WO2021100693A1

WO2021100693A1 - Method of evaluating continuous beams, program for evaluating continuous beams, and method for evaluating composite beams

Info

Publication number: WO2021100693A1
Application number: PCT/JP2020/042755
Authority: WO
Inventors: 政樹有田; 清水　信孝; 聡北岡
Original assignee: 日本製鉄株式会社
Priority date: 2019-11-21
Filing date: 2020-11-17
Publication date: 2021-05-27
Also published as: CN114945923A

Abstract

This method of evaluating continuous beams includes a solution determination step for finding, on the basis of given conditions, a plurality of bending moments at an intermediate fulcrum and a pair of end fulcrums, and a plurality of rotational angles at the pair of end fulcrums. The given conditions include: length and flexural rigidity of each of n beams; a plurality of rotational rigidities at the intermediate fulcrum and the pair of end fulcrums; vertical load acting on the n beams; and a plurality of perpendicular displacements at the intermediate fulcrum and the pair of end fulcrums. In the solution determination step: the plurality of bending moments and the plurality of rotational angles are defined as a plurality of unknowns; a relational expression for the plurality of perpendicular displacements, and relational expressions for the plurality of rotational rigidities, plurality of bending moments, and plurality of rotational angles are defined as a plurality of first boundary conditions of the same number as that of the plurality of unknowns; and at such time, the plurality of unknowns are solved so that the plurality of unknowns satisfy the plurality of first boundary conditions, thereby evaluating the bending moment and the sag distribution of the continuous beam.

Description

Evaluation method of continuous beam, evaluation program of continuous beam, and evaluation method of composite beam

The present invention relates to a continuous beam evaluation method, a continuous beam evaluation program, and a composite beam evaluation method.
The present application claims priority based on Japanese Patent Application No. 2019-210798 and Japanese Patent Application No. 2019-210811, which were filed in Japan on November 21, 2019, and the contents thereof are incorporated herein by reference.

Conventionally, continuous beams in which a plurality of beams are semi-rigidly joined to each other are known (see, for example,

Patent Documents

1 and 2 and Non-Patent Document 1).

On the other hand, conventionally, in a composite beam in which both ends are semi-rigidly joined, a method for evaluating a composite beam when the bending rigidity of forward bending and the bending rigidity of negative bending are different from each other is known (for example, Patent Document 3). reference).
Patent Document 3 solves the equation (positive function) for the bending moment and the deflection of the composite beam shown in the specification when a two-point concentrated load acts on the composite beam. Then, the maximum value of the bending moment at the end of the composite beam (the bending moment acting on the end of the composite beam) and the deflection is calculated.

Japanese Patent Application Laid-Open No. 2015-068001 Japanese Patent Application Laid-Open No. 2005-282019 Japanese Patent Application Laid-Open No. 2018-09410

However, in this type of continuous beam, a practical evaluation method of bending moment and deflection distribution generated when a load is applied to the continuous beam has not been provided.
Even if both ends are rigidly joined to the end support portion, for example, when the end support portion is a steel girder, the torsional resistance of the girder is insufficient to evaluate it as a rigid joint. Therefore, assuming rigid joints, the bending moment and deflection distribution are underestimated when loads of different sizes are applied to each beam of continuous beams. Therefore, there is a risk of evaluating continuous beams on the dangerous side.
Furthermore, for continuous beams with different bending stiffness between forward bending and negative bending, a practical evaluation method is provided to determine the bending moment and deflection distribution of the continuous beam in consideration of these bending stiffness and the joint state at both ends. There wasn't.

Conventionally, continuous beams are evaluated on the assumption that both ends of the continuous beams are supported by pin joints so that the continuous beams are not evaluated on the dangerous side. Therefore, the bending moment and the deflection distribution generated in the center of the continuous beam in the longitudinal direction are overestimated. As a result, it is determined that it is necessary to use a beam having a large cross section orthogonal to the longitudinal direction as a continuous beam, and a continuous beam that is more expensive than necessary is used.

In view of these problems, the first problem is to provide a continuous beam evaluation method and a continuous beam evaluation program that can more appropriately evaluate the bending moment and deflection distribution of a continuous beam according to the joint state at both ends. is there.

On the other hand, a case where an evenly distributed load acts on the composite beam over the entire length will be described. In this case, since this equation is an implicit function, it is necessary to perform convergence calculation in order to calculate the bending moment and deflection of the composite beam from the equation for the bending moment and deflection of the composite beam. Furthermore, for example, in order to dynamically analyze a structure having a plurality of composite beams, it is necessary to solve this by a convergence calculation by an implicit method. A large amount of time is required because the double convergence calculation of the composite beam and the structure is required.
Further, for example, the case where the optimization calculation for minimizing the objective function is performed with the cross-sectional area of the composite beam as the objective function and the design conditions such as the deflection and the bending moment as the constraint conditions will be described. In this case, a large amount of time is required because double convergence calculation is required for the optimization of the constraint condition and the objective function.

In view of these problems, there is a second problem of providing an evaluation method for a composite beam that can calculate the maximum values of bending moment and deflection at the end of a composite beam without performing convergence calculation.

The present invention has been made in view of the first and second problems, and provides a continuous beam evaluation method, a continuous beam evaluation program, and a composite beam evaluation method capable of appropriately evaluating deflection. The purpose is.

In order to solve the above problems, the present invention proposes the following means.
(1) The first aspect of the present invention is n, which is two or more natural numbers arranged side by side in the longitudinal direction and having end portions adjacent to each other in the longitudinal direction semi-rigidly joined to each other to form an intermediate fulcrum. A method for evaluating a continuous beam including a book beam in which both ends of the entire continuous beam are paired end fulcrums, wherein the intermediate fulcrum and the pair of end fulcrums are used. It has a solution determination step of obtaining a plurality of bending moments and a plurality of rotation angles at the pair of end fulcrums based on given conditions, and the given conditions are the length and bending rigidity of each of the n beams. And; including a plurality of rotational rigidity at the intermediate fulcrum and the pair of end fulcrums; a vertical load acting on the n beams; and a plurality of vertical displacements at the intermediate fulcrum and the pair of end fulcrums; In the solution determination step, the plurality of bending moments and the plurality of rotation angles are defined as a plurality of unknowns, the relational expressions of the plurality of rotational rigidity, the plurality of bending moments, and the plurality of rotation angles, and the said. When the relational expression of the plurality of vertical displacements is defined as the plurality of first fulcrums having the same number as the number of the plurality of unknowns, the plurality of unknowns satisfy the plurality of first fulcrums so as to satisfy the plurality of first fulcrums. The unknown number is solved to evaluate the bending moment and the deflection distribution of the continuous beam.

According to this aspect, the number of the plurality of unknowns and the number of the plurality of first boundary conditions are the same. Therefore, a plurality of first boundary conditions are set based on the beam length and flexural rigidity, a plurality of rotational rigidity, a vertical load, or a plurality of vertical displacements at an intermediate fulcrum and a pair of end fulcrums given as given conditions. Solve multiple unknowns to satisfy. By this method, a plurality of bending moments and a plurality of rotation angles included in a plurality of unknowns can be obtained.
Then, a plurality of vertical displacements are calculated from the obtained plurality of bending moments and a plurality of rotation angles based on the relational expressions of the plurality of vertical displacements. According to this method, for example, the bending moment and the deflection distribution of the continuous beam can be more appropriately evaluated according to the joint state at the pair of end fulcrums. Therefore, the deflection can be appropriately evaluated.

(2) In the evaluation method of the continuous beam according to the above (1), the solution determination step is the first design step of giving a design value including the plurality of vertical displacements; A second design step of imparting a second boundary condition to the pair of end fulcrums accordingly; includes the bending moment acting on the pair of end fulcrums and the intermediate fulcrum depending on the joining state at the pair of end fulcrums. A tentative design process that gives a tentative design value; a calculated value that includes a calculation result of vertical displacement at the intermediate fulcrum so as to satisfy the plurality of first boundary conditions and the second boundary condition based on the tentative design value. A solution calculation step for calculating; and a residual calculation step for obtaining a displacement residual which is a residual between the design value and the calculated value; whether or not the displacement residual is smaller than a predetermined threshold value. It may have a determination step for determining;
According to this aspect, the calculated value calculated based on the tentative design value is evaluated based on the design value and the displacement residual by determining whether or not the displacement residual is smaller than the threshold value in the determination step. be able to.

(3) In the evaluation method of the continuous beam according to the above (2), the solution determination step further includes a storage step of storing the temporary design value after the temporary design step, and in the determination step, the temporary design value is stored. When the displacement residual is equal to or greater than the threshold value, another new temporary design value is given in the temporary design process in place of the temporary design value stored in the storage process, and the new temporary design value is given. The solution calculation step, the residual calculation step, and the determination step are performed based on the above, and the determination step is repeated until the displacement residual is determined to be smaller than the threshold value in the determination step. The bending moment of the provisional design value when the displacement residual is determined to be smaller than the threshold value is defined as the bending moment acting on the intermediate fulcrum, and the plurality of bending moments are defined based on the bending moment. The rotation angle may be obtained.
According to this aspect, a new tentative design value is given until the displacement residual is determined to be smaller than the threshold value in the determination step, and the solution calculation step and the residual calculation are performed based on the new tentative design value. Repeat the process and the judgment process as a set. By this method, the bending moment can be calculated with arbitrary accuracy. Then, a plurality of rotation angles can be obtained based on this bending moment.

(4) In the method for evaluating a continuous beam according to (2) or (3), in the solution calculation step, the first end fulcrum, which is one of the continuous beams among the n beams, is the first end fulcrum. To the second end fulcrum, which is the other end fulcrum of the continuous beam, with respect to a natural number i of 1 or more and n or less, the intermediate fulcrum or the end on the first end fulcrum side of the i-th beam. The vertical displacement δ _0i (m) at the fulcrum is defined as the given condition; the calculation result of the vertical displacement at the intermediate fulcrum on the second end fulcrum side of the i-th beam or the end fulcrum δ _{i, The calc} (m) is calculated as being included in the calculated value by the equation (9) obtained from the equation (1) to the equation (8); (i + 1) for i of 1 or more and (n-1) or less. _{) The vertical displacement δ 0 (i + 1)} at the intermediate fulcrum on the first end fulcrum side of the second beam and the vertical displacement at the intermediate fulcrum on the second end fulcrum side of the i-th beam. When the residual of the calculation results δ _{i, calc} is defined as the i-th residual, in the residual calculation step, the intermediate is the sum of the first residual to the (n-1) residual. The residual is calculated; the calculation result of the vertical displacement δ _n _{at the second end fulcrum, which is the given condition, and the calculation result δ n, calc of the} vertical displacement at the second end fulcrum of the nth beam. The second end fulcrum residual, which is a residual, may be calculated; the displacement residual, which is the sum of the intermediate residual and the second end fulcrum residual, may be calculated.

However, with respect to 1 to n for a natural number i, the length of the beam of i-th L _{i (m),} in the beam of the i-th, the origin end of the first end supporting point side, the first end The coordinates defined when the direction from the fulcrum toward the second end fulcrum is positive are x _i (m), the vertical load acting on the _i-th beam is wi (N / m), and the i-th said. The rotational rigidity of the beam at the end on the first end fulcrum side is S _{jl, i (Nm / rad), and the rotational rigidity of the i-th} beam at the end on the second end fulcrum side is S _{jr, i} (Nm). / Rad), the bending rigidity of the forward bending of the i-th beam is EI _{s, i} (Nm ² ), and the bending rigidity of the negative bending of the i-th beam is EI _{h, i} (Nm ² ), the i-th The bending moment at the intermediate fulcrum or the end fulcrum on the second end fulcrum side of the beam _{is defined as M j, i} (Nm). _{The bending moment M j, 0} (Nm) at the first end fulcrum of the first beam included in the second boundary condition is 0 when the first end fulcrum is a pin joint, rigid joint or semi-rigid. In the case of joining, it is unknown, and the angle of rotation φ ₀₁ (rad) at the first end fulcrum of the first beam is 0 when the first end fulcrum is rigid joining, and it is pin joint or semi-rigid joint. The case is unknown.
According to this aspect, a plurality of vertical displacements at the intermediate fulcrum and the pair of end fulcrums can be evaluated with high accuracy using a mathematical formula.

(5) The second aspect of the present invention is n, which is two or more natural numbers, which are arranged side by side in the longitudinal direction and whose ends adjacent to each other in the longitudinal direction are semi-rigidly joined to each other to form an intermediate fulcrum. An evaluation program for an evaluation device for evaluating a continuous beam including a book beam in which both ends of the entire continuous beam are paired end fulcrums. A plurality of bending moments at the intermediate fulcrum and the pair of end fulcrums, and a plurality of rotation angles at the pair of end fulcrums are made to function as a solution determining unit obtained based on given conditions, and the given conditions are the n lines. With the length and bending rigidity of each of the beams; with the plurality of rotational rigidity at the intermediate fulcrum and the pair of end fulcrums; with the vertical load acting on the n beams; at the intermediate fulcrum and the pair of end fulcrums. The solution determination unit defines the plurality of bending moments and the plurality of rotation angles as a plurality of unknowns, and the plurality of rotational rigidity, the plurality of bending moments, and the plurality of rotation angles. When the relational expression of the rotation angle and the relational expression of the plurality of vertical displacements are defined as a plurality of first boundary conditions having the same number as the number of the plurality of unknowns, the plurality of unknowns are the plurality of firsts. The bending moment and the deflection distribution of the continuous beam are evaluated by solving the plurality of unknowns so as to satisfy the boundary condition.

(6) In the third aspect of the present invention, the bending rigidity of the forward bending and the bending rigidity of the negative bending are different from each other, and both ends are semi-rigidly joined to act on the end of the composite beam on which an evenly distributed load acts over the entire length. This is an evaluation method for a composite beam that calculates the bending moment at the end, which is the bending moment to be performed, and the maximum value of the deflection that occurs in the composite beam. The rotational rigidity of the composite beam at the end is determined by the unit length of the composite beam. When the value divided by the winning bending rigidity is defined as the dimensionless rotational rigidity, and the ratio of the forward bending bending rigidity of the composite beam to the negative bending bending rigidity of the composite beam is defined as the dimensionless bending rigidity. ,
The end bending moment and the maximum value of the deflection are calculated by an explicit function based on the non-dimensional rotational rigidity and the non-dimensional bending rigidity.

According to this aspect, the inventors evaluate the maximum value of the end bending moment and the deflection based on the non-dimensionalized rotational rigidity and the non-dimensionalized bending rigidity. By this method, it has been found that the maximum values of the end bending moment and the deflection can be calculated with high versatility and accuracy by an explicit function regardless of the specifications of the composite beam.
Based on the non-dimensional rotational rigidity and the non-dimensional bending rigidity, the maximum values of the end bending moment and the deflection of the composite beam are calculated by an explicit function. By this method, the maximum values of the end bending moment and the deflection of the composite beam can be calculated without performing the convergence calculation. Therefore, the deflection can be appropriately evaluated.

(7) In the method for evaluating a composite beam according to (6) above, the bending moment acting on the end of the composite beam when it is assumed that both ends are rigidly joined and an evenly distributed load acts over the entire length is rigid. The maximum value of the bending moment acting on the composite beam when it is assumed that the tangential moment is pin-joined at both ends and an evenly distributed load acts over the entire length is defined as the pin tangent moment, and both ends are defined as pin tangent moments. The bending moment acting on the end of the composite beam, which is semi-rigidly joined and an evenly distributed load acts over the entire length, is defined as the semi-rigid contact moment, and the value obtained by dividing the semi-rigid contact moment by the pin contact moment is dimensionless. When the value obtained by dividing the rigid contact moment by the pin contact moment is defined as the dimensionless rigid contact moment, the dimensionless rigid contact moment is defined based on the dimensionless flexural rigidity. Was calculated by an explicit function; the dimensionless joint moment was calculated by an explicit function based on the calculated dimensionless rigid contact moment, the dimensionless rotational rigidity, and the dimensionless flexural rigidity; The end bending moment and the maximum value of the deflection may be calculated by an explicit function based on the dimensionless joint moment.

According to this aspect, the dimensionless rigid contact moment is calculated by an explicit function based on the dimensionless flexural rigidity. Furthermore, the non-dimensional joint moment is calculated based on the calculated non-dimensional rigid contact moment, the non-dimensional rotational rigidity, and the non-dimensional bending rigidity, and the end bending is based on the calculated non-dimensional joint moment. The maximum values of moment and deflection are calculated by explicit functions, respectively.
In this way, the maximum values of the end bending moment and the deflection of the composite beam can be calculated without performing the convergence calculation.

(8) In the method for evaluating a composite beam according to (7), the non-dimensional _{rigid contact moment is β Mj, rigid} , the non-dimensional rotational rigidity is α _j , and the non-dimensional flexural rigidity is α _s . When specified, the dimensionless junction moment β _Mj may be calculated by Eq. (13) using Eqs. (10) to (12).

_{According to this aspect, the dimensionless joint moment β Mj} can be accurately calculated by the equation (13) using the equations (10) to (12) without performing the convergence calculation.

(9) In the method for evaluating a composite beam according to (8), when the dimensionless _{rigid contact moment β Mj, rigid} is 0.4 or less, the dimensionless rigid contact moment β is described in the equation (13). _mj, instead of _rigid, the dimensionless KaTsuyoshise' moment beta _mj, dimensionless KaTsuyoshise' moment beta _mj calculated by on the basis of the _rigid (14) _{equation, rigid,} may be used _Theo.

According to this aspect, when the dimensionless _{rigid contact moment β Mj, rigid} is 0.4 or less, the _{error of the dimensionless rigid contact moment β Mj, rigid} becomes large with respect to the exact solution. Even in this case, the dimensionless _rigid contact moment can be calculated more accurately by using the dimensionless _{rigid contact moment β Mj, rigid, Theo} instead of the dimensionless rigid contact moment β Mj, rigid. Can be done.

According to the continuous beam evaluation method, the continuous beam evaluation program, and the composite beam evaluation method according to each of the above aspects of the present invention, the deflection can be appropriately evaluated.

It is a top view of the building which uses the continuous beam to which the evaluation method of the continuous beam of 1st Embodiment of this invention is applied. It is sectional drawing of the cutting line A1-A1 in FIG. It is a schematic diagram explaining the specifications of a beam and the bending moment acting on a beam. It is a schematic diagram explaining an external force and a load acting on a beam. It is a schematic diagram explaining the shearing force acting on a beam. It is a schematic diagram explaining the continuous beam and the external force acting. It is a figure explaining the relationship between the rotational rigidity and the angle of rotation at the end of a pair of beams adjacent to each other in the longitudinal direction when the distance between the positions where rotational rigidity occurs is taken into consideration. It is a figure explaining the relationship between the rotational rigidity and the angle of rotation at the end of a pair of beams adjacent to each other in the longitudinal direction when the distance between the positions where rotational rigidity occurs is ignored. It is a block diagram of the evaluation apparatus of the continuous beam used to perform the evaluation method of the continuous beam. It is a flowchart which shows the evaluation method of the continuous beam. It is a figure which shows an example of the input sheet of the evaluation program of the continuous beam of 1st Embodiment of this invention. It is a figure which shows the relationship of the bending moment distribution acting on a continuous beam with respect to the distance from the 1st end fulcrum evaluated by using the evaluation program of the continuous beam of 1st Embodiment of this invention. It is a figure which shows the relationship of the rotation angle distribution with respect to the distance from the 1st end fulcrum evaluated by using the evaluation program of the continuous beam of 1st Embodiment of this invention. It is a figure which shows the relationship of the deflection distribution with respect to the distance from the 1st end fulcrum evaluated by using the evaluation program of the continuous beam of 1st Embodiment of this invention. It is a vertical sectional view of the building which uses the synthetic beam to which the evaluation method of the synthetic beam of the 2nd Embodiment of this invention is applied. It is a schematic front view which shows the composite beam together with the boundary condition. It is a front view which shows typically the region which is bent forward and the region which is bent negative in the composite beam. It is a figure which shows the relationship between the dimensionless rotational rigidity and the dimensionless joint moment in case 1 to case 5. It is a figure which shows the relationship of the trial calculation result of the dimensionless bending rigidity and the dimensionless rigid contact moment in a composite beam. It is a figure which shows the relationship between the approximate solution and the exact solution of a dimensionless rigid moment of force. It is a figure which shows the relationship between the dimensionless bending rigidity and a coefficient k. It is a figure which shows the relationship between the dimensionless flexural rigidity and the variables α _{j, T.} It is a figure which shows the relationship between the approximate solution and the exact solution of a dimensionless joint moment. It is a figure which shows the relationship between the approximate solution of the maximum value of deflection, and the exact solution.

(First Embodiment)
Hereinafter, the continuous beam evaluation method and the first embodiment of the continuous beam evaluation program according to the present invention will be described with reference to FIGS. 1 to 14.

[1. Continuous beam]
The continuous beam evaluation method of the present embodiment (hereinafter, also simply referred to as an evaluation method) is suitably used for evaluating the continuous beam 11 constituting the building 1 shown in FIGS. 1 and 2, for example. It should be noted that FIG. 1 shows the floor 17 described later through the floor 17, and FIG. 2 does not show the pillar 33 described later. To evaluate the continuous beam 11 referred to here, the bending moment, the angle of rotation, the deflection distribution, etc. of the continuous beam 11 are obtained, and for example, how much the bending strength of the continuous beam 11 has a margin with respect to the bending moment of the continuous beam 11. It means to evaluate if there is.
In this example, the continuous beam 11 includes n (two in this example) beams (small beams) 13 in which the ends adjacent to each other in the longitudinal direction are semi-rigidly joined to each other to form an intermediate fulcrum 12a. .. Note that n is a natural number of 2 or more.
In this example, the beam 13 includes a floor 17 and a beam body 18. The configuration of the beam 13 is not limited to this example.
The floor 17 is a so-called synthetic slab, which is supported from below by the beam body 18. The floor 17 includes a deck plate 20 and an RC (Reinforced Concrete) slab 21 arranged on the deck plate 20.
The uneven shape of the deck plate 20 extends in a direction along the horizontal plane and in a direction orthogonal to the direction in which the beam body 18 extends.
The RC slab 21 includes concrete 22 and reinforcing bars 23. The concrete 22 is formed in a plate shape in which the vertical direction is the thickness direction. The concrete 22 is supported from below by the deck plate 20.
The reinforcing bar 23 extends along the horizontal plane and is buried in the concrete 22. For example, the reinforcing bars 23 are arranged in a grid pattern in a plan view.
The n beams 13 are arranged side by side in the longitudinal direction of the beams 13.

The beam body 18 is made of steel H-section steel and extends along a horizontal plane. The lower end of the stud 26 is fixed to the upper flange of the beam body 18. The stud 26 penetrates the deck plate 20. The upper end of the stud 26 is embedded in the concrete 22.
The ends of the beam body 18 are semi-rigidly joined to the first girder (intermediate support portion) 27 extending along the horizontal plane at the intermediate fulcrum 12a. The intermediate fulcrum 12a is supported in the vertical direction by the first girder 27. The first girder 27 extends in a direction orthogonal to the beam body 18. The semi-rigid joint between the beam body 18 and the first girder 27 is performed by, for example, a shear plate 28 and a bolt 29.

The pair of end fulcrums 12b at both ends of the entire continuous beam 11 are supported in the vertical direction by the pair of second girders (end support portions) 31. The first girder 27 and the second girder 31 extend along the first direction along the horizontal plane. The state of joining the end of the beam body 18 and the second girder 31 at the end fulcrum 12b is not particularly limited, and may be pin joining, semi-rigid joining, or rigid joining.
The building 1 includes a third girder 32 that extends along a horizontal plane and along a second direction orthogonal to the first direction. The connecting portion between the first girder 27, the second girder 31, and the third girder 32 is supported in the vertical direction by the pillar 33.

The evaluation method of the continuous beam 11 configured in this way will be described below.

[2. Derivation of shear force distribution and moment distribution of continuous beams]
[2.1. Basic formula]
In the continuous beam 11 including the n (span) beams 13, the specifications of the arbitrary i-th beam (i is a natural number and 1 ≦ i ≦ n) counting from one end fulcrum 12b of the continuous beam 11. And the bending moment, the external force, the shearing force, etc. acting on the beam 13 are assumed as shown in FIGS. 3 to 5.
That is, as shown in FIG. 3, i-th beam 13 _i is set to extend along the horizontal plane, the length of the beam 13 _i is assumed to be L _{i (m).} When each beam 13 _i is not distinguished, it is also referred to as a beam 13.
In the beam _13i , it is assumed that the bending rigidity of the forward bending (convex downward) and the bending rigidity of the negative bending (convex upward) are different from each other. Positive bending the bending stiffness of the beam 13 _i and _{^{EI s, i (Nm 2)}} , the negative bending of the bending stiffness of the beam 13 _i _{EI h,} and i (Nm ^2).

As shown in FIG. 4 _{, the coordinates x i} (m) are defined when the right direction is positive along _{the beam 13 i.} The position of the left end of the beam 13 _i is defined as the origin of _{the coordinates x i} _{(the position of x i} = 0). Assume that the beam 13 _i, a uniformly distributed load of the downward-looking over the entire length (vertical _load) w i (N / _m) is applied. Beams 13 _i of the leftmost rotational stiffness _S jl, and i (Nm / rad), defines the rotational stiffness of the right end of the beam 13 _i _S jr, and i (Nm / rad).
As shown in FIG. 3, the absolute value of the bending moment (negative bending moment) acting on the left end of the _{beam 13 i} _{is M jl, i} (Nm), and the bending moment (negative bending moment) acting on the right end of the _{beam 13 i is defined as M jl, i (Nm).} The absolute value of is defined as M _{jr, i} (Nm). The bending moment shown by the curve L1 acts on the beam 13 _i. The bending moment is positive in the direction in which the downwardly convex bending occurs. In the beam 13 _i _{, the coordinates x i at} which the bending moment becomes 0 are defined as _{x h, i} (m), x _{s, i} (m) (0 ≦ x _{h, i} <x _{s, i} ≦ _Li ). ..
As shown in FIG. 5, _{the left end of the beam 13 i} _{is V jl, i} (N) as the shearing force (external force) supported from below by the

girders

27, 31 and the like _{, and the right end of the beam 13 i} is the

girder

27, 31 and the like. _{Let V jr, i} (N) be the shearing force supported from below. The beams 13 _i, shear force shown by the curve L2 acts.

At this time, the shear force distribution V (x _i ) (N) and the bending moment distribution M (x _i _{) (Nm) acting on the beam 13 i} can be expressed by Eqs. (21) and (22). The shear forces V _{jl, i} , V _{jr, i} , and the shear force distribution V (x _i ) are positive (+) in the direction in which clockwise rotation occurs.

(22) in the _expression, the use of boundary conditions of the bending moment in the x i = _{L i,} is obtained (23) and (24) below.

_{Here, the absolute values M jl, i} , M _{jr, i} of the bending moment in the equations (22) to (24) represent the magnitude of the negative bending moment, and in (24-1) and (24-2). Defined.

(21) by substituting the expression (24) into _equation, using the balance condition of the forces in the x i = _{L i,} is obtained (25) and (26) below.

Substituting (21) into equation (26), (27) for shear force distribution V _{(x i)} is obtained.

Substituting equation (24) into equation (22) gives equation (28).

Using equation (22), _{the coordinates x i} (x _{h, i} , x _{s, i} ) _{when the bending moment distribution M (x i} ) = 0 are obtained. (29) is solved for _{x i.} Considering that (x _{h, i} <x _{s, i} ), equations (30) and (31) can be obtained.

[2.2. Angle of rotation distribution]
The curvature distribution of the beam 13 _i _{is defined as ρ (x i} ) (1 / m), where the downward convex case is positive (+). The vertical deflection (displacement) distribution of the beam 13 _i _{is defined as δ (x i} ) (m) with the vertical downward direction as positive (+). At this time, the curvature distribution [rho _{(x i)} is expressed by equation (34).

[2.2.1. Rotation angle distribution of interval [a] = [0, x _{h, i]]}
When the coordinates x _i are 0 or more and x _{h, i} or less, the rotation angle distribution of the _{beam 13 i} _{is defined as φ (x i} ) (rad: radian). Rotation angle distribution phi (x _i), the direction of rotation from the horizontal line defining the case of clockwise as positive (+). The angle of rotation distribution φ (x _i ) expresses the curvature distribution ρ (x _i ) by the equations (28) and (34), and further integrates the _{curvature distribution ρ (x i} ) in the interval [0, x _i]. , By using the equation (35), it can be obtained as in the equation (36). Consider when determining the rotation angle distribution φ _{(x _i),} the boundary condition at _x i = 0, the rotation angle distribution phi a _{(x i = 0) = φ} 0i. That is, the rotation angle of the left (first end supporting point 12b ₁ side to be described later) in the beam 13 _i is phi _0i.
The rotation angle distribution φ (x _i ) in the interval [a] is also referred to as a rotation angle distribution φ _a (x _i).

[2.2.2. Rotation angle distribution of interval [b] = [x _{h, i} , x _{s, i]]}
When the coordinates x _i are x _{h, i} or more and x _{s, i} or less, the rotation angle distribution φ (x _i _{) of the beam 13 i} expresses the curvature distribution ρ (x _i ) by the equations (28) and (34). , Integrate over the interval [x _{h, i} , x _i ] and obtain as in Eq. (37). When finding the rotation angle distribution φ (x _i ), consider the boundary conditions at _{x i} = x _{h, i.}
The rotation angle distribution φ (x _i ) in the section [b] is also referred to as a rotation angle distribution φ _b (x _i).

[2.22.3. Rotation angle distribution of interval [c] = [x _{s, i} , _Li]]
When the coordinates _{x i} is _{x s,} the following more _{L i} _i, rotation angle distribution phi _{(x i)} of the beam 13 _i denotes curvature distribution ρ a _{(x i)} (28) and equation (34) in equation section Integrate with [x _{s, i} , x _i ] and obtain as in Eq. (38). When finding the rotation angle distribution φ (x _i ), consider the boundary conditions at _{x i} = x _{s, i.}
The rotation angle distribution φ (x _i ) in the interval [c] is also referred to as a rotation angle distribution φ _c (x _i).

[2.3. Deflection distribution]
[2.3.1. Deflection distribution of interval [a] = [0, x _{h, i]]}
When the coordinates x _i are 0 or more and x _{h, i} or less, the deflection distribution δ (x _i _{) of the beam 13 i} is obtained by integrating the rotation angle distribution φ (x _i ) in the interval [0, x _i ] and (41. ) Is used to obtain the equation (42). When determining the deflection distribution δ _{(x _i),} the boundary condition at _x i = 0, vertical displacement _{δ (x i = 0) =} δ consider _0i.
Incidentally, the deflection distribution [delta] _{(x i)} of the interval [a], the deflection distribution [delta] _a is also referred to as _{(x i).}

[2.3.3. Deflection distribution of interval [b] = [x _{h, i} , x _{s, i]]}
When the coordinates x _i are x _{h, i} or more and x _{s, i} or less, the deflection distribution δ (x _i ) is obtained by integrating the rotation angle distribution φ (x _i ) in the interval [x _{h, i} , x _i]. It is obtained as in Eq. (43). When determining the deflection distribution δ (x _i ), consider the boundary conditions at _{x i} = x _{h, i.}
Incidentally, the deflection distribution [delta] _{(x i)} in the interval [b], the deflection distribution [delta] _b is also referred to as _{(x i).}

[2.3.3. Interval _{[c] = [x s,} i, L i] deflection distribution]
When the coordinates _{x i} is _{x s,} the following more _{L i} _i, deflection distribution [delta] _{(x i),} the rotation angle distribution phi _{(x i)} the interval _{[x s,} i, _{x i]} by integrating with, (44 ) Is calculated as in the formula. When determining the deflection distribution δ (x _i ), consider the boundary conditions at _{x i} = x _{s, i.}
Incidentally, the deflection distribution [delta] _{(x i)} in the interval [c], the deflection distribution [delta] _c is also referred to as _{(x i).}

[3. Calculation procedure in the evaluation method of continuous beams]
_{In the evaluation method, a plurality of bending moments M j and i} at the intermediate fulcrum 12a and the pair of end fulcrums 12b, and a plurality of rotation angles φ ₀₁ and φ _c at the pair of end fulcrums 12b are obtained based on the given conditions.
Before explaining the given conditions mentioned here, a plurality of vertical displacements and a plurality of rotational rigidity will be described.
First, as shown in FIG. 6, of the pair of end fulcrums 12b, the left end fulcrum 12b (one end fulcrum) is also referred _{to as the first end fulcrum 12b 1.} Of the pair of end fulcrums 12b, the right end fulcrum 12b (the other end fulcrum) is also referred to as a _{second end fulcrum 12b 2.} In FIG. 6, the coordinates x _i of the _{i-th beam 13 i are shown as an example of the coordinates.}
The coordinates x _i have the origin on the first end fulcrum 12b ₁ side of the beam 13 _i , and the direction from the first end fulcrum 12b ₁ to the second end fulcrum 12b _{2 is positive.} Hereinafter, among the n number of beams 13 _i, the beam 13 _i of the i-th from the first end supporting point 12b ₁ toward the second end supporting point 12b _2, also referred to as the i-beam 13 _i. For example, the beam 13 of the most first end supporting point 12b ₁ side of the n of beams 13 is a first beam 13 _1.
The rotational stiffness _{S jl, i} is the rotational stiffness of the end of the first end supporting point 12b ₁ side in the i beam 13 _i (intermediate fulcrum 12a). The rotational stiffness _{S jr, i} is the rotational stiffness of the second end supporting point 12b ₂ side end of the i-th beam 13 _i (intermediate fulcrum 12a).

The deflection of the vertical at a first end supporting point 12b ₁ of the first beam 13 _1, a vertically downward as a positive (+), and [delta] ₀ (m). For one or more (n-1) following i, the vertical displacement of the second end supporting point 12b ₂ side of the intermediate supporting point 12a in the i beam 13 _i, and [delta] _i (m). The vertical displacement of the n-th beam 13 _n at the second end fulcrum 12b ₂ is defined as δ _n (m). _{The plurality of vertical displacements under the given conditions are vertical displacements δ i} with respect to a natural number i of 0 or more and n or less.
_{For example, the vertical displacement δ i} at the intermediate fulcrum 12a of the continuous beam 11 shown in FIG. 1 may be obtained from the range R of the distributed load defined to act on the first girder 27 by a known method. _{Then, the vertical displacement δ i} at the intermediate fulcrum 12a is obtained from the distributed load acting in the range R, the bending rigidity of the first girder 27, and the like.

Bending moments due to, for example, rotational resistance of the girder do not act on the

fulcrums

12a and 12b, and the bending moments acting on the ends of the beams 13 adjacent to each other in the longitudinal direction have the same value. That is, for example, the absolute value _{M jr} bending moment acting on the right end of the i beam 13 _{_i,} and _i, the (i + 1) beams _{13 i + 1} of the absolute value _{M jl} bending moment acting on the left _end, and _{i + 1,} is equal to .. Let the equal values be M _{j, i} (Nm). Bending moment _{M j, i,} to the one or more (n-1) following i, a bending moment in the second end supporting point 12b ₂ side of the intermediate supporting point 12a in the i beam 13 _i.
Incidentally, the bending moment _{M j, 0} is equal to the absolute value _{M jl} bending moment acting on the left end of the first beam 13 _{_1, 1} (bending moment in the first end supporting point 12b ₁ of the first beam 13 _1). Bending moment _{M j, n} is the absolute value _{M jr} bending moment acting on the right end of the n beam 13 _{_n,} equal to _n (bending moment in the second end supporting point 12b ₂ in the n beam 13 _n).

In this way, a plurality of (n + 1) bending moments M _{j, 0} , ..., M _{j, n at the} intermediate fulcrum 12a and the pair of end fulcrums 12b in the continuous beam 11 are defined.

The given condition, the plurality of vertical displacement, and a plurality of rotational stiffness S _{j, i,} each of a length L _i beam 13 _i of the n, n of beams 13 _i each flexural rigidity (positive bending including flexural rigidity _{EI s, i} and negative bending flexural rigidity _{EI h, i)} and the uniformly distributed load _{w i} acting on the n beams 13 _i each, a.

At this time, the rotation angle of the beam 13 _i in each

support point

12a, 12b can be expressed by using the rotational stiffness of the beam 13 _i in each

support point

12a, 12b, bending the absolute value _M jl moment, _{i, M jr,} the _i.
Here, the first beam 13 ₁ of the left end of the rotational stiffness _{S jl, 1} to _{S j, 0,} the rotational stiffness in a representative point of the intermediate support point 12a and _{S j, i.}
As shown in FIG. 6, the continuous beam 11 includes n beams 13 _i . The intermediate fulcrum 12a of the continuous beam 11 is semi-rigidly joined to the first girder 27, and the end fulcrum 12b of the continuous beam 11 is supported by the second girder 31.

As shown in FIG. 7, consider the joint between the (i-1) th (i-1) th beam 13 i-1 and the i _{-th i-} _{th beam 13 i.}
The angle of rotation of the i-th beam 13 _{i at} the left end is defined as φ _{jl, i} (rad), and the rotation angle of the i-th beam 13 _{i at} the right end is defined as φ _{jr, i} (rad). In this case, the rotation angle at the right end of the (i-1) th (i-1) beam 13 _i-1 _{is φ jr, i-1} (rad). The angle of rotation of the i-th beam 13 _i with respect to the horizontal plane at the left end is defined as φ _{l, i} (rad). The rotation angle at the right end of the i beam 13 _i with respect to the horizontal plane, is defined as φ _r, i (rad). In this case, the rotation angles of the first (i-1) beam 13 _i-1 _{with respect to the horizontal plane at the right end are φ r, i-1} (rad).
The distance between the position _{where the rotational rigidity S jr, i-1} is generated at the right end of the first (i-1) beam 13 _i-1 and the position where the rotational rigidity S _{jl, i} is generated at the left end of the i-th beam 13 _{i is extremely small.} Suppose it is a length.
Eqs. (47) and (48) can be obtained from the definitions of each variable in consideration of the positive and negative of the rotation angle distribution.

Equation (49) can be obtained from the geometrical relationship of the deformed state in FIG.

In this case, it is assumed that the minimum length is negligible. At this time, as shown in FIG. 8, the (i-1) beams _{13 i-1} of rotational stiffness _{S jr, i-1} and rotational stiffness _{S jl} i-th beam 13 _{_i, i} is the rotational stiffness _{S j,} It is represented by _i-1. The rotational rigidity S _{j, i-1} is obtained from the equations (50) and (51) in which the equations (47) and (48) are substituted into the equation (49).

Therefore, the rotational rigidity S _{j, i-1} can be expressed by the series connection of the rotating springs of the left and right joints sandwiching each intermediate fulcrum 12a.
Here, for i = 2 to (n-1), φ _0i in the equation (38) is obtained as follows. From FIG. 8, the _{rotation angles φ 0i} = φ _l, _{i at} the left end intermediate fulcrum 12a of the i-th beam 13i are related to the _{rotation angles φ r, i-1 at the} right end intermediate fulcrum 12a using the equation (49). Be attached. In equation (49), the absolute values M _{j, i-1} of the moment at the intermediate fulcrum 12a and the rotational stiffness S _{j, i-1} are used.
Moreover, phi _r, since _i-1 is a rotation angle in the (i-1) beams _{13 i-1} at the right end, the (i-1) to (38) below the beams _{13 i-1} _{x i-1} = L It is equal to the value to which _{i-1 is substituted.}
Therefore, the equation (38) is substituted into the equation (49), and the equation (8) is derived. Given the initial rotation angle φ _{01 of} the first end fulcrum 12b ₁ and the moments of each intermediate fulcrum 12a and both end fulcrums 12b, φ _0i for i = 2 to (n-1) is sequentially obtained by Eq. (8). ..
When evaluating the continuous beam 11, for the bending moment, (n + 1) unknowns, which are _{bending moments M j and i} (plural bending moments) with respect to a natural number i of 0 or more and n or less, are defined.

Although it depends on the joint state at both end fulcrums 12b of the continuous beam 11, the following can be generally said.
At the bending moments M _{j, 0} , M _j, ₁ _{at both ends of the first beam 131, and} the rotation angle φ ₀₁ (φ jl, 1) of the first beam 13 ₁ at the first end fulcrum 12b _{1, which is an unknown number.} Give initial values for each. _Then, x 1 = angle of rotation of the equation (38) in _{L 1} distribution phi _c _{(x i)} first rotation angle of the beam 13 ₁ by phi _{r, 1} is determined. (49) by the equation, the second beam 13 _{and second} rotation angles of the two eyes adjacent to the first beam 13 ₁ phi _{l, 2} is determined.
Rotation angle phi _{l, 2} corresponds to phi ₀₂ is a value in _x 2 = 0 of the rotational angle distribution phi _{(x 2)} when applied to the second beam 13 ₂ (38) below.
_{Similarly, when the bending moments M j, i-2} , M _{j, and i-1} of the joints at both ends of _{the (i-1) beam 13 i-1} are given to i which is 3 or more and n or less, x _i-1 _{= L} in _i-1 by (38) the rotation angle of formula distribution phi _c _{(x i-1),} the (i-1) beam ₁₃ rotation angle at the _i-1 of the rightmost phi _{r, i-1} Is decided. _{The rotation angles φ l and i} at the left end of the _i- th beam 13 i adjacent to the (i-1) beam 13 _i-1 are determined by the equation (49). The rotation angle phi _{l, i} at the left end of the i beam 13 _i is in _x i = 0 of the rotational angle distribution phi (x _i) of applying the i-th beam 13 _i relative to (38) below It corresponds to the value φ _0i.

In this way, the bending moments M _{j, i-1} , M _{j, and i} are assumed from i = 1 to n, and the corresponding rotation angles φ _{r, i-1} and rotation angles φ _{l, i} are obtained in this order. _Then, the x i = _{L i} in equation (44) of the deflection distribution [delta] _c _{(x i),} each supporting point 12a, the vertical displacement [delta] _0i at 12b obtained.
The rotation angles φ _{r and n} of the nth beam 13 _n at the second end fulcrum 12b ₂ are also defined as unknowns. That is, with respect to the rotation angles, the two rotation angles φ ₀₁ and the rotation angles φ _{r and n} are unknown.
Thus, the unknown numbers are, for example, the bending moments M _{j and i} at the

respective fulcrums

12a and 12b, and the rotation angle (n + 3) at both end fulcrums 12b.

On the other hand, a plurality of first boundary conditions for solving a plurality of unknowns are defined as follows.
From the vertical displacement δ ₀₁ at the first end fulcrum 12b ₁ , which is the given condition, and _{equation (44) at x i} = Li for _i of 1 or more and n or less, a plurality of orders for i of 0 or more and n or less. From Eq. (54-1), which is one boundary condition, (n + 1) relational expressions of Eq. (54- (n + 1)) can be obtained. These (n + 1) relational expressions are relational expressions of a plurality of vertical displacements δ _i .

For example, Eq. (54-2) is an equation for bending moments M _{j, 0} , M _{j, 1} . Equation (54-3) is an equation for bending moments M _{j, 0} , M _{j, 1} , M _{j, 2} . Equation (54- (n + 1)) is an equation for bending moments M _{j, 0} , M _{j, 1} , M _{j, 2} , ..., M _{j, n} .

From the relational expressions of the rotational rigidity, the bending moment, and the rotation angle at the fulcrums 12b at both ends of the continuous beam 11, two relational expressions (55-1) and (55-2) can be obtained.

Equations (55-1) and (55-2) are obtained according to the joint state at both end fulcrums 12b of the continuous beam 11. For example, if the left end fulcrum 12b (first end fulcrum 12b ₁ ) of the continuous beam 11 is a pin joint, the bending moments M _{j and 0} are 0.
Thus, the number of first boundary conditions is (n + 3) of (n + 3) relational expressions. That is, the number of the plurality of first boundary conditions is the same as the number of the plurality of unknowns.
Therefore, if the plurality of unknowns are solved so that the plurality of unknowns satisfy the plurality of first boundary conditions _{, the combination of (n + 1) bending moments Mj and i} is determined to be one.

_{Here, the values of the bending moments Mj and i} at all the

fulcrums

12a and 12b are assumed. _{Let δ i, calc} (m) be the calculation result of the vertical displacement corresponding to _{the vertical displacement δ i} at each

fulcrum

12a, 12b obtained from the assumed bending moments M _{j, i.} _{Assuming the values of bending moments M j and i} at all

fulcrums

12a and 12b, the vertical displacement calculation results δ _{i and calc} obtained from the assumed bending moments M _{j and i} and the vertical determined by the conforming conditions of deformation. The sum of the squares of the difference from the displacement δ _{i at} all the

fulcrums

12a and 12b (value according to equation (57), displacement residual) is defined as the objective function. Then, it is possible to identify the objective function is minimized bending moment M _j, the optimization calculation to find a combination of _i, all the fulcrum 12a, the bending moment M _j at _12b, and _i.
A known differential evolution method or the like can be used for the optimization calculation.
The calculation procedure in the evaluation method of the continuous beam according to the joint state at both end fulcrums 12b of the continuous beam 11 will be described later.

[4. Evaluation device used for evaluation method of continuous beam]
In order to evaluate the continuous beam 11 by performing the evaluation method of the continuous beam, for example, the evaluation device 101 shown in FIG. 9 is used. The evaluation device 101 is a computer, and includes a CPU (Central Processing Unit) 111, a main storage device 125, an auxiliary storage device 126, an input / output interface (IO / I / F) 131, and a recording / playback device 136. I have.

The main storage device 125 is a RAM (Random Access Memory) or the like that serves as a work area or the like of the CPU 111.
The input / output interface 131 is connected to an input device 132 such as a keyboard and a mouse, and a display device 133.
The recording / reproducing device 136 records and reproduces data on a recording medium 137 such as a disc type such as a CD or a DVD.

The auxiliary storage device 126 is a hard disk drive device or the like that stores various data, programs, and the like. The auxiliary storage device 126 stores a continuous beam evaluation program (hereinafter, simply referred to as an evaluation program) 127 for causing the computer to function as the evaluation device 101, various programs such as an OS program, a predetermined threshold value, and the like. Has been done. Various programs including the evaluation program 127 are taken into the auxiliary storage device 126 from the recording medium 137 via the recording / playback device 136. The evaluation program and the like are stored in the recording medium 137.
Note that these programs may be incorporated into the auxiliary storage device 126 from an external device via a portable memory such as a flash memory or a communication device (not shown).

The auxiliary storage device 126 is further provided with a temporary setting value file 128 in the process of executing the evaluation program 127. Temporary design values, which will be described later, are stored in the temporary setting value file 128.

The CPU 111 executes various arithmetic processes.
The CPU 111 functionally includes a solution determination unit 112 that obtains a plurality of bending moments and a plurality of rotation angles based on given conditions. Further, the solution determination unit 112 functionally includes a first design unit 113, a second design unit 114, a temporary design unit 115, a storage unit 116, a solution calculation unit 117, and a residual calculation unit 118. A determination unit 119 and a solution setting unit 120 are provided.
The first design unit 113, the second design unit 114, the temporary design unit 115, the storage unit 116, the solution calculation unit 117, the residual calculation unit 118, the determination unit 119, and the solution setting unit 120, which are the functional components of the CPU 111. All function when the CPU 111 executes the evaluation program 127 or the like stored in the auxiliary storage device 126. The evaluation program 127 and the like are programs for the evaluation device 101. The evaluation program 127 causes the evaluation device 101 to function as the solution determination unit 112.

[5. Evaluation method for continuous beams]
Next, the evaluation operation (evaluation method) of the evaluation device 101 will be described. As shown in Table 1, there _{are three joining states of the first end fulcrum 12b 1} : pin joining, semi-rigid joining, and rigid joining. There are three joining states of the second end fulcrum 12b ₂ , pin joining, semi-rigid joining, and rigid joining. Therefore, there are a total of nine joint states at the pair of end fulcrums 12b of the continuous beam 11.

However, for example, when the first end fulcrum 12b ₁ is a pin joint and the second end fulcrum 12b ₂ is a semi-rigid joint, and when the first end fulcrum 12b ₁ is a semi-rigid joint and the second end fulcrum 12b ₂ is a pin joint. Then, if the directions of the continuous beams 11 are reversed, the same configuration can be obtained. Therefore, _{the case where the fulcrums 12b 1} and 12b _{2 at} both ends are pin-joined will be described as Case 1. The case where the

fulcrums

12b ₁ and 12b _{2 at} both ends are semi-rigidly joined will be described as case 2, and the case where the

fulcrums

12b ₁ and 12b _{2 at} both ends are rigidly joined will be described as case 3. Case 4A when the first end fulcrum 12b ₁ is a pin joint and the second end fulcrum 12b ₂ is a semi-rigid joint, and case 5A when the first end fulcrum 12b ₁ is a pin joint and the second end fulcrum 12b _{2 is a rigid joint.} It is explained as. A case where the first end fulcrum 12b ₁ is a rigid joint and the second end fulcrum 12b ₂ is a semi-rigid joint will be described as Case 6A.
In the case 4B in which the first end fulcrum 12b ₁ is semi-rigidly joined and the second end fulcrum 12b ₂ is pin-joined, the continuous beam 11 has the same configuration as the case 4A, and thus the description thereof will be omitted. The case 5B in which the first end fulcrum 12b ₁ is a rigid joint and the second end fulcrum 12b ₂ is a pin joint is also omitted in the same manner. Case 6B in the case where the first end fulcrum 12b ₁ is a semi-rigid joint and the second end fulcrum 12b ₂ is a rigid joint is also omitted in the same manner.

[5.1. Evaluation method of continuous beam in case 1 (a pair of end fulcrums are pin-joined)]
FIG. 10 is a flowchart showing the evaluation method S11 of the present embodiment when the continuous beam 11 is the case 1. The evaluation method S11 has a solution determination step (step S12 shown in FIG. 10) of obtaining a plurality of bending moments and a plurality of rotation angles based on given conditions.
In the solution determination step S12, first, the first design unit 113 performs the first design step (step S14) (the solution determination step S12 has the first design process S14). In the first design step S14, a design value including _{a plurality of vertical displacements δ i} at the pair of end fulcrums 12b and the intermediate fulcrums 12a of the continuous beam 11 is given. The plurality of vertical displacements δ _i are given as the given conditions. In the first design step S14, the design value, the plurality of rotational stiffness, and n of beams 13 _i each length L _i, and the beam 13 _i of each flexural rigidity of the n, n of beams 13 _i comprising a uniformly distributed load _{w i} acting on each of the.
When the first design step S14 is completed, the process proceeds to step S16.

Next, in the second design step S16, the second design unit 114 gives a second boundary condition to the pair of end fulcrums 12b according to the joining state at the pair of end fulcrums 12b. In the case of Case 1, the second boundary condition is that the bending moments M _{j, 0} , M _{j, and n} are 0 Nm, respectively. That is, the second boundary condition includes that the rotation angle φ ₀₁ is unknown.
When the second design step S16 is completed, the process proceeds to step S18.
Next, in the temporary design step S18, the temporary design unit 115 includes a temporary design value _{M j, i} acting on the pair of end fulcrums 12b and the intermediate fulcrum 12a according to the joining state at the pair of end fulcrums 12b. give. If continuous beam 11 of the case 1 is tentatively designed section 115 further gives the rotation angle phi ₀₁ of the beam 13 ₁ in the first end supporting point 12b ₁ as a temporary design value.
When the tentative design step S18 is completed, the process proceeds to step S20.

Next, in the storage step S20, the storage unit 116 stores the temporary design value in the temporary setting value file 128 after the temporary design step S18. When the storage step S20 is completed, the process proceeds to step S22.
Next, in the solution calculation step S22, the solution calculation unit 117 determines the intermediate fulcrum 12a so as to satisfy the plurality of first boundary conditions and the second boundary conditions based on the temporary design values stored in the temporary setting value file 128. Calculation result of vertical displacement in δ _i, calculate the calculated value including cal. _{The calculated value includes the calculation result δ n, calc} of the vertical displacement, which is the calculation result of the vertical displacement at the second end fulcrum 12b ₂ . Calculation results of vertical displacement δ _{i, calc} are obtained based on n values from vertical displacement δ ₁ _{to vertical displacement δ n in} equations (54-2) to (54- (n + 1)).
Since the vertical displacement calculation result δ _{0, calc} vertical displacement δ ₀₁ (vertical displacement δ ₀ ) is one element of a plurality of vertical displacements given as given conditions, the value is newly calculated in the solution calculation step S22. Not done. Calculation result of vertical displacement δ _{i, calc} is calculated in the order of _{δ 1, calc} , δ _{2, calc} , ..., δ _{n, calc.}
When the solution calculation step S22 is completed, the process proceeds to step S24.

Next, in the residual calculation step S24, the residual calculation unit 118 obtains the displacement residual, which is the residual between the design value and the calculated value. _{The displacement residual is determined by using the calculation results δ i, calc} of the plurality of vertical displacements calculated in the solution calculation step S22 _{and the plurality of vertical displacements δ i} given as the given conditions in the first design step S14. It is calculated by the formula 57).
The displacement residual will be described in more detail. Wherein one or more (n-1) for the following i, the beam _{13 (i + 1)} vertical displacement of the first end supporting point 12b ₁ of the intermediate fulcrum 12a in δ _{0 (i + 1),} the second end of the beam 13 _i _{The residual with the calculation result δ i and calc} of the vertical displacement at the intermediate fulcrum 12a on the fulcrum 12b ₂ side is defined as the i-th residual.
In the residual calculation step S24, the intermediate residual, which is the sum of the first residual to the (n-1) residual, is calculated. The residual of the second end fulcrum, which is the residual of the vertical displacement δ _n at the second end fulcrum 12b ₂ which is the given condition, and the calculation result δ _{n, calc} of the vertical displacement at the second end fulcrum 12b ₂ of the beam 13 _n. Calculate the difference. At this time, the displacement residual is the sum of the intermediate residual and the second end fulcrum residual. The residual may be either positive or negative. In order to appropriately accumulate the sum of the residuals as an error, the sum of the absolute values of each residual or the sum of the squares of each residual is defined as the displacement residual.
In the residual calculation step S24, the rotation angle distribution φ (x _i ), the bending moment distribution M (x _i ) M _{j, i} , the bending rigidity EI _i , the bending moment M _{j, i} , the rotational rigidity S _{j, i} , It is determined whether or not the conforming conditions of Eqs. (60) and (61) relating to the rotation angles φ _{l, i} , φ _{r, and i-1 are satisfied.} Bending moment distribution M _{(x i)} is determined from equation (22).

However, the bending stiffness EI _i is the bending stiffness EI _{s, i} of the forward bending in the case of the forward bending, and the bending stiffness EI _{h, i} of the negative bending in the case of the negative bending.
When the residual calculation step S24 is completed, the process proceeds to step S26.

Next, in the determination step S26, the determination unit 119 determines whether or not the displacement residual is smaller than the threshold value. When it is determined in the determination step S26 that the displacement residual is smaller than the threshold value (Yes), the process proceeds to step S28. On the other hand, when it is determined in the determination step S26 that the displacement residual is equal to or greater than the threshold value (No), the process proceeds to step S18.

In the solution setting step S28, the solution setting unit 120 defines the bending moment of the temporary design value stored in the temporary setting value file 128 as the bending moment acting on the intermediate fulcrum 12a. Then, a plurality of rotation angles and deflection distributions are obtained based on this bending moment.
This completes all the steps in the evaluation method S11.

In the temporary design step S18, another new temporary design value is given in place of the temporary design value stored in the temporary setting value file 128 in the storage step S20. In the storage step S20 shifted from the temporary design step S18, the other new temporary design value is stored in the temporary setting value file 128. Then, the storage step S20, the solution calculation step S22, the residual calculation step S24, and the determination step S26 are performed based on the other new provisional design values.
As described above, when it is determined in the determination step S26 that the displacement residual is equal to or greater than the threshold value (No), another new provisional design value is given in the storage step S20, and the other new provisional design. Based on the values, the storage step S20, the solution calculation step S22, the residual calculation step S24, and the determination step S26 are performed as a set. Then, this set of steps is repeated until it is determined in the determination step S26 that the displacement residual is smaller than the threshold value. In the second and subsequent tentative design steps S18, for example, the tentative design values are reset according to the differential evolution method, and the tentative design steps S18 to the determination step S26 are calculated to effectively calculate the convergence of the displacement residuals. be able to.

[5.2. Evaluation method of continuous beam in case 2 (a pair of end fulcrums are semi-rigid joints)]
In the evaluation method S36 in this case, the solution determination step S37 for obtaining a plurality of bending moments and a plurality of rotation angles based on the given conditions is performed.
In the solution determination step S37, first, the first design step S14 is performed. When the first design step S14 is completed, the process proceeds to step S39.
Next, in the second design step S39, the second design unit 114 gives a second boundary condition according to the joining state at the pair of end fulcrums 12b. In the case of Case 2, the second boundary conditions are the equations (64) and (65) _{at the end fulcrums 12b 1} and 12b _{2 of the continuous beam 11.} That is, the bending moments M _{j, 0} , M _{j, n} are unknown. The second boundary condition includes that the rotation angle φ ₀₁ is unknown.

Since the rotational rigidity S _{j, 0} , S _{j, and i} are constants, for example, in equation (64), if one of the bending moment M _{j, 0} and the angle of rotation φ _{jl, 1} is obtained, equation (64) is obtained. The other of the bending moment M _{j, 0} and the rotation angle φ _{jl, 1} can be obtained from. The equation (65) is the same as that of the equation (64). When the second design step S39 is completed, the process proceeds to step S41.
Next, in the temporary design step S41, the temporary design unit 115 gives a temporary design value including _{bending moments M j and i} acting on the intermediate fulcrum 12a according to the joining state at the pair of end fulcrums 12b. When the continuous beam 11 is the case 2 _{, one of the bending moment M j, 0} and the rotation angle φ _{jl, 1} is further given as a provisional design value.
When the tentative design step S41 is completed, the storage step S20, the solution calculation step S22, the residual calculation step S24, and the determination step S26 are performed, and the process proceeds to the solution setting step S28 or the tentative design step S41 based on the determination in the determination step S26. ..

In the solution setting step S28 in this case, the _{bending moments M j and n} are calculated from the equation (65) using the _{calculated rotation angles φ r and n.}

[5.3. Evaluation method of continuous beam in case 3 (a pair of end fulcrums are rigid joints)]
In the evaluation method S46 in this case, the solution determination step S47 for obtaining a plurality of bending moments and a plurality of rotation angles based on the given conditions is performed.
In the solution determination step S47, first, the first design step S14 is performed. When the first design step S14 is completed, the process proceeds to step S49.
Next, in the second design step S49, the second design unit 114 gives a second boundary condition according to the joining state at the pair of end fulcrums 12b. In the case of Case 3, the second boundary condition is that the rotation angles φ _{jl, 1} (φ _{l, 1} , φ ₀₁ ) and the rotation angles φ _{jr, n} (φ _{r, n} ) are 0 rad, respectively. That is, the bending moments M _{j, 0} , M _{j, n} are unknown.
When the second design step S49 is completed, the process proceeds to step S51.

Next, in the temporary design step S51, the temporary design unit 115 gives a temporary design value including _{bending moments M j and i} acting on the intermediate fulcrum 12a according to the joining state at the pair of end fulcrums 12b. _{When the continuous beam 11 is the case 3, bending moments M j, 0} , M _{j, n} acting on the pair of end fulcrums 12b are further given as temporary design values.
When the tentative design step S51 is completed, the storage step S20, the solution calculation step S22, the residual calculation step S24, and the determination step S26 are performed, and the process proceeds to the solution setting step S28 or the tentative design step S51 based on the determination in the determination step S26. ..
However, in the residual calculation step S24, the vertical displacement calculation result δ _{0, calc} is not newly calculated, and the vertical displacement calculation result δ _{0, calc} is the value of the _{vertical displacement δ 0} given under the given conditions. To do. It is assumed that the rotation angles φ _{l and 1} are 0 rad without new calculation.
If continuous beam 11 of the case 3, the residual calculation step S24, in consideration of the phi _r, n = 0 is given condition, from _x i = I was _{L i} (68) In the formula (38) below The rotation angles φr _{and n} are calculated, and the square of the rotation angles φr _{and n} is multiplied by an appropriate weighting coefficient (positive value) and added to the displacement residual.

[5.4. Evaluation method of continuous beam in case 4A (first end fulcrum is pin joint and second end fulcrum is semi-rigid joint)]
In the evaluation method S56 in this case, the solution determination step S57 for obtaining a plurality of bending moments and a plurality of rotation angles based on the given conditions is performed.
In the solution determination step S57, first, the first design step S14 is performed. When the first design step S14 is completed, the process proceeds to step S59.
Next, in the second design step S59, the second design unit 114 gives a second boundary condition according to the joining state at the pair of end fulcrums 12b. In the case of Case 4A, the second boundary condition is that the bending moments M _{j and 0} are 0 Nm and the equation (65). That is, the second boundary condition includes that the rotation angle φ ₀₁ is unknown.
When the second design step S59 is completed, the process proceeds to step S61.

Next, in the temporary design step S61, the temporary design unit 115 gives a temporary design value including _{bending moments M j and i} acting on the intermediate fulcrum 12a according to the joining state at the pair of end fulcrums 12b. When the continuous beam 11 is the case 4A, the rotation angles φ _{l, 1} _{at the first end fulcrum 12b 1} and the bending moments M _{j, n} acting on the second end fulcrum 12b ₂ are further given as temporary design values.
When the tentative design step S61 is completed, the storage step S20, the solution calculation step S22, the residual calculation step S24, and the determination step S26 are performed, and the process proceeds to the solution setting step S28 or the tentative design step S61 based on the determination in the determination step S26. ..

[5.5. Evaluation method of continuous beam in case 5A (first end fulcrum is pin joint and second end fulcrum is rigid joint)]
In the evaluation method S66 in this case, the solution determination step S67 for obtaining a plurality of bending moments and a plurality of rotation angles based on the given conditions is performed.
In the solution determination step S67, first, the first design step S14 is performed. When the first design step S14 is completed, the process proceeds to step S69.
Next, in the second design step S69, the second design unit 114 gives a second boundary condition according to the joining state at the pair of end fulcrums 12b. In the case of Case 5A, the second boundary condition is that the bending moments M _{j and 0} are 0 Nm and the rotation angles φ _{r and n} are 0 rad. That is, the second boundary condition includes that the rotation angle φ ₀₁ is unknown. When the second design step S69 is completed, the temporary design step S61 is performed.
From this point onward, since the joining state at the pair of end fulcrums 12b is the same as in the case of case 4A, the description thereof will be omitted. However, when the continuous beam 11 is the case 5A, the residual calculation step S24 performs the following steps when determining whether or not the conforming condition is satisfied. That is, in the residual calculation step S24, in consideration of the phi _r, n = 0 is given condition, and calculates the rotation angle phi _{r, n} from _x i = was _{L i} (68) In the formula (38) below .. Then, the square of the rotation angles φ _{r and n} is multiplied by an appropriate weighting coefficient (positive value) and added to the displacement residual.

[5.6. Evaluation method of continuous beam in case 6A (first end fulcrum is rigid joint and second end fulcrum is semi-rigid joint)]
In the evaluation method S76 in this case, the solution determination step S77 for obtaining a plurality of bending moments and a plurality of rotation angles based on the given conditions is performed.
In the solution determination step S77, first, the first design step S14 is performed. When the first design step S14 is completed, the process proceeds to step S79.
Next, in the second design step S79, the second design unit 114 gives a second boundary condition according to the joining state at the pair of end fulcrums 12b. In the case of case 6A, the second boundary condition is that the rotation angles φ _{l and 1} are 0 rad and the equation (65). That is, the bending moments M _{j and 0} are unknown. When the second design step S79 is completed, the process proceeds to step S81.

Next, in the temporary design step S81, the temporary design unit 115 gives a temporary design value including _{bending moments M j and i} acting on the intermediate fulcrum 12a according to the joining state at the pair of end fulcrums 12b. _{When the continuous beam 11 is the case 6A, bending moments M j, 0} , M _{j, and n} acting on the pair of end fulcrums 12b are further given as temporary design values.
When the tentative design step S81 is completed, the storage step S20, the solution calculation step S22, the residual calculation step S24, and the determination step S26 are performed, and the process proceeds to the solution setting step S28 or the tentative design step S81 based on the determination in the determination step S26. .. However, when the continuous beam 11 is the case 6A, the residual calculation step S24 performs the following steps when determining whether or not the conforming condition is satisfied. _{That is, the rotation angles φr, n} obtained in the solution calculation step S22, and the rotation angles φr _{, n} calculated by the equation (65) included in the second boundary condition and the tentative design values _{Mj, n} . Multiply the square of the difference by an appropriate weighting factor (positive value) and add it to the displacement residuals.

As described above, in the continuous beam 11 including the n beams 13, the number of fulcrums (intermediate fulcrum 12a and end fulcrum 12b) included in the continuous beam 11 is (n + 1). Since the moment and the angle of rotation are unknown at each fulcrum, there are 2 (n + 1) unknowns according to the equation (n + 1) × 2. That is, the number of the plurality of unknowns is 2 (n + 1).
On the other hand, the conditional expression (n + 1) is given as the relational expression of the plurality of rotational rigidity, the plurality of bending moments, and the plurality of rotation angles.

When the second boundary condition is given, it becomes as follows.
(1) When at least one of the end fulcrums 12b of the continuous beam 11 is pin-joined Since the bending moment at the pin-joined end fulcrum 12b is 0, the number of pin-joined end fulcrums 12b (0 or more 2). The unknown number decreases by the following). As a result, the equations of the angle of rotation, the bending moment, and the rotational rigidity in the pin joint also _{become M j} = S _j = 0 in the equation of _{M j} = S _j × φ _j. Therefore, the rotation angle φ _j becomes indefinite, and the conditional expression based on the first boundary condition is reduced by the number of pin-joined end fulcrums 12b.
(2) When at least one of the end fulcrums 12b of the continuous beam 11 is rigidly joined Since the rotation angle at the rigidly joined end fulcrum 12b is 0, the number of rigidly joined end fulcrums 12b (0 or more 2). The unknown number decreases by the following). As a result, the equations of the angle of rotation, the bending moment, and the rotational rigidity in the rigid joint also become _{φ j} = 0 and S _j = ∞ (infinity) in the equation of _{M j} = S _j × φ _j. Therefore, the bending moment M _j becomes indefinite, and the conditional expression based on the first boundary condition is reduced by the number of rigidly joined end fulcrums 12b.

Further, as a condition for giving the vertical displacement of each fulcrum, the conditional expression (n + 1) is given.
By giving the second boundary condition that the end fulcrum 12b is a pin joint or a rigid joint, a plurality of unknowns are reduced by the number of the second boundary conditions. However, the conditional expression becomes indefinite, and the number of first boundary conditions decreases.
As a result, the condition that the number of the plurality of unknowns and the sum of the number of the plurality of first conditional expressions and the number of the plurality of second conditional expressions are the same (constant) is maintained.

By performing the evaluation method when the continuous beam 11 is the case 1 to the case 6 described above, a plurality of bending moments and a plurality of rotation angles can be obtained regardless of the joint state at the pair of end fulcrums 12b of the continuous beam 11. It can be obtained based on the given conditions.
The bending moment of the continuous beam 11 is obtained by the equation (70) from _{the shearing force V jl, i} , the bending moment M _{j, i, etc. obtained by the equation (24) in the equation (22).}

The deflection distribution of the continuous beam 11 can be obtained from Eqs. (42) to (44) according to the interval of the _{coordinates x i.}
Further, the evaluation method of the present embodiment may be used to perform a continuous beam design method for designing the continuous beam 11.

[6. Evaluation example 1]
The inventors performed an evaluation method using the differential evolution method, and created an evaluation program for obtaining a plurality of bending moments and a plurality of rotation angles based on given conditions. The input sheet C of the evaluation program is shown in FIG. The specifications and the like of the main continuous beams 11 to be input to the input sheet C will be described.
In the input sheet C, the number of continuous beam span n of the continuous beam 11 is input to the cell C1. In this example, n was set to 5. The threshold value for ending convergence used for the convergence calculation is input to cell C2. In this example, the threshold is 1.38 × 10 ⁻² mm ² .
Length of the beam 13 ₅ from the beam 13 ₁ (Span) _{L i,} respectively inputted to the cell C3. In this example, the length _{L i} of the beam 13 ₅ from the beam 13 _1, and a 13800mm (13.8m), respectively.

The uniformly distributed load (Composite stage load) _{w i} applied from the beam 13 ₁ to the beam 13 _5, respectively entered in the cell C6. In this example, a uniformly distributed load _{w i} are equal to each other acting on the beam 13 ₅ from the beam 13 _1, it was 28.56kN / m.
_{The vertical displacement δ i} at each intermediate fulcrum 12a was set to 0 m.
In this example, it is assumed that the weight of the slab concrete and the steel frame (structural mass × gravitational acceleration, ww_SW) is supported by the beam 13 of the pure steel frame frame pin-joined at the fulcrums 12b at both ends and the intermediate fulcrum 12a. This value was entered in cell C7.
Then, the load loaded on the beam 13 after the continuous beam 11 is put into service is the joint portion (intermediate fulcrum 12a) that behaves as a semi-rigid joint after the concrete is hardened, and the composite beam (beam) supported by the pin joint at both end fulcrums 12b. It is assumed that the main body 18 and the floor 17 are supported by a beam 13) that behaves integrally.

Figure 12 was evaluated using the evaluation program, shows the bending relationship moment distribution acting on the continuous beam 11 with respect to the distance from the first end supporting point 12b _1. In FIG. 12, the horizontal axis represents the distance (mm) moved _{from the first end fulcrum 12b 1} to the second end fulcrum 12b _{2 in the continuous beam 11.} The vertical axis represents the bending moment distribution (kNm) acting on the continuous beam 11.
The curve L4 shown by the solid line represents the bending moment distribution with respect to the load (equally distributed load) after the concrete is hardened. The curve L5 shown by the dotted line shows a value in consideration of the bending moment distribution due to the own weight of the slab concrete (concrete 22) and the steel frame (beam body 18) in addition to the value shown by the curve L4. That is, the curve L4 represents the bending moment distribution when the mass of the continuous beam 11 which is a structure is not taken into consideration.

The white triangle mark (Δ) with a sharp upper part and the white triangle mark (▽) with a sharp lower part represent the elastic limiting bending strength of the beam 13. The white square marks (□) represent the elastic limiting bending strength of the joints (intermediate fulcrum 12a and end fulcrum 12b).
It was found that both bending moment distributions were equal to or less than the elastic limiting bending strength of the beam 13 and the joint.

Figure 13 was evaluated using the evaluation program, shows the relationship between the rotational angle distribution of the continuous beam 11 with respect to the distance from the first end supporting point 12b _1. In FIG. 13, the horizontal axis represents the distance (mm) moved _{from the first end fulcrum 12b 1} to the second end fulcrum 12b _{2 in the continuous beam 11.} The vertical axis represents the rotation angle distribution (rad) of the continuous beam 11.
In the continuous beam 11, a rotation angle corresponding to the rotational rigidity is generated at the intermediate fulcrum 12a which is semi-rigidly joined. Therefore, the rotation angle distribution is discontinuous at the intermediate fulcrum 12a. For example, in FIG. 13, showing a rotational angle phi _{l, 4} in the left end of the third beam 13 ₃ of the rotation angle of the right end phi _{r, 3,} and the fourth beam 13 _4. The rotation angles φ _{r and 3} and the rotation angles φ _{l and 4} are discontinuous.
The difference in the absolute values of the rotation angles of the adjacent beams 13 at the intermediate fulcrum 12a is caused by the ratio of the bending moment of the joint to the rotational rigidity. Specifically, the smaller the rotational rigidity and the larger the bending moment, the larger the sum of the absolute values of the rotation angles of the adjacent beams 13 at the intermediate fulcrum 12a.

Figure 14 was evaluated using the evaluation program, shows the relationship between the deflection distribution of the continuous beam 11 with respect to the distance from the first end supporting point 12b _1. In FIG. 14, the horizontal axis represents the distance (mm) moved _{from the first end fulcrum 12b 1} to the second end fulcrum 12b _{2 in the continuous beam 11.} The vertical axis represents the deflection distribution (mm) of the continuous beam 11. One end is the maximum of the first beam 13 ₁ and the fifth beam 13 ₅ as a pin joint deflection, the second beam 13 _2, third beam 13 _3, and a fourth beam 13 is larger than the _fourth maximum deflection.

[7. Evaluation example 2]
The evaluation method of the continuous beam was performed on the continuous beams of Comparative Examples shown in Tables 2 to 8 and Examples 1 to 3.

As shown in Table 2, as the characteristics of the continuous beams, the number n of the beams included in the continuous beams of Comparative Examples and Examples 1 to 3 was set to 4, respectively. Hereinafter, the i-th beam counting from the first end fulcrum side is also referred to as an i-beam. The i-th intermediate fulcrum counted from the first end fulcrum side is also called the i-th intermediate fulcrum.
The length of the beam was set to 15500 mm (15.5 m) for each of the beams in Comparative Examples, Examples 1 and 2. In Example 3, the lengths of the first beam to the fourth beam were 13000 mm, 15500 mm, 14000 mm, and 13000 mm, respectively.
In Comparative Example and Example 1, the beam length was assumed to be 617 mm. The flange width is 230 mm, the web thickness is 13.1 mm, the flange thickness is 22.1 mm, the moment of inertia of area around the strong axis is 1120000000 mm ⁴ (0.00112 m ⁴ ), and the mass per unit length of the beam is 140 kg / m. The yield strength of the steel material of the beam was assumed to be ^{345 N / mm 2.}
In Examples 2 and 3, the beam length was assumed to be 600 mm. Flange width is 250 mm, web thickness is 9.0 mm, flange thickness is 16.0 mm, moment of inertia of area around the strong axis is 831000000 mm ⁴ , mass per unit length of the beam is 104 kg / m, yield strength of the steel material of the beam. Was assumed to be 355 N / mm ^2.

As shown in Table 3, as the characteristics of the slab (RC slab), it was assumed that the slab thickness was 130 mm and the deck plate height was 52 mm in both the comparative beam and the continuous beams of Examples 1 to 3.
In Comparative Examples, Examples 1 and 3, reinforcing bars having a diameter of D10 were arranged at a pitch of 100 mm as reinforcing bars for the portion where the beam is negatively bent (both ends of the beam). In Example 2, reinforcing bars having a diameter of D16 were arranged at a pitch of 100 mm.
In Examples 1 and 3, reinforcing bars having a diameter of D10 were arranged at a pitch of 100 mm as reinforcing bars at the joint where the beams are joined. In Example 2, reinforcing bars having a diameter of D16 were arranged at a pitch of 100 mm. In the comparative example, no reinforcing bar was placed at the joint.
The yield strength of the steel material of the reinforcing bar used in Comparative Examples and Examples 1 to 3 was 435 N / mm ² .

The effective width of the slab was determined based on the British Standards "Eurocode 4: Design of composite steel and concrete structures --Part 1-1: General rules and rules for buildings". The definition of the effective width of the slab is not limited to this, and may be defined based on, for example, "Various Composite Structure Design Guidelines / Explanation", Architectural Institute of Japan, and the like.
In this example, for example, in the continuous beam of the comparative example, the effective width of the slab with respect to the first beam to the fourth beam was set to 3294 mm, respectively. In the continuous beam of Example 1, the effective widths of the slabs from the first beam to the fourth beam were set to 3294 mm, 2713 mm, 2713 mm, and 3294 mm.

As shown in Table 4, as the support conditions at each fulcrum of the continuous beam, in the continuous beam of the comparative example, pin joining was performed at each end fulcrum and each intermediate fulcrum. In the case of pin joining, the rotational rigidity at each fulcrum becomes 0.
In the continuous beams of Examples 1 to 3, pin joints were made at each end fulcrum, and semi-rigid joints were made at each intermediate fulcrum. The rotational rigidity of semi-rigid joints is based on the reinforcement arrangement and effective width in Table 3, as described in British Standards "Eurocode 4: Design of composite steel and concrete structures --Part 1-1: General rules and rules for buildings". Was calculated as follows. In the continuous beam of Example 1, the rotational rigidity from the first intermediate fulcrum to the third intermediate fulcrum was calculated to be 108 kNm / mrad, 116 kNm / mrad, and 108 kNm / mrad, respectively. In the continuous beam of Example 2, the rotational rigidity from the first intermediate fulcrum to the third intermediate fulcrum was calculated to be 192 kNm / mrad, 242 kNm / mrad, and 192 kNm / mrad, respectively. In the continuous beam of Example 3, the rotational rigidity from the first intermediate fulcrum to the third intermediate fulcrum was calculated to be 103 kNm / mrad, 116 kNm / mrad, and 102 kNm / mrad, respectively.

As shown in Table 5, the flexural rigidity of the composite beam for forward bending and negative bending was calculated as follows, based on the cross-sectional dimensions of the beam body in Table 2 and the effective width in Table 3.
In the continuous beams of the comparative example, it was assumed that the bending rigidity of the forward bending was 555,000 kNm ² ^{and the bending rigidity of the negative bending was 308,000 kNm 2 in all of the first to fourth beams.}
The continuous beam of Example 1, the first beam and the fourth beam, flexural rigidity of the positive bending 555000KNm ^2, the bending stiffness of the negative bending was calculated as 308000kNm ^2. In a second beam and a third beam, the bending rigidity of the positive bending 525000KNm ^2, the bending stiffness of the negative bending was calculated as 292000KNm ^2.
The continuous beam of Example 2, the first beam and the fourth beam, flexural rigidity of the positive bending 433594KNm ^2, the bending stiffness of the negative bending was calculated as 241305KNm ^2. In a second beam and a third beam, the bending rigidity of the positive bending 413095KNm ^2, the bending stiffness of the negative bending was calculated as 232307KNm ^2.
The continuous beam of Example 3, the first beam and the fourth beam, flexural rigidity of the positive bending 415000KNm ^2, the bending stiffness of the negative bending was calculated as 232000kNm ^2. In a second beam, the bending rigidity of the positive bending 413000KNm ^2, the bending stiffness of the negative bending was calculated as 232000KNm ^2. In the third beam, the bending rigidity of the positive bending 402000KNm ^2, the bending stiffness of the negative bending was calculated as 227000kNm ^2.

As shown in Table 6, the yield bending strength of the beam and the joint was set. For example, in the continuous beam of the comparative example, the yield bending strength of the first beam was calculated to be 1750 kNm and the yield bending strength of the negative bending was 1415 kNm. In the continuous beam of Example 1, the yield bending strength of the first beam was calculated to be 1750 kNm and the yield bending strength of the negative bending was 1415 kNm. The yield strength of the negative bending at the first intermediate fulcrum was calculated to be 474 kNm.
As the load, the SDL (dead load excluding the structure) of the continuous beams of Comparative Examples and Examples 1 to 3 was calculated to ^{be 1.0 kN / m 2, respectively.} The LL (live load) was calculated to ^{be 4.5 kN / m 2 respectively.}

As a result of performing the evaluation method of the continuous beam for the continuous beams of Comparative Examples and Examples 1 to 3, the results shown in Tables 7 and 8 were obtained.

As shown in Table 7, for example, in the continuous beam of the comparative example, the maximum generated bending moment was 1400 kNm in the forward bending and 0 kNm in the negative bending in all of the first to fourth beams. The bending moment of negative bending did not act on each intermediate fulcrum.
In the continuous beam of Example 1, a bending moment of 1174 kNm was applied to the first beam in the forward bending, and a bending moment of 471 kNm was applied in the negative bending. A bending moment of 471 kNm acted on the first intermediate fulcrum by negative bending.
The maximum deflection of the continuous beam was 95 mm for the continuous beam of the comparative example. In the continuous beams of Examples 1 to 3, they were 82 mm, 104 mm, and 96 mm, respectively.

The ratio of the generated bending moment to the bending proof stress shown in Table 8 means that there is no margin for the bending proof stress as the value approaches from 0% to 100%. In the continuous beam of the comparative example, this ratio was 80% in the forward bending of the beam. In the continuous beam of Example 1, the forward bending of the beam was 68%, the negative bending of the beam was 34%, and the negative bending of the joint between the beams was 99%. In the continuous beam of Example 2, the forward bending of the beam was 83%, the negative bending of the beam was 49%, and the negative bending of the joint between the beams was 66%. In the continuous beam of Example 3, the forward bending of the beam was 58%, the negative bending of the beam was 37%, and the negative bending of the joint between the beams was 88%.
The maximum deflection ratio of the beam based on the continuous beam of the comparative example means that the closer the value is from 100% to 0%, the smaller the maximum deflection is with respect to the continuous beam of the comparative example. This ratio was 86% for the continuous beam of Example 1. It was 110% for the continuous beam of Example 2 and 101% for the continuous beam of Example 3.
As for the mass ratio of the beam (steel) based on the continuous beam of the comparative example, the closer the value is from 100% to 0%, the smaller the mass of the steel material required to form the beam with respect to the continuous beam of the comparative example. That is, it means that it is an economical design.
This ratio was 100% for the continuous beam of Example 1. In the continuous beams of Examples 2 and 3, it was 74%.

As a result of evaluating the continuous beam using this evaluation program, the mass of the beam does not change in the continuous beam of Example 1 as compared with the continuous beam of the comparative example, but the maximum deflection of the continuous beam is (82-95) / 95. It was found that it was reduced by 14% from the formula of.
In the continuous beam of Example 2, the maximum deflection of the continuous beam is increased by 10% from the equation of (104-95) / 95 as compared with the continuous beam of the comparative example, and the ratio of the generated bending moment to the bending strength is 80 to 83%. Equivalent, but the beam mass was found to be reduced by 26% from the equation (104-140) / 140.
In the continuous beam of Example 3, the maximum deflection of the continuous beam is equivalent to 95 to 96 mm as compared with the continuous beam of the comparative example, but the mass of the beam can be reduced by 26% from the equation (104-140) / 140. Do you get it.

Beams (synthetic beams) in which different materials such as steel beams and concrete slabs are joined differ in rigidity and yield strength depending on the bending direction. The reason for this is that the stress-strain relationship between concrete and steel is different, the tensile strength of concrete is smaller than the compressive strength, and the material properties depend on the direction of the force. When evaluating the bending moment distribution and proof stress in consideration of the rigidity of such a beam, an explicit solution could not be obtained, and it was necessary to obtain it by convergence calculation. Moreover, if the ends of the continuous beam are supported by semi-rigid joints, the equation for identifying the solution becomes more complicated. Therefore, there is a problem that it is too complicated to be used in design practice.
Therefore, in the evaluation of many continuous beams, it is assumed that the joint state at the end fulcrum is pin joint or rigid joint. Then, the rigidity of the beam (composite effect of the slab) is not taken into consideration, or even if it is taken into consideration, there is no accurate evaluation formula. For this reason, there is a problem that a large safety factor is taken in the design of the continuous beam and it is difficult to obtain an economic effect when designing the continuous beam.

On the other hand, according to the evaluation method and the evaluation program 127 of the present embodiment, the number of the plurality of unknowns and the number of the plurality of first boundary conditions are the same. Therefore, a plurality of unknowns are solved so as to satisfy a plurality of first boundary conditions based on the length and flexural rigidity of the beam 13, a plurality of rotational rigidity, a plurality of vertical loads, and a plurality of vertical displacements given as given conditions. By this method, a plurality of bending moments and a plurality of rotation angles included in a plurality of unknowns can be obtained.
Then, a plurality of vertical displacements are calculated from the obtained plurality of bending moments and a plurality of rotation angles based on the relational expressions of the plurality of vertical displacements. By this method, the bending moment and the deflection distribution of the continuous beam 11 can be evaluated more appropriately according to the joint state at the pair of end fulcrums 12b. Therefore, the deflection can be appropriately evaluated.
By the above steps, it is possible to prevent the evaluation of the continuous beam 11 from being overestimated and underestimated.

In the present embodiment, the continuous beam 11 composed of a concrete slab and a steel beam and having different bending rigidity between normal bending and negative bending has a joint portion (semi-rigid joint) having a finite rotational rigidity at the intermediate fulcrum 12a. A method for evaluating the case (semi-rigid continuous beam) was proposed. According to this evaluation method, the bending moment distribution, the rotation angle distribution, and the deflection distribution at an arbitrary position are represented by using the value of the bending moment at the joint. Then, the value of the bending moment of the joint is identified by the optimization calculation considering the first boundary condition, and the required performance in the design (bending moment distribution for design, rotation angle of the joint) and the deflection distribution of the beam are derived. can do.

In the solution determination process, the first design process S14, the second design process, the temporary design process, the solution calculation process S22, the residual calculation process S24, and the determination process S26 are performed. By determining whether or not the displacement residual is smaller than the threshold value in the determination step S26, the calculated value calculated based on the provisional design value can be evaluated based on the design value and the displacement residual.
In the determination step S26, when the displacement residual is equal to or greater than the threshold value, another new temporary design value is given in the temporary design step instead of the temporary design value stored in the storage step S20, and this new temporary design The solution calculation step S22, the residual calculation step S24, and the determination step S26 are performed based on the values. Then, this set is repeated until it is determined in the determination step S26 that the displacement residual is smaller than the threshold value. Therefore, the bending moment can be calculated with high accuracy, and a plurality of rotation angles can be obtained based on the bending moment.

The calculation results [delta] _{i, calc} of vertical displacement, with (24) or the like, is calculated by the equation which was _x i = _{L i} In equation (44). As a result, a plurality of vertical displacements at the intermediate fulcrum 12a and the pair of end fulcrums 12b can be evaluated with high accuracy using a mathematical formula.

Although the first embodiment of the present invention has been described in detail with reference to the drawings, the specific configuration is not limited to this embodiment, and changes and combinations of configurations within a range that does not deviate from the gist of the present invention, Deletion etc. are also included. Further, it goes without saying that each of the configurations shown in each embodiment can be used in combination as appropriate.
For example, in the evaluation method of the above-described embodiment, it is not necessary to shift to the solution setting step S28 or the tentative design step based on the determination in the determination step S26. In this case, the storage step S20 is not performed by the evaluation method.

In the beam 13, the bending rigidity of the forward bending and the bending rigidity of the negative bending may be equal to each other.
Although the end support portion is assumed to be the second girder 31, the end support portion may be a rigid body such as another building.

(Second Embodiment)
Next, the evaluation method of the composite beam according to the second embodiment of the present invention will be described with reference to FIGS. 15 to 24, but the same parts as those in the above embodiment are designated by the same reference numerals and the description thereof will be described. It will be omitted and only the differences will be explained.

[1. Synthetic beam with semi-rigid joints at both ends]
The synthetic beam evaluation method of the present embodiment is used, for example, to evaluate the synthetic beam 211 constituting the building 201 shown in FIG.
In this example, the composite beam 211 includes a floor 212 and a beam (small beam) 213. The configuration of the composite beam 211 is not limited to this example.
The floor 212 is a so-called synthetic slab and is supported from below by the beams 13. The floor 212 includes a deck plate 216 and an RC (Reinforced Concrete) slab 217 arranged on the deck plate 216.
The uneven shape of the deck plate 216 extends in a direction along the horizontal plane and in a direction orthogonal to the direction in which the beam 213 extends.
The RC slab 217 includes concrete 218 and reinforcing bars 219. The concrete 218 is formed in a plate shape in which the vertical direction is the thickness direction. The concrete 218 is supported from below by the deck plate 216.
Reinforcing bars 219 extend along a horizontal plane and are buried in concrete 218. For example, the reinforcing bars 219 are arranged in a grid pattern in a plan view.

The beam 213 is made of H-section steel and extends along a horizontal plane. The lower end of the stud 221 is fixed to the upper flange of the beam 213. The stud 221 penetrates the deck plate 216. The upper end of the stud 221 is embedded in concrete 218.
Both ends of the beam 213 are semi-rigidly joined to the girder 224 extending along the horizontal plane. The girder 224 extends in a direction orthogonal to the beam 213. The semi-rigid joint between the beam 213 and the girder 224 is performed by, for example, a shear plate 225 and a bolt 226.
The end of the girder 224 is supported from below by columns 228.

The evaluation method of the composite beam 211 configured in this way will be described below.

[2. Differential equation of deflection of synthetic beam with semi-rigid joint at both ends (exact solution)]
The following studies are all limited to the elastic range of synthetic beams.
As shown in FIG. 16, the composite beam is modeled.
The coordinates x (mm) are defined to the right along the composite beam. The position of the left end of the composite beam is defined as the origin of the coordinates x. It is assumed that the length of the composite beam is L (mm). As a boundary condition (precondition), it is assumed that both ends of the composite beam are semi-rigidly joined. It is assumed that a downward evenly distributed load w (N / mm: Newton per millimeter) acts on the composite beam over the entire length.
The rotational rigidity of the joints at both ends of the composite beam is S _j (Nmm / rad: Newton millimeter per radian), the rotation angle at the end of the composite beam is θ _j (positive in the front view shown in FIG. 16). rad: Radian).
At this time, the absolute value of the bending moment at the end of the composite beam (the moment of the semi-rigid joint at the end; hereinafter referred to as the semi-rigid contact moment) M _j (N mm) is expressed by the equation (111).

The distribution of bending moments M (x) (Nmm) along the coordinates x of the composite beam is (112) from the force balance condition, assuming that the bending moment when tensile stress acts on the lower flange of the composite beam is positive. It is represented by an expression.

The curvature φ (rad / mm) of the composite beam can be expressed by using the bending moment M (N mm) acting on the composite beam and the flexural rigidity EI (Nmm ^{2) of the composite beam.} Due to the action of concrete, the flexural rigidity of forward bending (convex downward) and the bending rigidity of negative bending (convex upward) of synthetic beams are different from each other. Therefore, the bending rigidity of the _{forward bending of the composite beam is defined as EI s} (N mm ² ), and the bending rigidity of the negative bending of the composite beam is _{defined as EI h} (N mm ² ). Then, as shown in FIG. 17, it is assumed that the composite beam is negatively bent in the range where _{the coordinates x are 0 or more and L 1} or less, and L _{2 or more and L or less.} However, L ₁ is greater than 0 and L ₂ is _{greater than L 1 and} less than L. It is assumed that the composite beam is bent forward in the range where the coordinates x are L ₁ or more and L _{2 or less.}
At this time, the solutions of x when Eq. (112) becomes 0 are L ₁ and L ₂ .
The curvature φ of the composite beam is divided into a region where the composite beam is bent forward and a region where the composite beam is negatively bent, and is expressed by equations (113) and (114).

When the equation (112) in which the right side is equal to 0 is solved for x, L ₁ and L ₂ can be obtained as in equations (115) and (116).

Next, the rotation angle θ (x) (rad) of the composite beam will be described with the clockwise rotation with respect to the horizontal plane as positive (+). The rotation angle θ (x) can be obtained by integrating the curvature φ of the equations (113) and (114) with the coordinates x and the like. Further, the rotation angle θ (x) is calculated from the equations (120) to (122) by using the equation (119) in consideration of the boundary condition that the _{rotation angle θ becomes θ j when the coordinate x is 0.} It is expressed as.

Next, the deflection δ (x) (mm) generated in the composite beam will be described with the vertical downward direction as positive (+). The deflection δ (x) can be obtained by integrating the rotation angles θ of the equations (120) to (122) with the coordinates x and the like. Further, the deflection δ (x) is obtained from the equation (127) by using the equations (125) and (126) in consideration of the boundary condition that the deflection δ (x) becomes 0 when the coordinate x is 0. It is expressed as in equation (129).

Equations (127) to (129) obtained above are expressed in a form including _{a semi-rigid contact moment M j} and a rotation angle θ _j. However, as it is, the semi- _{rigid contact moment M j} and the rotation angle θ _j cannot be uniquely determined with _{respect to the arbitrary rotational rigidity S j.} Further, if the conforming condition of the deformation that the rotation angle θ becomes 0 rad when the coordinate x is (L / 2) is used, the rotation rigidity S _j and the rotation angle θ can be obtained from the equations (111) and (121). _{The relationship with j} is expressed as in Eq. (130).

Further, when the coordinate x is L, the deflection δ (x) becomes 0, which is the conformity condition of the deformation. Then, from the equations (111) and (129) _{, the relationship between the rotational rigidity S j} and the rotation angle θ _j is expressed as in the equation (131).
The equation (130) is equivalent to the equation (131).

_{The maximum value δ max of the} deflection generated in the composite beam is the value when the coordinate x in the equation (128) is (L / 2).
_{In order to calculate the maximum value δ max of the} deflection using the above equation (130) or (131), the convergence calculation is performed using the equation (130) or the equation (131) to calculate the _{rotation angle θ j.} There is a need. Further, it is necessary to calculate the deflection δ (x) when the coordinate x is (L / 2) by using the equation (128).

[3. Approximate formula for deflection of synthetic beams with semi-rigid joints at both ends]
_{The inventors have found that by evaluating the maximum value δ max of the} deflection based on the non-dimensionalized rotational rigidity and the non-dimensionalized flexural rigidity, the following results are obtained. That is, regardless of the specifications of the composite beam, the maximum value δ _max of the end bending moment and the deflection can be calculated with high versatility and with high accuracy without performing the convergence calculation.
In the following, an approximate expression that can calculate the maximum values of the end bending moment and the deflection by an explicit function will be described.

As in Eqs. (135) and (136), the dimensionless flexural rigidity α _s and the dimensionless rotational rigidity α _j are defined.

That is, the non-dimensional bending rigidity α _s is defined as the ratio of the bending rigidity of the forward bending of the synthetic beam to the bending rigidity of the negative bending of the synthetic beam. In this example, the non-dimensional bending rigidity α _s is defined as the ratio of _{the bending rigidity EI s} of the positive bending of the synthetic beam to the bending rigidity EI _{h of the negative bending of the synthetic beam.} In this case, the dimensionless flexural rigidity α _s generally takes a value of 1 or more. This is because the floor is usually above the vertical direction of the beam, and the concrete of the floor has a higher compressive resistance than the tensile resistance. For this reason, the resistance of the floor to the positive bending that is compressed is larger than that of the negative bending in which the concrete of the floor is pulled. In the specifications of a general composite beam, the dimensionless bending rigidity α _s is 10 or less, and in a more commonly used composite beam, the dimensionless bending rigidity α _s is 3 or less.
The non-dimensional bending rigidity α _s may be defined as the ratio of _{the bending rigidity EI h} of the negative bending of the synthetic beam to the bending rigidity EI _{s of the forward bending of the synthetic beam.}
The non-dimensional rotational rigidity α _j is a value obtained by dividing the rotational rigidity at the end of the synthetic beam by the bending rigidity per unit length of the synthetic beam.

Further, defined pin contact moment _{M o,} HanTsuyoshise' moment _{M j,} rigid connection moment _{M jr,} dimensionless joint moment beta _Mj, and dimensionless KaTsuyoshise' moment beta _Mj, the _rigid.
The pin contact moment _Mo is defined when it is assumed that in a composite beam in which the bending rigidity of the forward bending and the bending rigidity of the negative bending are different from each other, both ends are pin-joined and an evenly distributed load acts over the entire length. The pin contact moment _Mo means the maximum value of the bending moment acting on the composite beam. The pin contact moment _Mo is expressed by the equation (137).
The pin contact moment _Mo is equal to the following value in a beam in which the bending rigidity of the forward bending and the bending rigidity of the negative bending are equal to each other. The pin contact moment _Mo is equal to the bending moment acting on the end of the beam when both ends are pin-joined and an evenly distributed load acts over the entire length. Specifically, the pin contact moment _{M o} is the value according to equation ^(wL 2/8).

As described above, the semi-rigid contact moment M _j is the absolute value of the bending moment at the end of the composite beam. More specifically, the semi-rigid contact moment M _j is a composite beam in which both ends are semi-rigidly joined and an evenly distributed load acts over the entire length in a composite beam in which the bending rigidity of forward bending and the bending rigidity of negative bending are different from each other. It means the bending moment acting on the end of the beam. The semi-rigid moment of a sequence M _j is expressed by the equation (138) using the _{rotation angle θ j} calculated by performing the convergence calculation using the equation (130) or the equation (131).

The rigid contact moment M _jr means the following values in a composite beam in which the bending rigidity of the forward bending and the bending rigidity of the negative bending are different from each other. The rigid contact moment M _jr means a bending moment acting on the end of the composite beam when both ends are rigidly joined and an evenly distributed load acts over the entire length.
The composite beam of the present embodiment is a beam in which the bending rigidity of the forward bending and the bending rigidity of the negative bending are different from each other. For convenience of explanation, a beam of a comparative example in which the bending rigidity of the forward bending and the bending rigidity of the negative bending are equal to each other is assumed. In the beam of the comparative example, when both ends of which are that the uniformly distributed load over the entire length are respectively rigidly joined to act, rigid connection moment M _jr is the value according to equation (wL ^2/12).
Dimensionless joint moment beta _Mj is a value obtained by dividing the HanTsuyoshise' moment M _j with a pin contact moment M _o, represented by (139) below.

Dimensionless KaTsuyoshise' moment beta _{Mj, rigid} is a value obtained by dividing the rigid connection moment _{M jr} pin contact moment _{M o,} represented by (140) below.

Here, indexes such as the dimensionless rotational rigidity α _j and the dimensionless bending rigidity α _{s were evaluated.} Therefore, the non-dimensional joint moment β _Mj is calculated _{by changing the non-dimensional rotational rigidity α j} and the non-dimensional flexural rigidity α _s for the composite beams of the specifications of Cases 1 to 6 shown in Table 9. did.
In Cases 1 to 5, both ends of the composite beam are semi-rigidly joined, and in Case 6, both ends of the composite beam are rigidly joined.

In Case 1, the length L of the composite beam was assumed to be 10.0 m (10000 mm). 229397KNm ² negative bending flexural rigidity EI _h, when a uniformly distributed load w assuming 28.6kN / m (28.6N / mm) , a pin contact moment _{M o} was calculated as 357kNm from (137) below. In Case 1, the dimensionless rotational rigidity α _j was changed from a minimum value of 0.00 to a maximum value of 50.00 in 1.00 increments to 51 different values. That is, the dimensionless rotational rigidity α _j was set to a value of 0.00, 1.00, 2.00, ..., 50.00. The dimensionless flexural rigidity α _s was changed from a minimum value of 1.00 to a maximum value of 6.00 in 0.10 increments to 51 different values. That is, the dimensionless flexural rigidity α _s was set to a value of 1.00, 1.10, 1.20, ..., 6.00. In Case 1, 2601 cases were calculated according to the equation (51 × 51) in which the values of the _{dimensionless rotational rigidity α j} and the dimensionless bending rigidity α _{s were changed.}

In the case 2, the length L of the composite beam was changed to 15.0 m based on the case 1. Pin contact moment _{M o} was calculated to 803kNm from (137) below. In case 2, the dimensionless rotational rigidity α _j and the dimensionless bending rigidity α _s were changed in the same manner as in case 1, and 2601 cases were calculated.
In the case 3, in the case 2, the step of the _{dimensionless rotational rigidity α j} and the maximum value and the step of the _{dimensionless bending rigidity α s were changed.} That is, in Case 3, the dimensionless rotational rigidity α _j was changed from a minimum value of 0.00 to a maximum value of 50.00 in increments of 0.01 to 5001 types of values. That is, the dimensionless rotational rigidity α _j was set to a value of 0.00, 0.01, 0.02, ..., 50.00. The dimensionless flexural rigidity α _s was changed from a minimum value of 1.00 to a maximum value of 1.06 in 0.01 increments to seven different values. That is, the dimensionless flexural rigidity α _s was set to a value of 1.00, 1.01, 1.02, ..., 1.06. In Case 3, a trial calculation was made for 35007 cases according to the equation (5001 × 7) in which the values of the _{dimensionless rotational rigidity α j} and the dimensionless bending rigidity α _{s were changed.} In Case 3, a more detailed trial calculation was made for a part of the range of the _{dimensionless rotational rigidity α j} and the dimensionless bending rigidity α _{s in Case 2.}

In case 4, the length L of the composite beam was changed to 8.4 m with reference to case 1. When a negative bending of the flexural rigidity EI _h Assuming 214311kNm ^2, pin contact moment _{M o} was calculated as 252kNm from (137) below. In Case 4, the dimensionless rotational rigidity α _j was changed from a minimum value of 0.00 to a maximum value of 100.00 in 0.50 increments to 201 kinds of values. That is, the dimensionless rotational rigidity α _j was set to a value of 0.00, 0.50, 1.00, ..., 100.00. The dimensionless flexural rigidity α _s was changed from a minimum value of 1.00 to a maximum value of 1.30 in 0.05 increments to seven different values. That is, the dimensionless flexural rigidity α _s was set to a value of 1.00, 1.05, 1.10, ..., 1.30. In Case 4, 1407 cases according to the equation (201 × 7) in which the values of the _{dimensionless rotational rigidity α j} and the dimensionless bending rigidity α _{s were changed were calculated.}

In case 5, the length L of the composite beam was changed to 13.8 m with reference to case 1. Pin contact moment _{M o} was calculated to 680kNm from (137) below. In case 5, _{the value of the dimensionless rotational rigidity α j} was changed in the same manner as in case 4. The dimensionless flexural rigidity α _s was changed from a minimum value of 1.00 to a maximum value of 4.00 in 0.50 increments to seven different values. That is, the dimensionless flexural rigidity α _s was set to a value of 1.00, 1.50, 2.00, ..., 4.00. In Case 5, 1407 cases according to the equation (201 × 7) in which the values of the _{dimensionless rotational rigidity α j} and the dimensionless bending rigidity α _{s were changed were calculated.}
In Cases 1 to 5, a total of 43023 cases were calculated.

In case 6, in case 2, the maximum values of the _{dimensionless rotational rigidity α j} and the dimensionless bending rigidity α _{s were changed.} In Case 6, the dimensionless rotational rigidity α _{j is set} to infinity (∞), that is, the rotational rigidity S _{j is set} to infinity, and both ends of the composite beam are rigidly joined. The dimensionless flexural rigidity α _s was changed from a minimum value of 1.00 to a maximum value of 51.00 in 0.10 increments to 501 types of values. That is, the dimensionless flexural rigidity α _s was set to a value of 1.00, 1.10, 1.20, ..., 51.00.

FIG. 18 shows the relationship between the _{dimensionless rotational rigidity α j} and the dimensionless joint moment β _Mj in cases 1 to 5. In FIG. 18, the horizontal axis represents the dimensionless rotational rigidity α _j , and the vertical axis represents the dimensionless joint moment β _Mj .
_{The straight line L11 is a dimensionless rigid contact moment β Mj, rigid, u} in a normal beam in which the bending rigidity of the forward bending and the bending rigidity of the negative bending are equal to each other. In a normal beam, the length is L (mm) and the evenly distributed load is w (N / mm). In this case, in the normal beam, the rigid connection moment _{^{M jr (wL 2/12)}} , since the pin contact moment _{M o} is ^(wL 2/8), dimensionless KaTsuyoshise' moment beta _{Mj, rigid, u} is the formula of ^{{(wL 2/12) /} (wL 2/8)}, a value of 2/3 (about 0.667).

The trial calculation result of Case 1 is represented by a white square mark. Similarly, the trial calculation result of Case 2 is represented by a white triangle mark, and the trial calculation result of Case 3 is represented by a white round mark. The trial calculation result of Case 4 is represented by a white diamond mark, and the trial calculation result of Case 5 is represented by a cross mark.
As the dimensionless rotational rigidity α _{j on the} horizontal axis increases, the joints at both ends of the composite beam approach rigid joints. As the dimensionless flexural rigidity α _s approaches 1, the dimensionless joint moment β _Mj approaches the curve L12, which is the upper envelope of the cases 1 to 5.
Further, as the dimensionless rotational rigidity α _j increases, the joints at both ends of the composite beam approach rigid joints. Then, the dimensionless junction moment β _Mj converges to the upper limit value of the dimensionless _{rigid contact moment β Mj, rigid represented by the straight line L11.}
The non-dimensionalized joint moment β _Mj represented by the curve L12 considers the shape of the function and the limit of the non-dimensionalized joint moment β _Mj _{when the non-dimensionalized rotational rigidity α j becomes 0 and infinity.} Therefore, it is considered that it can be approximated by Eq. (142). However, e is the number of Napiers (2.718 ...).

In equation (142), k is a coefficient. Variable alpha _{j, T} is a dimensionless rotational stiffness alpha _j when dimensionless joint moment beta _Mj dimensionless KaTsuyoshise' moment beta _Mj, take half of the _rigid. In the following, we propose a method for identifying the _{dimensionless rigid contact moment β Mj, rigid} , coefficient k, and variables α _{j, T.}

[4. Identification of approximate expression]
[4.1. Control variable of dimensionless rigid moment of force]
In equation (130), when both ends of the composite beam are rigidly joined, it is _{assumed that the rotational rigidity S j} is infinite (the non-dimensional rotational rigidity α _j is infinite). Further equations dimensionless KaTsuyoshise' moment β _{Mj, rigid,} non-dimensional stiffness alpha _s, to represent the formula with a pin contact moment _{M o,} it is as (146) below.

Equation (146) is modified as equations (147) and (148). Further, the equations (147) and (148) are modified as the equation (150) by using the equation (149).

From the above, _{it was found that the dimensionless rigid contact moment β Mj, rigid} is, strictly speaking, a solution of the cubic equation by Eq. (150). Furthermore, it was found that the solution of the dimensionless rigid contact moment β _{Mj, rigid} _{depends only on the dimensionless flexural rigidity α s} , and does not depend on the length L of the composite beam, the evenly distributed load w, and the like. Therefore, the solution of Eq. (50) can be obtained by using Cardano's formula, which is a solution of the cubic equation, and the exact solution of the _{dimensionless rigid contact moment β Mj, rigid can be obtained from the real number solution.}
The relationship between the dimensionless flexural rigidity α _s and the dimensionless _{rigid contact moments β Mj and rigid} for Case 6 is shown in FIG. In FIG. 19, the horizontal axis represents the dimensionless flexural rigidity α _s . The vertical axis represents the dimensionless _{rigid contact moment β Mj, rigid} .

[4.2. Approximate formula of dimensionless rigid moment of force]
Strictly speaking, the dimensionless rigid contact moment β _{Mj, rigid} can be obtained as a real number solution of the cubic equation by Eq. (150). However, as shown in FIG. 19, in _{the range where the non-dimensional flexural rigidity α s} is about 10 or less, the non-dimensional _{rigid contact moments β Mj and rigid} are in the common logarithm linear form of the non-dimensional flexural rigidity α _s. It is thought that it can be approximated.
As described above, the dimensionless flexural rigidity α _s is 10 or less if the slab thickness of a general composite beam is used. Therefore, in FIG. 19, _{the range in which the dimensionless flexural rigidity α s} is 10 or less is approximated in a linear form, and the dimensionless _{rigid contact moments β Mj and rigid} are approximated by the equation (153).

In equation (153), the dimensionless rigid contact moments β _{Mj and rigid} are calculated by an explicit function _{based on the dimensionless flexural rigidity α s.}
Equation (153) is shown by a straight line L14 in FIG. In the range where the dimensionless flexural rigidity α _s is 10 or less, the straight line L14 overlaps with the trial calculation result.
In case 6, _{FIG. 20 shows a comparison between the approximate solution of the dimensionless rigid contact moment β Mj and rigid according to} the equation (153) and the exact solution according to the equation (150). In FIG. 20, the horizontal axis _{represents an approximate solution of the dimensionless rigid contact moment β Mj, rigid} according to equation (153), and the vertical axis represents the exact solution of the _{dimensionless rigid contact moment β Mj, rigid} according to equation (150). Represents the dimensionless _{rigid contact moment β Mj, rigid, Theo} ). If the approximate and exact solutions of the dimensionless rigid contact moments β _{Mj and rigid} match each other, the plot of the trial calculation result is arranged on the straight line L16.
Dimensionless KaTsuyoshise' moment beta _Mj, the approximate solution and the exact solution of the _rigid, when the dimensionless KaTsuyoshise' moment beta _Mj, approximate solution of _rigid exceeds 0.4 is generally consistent. However, when the approximate solution of the dimensionless rigid-contact momet β _{Mj, rigid} is 0.4 or less, the dimensionless _{rigid-joint momet β Mj, rigid} may be corrected by the equation (154). Equation (154) is shown by a curve L17 in FIG. 20, with the dimensionless _{rigid contact moments β Mj and rigid} on the horizontal axis and the dimensionless _{rigid contact moments β Mj, rigid and Theo on the vertical axis.} Dimensionless KaTsuyoshise' moment β _{Mj, rigid, Theo} is, dimensionless KaTsuyoshise' Mometo β _Mj, and overlaps with the exact solution of _rigid.

That is, _{when the approximate solution of the dimensionless rigid contact moment β Mj, rigid} _{is 0.4 or less, the dimensionless rigid contact moment β Mj, rigid} is replaced with the dimensionless rigid contact moment β in equation (142). _The _{dimensionless rigid contact moment β Mj, rigid, Theo} calculated by Eq. (154) based on Mj, rigid may be used.
Next, the approximate expressions of the coefficient k and the variables α _{j and T are obtained.}
For each dimensionless flexural rigidity α _s , the coefficient k was obtained by the differential evolution method. The relationship between the obtained coefficient k and the dimensionless flexural rigidity α _s is shown in FIG. The coefficient k depends on the dimensionless flexural rigidity α _s in a higher order and is approximated by Eq. (155).
Similarly, for each dimensionless flexural rigidity α _s , the variables α _{j and T} were obtained by the differential evolution method. FIG. 22 shows the relationship between the obtained variables α _{j and T} and the dimensionless flexural rigidity α _s. The variables α _{j and T} depend on the dimensionless flexural rigidity α _s in a higher order and are approximated by Eq. (156).

Equation (155) is shown by curve L18 in FIG. In the study range, the curve L18 is a good approximation of the trial calculation results. Similarly, equation (156) is shown by curve L19 in FIG. In the examination range, the curve L19 is a good approximation of the trial calculation result.
As described above, when the dimensionless _{rigid contact moment β Mj, rigid} , the coefficient k, and the variables α _{j, T} are calculated by the dimensionless flexural rigidity α _s _{, the dimensionless joint moment β Mj} is calculated by Eq. (142). Is calculated. That is, based on the calculated non-dimensional flexural rigidity α _s , the non-dimensional rotational rigidity α _j , and the non-dimensional _{rigid contact moment β Mj, rigid} , the non-dimensional joint moment β _Mj is calculated by the positive equation (142). Calculated by a function. More specifically, the dimensionless joint moment β _Mj is calculated by Eq. (142) using Eqs. (153), (155), and (156). Or, (153) instead of equation (153) and (154) dimensionless KaTsuyoshise' moment beta _Mj was calculated using _{equation rigid,} dimensionless KaTsuyoshise' Mometo of _Theo (142) below beta _{Mj , Rigid} may be substituted to calculate the _{dimensionless junction moment β Mj.}

[5. Calculation formula for the maximum value of deflection using an approximate formula]
Then, using the derived deflection function, the maximum deflection [delta] _max (the maximum value of deflection) in the middle of the composite beam, it converts the expression of the dimensionless joint moments beta _Mj and pin contact moment M _o.
First, the deflection function of the forward bending region including the center of the composite beam is given by the above equation (128). The maximum value δ _{max of the} deflection is a value when x = (L / 2) in the equation (128).
_{Here, L 1} and L ₂ in the equations (115) and (116) _{are the dimensionless joint moment β Mj} , the pin contact moment _Mo , and the half as in the equations (160) and (161). It can be expressed by the equation of rigid contact moment M _j. However, L ₁ and L ₂ are points where the bending moment of the composite beam becomes zero, and satisfies Eq. (162).

When equation (128) _{is expressed by the dimensionless flexural rigidity α s} , the dimensionless rotational rigidity α _j , and the dimensionless joint moment β _Mj , it becomes equation (164).

Substituting Eqs. (160) into Eqs. (119) and (126) gives Eqs. (166) and (167). Substituting x = (L / 2) into equation (125) gives equation (168).

By setting x = (L / 2) in the equation (164) and using the equations (168) from the (166) equation, the maximum value δ _max of the deflection can be obtained by the equation (170).

Using equation (170), the maximum value of deflection δ _max is calculated by an explicit function based on the calculated non-dimensional flexural rigidity α _s , non-dimensional rotational rigidity α _j , non-dimensional joint moment β _{Mj, etc.} be able to.
_{On the other hand, by substituting the obtained bending moment M j} (semi-rigid contact moment M _j ) at the end of the composite beam into the above equation (112), a function of the moment distribution of the composite beam can be obtained.
The end bending moment, which is the bending moment acting on the end of the composite beam, is a value when x = 0, L is substituted in the function of the moment distribution of the composite beam. _{Specifically, the bending moment M j} can be directly obtained from the equation (171) which is a modification of the equation (139).

The non-dimensional joint moment β _Mj and the pin contact moment _Mo are the given conditions of the non-dimensional rotational rigidity α _j , the non-dimensional bending rigidity α _s , and the composite beam which is the design requirement (given condition). It can be calculated by the explicit functions of the above equations (142) and (137) using the length L of the above and the evenly distributed load w.
Further, the coefficient k, the variables α _{j, T} , and the dimensionless _{rigid contact moment β Mj, rigid} in the equation (142) are calculated by the explicit functions of the equations (155), (156), and (153). it can.
_{When the dimensionless rigid contact moment β Mj, rigid} calculated by the equation (153) is 0.4 or less, the dimensionless _{rigid contact moment β Mj, rigid} is replaced with the dimensionless rigid contact moment β Mj, rigid in the equation (142). _{The dimensionless rigid contact moment β Mj, rigid, Theo} calculated by Eq. (154) based on the rigid contact moment β _{Mj, rigid} may be used.
_{The maximum value δ max} of the deflection, which is the maximum deflection at the center of the composite beam, _{can be obtained explicitly by substituting the dimensionless joint moment β Mj} calculated by Eq. (142) and the given conditions into Eq. (170). it can.

As described above, in the evaluation method of the composite beam of the present embodiment, the maximum value δ _max of the end bending moment and the deflection is set by an explicit function based on the non-dimensional rotational rigidity α _j and the non-dimensional bending rigidity α _s. calculate.

[6. Accuracy verification of approximate expression]
For case 5 from the case 1 of Table 9, (142) approximate solution of the non-dimensional joint moment beta _Mj by equation (153) below, (155) type, (156) dimensionless joint moment by Formula beta _Mj The exact solutions of are compared and shown in FIG. In FIG. 23, the horizontal axis represents the exact solution of the _{dimensionless junction moment β Mj.} The vertical axis represents an approximate solution of the dimensionless joint moment β _Mj. If the approximate solution and the exact solution of the dimensionless junction moment β _Mj match each other, the plot of the trial calculation result is arranged on the straight line L21. From the results shown in FIG. 23, it can be seen that the approximate solution and the exact solution of the _{dimensionless joint moment β Mj match with sufficient accuracy for practical use.}
Further, regarding the maximum value δ _max of the deflection, the approximate solution by Eq. (170) using the approximate solution of the dimensionless joint moment β _{Mj and the exact solution are compared and shown in FIG. 24.} In FIG. 24, the trial calculation result of Case 6 is also shown. The trial calculation result of Case 6 is represented by a white rectangular mark. From the results shown in FIG. 24, it can be seen that the approximate solution and the exact solution of _{the maximum value of deflection δ max match with sufficient accuracy for practical use.}
As described above, the maximum value δ _{max of the} deflection was evaluated based on the non-dimensional rotational rigidity α _j and the non-dimensional bending rigidity α _s. _{As a result, it was found that the maximum value δ max of the} deflection can be calculated with high versatility and accuracy regardless of the specifications of the composite beam 211.

A new composite beam may be designed based on the maximum values of the end bending moment and the deflection calculated by the composite beam evaluation method of the present embodiment. That is, the composite beam design method may be performed using the composite beam evaluation method of the present embodiment.

As described above, in the method for evaluating the composite beam of the present embodiment, the inventors have found that the maximum values of the end bending moment and the deflection are δ _max _{based on the non-dimensional rotational rigidity α j} and the non-dimensional bending rigidity α _s. Was evaluated. _{As a result, it was found that the maximum value δ max} of the end bending moment and the deflection can be calculated with high versatility and accuracy by the explicit function regardless of the specifications of the composite beam 211.
The end bending moment of the composite beam 211 and the maximum bending moment δ _max of the composite beam 211 are calculated by an explicit function based on the dimensionless rotational rigidity α _j and the dimensionless bending rigidity α _s. And the maximum value of deflection δ _max can be calculated without performing convergence calculation.

Further, the non-dimensionalized flexural rigidity α _s _{and the non-dimensionalized rigid-contact moment β Mj, rigid} are calculated by an explicit function based on the equation (153), and the calculated non-dimensional _{rigid-contact moment β Mj, rigid is} made non-dimensional. The non-dimensionalized joint moment β _Mj is calculated by an explicit function based on the rotational rigidity α _j , the non-dimensionalized bending rigidity α _{s, and the equation (142).} _{Further, the maximum value δ max} of the end bending moment and the deflection is calculated by an explicit function based on the calculated dimensionless joint moment β _{Mj and the equation (170).} _{In this way, the maximum value δ max} of the end bending moment and the deflection of the composite beam 211 can be calculated without performing the convergence calculation.
The dimensionless joint moment β _Mj is calculated by Eq. (142) using Eqs. (153), (155), and (156). Therefore, the dimensionless joint moment β _Mj can be calculated accurately by these equations without performing the convergence calculation. Therefore, the deflection can be appropriately evaluated.

When the dimensionless _{rigid contact moment β Mj, rigid} is 0.4 or less, in equation (142), the dimensionless _{rigid contact moment β Mj, rigid} is used instead of the dimensionless _{rigid contact moment β Mj, rigid} . _{The dimensionless rigid contact moment β Mj, rigid, and Theo} calculated by Eq. (154) are used. When the dimensionless _{rigid contact moment β Mj, rigid} is 0.4 or less, the _{error of the dimensionless rigid contact moment β Mj, rigid} becomes large with respect to the exact solution. Even in this case, the dimensionless _rigid contact moment can be calculated more accurately by using the dimensionless _{rigid contact moment β Mj, rigid, Theo} instead of the dimensionless rigid contact moment β Mj, rigid. Can be done.
Further, for example, in the convergence calculation of the dynamic analysis of a structure having a plurality of composite beams, the convergence calculation for calculating the bending moment of the composite beam which is a member constituting the structure becomes unnecessary. Therefore, it is not necessary to perform double convergence calculation for dynamic analysis of the structure, and the calculation time can be significantly shortened. Further, for example, when the cross-sectional area of the composite beam is set as the objective function and the design conditions such as the deflection and the bending moment are set as the constraint conditions, and the optimization calculation for minimizing the objective function is performed, there are the following advantages. That is, since the convergence calculation for calculating the deflection and the bending moment, which are the constraint conditions, is not required, it is not necessary to perform the double convergence calculation for the optimization of the objective function, and the calculation time can be significantly shortened.

Although the second embodiment of the present invention has been described in detail with reference to the drawings, the specific configuration is not limited to this embodiment, and changes and combinations of configurations within a range that does not deviate from the gist of the present invention, Deletion etc. are also included.
For example, in the method for evaluating a composite beam of the above embodiment, if the bending moment M _j and the maximum value δ _{max of the} deflection are calculated by an explicit function based on the non-dimensional rotational rigidity α _j and the non-dimensional bending rigidity α _s. Good.
At that time, the following procedure may be performed. That is, the dimensionless rigid contact moments β _{Mj and rigid} are calculated by an explicit function _{based on the dimensionless flexural rigidity α s.} Based on the calculated non-dimensional _{rigid contact moment β Mj, rigid} , non-dimensional rotational rigidity α _j , and non-dimensional bending rigidity α _s , the non-dimensional joint moment β _Mj is calculated by an explicit function. Then, based on the calculated dimensionless joint moment β _Mj , the bending moment M _j and the maximum value δ _{max of the} deflection are calculated by an explicit function.

Although the first embodiment and the second embodiment of the present invention have been described in detail with reference to the drawings, the specific configuration is not limited to this embodiment, and the configuration does not deviate from the gist of the present invention. Changes, combinations, deletions, etc. of are also included. Further, it goes without saying that each of the configurations shown in each embodiment can be used in combination as appropriate.

According to the present invention, it is possible to provide an evaluation method for continuous beams, an evaluation program for continuous beams, and an evaluation method for composite beams. Therefore, the industrial applicability is great.

11 Continuous beam 12a Intermediate fulcrum 12b End fulcrum 13 Beam (small beam)
27 1st girder (intermediate support)
31 Second girder (end support)
101 Evaluation apparatus 112 solution determining unit 211 composite beam M _j HanTsuyoshise' moment M _jr rigid connection moment M _o pin contact moment S11, S36, S46, S56, S66, S76 evaluation method (evaluation method of continuous beam)
S12, S37, S47, S57, S67, S77 Solution determination process S14 First design process S16, S39, S49, S59, S69, S79 Second design process S18, S41, S51, S61, S81 Temporary design process S20 Storage process S22 Solution calculation process S24 Residual calculation process S26 Judgment process α _j Non-dimensional rotational rigidity α _s Non-dimensional flexural rigidity β _Mj Non-dimensional joint moment β _{Mj, rigid} , β _{Mj, rigid, Theo} Non-dimensional rigid contact moment

Claims

A continuous beam having two or more natural number n beams arranged side by side in the longitudinal direction and having end portions adjacent to each other in the longitudinal direction semi-rigidly joined to each other as an intermediate fulcrum. A method for evaluating a continuous beam in which both ends of the entire continuous beam are paired end fulcrums.
It has a solution determination step of obtaining a plurality of bending moments at the intermediate fulcrum and the pair of end fulcrums, and a plurality of rotation angles at the pair of end fulcrums based on given conditions.
The above-mentioned given conditions
With the length and flexural rigidity of each of the n beams;
With a plurality of rotational rigidity at the intermediate fulcrum and the pair of end fulcrums;
With the vertical load acting on the n beams;
Including a plurality of vertical displacements at the intermediate fulcrum and the pair of end fulcrums;
In the solution determination step,
The plurality of bending moments and the plurality of rotation angles are defined as a plurality of unknowns.
The relational expression of the plurality of rotational rigidity, the plurality of bending moments, and the plurality of rotation angles and the relational expression of the plurality of vertical displacements are combined with the plurality of first boundary conditions having the same number as the number of the plurality of unknowns. When specified,
A method for evaluating a continuous beam in which the plurality of unknowns are solved so that the plurality of unknowns satisfy the plurality of first boundary conditions, and the bending moment and the deflection distribution of the continuous beam are evaluated.
The solution determination step
With the first design step of giving a design value including the plurality of vertical displacements;
With the second design step of giving the second boundary condition to the pair of end fulcrums according to the joining state at the pair of end fulcrums;
A tentative design step of giving a tentative design value including the bending moment acting on the pair of end fulcrums and the intermediate fulcrum according to the joining state at the pair of end fulcrums;
A solution calculation step of calculating a calculated value including a calculation result of vertical displacement at the intermediate fulcrum so as to satisfy the plurality of first boundary conditions and the second boundary condition based on the tentative design value.
With the residual calculation process for obtaining the displacement residual, which is the residual between the design value and the calculated value;
A determination step for determining whether or not the displacement residual is smaller than a predetermined threshold value;
The evaluation method for a continuous beam according to claim 1.
The solution determination step further includes a storage step of storing the tentative design value after the tentative design step.
In the determination step, when the displacement residual is equal to or greater than the threshold value, another new tentative design value is given in the tentative design step in place of the tentative design value stored in the storage step, and this By combining the solution calculation step, the residual calculation step, and the determination step based on the new provisional design value, until the displacement residual is determined to be smaller than the threshold value in the determination step. repetition,
The bending moment of the provisional design value when the displacement residual is determined to be smaller than the threshold in the determination step is defined as the bending moment acting on the intermediate fulcrum, and is based on this bending moment. The method for evaluating a continuous beam according to claim 2, wherein the plurality of rotation angles are obtained.
In the solution calculation step,
Of the n beams, a natural number i of 1 or more and n or less from the first end fulcrum which is one of the continuous beams to the second end fulcrum which is the other end fulcrum of the continuous beam. On the other hand, the vertical displacement δ 0i (m) at the intermediate fulcrum on the first end fulcrum side or the end fulcrum of the i-th beam is defined as the given condition;
The calculation result δ i, calc (m) of the vertical displacement at the intermediate fulcrum on the second end fulcrum side or the end fulcrum of the i-th beam is obtained from Eqs. (1) to (8). Calculated as being included in the calculated value by the formula (9).
The vertical displacement δ 0 (i + 1) at the intermediate fulcrum on the first end fulcrum side of the (i + 1) th beam with respect to i of 1 or more (n-1) or less, and the first beam of the i-th beam. When the residual of the vertical displacement calculation result δ i, calc at the intermediate fulcrum on the two-end fulcrum side is defined as the i-th residual.
In the residual calculation step,
The intermediate residual, which is the sum of the first residual to the (n-1) residual, is calculated;
The second is the residual of the vertical displacement δ n at the second end fulcrum, which is the given condition, and the calculation result δ n, calc of the vertical displacement at the second end fulcrum of the nth beam. Calculate the end fulcrum residuals;
The displacement residual, which is the sum of the intermediate residual and the second end fulcrum residual, is calculated;
The method for evaluating a continuous beam according to claim 2 or 3.

However, with respect to 1 to n for a natural number i, the length of the beam of i-th L i (m), in the beam of the i-th, the origin end of the first end supporting point side, the first end The coordinates defined when the direction from the fulcrum toward the second end fulcrum is positive are x i (m), the vertical load acting on the i-th beam is wi (N / m), and the i-th said. The rotational rigidity of the beam at the end on the first end fulcrum side is S jl, i (Nm / rad), and the rotational rigidity of the i-th beam at the end on the second end fulcrum side is S jr, i (Nm). / Rad), the bending rigidity of the forward bending of the i-th beam is EI s, i (Nm 2 ), and the bending rigidity of the negative bending of the i-th beam is EI h, i (Nm 2 ), the i-th The bending moment at the intermediate fulcrum or the end fulcrum on the second end fulcrum side of the beam is defined as M j, i (Nm). The bending moment M j, 0 (Nm) at the first end fulcrum of the first beam included in the second boundary condition is 0 when the first end fulcrum is a pin joint, rigid joint or semi-rigid. In the case of joining, it is unknown, and the angle of rotation φ 01 (rad) at the first end fulcrum of the first beam is 0 when the first end fulcrum is rigid joining, and it is pin joint or semi-rigid joint. The case is unknown.
A continuous beam having two or more natural number n beams arranged side by side in the longitudinal direction and having end portions adjacent to each other in the longitudinal direction semi-rigidly joined to each other as an intermediate fulcrum. An evaluation program for a continuous beam for an evaluation device that evaluates the continuous beam having both ends of the entire continuous beam as a pair of end fulcrums.
A plurality of bending moments at the intermediate fulcrum and the pair of end fulcrums, and a plurality of rotation angles at the pair of end fulcrums are made to function as a solution determination unit for obtaining based on given conditions.
The given conditions are
With the length and flexural rigidity of each of the n beams;
With a plurality of rotational rigidity at the intermediate fulcrum and the pair of end fulcrums;
With the vertical load acting on the n beams;
Including a plurality of vertical displacements at the intermediate fulcrum and the pair of end fulcrums;
The solution determination unit
The plurality of bending moments and the plurality of rotation angles are defined as a plurality of unknowns.
The relational expression of the plurality of rotational rigidity, the plurality of bending moments, and the plurality of rotation angles and the relational expression of the plurality of vertical displacements are combined with the plurality of first boundary conditions having the same number as the number of the plurality of unknowns. When specified,
An evaluation program for a continuous beam that solves the plurality of unknowns so that the plurality of unknowns satisfy the plurality of first boundary conditions, and evaluates the bending moment and the deflection distribution of the continuous beam.
The bending rigidity of the forward bending and the bending rigidity of the negative bending are different from each other, and the end bending moment, which is the bending moment acting on the end of the composite beam in which both ends are semi-rigidly joined and an evenly distributed load acts over the entire length, and the above This is an evaluation method for composite beams that calculates the maximum value of deflection that occurs in composite beams.
The value obtained by dividing the rotational rigidity at the end of the composite beam by the bending rigidity per unit length of the synthetic beam is defined as the non-dimensional rotational rigidity.
When the ratio of the bending rigidity of the forward bending of the composite beam to the bending rigidity of the negative bending of the composite beam is defined as the non-dimensional bending rigidity,
A method for evaluating a composite beam in which the end bending moment and the maximum value of the deflection are calculated by an explicit function based on the non-dimensional rotational rigidity and the non-dimensional bending rigidity.
The bending moment acting on the end of the composite beam when it is assumed that both ends are rigidly joined and an evenly distributed load acts over the entire length is defined as the rigid contact moment.
The maximum value of the bending moment acting on the composite beam when it is assumed that both ends are pin-joined and an evenly distributed load acts over the entire length is defined as the pin contact moment.
The bending moment acting on the end of the composite beam on which the both ends are semi-rigidly joined and an evenly distributed load acts over the entire length is defined as the semi-rigid contact moment.
The value obtained by dividing the semi-rigid contact moment by the pin contact moment is defined as the dimensionless joint moment.
When the value obtained by dividing the rigid contact moment by the pin contact moment is defined as the dimensionless rigid contact moment,
The dimensionless rigid contact moment is calculated by an explicit function based on the dimensionless flexural rigidity;
Based on the calculated non-dimensional rigid contact moment, the non-dimensional rotational rigidity, and the non-dimensional flexural rigidity, the non-dimensional joint moment is calculated by an explicit function;
Based on the calculated dimensionless joint moment, the end bending moment and the maximum value of the deflection are calculated by an explicit function;
The method for evaluating a composite beam according to claim 6.
When the dimensionless rigid contact moment is defined as β Mj, rigid , the dimensionless rotational rigidity is defined as α j , and the dimensionless bending rigidity is defined as α s , the dimensionless joint moment β Mj is defined as (. The method for evaluating a composite beam according to claim 7, which is calculated by the formula (13) using the formulas (12) from the formula (10).
When the dimensionless rigid contact moment β Mj, rigid is 0.4 or less, in the equation (13), the dimensionless rigid contact moment β Mj, instead of the dimensionless rigid contact moment β Mj, rigid , the dimensionless rigid contact moment β Mj, The method for evaluating a composite beam according to claim 8, wherein the dimensionless rigid contact moment β Mj, rigid, and Theo calculated by the equation (14) based on the rigid is used.