CN110032829B - Stress calculation method of steel-concrete composite beam - Google Patents

Stress calculation method of steel-concrete composite beam Download PDF

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CN110032829B
CN110032829B CN201910416635.2A CN201910416635A CN110032829B CN 110032829 B CN110032829 B CN 110032829B CN 201910416635 A CN201910416635 A CN 201910416635A CN 110032829 B CN110032829 B CN 110032829B
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刘洋
周世军
高艳芹
郑光
常鸣
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Chengdu Univeristy of Technology
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Abstract

The invention provides a stress calculation method of a steel-concrete composite beam, which relates to the field of bridge construction, and comprises the steps of dividing the composite beam into a plurality of beam section units, wherein each beam section unit comprises at least two nodes; acquiring all generalized forces including axial force and two shearing force hysteresis unit node forces on each node on a beam section unit, and forming a node force matrix; constructing a node displacement matrix corresponding to the node moment matrix; obtaining a balance equation comprising a node force matrix and a node displacement matrix based on the node force matrix and the node displacement matrix; obtaining a node displacement matrix with known node displacement based on a balance equation; and calculating the stress of the beam section unit based on the node displacement matrix. The stress calculation method solves the problem that the stress change of the main beam of the combined beam cable-stayed bridge caused by the factors such as sudden change of the shear hysteresis displacement boundary condition, continuous change of the load and the shear hysteresis displacement boundary condition and the like in the construction process can not be effectively analyzed by the stress calculation method in the prior art.

Description

Stress calculation method of steel-concrete composite beam
Technical Field
The invention relates to the technical field of bridge construction, in particular to a stress calculation method of a steel-concrete composite beam.
Background
The cable-stayed bridge is mainly divided into a concrete cable-stayed bridge, a steel cable-stayed bridge, a combined beam cable-stayed bridge and a mixed beam cable-stayed bridge according to the difference of materials of the main beam. The composite beam cable-stayed bridge is a bridge form in which a main beam is of a steel structure and a bridge deck is of a concrete structure, and the main beam and the bridge deck are stressed together through a shear connector, and is also commonly called as a composite beam cable-stayed bridge or a composite beam cable-stayed bridge. The composite beam cable-stayed bridge has the advantages that: compared with a steel cable-stayed bridge, the concrete compression can be fully utilized, so that the compression resistance of the main beam near the bridge tower is improved, the influence of larger axial force caused by stay cables and prestress is reduced, the bending resistance and the integral rigidity of the bridge are increased, and the problems of steel bridge deck fatigue of a steel box girder and easy damage of bridge deck pavement are avoided; compared with a concrete cable-stayed bridge, the cable-stayed bridge has the advantages of light self weight, stronger crossing capability and the like. Therefore, the method is considered to be a cable-stayed bridge structure scheme which integrates the advantages of the concrete cable-stayed bridge and the steel cable-stayed bridge, fills the span interval of the concrete cable-stayed bridge and the steel cable-stayed bridge with the unfavorable economic index of 400-700 m of span, and is a large-span bridge structure form with high competitiveness. With the continuous increase of bridge span and the wide application of composite structures, the composite beam cable-stayed bridge has wider application prospect.
The concrete slab in the composite beam is usually very wide, and the shear stress of the main beam is not uniform along the width direction of the concrete slab, which is called as shear hysteresis effect in engineering field, and neglecting the shear hysteresis effect underestimates the stress at the joint of the concrete slab and the steel beam, thereby causing the structure of the cable-stayed bridge to be unsafe. Therefore, the stress of the main beam under the influence of shear hysteresis in the construction process of the combined beam cable-stayed bridge is effectively analyzed, and the method has important significance on the structural safety of the cable-stayed bridge.
The width of a bridge deck (namely a concrete slab) of the composite beam cable-stayed bridge is usually very wide, and the eccentric anchoring of a stay cable and a prestressed tendon which cause the sudden change of a shearing force hysteresis displacement boundary condition caused by the fact that a main beam bears concentrated bending moment and axial force is also required to be arranged on the bridge deck. The existing method generally used for calculating the stress of the main beam in the construction process of the composite beam cable-stayed bridge is a limited beam section method, wherein the limited beam section method is to divide the main beam into a plurality of beam section units along the axial direction and then calculate the stress of each beam section unit respectively. However, the limited beam section method only considers one shear hysteresis degree of freedom for each beam section unit, cannot well adapt to various shear hysteresis analysis boundary conditions, and cannot effectively analyze the sudden change of the shear hysteresis displacement boundary conditions caused by concentrated bending moment and axial force in the construction process of the combined beam cable-stayed bridge, and the complicated main beam stress change problems caused by the continuous change of the load and the shear hysteresis displacement boundary conditions in the construction process.
Disclosure of Invention
The invention aims to provide a stress calculation method of a steel-concrete composite beam, which is used for relieving the technical problems that the prior stress calculation method can not effectively analyze the sudden change of the shear hysteresis displacement boundary condition caused by concentrated bending moment and axial force in the construction process and the stress change of a main beam of a composite beam cable-stayed bridge caused by the continuous change of the load and the shear hysteresis displacement boundary condition in the construction process and other factors in the construction process of the composite beam cable-stayed bridge.
The invention provides a stress calculation method of a steel-concrete composite beam, which comprises the following steps:
dividing the composite beam into a plurality of beam section units, wherein each beam section unit comprises at least two nodes; acquiring all generalized forces including axial force and two shearing force hysteresis unit node forces on each node on a beam section unit, and forming a node force matrix of the beam section unit according to all generalized force data acquired on each node; constructing a node displacement matrix corresponding to the node moment matrix, wherein node displacements in the node displacement matrix are all unknown quantities;
calculating the total potential energy of the beam section unit based on the node force matrix and the node displacement matrix, and performing first-order variation on the total potential energy of the beam section unit to obtain a balance equation comprising the node force matrix and the node displacement matrix;
calculating the unknown quantity of the node displacement matrix based on a balance equation to obtain a node displacement matrix with known node displacement;
and calculating the stress of the beam section unit based on the node displacement matrix with known node displacement.
Further, the beam section unit comprises an i node and a j node, and the node force matrix is as follows:
{F}＝[Ni Qi Mi Si Ti' Nj Qj Mj Sj Tj']T
n, Q, M is axial force, shearing force and bending moment on the beam section unit respectively; s and T are both generalized shear hysteresis unit node forces on the beam section unit;
the node displacement matrix with unknown node displacement is as follows:
Figure GDA0002583921630000031
wherein u is an axial displacement corresponding to the axial force; v is vertical displacement corresponding to the shearing force; theta is the section corner displacement corresponding to the bending moment, the section is the longitudinal section of the beam section unit along the width direction and the height direction of the combined beam, and theta is-v';
Figure GDA0002583921630000032
the generalized shear hysteresis displacement is corresponding to the node force of the generalized shear hysteresis unit.
Further, a coordinate system is established by taking the centroid of the composite beam as an original point, the coordinate system takes the axial direction of the composite beam as the x-axis direction, the height direction of the composite beam as the z-axis direction and the width direction of the composite beam as the y-axis direction;
the longitudinal displacement function of the steel beam in the constructed composite beam is as follows:
uw(x,z)＝u(x)+zθ(x)
wherein u isw(x, z) is the longitudinal displacement of the steel beam; u (x) is the axial displacement of the beam section unit; z is a z coordinate of a position to be subjected to stress on the composite beam; theta (x) is the corner displacement of the cross section, theta (x) is-v' (x), and v (x) is the vertical displacement of the beam section unit;
the longitudinal displacement function of a concrete slab in a constructed composite beam is:
Figure GDA0002583921630000041
wherein u isf(x, y, z) is the longitudinal displacement of the concrete slab;
Figure GDA0002583921630000042
the k-th shear hysteresis generalized displacement is obtained, and k is the position of the composite beam where the stress is to be solved; omegak(y) is a shear hysteresis warping function corresponding to the kth shear hysteresis generalized displacement,
Figure GDA0002583921630000043
wherein y is a y coordinate of a position to be subjected to stress on the beam section unit, and b is a larger value of the net width of a cantilever wing plate of the concrete slab or half of the net distance between webs; xikCorrecting a factor for the relative width of the concrete slab cantilever; zetakFor the shear deformation correction factor of the composite beam,
Figure GDA0002583921630000044
nktaking 1.2 of the inner concrete wing plate and 1.0 of the outer concrete wing plate;
calculating the strain energy U of the beam section unit based on a longitudinal displacement formula of a steel beam in the composite beam and a longitudinal displacement formula of a concrete slab in the composite beam as follows:
Figure GDA0002583921630000045
wherein E issIs the modulus of elasticity of the steel beam, EsIs a known amount; ecBeing the modulus of elasticity of the concrete slab, EcIs a known amount; gcIs the shear modulus, G, of the concrete slabcIs a known amount;w、fthe x-direction positive strain of the steel beam and the concrete slab respectively,
Figure GDA0002583921630000046
γfis the shear strain in the x-y direction of the concrete slab,
Figure GDA0002583921630000047
further, the stress calculation method further includes:
acquiring a vertically distributed load and an axially distributed load on the beam section unit, and calculating the external load potential energy V of the beam section unit based on the vertically distributed load and the axially distributed load as follows:
Figure GDA0002583921630000048
wherein q isν(x) Vertically distributing loads on the beam section units;qu(x) Axially distributing the load; l is the length of the beam section unit;
calculating the total potential energy of the beam section unit to be equal to based on the strain energy U of the beam section unit and the external load potential energy V of the beam section unit:
∏＝U+V
wherein pi is the total potential energy of the beam section unit.
Further, carrying out first-order variation on the total potential energy pi of the beam section unit to obtain:
∏＝0
substituting the strain energy U of the beam section unit and the external load potential energy V of the beam section unit into pi ═ 0 to obtain a balance equation as follows:
Figure GDA0002583921630000051
wherein A is the area of the cross section at the position of the composite beam where stress is to be solved,
Figure GDA0002583921630000052
u' is the first derivative of the axial displacement u of the beam section unit; i is the moment of inertia of the cross-section of the composite beam at the location of the desired stress to the neutral axis of the composite beam,
Figure GDA0002583921630000053
v' is the first derivative of the vertical displacement v of the beam section unit, and v "is the second derivative of v;
Figure GDA0002583921630000054
is the shear hysteresis warp area moment of the concrete slab to the y-axis,
Figure GDA0002583921630000055
Figure GDA0002583921630000056
Figure GDA0002583921630000057
is the shear hysteresis warpage inertia of the concrete slab to the y axisThe moment of the sex is determined by the ratio of the sexual moment,
Figure GDA0002583921630000058
Figure GDA0002583921630000059
is shear hysteresis cross warp moment of inertia of the concrete slab to the y-axis
Figure GDA00025839216300000510
Figure GDA00025839216300000511
Shear hysteresis displacement of the beam section unit
Figure GDA0002583921630000061
The first derivative of (a);
Figure GDA0002583921630000062
is the shear hysteresis buckling cross-sectional area of the concrete slab,
Figure GDA0002583921630000063
ω′iis omegaiOf the first derivative, ω'jIs omegajThe first derivative of (a).
Further, axial displacement u and shear hysteresis displacement
Figure GDA0002583921630000064
First-order Lagrange interpolation polynomials are adopted to obtain:
Figure GDA0002583921630000065
and (3) adopting a first-order Hermite interpolation polynomial to the vertical displacement v to obtain:
v(ξ)＝(1-3ξ2+2ξ3)vi+(3ξ2-2ξ3)vj-(ξ-2ξ2+ξ3)lθi-(ξ3-ξ2)lθj
opposite beamAxial displacement of segment unit introduces shape function matrix [ N ]u]Introducing shape function matrix [ N ] to axial displacement and vertical displacement of beam section unitv]Introducing shape function matrix to shear hysteresis displacement of beam section unit
Figure GDA0002583921630000066
Obtaining a relation formula of a node displacement matrix of unknown node displacement and axial displacement of the beam section unit as follows:
u＝[Nu]{Δ}
obtaining a relational expression of a node displacement matrix of unknown node displacement and vertical displacement of the beam section unit as follows:
v＝[Nv]{Δ}
obtaining a relation formula of a node displacement matrix of unknown node displacement and shear hysteresis displacement of the beam section unit as follows:
Figure GDA0002583921630000067
wherein N isu＝[N1 0 0 0 0 N4 0 0 0 0]，
Nv＝[0 N2 N3 0 0 0 N5 N6 0 0]，
Figure GDA0002583921630000068
Wherein N is1＝1-ξ；N2＝1-3ξ2+2ξ3；N3＝-(ξ-2ξ2+ξ3)l；N4＝ξ；N5＝3ξ2-2ξ3；N6＝-(ξ3-ξ2)l；ξ＝(x-x0)/(xl-x0)∈[0,1](ii) a x is an x coordinate value of the position of the stress to be solved on the composite beam, and x is an unknown quantity; x is the number of0X coordinate value of the starting point of the beam segment unit, x0Is a known amount; x is the number oflIs the x coordinate value of the end point of the beam section unit, xlIs a known amount;
substituting the relational expression of the axial displacement and node displacement matrix of the beam section unit, the relational expression of the vertical displacement and node displacement matrix of the beam section unit and the relational expression of the axial force and node displacement matrix of the beam section unit into a balance equation to obtain the relational expression of the rigidity matrix [ K ] and the node displacement matrix { delta } of the beam section unit:
[K]{Δ}＝{P}
wherein [ K ]]＝[Ke]+[Ks]，
Figure GDA0002583921630000071
Wherein [ K ]e]Is a matrix of elastic stiffness of the elementary beam units,
Figure GDA0002583921630000072
N′uis NuFirst derivative of (1), N ″)vIs NvThe second derivative of (a);
Figure GDA0002583921630000073
[Ks]in order to influence the rigidity matrix of the lower beam section units by shear hysteresis,
Figure GDA0002583921630000074
is composed of
Figure GDA0002583921630000075
The first derivative of (a);
and acquiring an elastic rigidity matrix of the primary beam unit and a rigidity matrix of the beam section unit.
Further, acquiring [ K ] based on the elastic stiffness matrix of the elementary beam unit and the stiffness matrix of the beam section unit; substituting [ K ] into the balance equation and obtaining a node displacement matrix { delta } with known node displacement.
Further, the stress of the steel beam of the beam section unit is obtained based on the node displacement matrix { delta } with known node displacement as follows:
σw＝Es[N′u]{Δ}-Esz[N″v]{Δ}
wherein σwStress of the steel beam being a beam section unit;
acquiring the stress of the concrete plate of the beam section unit based on the node displacement matrix { delta } with known node displacement as follows:
Figure GDA0002583921630000081
wherein σfIs the stress of the concrete slab of the beam section unit.
The stress calculation method of the steel-concrete composite beam provided by the invention can produce the following beneficial effects:
the stress calculation method of the steel-concrete composite beam provided by the invention is improved based on the existing finite element beam section analysis method, and each beam section unit comprises at least two nodes. All generalized forces including axial forces and two shear hysteresis unit node forces on each node on the beam section unit are obtained, all the generalized forces are combined into an n x 1 matrix according to all the generalized force data obtained on each node, the n x 1 matrix is a node force matrix of the beam section unit, and values of various node forces in the node force matrix can be measured or calculated based on measured known quantities, so that the node forces are all known quantities. And constructing a node displacement matrix corresponding to the node moment matrix, wherein various node displacements in the node displacement matrix correspond to various node force positions in the node force matrix, and the node displacements in the node displacement matrix are all unknown quantities. And then calculating the total potential energy of the beam section unit based on the node force matrix and the node displacement matrix, and performing first-order variation on the total potential energy of the beam section unit to obtain a balance equation comprising the node force matrix and the node displacement matrix. The balance equation comprises a node force matrix with a known quantity and a node displacement matrix with an unknown quantity, so that a node displacement matrix with known node displacement can be obtained based on the balance equation. And finally, calculating the stress values of the beam section units according to the existing stress calculation formula and the node displacement matrix with known node displacement, and combining and analyzing the stress values of each beam section unit to obtain the stress value change conditions of all parts of the combined beam.
According to the stress calculation method of the steel-concrete composite beam, in the process of calculating the stress of each beam section unit, the displacement analysis of the beam section unit is carried out on the basis of the shear hysteresis displacement caused by the node force of each node comprising two shear hysteresis units, namely, the calculation method is carried out on the basis of each node comprising two shear hysteresis degrees of freedom, and can be well adapted to the boundary conditions of various shear hysteresis analyses. In addition, the displacement caused by the shearing force and the bending moment is considered in the calculation method when the displacement of the beam section unit is calculated, and the displacement caused by the axial force is also considered. Compared with the prior art, the stress calculation method for the steel-concrete composite beam can effectively analyze the sudden change of the shear hysteresis displacement boundary condition caused by concentrated bending moment and axial force in the construction process and the stress change problem of the main beam of the cable-stayed bridge of the composite beam caused by the continuous change of the load and the shear hysteresis displacement boundary condition in the construction process.
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In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the embodiments or the prior art will be briefly described below, and it is obvious that the drawings in the following description are some embodiments of the present invention, and other drawings can be obtained by those skilled in the art without creative efforts.
Fig. 1 is a longitudinal sectional view of a beam section unit in a width direction and a height direction of a composite beam according to a first embodiment of the present invention;
FIG. 2 is a side view in longitudinal section of FIG. 1;
FIG. 3 is an analysis of generalized forces on the beam segment unit of FIG. 2;
fig. 4 is a schematic structural diagram of a cable-stayed bridge according to a second embodiment of the present invention;
fig. 5 is a longitudinal cross-sectional view of the cable-stayed bridge in fig. 4, taken in the width direction and the height direction of the cable-stayed bridge;
FIG. 6 is a schematic structural diagram of a cross-section measuring point provided in the second embodiment of the present invention;
FIG. 7 is a graph of stress difference between the calculated stress and the actual stress provided by the second embodiment of the present invention;
fig. 8 is a stress variation curve of a cable-stayed bridge test section No. S7 according to the second embodiment of the present invention.
Icon: 1-a concrete slab; 2-a steel beam; 3-east pylon; 4-west pylon; 5-section measurement point.
Detailed Description
The technical solutions of the present invention will be described clearly and completely with reference to the following embodiments, and it should be understood that the described embodiments are some, but not all, embodiments of the present invention. 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.
The first embodiment is as follows:
the stress calculation method for the steel-concrete composite beam provided by the embodiment comprises the following steps: the composite beam is divided into a plurality of beam section units, and each beam section unit comprises at least two nodes. All generalized forces including axial force and two shearing force hysteresis unit node forces on each node on the beam section unit are obtained, and a node force matrix of the beam section unit is formed according to all generalized force data obtained on each node. And constructing a node displacement matrix corresponding to the node moment matrix, wherein the node displacements in the node displacement matrix are all unknown quantities. And calculating the total potential energy of the beam section unit based on the node force matrix and the node displacement matrix, and performing first-order variation on the total potential energy of the beam section unit to obtain a balance equation comprising the node force matrix and the node displacement matrix. And calculating the unknown quantity of the node displacement matrix based on a balance equation to obtain the node displacement matrix with known node displacement. And calculating the stress of the beam section unit based on the node displacement matrix with known node displacement.
The beam section unit is arranged along the axial direction of the cable-stayed bridge combined structure, namely the combined beam is divided into a plurality of sections along the axial direction of the combined beam, and each section of the combined beam is the beam section unit.
The stress calculation method of the steel-concrete composite beam provided by the embodiment is an improvement on the existing finite element beam section analysis method. The stress calculation method for the steel-concrete composite beam provided by the embodiment sets each beam section unit to comprise at least two nodes. All generalized forces including axial forces and two shearing force hysteresis unit node forces on each node on the beam section unit are obtained, all generalized forces are combined into an n x 1 matrix according to all generalized force data obtained on each node, the n x 1 matrix is a node force matrix of the beam section unit, values of various node forces in the node force matrix can be measured by an existing measuring instrument or calculated based on measured known quantities, and therefore all node forces in the node force matrix are known quantities.
And then, a node displacement matrix corresponding to the node moment matrix can be constructed, various node displacements in the node displacement matrix correspond to various node force positions in the node force matrix, the node displacement matrix is also an n x 1 matrix, and the node displacements in the node displacement matrix are all unknown quantities.
And then calculating the total potential energy of the beam section unit based on the node force matrix and the node displacement matrix, performing first-order variation on the total potential energy of the beam section unit, and obtaining a balance equation comprising the node force matrix and the node displacement matrix according to a minimum potential energy principle. The balance equation comprises a node force matrix with a known quantity and a node displacement matrix with an unknown quantity, so that a node displacement matrix with known node displacement can be obtained based on the balance equation. And finally, calculating the stress values of the beam section units according to the existing stress calculation formula and the node displacement matrix with known node displacement, and combining and analyzing the stress values of each beam section unit to obtain the stress value change conditions of all parts of the combined beam.
In the stress calculation method for the steel-concrete composite beam provided by the embodiment, in the process of calculating the stress of each beam section unit, the displacement analysis of the beam section unit is performed based on the shear hysteresis displacement caused by the node force of each node including two shear hysteresis units, namely, the calculation method is performed based on the node force including two shear hysteresis degrees of freedom, and can be well adapted to the boundary conditions of various shear hysteresis analyses. In addition, when the displacement of the beam section unit is calculated, the calculation method not only considers the displacement caused by the shearing force and the bending moment, but also considers the displacement caused by the axial force in order to prevent the longitudinal section of the beam section along the width direction and the height direction of the combined beam from bending around the shape-center axis of the beam section unit to generate errors, so that the influence of the shear-lag effect on the combined beam can be more accurately analyzed.
Therefore, compared with the prior art, the stress calculation method for the steel-concrete composite beam provided by the embodiment can effectively analyze the sudden change of the shear hysteresis displacement boundary condition caused by concentrated bending moment and axial force in the construction process and the stress change problem of the main beam of the cable-stayed bridge of the composite beam caused by the continuous change of the load and the shear hysteresis displacement boundary condition in the construction process. How to calculate the stress of the composite beam by the method for calculating the stress of the steel-concrete composite beam according to the present embodiment is described below. As shown in fig. 1, the present embodiment takes a composite beam formed by combining a concrete slab and a double-H-shaped steel beam as an example, and describes the calculation steps of the composite beam. Further, as shown in fig. 3, the beam segment unit includes an i-node and a j-node, and the node force matrix is:
{F}＝[Ni Qi Mi Si Ti' Nj Qj Mj Sj Tj']T (1)
n, Q, M is axial force, shearing force and bending moment on the beam section unit respectively; s and T are generalized shear hysteresis unit node forces on the beam section units.
Wherein, the i node and the j node can be two ends of the beam section unit respectively.
The node displacement matrix with unknown node displacement is as follows:
Figure GDA0002583921630000121
wherein u is an axial displacement corresponding to the axial force; v is vertical displacement corresponding to the shearing force; θ is a section corner displacement corresponding to the bending moment, as shown in fig. 1, the section is a longitudinal section of the beam section unit along the width direction and the height direction of the composite beam, and θ ═ ν';
Figure GDA0002583921630000122
are all units of generalized shear hysteresisAnd generalized shear hysteresis displacement corresponding to node force.
Further, as shown in fig. 1 and 2, a coordinate system is established with the centroid of the composite beam as an origin, the coordinate system takes the axial direction of the composite beam as the x-axis direction, the height direction of the composite beam as the z-axis direction, and the width direction of the composite beam as the y-axis direction.
In this embodiment, before calculating the stress of the composite beam, the vertical crush deformation of the upper and lower flanges of the i-beam is assumed assuming that the steel beam and the concrete slab in the composite beam are both ideal linear elastomerszShear deformation gamma of steel beam from upper flange plane to lower flange planexz、γyzThe transverse bending and the transverse strain are all neglected in a trace manner, and the combined beam respectively conforms to the assumption of a flat section before and after deformation.
Further, the longitudinal displacement function of the steel beam in the construction of the composite beam is as follows:
uw(x,z)＝u(x)+zθ(x) (3)
wherein u isw(x, z) is the longitudinal displacement of the steel beam; u (x) is the axial displacement of the beam section unit; z is a z coordinate of a position to be subjected to stress on the composite beam; and theta (x) is the corner displacement of the section, theta (x) is-v' (x), and v (x) is the vertical displacement of the beam section unit.
Wherein, the z coordinate is a known quantity, and u (x), theta (x) and nu (x) are unknown quantities.
The longitudinal displacement function of a concrete slab in a constructed composite beam is:
Figure GDA0002583921630000131
wherein u isf(x, y, z) is the longitudinal displacement of the concrete slab;
Figure GDA0002583921630000132
the k-th shear hysteresis generalized displacement is obtained, and k is the position of the beam section unit where stress is to be solved; omegak(y) is a shear hysteresis warping function corresponding to the kth shear hysteresis generalized displacement,
Figure GDA0002583921630000133
the cantilever wing plate of the concrete slab is a part on the concrete slab, which goes over the steel beam and extends outwards, and the web plate is the web plate on the steel beam; xikCorrecting a factor for the relative width of the concrete slab cantilever; zetakFor the shear deformation correction factor of the composite beam,
Figure GDA0002583921630000134
nkthe inner concrete wing plate is 1.2, the outer concrete wing plate is 1.0, the inner concrete wing plate is a part of the concrete slab between two steel beams, and the outer concrete wing plate is a part of the concrete slab extending beyond the steel beams. The other symbol meanings are the same as those in the formula (3). It should be noted that the shear hysteresis warping function ω corresponding to the k-th shear hysteresis generalized displacement iskThe values of (y) refer to the book of box-type thin-wall beam shear hysteresis effect, edited by Zhang Teze, published by people's traffic press in 1998.
Wherein,
Figure GDA0002583921630000135
as unknown, y-coordinate, b, nk、ξkAnd ximaxAll are known quantities, and ω can be obtained based on the known quantitieskThe value of (y), thus ωk(y) is also a known amount.
For displacements in equations (3) and (4), where the linear displacement is positive along the coordinate axis, the angular displacement rotates positive about the coordinate axis.
Further, the strain energy U of the beam section unit is calculated based on a longitudinal displacement formula of a steel beam in the composite beam and a longitudinal displacement formula of a concrete slab in the composite beam as follows:
Figure GDA0002583921630000136
wherein E issIs the modulus of elasticity of the steel beam, EsIs a known amount; ecBeing the modulus of elasticity of the concrete slab, EcIs a known amount; gcIs the shear modulus, G, of the concrete slabcIs a known amount;w、fthe x-direction positive strain of the steel beam and the concrete slab respectively,
Figure GDA0002583921630000141
γfis the shear strain in the x-y direction of the concrete slab,
Figure GDA0002583921630000142
wherein, the formula (5) is obtained according to the definition of strain energy; es、Es、Ec、GcAll can be found according to the existing engineering material handbooks, thus all are known quantities. In addition, in this case, u (x), θ (x), and ν (x) in the formula (3) and the formula (4) are u, (x) θ, and ν (x) ν. Due to uw(x,z)、uf(x, y, z) are all unknown quantities, and thusw、f、γfSpecific numerical values cannot be calculated, and the numerical values are unknown quantities; dVwVolume of steel beams in beam section units, dVfIs the volume of the concrete slab in the beam section unit.
Further, substituting equation (3) and equation (4) into equation (5) may expand equation (5) to equation (6) as follows:
Figure GDA0002583921630000143
wherein A is the area of the cross section at the position of the composite beam where stress is to be solved,
Figure GDA0002583921630000144
u' is the first derivative of the axial displacement u of the beam section unit; i is the moment of inertia of the cross-section of the composite beam at the location of the desired stress to the neutral axis of the composite beam,
Figure GDA0002583921630000145
v' is the first derivative of the vertical displacement v of the beam section unit, and v "is the second derivative of v;
Figure GDA0002583921630000146
is the shear hysteresis warp area moment of the concrete slab to the y-axis,
Figure GDA0002583921630000147
Figure GDA0002583921630000148
Figure GDA0002583921630000149
is the shear hysteresis warp moment of inertia of the concrete slab to the y-axis,
Figure GDA00025839216300001410
Figure GDA00025839216300001411
is shear hysteresis cross warp moment of inertia of the concrete slab to the y-axis
Figure GDA00025839216300001412
Figure GDA00025839216300001413
Shear hysteresis displacement of the beam section unit
Figure GDA0002583921630000151
The first derivative of (a);
Figure GDA0002583921630000152
is the shear hysteresis buckling cross-sectional area of the concrete slab,
Figure GDA0002583921630000153
ω′iis omegaiOf the first derivative, ω'jIs omegajThe first derivative of (a).
The meaning of the rest symbols in the formula (6) is the same as that of the symbols in the formula (1) to the formula (5).
The unknowns in equation (6) are u ', θ'),
Figure GDA0002583921630000154
The numerical values represented by the remaining symbols in formula (6) are known quantities, or specific numerical values may be found from the known quantities.
Further, the stress calculation method provided in this embodiment further includes:
acquiring a vertically distributed load and an axially distributed load on the beam section unit, and calculating the external load potential energy V of the beam section unit based on the vertically distributed load and the axially distributed load as follows:
Figure GDA0002583921630000155
wherein q isν(x) Vertically distributing loads on the beam section units; q. q.su(x) Axially distributing load on the beam section unit; l is the beam section unit length.
Wherein q isν(x) And q isu(x) The loads are all loads borne in the construction process of the cable-stayed bridge and are all known quantities; l can be obtained from a tool measuring the length or from a preset cut-out length, also a known quantity. The other symbol meanings are the same as those in the formula (4).
Where the unknowns in equation (7) are { Δ }, { F }, v (x), and u (x).
Further, calculating the total potential energy pi of the beam section unit to be equal to based on the strain energy U of the beam section unit and the external load potential energy V of the beam section unit:
∏＝U+V (8)
further, carrying out first-order variation on the total potential energy pi of the beam section unit to obtain:
∏＝0 (9)
where, pi ═ U + V ═ 0, equation (6) and equation (7) are substituted into equation (9), and the equilibrium equation is obtained as:
Figure GDA0002583921630000161
wherein v 'is a first derivative of the vertical displacement v of the beam section unit, and v' is a second derivative of v; the meanings of the remaining symbols are the same as those of the symbols in the formula (6) and the formula (7).
The unknowns in equation (10) are u ', θ'),
Figure GDA0002583921630000162
{ Δ }, { F }, v (x), and u (x).
Further, axial displacement u and shear hysteresis displacement
Figure GDA0002583921630000163
First-order Lagrange interpolation polynomials are adopted to obtain:
Figure GDA0002583921630000164
and (3) adopting a first-order Hermite interpolation polynomial to the vertical displacement v to obtain:
v(ξ)＝(1-3ξ2+2ξ3)vi+(3ξ2-2ξ3)vj-(ξ-2ξ2+ξ3)lθi-(ξ3-ξ2)lθj (12)
introducing shape function matrix [ N ] into axial displacement of beam section unitu]Introducing shape function matrix [ N ] to axial displacement and vertical displacement of beam section unitv]Introducing shape function matrix to shear hysteresis displacement of beam section unit
Figure GDA0002583921630000165
Obtaining a relation formula of a node displacement matrix of unknown node displacement and axial displacement of the beam section unit as follows:
u＝[Nu]{Δ} (13)
obtaining a relational expression of a node displacement matrix of unknown node displacement and vertical displacement of the beam section unit as follows:
v＝[Nv]{Δ} (14)
obtaining a relation formula of a node displacement matrix of unknown node displacement and shear hysteresis displacement of the beam section unit as follows:
Figure GDA0002583921630000171
wherein, the formula (13), the formula (14) and the formula (15) are vector forms obtained by combining the formula (11) and the formula (12) with respective shape function matrixes.
Wherein, the formula (13) and the formula (14) after the formula (11) and the formula (12) are combined with the respective shape function matrixes are in a simplified form; in the formula (13)
Nu＝[N 1 0 0 0 0 N 4 0 0 0 0]In formula (14)
Nv＝[0 N2 N3 0 0 0 N5 N6 0 0]In equation (15)
Figure GDA0002583921630000172
Wherein N is1＝1-ξ；N2＝1-3ξ2+2ξ3；N3＝-(ξ-2ξ2+ξ3)l；N4＝ξ；N5＝3ξ2-2ξ3；N6＝-(ξ3-ξ2)l；ξ＝(x-x0)/(xl-x0)∈[0,1](ii) a x is an x coordinate value of the position of the stress to be solved on the combined beam, and x is an unknown quantity; x is the number of0An x-coordinate value, x, of the origin of the beam segment unit0Is a known amount; x is the number oflIs the x coordinate value of the end point of the beam section unit, xlIn known amounts.
According to x, x0And xlXi expressed in x can be solved, and then substituted into N1、N2、N3、N4、N5And N6In (3), the expression of [ N ] by x can be obtainedu]、[Nv]And
Figure GDA0002583921630000173
further, substituting the formula (13), the formula (14) and the formula (15) into the formula (10), i.e. the equilibrium equation, the following formula (16) is obtained by arranging:
Figure GDA0002583921630000174
wherein, the meaning of each symbol in the formula (16) is the same as that of each symbol in the formula (10).
Since { Δ } may be an arbitrary value, equation (16) may be reduced to equation (17) below:
[K]{Δ}＝{P} (17)
and the formula (17) is a relational expression of the rigidity matrix [ K ] of the beam section unit and the node displacement matrix [ delta ].
Wherein [ K ]]＝[Ke]+[Ks]，
Figure GDA0002583921630000181
{ P } is an equivalent node load array, for { P }, according to [ N expressed by xu]And [ Nv]And according to known quantities { F }, q }ν(x)、qu(x) And l can obtain a specific value of { P }.
[Ke]Is the elastic stiffness matrix of the elementary beam unit, [ Ke]Can be expressed by the following formula (18):
Figure GDA0002583921630000182
wherein, N'uIs NuFirst derivative of (1), N ″)vIs NvThe second derivative of (a). According to [ N ] expressed with xu]And
Figure GDA0002583921630000183
from A and I, from which specific values can be found, and from the known quantity EcAnd l can be obtained as [ Ke]The specific numerical value of (1).
[Ks]For shear hysteresis to affect the stiffness matrix of the lower beam section units, [ K ]s]Can be expressed by the following formula (19):
Figure GDA0002583921630000184
wherein,
Figure GDA0002583921630000191
is composed of
Figure GDA0002583921630000192
The first derivative of (a); according to [ N ] expressed with xu]、[Nv]And
Figure GDA0002583921630000193
from A and I, from which specific values can be found, and from the known quantity Ec、GcAnd l can be obtained as [ Ke]The specific numerical value of (1).
Thus, the elastic stiffness matrix [ K ] of the elementary beam unitse]And stiffness matrix [ K ] of beam section unitss]May be acquired. Obtaining an elastic stiffness matrix [ K ] of the primary equal beam unite]And stiffness matrix [ K ] of beam section unitss]After being obtained, [ K ] is addede]And [ K ]s]Add to obtain [ K]The specific numerical value of (1). Then can be converted into [ K ]]And substituting the obtained data into the formula (10), namely substituting the obtained data into a balance equation and calculating a node displacement matrix { delta } with known node displacement.
Further, the stress calculation formula is a stress that is an elastic modulus × strain, and the stress σ of the beam section unit can be obtained by combining the formula (3), the formula (13), and the formula (14)wThe expression of (a) is:
σw＝Es[N′u]{Δ}-Esz[N″v]{Δ} (20)
will be expressed by x [ Nu]And [ Nv]A node displacement matrix { Δ } for which the node displacement is known, and a known quantity EsThe stress sigma of the steel beam at any position on the beam section unit can be obtained by substituting the formula (20)wThe specific numerical value of (1).
Further, the stress calculation formula is stress ═ elastic modulus × strain, and the formula (c) is combined4) And the formula (13), the formula (14) and the formula (15) to obtain the stress sigma of the concrete plate of the beam section unitfThe expression of (a) is:
Figure GDA0002583921630000194
will be expressed by x [ Nu]、[Nv]And
Figure GDA0002583921630000195
a node displacement matrix { Δ } in which the node displacement is known, and a known quantity EcThe stress sigma of the steel beam at any position on the beam section unit can be obtained by substituting the formula (21)fThe specific numerical value of (1).
In summary, it can be known that the stress of the beam section unit obtained according to the formula (20) and the formula (21) in this embodiment is calculated based on the shear hysteresis displacement caused by the node force of the two shear hysteresis units included in each node on the beam section unit and the stress calculated by the displacement caused by the axial force, and compared with the existing stress calculation method, the stress calculated by the stress calculation method provided by the present invention is closer to the actual stress value of the composite beam.
In addition, the stress calculation method for the steel-concrete composite beam provided by the embodiment can be further compiled into a program for calculating stress, and simulation analysis software for calculating stress is formed. After the beam section units of the combined beam are divided and the parameters are numbered, the original information of the combined beam is arranged to form a data file, then the rigidity matrix of the beam section units and the stress on the beam section units can be calculated according to simulation analysis software, and finally finite element analysis results of all stages in the construction process of the combined beam cable-stayed bridge can be formed.
When the beam section units of the combined beam are divided, the larger the number of the divided beam section units is, the more the finite element analysis result is attached to the actual stress state of the combined beam, but the calculation amount is huge. The embodiment provides a method for calculating the stress of the composite beam by using simulation analysis software, which can greatly save manpower and improve the working efficiency.
Example two:
in this embodiment, the stress calculation method for a steel-concrete composite beam provided in the first embodiment is applied to the stress calculation process for a composite beam in an actual composite beam cable-stayed bridge construction process, and then the stress calculated by the stress calculation method for a steel-concrete composite beam provided in the first embodiment and the stress calculated by the existing stress calculation method are respectively compared with the actually measured stress for a composite beam. According to the comparison result, the stress calculation method for the steel-concrete composite beam provided by the embodiment has smaller error compared with the existing stress calculation method, and can effectively analyze the sudden change of the shear hysteresis displacement boundary condition caused by concentrated bending moment and axial force in the construction process and the stress change problem of the main beam of the cable-stayed bridge of the composite beam caused by the continuous change of the load and the shear hysteresis displacement boundary condition in the construction process.
As shown in fig. 4, in this embodiment, the stress of the double-tower double-cable-side composite beam cable-stayed bridge in the construction process is calculated by taking the double-tower double-cable-side composite beam cable-stayed bridge as an example. The double-tower double-cable-plane combined beam cable-stayed bridge is a 714 m-full-length double-tower (99m reinforced concrete A-type tower) double-cable-plane combined beam cable-stayed bridge, and the double towers form a main tower of the cable-stayed bridge and comprise an east cable tower and a west cable tower.
The span combination of the main bridge of the cable-stayed bridge is 77m +100m +360m +100m +77 m. Wherein, the main beam is an I-shaped steel-concrete combined beam, and the cross section of the main beam is II-shaped as shown in figure 5. The steel beam is a welded I-beam and is made of Q370Q-E-Z35 steel; the bridge deck is made of precast concrete plates with the strength of C55, and the square amount of the bridge deck is 4.65m2The whole bridge comprises 704 blocks, and each block weighs 12.1 t; the stay cable is a low-relaxation steel wire bundle with the diameter of 7mm, and the specifications of the stay cable adopted by the cable-stayed bridge are 5 in total. And 14 stay cables (14 multiplied by 2) are symmetrically arranged on both sides of the east cable tower and both sides of the west cable tower in each main tower, 112 stay cables are arranged on each main tower in total, the maximum cable length is 185.571m, the maximum cable weight is 17t, and the standard distance between every two adjacent stay cables is 12.0 m. The standard tensile strength of the prestressed stranded wire adopted by the stay cable is 1860MPa, the nominal diameter is 15.2mm, and the nominal area is 139mm2Modulus of elasticity EpThe relaxation rate was 3.5% at 1.95 × 105 MPa.
The main beam of the cable-stayed bridge is constructed by adopting a cantilever method and a support method, two side beam sections adjacent to the east cable tower, two side beam sections adjacent to the west cable tower, a side span beam section on one side of the east cable tower far away from the west cable tower, and a side span beam section on one side of the west cable tower far away from the east cable tower are constructed by adopting the support method, and the rest beam sections are constructed by adopting a cantilever assembly method. The main construction sequence of the standard beam section is as follows: the method comprises the steps of steel beam installation, one stay cable (first stay cable tensioning), bridge deck installation and wet joint construction, two stay cables (second stay cable tensioning) and forward movement of a bridge deck crane.
In the construction process of the cable-stayed bridge, the actual stress test process of the composite beam is as follows: as shown in FIG. 4, the full bridge has 16 test sections (S) according to the corresponding relationship between east and west pylons1～S16) Correspondingly, the full bridge is divided into 16 beam section units, the arrangement of the section measuring points on each test section is shown in fig. 6, the stress of the section measuring points on the bridge deck is tested by using the existing steel bar stress meter, and the stress of the section measuring points on the steel beam is tested by using the existing surface strain meter. The actual stress value of the composite beam in the construction process of the cable-stayed bridge can be measured through the steel bar stress meter and the surface strain meter.
Based on the method for calculating the stress of the steel-concrete composite beam in the first embodiment, a Simulation Analysis software (CSB) for the structure of the cable-stayed Bridge Construction Process can be generated. The software adopts a plate-beam model, and both a concrete plate and a steel beam are simulated by using a double-layer beam model; simulating the shear nails and the bridge tower by using the plane beam units; the influence of the sag effect is considered for the stay cable, and a limited number of straight rod units are adopted for simulation. The CSB analysis of the stress of the combined beam in the construction process of the cable-stayed bridge mainly comprises the following steps: discretizing the upper structure of the cable-stayed bridge to finish beam section unit division and parameter numbering; arranging the original information of the combined beam to form a data file; thirdly, carrying out beam section unit analysis and integral analysis and forming a rigidity matrix; fourthly, load vectors are added according to the actual construction process, and the displacement matrix of the equation set is modified and perfected by combining the actual boundary conditions; and fifthly, solving by using a displacement matrix equation to obtain the displacement of each node and calculating the stress and the internal force of the beam section unit so as to form a finite element analysis result of each stage in the construction process.
For the twin-tower twin-cable-plane composite beam cable-stayed bridge in the present embodiment, when the CSB software is used to calculate the stress of the composite beam, the CSB model thereof comprises 884 nodes (the nodes comprise nodes on concrete plates and nodes on steel beams), 1245 units (358 concrete plates, 358 steel beams, 56 stay cables, 104 bridge towers, 10 tower-beam connections, 359 shear connectors) and 124 construction stages.
By using the CSB software, the stress value of the composite beam calculated by using the method for calculating the stress of the steel-concrete composite beam in the first embodiment can be obtained, and the stress value of the composite beam is a theoretical solution obtained by using the CSB software.
And subtracting a theoretical solution calculated by CSB software from the stress value actually measured by the steel bar stress meter and the surface strain meter to obtain a result, namely the stress difference. And the stress difference values of the above 16 test sections are shown in fig. 7. Through the stress difference curve diagram shown in fig. 7, it can be seen that the stress difference between the theoretical solution calculated by using the CSB software and the actually measured stress value is ± 3MPa, the relative error is not more than 5%, and the positive and negative values are staggered and present a certain randomness, which indicates that no system error exists, and verifies that the theoretical solution calculated by using the CSB software has higher precision and accuracy.
In addition, the present embodiment is directed to S7The section measuring points on the steel beam in the test section are analyzed when the bridge crane goes from S7The test section is moved to S6When corresponding No. 6 beam section unit, S7The stress change at the section test point on the test section is shown in fig. 8, where the upper edge is the upper flange of the i-beam and the lower edge is the lower flange of the i-beam. As can be seen from fig. 8, considering that the stress under the influence of shear hysteresis is well matched with the measured value, the theoretical solution calculated by the CSB software, that is, the stress distribution of the cross section of the composite beam can be effectively solved by using the method for calculating the stress of the steel-concrete composite beam in the first embodiment.
In order to compare the conventional calculation results of Multi-tier Distributed Applications Services (MIDAS) of the existing general finite element analysis software, a MIDAS space finite element structure model is also established for the double-tower double-cable-surface composite beam cable-stayed bridge, and 1303 units are calculated in total.
By using MIDAS software, the stress value of the composite beam calculated by using the existing composite beam stress calculation method can be obtained, and the stress value of the composite beam is a theoretical solution obtained by using the MIDAS software.
During construction, the place with large stress change is the tower root, the auxiliary pier, the front end of the cantilever and the like of the cable-stayed bridge. In this embodiment, the CSB software and the MIDAS software are used to calculate the stress of the upper edge of the east pylon steel beam (the pylon root and the auxiliary pier) and the general construction site (the stay cable) at different construction stages, respectively, and the stress value of the above sites is measured by using a steel bar stress meter and a surface strain gauge.
Then, the theoretical solution calculated by the CSB software is subtracted from the actual measurement stress value to obtain the relative difference between the theoretical solution and the actual measurement value calculated by the CSB software, as shown in table 1, table 1 is a data table of the theoretical solution and the actual measurement value calculated by the CSB software and the relative difference between the theoretical solution and the actual measurement value.
TABLE 1
Figure GDA0002583921630000231
Note: the working conditions of the largest double cantilever in table 1 are: after the construction of the No. 7 beam section unit is finished, the bridge deck crane moves forwards to the No. 7 beam section unit; the working conditions of the maximum single cantilever are as follows: the side spans of the cable-stayed bridge are closed, and the bridge deck crane is moved to a 14-number beam section unit and is ready for mid-span closing.
The theoretical solution calculated by the MIDAS software is subtracted from the actual measurement stress value to obtain the relative difference between the theoretical solution calculated by the MIDAS software and the actual measurement value, as shown in table 2, table 2 is a data table of the theoretical solution and the actual measurement value calculated by the MIDAS software and the relative difference between the theoretical solution and the actual measurement value.
TABLE 2
Figure GDA0002583921630000241
Note: the working conditions of the largest double cantilever in table 2 are: after the construction of the No. 7 beam section unit is finished, the bridge deck crane moves forwards to the No. 7 beam section unit; the working conditions of the maximum single cantilever are as follows: the side spans of the cable-stayed bridge are closed, and the bridge deck crane is moved to a 14-number beam section unit and is ready for mid-span closing.
The following can be seen from tables 1 and 2: the relative difference between the measured value and the CSB calculation result is not more than 3%, and the result calculated by the steel-concrete composite beam stress calculation method provided by the embodiment has higher precision and accuracy. The relative difference of the MADIS calculation result even exceeds 10% in a part of construction stages of parts with large stress mutation, the minimum deviation is also more than 7%, but the relative difference of the stress calculation of the parts with small mutation does not exceed 5%, and the result error calculated by the existing stress calculation method is verified to be large.
In conclusion, compared with the existing stress calculation method, the stress calculation method for the steel-concrete composite beam provided by the invention can effectively solve the stress of the composite beam under the influence of factors such as concentrated load sudden change, boundary condition change and the like in the construction process of the cable-stayed bridge, and has the advantages of better practicability and higher precision.
Finally, it should be noted that: the above embodiments are only used to illustrate the technical solution of the present invention, and not to limit the same; while the invention has been described in detail and with reference to the foregoing embodiments, it will be understood by those skilled in the art that: the technical solutions described in the foregoing embodiments may still be modified, or some or all of the technical features may be equivalently replaced; and the modifications or the substitutions do not make the essence of the corresponding technical solutions depart from the scope of the technical solutions of the embodiments of the present invention.

Claims (4)

1. A stress calculation method of a steel-concrete composite beam is characterized by comprising the following steps:
dividing the composite beam into a plurality of beam section units, wherein each beam section unit comprises at least two nodes;
acquiring all generalized forces including axial force and two shearing force hysteresis unit node forces on each node on the beam section unit, and forming a node force matrix of the beam section unit according to all generalized force data acquired on each node;
constructing a node displacement matrix corresponding to the node moment matrix, wherein node displacements in the node displacement matrix are all unknown quantities;
calculating the total potential energy of the beam section unit based on the node force matrix and the node displacement matrix, and performing first-order variation on the total potential energy of the beam section unit to obtain a balance equation comprising the node force matrix and the node displacement matrix;
calculating the unknown quantity of the node displacement matrix based on the balance equation to obtain the node displacement matrix with known node displacement;
calculating the stress of the beam section unit based on the node displacement matrix with known node displacement;
the beam section unit comprises an i node and a j node, and the node force matrix is as follows:
{F}＝[Ni Qi Mi Si Ti' Nj Qj Mj Sj T′j]T
n, Q, M is axial force, shearing force and bending moment on the beam section unit respectively; s and T are both generalized shear hysteresis unit node forces on the beam section unit;
the node displacement matrix with unknown node displacement is as follows:
Figure FDA0002583921620000011
wherein u is an axial displacement corresponding to the axial force; v is a vertical displacement corresponding to the shear force; theta is the section corner displacement corresponding to the bending moment, the section is a longitudinal section of the position to be subjected to stress on the combined beam along the width direction and the height direction of the combined beam, and theta is-v';
Figure FDA0002583921620000021
generalized shear hysteresis displacement corresponding to the node force of the generalized shear hysteresis unit is adopted;
establishing a coordinate system by taking the centroid of the combined beam as an original point, wherein the coordinate system takes the axial direction of the combined beam as the x-axis direction, the height direction of the combined beam as the z-axis direction and the width direction of the combined beam as the y-axis direction;
constructing a longitudinal displacement function of the steel beam in the combined beam as follows:
uw(x,z)＝u(x)+zθ(x)
wherein u isw(x, z) is the longitudinal displacement of the steel beam; u (x) is the axial displacement of the beam section unit; z is a z coordinate of a position to be subjected to stress on the composite beam; theta (x) is the corner displacement of the section, theta (x) is-v' (x), and v (x) is the vertical displacement of the beam section unit;
the longitudinal displacement function of the concrete slab in constructing the composite beam is as follows:
Figure FDA0002583921620000022
wherein u isf(x, y, z) is the longitudinal displacement of the concrete slab;
Figure FDA0002583921620000023
the k-th shear hysteresis generalized displacement is obtained, and k is the position of the beam section unit where stress is to be solved; omegak(y) is a shear hysteresis warping function corresponding to the kth shear hysteresis generalized displacement,
Figure FDA0002583921620000024
y is a y coordinate of a position to be subjected to stress on the composite beam, and b is a larger value of the clear width of a cantilever wing plate of the concrete slab or half of the clear distance between webs; xikCorrecting a factor for the concrete slab cantilever relative width; zetakFor the shear deformation correction factor of the composite beam,
Figure FDA0002583921620000025
nk1.2 is taken for the inner side concrete wing plate, and 1.0 is taken for the outer side concrete wing plate;
calculating strain energy U of the beam section unit based on a longitudinal displacement formula of a steel beam in the composite beam and a longitudinal displacement formula of a concrete plate in the composite beam as follows:
Figure FDA0002583921620000026
wherein E issIs the modulus of elasticity of the steel beam, EsIs a known amount; ecBeing the modulus of elasticity of the concrete slab, EcIs a known amount; gcIs the shear modulus, G, of the concrete slabcIs a known amount;w、fthe x-direction positive strain of the steel beam and the concrete slab respectively,
Figure FDA0002583921620000031
γffor x-y shear strains of the concrete slab,
Figure FDA0002583921620000032
the stress calculation method further includes:
acquiring a vertically distributed load and an axially distributed load on the beam section unit, and calculating the external load potential energy V of the beam section unit based on the vertically distributed load and the axially distributed load as follows:
Figure FDA0002583921620000033
wherein q isν(x) Vertically distributing loads on the beam section units; q. q.su(x) Axially distributing the load; l is the length of the beam section unit;
calculating the total potential energy of the beam section unit to be equal to based on the strain energy U of the beam section unit and the external load potential energy V of the beam section unit:
∏＝U+V
II, wherein pi is the total potential energy of the beam section unit;
carrying out first-order variation on the total potential energy pi of the beam section unit to obtain:
∏＝0
substituting the beam section unit strain energy U and the beam section unit external load potential energy V into pi ═ 0 to obtain the balance equation as follows:
Figure FDA0002583921620000034
wherein A is the area of the cross section at the position of the composite beam where stress is to be solved,
Figure FDA0002583921620000041
u' is the first derivative of the axial displacement u of the beam section unit; i is the moment of inertia of the cross-section of the composite beam at the location of the desired stress to the neutral axis of the composite beam,
Figure FDA0002583921620000042
v' is the first derivative of the vertical displacement v of the beam section unit, and v "is the second derivative of v;
Figure FDA0002583921620000043
is the shear hysteresis warp area moment of the concrete slab to the y-axis,
Figure FDA0002583921620000044
Figure FDA0002583921620000045
Figure FDA0002583921620000046
is the shear hysteresis warp moment of inertia of the concrete slab to the y-axis,
Figure FDA0002583921620000047
Figure FDA0002583921620000048
is shear hysteresis cross warp moment of inertia of the concrete slab to the y-axis
Figure FDA0002583921620000049
Figure FDA00025839216200000410
Shear hysteresis displacement of the beam section unit
Figure FDA00025839216200000411
The first derivative of (a);
Figure FDA00025839216200000412
is the shear hysteresis buckling cross-sectional area of the concrete slab,
Figure FDA00025839216200000413
ω′iis omegaiOf the first derivative, ω'jIs omegajThe first derivative of (a).
2. The stress calculation method of claim 1, wherein the axial displacement u and the shear hysteresis displacement are measured
Figure FDA00025839216200000414
First-order Lagrange interpolation polynomials are adopted to obtain:
Figure FDA00025839216200000415
and adopting a first-order Hermite interpolation polynomial to the vertical displacement v to obtain:
v(ξ)＝(1-3ξ2+2ξ3)vi+(3ξ2-2ξ3)vj-(ξ-2ξ2+ξ3)lθi-(ξ3-ξ2)lθj
introducing a shape function matrix [ N ] into the axial displacement of the beam section unitu]Introducing a shape function matrix [ N ] into the axial displacement and vertical displacement of the beam section unitv]Introducing a shape function matrix to the shear hysteresis displacement of the beam section unit
Figure FDA00025839216200000416
Obtaining a relational expression of the axial displacement of the beam section unit and the node displacement matrix of which the node displacement is unknown, wherein the relational expression is as follows:
u＝[Nu]{Δ}
obtaining a relational expression of the vertical displacement of the beam section unit and the node displacement matrix with unknown node displacement as follows:
v＝[Nv]{Δ}
obtaining a relation between the shear hysteresis displacement of the beam section unit and the node displacement matrix with unknown node displacement as follows:
Figure FDA0002583921620000051
wherein N isu＝[N1 0 0 0 0 N4 0 0 0 0]，
Nv＝[0 N2 N3 0 0 0 N5 N6 0 0]，
Figure FDA0002583921620000052
Wherein N is1＝1-ξ；N2＝1-3ξ2+2ξ3；N3＝-(ξ-2ξ2+ξ3)l；N4＝ξ；N5＝3ξ2-2ξ3；N6＝-(ξ3-ξ2)l；ξ＝(x-x0)/(xl-x0)∈[0,1](ii) a x is x at the position of the stress to be solved on the composite beamCoordinate values, x being an unknown quantity; x is the number of0An x-coordinate value, x, of the origin of the beam segment unit0Is a known amount; x is the number oflIs the x coordinate value of the end point of the beam section unit, xlIs a known amount;
substituting the relational expression of the axial displacement of the beam section unit and the node displacement matrix, the relational expression of the vertical displacement of the beam section unit and the node displacement matrix, and the relational expression of the axial force of the beam section unit and the node displacement matrix into the balance equation to obtain the relational expression of the stiffness matrix [ K ] of the beam section unit and the node displacement matrix { delta }:
[K]{Δ}＝{P}
wherein [ K ]]＝[Ke]+[Ks]，
Figure FDA0002583921620000053
Wherein [ K ]e]Is a matrix of elastic stiffness of the elementary beam units,
Figure FDA0002583921620000054
N'uis NuFirst derivative of (1), N ″)vIs NvThe second derivative of (a);
Figure FDA0002583921620000061
[Ks]for the stiffness matrix of the beam section units under the influence of shear hysteresis,
Figure FDA0002583921620000062
is composed of
Figure FDA0002583921620000063
The first derivative of (a);
and acquiring an elastic rigidity matrix of the elementary beam unit and a rigidity matrix of the beam section unit.
3. The stress calculation method according to claim 2, wherein the [ K ] is obtained based on an elastic stiffness matrix of the primary beam unit and a stiffness matrix of the beam section unit; and substituting the [ K ] into the balance equation and obtaining a node displacement matrix { delta } with known node displacement.
4. The stress calculation method according to claim 3, wherein the stress of the steel beam of the beam section unit is obtained based on a node displacement matrix { Δ } in which the node displacement is known as:
σw＝Es[N′u]{Δ}-Esz[N″v]{Δ}
wherein σwStress of a steel beam being the beam section unit;
acquiring the stress of the concrete plate of the beam section unit based on the node displacement matrix { delta } with the known node displacement as follows:
Figure FDA0002583921620000064
wherein σfIs the stress of the concrete slab of the beam section unit.
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