce the number of iteration steps, and accelerate the convergence speed.

[0064] The specific deformation span ratio threshold and shape coefficient range can be selected based on the actual span of the project, the structural cross-section form, and the adaptability of the load conditions. This can provide a mature and traceable engineering judgment approach and reference for the initial selection of form-finding parameters for similar large-span suspension beams, stiffened cables, and catenary beams.

[0065] The overall structural layout described in this embodiment is as follows: Figure 5 As shown, it includes: a suspended grid roof structure 1; a suspended beam 2; a bottom support 3; a top support 4; and purlins 5. The obtained zero-state installation linear curve characteristics, three-dimensional coordinates of control points, and bending processing parameters can be directly used for bending of stiffening cable components, factory pre-assembly, and on-site construction and installation positioning, with a simple process and strong practicality.

[0066] The above description is merely a preferred embodiment of the present invention. The scope of protection of the present invention is not limited to the above embodiments. All technical solutions falling within the scope of the present invention's concept are within the scope of protection of the present invention. It should be noted that for those skilled in the art, any improvements and modifications made without departing from the principles of the present invention should also be considered within the scope of protection of the present invention.

Claims

1. A zero-state form-finding method for a suspended beam, characterized in that, Includes the following steps: (1) Obtain the design target line of the suspended beam under dead load, connect the endpoints of the fixed supports at both ends to construct the reference chord, and set up control points along the design target line; (2) Define a dimensionless shape coefficient and establish a parametric geometric mapping model between the reference chord, the design target line shape and the zero-state installation line shape; the parametric geometric mapping model realizes the overall shape adjustment of the suspension beam by continuously adjusting the shape coefficient, keeping the geometric position of the two ends of the suspension beam constant, and the curvature of the whole curve is continuous and smooth. (3) The root mean square error of the control point position is used as the optimization objective function, and the minimum of the objective function is used as the judgment objective for iterative optimization; (4) Set the initial value and convergence condition of the shape coefficient, and use an intelligent optimization algorithm to iteratively optimize the shape coefficient; (5) Substitute the shape coefficient of the current iteration into the parameterized geometric mapping model to generate the initial installation line shape in the zero state, establish a finite element model and apply a constant load to simulate the entire construction process, extract the coordinates of the control points after deformation and calculate the objective function, and feed the deviation results back to the optimization algorithm; (6) After iterating until the convergence condition is met, the optimal shape coefficient is output, the optimal zero-state installation line of the suspended beam is reconstructed, and the construction guidance data is output.

2. The zero-state form-finding method for a suspended beam according to claim 1, characterized in that, The shape factor ranges from 0 to 1; when the shape factor is 0, the zero-state installation line is the reference chord; when the shape factor is 1, the zero-state installation line coincides with the design target line.

3. The zero-state form-finding method for a suspended beam according to claim 1, characterized in that, Before iterative optimization, the optimal search range of the shape coefficient is defined based on the ratio of the maximum vertical deformation of the suspended beam under dead load to the calculated span, thus narrowing the optimization range of the algorithm.

4. The zero-state form-finding method for a suspended beam according to claim 1, characterized in that, The dead loads applied in the simulation of the entire construction process in step (5) include the self-weight of the structure and the uniformly distributed dead loads of the design.

5. The zero-state form-finding method for a suspended beam according to claim 1, characterized in that, The expression for calculating the root mean square error of the control point location in step (3) is as follows: ； in, Objv To optimize the objective function, its value is equal to the root mean square error of the control point locations. RMSE That is, the root mean square error of the spatial distance between each control point and the corresponding point of the design target alignment; η The shape factor; N Number of control points; P z,i For the first i The coordinates of each control point on the zero-state installation line; P d,i For the first i The coordinates of each control point on the target alignment.

6. The zero-state form-finding method for a suspended beam according to claim 1, characterized in that, The construction guidance data mentioned in step (6) includes the zero-state curve morphological characteristics, component processing arc parameters, and three-dimensional installation coordinates of control points.

7. The zero-state form-finding method for a suspended beam according to claim 1, characterized in that, The suspended beams include stiffened cables, catenary beams, large-span irregular curved beams, and curved surface suspended grid beams, which are beam-type components with constant geometric positions at both ends and smooth downward deflection curves under their own weight.

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