CN110793737A - Beam bridge damage detection method based on elastic constraint supporting beam deflection influence line - Google Patents

Beam bridge damage detection method based on elastic constraint supporting beam deflection influence line Download PDF

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CN110793737A
CN110793737A CN201911030935.3A CN201911030935A CN110793737A CN 110793737 A CN110793737 A CN 110793737A CN 201911030935 A CN201911030935 A CN 201911030935A CN 110793737 A CN110793737 A CN 110793737A
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damage
influence line
deflection influence
deflection
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CN110793737B (en
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周宇
李宁
吴侃
黄浩
刘金澳
万杨旭
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Anhui Jianzhu University
Anhui University of Architecture
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01M—TESTING STATIC OR DYNAMIC BALANCE OF MACHINES OR STRUCTURES; TESTING OF STRUCTURES OR APPARATUS, NOT OTHERWISE PROVIDED FOR
    • G01M5/00—Investigating the elasticity of structures, e.g. deflection of bridges or air-craft wings
    • G01M5/0008—Investigating the elasticity of structures, e.g. deflection of bridges or air-craft wings of bridges
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01M—TESTING STATIC OR DYNAMIC BALANCE OF MACHINES OR STRUCTURES; TESTING OF STRUCTURES OR APPARATUS, NOT OTHERWISE PROVIDED FOR
    • G01M5/00—Investigating the elasticity of structures, e.g. deflection of bridges or air-craft wings
    • G01M5/0041—Investigating the elasticity of structures, e.g. deflection of bridges or air-craft wings by determining deflection or stress
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01M—TESTING STATIC OR DYNAMIC BALANCE OF MACHINES OR STRUCTURES; TESTING OF STRUCTURES OR APPARATUS, NOT OTHERWISE PROVIDED FOR
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Abstract

The invention discloses a beam bridge damage detection method based on elastic constraint supporting beam deflection influence lines, which comprises the following steps: model establishment, deflection influence line analysis and damage identification. In the invention, for the defects of the assembled railway bridge model, an elastic support boundary beam model with rotation constraint is established by combining the elastic modulus of the initial material of the main beam and the uncertainty of the section size, and is used for simulating the main beam structure of the assembled railway bridge; by deducing an analytic expression of the model deflection influence line and combining finite element example analysis, the proposed deflection influence line method is used for damage identification research of the fabricated beam structure, the influence of the influence line measuring point position, the local damage position and degree and the test noise on the identification result is researched, and a method reference and research thought is provided for damage identification and service performance evaluation of the existing fabricated railway bridge structure; the method can be used for identifying the occurrence of local damage of the elastic supporting beam with rotation constraint, accurately positioning the damage position and realizing accurate quantitative damage.

Description

Beam bridge damage detection method based on elastic constraint supporting beam deflection influence line
Technical Field
The invention belongs to the field of bridge damage detection, and particularly relates to a bridge damage detection method based on an elastic constraint supporting beam deflection influence line.
Background
In recent years, fabricated railroad bridges are widely adopted due to the convenience in pouring and mounting, the actual supporting conditions of the fabricated railroad bridges are different from theoretical boundaries to a certain extent, finite element software is difficult to simulate interfaces at prefabricated assembly nodes, and large deviation is introduced for theoretical calculation and actual engineering. Uncertain factors such as ambiguity of construction materials, randomness of construction errors, discreteness of section sizes and the like are not negligible, so that an initial model and boundary conditions of the railway bridge are not ideal, ideal hinging and rigid connection are difficult to achieve in engineering, and therefore difficulty is brought to accurate analysis and damage identification of an assembled railway bridge structure, and meanwhile, when the railway bridge is subjected to simulation analysis and damage identification, interference brought to a final result by uncertainty of the initial model and the boundary is not negligible. Compared with most damage identification methods based on dynamic characteristics, the damage identification method based on dynamic characteristics relies on parameter identification results, the influence line theory and the mobile loading mode have the characteristic of global loading single-point response, and related damage identification research has been developed initially. The influence of beam end vertical elastic constraint and uncertain factors of a structural initial model is not considered in the conventional method, and the influence of a non-ideal supporting boundary of an assembled beam model is not considered in related researches, so that the influence is brought to the bridge damage detection precision.
Disclosure of Invention
The invention aims to overcome the problems in the prior art and provides a beam bridge damage detection method based on an elastic constraint supporting beam deflection influence line.
In order to achieve the technical purpose and achieve the technical effect, the invention is realized by the following technical scheme:
a beam bridge damage detection method based on elastic constraint supporting beam deflection influence lines comprises the following steps:
the method comprises the following steps: the spring is used for replacing an ideal chain rod of a simply supported beam structure, and the rotation of the cross section of the beam end is restrained by the spring to obtain a rotation restrained elastic supporting beam model which is used for simulating a railway bridge structure;
step two: springs which are parallel to the normal direction of the cross section of the beam end are arranged on the upper edge and the lower edge of the cross section of the beam end and are used for restraining the rotation of the fulcrum;
step three: carrying out influence line loading on the rotation constraint elastic supporting beam, and introducing discrete parameters which are subjected to lognormal distribution;
step four: sliding supports are arranged at two ends of the beam, and deformation of the main beam caused by translational coupling and rotational coupling of the cross section of the beam end is ignored;
step five: a spring parallel to the normal direction of the cross section of the beam end is equivalent to an elastic rotation constraint effect, and a formula is established according to internal force balance and deformation conditions;
step six: calculating the deflection of any section of the main beam by graph multiplication, adding virtual unit force at the position of the section by utilizing the virtual work principle, and establishing a section bending moment expression in a virtual state;
step seven: calculating a deflection influence line analytic expression of any beam end cross section, and performing piecewise graph multiplication according to the main beam rigidity and the inflection point of the bending moment graph;
step eight: and calculating and analyzing the difference value of the deflection influence lines before and after the beam end cross section is damaged, wherein when the structure is not damaged, the difference value of the deflection influence lines is zero, and judging whether the structure is damaged or not according to the difference value of the deflection influence lines.
Further, the detection method also introduces a damage identification index of the deflection influence line difference curvature:
when x' is equal to [0, d- ξ ] ∪ [ d + ξ, l ], DILDC is equal to 0;
when x' is e [ d- ξ, d + ξ]When the temperature of the water is higher than the set temperature,
Figure BDA0002250122810000021
when the moving concentrated force is positioned outside the main beam damage area, the deflection influence line difference curvature is zero, and when the moving concentrated force enters the damage area, the deflection influence line difference curvature is not zero, so that the damage position of the fabricated beam is judged.
Further, when the section bending rigidity is determined, the differential curvature of the deflection influence line is calculated according to the actually measured deflection influence line, so that the section bending rigidity degradation degree is obtained, and the bending rigidity residual rate and the damage degree are defined.
Further, the specific steps for judging the structural damage are as follows:
s1: loading an influence line on a newly-built assembled railway bridge structure to obtain an initial deflection influence line;
s2: under the condition that the loading step length, the moving load size and the measuring point position are the same, carrying out deflection influence line test on the girder structure in service period to obtain a deflection influence line in service period;
s3: carrying out smoothing treatment on discrete data of the deflection influence line and the initial deflection influence line in the service life through the sliding average;
s4: calculating the difference value of the deflection influence lines, and qualitatively judging whether damage occurs or not;
s5: and (4) carrying out damage positioning and quantification by calculating the difference curvature of the deflection influence lines.
The invention has the beneficial effects that:
1. in the invention, for the defects of the assembled railway bridge model, an elastic support boundary beam model with rotation constraint is established by combining the elastic modulus of the initial material of the main beam and the uncertainty of the section size, and is used for simulating the main beam structure of the assembled railway bridge; by deducing an analytic expression of the model deflection influence line and combining finite element example analysis, the proposed deflection influence line method is used for damage identification research of the fabricated beam structure, the influence of the influence line measuring point position, the local damage position and degree and the test noise on the identification result is researched, and a method reference and research thought is provided for damage identification and service performance evaluation of the existing fabricated railway bridge structure;
2. the lifted elastic support boundary girder model with the rotation constraint is reasonable and feasible in static analysis of the fabricated railway bridge structure, and can well reflect the non-ideal support conditions of the actual fabricated boundary;
3. the method can be used for identifying the occurrence of local damage of the elastic supporting beam with rotation constraint, accurately positioning the damage position and realizing accurate quantitative damage.
4. The method is known by combining field actual measurement deflection influence line data and model example analysis, and the strength of the continuity of the support in the continuous beam bridge can be evaluated based on the actual measurement influence line, so that reference is provided for quantitatively evaluating the bearing capacity of the fabricated railway bridge.
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The accompanying drawings, which are included to provide a further understanding of the invention and are incorporated in and constitute a part of this application, illustrate embodiment(s) of the invention and together with the description serve to explain the invention without limiting the invention. In the drawings:
FIG. 1 is a diagram of an elastically constrained boundary beam model;
FIG. 2 is a schematic view of a deformed exploded view of a resilient support beam with rotational restraint;
FIG. 3 is a schematic representation of the rigid displacement of the main beam;
FIG. 4 is a graphical representation of the deflection of the main beam under the force of the movement;
fig. 5 is a schematic view of the bending and deflection of the main beam in a virtual state.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
A beam bridge damage detection method based on elastic constraint supporting beam deflection influence lines comprises the following steps:
the method comprises the following steps: the method comprises the steps that ideal chain rods of a simply supported beam structure are replaced by springs, and rotation of the cross section of the beam end is restrained by the springs to obtain a rotation restraint elastic supporting beam model with quantifiable rigidity, which is used for simulating a railway bridge structure and is shown in figure 1;
step two: springs parallel to the normal direction of the cross section are arranged on the upper edge and the lower edge of the section of the beam end and used for restraining the rotation of a fulcrum, and the rigidity is K1Two parallel springs restrain the rotation of the A end section of the beam and the rigidity K2The two parallel springs restrain the section of the B end of the beam to rotate; the spring is used for replacing an ideal simply supported beam vertical chain rod and simulating vertical deformation of an assembled beam end caused by construction defects, stress or damage, and the vertical elastic supporting rigidity of each of the two supporting points is k1And k2And is not zero;
thirdly, loading an influence line on the rotationally constrained elastic supporting beam, wherein a variable x' is used for describing the position of a loading concentrated force, the distance D from the central point D of a damaged area to a beam end, the length of a local damaged area is 2 ξ, and the distance C from a deflection measuring point C of any section to the beam end A is C;
step four: the uneven vertical deformation of the pivot causes the additional internal force of the main beam, and in order to facilitate the analysis of the stress and deformation of the assembled beam, sliding supports are arranged at two ends of the beam, and the deformation of the main beam caused by the translational coupling and the rotational coupling of the section of the beam end is not considered; the additional bending of the actual fabricated beam caused by uneven settlement is assumed to be smaller than the bending effect generated by concentrated force, namely the bending stress of the main beam caused by uneven settlement is ignored; in addition, the axial deformation caused by non-load factors is not considered in the model analysis process;
step five: the spring parallel to the normal direction of the cross section of the beam end is equivalent to an elastic rotation restriction effect, for exampleIf the elastic internal force of the horizontal spring is f, the expansion deformation of the two horizontal springs is α, and the section corner of the A end is
Figure BDA0002250122810000061
The height h of the cross section of the beam end and the rotation restraint internal force (bending moment) of the beam end are MAThe rotational stiffness of the A end is KaEstablishing a formula according to the internal force balance and the deformation condition:
f＝α×K1
MA+f·h＝0
the relation (2) of the horizontal spring stiffness and the rotation constraint stiffness is simplified and obtained:
Figure BDA0002250122810000063
under the action of the concentrated force F ═ P (N), the elastic support and the rotation constraint are stressed and deformed, the beam body generates bending deformation, and the deformation decomposition of the rotation constraint elastic support beam is schematically shown in figure 2;
the deflection D of any section of the model can be obtained by analysisC(x') can be decomposed into two portions of translational spring with limited rigidity and rotary spring with random rigidity, in which the former is formed from main beam rigid body deflection to make main beam section displacement DAB(x'), the latter causing a section deflection D due to the superposition of the effects of the flexural stiffness of the main beam and the rotational constraintC(x′)；
Under the action of the moving concentration force, the vertical elastic deformation of the cast-in-place fulcrum of the main beam structure is the uneven settlement of the main beam, and the rigid body displacement of the main beam is set as DAB(x'), as shown in FIG. 3.
Along with the change of x', the vertical elastic counter force of the beam end support can be calculated, and the elastic deformation is respectively dA(x') and dB(x'), easily obtainable DAB(x'), as shown in formula (3):
Figure BDA0002250122810000064
according to formula (3), DAB(x ') is in linear relation with the moving load position x', and when the spring stiffness of the vertical elastic support is infinite, the rigid body deflection D of the main beam isAB(x') is zero, i.e., is an ideal support; it is worth mentioning that
Figure BDA0002250122810000065
And when the vertical supporting spring is in a small rigidity, the main beam is uniformly settled, and the situation that the rigidity of the vertical supporting spring is too low is not considered.
Let the deflection of the section of the main beam C caused by the superposition of rotation constraint and self bending effect be dC(x') replacing elastic rotation constraint of the beam end with two equivalent bending moments M based on the force method principleAAnd MBEstablishing a displacement balance equation as shown in formula (4):
Figure BDA0002250122810000071
in the formula ofijIs the i-position corner caused by j-position unit force, and has delta according to the mutual theorem of displacementAB＝δBA，ΔiPThe angle is a corner at the position i caused by an external force P; coefficient of equation deltaij、ΔiPAll can be obtained by solving through graph multiplication, as shown in formulas (5) to (9):
Figure BDA0002250122810000073
Figure BDA0002250122810000075
Figure BDA0002250122810000076
analytical formulae (5) to (9), coefficient. deltaijThe flexibility main coefficient and the flexibility auxiliary coefficient are not changed along with the change of the external force, and are only related to the damage position, the damage range and the damage degree; z in the formulae (7) and (8)1、Z2、Z3、Z4、Z5、Z6Expressed by formulae (10) to (15):
Figure BDA0002250122810000077
Figure BDA0002250122810000081
Figure BDA0002250122810000082
Figure BDA0002250122810000084
as shown in FIG. 4, when x' is located in the lesion region [ d- ξ, d + ξ ] as shown in the analysis of the formulae (11) and (14)]When P is located in the lesion field, the free term ΔiPThe position x' of the concentration force is in a cubic function relation; as is apparent from the formulae (10), (12), (13) and (15), when the movement concentration force P acts outside the lesion field, ΔiPIs in a linear function relation with x'; substituting formulas (5) - (9) into formula (4), simplifying the structure to obtain the influence line M of the internal force of the elastic rotation constraintA(x′)、MB(x′)：
Figure BDA0002250122810000085
Analytical formulae (5) to (9) and (16) to (17) analysis, MA(x′)、MB(x ') the denominator does not contain the concentration position quantity x ', the number of times x ' in the numerator and the number of times deltaiPThe consistency is achieved; it is easy to know that when the moving concentrated force is positioned in the damage area, the fulcrum rotation constraint internal force and x 'are in a cubic function relationship, and when the moving concentrated force is positioned outside the damage area, the fulcrum rotation constraint bending moment and x' are in a linear function relationship; the obtained rotation constraint bending moment MA(x′)、MB(x'), an expression of the influence line of the bending moment of any section of the main beam in the figure 3 can be established:
Figure BDA0002250122810000087
step six: according to the analysis formula (18), M (x) and x' are related to MA(x′)、MB(x ') is consistent with the relationship x'; calculating the deflection of any section C of the assembled main beam by using graph multiplication, adding virtual unit force at the calculated section by using a virtual work principle, and establishing a bending moment expression of each section of the basic beam structure in a virtual state as shown in a formula (19) by using a deflection curve in the virtual state as shown in a graph 5:
Figure BDA0002250122810000091
step seven: multiplying the graphs of the formula (18) and the formula (19), calculating a deflection influence line analytical formula of any section C, and performing subsection graph multiplication according to the main beam rigidity and the inflection point of the bending moment graph;
when x' is equal to 0, c,
Figure BDA0002250122810000092
when x' is ∈ [ c, d- ξ ],
Figure BDA0002250122810000093
when x' ∈ [ d- ξ, d + ξ ],
Figure BDA0002250122810000094
when x' ∈ [ d + ξ, l ],
Figure BDA0002250122810000101
dC(x') M contained in the formulaA(x′)、MB(x') are all piecewise functions, and the deflection influence line is also piecewise function and is limited to space and is not expanded any more; the piecewise functions (20) - (23) are respectively superposed with the formula (3) to obtain the influence line D of the deflection of the main beam of the fabricated railroad bridgeC(x') see formula:
Figure BDA0002250122810000102
step eight: calculating the difference value of the Deflection Influence Lines (DILD) of the sections C before and after the damage and analyzing the difference value:
when x' is equal to 0, d- ξ,
Figure BDA0002250122810000103
when x' ∈ [ d- ξ, d + ξ ],
when x' ∈ [ d + ξ, l ],
analyzing the formulas (25) - (27), wherein DILDs in the formulas (25) - (27) are zero when the structure is not damaged, so that the DILDs can be used for judging whether the structure is damaged or not, when the moving concentration force is positioned outside a damaged area, the DILDs are linear functions of x ', when x ' belongs to [ d- ξ, d + ξ ], the DILDs are cubic functions of x ', the structure is damaged, the continuity of adjacent section materials and sections is changed, the continuity change of a line difference curve is influenced, the curvature of the DILD curve is calculated to position the damage due to the fact that the curvature can amplify the singular point of the curve change rate, the denominator of a curvature formula is approximate to 1, the curvature of the curve is approximately equal to the second derivative of the curve in the engineering, and the formula is shown as follows:
Figure BDA0002250122810000111
further, by introducing a damage identification index of a Deflection Influence Line Difference Curvature (DILDC) of the main beam of the fabricated railroad bridge, the damage identification indexes can be obtained by analysis formulas (25) to (27):
when x' is equal to 0, d- ξ to ∪ d + ξ, l,
DILDC＝0 (29)
when x' ∈ [ d- ξ, d + ξ ],
Figure BDA0002250122810000112
as is apparent from the formulas (29) to (30), when the moving concentration force is positioned outside the main beam damage area, the deflection influence line difference curvature is zero, and when the moving concentration force enters the damage area, the deflection influence line difference curvature is not zero, so that the damage position of the assembled beam is judged.
When the bending rigidity of the section is determined, EI delta (x) is a known quantity, and the differential curvature of the deflection influence line is obtained according to the actually measured deflection influence line so as to obtain the degradation degree of the bending rigidity of the section;
the Residual bending stiffness (EI Residual (DILDC), EIR (DILDC)) is defined and shown in the following formula:
the degree of Damage (Damage extend, DE) can be defined by equation (31):
Figure BDA0002250122810000114
the method for identifying the damage based on the deflection influence line comprises the following steps:
s1: loading an influence line on the newly-built assembled railway bridge structure to obtain an initial DIL;
s2: under the condition that the loading step length, the moving load size and the measuring point position are the same, carrying out DIL test on the girder structure in the service period to obtain DIL in the service period;
s3: smoothing discrete data of the DIL in the active period and the initial DIL through the moving average;
s4: calculating and solving DILD, and qualitatively judging whether damage occurs or not;
s5: lesion localization and quantification was performed by calculating DILDC.
In the description herein, references to the description of "one embodiment," "an example," "a specific example" or the like are intended to mean that a particular feature, structure, material, or characteristic described in connection with the embodiment or example is included in at least one embodiment or example of the invention. In this specification, the schematic representations of the terms used above do not necessarily refer to the same embodiment or example. Furthermore, the particular features, structures, materials, or characteristics described may be combined in any suitable manner in any one or more embodiments or examples.
The foregoing shows and describes the general principles, essential features, and advantages of the invention. It will be understood by those skilled in the art that the present invention is not limited to the embodiments described above, which are described in the specification and illustrated only to illustrate the principle of the present invention, but that various changes and modifications may be made therein without departing from the spirit and scope of the present invention, which fall within the scope of the invention as claimed.

Claims (4)

1. A beam bridge damage detection method based on elastic constraint supporting beam deflection influence lines is characterized by comprising the following steps: the method comprises the following steps:
the method comprises the following steps: the spring is used for replacing an ideal chain rod of a simply supported beam structure, and the rotation of the cross section of the beam end is restrained by the spring to obtain a rotation restrained elastic supporting beam model which is used for simulating a railway bridge structure;
step two: springs which are parallel to the normal direction of the cross section of the beam end are arranged on the upper edge and the lower edge of the cross section of the beam end and are used for restraining the rotation of the fulcrum;
step three: carrying out influence line loading on the rotation constraint elastic supporting beam, and introducing discrete parameters which are subjected to lognormal distribution;
step four: sliding supports are arranged at two ends of the beam, and deformation of the main beam caused by translational coupling and rotational coupling of the cross section of the beam end is ignored;
step five: a spring parallel to the normal direction of the cross section of the beam end is equivalent to an elastic rotation constraint effect, and a formula is established according to internal force balance and deformation conditions;
step six: calculating the deflection of any section of the main beam by graph multiplication, adding virtual unit force at the position of the section by utilizing the virtual work principle, and establishing a section bending moment expression in a virtual state;
step seven: calculating a deflection influence line analytic expression of any beam end cross section, and performing piecewise graph multiplication according to the main beam rigidity and the inflection point of the bending moment graph;
step eight: and calculating and analyzing the difference value of the deflection influence lines before and after the beam end cross section is damaged, wherein when the structure is not damaged, the difference value of the deflection influence lines is zero, and judging whether the structure is damaged or not according to the difference value of the deflection influence lines.
2. The beam bridge damage detection method based on the elastic constraint supporting beam deflection influence line as claimed in claim 1, characterized in that: the detection method also introduces a damage identification index of the deflection influence line difference curvature:
when x' is equal to [0, d- ξ ] ∪ [ d + ξ, l ], DILDC is equal to 0;
when x' is e [ d- ξ, d + ξ]When the temperature of the water is higher than the set temperature,
Figure FDA0002250122800000021
when the moving concentrated force is positioned outside the main beam damage area, the deflection influence line difference curvature is zero, and when the moving concentrated force enters the damage area, the deflection influence line difference curvature is not zero, so that the damage position of the fabricated beam is judged.
3. The beam bridge damage detection method based on the elastic constraint supporting beam deflection influence line as claimed in claim 1, characterized in that: when the bending rigidity of the section is determined, calculating the differential curvature of the deflection influence line according to the actually measured deflection influence line, thereby obtaining the degradation degree of the bending rigidity of the section, and defining the residual rate and the damage degree of the bending rigidity.
4. The beam bridge damage detection method based on the elastic constraint supporting beam deflection influence line as claimed in claim 1, characterized in that: the specific steps for judging the structural damage are as follows:
s1: loading an influence line on a newly-built assembled railway bridge structure to obtain an initial deflection influence line;
s2: under the condition that the loading step length, the moving load size and the measuring point position are the same, carrying out deflection influence line test on the girder structure in service period to obtain a deflection influence line in service period;
s3: carrying out smoothing treatment on discrete data of the deflection influence line and the initial deflection influence line in the service life through the sliding average;
s4: calculating the difference value of the deflection influence lines, and qualitatively judging whether damage occurs or not;
s5: and (4) carrying out damage positioning and quantification by calculating the difference curvature of the deflection influence lines.
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CN111337212A (en) * 2020-03-31 2020-06-26 广西交科集团有限公司 Method for measuring maximum deflection of simply supported beam based on corner under unknown state
CN111460717A (en) * 2020-03-31 2020-07-28 广西交科集团有限公司 Influence line-based method for counting passing equivalent times of overloaded vehicle of simply supported beam bridge
CN111581723A (en) * 2020-04-30 2020-08-25 哈尔滨工业大学 Method for quickly extracting influence line of vehicle slowly passing through midspan displacement of simply supported beam bridge
CN111879626A (en) * 2020-07-31 2020-11-03 广西交科集团有限公司 Method for testing actual rigidity static force of bridge rubber support
CN116579193A (en) * 2022-12-20 2023-08-11 湘潭大学 Deflection strain energy equivalent beam structure local equivalent stiffness calculation method
CN116839659A (en) * 2023-06-21 2023-10-03 中交三航局第三工程有限公司 Method and system for identifying damage of girder of bridge girder erection machine

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CN111337212A (en) * 2020-03-31 2020-06-26 广西交科集团有限公司 Method for measuring maximum deflection of simply supported beam based on corner under unknown state
CN111460717A (en) * 2020-03-31 2020-07-28 广西交科集团有限公司 Influence line-based method for counting passing equivalent times of overloaded vehicle of simply supported beam bridge
CN111460717B (en) * 2020-03-31 2022-04-08 广西交科集团有限公司 Influence line-based method for counting passing equivalent times of overloaded vehicle of simply supported beam bridge
CN111581723A (en) * 2020-04-30 2020-08-25 哈尔滨工业大学 Method for quickly extracting influence line of vehicle slowly passing through midspan displacement of simply supported beam bridge
CN111581723B (en) * 2020-04-30 2020-11-24 哈尔滨工业大学 Method for quickly extracting influence line of vehicle slowly passing through midspan displacement of simply supported beam bridge
CN111879626A (en) * 2020-07-31 2020-11-03 广西交科集团有限公司 Method for testing actual rigidity static force of bridge rubber support
CN111879626B (en) * 2020-07-31 2022-05-31 广西交科集团有限公司 Method for testing actual rigidity static force of bridge rubber support
CN116579193A (en) * 2022-12-20 2023-08-11 湘潭大学 Deflection strain energy equivalent beam structure local equivalent stiffness calculation method
CN116579193B (en) * 2022-12-20 2024-03-19 湘潭大学 Deflection strain energy equivalent beam structure local equivalent stiffness calculation method
CN116839659A (en) * 2023-06-21 2023-10-03 中交三航局第三工程有限公司 Method and system for identifying damage of girder of bridge girder erection machine

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