CN109252441B - Analysis method for shear hysteresis effect of variable cross-section box beam - Google Patents
Analysis method for shear hysteresis effect of variable cross-section box beam Download PDFInfo
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Abstract
The invention relates to an analysis method for a shear hysteresis effect of a variable cross-section box beam, and belongs to the technical field of bridge construction. The method comprises the following steps: s1: the variable cross-section box girder model is equivalent to a structural system consisting of ideal stiffening rods only bearing axial force and thin plates only transmitting in-plane shear force between the ideal stiffening rods by using a simulation rod equivalent model, and the equivalent cross-section area of the ideal stiffening rods and the equivalent thickness of the thin plates are calculated; s2: establishing a shear hysteresis effect control equation set according to a balance condition and a deformation coordination condition between the ideal stiffening rod and the wing plate; s3: calculating the horizontal shear flow at the junction of the variable cross-section beam web plate and the top plate, solving a shear hysteresis effect control equation set, and obtaining a stiffening rod axial force representing the bending normal stress in the wing plate; s4: and calculating the shear hysteresis coefficient of the box girder by using the axial force of the stiffening rod. According to the invention, by analyzing the distribution rule of the shear flow in the variable cross-section box girder top plate and the web plate along the span length, a basis is provided for the reinforcement design and the checking calculation of the main girder.
Description
Technical Field
The invention belongs to the technical field of bridge construction, and relates to an analysis method for a shear hysteresis effect of a variable cross-section box beam.
Background
When the thin-wall box girder is subjected to bending deformation under the action of vertical load, the section of the box girder can not keep flat section deformation due to the influence of shearing deformation of the upper wing plate and the lower wing plate, so that the longitudinal bending normal stress of the wing plates is unevenly distributed along the cross section, and the phenomenon of shear hysteresis is generated. However, the design method of the existing beam bridge is still based on the analysis result of the finite element software of the rod system, and reinforcement design and checking calculation are carried out on each component forming the bridge structure according to the combination effect of different design load working conditions. Due to the limitation of the assumption of a flat section, the structural internal force analysis result based on a rod system finite element cannot reflect the shear hysteresis effect of the box beam, so the design specification of the highway bridge provides a concept of the effective distribution width of the wing plate to consider the shear hysteresis effect of the wing plate, but the specification only provides values of the effective distribution width of the wing plate of the simple supporting beam, the cantilever beam and the continuous beam of the constant section, and does not consider the influence of the section change on the effective distribution width of the wing plate.
At present, most of analytical methods for calculating the shear hysteresis effect of the section of the box beam are only suitable for analyzing relatively simple structures such as equal-section box beams, and the shear hysteresis effect of the variable-section box beam is calculated and analyzed by a three-dimensional finite element method, so that the time is long, the qualitative analysis of the design is not facilitated, and the analytical analysis of the shear hysteresis effect of the variable-section box beam is helpful for deep cognition and general regularity description of the mechanical essence of the variable-section box beam. In view of this, some scholars have studied the analytical method of shear hysteresis effect of the variable section beam by using a finite-section method, a finite difference method or an approximation method based on the equal section theory, have derived a box beam shear hysteresis effect control differential equation by using an energy variation method, such as a compass, and have established a finite-section model considering shear hysteresis shear effect, and have compared the finite-section model result with the organic glass model test result and the numerical result of the entity finite element method, and the results are in good agreement. Aiming at the shear hysteresis effect of the variable-section box girder, such as Zhang Yuan sea, an improved girder segment finite element method is provided on the basis of introducing shear hysteresis generalized moment and warping geometric characteristics, a variable-height organic glass box girder model is calculated and compared with an actual measurement value, and the reliability of the proposed girder segment unit for analyzing the shear hysteresis effect of the variable-section box girder is verified. A transfer matrix method for researching the shear hysteresis effect of a variable-section continuous box girder bridge is provided by Weichenglong and the like based on an energy principle. And a corresponding field matrix and a point matrix are established, so that one-dimensional recursive solution of the internal force, the stress and the displacement of the variable-section continuous box girder bridge is realized. The finite-section method and the transfer matrix method can accurately calculate the shear hysteresis effect of the variable-section box girder, but the theory is relatively complex, the calculation process is complicated, and the method is inconvenient to be applied to the popularization and application of the engineering.
Compared with other methods, the simulation rod method is definite in mechanical concept, simple in model, high in calculation precision and convenient to apply to actual engineering. A simulation rod method is applied to Cheng Xiang Yun and the like, a differential equation for controlling the shear hysteresis effect of the single-box single-chamber box beam is deduced, the shear hysteresis effect of the single-box single-chamber box beam under vertical symmetrical load is analyzed, and the result shows that the calculation accuracy of the simulation rod method is good. The shear hysteresis effect of the simply supported beam and the cantilever beam is analyzed by adopting a simulation rod method in Zhao Shifeng and the like, and compared with an organic glass model test solution and a plate shell finite element solution, the result shows that the simulation rod method has better calculation precision, but at present, the research aiming at the simulation rod mostly takes an equal-section structure as an object.
Therefore, the invention provides an analysis method for the shear hysteresis effect of the variable cross-section box beam, which systematically discusses the influence of the change of the height of the box beam and the thickness of a web plate along the span length on the shear hysteresis effect of the box beam and provides a basis for reinforcement design and checking calculation of each component forming a bridge structure in bridge design.
Disclosure of Invention
In view of the above, the invention aims to provide an analysis method for a shear hysteresis effect of a variable cross-section box beam, which is used for solving the defect that the traditional analog rod method can only analyze the shear hysteresis effect of the constant cross-section box beam, deducing the equivalent area of a stiffening rod of the variable cross-section box beam and a shear hysteresis effect differential equation, checking the correctness of the algorithm by using the experimental result of an organic glass cantilever beam model, discussing the influence of the height and the thickness change of a web plate of the box beam on the shear hysteresis effect of the box beam, disclosing the reason of the weakening of the shear hysteresis effect of the variable cross-section box beam by analyzing the distribution rule of shear flow in a top plate and the web plate of the box beam along the span length, and providing a basis for the reinforcement design and checking calculation of a.
In order to achieve the purpose, the invention provides the following technical scheme:
a method for analyzing shear hysteresis effect of a variable cross-section box beam specifically comprises the following steps:
s1: the variable cross-section box girder model is equivalent to a structural system consisting of an ideal stiffening rod only bearing axial force and thin plates only transmitting in-plane shear force between the ideal stiffening rod and the thin plates by using a simulation rod equivalent model, and the equivalent cross-section area of the ideal stiffening rod and the equivalent thickness of the thin plates are calculated on the basis of the principle that the bending stress of the top plate and the bending stress of the bottom plate of the box girder are equal;
s2: establishing a shear hysteresis effect control equation set according to a balance condition and a deformation coordination condition between the ideal stiffening rod and the wing plate;
s3: calculating the horizontal shear flow at the junction of the variable cross-section beam web plate and the top plate, solving a shear hysteresis effect control equation set, and obtaining the axial force of a stiffening rod representing the bending normal stress in the wing plate;
s4: and calculating the shear hysteresis coefficient of the box girder by using the axial force of the stiffening rod.
Further, in step S1, when the cross section of the box girder changes along the span direction (x direction) of the bridge, the equivalent thickness of the stiffening sheet will change along the span direction of the bridge, and the equivalent thickness of the stiffening sheet is expressed as:
wherein x represents the position of the cross section of the box girder in the span direction, t1(x) Indicates the thickness of the top plate of the box girder, t2(x) Indicates the thickness of the bottom plate of the box girder, h1(x)、h2(x) Respectively represents the vertical distance t from the top edge of the top plate and the bottom edge of the bottom plate of the box girder to the centroid of the cross section1e(x) Denotes the equivalent thickness, t, of the roof of the box girder2e(x) Representing the box girder floor equivalent thickness.
Further, in the step S1,
for the stiffeners i not intersecting the web (i is 1,2,4,5,6,8,9,11,12,13), "condensing" the wing area represented by the stiffeners i to the stiffeners i, and calculating the equivalent cross-sectional area a of the ith stiffener in the top and bottom plates of the box girder using the equations (3) and (4), respectively1ei(x) And A2ei(x)：
A1ei(x)＝t1e(x)(yi+1-yi-1)/2 (3)
A2ei(x)＝t2e(x)(yi+1-yi-1)/2 (4)
In the formula, yiRepresenting the horizontal distance of the ith stiffener from the centroid of the cross-section.
Further, in the step S1,
for the stiffening rods j (j is 3,7,10 and 14) intersected with the web plate, in addition to the area of the wing plate of the area, the area of the web plate area of the box girder is proportionally distributed to the stiffening rods at the position according to the principle of equivalent bending stiffness, and the equivalent area A of the jth stiffening rod in the top plate and the bottom plate of the box girder is calculated by using the formulas (5) and (6) respectively1ej(x) And A2ej(x)：
A1ej(x)＝2α1(x)H(x)tw(x)+t1e(x)(yi+1-yi-1)/2 (5)
A2ej(x)＝2α2(x)H(x)tw(x)+t2e(x)(yi+1-yi-1)/2 (6)
Wherein H (x) is the beam height, tw(x) Is web thickness, α1(x) Is the upper panel area equivalent coefficient of the web, α2(x) Is the lower web area equivalent coefficient of the web, I (x) is the main inertia moment of the box girder, a1Is the horizontal distance from the free end of the cantilever of the top plate of the box girder to the center of the web plate, a2Is the sum of the horizontal distance between two webs and the thickness of one web.
Further, in step S2, a balance equation as shown in formula (9) is established according to the balance relationship between the axial force on each stiffener and the shear force in the two side sheets, and a coordination equation as shown in formula (10) is established according to the coordination condition between the axial deformation of the stiffener and the shear deformation of the two side sheets;
in the formula, the subscript i represents the ith (i is 1,2, …,5) stiffening rod in the top plate, and when i-1 is 0, the i-1 rod does not exist, and the variable related to the i-1 rod is 0; e represents the modulus of elasticity, Ni(x) Showing the axial force of the ith stiffener of the roof, A1ei(x) Representing the equivalent cross-sectional area of the ith stiffener in the roof, di,i-1Representing the horizontal distance between the i-th and i-1-th stiffeners in the roof, the distance between adjacent stiffeners at the cross-sectional web should be reduced by half the width of the webThe final distance between the stiffening rod rods; q. q.si,i-1(x) And gammai,i-1(x) Q represents the in-plane shear flow function and shear strain function, respectively, of the sheet between the ith and (i-1) th stiffeners in the top plateEi(x) Representing the shear flow function in the web of the box girder, q when the stiffeners do not intersect the webEi(x)＝0。
Further, the functions of the in-plane shear flow and the shear strain of the thin plate between the ith stiffening rod and the (i-1) th stiffening rod in the top plate satisfy the physical equation shown in the formula (11),
qi,i-1(x)＝γi,i-1(x)t1e(x)G (11)
wherein G represents a shear modulus.
Further, let i equal to 1,2, …,5, the stiffener intersects the web when i equal to 3; and obtaining the control equation set of the shear hysteresis effect of the top plate by combining equations (9) to (11) as follows:
the solving process of the equation (12) is as follows:
(1) dividing the box girder into n sections along the span length, and adopting a segmented spline function to approximately express the distribution of the shear flow of the stiffening sheet in each section;
(2) converting the roof shear hysteresis effect control equation set into an algebraic equation set, and solving the algebraic equation set by combining boundary conditions;
(3) and integrating the shear flow in the stiffening sheet to obtain a distribution function of the axial force of each stiffening rod along the span length.
Furthermore, for the free end (such as the free end of a cantilever beam) without external constraint and load action, the axial force N on each stiffening rodi(x) Is 0;
for the embedded end (such as the cantilever beam embedded end), each stiffening sheet cannot be subjected to shear deformation due to external constraint, and the boundary conditions are as follows from the formula (11):
qi,i-1(x)＝0 (13)
for the articulated boundary conditions, the support reaction force is determined and then applied to the structure in the opposite direction in the form of an external load.
Further, step S3 specifically includes:
the total axial force borne by each stiffening rod of the top plate under the action of uniformly distributed load of the cantilever beam is N (x), and the horizontal shear flow of the web plate acting on the top plate is qEi(x) From the balance of forces in the horizontal direction, it can be seen that:
in the formula, σxRepresents the bending normal stress of the section of the box girder, Ae1iRepresenting the area of the ith stiffener; the moment balance equation is established for point O:
M(x)＝N(x)H(x)＝q(x)x2/2 (15)
wherein M (x) represents the external moment of an external load to a point O, and q (x) represents the shear flow of the section of the box girder;
the horizontal shear flow q at the web-to-roof interface can be obtained from the equations (14) and (15)Ei(x)：
Wherein Q (x), S1(x) Respectively representing the cross-sectional shear and the area moment.
Further, the step S4 specifically includes: in order to quantitatively evaluate the uneven distribution condition of the bending normal stress along the flange of the wide box girder, a shear hysteresis coefficient lambda of a web plate is defined as follows:
wherein σwThe bending positive stress at a certain height of the position of the box girder section web plate considering the shear hysteresis effect;
the bending normal stress at the corresponding section height of the box girder is calculated based on the Euler girder theory; for a solid finite element numerical solution,
the average value of the bending normal stress at the corresponding height position of the section of the box girder is obtained; in the case of the analog rod method,
for the average axial normal stress of each stiffener, the equation (18) is used to solve
Wherein A isiDenotes the area of the i-th stiffener, σiRepresenting the axial positive stress of the ith stiffener.
The invention has the beneficial effects that: aiming at the defect that the traditional analog rod method can only analyze the shear hysteresis effect of the box girder with the equal cross section, the invention deduces the equivalent area of the stiffening rod of the box girder with the variable cross section and a control equation set of the shear hysteresis effect again, discusses the influence of the change of the height of the box girder and the thickness of the web plate on the shear hysteresis effect of the box girder, reveals the reason of the weakening of the shear hysteresis effect of the box girder with the variable cross section by analyzing the distribution rule of the shear flow in the top plate and the web plate of the box girder along the span length, and provides a basis for the reinforcement design and the checking calculation of each component forming the bridge structure in the bridge design.
Drawings
In order to make the object, technical scheme and beneficial effect of the invention more clear, the invention provides the following drawings for explanation:
FIG. 1 is a cross section of a single box single chamber box girder;
FIG. 2 is a stiffener-sheet equivalent structure system;
FIG. 3 is a schematic view of the stiffener-sheet structure under stress;
FIG. 4 is a schematic view of a cantilever beam construction and spacer stressing;
FIG. 5 is a cross-sectional view (mm) of the test beam;
FIG. 6 is a graph comparing the normal bending stress of the top plate of the experimental test section;
FIG. 7 is a cross-sectional view (cm) of a numerical example;
FIG. 8 is a graph comparing the stress distribution of the top edge of the top plate at different positions according to numerical calculations;
FIG. 9 is a shear flow q between stiffeners 1,221Along a cross-direction distribution diagram;
FIG. 10 is a graph of the effect of beam height variation on the shear hysteresis factor of the top edge of the roof panel;
FIG. 11 is a graph of shear hysteresis coefficient of a top plate edge versus web thickness variation;
FIG. 12 shows shear flow q in the web of a lower box girder in different structural formsEDistributing the graph along the span length;
FIG. 13 shows the shear force q in the lower top plate in different structural forms21Distributed along the span length.
Detailed Description
Preferred embodiments of the present invention will be described in detail below with reference to the accompanying drawings.
A method for analyzing shear hysteresis effect of a variable cross-section box beam specifically comprises the following steps:
s1: the box girder model is equivalent to a structural system consisting of ideal stiffening rods only bearing axial force and thin plates only transmitting in-plane shearing force between the ideal stiffening rods by using a simulation rod equivalent model, namely the box girder shown in the figure 1 is equivalent to a stiffening rod-thin plate structural system shown in the figure 2 (for convenience of description, the upper flange plate and the lower flange plate are respectively equivalent to 9 stiffening rods and 5 stiffening rods). Calculating the equivalent cross section area of the ideal stiffening rod and the equivalent thickness of the thin plate on the basis of the principle that the bending stresses of the top plate and the bottom plate of the box girder are equal;
solving the equivalent thickness of the thin plate of the stiffening rod according to the principle that the bending stiffness of the top plate and the bottom plate of the box girder relative to the centroid of the cross section is equal, and then calculating the equivalent coefficient of the web according to the principle that the bending stress of the top plate and the bottom plate of the box girder is equal under the condition that the equivalent thickness of the stiffening thin plate is known (α)1、α2) To obtain equivalent areas of the stiffeners at different locations. When the section of the box girder changes along the span direction (x direction) of the bridge, the equivalent thickness of the stiffening sheet changes along the span direction of the bridge, and the equivalent thickness t of the stiffening sheet1e(x)、t2e(x) Can be expressed as:
wherein x represents the position of the cross section of the box girder in the span direction, t1(x) Indicates the thickness of the top plate of the box girder, t2(x) Indicates the thickness of the bottom plate of the box girder, h1(x)、h2(x) Respectively represents the vertical distance t from the top edge of the top plate and the bottom edge of the bottom plate of the box girder to the centroid of the cross section1e(x) Denotes the equivalent thickness, t, of the roof of the box girder2e(x) Representing the box girder floor equivalent thickness.
For stiffeners i that do not intersect the web (i is 1,2,4,5,6,8,9,11,12,13), the area of the web represented by it is simply "gathered" into the stiffener i, and its equivalent area a is calculated using equations (3) (4)1ei(x) And A2ei(x)：
A1ei(x)＝t1e(x)(yi+1-yi-1)/2 (3)
A2ei(x)＝t2e(x)(yi+1-yi-1)/2 (4)
In the formula, yiRepresenting the horizontal distance of the ith stiffener from the centroid of the cross-section.
For the stiffeners j (j is 3,7,10,14) intersecting the web, in addition to the area of the wing plate in the 'condensed' area, the area of the web area of the box girder needs to be proportionally distributed to the stiffeners at the position according to the bending stiffness equivalence principle, and the equivalent areas A are respectively calculated by using the equations (5) and (6)1ej(x) And A2ej(x)：
A1ej(x)＝2α1(x)H(x)tw(x)+t1e(x)(yi+1-yi-1)/2 (5)
A2ej(x)＝2α2(x)H(x)tw(x)+t2e(x)(yi+1-yi-1)/2 (6)
Wherein H (x) is the beam height, tw(x) Is web thickness, α1(x) Is the upper panel area equivalent coefficient of the web, α2(x) Is the lower web area equivalent coefficient of the web, I (x) is the main inertia moment of the box girder, a1Horizontal distance from free end of cantilever of top plate of box girder to center of web plate, a2The sum of the horizontal distance between two webs and the thickness of one web is labeled in detail in fig. 1.
S2: establishing a shear hysteresis effect control equation set according to a balance condition and a deformation coordination condition between an ideal stiffening rod and a wing plate, and representing the bending normal stress in the wing plate through the axial force of the stiffening rod;
when the shear hysteresis effect control equation set is established by the traditional analog rod method, a balance equation, a coordination equation and a physical equation are simultaneously established to derive a second-order differential equation set related to a shear flow function of the wing plate. When the cross section of the box girder changes along the span direction of the bridge, the area of each stiffening rod changes along the longitudinal direction, a differential form related to an area function is generated when a differential equation is established, redundant unknowns cannot be eliminated, and the solution difficulty is caused.
Taking the above flange plate as an example, as shown in fig. 3, a balance equation as shown in equation (9) is established according to the balance relationship between the axial force on each stiffening rod and the shear force in the thin plates on both sides, and a coordination equation as shown in equation (10) is established according to the coordination condition between the axial deformation of the stiffening rod and the shear deformation of the thin plates on both sides.
In the formula, the subscript i represents the ith (i is 1,2, …,5) stiffener in the top plate, and when i-1 is 0The variable related to the absence of the rod i-1 is 0; e represents the modulus of elasticity, Ni(x) Showing the axial force of the ith stiffener of the roof, A1ei(x) Representing the equivalent cross-sectional area of the ith stiffener in the roof, di,i-1Representing the horizontal distance between the ith stiffening rod and the (i-1) th stiffening rod in the top plate, and taking the distance between the adjacent stiffening rods at the cross-section web plate minus half the width of the web plate as the final spacing between the stiffening rods; q. q.si,i-1(x) And gammai,i-1(x) Q represents the in-plane shear flow function and shear strain function, respectively, of the sheet between the ith and (i-1) th stiffeners in the top plateEi(x) Representing the shear flow function in the web of the box girder, q when the stiffeners do not intersect the webEi(x) 0. The functions of the in-plane shear flow and the shear strain of the thin plate between the ith stiffening rod and the (i-1) th stiffening rod in the top plate satisfy a physical equation shown as a formula (11),
qi,i-1(x)＝γi,i-1(x)t1e(x)G (11)
wherein G represents a shear modulus.
The control equation set of the shear hysteresis effect of the top plate obtained by the simultaneous equations (9) to (11) is as follows:
the shear flow of the web plate in the control equation set of the shear hysteresis effect of the top plate can be directly solved, compared with a beam with equal section, the influence of additional shear stress caused by bending moment in the beam with variable section on the shear stress of the web plate is obvious, and in consideration of the particularity of the equivalent beam with the variable section simulation rod, the invention redevelops the shear flow formula by using the moment balance equation of the equivalent beam with the variable section simulation rod, wherein the moment balance equation is used for deducing the shear flow formula, the number ① of the isolation body from the free end of the equivalent beam with the variable section simulation rod to the position x is taken, the acting forces of the isolation body are shown in figure 4, the total axial force borne by each stiffening rod of the top plate under the action of uniform load of the cantilever beam is N (xEi(x) From the balance of forces in the horizontal direction, it can be seen that:
in the formula, σxRepresents the bending normal stress of the section of the box girder, Ae1Representing the area of the ith stiffener;
the moment balance equation is established for point O:
M(x)＝N(x)H(x)＝q(x)x2/2 (14)
wherein M (x) represents the external moment of an external load to a point O, and q (x) represents the shear flow of the section of the box girder;
the horizontal shear flow q at the interface of the web and the top plate can be obtained by the formulas (13) and (14)Ei(x)：
Wherein Q (x), S1(x) Respectively representing the cross-sectional shear and the area moment. The formula is consistent with a shear flow formula, when the section is not changed, the spalling is converted into a traditional shear stress calculation formula in material mechanics, and meanwhile, the correctness of the method for calculating the horizontal shear flow of the web is explained.
For the free end (such as the free end of a cantilever beam) without external constraint and load action, the axial force N on each stiffening rodi(x) Is 0;
for the embedded end (such as the cantilever beam embedded end), each stiffening sheet cannot be subjected to shear deformation due to external constraint, and the boundary conditions are as follows from the formula (11):
qi,i-1(x)＝0 (16)
for the articulated boundary conditions, the support reaction force is determined and then applied to the structure in the opposite direction in the form of an external load.
S3: solving the shear flow of the web plate according to the shear hysteresis control equation established in the step S2;
the solution of the differential equation set shown in the formula (12) belongs to the boundary value problem of a first-order constant coefficient differential equation, the invention adopts a spline function approximation method to solve, and the specific solution thought is as follows: the method comprises the steps of firstly, approximately expressing a shear flow function in a stiffening sheet by using a 3-time spline interpolation function meeting boundary conditions, converting a differential equation set into an algebraic equation set to solve, and integrating the shear flow in the stiffening sheet to obtain a distribution function of the axial force of each stiffening rod along the span length. In order to improve the solving precision, the box girder can be divided into n sections along the span length, and the distribution of the shear flow of the stiffening sheet in each section is expressed by adopting a segmented spline function.
S4: calculating the shear hysteresis coefficient of the web plate;
in order to quantitatively evaluate the uneven distribution condition of the bending normal stress along the flange of the wide box girder, a shear hysteresis coefficient lambda of a web plate is defined as follows:
wherein σwThe bending positive stress at a certain height of the position of the box girder section web plate considering the shear hysteresis effect;
the bending normal stress at the corresponding section height of the box girder is calculated based on the Euler girder theory; for a solid finite element numerical solution,
the average value of the bending normal stress at the corresponding height position of the section of the box girder is obtained; in the case of the analog rod method,
for the average axial normal stress of each stiffener, the equation (18) is used to solve
Wherein A isiDenotes the area of the i-th stiffener, σiRepresenting the axial positive stress of the ith stiffener.
Numerical calculation example:
1. verification of reasonability of analysis of shear hysteresis effect of variable cross-section box girder by using simulation rod method
The cantilever box girder made of organic glass is used as a test model, the span l is 46cm, and the height of the box girder changes according to a quadratic parabola along the unfolding direction (self-parabolic)Height H of end beama4cm, embedded and fixed end beam height Hb8cm), a beam body with the length of 6cm close to the free end of the cantilever beam adopts an equal section, the cross section size and the measuring point position are shown in fig. 5, a symmetrical and uniformly distributed load (q is 10N/cm) acts at the position close to the web plate during test, the box body elastic modulus E is 2600MPa, and the Poisson ratio mu is 0.4. The ANSYS model adopts Solid95 units and is divided into hexahedron mapping grids, and the boundary conditions and the loads are calculated and applied to the grids and are the same as those in the test. Fig. 6 shows the experimental test values of the test section (3 cm from the embedded end), the finite segment calculation results, the theoretical solution of the present invention and the finite element numerical solution, and the calculation results all correspond to the normal stress at the height of the center line of the top plate of the box girder.
As can be seen from FIG. 6, under the action of symmetrically and uniformly distributed loads, the normal bending stress of the top plate of the section of the cantilever beam close to the embedding end presents a positive shear hysteresis effect, and the results obtained by the four calculation methods are well matched, so that the correctness of the simulation rod equivalent area and calculation method and the ANSYS entity finite calculation model are verified, and the fact that the simulation rod analysis result is checked by using the calculation result of the entity finite element model is feasible.
2. Analyzing shear hysteresis effect of cantilever beam under action of dead weight
In actual engineering, the height of the box girder and the web plate can be changed along the bridge span direction, the self weight is used as a main load form of the structure, and the research on the shear hysteresis effect of the variable cross-section box girder under the action of the self weight has more guiding significance on the actual engineering. For this purpose, a cantilever bridge with a span l of 15m is designed with reference to the common structural dimensions of a variable-section cantilever box bridge, the box beam height varies according to a quadratic parabola in the span direction (free-end beam height H)a1.2m, height of embedded end beam Hb2.0m), the thickness of the web plate is linearly reduced from 0.4m at the embedded end to 0.2m at the free end, the cross section size and the measuring point position are shown in fig. 7, the beam body material is C50 concrete, the elastic modulus E is 3450MPa, the Poisson ratio mu is 0.2, a Solid95 unit is adopted in an ANSYS finite element model and is divided into hexahedron mapping grids, and the boundary condition of the embedded end of the cantilever beam is simulated by using node full constraint.
FIG. 8 shows the cantilever under the action of its own weightThe bending normal stress distribution of the top plate top edge at different positions of the beam along the longitudinal direction (the root section of the cantilever beam, the section 1m away from the root of the cantilever beam, the section 3/4, the section 1/2 and the section 1/4), and it can be seen from fig. 8 that the analog rod analytic solution of the invention is well matched with the entity finite element numerical solution at the sections 1/2, 3/4 and the section 1m away from the root, which shows that the analog rod algorithm of the invention is used for calculating the reliability of the shear hysteresis effect of the top plate of the cantilever box beam with the height of the beam and the thickness of the web changing along the span direction; the difference between the finite element calculation result and the analysis solution of the analog rod at the root section of the cantilever beam is obvious, which is probably because the root section of the finite element model has obvious stress concentration phenomenon, so that the integral stress level of the top edge of the top plate of the section is higher; the positive stress distribution of the top plate of the box girder in the section from the embedded end to the 1/2 shows a positive shear hysteresis phenomenon, and the positive stress distribution and the negative shear hysteresis phenomenon of the top plate in the section from the 1/2 section to the free end of the cantilever. The uneven distribution of horizontal shear stress in the flange plate and the change of the acting direction are the root causes of the shear hysteresis phenomenon of the box girder, so for further researching the reasons, the shear flow q between the stiffening rods 1 and 2 is shown in FIG. 921Along the span distribution, it can be found from fig. 9 that: q when approaching from the free end of the cantilever to the embedded end21The rule of increasing and then reducing is presented, and the change facilitates the conversion of the positive bending stress of the top plate of the cantilever box girder from negative shear hysteresis to positive shear hysteresis; in addition, the difference of the analytic solutions of the solid finite element and the analog rod is more obvious when the solid finite element is closer to the embedded end, and the q is based on the solid finite element21The sudden change occurs near the embedded end, which is probably caused by stress concentration generated by the fixed constraint of the embedded end in a limited model, and the reason that the finite element numerical solution of the bending stress of the top plate of the section of the embedded end of the cantilever beam is greatly different from the analog rod analytic solution can be explained.
3. Analyzing the influence of the change of the height of the box girder and the thickness of the web plate on the shear hysteresis coefficient
In order to further explore the influence of the change of the height of the box girder and the thickness of the web plate on the shear hysteresis coefficient, the variable cross-section cantilever beam calculation example is taken as a blue book, and the influence of the change of the height of the box girder and the thickness of the web plate on the shear hysteresis coefficient is revealed by changing the change law of the height of the girder (the height of the girder which changes along the secondary parabola in the spanwise direction is adjusted into three working conditions of linear change in the spanwise direction, equal cross section with the height of 1.2m and equal cross section with the height of 2.0m) and the change law of the thickness of the web plate of the box girder (the structure of the web plate which changes from 0.2m at the free end to 0.3m at the embedded end is adjusted into a structure with the equal thickness of the web plate.
Fig. 10 and 11 respectively show the change rule of the bending normal stress shear hysteresis coefficient of the top plate from the position 3m away from the free end to the embedded end when the beam height and the web thickness change. It can be known from the figure that the cantilever beam in various structural forms presents a negative shear hysteresis phenomenon at the position close to the free end, and presents a positive shear hysteresis phenomenon at the position close to the embedded end, and the more the negative shear hysteresis effect is close to the free end, the more the positive shear hysteresis effect is close to the embedded end; the change of the height of the box girder and the thickness of the web plate can reduce the shear hysteresis effect of the bending stress of the top plate, but the influence degree of the thickness of the web plate is relatively weak; the change of the height of the cross section beam of the box beam can cause the change of the junction position of positive and negative shear hysteresis, particularly, the length of the positive shear hysteresis section of the variable cross section cantilever box beam is obviously larger than that of the constant cross section box beam, and the positive shear hysteresis section of the cantilever beam is also slightly larger than that of the box beam with the linearly changed beam height when the secondary parabola of the beam height is changed, but the change of the web thickness can not obviously influence the junction position of the positive and negative shear hysteresis.
To further explain the above phenomenon, fig. 12 and 13 show the shear flow q of the box girder web plate under different structural formsEAnd horizontal shear flow q between the stiffeners 1 and 221The change in direction along the bridge span, from which can be seen: the structure that the beam height is gradually increased from the free end to the embedded end can obviously reduce the level of shear stress in a box girder web plate, although the shear force of a heightened cantilever box girder (the minimum section is 1.2m high) at the same position along the bridge span is larger than that of a constant-section cantilever box girder with the beam height of 1.2m, the shear force flow in the web plate is obviously smaller than that of the constant-section cantilever box girder with the beam height of 1.2m, which is the most essential reason that the shear hysteresis effect of the heightened box girder is smaller than that of the constant-height box girder; under the same condition, the horizontal shear flow in the top plate of the box girder with large aspect ratio is larger, which can eliminate the shear flow in the top plate and the box girder web plate with the girder height of 1.2m from the other construction formsThe long comparison shows that the shear hysteresis effect of the box girder with the equal section and the height of 1.2m is the most obvious reason; relative to the linear change of the beam height, the horizontal shear flow q in the top plate is changed when the beam height changes in a parabola21The shear hysteresis effect of the cantilever box girder with parabola change is weaker than that of the cantilever box girder with the height linearly changed; horizontal shear flow q in the top plate when approaching from the free end of the cantilever to the embedded end21The rule of increasing firstly and then reducing is presented, and the curve inflection point is just the junction position of the positive and negative shear hysteresis of the box girder, which shows that the change of the horizontal shear flow in the top plate causes the positive and negative shear hysteresis of the cantilever box girder, and when the height of the girder changes in a parabola, q is21The curve inflection point which is increased and then decreased is closer to the free end of the cantilever, so the length of a positive shear hysteresis zone of the variable-section cantilever box beam is obviously larger than that of the constant-section box beam.
4. Conclusion
On the basis of a traditional analog rod method, by deducing an equivalent area and a shear hysteresis effect analog rod control equation of a stiffening rod of a variable cross-section box beam, the invention provides a method suitable for shear hysteresis effect analysis of the variable cross-section box beam, a cantilever box beam model is taken as an example, the correctness of the calculation method is verified, and parameter analysis finds that:
1) the shear hysteresis effect of the bending stress of the top plate can be reduced by the change of the height of the box girder and the thickness of the web plate, but the influence degree of the thickness of the web plate is relatively weak, and the length of a positive shear hysteresis section of the variable-section cantilever box girder is obviously larger than that of the constant-section box girder;
2) the structure that the beam height is gradually increased from the free end to the embedded end can obviously reduce the shear stress level in a box girder web plate, and the horizontal shear flow in a box girder top plate with large width-height ratio under the same condition is larger, which is the most essential reason that the shear hysteresis effect of the high-height box girder is smaller than that of the box girder with equal height;
3) the change of horizontal shear flow in the top plate of the box girder along the span length which is firstly increased and then reduced leads to the positive and negative shear hysteresis phenomena of the cantilever box girder, when the height of the girder changes in a parabola, the curve inflection point of the shear force in the top plate which is firstly increased and then reduced is closer to the free end of the cantilever, so the length of the positive shear hysteresis section of the variable-section cantilever box girder is obviously larger than that of the constant-section box girder.
Finally, it is noted that the above-mentioned preferred embodiments illustrate rather than limit the invention, and that, although the invention has been described in detail with reference to the above-mentioned preferred embodiments, it will be understood by those skilled in the art that various changes in form and detail may be made therein without departing from the scope of the invention as defined by the appended claims.
Claims (9)
1. The method for analyzing the shear hysteresis effect of the variable cross-section box beam is characterized by comprising the following steps of:
s1: the variable cross-section box girder model is equivalent to a structural system consisting of an ideal stiffening rod only bearing axial force and thin plates only transmitting in-plane shear force between the ideal stiffening rod and the thin plates by using a simulation rod equivalent model, and the equivalent cross-section area of the ideal stiffening rod and the equivalent thickness of the thin plates are calculated on the principle that the bending stress of the top plate and the bending stress of the bottom plate of the box girder are equal:
when the box girder cross-section changes along the bridge spanwise, the equivalent thickness of the thin plate of putting more energy into will change along the bridge spanwise, the equivalent thickness of the thin plate of putting more energy into expresses as:
wherein x represents the position of the cross section of the box girder in the span direction, t1(x) Indicates the thickness of the top plate of the box girder, t2(x) Indicates the thickness of the bottom plate of the box girder, h1(x)、h2(x) Respectively represents the vertical distance t from the top edge of the top plate and the bottom edge of the bottom plate of the box girder to the centroid of the cross section1e(x) Denotes the equivalent thickness, t, of the roof of the box girder2e(x) Representing the equivalent thickness of the bottom plate of the box girder;
s2: establishing a first-order differential equation set of a shear hysteresis effect according to a balance condition and a deformation coordination condition between the ideal stiffening rod and the wing plate;
s3: calculating horizontal shear flow in the thin plate at the junction of the variable cross-section beam web plate and the top plate, solving a first-order differential equation set of a shear hysteresis effect, and directly obtaining the axial force of a stiffening rod representing the bending normal stress in the wing plate;
s4: and calculating the shear hysteresis coefficient of the box girder by using the axial force of the stiffening rod.
2. The method for analyzing shear hysteresis effect of a variable cross-section box beam as claimed in claim 1, wherein in step S1, for the stiffener i not intersecting with the web, the equivalent cross-sectional area a of the ith stiffener in the top plate and the bottom plate of the box beam is calculated by using equations (3) and (4) respectively1ei(x) And A2ei(x)：
A1ei(x)＝t1e(x)(yi+1-yi-1)/2 (3)
A2ei(x)＝t2e(x)(yi+1-yi-1)/2 (4)
In the formula, yiRepresenting the horizontal distance of the ith stiffener from the centroid of the cross-section.
3. The method for analyzing shear hysteresis effect of a box beam with variable cross section as claimed in claim 2, wherein in step S1, for the stiffener j intersecting the web, the equivalent area a of the jth stiffener in the top plate and the bottom plate of the box beam is calculated by using equations (5) and (6) respectively1ej(x) And A2ej(x)：
A1ej(x)＝2α1(x)H(x)tw(x)+t1e(x)(yi+1-yi-1)/2 (5)
A2ej(x)＝2α2(x)H(x)tw(x)+t2e(x)(yi+1-yi-1)/2 (6)
Wherein H (x) is the beam height, tw(x) Is web thickness, α1(x) Is the upper panel area equivalent coefficient of the web, α2(x) Is the lower web area equivalent coefficient of the web, I (x) is the main inertia moment of the box girder, a1Horizontal distance from free end of cantilever of top plate of box girder to center of web plate, a2Is the sum of the horizontal distance between two webs and the thickness of one web.
4. The method for analyzing shear hysteresis effect of a variable cross-section box beam according to claim 3, wherein in step S2, a balance equation as shown in formula (9) is established according to the balance relationship between the axial force on each stiffening rod and the shear force in the thin plates at two sides, and a coordination equation as shown in formula (10) is established according to the coordination condition between the axial deformation of the stiffening rod and the shear deformation of the thin plates at two sides;
in the formula, the subscript i represents the ith stiffening rod in the top plate, and when i-1 is equal to 0, the i-1 rod is not existed, and the variable related to the i-1 rod is 0; e represents the modulus of elasticity, Ni(x) Showing the axial force of the ith stiffener of the roof, A1ei(x) Representing the equivalent cross-sectional area of the ith stiffener in the roof, di,i-1Representing the horizontal distance between the ith stiffening rod and the (i-1) th stiffening rod in the top plate, and taking the distance between the adjacent stiffening rods at the cross-section web plate minus half the width of the web plate as the final spacing between the stiffening rods; q. q.si,i-1(x) And gammai,i-1(x) Q represents the in-plane shear flow function and shear strain function, respectively, of the sheet between the ith and (i-1) th stiffeners in the top plateEi(x) Representing the shear flow function in the web of the box girder, q when the stiffeners do not intersect the webEi(x)＝0。
5. The method for analyzing shear hysteresis effect of variable cross-section box girder according to claim 4, wherein the in-plane shear flow and shear strain functions of the thin plate between the ith and (i-1) th stiffeners in the top plate satisfy the physical equation shown in formula (11),
qi,i-1(x)＝γi,i-1(x)t1e(x)G (11)
wherein G represents a shear modulus.
6. The method for analyzing the shear hysteresis effect of the variable cross-section box girder according to claim 5, wherein i is 1,2, …,5, and when i is 3, the stiffening rods and the web plates are crossed; then the equations (9) - (11) are combined to obtain a first order differential equation system of the shear hysteresis effect, which is:
the solving process of the equation (12) is as follows:
(1) dividing the box girder into n sections along the span length, and adopting a segmented spline function to approximately express the distribution of the shear flow of the stiffening sheet in each section;
(2) converting the roof shear hysteresis effect control equation set into an algebraic equation set, and solving the algebraic equation set by combining boundary conditions;
(3) and integrating the shear flow in the stiffening sheet to obtain a distribution function of the axial force of each stiffening rod along the span length.
7. The method for analyzing shear hysteresis effect of variable cross-section box girder according to claim 6, wherein the axial force N on each stiffening rod is applied to the free end which is free from external constraint and load-freei(x) Is 0;
for the embedded end, each stiffening sheet cannot generate shear deformation due to external constraint, and the boundary conditions are shown in the formula (11):
qi,i-1(x)＝0 (13)
for the articulated boundary conditions, the support reaction force is determined and then applied to the structure in the opposite direction in the form of an external load.
8. The method for analyzing the shear hysteresis effect of the variable cross-section box beam according to claim 7, wherein the step S3 specifically comprises: the total axial force borne by each stiffening rod of the top plate under the action of uniformly distributed load of the cantilever beam is N (x), and the horizontal shear flow of the web plate acting on the top plate is qEi(x) From the balance of forces in the horizontal direction, it can be seen that:
in the formula, σxRepresents the bending normal stress of the section of the box girder, Ae1Representing the area of the ith stiffener; the moment balance equation is established for point O:
M(x)＝N(x)H(x)＝q(x)x2/2 (15)
wherein M (x) represents the external moment of an external load to a point O, and q (x) represents the shear flow of the section of the box girder;
the horizontal shear flow q at the web-to-roof interface can be obtained from the equations (14) and (15)Ei(x)：
Wherein Q (x), S1(x) Respectively representing the cross-sectional shear and the area moment.
9. The method for analyzing the shear hysteresis effect of the variable cross-section box beam according to claim 8, wherein the step S4 specifically comprises: in order to quantitatively evaluate the uneven distribution condition of the bending normal stress along the flange of the wide box girder, a shear hysteresis coefficient lambda of a web plate is defined as follows:
wherein σwThe bending positive stress at a certain height of the position of the box girder section web plate considering the shear hysteresis effect;
the bending normal stress at the corresponding section height of the box girder is calculated based on the Euler girder theory; for a solid finite element numerical solution,
the average value of the bending normal stress at the corresponding height position of the section of the box girder is obtained; in the case of the analog rod method,
for the average axial normal stress of each stiffener, the equation (18) is used to solve
Wherein A isiDenotes the area of the i-th stiffener, σiRepresenting the axial positive stress of the ith stiffener.
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