Calculation method suitable for hinged shaft rotation angle of two-hinged arch bridge

Abstract

The invention discloses a calculation method suitable for a hinged shaft rotation angle of a two-hinged arch bridge. The calculation method comprises the steps of arch section bending moment calculation of the two-hinged arch bridge under the vehicle load effect and arch section bending moment calculation of the two-hinged arch bridge under the overall heating and cooling effect. The hinge shaft rotation angle analytical expression of the two-hinged arch bridge under the action of the vehicle load and the overall temperature change is deduced, the method is suitable for calculation of the hinge shaft rotation range in the preliminary design stage of the two-hinged arch bridge, the hinge shaft rotation range can be rapidly and simply determined, and design of the hinge shaft of the two-hinged arch bridge is guided.

Description

Technical Field

[0001] The present invention relates to the field of hinge axis rotation angle calculation of two-hinged arch bridges, and in particular to a calculation method for the hinge axis rotation angle of two-hinged arch bridges, which can calculate the limit rotation angle of the hinge axis under the action of vehicle load and temperature change. Background Art

[0002] At present, most of the arch bridges built in the world adopt hingeless arches. However, the hingeless arch has a high degree of static indeterminacy. Under the influence of factors such as temperature change, concrete shrinkage creep and foundation displacement, it will produce large additional stress, which is not conducive to the stress and safety performance of the arch bridge structure. Figure 1 As shown. The two-hinged arch bridge can reduce the additional internal forces caused by factors such as foundation displacement, temperature change, concrete shrinkage and creep, and the hinge axis of the arch rib at the arch foot can rotate in the plane, which can release the in-plane bending moment generated by the arch rib during the operation stage, so as to improve the stress of the bridge structure and improve the economic and safety performance. However, the two-hinged arch bridge needs to be equipped with a hinge axis at the end, which will produce an in-plane rotation angle under the influence of factors such as vehicle load and ambient temperature changes. The rotation range of the hinge axis needs to be clarified during the design as the basis for the design of the hinge axis device. Therefore, it is necessary to propose a calculation method for the hinge axis rotation angle of the two-hinged arch bridge to guide the design of this type of bridge. Summary of the invention

[0003] In order to solve the problems existing in the above-mentioned prior art, the purpose of the present invention is to provide a calculation method for the hinge axis rotation angle of a two-hinged arch bridge, which can calculate the limit rotation angle of the hinge axis under the action of vehicle load and temperature change.

[0004] In order to achieve the above-mentioned invention objectives, the present invention provides the following technical solutions: a calculation method suitable for the hinge axis rotation angle of a two-hinged arch bridge, including the calculation of the arch section bending moment of the two-hinged arch bridge under vehicle load and the calculation of the arch section bending moment of the two-hinged arch bridge under the action of overall temperature rise and fall.

[0005] Furthermore, the calculation of the bending moment of the arch section of the two-hinged arch bridge under the vehicle load includes:

[0006] 1) Calculate the bending moment of each arch rib section along the arch axis of a two-hinged arch bridge under the action of concentrated force P, using the force method;

[0007] 2) Calculate the bending moment M of any arch rib section under the action of concentrated force P p , assuming that the arch axis is the catenary equation:

[0008]

[0009] Where y is the vertical distance from any section of the arch axis to the arch top; ξ is the ratio of the horizontal coordinate x of any arch section to half of the span of the arch bridge L / 2; f is the sagittal height of the arch axis; m is the arch axis coefficient;

[0010]

[0011] 3) Assuming that the concentrated force P acts on the left half span, the bending moment of the arch rib section of the left half span, M pl (x), is calculated as follows:

[0012]

[0013] in, x 0 is the horizontal distance between the concentrated force P and the dome; 0 For x 0 and The ratio of; x is the distance from any section of the arch rib to the arch crown;

[0014] Bending moment of the arch rib section of the right half span, M pr (x), is calculated as follows:

[0015]

[0016] 4) Remove the horizontal constraint of the right arch hinge and apply a unit force 1 in the horizontal direction. Calculate the arch axis section bending moment M under the unit force. 1 :

[0017]

[0018] 5) Calculated according to the force method:

[0019]

[0020]

[0021] Among them, δ 11 is the displacement of the position under the action of unit force; Δ 1p is the displacement at the unit force position under the action of concentrated force P; y′ is the first derivative of the arch axis equation y(x); EI is the bending stiffness of the arch section;

[0022] 6) Establish the force balance equation:

[0023] δ 11 X 1 +Δ 1p =0 (8)

[0024] By solving the equilibrium equation (8), we can obtain X 1 ;

[0025] 7) Calculate the bending moment distribution at any cross-section position of the arch axis under the action of concentrated force P:

[0026] Left half span bending moment M l (x):

[0027]

[0028] Right half span bending moment M r (x):

[0029]

[0030] 8) When a unit rotation is applied to the left side of the two-hinged arch basic system, the arch axis bending moment is calculated as follows:

[0031] Left half span bending moment M 2l (x):

[0032]

[0033] Right half span bending moment M 2r (x):

[0034]

[0035] 9) Calculate the rotation angle θ of the left arch foot of the two-hinged arch left :

[0036]

[0037]

[0038] 10) When a unit rotation is applied to the right side of the two-hinged arch basic system, the arch axis bending moment is calculated as follows:

[0039] Left half span bending moment M 3l (x):

[0040]

[0041] Right half span bending moment M 3r (x):

[0042]

[0043] 11) Calculate the rotation angle θ of the right arch foot of the two-hinged arch right :

[0044]

[0045] Furthermore, the calculation of the bending moment of the arch section of the two-hinged arch bridge under the action of overall temperature rise and fall includes:

[0046] 1) The original arch axis equation y is shown in formula (1). Assuming that the arch axis equation after the overall temperature rise Δt is y up , the arch axis equation after the overall temperature drop Δt is y dn , then the arch axis equation after heating satisfies:

[0047]

[0048] The arch axis equation after cooling satisfies:

[0049]

[0050] Where α is the linear expansion coefficient of the arch ring material, y′, y u ' p , y′ dn represents the first derivative of the arch axis equation;

[0051] 2) By solving formulas (17) and (18), the derivative y of the arch axis equation after overall temperature rise and temperature drop Δt is obtained respectively: u ' p and y′ dn as follows:

[0052]

[0053] Among them, ln() represents the natural logarithm; sinh() represents the hyperbolic sine function;

[0054] 3) By differentiating formula (1), we get the derivative y′ of the original arch axis equation:

[0055]

[0056] 4) Change in the angle of the arch foot after overall heating, Δθ 1 for:

[0057] Δθ 1 =arctan(y′ up )-arctan(y′) (22)

[0058] The change of the rotation angle at the arch foot after overall cooling, Δθ 2 for:

[0059] Δθ 2 =arctan(y′ dn )-arctan(y′) (23).

[0060] Compared with the prior art, the present invention has at least the following beneficial effects:

[0061] (1) The prior art lacks a simple calculation method specifically for the hinge axis rotation angle of a two-hinged arch bridge.

[0062] (2) The present invention derives an analytical expression for the hinge angle of a two-hinged arch bridge under the action of vehicle load and overall temperature change, which is suitable for calculating the hinge rotation range in the preliminary design stage of a two-hinged arch bridge. It can quickly and easily determine the hinge rotation range and guide the design of the hinge axis of the two-hinged arch bridge. BRIEF DESCRIPTION OF THE DRAWINGS

[0063] The drawings described herein are used to provide a further understanding of the present application and constitute a part of the present application. The illustrative embodiments of the present application and their descriptions are used to explain the present application and do not constitute an improper limitation on the present application. In the drawings:

[0064] Figure 1 It is a schematic diagram of a hingeless arch and a two-hinged arch;

[0065] Figure 2 It is a schematic diagram of a two-hinged arch structure (acting concentrated force);

[0066] Figure 3 It is the basic system for force method calculation structure;

[0067] Figure 4 M pl (x), M pr (x) Calculation diagram;

[0068] Figure 5 M 1 (x) Calculation diagram;

[0069] Figure 6 M 2l (x), M 2r (x) Calculation diagram;

[0070] Figure 7 M 3l (x), M 3r (x) Calculation diagram;

[0071] Figure 8 is the rotation angle θ left The influence line of

[0072] Fig. 9 is the rotation angle θ left The most unfavorable lane load arrangement for the positive turning angle (single lane standard value);

[0073] Fig.10 is the rotation angle θ left The most unfavorable lane load arrangement for the reverse turning angle (single lane standard value);

[0074] Fig.11 Schematic diagram of the change of the arch axis after overall heating and cooling. DETAILED DESCRIPTION

[0075] In order to facilitate ordinary technicians in the field to understand and implement the present invention, the present invention is further described in detail below in conjunction with the accompanying drawings and implementation examples. It should be understood that the implementation examples described herein are only used to illustrate and explain the present invention, and are not used to limit the present invention.

[0076] A calculation method for the hinge axis rotation angle of a two-hinged arch bridge, comprising:

[0077] 1. Calculation method of arch section bending moment of two-hinged arch bridge under vehicle load:

[0078] 1) Calculate the two-hinged arch bridge under the action of concentrated force P, such as Figure 2 As shown in the figure, the bending moment of each arch rib section along the arch axis (span is L). The force method is used for calculation. At this time, the calculation diagram of the structure is as follows: Figure 3 shown.

[0079] 2) Calculate the bending moment M of any arch rib section under the action of concentrated force P p , the calculation diagram is as follows Figure 4 As shown, assuming that the arch axis is the catenary equation:

[0080]

[0081] Where y is the vertical distance from any section of the arch axis to the arch top; ξ is the ratio of the horizontal coordinate x of any arch section to half of the span of the arch bridge L / 2; f is the sagittal height of the arch axis; m is the arch axis coefficient;

[0082]

[0083] 3) Assuming that the concentrated force P acts on the left half span, the bending moment of the arch rib section of the left half span, M pl (x), is calculated as follows:

[0084]

[0085] in, x 0 is the horizontal distance between the concentrated force P and the dome; 0 For x 0 and The ratio of; x is the distance from any section of the arch rib to the arch crown;

[0086] Bending moment of the arch rib section of the right half span, M pr (x), is calculated as follows:

[0087]

[0088] 4) If Figure 5 As shown, the horizontal constraint of the right arch hinge is removed, and a unit force 1 is applied in the horizontal direction. The bending moment M of the arch axis section under the unit force is calculated.1 as follows:

[0089]

[0090] 5) Calculated according to the force method:

[0091]

[0092] Among them, δ 11 is the displacement of the position under the action of unit force; Δ 1p is the displacement at the unit force position under the action of concentrated force P; y′ is the first derivative of the arch axis equation y(x); EI is the bending stiffness of the arch section;

[0093] 6) Establish the force balance equation as follows:

[0094] δ 11 X 1 +Δ 1p =0 (8)

[0095] Among them, X 1 for Figure 5 The magnitude of the unit force in the horizontal direction.

[0096] By solving the equilibrium equation (8), we can obtain X 1 .

[0097] 7) Calculate the bending moment distribution at any cross-section position of the arch axis under the action of concentrated force P:

[0098] Left half span bending moment M l (x):

[0099]

[0100] Right half span bending moment M r (x):

[0101]

[0102] 8) Further, a unit rotation is applied to the left side of the two-hinged arch basic system, as shown in the attached figure. Figure 6 As shown, the arch axis bending moment is calculated as follows:

[0103] Left half span bending moment M 2l (x):

[0104]

[0105] Right half span bending moment M 2r (x):

[0106]

[0107] 9) Further, calculate the rotation angle θ of the left arch foot of the two-hinged arch left as follows:

[0108]

[0109] 10) Further, a unit rotation is applied to the right side of the two-hinged arch basic system, as shown in the attached figure. Figure 7 As shown, the arch axis bending moment is calculated as follows:

[0110] Left half span bending moment M 3l (x):

[0111]

[0112] Right half span bending moment M 3r (x):

[0113]

[0114] 11) Further, calculate the rotation angle θ of the right arch foot of the two-hinged arch right as follows:

[0115]

[0116] 12) Since the two-hinged arch structure in the example is a symmetrical structure, the rotation angle θ of the left arch foot is left Take for example, calculate the range of the turning angle under the vehicle load. The structural parameters are shown in the following table.

[0117] Arrow height f(m) Span L(m) Arch axis coefficient m <![CDATA[Flexural rigidity EI (N·mm 2 )]]> 45 250 1.5 <![CDATA[9.7741×10 17 ]]>

[0118] 13) First, according to formula (13), the position of the unit force (10 kN concentrated force) along the arch axis is changed to obtain θ left The influence line is shown in the attached Figure 8 shown.

[0119] 14) Calculate the turning angle θ under the action of a single lane left The lane load is determined in accordance with the General Specification for Highway Bridge and Culvert Design (JTG-D60-2015). The standard value of concentrated force P k =360kN, standard value of uniformly distributed load q k =10.5kN / m. Figure 8 We can get: θ left The most unfavorable lane load arrangement for the positive turning angle is shown in the attached figure. Fig. 9 As shown, θ left The most unfavorable lane load arrangement for the reverse turning angle is shown in the attached figure. Fig.10 shown.

[0120] 15) By the attached Fig. 9It can be calculated (load × sum of the vertical coordinates of the influence line) to obtain, θ left The maximum positive turning angle under the most unfavorable arrangement of lane load is 0.041°; Fig.10 It can be calculated that θ left The maximum reverse rotation angle under the most unfavorable arrangement of lane load is -0.04°. Therefore, the rotation angle of the left arch foot of the two-hinged arch under the standard value of a single lane ranges from [-0.04° to 0.041°].

[0121] 2. Calculation method of arch section bending moment of two-hinged arch bridge under the action of overall temperature rise and fall:

[0122] 1) As attached Fig.11 As shown in the figure, the original arch axis equation y is still as shown in formula (1). Assuming that the arch axis equation after the overall temperature rise Δt is y up , the arch axis equation after the overall temperature drop Δt is y dn , then the arch axis equation after heating satisfies:

[0123]

[0124] The arch axis equation after cooling satisfies:

[0125]

[0126] Where α is the linear expansion coefficient of the arch ring material, y′, y u ' p , y′ dn Represents the first derivative of the arch axis equation.

[0127] 2) By solving formulas (17) and (18), the derivative y of the arch axis equation after overall temperature rise and temperature drop Δt is obtained respectively: u ' p and y′ dn as follows:

[0128]

[0129] Among them, ln() represents the natural logarithm; sinh() represents the hyperbolic sine function.

[0130] 3) By differentiating formula (1), the derivative y′ of the original arch axis equation can be obtained as follows:

[0131]

[0132] 4) Change in the angle of the arch foot after overall heating, Δθ 1 for:

[0133] Δθ 1 =arctan(y u 'p )-arctan(y′) (22)

[0134] The change of the rotation angle at the arch foot after overall cooling, Δθ 2 for:

[0135] Δθ 2 =arctan(y′ dn )-arctan(y ′ ) (twenty three)

[0136] 5) According to the sagittal height f = 45m, span L = 250m, arch axis coefficient m = 1.5, overall temperature rise and fall Δt = ±20℃, and the linear expansion coefficient of concrete material α = 1×10 -5 , the calculated rotation angle range of the arch foot position is: ±0.0055°.

[0137] The above embodiments are only used to illustrate the present invention and are not intended to limit the technical solutions described in the present invention. Although the present invention has been described in detail with reference to the above embodiments, the present invention is not limited to the above specific implementation methods. Therefore, any modification or equivalent replacement of the present invention; and all technical solutions and improvements thereof that do not depart from the spirit and scope of the invention are included in the scope of the claims of the present invention.

Claims

1. A method for calculating the hinge angle of a two-hinged arch bridge, characterized in that: It includes the calculation of the arch section bending moment of a two-hinged arch bridge under vehicle load and the calculation of the arch section bending moment of a two-hinged arch bridge under the action of overall temperature rise and fall.

2. The calculation method for the hinge axis rotation angle of a two-hinged arch bridge according to claim 1 is characterized in that: The calculation of the arch section bending moment of the two-hinged arch bridge under vehicle load includes: 1) Calculate the bending moment of each arch rib section along the arch axis of a two-hinged arch bridge under the action of concentrated force P, using the force method; 2) Calculate the bending moment M of any arch rib section under the action of concentrated force P p , assuming that the arch axis is the catenary equation: Where y is the vertical distance from any section of the arch axis to the arch top; ξ is the ratio of the horizontal coordinate x of any arch section to half of the span of the arch bridge L / 2; f is the sagittal height of the arch axis; m is the arch axis coefficient; 3) Assuming that the concentrated force P acts on the left half span, the bending moment of the arch rib section of the left half span, M pl (x), is calculated as follows: in, x0 is the horizontal distance between the concentrated force P and the dome; ξ0 is the distance between x0 and The ratio of; x is the distance from any section of the arch rib to the arch crown; Bending moment of the arch rib section of the right half span, M pr (x), is calculated as follows: 4) Remove the horizontal constraint of the right arch hinge and apply a unit force 1 in the horizontal direction. Calculate the arch axis section bending moment M1 under the unit force: 5) Calculated according to the force method: Among them, δ 11 is the displacement of the position under the action of unit force; Δ 1p is the displacement at the unit force position under the action of concentrated force P; y′ is the first derivative of the arch axis equation y(x); EI is the bending stiffness of the arch section; 6) Establish the force balance equation: d 11 X1+D 1p =0 (8) By solving the equilibrium equation (8), we obtain X1; 7) Calculate the bending moment distribution at any cross-section position of the arch axis under the action of concentrated force P: Left half span bending moment M l (x): Right half span bending moment M r (x): 8) When a unit rotation is applied to the left side of the two-hinged arch basic system, the arch axis bending moment is calculated as follows: Left half span bending moment M 2l (x): Right half span bending moment M 2r (x): 9) Calculate the rotation angle θ of the left arch foot of the two-hinged arch left : 10) When a unit rotation is applied to the right side of the two-hinged arch basic system, the arch axis bending moment is calculated as follows: Left half span bending moment M 3l (x): Right half span bending moment M 3r (x): 11) Calculate the rotation angle θ of the right arch foot of the two-hinged arch right :

3. The calculation method for the hinge axis rotation angle of a two-hinged arch bridge according to claim 1 is characterized in that: The calculation of the arch section bending moment of the two-hinged arch bridge under the action of overall temperature rise and fall includes: 1) The original arch axis equation y is shown in formula (1). Assuming that the arch axis equation after the overall temperature rise Δt is y up , the arch axis equation after the overall temperature drop Δt is y dn , then the arch axis equation after heating satisfies: The arch axis equation after cooling satisfies: Where α is the linear expansion coefficient of the arch ring material, y′, y u ' p , y′ dn represents the first derivative of the arch axis equation; 2) By solving formulas (17) and (18), the derivative y of the arch axis equation after overall temperature rise and temperature drop Δt is obtained respectively: u ' p and y′ dn as follows: Among them, ln() represents the natural logarithm; sinh() represents the hyperbolic sine function; 3) By differentiating formula (1), we get the derivative y′ of the original arch axis equation: 4) The change in the rotation angle at the arch foot after overall heating, Δθ1, is: Δθ1=arctan(and u ′ p )-arctan(y′) (22) The change in the rotation angle at the arch foot after overall cooling, Δθ2, is: Δθ2=arctan(y′ dn )-arctan(y′) (23).

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