JP7348510B2 - Evaluation method for composite beams - Google Patents

Description

The present invention relates to a method for evaluating composite beams.

Conventionally, there has been known a method for evaluating a composite beam in which the bending rigidity in positive bending and the bending rigidity in negative bending are different from each other in a composite beam whose both ends are semi-rigidly joined (for example, see Patent Document 1).
In Patent Document 1, when a two-point concentrated load acts on a composite beam, the end bending moment ( The bending moment acting on the end of the composite beam) and the maximum value of deflection are calculated.

JP2018-09410A

However, when a uniformly distributed load acts on a composite beam over its entire length, 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, this equation is an implicit function. In order to solve this problem, it is necessary to perform a convergence calculation. Furthermore, for example, in order to dynamically analyze a structure with multiple composite beams, it is necessary to solve this using an implicit method using convergent calculations, which requires double convergence calculations for the composite beams and the structure. It requires a lot of time.
Furthermore, when performing optimization calculations to minimize the objective function, for example, using the cross-sectional area of a composite beam as the objective function and design conditions such as deflection and bending moment as constraints, it is necessary to Since double convergence calculations are required, a large amount of time is required.

The present invention has been made in view of these problems, and it is an object of the present invention to provide a composite beam evaluation method that can calculate the maximum values of the end bending moment and deflection of the composite beam without performing convergence calculations. purpose.

In order to solve the above problems, the present invention proposes the following means.
The evaluation method of a composite beam of the present invention is a composite beam in which the bending stiffness in positive bending and the bending rigidity in negative bending are different from each other, and both ends are semi-rigidly connected and uniformly distributed loads are applied over the entire length. A composite beam evaluation method that calculates the end bending moment, which is a moment, and the maximum value of the deflection that occurs in the composite beam, the rotational stiffness at the end of the composite beam being calculated by calculating the rotational stiffness per unit length of the composite beam. When the value divided by the bending stiffness is taken as the non-dimensional rotational stiffness, and the ratio of the bending stiffness of the positive bending of the composite beam and the bending rigidity of the negative bending of the composite beam is the non-dimensional bending stiffness, the end bending The present invention is characterized in that the moment and the maximum value of the deflection are calculated by explicit functions based on the dimensionless rotational stiffness and the dimensionless bending stiffness.

According to this invention, the inventors evaluated the maximum value of the end bending moment and deflection based on the nondimensionalized rotational stiffness and nondimensionalized bending stiffness, which are nondimensionalized values. We have found that the maximum values of end bending moment and deflection can be calculated using explicit functions with high versatility and accuracy, regardless of specifications.
Convergently calculate the maximum end bending moment and deflection of the composite beam by calculating the maximum end bending moment and deflection of the composite beam using an explicit function based on the nondimensional rotational stiffness and nondimensional bending stiffness. It can be calculated without doing.

Further, in the above method for evaluating a composite beam, the bending moment acting on the end of the composite beam when both ends are rigidly connected and a uniformly distributed load is applied over the entire length is defined as the rigid contact moment, and The pin contact moment is the maximum value of the bending moment that acts on the composite beam when both ends are connected with pins and a uniformly distributed load acts over the entire length, and both ends are semi-rigidly connected and an evenly distributed load acts over the entire length. The bending moment acting on the end of the composite beam is defined as a semi-rigid moment, the value obtained by dividing the semi-rigid moment by the pin contact moment is defined as a dimensionless joint moment, and the rigid moment is defined as the pin contact moment. If the value divided by The non-dimensional joint moment is calculated by an explicit function based on the non-dimensional rotational stiffness and the non-dimensional bending stiffness, and the end bending moment and the deflection are calculated based on the calculated non-dimensional joint moment. The maximum value of may be calculated using an explicit function.

According to this invention, the non-dimensional rigid contact moment is calculated by an explicit function based on the non-dimensional bending stiffness. Furthermore, a nondimensional joint moment is calculated based on the calculated nondimensional rigid contact moment, nondimensional rotational stiffness, and nondimensional bending stiffness, and end bending is performed based on the calculated nondimensional joint moment. The maximum values of moment and deflection are each calculated using explicit functions.
In this way, the maximum values of the end bending moment and deflection of the composite beam can be calculated without performing convergence calculations.

In addition, in the above composite beam evaluation method, when the non-dimensional rigid moment is β _Mj,rigid , the non-dimensional rotational stiffness is α _j and the non-dimensional bending stiffness is α _s , The dimensional joint moment β _Mj may be calculated by equation (4) using equations (1) to (3).

According to the present invention, the dimensionless joint moment β _Mj can be calculated with high precision without performing a convergence calculation using equation (4) using equations (1) to (3).

In addition, in the above composite beam evaluation method, when the non-dimensional rigid contact moment β _Mj,rigid is 0.4 or less, in equation (4), instead of the non-dimensional rigid contact moment β _{Mj, rigid,} , the non-dimensionalized rigid contact moment β _{Mj,rigid,Theo} calculated by equation (5) based on the non-dimensionalized rigid contact moment β _Mj,rigid may be used.

According to this invention, when the nondimensional rigid moment β _Mj,rigid is 0.4 or less, the error of the nondimensional rigid moment β _Mj,rigid becomes large with respect to the exact solution. Even in this case, the non-dimensional rigid moment β _{Mj, rigid} can be replaced with the non-dimensional rigid moment β _{Mj, rigid, Theo} to calculate the non-dimensional rigid moment with higher accuracy. I can do it.

According to the composite beam evaluation method of the present invention, the maximum values of the end bending moment and deflection of the composite beam can be calculated without performing convergence calculations.

1 is a longitudinal cross-sectional view of a building in which a composite beam is used, to which the composite beam evaluation method of the first embodiment of the present invention is applied. FIG. 2 is a schematic front view showing the composite beam together with boundary conditions. FIG. 3 is a front view schematically showing a positively bent region and a negatively bent region in the composite beam. FIG. 6 is a diagram showing the relationship between dimensionless rotational stiffness and dimensionless joint moment in Cases 1 to 5; It is a figure which shows the relationship of the trial calculation result of the non-dimensional bending stiffness and non-dimensional rigid contact moment in a composite beam. FIG. 7 is a diagram showing the relationship between an approximate solution and an exact solution of a dimensionless rigid moment. FIG. 3 is a diagram showing the relationship between dimensionless bending stiffness and coefficient k. FIG. 3 is a diagram showing the relationship between dimensionless bending stiffness and variables α _j,T . FIG. 3 is a diagram showing the relationship between an approximate solution and an exact solution of a dimensionless joint moment. FIG. 7 is a diagram showing the relationship between an approximate solution and an exact solution for the maximum value of deflection.

An embodiment of the composite beam evaluation method according to the present invention will be described below with reference to FIGS. 1 to 10.

[1. Composite beam with semi-rigid connections at both ends]
The composite beam evaluation method of this embodiment is used, for example, to evaluate the composite beam 11 that constitutes the building 1 shown in FIG. 1.
In this example, the composite beam 11 includes a floor 12 and a beam (small beam) 13. Note that the configuration of the composite beam 11 is not limited to this example.
The floor 12 is a so-called composite slab, and is supported from below by beams 13. The floor 12 includes a deck plate 16 and an RC (Reinforced Concrete) slab 17 placed on the deck plate 16.
The uneven shape of the deck plate 16 extends in a direction along the horizontal plane and perpendicular to the direction in which the beam 13 extends.
The RC slab 17 includes concrete 18 and reinforcing bars 19. The concrete 18 is formed into a plate shape whose thickness direction is the vertical direction. Concrete 18 is supported from below by deck plate 16.
The reinforcing bars 19 extend along the horizontal plane and are buried in the concrete 18. For example, the reinforcing bars 19 are arranged in a grid pattern in plan view.

The beam 13 is made of H-beam steel and extends along a horizontal plane. The lower end of the stud 21 is fixed to the upper flange of the beam 13. Stud 21 passes through deck plate 16. The upper end of the stud 21 is buried in the concrete 18.
Both ends of the beam 13 are semi-rigidly connected to a large beam 24 extending along a horizontal plane. The girder 24 extends in a direction perpendicular to the beam 13. The semi-rigid connection between the beam 13 and the girder 24 is achieved by, for example, shear plates 25, bolts 26, and the like.
The ends of the girders 24 are supported from below by columns 28.

Below, a method for evaluating the composite beam 11 configured in this manner will be described.

[2. Differential equation of deflection of a composite beam with both ends semi-rigidly connected (exact solution)]
The following discussion is limited to the elastic range of composite beams.
As shown in FIG. 2, a composite beam is schematically illustrated.
Define the coordinate x (mm) to the right along the composite beam. Let the left end position of the composite beam be the origin of the coordinate x. Assume that the length of the composite beam is L (mm). As a boundary condition (precondition), it is assumed that the composite beam is semi-rigidly connected at both ends. It is assumed that a uniformly distributed downward load w (N/mm: Newton per millimeter) is applied to the composite beam over its entire length.
The rotational rigidity of the joint at both ends of the composite beam is S _j (Nmm/rad: Newton millimeter per radian), and the rotation angle at the end of the composite beam is θ _j (with clockwise rotation as seen from the front shown in FIG. 2 being positive). 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 moment) M _j (Nmm) is expressed by equation (11).

The bending moment distribution M(x) (Nmm) along the coordinate x of the composite beam is given by the force balance condition (12), assuming that the bending moment when tensile stress acts on the lower flange of the composite beam is positive. Expressed by the formula.

The curvature φ (rad/mm) of the composite beam can be expressed using the bending moment M (Nmm) acting on the composite beam and the bending rigidity EI (Nmm ² ) of the composite beam. The composite beam has different bending rigidity in positive bending (convex downward) and in bending rigidity in negative bending (convex upward) due to the action of concrete. Therefore, the bending rigidity of the composite beam in positive bending is set as EI _s (Nmm ² ), and the bending rigidity of the composite beam in negative bending is set as EI _h (Nmm ² ). As shown in FIG. 3, it is assumed that the composite beam is negatively bent in a range where the coordinate x is 0 or more and _L1 or less and L2 or _more and L or less. However, L ₁ is larger than 0, and L ₂ is larger than L ₁ and smaller than L. It is assumed that the composite beam is normally bent within the range of the coordinate x from L1 _{to L2} .
At this time, the solutions of x when equation (12) 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 positively bent and a region where the composite beam is negatively bent, and is expressed by equations (13) and (14).

When equations (13) and (14) are solved for L ₁ and L ₂ , equations (15) and (16) are obtained.

Next, the rotation angle θ(x) (rad) of the composite beam will be explained assuming that clockwise rotation with respect to the horizontal plane is positive (+). The rotation angle θ(x) is calculated by integrating the curvature φ in equations (13) and (14) with the coordinate x, and further considering the boundary condition that the rotation angle θ becomes θ _j when the coordinate x is 0. , (19) can be expressed as Equations (20) to (22).

Next, the deflection δ(x) (mm) generated in the composite beam will be explained assuming that the vertically downward direction is positive (+). The deflection δ(x) is calculated by integrating the rotation angle θ of equations (20) to (22) with the coordinate x, and further considering the boundary condition in which the deflection δ(x) becomes 0 when the coordinate x is 0. Using equations (25) and (26), it can be expressed as equations (27) to (29).

Equations (27) to (29) obtained above are expressed in a form that includes a semi-rigid moment M _j and a rotation angle θ _j , but as they are, for any rotational stiffness S _j , The rigid contact moment M _j and the rotation angle θ _j are not uniquely determined. Here, if we further use the transformation compatibility condition that the rotation angle θ becomes 0 rad when the coordinate x is (L/2), from equations (11) and (21), the rotational stiffness S _j and the rotation angle θ The relationship with _j is expressed as in equation (30).

Furthermore, if we use the deformation compatibility condition that the deflection δ(x) becomes 0 when the coordinate x is L, then from equations (11) and (29), the relationship between the rotational stiffness S _j and the rotation angle θ _j is expressed as equation (31).
Note that equation (30) is equivalent to equation (31).

The maximum value δ _max of deflection occurring in the composite beam is the value when the coordinate x in equation (28) is (L/2).
In order to calculate the maximum deflection value δ _max using the above equation (30) or (31), the rotation angle θ _j is calculated by performing a convergence calculation using the equation (30) or (31). Furthermore, it is necessary to use equation (28) to calculate the deflection δ(x) when the coordinate x is (L/2).

[3. Approximate equation for the deflection of a composite beam with both ends semi-rigidly connected]
The inventors evaluated the maximum deflection value δ _max based on nondimensional rotational stiffness and nondimensional bending stiffness, which are nondimensional values. It has been found that the maximum value δ _max of moment and deflection can be calculated with high versatility and with high precision without performing convergence calculations.
Below, an approximate expression that can calculate the maximum value of the end bending moment and deflection using an explicit function will be described.

A nondimensional bending stiffness α _s and a nondimensional rotational stiffness α _j were defined as in equations (35) and (36).

That is, the nondimensionalized bending stiffness α _s is the ratio of the bending stiffness of the composite beam in positive bending to the bending rigidity of the composite beam in negative bending. In this example, the nondimensionalized bending stiffness α _s is the ratio of the positive bending bending stiffness EI _s of the composite beam to the negative bending bending stiffness EI _h of the composite beam. In this case, the dimensionless bending stiffness α _s generally takes a value of 1 or more. This is because the floor is usually located vertically above the beam, and the concrete of the floor has greater compression resistance than tensile resistance. For this reason, the resistance of the floor is greater in positive bending, in which the concrete on the floor is compressed, than in negative bending, in which the concrete on the floor is stretched. In the specifications of general composite beams, the non-dimensional bending stiffness α _s is 10 or less, and in more commonly used composite beams, the non-dimensional bending stiffness α _s is 3 or less.
Note that the nondimensional bending stiffness α _s may be a ratio of the bending stiffness EI _h of the composite beam in negative bending to the bending rigidity EI _s of the composite beam in positive bending.
The nondimensional rotational stiffness α _j is the value obtained by dividing the rotational stiffness at the end of the composite beam by the bending rigidity per unit length of the composite beam.

Furthermore, a pin contact moment M _o , a semi-rigid contact moment M _j , a rigid contact moment M _jr , a non-dimensional joint moment β _Mj , and a non-dimensional rigid contact moment β _Mj,rigid were defined.
The pin contact moment M _o is the bending moment that acts on a composite beam when the bending stiffness in positive bending and the bending rigidity in negative bending are different from each other, and both ends are pin-joined and a uniformly distributed load is applied over the entire length. It means the maximum value of moment. The pin contact moment M _o is expressed by equation (37).
In addition, the pin contact moment M _o is the amount of the pin contact moment M o at the end of a beam when the bending stiffness in positive bending and the bending stiffness in negative bending are equal to each other, when both ends are connected with pins and a uniformly distributed load is applied over the entire length. equal to the acting bending moment. Specifically, the pin contact moment M _o is a value based on the formula (wL ² /8).

As mentioned above, the semi-rigid moment M _j is the absolute value of the bending moment at the end of the composite beam, and more specifically, in a composite beam where the bending stiffness of positive bending and the bending stiffness of negative bending are different from each other. , refers to the bending moment that acts on the ends of a composite beam where both ends are semi-rigidly joined and a uniformly distributed load is applied over the entire length. The semi-rigid moment M _j is expressed by Equation (38) using rotation angle θ _j calculated by performing convergence calculation using Equation (30) or Equation (31).

The rigid contact moment M _jr is the amount that acts on the end of a composite beam when the bending stiffness in positive bending and the bending stiffness in negative bending are different from each other, when both ends are rigidly connected and a uniformly distributed load is applied over the entire length. It means the bending moment.
Note that the composite beam of this embodiment is a beam in which the bending rigidity in positive bending and the bending rigidity in negative bending are different from each other. For convenience of explanation, a comparative example beam is assumed in which the bending rigidity in positive bending and the bending rigidity in negative bending are equal to each other. In the beam of the comparative example, when both ends are rigidly connected and a uniformly distributed load is applied over the entire length, the rigid contact moment M _jr is a value based on the formula (wL ² /12).
The dimensionless joint moment β _Mj is a value obtained by dividing the semi-rigid moment M _j by the pin contact moment _Mo , and is expressed by equation (39).

The dimensionless rigid contact moment β _Mj,rigid is a value obtained by dividing the rigid contact moment M _jr by the pin contact moment _Mo , and is expressed by equation (40).

In order to evaluate the indices such as the non-dimensional rotational stiffness α _j and the non-dimensional bending stiffness α _s , the non-dimensional rotation stiffness α _j , the non-dimensional joint moment β _Mj was estimated by changing the non-dimensional bending stiffness α _s .
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 is 10.0 m (10000 mm), the bending stiffness EI _h of negative bending is 229397 kNm ² , the uniformly distributed load w is 28.6 kN/m (28.6 N/mm), and the pin connection The moment M _o was set to 357 kNm from equation (37). In case 1, the dimensionless rotational stiffness α _j was changed to 51 different values from a minimum value of 0.00 to a maximum value of 50.00 in steps of 1.00. That is, the dimensionless rotational stiffness α _j was set to values of 0.00, 1.00, 2.00, . . . , 50.00. The dimensionless bending stiffness α _s was changed to 51 different values from a minimum value of 1.00 to a maximum value of 6.00 in increments of 0.10. That is, the dimensionless bending stiffness α _s was set to values of 1.00, 1.10, 1.20, . . . , 6.00. In Case 1, 2601 cases were calculated using the formula (51×51) in which the values of the nondimensional rotational stiffness α _j and the nondimensional bending stiffness α _s were changed.

In case 2, in case 1, the length L of the composite beam was set to 15.0 m, and the pin contact moment M _o was set to 803 kNm from equation (37). In Case 2, as in Case 1, the nondimensional rotational stiffness α _j and the nondimensional bending stiffness α _s were varied, and 2601 cases were estimated.
In Case 3, in Case 2, the increments of the non-dimensional rotational stiffness α _j and the maximum value and increments of the non-dimensional bending stiffness α _s were changed. That is, in case 3, the dimensionless rotational stiffness α _j was changed to 5001 different values from the minimum value 0.00 to the maximum value 50.00 in steps of 0.01. That is, the dimensionless rotational stiffness α _j was set to values of 0.00, 0.01, 0.02, . . . , 50.00. The dimensionless bending stiffness α _s was changed to seven types of values from a minimum value of 1.00 to a maximum value of 1.06 in increments of 0.01. That is, the dimensionless bending stiffness α _s was set to a value of 1.00, 1.01, 1.02, . . . , 1.06. In case 3, 35007 cases were calculated using the formula (5001×7) in which the values of the nondimensional rotational stiffness α _j and the nondimensional bending stiffness α _s were changed. In Case 3, a more detailed trial calculation was performed for a part of the range of the non-dimensional rotational stiffness α _j and the non-dimensional bending stiffness α _s in Case 2.

In case 4, in case 1, the length L of the composite beam was 8.4 m, the bending rigidity EI _h of negative bending was 214311 kNm ² , and the pin contact moment M _o was 252 kNm from equation (37). In case 4, the dimensionless rotational stiffness α _j was changed to 201 different values from the minimum value 0.00 to the maximum value 100.00 in steps of 0.50. That is, the dimensionless rotational stiffness α _j was set to values of 0.00, 0.50, 1.00, . . . , 100.00. The dimensionless bending stiffness α _s was changed to seven different values from a minimum value of 1.00 to a maximum value of 1.30 in steps of 0.05. That is, the dimensionless bending stiffness α _s was set to values of 1.00, 1.05, 1.10, . . . , 1.30. In case 4, 1407 cases were calculated using the formula (201×7) in which the values of the nondimensional rotational stiffness α _j and the nondimensional bending stiffness α _s were changed.

In case 5, in case 1, the length L of the composite beam was set to 13.8 m, and the pin contact moment M _o was set to 680 kNm from equation (37). In case 5, the value of the dimensionless rotational stiffness α _j was changed in the same way as in case 4. The dimensionless bending stiffness α _s was changed to seven types of values from a minimum value of 1.00 to a maximum value of 4.00 in increments of 0.50. That is, the dimensionless bending stiffness α _s was set to values of 1.00, 1.50, 2.00, . . . , 4.00. In case 5, 1407 cases were calculated using the formula (201×7) in which the values of the nondimensional rotational stiffness α _j and the nondimensional bending stiffness α _s were changed.
For Cases 1 to 5, a total of 43,023 cases were calculated.

In Case 6, in Case 2, the maximum values of the nondimensional rotational stiffness α _j and the nondimensional bending stiffness α _s were changed. In Case 6, the non-dimensional rotational stiffness α _j is set to infinity (∞), that is, the rotational stiffness S _j is set to infinity, and both ends of the composite beam are rigidly connected. The dimensionless bending stiffness α _s was changed to 501 different values from a minimum value of 1.00 to a maximum value of 51.00 in increments of 0.10. That is, the dimensionless bending stiffness α _s was set to values of 1.00, 1.10, 1.20, . . . , 51.00.

FIG. 4 shows the relationship between the nondimensional rotational stiffness α _j and the nondimensional joint moment β _Mj in Cases 1 to 5. In FIG. 4, the horizontal axis represents the non-dimensional rotational stiffness α _j and the vertical axis represents the non-dimensional joint moment β _Mj .
The straight line L1 is a non-dimensional rigid contact moment β _Mj,rigid,u in a normal beam in which the bending stiffness in positive bending and the bending stiffness in negative bending are equal to each other. In a normal beam, the length is L (mm) and the uniformly distributed load is w (N/mm). In this case, in a normal beam, the rigid moment M _jr is (wL ² /12) and the pin contact moment M _o is (wL ² /8), so the nondimensional rigid moment β _Mj,rigid,u is The formula {(wL ² /12)/(wL ² /8)} gives a value of approximately 0.667.

The trial calculation result for Case 1 is represented by an open square mark. Similarly, the trial calculation results for Case 2 are represented by a white triangle mark, the trial calculation results for Case 3 are represented by a white circle mark, the trial calculation results for Case 4 are represented by a white diamond mark, and the trial calculation results for Case 5 are represented by a white diamond mark. The result is represented by a cross.
As the dimensionless rotational stiffness α _j on the horizontal axis increases, the joint at both ends of the composite beam approaches a rigid joint. As the dimensionless bending stiffness α _s approaches 1, the dimensionless joint moment β _Mj approaches the curve L2, which is the upper limit envelope of cases 1 to 5.
Furthermore, as the non-dimensional rotational stiffness α _j increases, the joint at both ends of the composite beam approaches a rigid joint, and the non-dimensional joint moment β _Mj is an upper limit value of the non-dimensional rigid joint represented by the straight line L1. The moment β converges to the value of _{Mj, rigid} .
The nondimensional joint moment β _Mj represented by the curve L2 takes into account the shape of the function and the limit of the nondimensional joint moment β _Mj when the nondimensional rotational stiffness α _j becomes 0 and infinity. Therefore, it is thought that it can be approximated by equation (42).

In equation (42), k is a coefficient. The variable α _j,T is the non-dimensional rotational stiffness α _j when the non-dimensional joint moment β Mj takes a value half of the non-dimensional rigid moment β _{Mj, rigid .} Below, we will propose a method for identifying the dimensionless rigid moment β _Mj,rigid , the coefficient k, and the variable α _j,T .

[4. Identification of approximate formula]
[4.1. Controlling variables of non-dimensional rigid moment]
In equation (30), when both ends of the composite beam are rigidly connected, the rotational stiffness S _j is set to infinity (the non-dimensionalized rotational stiffness α _j is infinite), and the equation is further changed to the non-dimensionalized rigid moment β _Mj, When expressed using _rigid , nondimensional bending stiffness α _s , and pin contact moment M _o , the equation is as shown in equation (46).

Equation (46) is transformed into equations (47) and (48). Furthermore, using equation (49), it is transformed into equation (50).

From the above, the non-dimensional rigid moment β _Mj,rigid is strictly speaking the solution of the cubic equation according to equation (50), and the solution to the non-dimensional rigid moment β _{Mj, rigid} is the non-dimensional bending stiffness It was found that it depends only on α _s and does not depend on the length L of the composite beam, uniformly distributed load w, etc. Therefore, the solution to equation (50) is obtained using Cardano's formula, which is a method for solving cubic equations, and the exact solution of the dimensionless rigid moment β _Mj,rigid can be obtained by using the real number solution.
The relationship between the trial calculation results of the non-dimensional bending stiffness α _s and the non-dimensional rigid contact moment β _Mj,rigid for Case 6 is shown in FIG. In FIG. 5, the horizontal axis represents the non-dimensional bending stiffness α _s , and the vertical axis represents the non-dimensional rigid contact moment β _Mj,rigid .

[4.2. Approximate formula for non-dimensional rigid moment]
Strictly speaking, the dimensionless rigid moment β _Mj,rigid can be obtained as a real number solution of the cubic equation according to equation (50). However, as shown in FIG. 5, in a range where the nondimensional bending stiffness α _s is approximately 10 or less, the nondimensional rigid contact moment β _Mj,rigid is expressed as a linear form of the common logarithm of the nondimensional bending stiffness α _s . It is thought that it can be approximated.
As mentioned above, the dimensionless bending stiffness α _s is 10 or less if the slab thickness of a general composite beam is used. Therefore, in FIG. 5, the range in which the non-dimensional bending stiffness α _s is 10 or less is approximated in a linear form, and the non-dimensional rigid moment β _Mj,rigid is approximated by equation (53).

In equation (53), the nondimensional rigid moment β _Mj,rigid is calculated by an explicit function based on the nondimensional bending stiffness α _s .
Equation (53) is shown in FIG. 5 by a straight line L4. In a range where the dimensionless bending stiffness _αs is 10 or less, the straight line L4 overlaps with the trial calculation result.
For Case 6, FIG. 6 shows a comparison between the approximate solution of the dimensionless rigid moment β _Mj,rigid based on equation (53) and the exact solution based on equation (50). In FIG. 6, the horizontal axis represents the approximate solution of the nondimensional rigid moment β _{Mj,rigid according} to equation (53), and the vertical axis represents the exact solution ( represents the dimensionless rigid moment β _{Mj, rigid, Theo} ). If the approximate solution and exact solution of the dimensionless rigid moment β _Mj,rigid match each other, the plot of the trial calculation result is placed on the straight line L6.
The approximate solution and exact solution of the non-dimensional rigid moment β _Mj,rigid generally match when the approximate solution of the non-dimensional rigid moment β _{Mj, rigid} exceeds 0.4. However, when the approximate solution of the nondimensional rigid moment β _Mj,rigid is 0.4 or less, the nondimensional rigid moment β _Mj,rigid may be corrected using equation (54). Equation (54) is shown by a straight line L7 in FIG. 6, with the non-dimensional rigid moment β _{Mj, rigid} on the horizontal axis and the non-dimensional rigid moment β _{Mj, rigid, Theo} on the vertical axis. The non-dimensional rigid moment β _{Mj,rigid,Theo} overlaps with the exact solution of the non-dimensional rigid moment β _Mj,rigid .

That is, when the approximate solution of the non-dimensional rigid moment β _Mj,rigid is 0.4 or less, in equation (42), instead of the non-dimensional rigid moment β _{Mj, rigid} , the non-dimensional rigid moment β A dimensionless rigid moment β Mj _{, rigid, Theo calculated by equation (54) based on Mj, rigid} may be used.
Next, approximate expressions for coefficient k and variables α _{j and T} are determined.
For each dimensionless bending stiffness α _s , the coefficient k was determined by the differential evolution method. The relationship between the determined coefficient k and the dimensionless bending stiffness _αs is shown in FIG. The coefficient k is highly dependent on the dimensionless bending stiffness _αs , and was approximated by equation (55).
Similarly, for each dimensionless bending stiffness α _s , variables α _j,T were determined by the differential evolution method. The relationship between the determined variables α _j,T and the dimensionless bending stiffness α _s is shown in FIG. 8 . The variables α _j,T are highly dependent on the dimensionless bending stiffness α _s and were approximated by equation (56).

Equation (55) is shown in FIG. 7 by curve L8. In the study range, the curve L8 closely approximates the trial calculation results. Similarly, equation (56) is shown by curve L9 in FIG. In the study range, the curve L9 closely approximates the trial calculation results.
As described above, when the non-dimensional rigid contact moment β _Mj,rigid , the coefficient k, and the variable α _j,T are calculated using the non-dimensional bending stiffness α _s , the non-dimensional joint moment β _Mj is calculated by equation (42). is calculated. That is, based on the calculated non-dimensional bending stiffness α _s , non-dimensional rotational stiffness α _j , and non-dimensional rigid contact moment β _Mj,rigid , the non-dimensional joint moment β _Mj is expressed explicitly in equation (42). Calculate using a function. To explain in more detail, the dimensionless joint moment β _Mj is calculated by equation (42) using equations (53), (55), and (56). Alternatively, instead of formula (53), the nondimensional rigid moment β _{Mj, rigid, Theo} calculated using formulas (53) and (54) is converted to the nondimensional rigid moment β _Mj of formula (42). _{, rigid} to calculate the dimensionless joint moment β _Mj .

[5. Calculation formula for maximum value of deflection using approximate formula]
Next, using the derived deflection function, the maximum deflection (maximum value of deflection) δ _max at the center of the composite beam is converted into expressions for the dimensionless joint moment β _Mj and pin contact moment M _o .
First, the deflection function of the positive bending region including the center of the composite beam is expressed by the above equation (28). The maximum value of deflection δ _max is the value when x=(L/2) in equation (28).
Here, L ₁ and L ₂ in equations (15) and (16) are the dimensionless joint moment β _Mj , pin contact moment M _o , and half It can be expressed by the formula of rigid contact moment _Mj . However, L ₁ and L ₂ are points where the bending moment of the composite beam becomes zero, and satisfy equation (62).

Expressing equation (28) in terms of nondimensional bending stiffness α _s , nondimensional rotational stiffness α _j , and nondimensional joint moment β _Mj gives equation (64).

Substituting equation (60) into equations (19) and (26) results in equations (66) and (67). Substituting x=(L/2) into equation (25) yields equation (68).

Based on Equations (28) and Equations (66) to (68), the maximum value δ _max of deflection can be determined using Equation (70).

Equation (70) calculates the maximum deflection value δ _max using an explicit function based on the calculated non-dimensional bending stiffness α _s , non-dimensional rotational stiffness α _j , non-dimensional joint moment β _Mj , etc.
On the other hand, by substituting the obtained bending moment M _j (semi-rigid moment M _j ) at the end of the composite beam into Equation (12), a function of the moment distribution of the composite beam can be determined.
The end bending moment, which is the bending moment acting on the end of the composite beam, is the value obtained by substituting x=0 and L in the moment distribution function of the composite beam. Specifically, the bending moment M _j can be directly determined from equation (71), which is a modification of equation (39).

Note that the nondimensional joint moment β _Mj and the pin contact moment _Mo are the nondimensional rotational stiffness α _j which is a given condition, the nondimensional bending stiffness α _s which is a given condition, and the composite beam which is a design requirement value (given condition). It can be calculated using the length L and the uniformly distributed load w by an explicit function such as the above equations (42) and (37).
Furthermore, the coefficient k, variable α _j,T , and dimensionless rigid moment β _Mj,rigid in equation (42) are calculated by explicit functions of equations (55), (56), and (53). can.
Note that when the nondimensional rigid moment β _Mj,rigid calculated by equation (53) is 0.4 or less, in equation (42), instead of the nondimensional rigid moment β _{Mj, rigid} , A dimensionless rigid moment β _{Mj, rigid} , Theo calculated by equation (54) based on the rigid moment β _{Mj, rigid} may be used.
The maximum deflection value δ _max , which is the maximum deflection at the center of the composite beam, can be explicitly determined by substituting the dimensionless joint moment β _Mj calculated by equation (42) and the given conditions into equation (70). can.

As described above, in the composite beam evaluation method of this embodiment, the maximum value δ _max of the end bending moment and deflection is determined by an explicit function based on the nondimensional rotational stiffness α _j and the nondimensional bending stiffness α _s . calculate.

[6. Accuracy verification of approximate formula]
For cases 1 to 5 in Table 1, approximate solutions of non-dimensionalized joint moment β Mj _by equation (42), non-dimensionalized joint moment β _Mj by equations (53), (55), and (56) Figure 9 shows a comparison of the exact solutions. In FIG. 9, the horizontal axis represents an exact solution of the non-dimensionalized joint moment β _Mj , and the vertical axis represents an approximate solution of the non-dimensionalized joint moment β _Mj . If the approximate solution and exact solution of the dimensionless joint moment β _Mj agree with each other, the plot of the trial calculation result is placed on the straight line L11. From the results shown in FIG. 9, it can be seen that the approximate solution and exact solution of the dimensionless joint moment β _Mj match with sufficient accuracy for practical use.
Further, regarding the maximum value δ _max of deflection, FIG. 10 shows a comparison between an approximate solution based on equation (70) using an approximate solution of the dimensionless joint moment β _Mj and an exact solution. In FIG. 10, the trial calculation results for case 6 are also shown. The trial calculation result for Case 6 is represented by a white rectangle. From the results shown in FIG. 10, it can be seen that the approximate solution and exact solution for the maximum deflection value δ _max match with sufficient accuracy for practical use.
As explained above, by evaluating the maximum value δ _max of deflection based on the non-dimensional rotational stiffness α _j and the non-dimensional bending stiffness α _s , the maximum value δ max of deflection can be determined regardless of the specifications of the composite beam 11. It was found that _max can be calculated with high versatility and accuracy.

Note that a new composite beam may be designed based on the maximum values of the end bending moment and deflection calculated by the composite beam evaluation method of this embodiment. That is, the method for evaluating a composite beam of this embodiment may be used to design a composite beam.

As explained above, in the composite beam evaluation method of this embodiment, the inventors calculated the maximum value δ _max of the end bending moment and deflection based on the nondimensional rotational stiffness α _j and the nondimensional bending stiffness α _s . By evaluating , it was found that the end bending moment and the maximum value δ _max of deflection can be calculated with high versatility and accuracy using an explicit function, regardless of the specifications of the composite beam 11.
The end bending moment of the composite beam 11 is calculated by calculating the maximum value δ _max of the end bending moment and deflection of the composite beam 11 using an explicit function based on the nondimensional rotational stiffness α _j and the nondimensional bending stiffness α _s . and the maximum value of deflection δ _max can be calculated without performing convergence calculations.

In addition, the nondimensional bending stiffness α _s and the nondimensional rigid contact moment β _Mj,rigid are calculated by an explicit function based on the equation (53), and the calculated nondimensional rigid contact moment β _Mj, rigid is The dimensionless joint moment β _Mj is calculated by an explicit function based on the rotational stiffness α _j , the dimensionless bending stiffness α _s , and equation (42). Furthermore, the maximum value δ _max of the end bending moment and deflection is calculated using an explicit function based on the calculated dimensionless joint moment β _Mj and equation (70). In this way, the maximum value δ _max of the end bending moment and deflection of the composite beam 11 can be calculated without performing convergence calculation.
The dimensionless joint moment β _Mj is calculated by equation (42) using equations (53), (55), and (56). Therefore, the dimensionless joint moment β _Mj can be calculated with high precision using these equations without performing convergence calculations.

When the non-dimensional rigid moment β _Mj,rigid is 0.4 or less, in equation (42), instead of the non-dimensional rigid moment β _{Mj, rigid} , the non-dimensional rigid moment β _{Mj, rigid} is used. The dimensionless rigid moment β _{Mj, rigid, Theo} calculated by equation (54) is used. When the nondimensional rigid moment β _Mj,rigid is 0.4 or less, the error of the nondimensional rigid moment β _Mj,rigid becomes large compared to the exact solution. Even in this case, the non-dimensional rigid moment β _{Mj, rigid} can be replaced with the non-dimensional rigid moment β _{Mj, rigid, Theo} to calculate the non-dimensional rigid moment with higher accuracy. I can do it.
Furthermore, for example, in convergent calculations for dynamic analysis of a structure with multiple composite beams, convergence calculations for calculating the bending moment of the composite beams, which are the members that make up the structure, are no longer required. There is no need to perform double convergence calculations for dynamic analysis, and calculation time can be significantly reduced. Furthermore, when performing optimization calculations to minimize the objective function, for example, using the cross-sectional area of a composite beam as the objective function and design conditions such as deflection and bending moments as constraints, the constraints such as deflection and bending moments can be calculated. Since there is no need to perform convergence calculation for optimization of the objective function, there is no need to perform double convergence calculation for optimization of the objective function, and calculation time can be significantly reduced.

Although one embodiment of the present invention has been described above in detail with reference to the drawings, the specific configuration is not limited to this embodiment, and modifications, combinations, and deletions of the configuration within the scope of the gist of the present invention. etc. are also included.
For example, in the composite beam evaluation method of the embodiment, if the bending moment M _j and the maximum value δ _max of deflection are calculated by an explicit function based on the nondimensional rotational stiffness α _j and the nondimensional bending stiffness α _s , good.
In addition, at that time, the nondimensional rigid contact moment β _Mj,rigid is calculated by an explicit function based on the nondimensional bending stiffness α _s , and the calculated nondimensional rigid contact moment β _Mj,rigid is The non-dimensional joint moment β _Mj is calculated by an explicit function based on the stiffness α _j and the non-dimensional bending stiffness α _s , and the bending moment M j and the bending moment β _Mj are calculated based on the calculated non-dimensional joint moment β _Mj . The maximum value δ _max may be calculated using an explicit function.

11 Composite beam M _j semi-rigid moment M _jr rigid moment M _o pin moment α _j non-dimensional rotational stiffness α _s non-dimensional bending stiffness β _Mj non-dimensional joint moment β _{Mj, rigid} , β _{Mj, rigid , Theo} dimensionless rigid moment

Claims (4)

An end bending moment that is a bending moment that acts on the end of a composite beam in which the bending stiffness of positive bending and the bending stiffness of negative bending are different from each other, both ends are semi-rigidly joined, and a uniformly distributed load acts over the entire length, and An evaluation method for a composite beam that calculates the maximum value of deflection that occurs in the composite beam, the method comprising:
A value obtained by dividing the rotational rigidity at the end of the composite beam by the bending rigidity per unit length of the composite beam is defined as a nondimensional rotational rigidity,
When the ratio of the bending rigidity of the composite beam in positive bending and the bending rigidity of negative bending of the composite beam is taken as the nondimensionalized bending rigidity,
An evaluation method for a composite beam, in which the maximum value of the end bending moment and the deflection is calculated by an explicit function based on the nondimensional rotational stiffness and the nondimensional bending stiffness. The bending moment that acts on the end of the composite beam when both ends are rigidly connected and a uniformly distributed load acts over the entire length is defined as a rigid contact moment,
The pin contact moment is the maximum value of the bending moment that acts on the composite beam when both ends are pin-joined and a uniformly distributed load is applied over the entire length,
A bending moment acting on the ends of the composite beam, both ends of which are semi-rigidly connected and a uniformly distributed load acts over the entire length, is defined as a semi-rigid contact moment;
The value obtained by dividing the semi-rigid contact moment by the pin contact moment is set as a dimensionless joint moment,
When the value obtained by dividing the rigid contact moment by the pin contact moment is defined as the nondimensional rigid contact moment,
Calculating the non-dimensional rigid contact moment by an explicit function based on the non-dimensional bending stiffness,
Calculating the non-dimensional joint moment by an explicit function based on the calculated non-dimensional rigid contact moment, the non-dimensional rotational stiffness, and the non-dimensional bending stiffness,
The method for evaluating a composite beam according to claim 1, wherein the maximum value of the end bending moment and the deflection is calculated using an explicit function based on the calculated dimensionless joint moment. When the nondimensional rigid contact moment is β _Mj,rigid , the nondimensional rotational stiffness is α _j , and the nondimensional bending stiffness is α _s , the nondimensional joint moment β _Mj is expressed as (1 3. The method for evaluating a composite beam according to claim 2, wherein the evaluation method is performed using equation (4) using equation (3) from equation ().

When the nondimensional rigid contact moment β _Mj,rigid is 0.4 or less, in equation (4), instead of the nondimensional rigid contact moment β _Mj,rigid , the nondimensional rigid contact moment β _Mj, 4. The method for evaluating a composite beam according to claim 3, wherein the non-dimensional rigid moment β _{Mj, rigid, Theo} calculated by equation (5) based on _rigid is used.

Images