CN111337212A - Method for measuring maximum deflection of simply supported beam based on corner under unknown state - Google Patents
Method for measuring maximum deflection of simply supported beam based on corner under unknown state Download PDFInfo
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Abstract
The invention discloses a method for measuring the maximum deflection of a simply supported beam based on a corner in an unknown state, which comprises the steps of firstly segmenting a beam body, distributing an inclination angle sensor at the segmentation position, testing the corner of the beam structure under the action of known load, identifying the bending rigidity of each segment of the beam body according to a corner test value, and then obtaining the equivalent bending rigidity of the beam body based on the principle of equal bending strain energy; then, based on the corner test value, determining the specific position of the maximum deflection according to the relation between the beam corner curve and the structural bending moment; and finally, calculating and determining the maximum deflection value of the beam structure by adopting a virtual cantilever beam method. The measuring method provided by the invention can realize the measurement of the maximum deflection value of the simply supported beam structure under the condition of unknown initial state through the structure corner test, not only solves the limitation of the conventional deflection test method, but also better fits the engineering practice of unknown initial state of the in-service beam structure.
Description
Technical Field
The invention belongs to the technical field of civil engineering, relates to a beam structure, and particularly relates to a method for measuring the maximum deflection of a simply supported beam based on a corner in an unknown state.
Background
The simple beam structure is one of the most common structures in civil engineering, and the structural materials of the simple beam structure are (prestressed) reinforced concrete, steel, reinforced concrete composite materials and the like. In bridge engineering, the simple supported beam is widely used. Because of new types of steel-concrete combination and the like and the application of new materials, the simply supported beam bridge is lighter and lighter, and the span is larger and larger. In some cases, the structural rigidity exceeds the strength, and becomes a key for controlling the design. Especially for railway bridges, the rigidity requirement on the beam structure is stricter in order to ensure the safety of train operation, meet the comfort level of passenger train riding, ensure the stability of tracks and other equipment on the bridge and the like. For a beam structure, the deflection refers to the size of longitudinal displacement of a centroid of a certain section in the direction perpendicular to an axis under the action of load, and is visual representation of the overall vertical rigidity of the bridge, and the maximum deflection value is most representative.
And for the maximum deflection of the simply supported beam structure determined under the known structure state information, solving according to basic mechanics knowledge. However, for an active simple beam structure, its initial state information is unknown. In this case, it is very difficult to determine the location and magnitude of maximum deflection under load. The existing method generally assumes that the structure is not damaged, namely, the solution is carried out under the design state information, so that the difference between the actual and theoretical conditions cannot be solved. In addition, in the method for measuring the deflection, a total station method, a precision level method, a dial gauge method, a displacement gauge method, a GPS method, a communicating tube method, a high-definition camera method, and the like are mainly used at present. However, the above methods have their limitations. For example, the precision level method has slow testing speed, needs reference points and cannot carry out real-time online long-term observation; the hundred (thousand) minute meter method needs to set up a fixed support and cannot observe in real time; the installation and construction of the communicating pipe method are complex, and the range is limited; the GPS method has lower precision and the like.
In order to solve the problems, the method can identify the maximum deflection of the simply supported beam under the unknown structure initial state information, and has important significance for evaluating the structural rigidity of the in-service beam, identifying structural damage, monitoring health and the like.
Disclosure of Invention
In view of the above, it is necessary to provide a method for determining maximum deflection of a simply supported beam based on a corner under an unknown state, in which a beam body is segmented, an inclination angle sensor is arranged at the segmented position, the corner of the beam structure under a known load is tested, the bending stiffness of each segment of the beam body is identified according to a corner test value, and then the equivalent bending stiffness of the beam body is obtained based on the principle that bending strain energy is equal; then, based on the corner test value, determining the specific position of the maximum deflection according to the relation between the beam corner curve and the structural bending moment; and finally, calculating and determining the maximum deflection value of the beam structure by adopting a virtual cantilever beam method.
Based on the principle, the invention specifically adopts the technical scheme that:
the method for measuring the maximum deflection of the simply supported beam based on the corner under an unknown state comprises the following steps:
the method comprises the following steps of firstly, identifying the bending rigidity of each section of beam body according to the beam structure corner under the known load action to obtain the equivalent bending rigidity of the beam structure, wherein the specific method comprises the following steps:
(a) segmenting the beam structure on the concerned section, specifically, dividing the beam structure into eight equal parts according to the span l, and numbering the beam structures from the left side to the right side as the 1 st section to the 8 th section; the bending rigidity of each section of beam body in the subsection is a certain value, and the bending rigidity of the 1 st section to the 8 th section of beam body is EIr1、(k2)-1EIr1、(k3)-1EIr1、(k4)-1EIr1、(k5)-1EIr1、(k6)-1EIr1、(k7)- 1EIr1、(k8)-1EIr1Wherein k is2、k3、k4、k5、k6、k7、k8The reciprocal of the bending rigidity ratio of the 2 nd to 8 th sections of beam bodies to the 1 st section of beam body;
(b) applying load to the beam structure, and applying a concentrated force p in the span of the beam structure;
(c) the inclination angle sensors are arranged at the beam structure section and the left and right fulcrum sections, are used for testing the rotation angle of the beam body rotating around the transverse shaft, and are set to have a left fulcrum section testing rotation angle value theta0The section test corner value of the 1 st section and the 2 nd section of the beam body is theta1The section test corner value of the 2 nd section and the 3 rd section of the beam body is theta2By analogy, is theta3、θ4、θ5、θ6、θ7The right fulcrum section test rotation angle value is theta8；
(d) Testing the cross section of the above test to obtain a rotation angle value theta0～θ8Applying a concentration force p and substituting the span l into the following system of equations:
determine EIr1、k2、k3、k4、k5、k6、k7、k8The bending rigidity of the beam bodies from the 1 st section to the 8 th section is EIr1、(k2)-1EIr1、(k3)-1EIr1、(k4)-1EIr1、(k5)-1EIr1、(k6)-1EIr1、(k7)-1EIr1、(k8)-1EIr1；
(e) Substituting the result obtained in the previous step into the following formula to calculate the equivalent bending stiffness (EI) of the beam structuree：
Wherein the equivalent bending stiffness (EI) of the 1 st to i-th sections of the beam bodye ibComprises the following steps:
equivalent bending stiffness (EI) of i-th to 8-th sections of beamse iaComprises the following steps:
in the two formulas: i is the number of the beam section, and 1, 2, 3, 4, 5, 6, 7 and 8 are sequentially arranged from the left side to the right side; j is a coefficient; when i is 1, j is 1; when i is 2, j is 7; when i is 3, j is 19; when i is 4, j is 37; when i is 5, j is 37; when i is 6, j is 19; when i is 7, j is 7; when i is 8, j is 1;
secondly, determining the position of the maximum deflection according to the corner test value;
according to the relation between the deflection and the corner, the maximum deflection occurs in the section with the zero corner, and according to the corner test value of the middle section in the previous step, the beam section where the maximum deflection section is located is judged firstly by the positive and negative alternation of the corner and is set to occur in the ith section, so that the distance s between the maximum deflection section and the left beam end can be calculated according to the following formula:
in the formula: k is a radical ofiThe inverse of the bending rigidity ratio of the i-th section of beam body to the 1 st section of beam body is obtained by the previous step; thetaiTesting a turning angle value for the sections at the i-th section and the i + 1-th section of the beam body, and testing a turning angle value for the section of the right fulcrum when i is 8;
thirdly, calculating the maximum deflection omega of the simply supported beam under the load action by a virtual cantilever beam method according to the following formulamax：
In the formula: when 8s/l is an integer, i is 8 s/l; when 8s/l is not an integer, i ═ int (8s/l) + 1.
Further, in the first step (c), each section rotation angle measurement accuracy is not less than 0.001 °.
After the method is adopted, the invention has the following beneficial effects:
1. the method can realize the measurement of the maximum deflection value of the simply supported beam structure under the condition of unknown initial state through the structure corner test, not only solves the limitation of the conventional deflection test method, but also better fits the engineering practice that the initial state of the in-service beam structure is unknown.
2. The measuring method provided by the invention has universal applicability, and is characterized in that a complex finite element numerical model does not need to be established according to the structural characteristics of each beam, and multiple iterations are not needed; and the method is applicable to simple beam structures with uncertain materials and section geometric information under the condition of unknown initial states, such as (prestressed) reinforced concrete beams, steel beams or steel-concrete composite beams and the like.
3. The testing method provided by the invention is simple and convenient, and only the inclination angle sensor needs to be arranged on the concerned section of the beam structure in the testing process, so that the extra workload does not need to be increased; in addition, the method can be used for finding whether the bending rigidity of each section of the beam structure is reduced or not, positioning damage and determining damage amount, and evaluating the degradation condition of the structural rigidity and the like.
Drawings
FIG. 1 is a flow chart of the method of the present invention.
Fig. 2 is a schematic diagram of a segmented stiffness identification method.
Figure 3 is a virtual cantilever.
FIG. 4 is a schematic view (unit: cm) of the beam structure of example 1.
FIG. 5 is a finite element numerical model diagram of a beam structure of example 1.
FIG. 6 is a schematic view (unit: cm) of the beam structure of example 2.
FIG. 7 is a finite element numerical model diagram of a beam structure according to example 2.
Detailed Description
The present invention will be described in further detail with reference to the accompanying drawings and examples.
Referring to fig. 1, the method for determining the maximum deflection of a simply supported beam based on a corner in an unknown state according to the present invention includes the following steps:
the method comprises the following steps of firstly, identifying the bending rigidity of each section of beam body according to the beam structure corner under the known load action to obtain the equivalent bending rigidity of the beam structure, wherein the specific method comprises the following steps:
(a) segmenting the beam structure on the concerned section, specifically, dividing the beam structure into eight equal parts according to the span l, and numbering the beam structures from the left side to the right side as the 1 st section to the 8 th section; the bending rigidity of each section of beam body in the subsection is a certain value, and the bending rigidity of the 1 st section to the 8 th section of beam body is EIr1、(k2)-1EIr1、(k3)-1EIr1、(k4)-1EIr1、(k5)-1EIr1、(k6)-1EIr1、(k7)- 1EIr1、(k8)-1EIr1Wherein k is2、k3、k4、k5、k6、k7、k8The bending rigidity ratio of the 2 nd to 8 th sections of beam bodies to the 1 st section of beam body is reciprocal.
(b) A load is applied to the beam structure, and a concentrated force p is applied across the beam structure.
(c) The inclination angle sensors are arranged at the beam structure section and the left and right fulcrum sections, are used for testing the rotation angle of the beam body rotating around the transverse shaft, and are set to have a left fulcrum section testing rotation angle value theta0The section test corner value of the 1 st section and the 2 nd section of the beam body is theta1The section test corner value of the 2 nd section and the 3 rd section of the beam body is theta2By analogy, is theta3、θ4、θ5、θ6、θ7The right fulcrum section test rotation angle value is theta8. In the step, the testing precision of the rotation angle of each section is not lower than 0.001 degrees, and the testing precision of the rotation angle of each section is required to be as high as possible.
(d) Testing the cross section of the above test to obtain a rotation angle value theta0～θ8Applying a concentration force p and substituting the span l into the following system of equations:
determine EIr1、k2、k3、k4、k5、k6、k7、k8The bending rigidity of the beam bodies from the 1 st section to the 8 th section is EIr1、(k2)-1EIr1、(k3)-1EIr1、(k4)-1EIr1、(k5)-1EIr1、(k6)-1EIr1、(k7)-1EIr1、(k8)-1EIr1。
(e) Substituting the result obtained in the previous step into the following formula to calculate the equivalent bending stiffness (EI) of the beam structuree：
Wherein the equivalent bending stiffness (EI) of the 1 st to i-th sections of the beam bodye ibComprises the following steps:
equivalent bending stiffness (EI) of i-th to 8-th sections of beamse iaComprises the following steps:
in the two formulas: i is the number of the beam section, and 1, 2, 3, 4, 5, 6, 7 and 8 are sequentially arranged from the left side to the right side; j is a coefficient; when i is 1, j is 1; when i is 2, j is 7; when i is 3, j is 19; when i is 4, j is 37; when i is 5, j is 37; when i is 6, j is 19; when i is 7, j is 7; when i is 8, j is 1.
And secondly, determining the position of the maximum deflection according to the corner test value.
According to the relation between the deflection and the corner, the maximum deflection occurs in the section with the zero corner, and according to the corner test value of the middle section in the previous step, the beam section where the maximum deflection section is located is judged firstly by the positive and negative alternation of the corner and is set to occur in the ith section, so that the distance s between the maximum deflection section and the left beam end can be calculated according to the following formula:
in the formula: k is a radical ofiThe inverse of the bending rigidity ratio of the i-th section of beam body to the 1 st section of beam body is obtained by the previous step; thetaiAnd (3) testing the section of the ith section and the (i + 1) th section of the beam body, and testing the section of the right fulcrum when i is 8.
Thirdly, calculating the maximum deflection omega of the simply supported beam under the load action by a virtual cantilever beam method according to the following formulamax：
In the formula: when 8s/l is an integer, i is 8 s/l; when 8s/l is not an integer, i ═ int (8s/l) + 1.
The derivation of the correlation formula in the present invention is described in detail below with reference to fig. 2.
First, the derivation of the formula in the first step will be described.
In fig. 2, the known parameters are: span l, applying concentration force p, and testing rotation angle value theta of cross section at left end support0Section testing angle value theta of the 1 st section and the 2 nd section of the beam body at the section (section l/8)1Section test angle value l/4 section turn of 2 nd section and 3 rd section beam body subsection section (l/8 section)Angle theta2Section testing corner value theta of the 3 rd section and the 4 th section of the beam body at the section (3l/8 section)3Section testing angle value theta of beam body sections (l/2 sections) at sections 4 and 54Section testing corner value theta of the 5 th section and the 6 th section of the beam body at the section (5l/8 section)5Section testing corner value theta of the beam body sections (3l/4 sections) of the 6 th section and the 7 th section6Section testing corner value theta of the 7 th section and the 8 th section of the beam body at the section (7l/8 section)7Right end support section testing angle value theta8The unknown variables are: bending stiffness EI of 1 st section beam bodyr1The reciprocal k of the bending rigidity ratio of the beam bodies from the 2 nd section to the 8 th section to the beam body at the 1 st section2、k3、k4、k5、k6、k7、k8。
To solve the above unknown variables, a pulse function s (x) is used, the function expression being:
S(x)＝<x-a>n(1)
in the formula, the < > symbol is mecolline bracket, x is unknown variable, a is any constant, and n is exponential. When each variable takes a different value, the pulse function has a different form, which is as follows:
when n is more than or equal to 0,
when n is<At the time of 0, the number of the first,
due to the special form and definition of the pulse function, the solution of an integral constant can be avoided during calculus operation, and the workload of calculation is simplified. The pulse function calculus form is summarized as follows:
the bending stiffness for the beam structure shown in fig. 2 is expressed as an impulse function:
according to the Timoshenko beam theory, the basic differential equation system of the beam considering the influence of shear deformation is as follows:
wherein y is the deflection of the beam,
is the angle of the beam, C (x) is the shear stiffness of the beam, B (x) is the bending stiffness of the beam, and q (x) and m (x) are functions of the load density acting on the beam.
Referring to fig. 2, the load density function acting on the beam can be expressed as a pulse function:
m(x)＝0 (10)
substituting formula (9) for formula (7), and integrating formula (7) to obtain:
substituting formula (11) for formula (8), and integrating x to obtain:
integrating equation (12) yields the angle of rotation equation for the beam member:
the measured angle values at the left and right end supports and at the beam member segments are respectively substituted into equation (13), and the following equations can be listed:
as can be seen from the equation (14), the conditional number of the equation set is 8, which is exactly equal to the number (8) of the unknown variables, so that the bending stiffness of each section of the beam structure can be obtained by back-deducing from the measured corner value through the equation set. After the bending rigidity value of each subsection is obtained, the equivalent bending rigidity of the beam structure can be obtained according to the principle that the bending strain energy is equal, and the derivation process is as follows:
in the formula, M1(x) For equivalent bending moment, EI, of beams of constant cross-sectioneFor equivalent flexural rigidity of beams of constant cross-section, M2(x) B (x) is a bending moment of the actual beam member, and b (x) is a bending rigidity of the actual beam member (see formula (6)).
For the structure of fig. 2, the following is developed from equation (15):
solving according to the formula (16) to obtain the equivalent bending rigidity of the beam member:
similarly, equivalent bending stiffness (EI) of the 1 st to i th sections of the beam body can be derivede ib：
And the i-th to 8-th sections of the beam bodyEquivalent bending stiffness (EI)e ia：
In the formulas (18) and (19), i is the number of the beam section and is 1, 2, 3, 4, 5, 6, 7 and 8 in sequence from the left side to the right side; j is a coefficient. When i is 1, j is 1; when i is 2, j is 7; when i is 3, j is 19; when i is 4, j is 37; when i is 5, j is 37; when i is 6, j is 19; when i is 7, j is 7; when i is 8, j is 1.
Next, the derivation process of the formula in the second step of the present invention is described in detail with reference to FIG. 2.
From the approximate differential equation of the beam deflection line and the differential relationship between the shearing force and the bending moment, the beam corner equation theta (x), the bending moment equation M (x), the shearing force equation Q (x) and the deflection function y (x) have the following relations:
in the formula, EI is the bending stiffness of the beam structure.
From the equation (20), the rotation angle is zero when the deflection assumes the extreme value. In the structural state of fig. 2, the rotation angle test value θ is assumedi-1>0, and θi<0, judging that the maximum deflection value occurs at the ith section, wherein the bending moment equation of the beam body at the ith section is as follows:
from equation (21), the corner equation can be obtained by integrating the bending moment equation:
in the formula (24), c1And c2Is the undetermined coefficient. Testing the rotation angle theta of the right end of the ith sectioniSolving undetermined coefficient c by substituting formula (24)1And c2The values, namely:
when i is more than or equal to 1 and less than or equal to 4,
get it solved
When i is more than or equal to 5 and less than or equal to 8,
get it solved
C to be obtained1And c2Substituting the value into the formula (24) and making the corner zero, solving the distance between the corner and the beam end, namely:
when i is more than or equal to 1 and less than or equal to 4,
get it solved
When i is more than or equal to 5 and less than or equal to 8,
get it solved
Considering the number between the distance from the zero cross section to the beam end and the span l, the final calculation formula for determining the distance s from the cross section to the left beam end is as follows:
after determining the position where the rotation angle is zero according to equation (25), a rigid arm can be applied at the position, i.e., the simple cantilever beam of fig. 2 is converted into the virtual cantilever beam shown in fig. 3. As can be easily seen from FIG. 3, the deflection value of the cantilever end is equal to the maximum deflection value ω in FIG. 2maxThe calculation formula is as follows:
in the above formula: i is a beam section number, and when 8s/l is an integer, i is 8 s/l; when 8s/l is not an integer, i ═ int (8s/l) + 1.
The method of the present invention is described in detail below with reference to the finite element numerical analysis results by taking the simply supported beam in two cases as an example.
Example 1-simply-supported Beam with maximum deflection at left half span
The span of a concrete simply supported box girder is 20m, the concrete strength grade is C50, the height of the box girder is 1.3m, the width of the bottom plate is 1.4m, the width of the top plate is 2.4m, and the thicknesses of the web plate and the top and bottom plates are 0.2 m. Setting the initial state of the beam as follows: the bending rigidity of the 1 st section of beam body is damaged by 30%, the bending rigidity of the 2 nd section of beam body is damaged by 20%, the bending rigidity of the 3 rd section of beam body is damaged by 30%, the bending rigidity of the 4 th section of beam body is damaged by 50%, the bending rigidity of the 5 th section of beam body is 1.5 times of the design, the bending rigidity of the 6 th section of beam body is 1.3 times of the design, the bending rigidity of the 7 th section of beam body is 1.2 times of the design, and the bending rigidity of the 8 th section of beam body is 1.0 times of the design, at the moment, the structural schematic diagram is shown in figure 4, and the finite element.
The values of the structure angles in the structural state of fig. 4 are shown in table 1, based on the results of the finite element calculations.
Table 1 example 1 calculation of corner values for beam construction
Note: the rotation angle value is positive clockwise and negative counterclockwise.
The values in table 1 are substituted into the following system of equations of the present invention:
obtaining by solution:
therefore, the flexural rigidity of each section of the beam body identified by the corners is shown in Table 2, and for comparison, the flexural rigidity in the finite element model is also shown in the table.
Table 2 example 1 bending rigidity value of each section of beam body of simply supported beam
Note: this example uses C50 concrete, Ec＝3.45×104MPa, intact section moment of inertia I0＝0.2459298m4The intact section design bending stiffness was 3.45 × 107×0.2459298＝8484578.1kNm2。
As can be seen from Table 2, the bending stiffness of the beam identified by the testing method provided by the present invention, i.e., the corner, is different from the bending stiffness of the finite element model by 2.11% at most. Therefore, the determination method provided by the invention has high identification precision under the condition of ensuring the test precision. EI to be obtainedr1And k2～k8The value of the equivalent bending rigidity of the whole structure is obtained by substituting the value into the calculation formula of the invention:
the equivalent bending rigidity of the front 4 sections of the beam bodies is obtained as follows:
from Table 1, θ3＝0.0271>0, and θ4＝-0.0172<0, so the maximum deflection is judged to occur in the 4 th section of beam body. I is 4, k4＝1.40634920634929、EIr1＝5968310.36594645、θ4Substituted into-0.0003002 the following formula of the present invention:
the distance s between the maximum deflection section and the left beam end is 9.11116641295262m, and the maximum deflection section is substituted into the following formula of the invention:
the maximum deflection is obtained
Negative values indicate downward deflection. According to the finite element calculation result, in the structural state of FIG. 2, the distance between the maximum deflection section and the left side beam section is 9.25m, and the maximum deflection value is-7.633 mm. It can be known that the relative error of the maximum deflection position
Relative error of maximum deflection
Therefore, the measuring method provided by the invention can be used for measuring the maximum deflection of the simply supported beam structure with high precision, and is feasible.
Example 2-simply-supported Beam with maximum deflection at Right half span
The span of a concrete simply supported box girder is 20m, the concrete strength grade is C50, the height of the box girder is 1.3m, the width of the bottom plate is 1.4m, the width of the top plate is 2.4m, and the thicknesses of the web plate and the top and bottom plates are 0.2 m. Setting the initial state of the beam as follows: the bending rigidity of the 1 st section of beam body is 1.0 time of design, the bending rigidity of the 2 nd section of beam body is 1.2 times of design, the bending rigidity of the 3 rd section of beam body is 1.3 times of design, the bending rigidity of the 4 th section of beam body is 1.35 times of design, the bending rigidity damage of the 5 th section of beam body is 40%, the bending rigidity damage of the 6 th section of beam body is 50%, the bending rigidity damage of the 7 th section of beam body is 20%, and the bending rigidity damage of the 8 th section of beam body is 30%, at this time, the structural schematic diagram is shown in figure 6, and the finite element numerical model is shown in figure 7.
From the finite element calculation results, the values of the structural angle in the structural state of fig. 6 are shown in table 3.
Table 3 example 2 calculation of corner values for beam structures
Note: the rotation angle value is positive clockwise and negative counterclockwise.
The values in table 3 are substituted into the following system of equations of the present invention:
obtaining by solution:
therefore, the flexural rigidity of each section of the beam body identified by the corners is shown in Table 4, and for comparison, the flexural rigidity in the finite element model is also shown in the table.
TABLE 4 bending rigidity of each section of beam body of example 2 simply supported beam
Note: the present embodiment uses C50 concrete,Ec＝3.45×104MPa, intact section moment of inertia I0＝0.2459298m4The intact section design bending stiffness was 3.45 × 107×0.2459298＝8484578.1kNm2。
As can be seen from Table 4, the beam flexural rigidity identified by the measurement method of the present invention, i.e., the corner, differs by 2.11% at the maximum from the flexural rigidity in the finite element model. Therefore, the determination method provided by the invention has high identification precision under the condition of ensuring the test precision. EI to be obtainedr1And k2～k8The value of the equivalent bending rigidity of the whole structure is obtained by substituting the value into the calculation formula of the invention:
the equivalent bending rigidity of the rear 4 sections of the beam body is obtained as follows:
as can be seen from Table 3, θ4＝0.0161>0, and θ5＝-0.0209<0, so the maximum deflection is judged to occur in the 5 th section of the beam body. I is 5 and k5＝1.705069124423939、EIr1＝8663676.33766353、θ5Substituted into-0.00036477 the following formula of the present invention:
the distance s between the maximum deflection section and the left beam end is 11.0020791586023m, and the maximum deflection section is substituted into the following formula:
the maximum deflection is obtained
Negative valuesIndicating deflection down. According to the finite element calculation result, in the structural state of FIG. 6, the distance between the maximum deflection section and the left side beam section is 11.00m, and the maximum deflection value is-7.686 mm. From this, the relative error of the maximum deflection position
Relative error of maximum deflection
Therefore, the measuring method provided by the invention can be used for measuring the maximum deflection of the simply supported beam structure with high precision, and is feasible.
According to the method, the applied load can be changed at will according to the actual conditions (namely, any load form can be applied, such as uniform force, trapezoidal load, bending moment and the like), the load can be applied to any position on the beam, the number of the corner test sections can also be increased, namely, the number of the sections of the beam structure can also be increased (the more the sections are, the more the deflection maximum value is, the more accurate the measurement result is), but the method can be used for measuring the maximum deflection of the simply supported beam structure. The invention is only one of the common cases and any variation on the method according to the invention is within the scope of protection of the invention.
Claims (2)
1. The method for measuring the maximum deflection of the simply supported beam based on the corner under the unknown state is characterized by comprising the following steps of:
the method comprises the following steps of firstly, identifying the bending rigidity of each section of beam body according to the beam structure corner under the known load action to obtain the equivalent bending rigidity of the beam structure, wherein the specific method comprises the following steps:
(a) segmenting the beam structure on the concerned section, specifically, dividing the beam structure into eight equal parts according to the span l, and numbering the beam structures from the left side to the right side as the 1 st section to the 8 th section; the bending rigidity of each section of beam body in the subsection is a certain value, and the bending rigidity of the 1 st section to the 8 th section of beam body is EIr1、(k2)-1EIr1、(k3)-1EIr1、(k4)-1EIr1、(k5)-1EIr1、(k6)-1EIr1、(k7)-1EIr1、(k8)-1EIr1Wherein k is2、k3、k4、k5、k6、k7、k8The reciprocal of the bending rigidity ratio of the 2 nd to 8 th sections of beam bodies to the 1 st section of beam body;
(b) applying load to the beam structure, and applying a concentrated force p in the span of the beam structure;
(c) the inclination angle sensors are arranged at the beam structure section and the left and right fulcrum sections, are used for testing the rotation angle of the beam body rotating around the transverse shaft, and are set to have a left fulcrum section testing rotation angle value theta0The section test corner value of the 1 st section and the 2 nd section of the beam body is theta1The section test corner value of the 2 nd section and the 3 rd section of the beam body is theta2By analogy, is theta3、θ4、θ5、θ6、θ7The right fulcrum section test rotation angle value is theta8；
(d) Testing the cross section of the above test to obtain a rotation angle value theta0～θ8Applying a concentration force p and substituting the span l into the following system of equations:
determine EIr1、k2、k3、k4、k5、k6、k7、k8The bending rigidity of the beam bodies from the 1 st section to the 8 th section is EIr1、(k2)- 1EIr1、(k3)-1EIr1、(k4)-1EIr1、(k5)-1EIr1、(k6)-1EIr1、(k7)-1EIr1、(k8)-1EIr1；
(e) Substituting the result obtained in the previous step into the following formula to calculate the equivalent bending stiffness (EI) of the beam structuree：
Wherein the equivalent bending stiffness (EI) of the 1 st to i-th sections of the beam bodye ibComprises the following steps:
equivalent bending stiffness (EI) of i-th to 8-th sections of beamse iaComprises the following steps:
in the two formulas: i is the number of the beam section, and 1, 2, 3, 4, 5, 6, 7 and 8 are sequentially arranged from the left side to the right side; j is a coefficient; when i is 1, j is 1; when i is 2, j is 7; when i is 3, j is 19; when i is 4, j is 37; when i is 5, j is 37; when i is 6, j is 19; when i is 7, j is 7; when i is 8, j is 1;
secondly, determining the position of the maximum deflection according to the corner test value;
according to the relation between the deflection and the corner, the maximum deflection occurs in the section with the zero corner, and according to the corner test value of the middle section in the previous step, the beam section where the maximum deflection section is located is judged firstly by the positive and negative alternation of the corner and is set to occur in the ith section, so that the distance s between the maximum deflection section and the left beam end can be calculated according to the following formula:
in the formula: k is a radical ofiThe inverse of the bending rigidity ratio of the i-th section of beam body to the 1 st section of beam body is obtained by the previous step; thetaiTesting a turning angle value for the sections at the i-th section and the i + 1-th section of the beam body, and testing a turning angle value for the section of the right fulcrum when i is 8;
thirdly, calculating the load by a virtual cantilever beam method according to the following formulaMaximum deflection omega of lower simply supported beammax：
In the formula: when 8s/l is an integer, i is 8 s/l; when 8s/l is not an integer, i ═ int (8s/l) + 1.
2. The method for measuring the maximum deflection of the simply supported beam based on the rotation angle in the unknown state as claimed in claim 1, wherein in the step (c) of the first step, the test accuracy of each section rotation angle is not less than 0.001 °.
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