CN112906093A - Method for calculating variable cross-section height of arch ring of large-span arch bridge - Google Patents
Method for calculating variable cross-section height of arch ring of large-span arch bridge Download PDFInfo
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Abstract
The invention provides a method for calculating the variable cross-section height of an arch ring of a large-span arch bridge, which defines the cross-section height of the arch ring at any position of the arch ring of the large-span arch bridge to be calculated as follows: the ratio of the height of the arch crown section to the power of the cosine of the horizontal inclination angle of the arch axis of the section; the power coefficient can be uniquely determined according to the vault section height and the arch foot section height which are planned in advance; compared with the Litter formula, the calculation method is simpler, and the subsequent process of calculating the bridge parameters is simplified; and the height of the section of the arch ring at any position is calculated by using the formula provided by the invention, except that the heights of the sections of the arch foot and the arch crown are equal, the heights of the sections of the arch ring at other arbitrary positions calculated by the calculation method provided by the invention are all smaller than the height calculated by the Litter formula, so that the construction materials can be effectively reduced in the actual engineering, the budget can be reduced, and the economic benefit can be improved.
Description
Technical Field
The invention relates to the technical field of bridge construction, in particular to a method for calculating the height of a variable cross section of an arch ring of a large-span arch bridge.
Background
The large-span arch bridge is a bridge structure commonly used for spanning rivers and valleys, and hundreds of large-span reinforced concrete arch bridges, steel arch bridges and steel pipe concrete arch bridges are continuously built at home and abroad from the last 40 th century to the present. In order to make the distribution of the internal force of the arch ring section uniform, the arch ring section with the height varying along the span direction is often adopted, and the height of the variable section of the arch ring is often calculated by adopting a Litter formula (Ritter formula) in the engineering design.
The Lit formula assumes that the inertia moment of the arch crown and any section and the arch thickness coefficient are linearly changed, which causes the height of the section of the arch ring near the quarter span to be higher and the bending moment of the section of the arch foot to be larger. However, the quarter span is just positioned in the section positive and negative bending moment alternating region to mainly bear axial force, and the overlarge section not only increases the material consumption, but also increases the bending moment of the section of the arch springing.
The arch axis horizontal dip angle is related to the arch bridge sagittal cross ratio (f/L) and an arch axis coefficient m or an arch axis equation, and under the condition that the arch axis coefficient m or the arch axis equation is selected reasonably, the relationship between the height of any section of the arch ring and the horizontal dip angle can be directly established, so that the formula is simple, and the problems of the Litter formula can be avoided.
Disclosure of Invention
Aiming at the defects in the prior art, according to the embodiment of the invention, the calculation method for the variable cross-section height of the arch ring of the large-span arch bridge is provided, the calculation process for calculating the variable cross-section height of the arch ring at any position of the large-span arch bridge in the prior art is simplified, and the results of the embodiment prove that when the calculation method provided by the invention is compared with the calculation method of the Litter formula, the numerical values of the cross-section heights of the arch ring at other positions are small except that the heights of the arch foot and the arch crown are equal, so that the construction material and engineering investment are saved, and the construction requirements and the technical standards are met.
According to the embodiment of the invention, the method for calculating the variable section height of the arch ring of the large-span arch bridge comprises the following steps:
the arch ring section height h of the large-span arch bridge arch ring at any position needing to be calculated is defined as: height h of vault sectiondWith arch axis of the cross-sectionHorizontal inclination angleThe power of the cosine of (a); according to the pre-established height h of the cross section of the vaultdHeight h of arch foot sectionaDetermining a unique power coefficient d; drawing a formula:
drawing up the height h of the arch crown section of the large-span arch bridge to be calculateddArch foot section height ha;
Selecting the corresponding arch axis type of the large-span arch bridge to be calculated, and calculating the horizontal inclination angle of the arch axis of the section according to the arch axis equation corresponding to the selected arch axis typeAnd horizontal inclination of arch axis of arch foot section
The arch shaft horizontal inclination angle of the arch springing section obtained by calculationAnd dome cross-sectional height hdArch foot section height haSubstitution formula:
calculating to obtain a power coefficient d;
finally, the height h of the section of the vault is determineddHorizontal inclination angle of arch axis of cross sectionSubstituting the power coefficient d into the formula (1) to calculate to obtain the arch ring at any positionThe cross-sectional height h.
Preferably, when the arch axis is catenary, the horizontal inclination angle of the arch axis of the section isThe calculation formula of (2) is as follows:
when xi takes 1, the horizontal inclination angle of the arch axis of the arch springing section is calculated
Wherein: f is the calculated rise; m is the arch axis coefficient; k is a coefficient of the number of the elements,xi is a coefficient of the number of the magnetic poles,and L is the calculation span.
Preferably, when the arch axis type is a secondary or higher parabola, the arch foot is taken as the origin to establish a corresponding coordinate system, and the horizontal inclination angle of the arch axis of the section where the corresponding coordinate system is located isThe calculation formula of (2) is as follows:
in formula (4): n is more than or equal to 2; a is a coefficient related to f, L and n; f is the calculated rise; and L is the calculation span.
Selecting the specific value of n to obtain the horizontal inclination angle of the arch axis of the section where the corresponding parabola is locatedFor the arch foot section: taking x as 0, and then taking the x as 0,thenAccording to the corresponding coordinate system and the corresponding f and L values in the coordinate system, the arch axis horizontal inclination angle of the arch springing section is calculated
Preferably, when xi is 1, the compound is obtained by substituting formula (3) into formula (1);
to be provided withTaking logarithms of two sides of the formula (5) as a base to obtain a calculation expression of a power coefficient d:
wherein:the horizontal inclination angle of an arch shaft of the arch springing section is shown; h isdIs the height of the section of the vault; h isaIs the height of the arch springing section.
Preferably, when x is 0, i.e., ξ is 1, the horizontal inclination of the arch axis of the arch springing section is obtained by simplifying formula (4) in accordance with the specific value of n selectedThe formula (2) is obtained by substituting the formula (1) into the formula (1);
to be provided withTaking logarithms of two sides of the formula (4) as a base to obtain a calculation expression of a power coefficient d:
wherein:the horizontal inclination angle of an arch shaft of the arch springing section is shown; h isdIs the height of the section of the vault; h isaIs the height of the arch springing section.
Preferably, the type of arch ring variable cross-section comprises: rectangular solid sections, hollow box sections or truss-like sections.
The invention also discloses application of the method for calculating the variable cross-section height of the arch ring of the large-span arch bridge, and the method can be used for calculating the concrete-filled steel tube arch bridge, the stone arch bridge, the concrete arch bridge and the steel arch bridge according to the method for calculating the variable cross-section height of the arch ring of the large-span arch bridge.
Compared with the prior art, the invention has the following beneficial effects:
the invention discloses a method for calculating the height of an arch ring variable cross section at any position of a large-span arch bridge, which defines the height h of the arch ring cross section at any position of the arch ring of the large-span arch bridge to be calculated as follows: height h of vault sectiondHorizontal inclination angle of arch axis with sectionThe power of the cosine of (a); drawing up a formula to establish the height h of the variable section of the arch ring at any position and the horizontal inclination angle of the arch shaft of the sectionHeight h of vault sectiondCompared with a Lett formula, the calculation method is simpler, and the subsequent process of calculating the bridge parameters is simplified; and calculating arbitrary positions using the formula provided by the inventionThe height h of the section of the arch ring is equal to the height of the section of the arch foot and the arch crown, and the heights of the sections of the arch ring at any positions are calculated according to the calculation method provided by the invention and are all smaller than the height calculated by the Lit formula, so that the construction materials can be effectively saved in the actual engineering, the engineering budget is reduced, the economic benefit is improved, the actual bridge forming parameters calculated by the calculation formula provided by the invention meet the construction and safety standards of the large-span arch bridge through calculation and demonstration, and the authenticity and the feasibility of the invention are further proved.
Drawings
FIG. 1 is a schematic diagram of a computational model of formula (1) of the present invention;
FIG. 2 is a horizontal inclination angle of the arch axis of the cross section of the present inventionThe computational model map of (1);
FIG. 3 is a schematic diagram of a coordinate system model of quadratic and higher-order parabolas in the present invention;
FIG. 4 is a line drawing of the height values of each section calculated by the actual design drawing of the ebony river bridge, the Litter formula and the calculation formula provided by the invention;
fig. 5 is a line graph of the difference between the height of the variable cross section of the smoked plum river bridge calculated by the calculation formula provided by the present invention and the litter formula and the actual design drawing;
FIG. 6 is an arch ring elevation of the actual design drawing of the ebony river bridge;
FIG. 7 is a line drawing of the upper and lower chord axial forces of the arch rib under the action of the dead weight of the finished bridge, according to the actual design drawing of the dark plum river bridge, the Litter formula and the calculation formula provided by the invention;
fig. 8 is a comparison graph of the absolute value difference of the axial force of the ebony river bridge under the action of self weight after the bridge is formed by the actual design drawing, the litter formula and the calculation formula provided by the invention.
Detailed Description
In order to further clarify the objects, technical solutions and advantages of the present invention, the present invention will be described in further detail below with reference to the accompanying drawings and examples.
The invention aims to provide a novel method for calculating the variable cross-section height of an arch ring of a large-span arch bridge, which is used for replacing the traditional method for calculating the variable cross-section height of the arch ring of the large-span arch bridge by using a Litter formula (Ritter formula) or a fitting method.
The internal force of the section of the arch ring of the large-span arch bridge is mainly controlled by the self weight, usually, the arch crown bears positive bending moment, the arch foot bears negative bending moment, the positive bending moment and the negative bending moment alternately appear in a quarter-span section, and the absolute value of the bending moment in the section is relatively small. The cross section height is drawn up by adopting the Litter formula, and the cross section height of the quarter-span section is usually larger.
Therefore, the invention provides a novel method for calculating the variable section height of the arch ring of the large-span arch bridge, and the height of any section is defined as the ratio of the height of the arch section to the power of the cosine of the horizontal inclination angle of the arch axis of the section. According to the heights of the sections of the arch crown and the arch foot, a unique power coefficient can be determined, so that the height of the variable section of the arch ring at the corresponding position can be calculated, the height of the section of the quarter-span section calculated by the method is smaller than the calculation result of the Lett formula, the building materials can be saved, and the calculation process can be simplified.
In the invention, the most common large-span variable-height truss type concrete-filled steel tube arch bridge is taken as a calculation description;
the large-span concrete-filled steel tube arch bridge adopts a variable cross-section form, and the cross-section height is also determined by a Litter formula. When designing the steel pipe arch, the height h of the arch crown and arch foot section is determineddAnd haThe arch crown section inertia moment I under the condition of neglecting the influence of the web rod on the rigidity of the arch rib sectiondAnd arch foot section moment of inertia IaComprises the following steps:
in the formula: i isDIs the moment of inertia of the upper/lower chord tube to the self-form axis, ASThe upper/lower chord tube cross-sectional area.
Compared with the moment of inertia generated by the area distance of the chord tube to the whole section, the moment of inertia value of the chord tube per se is much smaller, and the arch thickness coefficient n and the center height h of any section can be determined by substituting the moment of inertia into the Litter formula under the condition of neglecting the influence:
in the litter formula, the arch thickness coefficient n is calculated as:
Substituting the formulas (6) and (7) into the formula (8) to obtain an expression of the arch thickness coefficient n of the large-span concrete-filled steel tube arch bridge:
in the Lit formula, the calculation formula of the arch ring variable section height h of the large-span concrete-filled steel tube arch bridge is as follows:
K1defined as the section height coefficient of the litter equation.
Due to the arch shaft with any sectionHorizontal inclination angleHas a cosine value of less than 1.0 (except for the dome cross-section), and thus hasFor bridges with small and medium spans, even if a section with a high height is adopted, the height change is not too large, so that the bridge can be determined by the formula (10).
The cross-sectional height of the large-span arch bridge is determined by the Lit formula, althoughThe cosine value of (A) is compared with the value after being squaredThe cosine value of (a) is large, but the requirement of the height change of the cross section cannot be met in some cases, and the cosine value needs to be determined by using the arch thickness coefficient n. As can be seen from the formulas (9) and (10), the smaller n is, the larger the change of the section height is, and the method is suitable for the requirement of the change of the section height of the steel pipe concrete arch bridge.
Further analysis shows that the height of the cross section obtained by the Litter formula changes more gradually in the whole bridge span range, and the height of the arch foot section changes more sharply due to the combined action of the position of the cross section and the horizontal inclination angle.
For a steel pipe concrete arch bridge with a truss type section, the chord pipe mainly bears axial force, the horizontal thrust is gradually increased from the arch top to the arch springing, and the section height obtained according to the Lit formula is usually larger.
In the litter formula, the calculation formula of the section moment of inertia is as follows:
As can be seen from the formulas (10) and (12), the calculation formulas of the section inertia moment I and the variable section height h both comprise the horizontal inclination angle of the arch axis of any section of the archThe inclination angle is related to the vector-span ratio and the arch axis coefficient, and also implies the relation between the arch ring axial force and the bending moment, so that the arch ring section height h at any position can be represented by the following formula under the condition that the arch axis coefficient is reasonably selected:
wherein: h isdIs the height of the section of the vault;the horizontal inclination angle of the arch axis with any section is shown; d is a power coefficient.
The specific position relationship of each parameter in the real bridge in the formula (1) is shown in fig. 1: wherein 1, the section of the arch ring at any position; 2. A dome cross section; 3. a spur section; 4. the horizontal inclination angle of the arch axis of the section is; 5. an arch axis;
selecting the corresponding arch axis type of the large-span arch bridge to be calculated, and calculating the horizontal inclination angle of the arch axis of the section to be calculated according to the corresponding arch axis equation of the selected arch axis typeAnd horizontal inclination of arch axis of arch foot section
When the arch axis is catenary, the horizontal inclination angle of the arch axis of the section isThe calculation formula of (2) is as follows:
when xi takes 1, the horizontal inclination angle of the arch axis of the arch springing section is calculated
When xi is 1, the formula (2) is substituted into the formula (1);
to be provided withTaking logarithms at two ends of the formula (4) as a base to obtain an expression of a power coefficient d:
the arch shaft horizontal inclination angle of the arch springing section obtained by calculationAnd dome cross-sectional height hdArch foot section height haSubstituting in an expression (2), and calculating to obtain a power coefficient d;
finally, the height h of the section of the vault is determineddHorizontal inclination angle of arch axis of cross sectionThe power coefficient d is substituted into the formula (1) to calculate, and the cross-sectional height h of an arbitrary position is obtained.
When the arch axis type is a quadratic or high-order parabola, the arch springing is taken as the origin, and a corresponding coordinate system is established, wherein the corresponding coordinate system is shown in fig. 3: its parabolic equation f (x) can be expressed by the following equation:
f(x)=anxn+an-1xn-1+...+a1x1 (13)
wherein: in the formula (13), n is more than or equal to 2;
then the horizontal inclination angle of the arch axis of the cross sectionThe calculation formula of (2) is as follows:
in formula (4): n is more than or equal to 2; a is a coefficient related to f, L and n; f is the calculated rise; and L is the calculation span.
Selecting a specific value of n to obtain a specific power number of a parabola, and calculating by taking a quadratic parabola as an example;
taking n as 2, substituting in formula (13), the expression of the quadratic parabolic equation is obtained as follows:
then the horizontal inclination angle of the arch axis of the cross sectionThe calculation formula of (2) is as follows:
for a pair-arch foot section: substituting x ═ 0, i.e., ξ ═ 1, into formula (15) yields:
namely, the horizontal inclination angle of the arch axis of the arch springing section can be calculated
Then, formula (16) is substituted into formula (1) to obtain:
to be provided withTaking logarithms of two sides of the formula (4) as a base to obtain a calculation expression of a power coefficient d:
wherein:the horizontal inclination angle of an arch shaft of the arch springing section is shown; h isdIs the height of the section of the vault; h isaIs the height of the arch springing section.
The above calculation is exemplified by using the arch axis as a quadratic parabola, and when the arch axis type is a high-order parabola of which the parabola coefficient is greater than that of the quadratic parabola, the calculation can also be deduced according to the above calculation process.
When the variable cross-section height at any position is determined by adopting a Lit formula, the same design method is adopted, namely, the formula (10) is replaced by the condition that the arch springsection height and the arch springsection arch axis horizontal inclination angle are known to determine the arch thickness coefficient n, so that the method is obtained:
or:
note that the terms on the right of (5) and equation (18) are the same, then the reduction is:
the arch thickness coefficient n obtained by continuing the simplified expression (12) is as follows:
the formula (20) establishes the relationship between the formula and the Litter formula in the invention; it should be noted that, although the relationship between the two is established, the height of the arch ring section h at any position is smaller than the height calculated by the litter formula except that the heights of the arch springing and the arch crown section are equal.
The authenticity and feasibility of the calculation method for the variable cross-section height of the arch ring of the large-span arch bridge provided by the invention are proved by carrying out real bridge substitution calculation.
Selecting a solid bridge: dark plum river bridge
Introduction to real bridge: the dark plum river bridge is a steel tube concrete overhead arch bridge which spans canyons and is marked by TJ12 from Guiyang to Huangping expressway, the calculated span L is 300m, the calculated rise f is 60m, the rise ratio f/L is 1/5, and the arch axis coefficient m is 1.55. The arch axis adopts a catenary, the half main arch ring adopts a 3-truss space truss structure, the cross section adopts a constant width and variable height, the height is changed from 5m of an arch crown to 9m of arch feet (from middle to middle), the transverse distance between each arch rib of the half main arch ring is 5m, the distance between two arch ribs is 7m, a cross beam is arranged between the arch ribs of the half main arch ring, and a K-shaped support is arranged between the two bridges. The upper and lower chord tubes of the arch rib adopt two steel tubes (arranged according to the stress requirement) of phi 1200 multiplied by 24mm and phi 1200 multiplied by 35mm, which are both made of Q345D material, and C55 self-compacting compensation shrinkage concrete is poured into the tubes.
Calculating parameters: h isd/ha5/9; calculated by using an arch shaft horizontal inclination angle formulaSubstituted typeSubstituting d into 2.09951004And n is 0.40835859. The design drawing adopts a fitting method to determine the height of the arch ring, and the data in the table 1 are the height of each section of the arch ring calculated by an actual design drawing, a Lit formula and a calculation formula provided by the invention (abbreviated as the invention formula in the table 1); FIG. 4 is a line graph of the height values of each section calculated from the actual design drawing, the Litter equation and the calculation equation provided by the present invention; fig. 5 is a line graph of the difference between the height of the variable cross section of the smoked plum river bridge calculated by the calculation formula provided by the present invention and the litter formula and the actual design drawing; as is clear from fig. 4 and 5, the arch heights (except for the arch crown and the arch foot) of the cross sections calculated by the calculation formula provided by the invention are smaller than those calculated by the design drawing and the litter formula, and are particularly obvious in the l/4 section.
TABLE 1 calculation of arch height (unit: m) by means of three arch height calculation modes of ebony river bridge
A full-bridge model is established by using Midas/Civil, and the construction stage of concrete pouring in a pipe is considered, so that the internal force of the chord pipe under the action of self weight of the finished bridge and the internal force difference between different design methods are obtained by using three arch height calculation methods as shown in the figures 7 and 8: as can be seen from fig. 7 and 8, the single tube axial forces calculated by the three methods have little difference; in the arch foot area with the largest eccentricity, the upper chord tube axial force obtained by the calculation method of the invention formula is increased, the lower chord tube axial force is reduced, the upper chord axial force obtained by the calculation of the arch height by the Lett formula is increased by 7.00 percent, and the lower chord is reduced by 1.71 percent; due to the change of the chord tube axial force obtained by the formula, the upper chord tube axial force and the lower chord tube axial force in the arch springing area are more uniform, the eccentricity of the section near the arch springing is smaller, and the mechanical property of the concrete filled steel tube is favorably exerted.
The table shows the maximum and minimum web member stresses of the dark plum river bridge, and the results obtained by comparing the three calculation modes are not very different and are far smaller than the designed strength value of Q345D steel. In the table, "-" represents compressive stress, and positive values represent tensile stress.
Stress | Design paper | Lie formula | Invention formula |
Maximum of | 73.9 | 72.3 | 70.6 |
Minimum size | -92.4 | -90.5 | -92.6 |
TABLE 2 stress values of web members of large bridge of dark plum river
By
The method has the advantages that the arch height is drawn by the calculation formula provided by the invention, and the dynamic characteristic difference of the full bridge calculated by the design drawing and the drawing drawn by the Litter formula is small. By
TABLE 3 comparison of the dynamic characteristics of the ebony river bridge (unit: Hz)
It can be seen that the stability coefficients are not very different, and are all larger than 7.8, which meets the requirement that the specification is larger than 4.0.
TABLE 3 comparison of the dynamic characteristics of the ebony river bridge (unit: Hz)
Instability mode | Design paper | Lie formula | Invention formula |
Out-of-first-order instability | 7.882917 | 7.888777 | 7.913372 |
First order in-plane instability | 20.158485 | 20.599416 | 19.60792 |
TABLE 4 dark plum river bridge flexion analysis
By
As can be seen from the table, the material consumption of the arch height arch ring material determined by the calculation formula provided by the invention is obviously less, the steel materials are reduced by 65.2t and 61.2t respectively compared with the design drawing and the Litter formula, and the economic benefit is obvious.
Weight (D) | Design paper | Lie formula | Invention formula |
t | 7135.3 | 7131.3 | 7070.1 |
TABLE 5 dosage of steel material for arch ring of Meihe bridge (unit: t)
Finally, the above embodiments are only used for illustrating the technical solutions of the present invention and not for limiting, although the present invention has been described in detail with reference to the preferred embodiments, it should be understood by those skilled in the art that modifications or equivalent substitutions can be made on the technical solutions of the present invention without departing from the spirit and scope of the technical solutions of the present invention, and all of them should be covered in the claims of the present invention.
Claims (7)
1. A method for calculating the variable cross-section height of an arch ring of a large-span arch bridge is characterized by comprising the following steps:
the arch ring section height h of the large-span arch bridge arch ring at any position needing to be calculated is defined as: height h of vault sectiondHorizontal inclination angle of arch axis with cross sectionThe power of the cosine of (a); according to the pre-established height h of the cross section of the vaultdHeight h of arch foot sectionaDetermining a unique power coefficient d; drawing a formula:
drawing up the height h of the arch crown section of the large-span arch bridge to be calculateddArch foot section height ha;
Selecting the corresponding arch axis type of the large-span arch bridge to be calculated, and calculating the horizontal inclination angle of the arch axis of the section according to the arch axis equation corresponding to the selected arch axis typeAnd horizontal inclination of arch axis of arch foot section
The arch shaft horizontal inclination angle of the arch springing section obtained by calculationAnd dome cross-sectional height hdArch foot section height haSubstituting into a formula:
calculating to obtain a power coefficient d;
2. The method for calculating the variable cross-section height of the arch ring of the large-span arch bridge as claimed in claim 1, wherein: when the arch axis is catenary, the horizontal inclination angle of the arch axis of the section isThe calculation formula of (2) is as follows:
when xi takes 1, the horizontal inclination angle of the arch axis of the arch springing section is calculated
3. The method for calculating the variable cross-section height of the arch ring of the large-span arch bridge as claimed in claim 1, wherein: when the arch axis type is a secondary or higher parabola, the arch foot is taken as the origin to establish a corresponding coordinate system, and the horizontal inclination angle of the arch axis of the section where the corresponding coordinate system is locatedThe calculation formula of (2) is as follows:
in formula (4): n is more than or equal to 2; a is a coefficient related to f, L and n; f is the calculated rise; and L is the calculation span.
Selecting the specific value of n to obtain the horizontal inclination angle of the arch axis of the section where the corresponding parabola is locatedFor the arch foot section: if x is 0, thenAccording to the corresponding coordinate system and the corresponding f and L values in the coordinate system, the arch axis horizontal inclination angle of the arch springing section is calculated
4. The method for calculating the variable cross-section height of the arch ring of the large-span arch bridge as claimed in claim 2, wherein: when xi takes 1, substituting formula (3) into formula (1) to obtain;
to be provided withTaking logarithms of two sides of the formula (5) as a base to obtain a calculation expression of a power coefficient d:
5. A method for calculating the variable cross-sectional height of the arch ring of the large-span arch bridge as claimed in claim 3, wherein: when x is equal to 0, according to the specific value of the selected n, the arch shaft horizontal inclination angle of the arch springing section is obtained by substituting formula (4) for simplificationThe formula (2) is obtained by substituting the formula (1) into the formula (1);
to be provided withTaking logarithms of two sides of the formula (4) as a base to obtain a calculation expression of a power coefficient d:
6. The method for calculating the variable cross-section height of the arch ring of the large-span arch bridge as claimed in claim 1, wherein: the type of the arch ring variable cross section comprises: rectangular solid sections, hollow box sections or truss-like lattice sections.
7. Use of a method according to claim 1 for calculating the variable section height of an arch ring of a long-span arch bridge in the construction of concrete-filled steel tube arch bridges, stone arch bridges, concrete arch bridges or steel arch bridges.
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