Calculation method for stress influence line and fatigue evaluation method for steel plate deck

Abstract

To provide a calculation method for stress influence line and a fatigue evaluation method for steel plate deck that only make a finite element analysis once for what is aimed at, and also need not correct a finite element model.SOLUTION: A calculation method for a stress influence line includes: a load setting step of setting, based upon a reciprocity theorem, a load corresponding to arbitrary stress at a point of focus of a finite element model, at a plurality of positions corresponding to the point of focus of the finite element model; and an influence line calculation step of performing finite element analysis for the finite element model having the load set so as to calculate a stress influence line of the arbitrary stress at the point of focus for a unit load.SELECTED DRAWING: Figure 11

Description

[Technical field]

[0001] The present invention relates to a method for calculating stress influence lines and a method for evaluating fatigue of a steel deck. [Background technology]

[0002] A technique for estimating fatigue through analysis in structures such as bridges has been disclosed (see, for example, Patent Document 1). [Prior art documents] [Patent documents]

[0003] [Patent Document 1] JP 2019-189009 A Summary of the Invention [Problem to be solved by the invention]

[0004] In the above analysis, it is essential to obtain an influence line that expresses the response of stress, etc., at the point of interest in the structure by the position of the load. Conventionally, a method is used in which many unit load positions are set in the finite element model and finite element analysis is performed for each load position, but this method requires performing many analysis cases in which the unit load position is changed. In addition, a method is used in which finite element analysis is performed in which a load simulating a tire model is applied in the finite element model, but it is necessary to perform element division according to the tire model, and when the tire model is changed, the element division must be changed. Furthermore, an analysis method based on the so-called Muller-Breslau principle is used, but this method requires modification of the finite element model, such as adding a node to the point of interest and modifying the element connectivity (double node), and this modification process must be performed for each point of interest.

[0005] The present invention has been made in consideration of the above, and aims to provide a method for calculating stress influence lines and a method for evaluating fatigue of a steel deck, which require only one finite element analysis for one point of interest and do not require modification of the finite element model. [Means for solving the problem]

[0006] The method for calculating a stress influence line according to the present invention includes a load setting step of setting loads corresponding to an arbitrary stress at a point of interest of a finite element model at multiple positions corresponding to the point of interest of the finite element model based on the reciprocity theorem, and an influence line calculation step of calculating a stress influence line of an arbitrary stress at the point of interest for a unit load by performing finite element analysis on the finite element model in which the loads have been set.

[0007] The fatigue evaluation method for a steel deck of the present invention includes the steps of generating the finite element model of the steel deck, setting a point of interest in the generated finite element model, calculating a stress influence line of the corrected structural hotspot stress of the point of interest for a unit load for the finite element model in which the point of interest has been set using the above-mentioned stress influence line calculation method, generating a stress waveform generated at the point of interest in the finite element model based on the calculated stress influence line, and calculating the fatigue of the point of interest based on the generated stress waveform. Effect of the Invention

[0008] According to the present invention, it is possible to provide a method for calculating stress influence lines and a method for evaluating fatigue of a steel deck, which require only one finite element analysis for one point of interest and do not require modification of the finite element model. [Brief description of the drawings]

[0009] [Figure 1] FIG. 1 is a diagram showing an example of analysis of the influence lines of deflection using the reciprocity theorem. [Diagram 2] FIG. 2 is a diagram showing an example of analysis of the influence lines of the deflection angle according to the reciprocity theorem. [Diagram 3]FIG. 3 is a diagram showing an example of analysis of the influence lines of bending moment according to the Muller-Breslau principle. [Figure 4] FIG. 4 is a diagram showing an example of analysis of the influence line of strain on an element side. [Diagram 5] FIG. 5 is a diagram showing an example of analysis of the influence lines of strain at a node. [Figure 6] FIG. 6 is a diagram showing an example of analysis of the influence lines of strain at the outermost node. [Figure 7] FIG. 7 is a diagram showing an example of analysis of the influence lines of stress at a node. [Figure 8] FIG. 8 is a diagram showing an example of an analysis of the influence lines of the displacement on the plate surface of a node. [Figure 9] FIG. 9 is a diagram showing an example of an analysis of the stress influence lines on the plate surface at a node. [Figure 10] FIG. 10 is a diagram showing an example of analysis of the influence lines of displacement at points on an element side. [Figure 11] FIG. 11 is a flowchart showing an example of a method for calculating a stress influence line according to this embodiment. [Figure 12] FIG. 12 is a flowchart showing an example of a method for evaluating fatigue of a steel deck according to this embodiment. [Figure 13] FIG. 13 is a diagram showing an example of a finite element model of a steel deck. [Figure 14] FIG. 14 is a diagram showing an example of a finite element model of a steel deck. [Figure 15] FIG. 15 is a diagram showing a part of a finite element model of a steel deck. [Figure 16] FIG. 16 is a diagram showing a schematic diagram of stress influence lines calculated for a finite element model of a steel deck. [Figure 17] FIG. 17 is a diagram showing an example of vehicle setting items in the stress waveform generating step. [Figure 18] FIG. 18 is a diagram showing an example of vehicle settings in the stress waveform generating step. [Figure 19] FIG. 19 is a diagram showing an example of vehicle setting items in the stress waveform generating step. [Figure 20] FIG. 20 is a diagram showing a schematic diagram of a part of the calculation procedure of the stress waveform. [Figure 21] FIG. 21 is a diagram showing examples of stress waveforms at a point of interest for a two-axle vehicle, a three-axle vehicle, and a four-axle vehicle. [Figure 22] FIG. 22 is a diagram showing an example of problem setting for a two-span continuous beam. [Diagram 23] FIG. 23 is a diagram showing an example of a finite element model in the vicinity of a point of interest. [Figure 24] FIG. 24 is a diagram showing an example of the state of deformation of the finite element model. [Diagram 25] FIG. 25 is a graph showing an example of the influence line of the strain of an element of interest at the upper edge of a beam. [Figure 26] FIG. 26 is a diagram showing the analysis results at the reference points of the influence line by the method according to this embodiment. [Figure 27] FIG. 27 is a diagram showing an example of a target structure based on two main composite girders. [Figure 28] FIG. 28 is a diagram showing an example of a target structure based on two main composite girders. [Figure 29] FIG. 29 is a diagram showing an example of material constants of steel and concrete in a model of two main composite girders. [Diagram 30] FIG. 30 is a cross-sectional view of the vicinity of a point of interest in a model of two main composite girders. [Diagram 31] FIG. 31 is a diagram showing an overall view of an analytical model and the positions of points of interest. [Diagram 32] FIG. 32 is a contour diagram of the influence lines of the x-direction vertical stress at a point of interest on the upper surface of the deck. [Diagram 33] FIG. 33 is a diagram summarizing the coordinates of the reference points, the stress at the point of interest when a unit load is applied to each reference point, and the values ​​of the influence lines at the reference points using the method according to this embodiment. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0010] Hereinafter, an embodiment of the method for calculating a stress influence line and the method for evaluating fatigue of a steel deck according to the present invention will be described with reference to the drawings. Note that the present invention is not limited to the embodiment. In addition, the components in the following embodiments include those that are replaceable and easy for a person skilled in the art, or those that are substantially the same.

[0011] [principle] The principle of the method for calculating the stress influence line according to this embodiment will be described. For example, when designing a structure that receives an uncertain moving load, such as a bridge, a function that expresses the response of the stress, etc. at a point of interest in the structure at the position where the load is applied is indispensable. In a finite element model, for example, when the loading position is one-dimensional, the function is called a stress influence line (or influence line). Note that when the loading position is two-dimensional, such as on a deck surface, the function may be called a stress influence surface (or influence surface). In this specification, the term stress influence line (or influence line) will be used consistently regardless of the dimension of the loading position.

[0012] In this embodiment, the stress influence line is efficiently calculated based on finite element analysis using the reciprocity theorem. The reciprocity theorem is a theorem that considers two systems, considers a complementary virtual work equation in each system as a balanced system of external and internal forces of different systems, and states that in a linear elastic body, the complementary virtual work of the internal forces in the two systems is equal, and therefore the complementary virtual work of the external forces in the two systems is equal. If the displacement in system i is u i , the external force is q i Then, the reciprocity theorem is

number

[0013] (1) Analysis of influence lines in structural analysis a. Line of influence of deflection For example, the influence line of deflection in a beam structure can be obtained as follows. Figure 1 shows an example of analysis of the influence line of deflection using the reciprocity theorem. As shown in Figure 1, the external force of system 1 is a concentrated load P 1(ξ), and the external force of system 2 is the concentrated load P at the deflection point x. 2 (x). Let the deflection at x in system 1 and the deflection at ξ in system 2 be w 1 (x; ξ), w 2 (ξ;x). Here, the variable after ";" indicates the load application point. From the above, by the reciprocity theorem,

number

number

[0014] b. Line of influence of deflection angle Next, the influence line of the deflection angle in a beam structure can be obtained as follows. Figure 2 shows an example of analysis of the influence line of the deflection angle by the reciprocity theorem. As shown in Figure 2, the external force of system 1 is a concentrated load P 1 (ξ), and the external force of system 2 is the concentrated moment load M at the point x of the deflection angle. 2 (x). The deflection angle at x in system 1 is θ 1 (x;ξ), the deflection at ξ in system 2 is w 2 (ξ;x). From the above, by the reciprocity theorem,

number

number

[0015] c. Bending moment influence line The Muller-Breslau principle also allows the line of influence of an internal force, such as a bending moment in a beam structure, to be calculated. In this case, discontinuous deformation must be considered in order to extract the complementary virtual work of the internal force. Figure 3 shows an example of analysis of the line of influence of a bending moment using the Muller-Breslau principle. As shown in Figure 3, a discontinuous relative rotation △θ 2 Consider (x). Since we are considering a discontinuous deformation at x, θ 2 The left and right limits are as follows:

number

number

number

number

number

number

number

[0016] (2) Efficient analysis of influence lines in finite element analysis Next, an efficient finite element analysis method for influence lines based on the reciprocity theorem will be described.

[0017] a. Strain influence line FIG. 4 is a diagram showing an example of analysis of the influence line of strain on an element side. The normal strain (ε x ) ab Consider the following. In system 1, a unit load P in the y direction is applied to a node ξ at an arbitrary position. 1 ξ =1 is acting on system 2, and unit load P in the x direction is acting on nodes a and b in opposite directions. 2 a =-1, P 2 b = 1 is applied. The displacement of node ξ in the y direction is v ξ The displacement of nodes a and b in the x direction is u α , u b Then, by the reciprocity theorem,

number

number

number

[0018] Figure 5 shows an example of analysis of the influence line of strain at a node. Consider the finite element model shown in Figure 5. In system 1, a unit load P in the y direction is applied to a node ξ at an arbitrary position. 1 ξ =1 is acting on nodes a, b, and c in system 2, and the x-direction load P 2 α (α=a, b, c) is acting. The displacement of node ξ in the y direction is v ξ Let the displacement of nodes a, b, and c in the x direction be u α (α=a, b, c).

[0019] In designing structures, the stress and strain at a point of interest are an issue. In the normal finite element method, continuity of displacement between elements is imposed, but the strain, which is its derivative, does not have continuity. Therefore, the strain at a node is not generally defined, but some average of the strain at the two element sides that sandwich the node is often used. Therefore, as shown in Figure 5, the slope at b when the displacements of the three points of the node of interest b and its two neighboring nodes a and c are interpolated with a quadratic function rather than the interpolation function of the finite element method is considered to be the strain at b. Here, the length between a and b is defined as l ab , the length between bc is l bc Then, the normal strain (ε x ) b teeth,

number

number

number

number

[0020] Figure 6 shows an example of analysis of the influence line of strain at the outermost node. In many cases, the strain or stress at the outermost edge, such as point b of the finite element model shown in Figure 6, becomes a problem. In this case, the x-direction normal strain (ε x ) ab can be used.

[0021] b. Lines of influence of stress In linear problems, the stress and strain are in a linear relationship. For example, in a plate problem, the plate is in a plane stress state, and the constitutive relationship is normal stress σ x , σ y , the shear stress τ xy , normal strain is ε x , ε y , shear strain is ε xy Then,

number

[0022] FIG. 7 is a diagram showing an example of an analysis of the influence line of stress at a node. The normal stress (σ x ) bWe will consider how to find the influence lines of. For simplicity, all elements are assumed to be square bilinear quadrilateral elements with side length l. In other cases, the following method can be applied since the displacement at any point can be expressed as the nodal displacement as long as finite element discretization is used.

[0023] In system 1 in Fig. 7, a unit load P in the y direction is applied to a node ξ at an arbitrary position. 1 ξ In system 2 in Fig. 7, the x-direction load P is applied to nodes a and c adjacent to node b in the x-direction. 2 a , P 2 c is acting on nodes d and f adjacent to node b in the y direction, and a y-direction load Q 2 d , Q 2 f is at work.

[0024] The displacement of node ξ in the y direction is v ξ The displacement of nodes a and c in the x direction is u a , u c The displacement of nodes d and f in the y direction is v d , v f Applying the reciprocity theorem to systems 1 and 2, we get

number

number

number

number

number

[0025] x-direction normal stress σ x is expressed by Equation 20 as the normal strain in the x direction (ε x ) b and the normal strain in the y direction (ε y ) b Since it is expressed as a linear combination of

number

number

[0026] In three-dimensional problems, the influence line can be calculated using a similar method. The three-dimensional constitutive relationship is expressed as σ x , σ y , σ z and the shear stress is τ xy , τ yz , τ zx and the normal strain is ε x , ε y , ε z and the shear strain is γ xy , γ yz , γ zx Then,

number

[0027] Although the influence lines of normal strain and normal stress have been explained above, a similar concept can also be applied to the influence lines of shear strain and shear stress.

[0028] c. Influence lines in plane shell elements Consider a finite element model discretized with plane shell elements. In the plane shell elements, it is necessary to obtain the stress and strain on the plate surface to take into account the effects of bending.

[0029] Fig. 8 is a diagram showing an example of an analysis of the influence line of the displacement on the plate surface of a node. In the finite element model shown in Fig. 8, the x-direction displacement u 1 a,surface Let us consider how to find the influence line of the force P in the y direction at an arbitrary node ξ. 1 ξ =1 is acting on node a in system 2, and the x-direction load P 2 a and bending moment M about the y-axis 2 a The displacement of node ξ in the y direction is v ξ Let the displacement of node a in the x direction be u a , the rotation angle around the y-axis is θ a If the plate thickness is t, the displacement on the plate surface is u a,surface teeth,

number

number

number

[0030] The influence lines for strain and stress on the plate surface at the node can also be found using the same approach as for items a and b above. Figure 9 is a diagram showing an example of an analysis of the influence lines for stress on the plate surface at the node. In the finite element model shown in Figure 9, the normal stress (σ x ) b,surface This section describes how to find the influence line of the force. For simplicity, all elements are assumed to be square with side length l and plane shell elements with plate thickness t. In other cases, the following method can be applied since the displacement at any point can be expressed as the nodal displacement as long as finite element discretization is used. In system 1, a unit load P in the y direction is applied to a node ξ at an arbitrary position. 1 ξ =1 is acting on node a and node c adjacent to node b in the x direction in system 2, and a load P 2 a , P 2 c , bending moment about the y-axis M 2 a , M 2 c is acting on nodes d and f adjacent to node b in the y direction, and a y-direction load P 2 d , P 2 f , bending moment about the x-axis M 2 d , M 2 f The displacement of node ξ in the y direction is v ξ The displacement of nodes a and c in the x direction is u a , u c , the rotation angle around the y-axis is θ a , θ c , the displacement of nodes d and f in the y direction is v d , v f , the rotation angle around the x-axis is θ d , θ f Using the reciprocity theorem on systems 1 and 2, we get

number

number

number

number

[0031] d. Influence line of displacement at a point on an element edge FIG. 10 is a diagram showing an example of an analysis of the influence line of the displacement at a point on an element side. The normal strain (ε x ) d Let us consider the case where the influence line of is sought. For simplicity, all elements are assumed to be bilinear quadrilateral elements. In other cases, the following method can be applied since the displacement of any point can be expressed as a nodal displacement as long as finite element discretization is used. With node d as the origin, there is no node adjacent in the positive x-axis direction. In such a case, the x-direction displacement u at point b on the element side adjacent to node d is expressed as b is used.

[0032] First, the x-direction displacement u at point b on the element side b In the finite element model shown in Fig. 10, a unit load P in the y direction is applied to a node ξ at an arbitrary position in system 1. 1 ξ = 1 is acting on the element edge including node b in system 2, and x-direction load P 2 a , P 2 c The displacement of node ξ in the y direction is v ξ Let the displacement of nodes a, b, and c in the x direction be u α Let (α=a, b, c). The length between a and b is l ab , the length between bc is l bc Then, the displacement of point b in the x direction, u b teeth,

number

number

number

[0033] [Calculation method of stress influence line] Next, a method for calculating a stress influence line according to this embodiment will be described. Fig. 11 is a flowchart showing an example of a method for calculating a stress influence line according to this embodiment. As shown in Fig. 11, the method for calculating a stress influence line according to this embodiment includes a load setting step S101 and an influence line calculation step S102.

[0034] In the load setting step S101, a load corresponding to an arbitrary stress at a point of interest set in the finite element model is set in the vicinity of the point of interest. In the load setting step S101, for example, a point of interest is set in the finite element model shown in Fig. 4 to Fig. 10, and a load corresponding to an arbitrary stress is set in a node in the vicinity of the set point of interest.

[0035] In the influence line calculation step S102, a finite element analysis of the finite element model in which the load is set in the load setting step S101 is performed to calculate a stress influence line of an arbitrary stress at the point of interest for a unit load. The finite element analysis performed in the influence line calculation step S102 utilizes the reciprocity theorem, making it possible to calculate an influence line by performing only one analysis for one point of interest. In addition, since it is possible to calculate a stress influence line of an arbitrary stress that is not limited to the load model, and there is no restriction on the element division of the load acting surface, it is possible to obtain the response of the point of interest due to an arbitrary distributed load. Therefore, even if the load model, etc. is changed, it is not necessary to modify or add to the finite element model.

[0036] [Fatigue evaluation method for steel deck] Next, a method for evaluating fatigue of a steel deck according to this embodiment will be described. Fig. 12 is a flowchart showing an example of the method for evaluating fatigue of a steel deck according to this embodiment. As shown in Fig. 12, the method for evaluating fatigue of a steel deck according to this embodiment includes a model generation step S201, a focus point setting step S202, an influence line calculation step S203, a stress waveform generation step S204, and a fatigue calculation step S205.

[0037] The model generation step S201 generates a finite element model of a steel deck. Figures 13 and 14 are diagrams showing an example of a finite element model of a steel deck. The finite element model 100 of a steel deck shown in Figures 13 and 14 has a deck plate 10, a main girder 20, a horizontal rib (horizontal plate) 30, and a vertical rib (vertical plate) 40, and the horizontal rib 30 and the vertical rib 40 are welded all around by a weld 50. In Figure 13, some of the welds 50 are shown by dashed lines.

[0038] In the focus point setting step S202, a focus point is set for the generated finite element model. Fig. 15 is a diagram showing a part of the finite element model of a steel deck. In the above-mentioned example of the finite element model of a steel deck, as shown in Fig. 15, for example, a focus point can be set at the toe of the weld on the transverse rib side at the intersection of the transverse rib and the longitudinal rib.

[0039] In the influence line calculation step S203, for the finite element model in which the focus point is set, the stress influence line of the "corrected structural hotspot stress (hereinafter referred to as stress σ'h)" of the focus point against the unit load is calculated using the above-mentioned stress influence line calculation method (including steps S101 and S102).

[0040] Specifically, first, for a finite element model of a steel deck in which a point of interest has been set, a load corresponding to the stress σ′h at the point of interest is set at multiple positions of the finite element model (corresponding to load setting step S101).

[0041] Here, the load for determining the influence line of the stress σ'h will be explained. For simplicity, it is assumed that the elements near the target point are divided into square elements of length l as shown in FIG.

[0042] Structural hot spot stresses are

number

[0043] The structural hot spot stress taking into account the bending stress reduction correction and the strength reduction correction due to the increase in plate thickness (corrected structural hot spot stress: stress σ´h) is

number

[0044] From the above, the corrected tectonic hotspot stress is

number

[0045] The load to determine the influence line of the membrane stress in the vertical direction is the horizontal (y) load at points a and b in Figure 15, taking 0.4t as an example.

number

number

[0046] The load to determine the influence line of the vertical bending stress is 0.4t in Figure 15. The moment around the vertical (z) axis at points a and b is

number

number

[0047] Considering the load for determining the stress influence line at the 1.0t position in Figure 15 in the same way, and taking into account the coefficient of the above number 41, the load for determining the influence line of the corrected structural hot spot stress is

number

number

number

number

number

number

number

number

[0048] Next, a finite element analysis using the reciprocity theorem is performed on the finite element model of the steel deck with the load set, to calculate the stress influence line of the stress σ'h at the point of interest for the unit load (corresponding to the influence line calculation step S203).

[0049] Fig. 16 is a schematic diagram showing stress influence lines calculated for a finite element model of a steel deck. In the influence line calculation step S203, by utilizing the reciprocity theorem in the finite element analysis, it is possible to calculate influence lines as shown in Fig. 16 by performing only one analysis for one point of interest.

[0050] In the stress waveform generating step S204, a stress waveform occurring at the point of interest of the finite element model is generated based on the calculated stress influence line.

[0051] In the stress waveform generating step S204, a vehicle model and vehicle weight are set for a vehicle traveling on the steel deck, and the vehicle traveling position in the finite element model is set, and a stress waveform is generated based on these settings. Figures 17 to 19 are diagrams showing examples of vehicle settings in the stress waveform generating step S204. Figures 17 to 19 show examples of setting the axle distance, axle load ratio of each axle, and vehicle weight for each vehicle model. Figure 17 shows an example of a two-axle vehicle, Figure 18 shows an example of a three-axle vehicle, and Figure 19 shows an example of a four-axle vehicle. In addition, in the graph showing vehicle weight, the horizontal axis shows vehicle weight (tf) and the vertical axis shows the number of vehicles (number of vehicles traveling per 72 hours).

[0052] FIG. 20 is a diagram showing a schematic diagram of a part of the calculation procedure of the stress waveform. FIG. 20 shows an example in which a four-axle vehicle is used as the vehicle model. As shown in FIG. 20, in this embodiment, the stress influence line corresponding to the unit load of the stress σ′h at the point of interest is calculated, so that any tire model such as a single tire or a double tire can be handled. In the example shown in FIG. 20, a single tire is shown as a tire range T1, and a double tire is shown as a tire range T2. Note that FIG. 20 shows a state in which the tire range of the vehicle is superimposed on the stress influence line shown in FIG. 16. In this example, the area integrals of the tire ranges T1 and T2 are calculated while shifting the position in the bridge axis direction, so that the stress waveform at the point of interest can be calculated.

[0053] FIG. 21 is a diagram showing examples of stress waveforms at points of interest for a two-axle vehicle, a three-axle vehicle, and a four-axle vehicle. The horizontal axis of FIG. 21 shows the travel distance (mm), and the vertical axis shows the stress (N / mm 2 ) is shown. By performing the calculation for the area integral of the tire range while shifting the position in the bridge axis direction for each of the two-axle, three-axle, and four-axle vehicles, it is possible to calculate the stress waveforms as shown in Fig. 21.

[0054] In the fatigue calculation step S206, the fatigue of the point of interest is calculated based on the generated stress waveform. Specifically, first, a stress range frequency distribution for the point of interest is calculated based on the generated stress waveform. The stress range frequency distribution can be calculated, for example, by performing rainflow processing on continuous stress waveforms. Next, a cumulative fatigue damage ratio for the point of interest is calculated based on the calculated stress range frequency distribution. For example, the cumulative fatigue damage ratio D for a design life of 100 years can be calculated according to Miner's rule.

[0055] [Application to plane stress problems] We will explain the case where the finite element model is applied to a plane stress model. Below, we will explain the plane stress model by taking a model of a two-span continuous beam as an example. Figure 22 is a diagram showing an example of problem setting for a two-span continuous beam. We will consider the plane stress problem of the two-span continuous beam shown in Figure 22. The beam has a height of 2000 mm, a span of 30000 + 40000 mm, and is divided into bilinear quadrilateral elements of 100 mm square. In this case, the total number of elements is 14000 and the total number of nodes is 14742. The material is homogeneous and has Young's modulus E = 2 × 10 5 (N / mm 2 ), and Poisson's ratio ν = 0.3. The coordinate x is taken from the left support to the right, and the coordinate y is taken from the top edge of the beam downward. The point of interest is x = 10,000 mm.

[0056] Fig. 23 is a diagram showing an example of a finite element model in the vicinity of a point of interest. In the finite element model shown in Fig. 23, the validity of the method according to this embodiment is confirmed by the influence line of the x-direction normal strain of element side ab. As described above, the following x-direction loads are applied to node a and node b, respectively.

number

[0057] Figure 24 shows the deformation of the finite element model at this time. Figure 25 is a graph showing an example of the influence line of the strain of the element side of interest at the upper edge of the beam. The vertical axis of Figure 25 shows the magnitude of the displacement in the y direction, and the horizontal axis shows the position in the x direction. In a structure such as a bridge, a vertical external force acts only on the deck, so in Figure 25, the y direction displacement of the upper edge of the beam is continuously shown for any x. Figure 25 corresponds to the influence line in structural analysis of a frame. The validity of this analysis is confirmed by comparing this influence line with the strain at the point of interest for a unit load in a normal boundary value problem. A unit load was applied with the upper edges at x=10,000 mm, 20,000 mm, and 50,000 mm as reference points A, B, and C, respectively. Figure 26(a) shows the strain (ε x ) aband a diagram showing values ​​at reference points of the influence line by the method according to this embodiment. The element side strain when a unit load is applied is calculated based on the relative displacement of nodes a and b. The difference between the two is compared in terms of relative error, but since the solution by the unit load is also an approximate solution, it is expressed as a relative difference. The relative difference is defined as follows:

number

[0058] Next, the validity of the method according to this embodiment is confirmed for the influence line of strain at the node. In order to obtain the influence line of the x-direction vertical strain of node b in Figure 23, an analysis was performed by applying a load given by the above equation (19). The deformation of the finite element model and the distribution of the upper edge vertical displacement at this time are the same as those in Figures 24 and 25, respectively. Figure 26(b) is a diagram summarizing the strain when unit loads are applied to reference points A, B, and C, and the values ​​at the reference points of the influence line of strain according to this embodiment. Note that the strain at the node when a unit load is applied was obtained from the nodal displacement using the above equation (16). As shown in Figure 26(b), the method according to this embodiment is 10 times better than the solution obtained by the conventional solution method when a unit load is applied. -3 The agreement is within about % accuracy.

[0059] Next, the validity of the method according to this embodiment for the influence line of stress at the node is confirmed. In order to obtain the influence line of the x-direction normal stress at node b in FIG. 23, an analysis was performed by applying a load given by the above formula 26. The distribution of the deformation of the finite element model and the upper edge vertical displacement at this time are the same as those in FIG. 24 and FIG. 25, respectively. FIG. 26(c) is a diagram summarizing the stress when unit loads are applied to reference points A, B, and C, and the values ​​at the reference points of the influence line of stress according to this embodiment. Note that the stress at the node when a unit load is applied was obtained from the constitutive relationship between the x-direction normal strain and y-direction normal strain at node b obtained from the nodal displacement, and the plane stress. As shown in FIG. 26(c), the method according to this embodiment is 10 times better than the solution obtained by the conventional solution method when a unit load is applied. -3 The agreement is within about % accuracy.

[0060] Next, the validity of the method according to the present embodiment for stress due to distributed load is confirmed. In the beam shown in FIG. 22, a uniformly distributed load per unit depth q=1N / mm is applied from the left span x=0 to x=10000mm. 2 The normal stress in the x direction at node b is 1.13428×10 2 N / mm 2 The product of the stress influence line and the uniformly distributed load by the method according to this embodiment is integrated over the loading range to obtain 1.13436×10 2 N / mm 2 The relative difference is 6.6×10 -3 %.

[0061] In designing structures, the stress at the outermost edge, such as point f in Figure 23, is often an issue. In this case, by defining the x-direction vertical strain from the displacements of nodes g, f, and h, and defining the y-direction vertical strain from the displacements of nodes b and f, it is possible to obtain the influence line of the stress at node f in the same way as the stress at node b. Specifically, by applying an x-direction load P to nodes g and h, and a y-direction load Q to nodes b and f,

number

[0062] [Example of application to 3D problems] A case where the finite element model is applied to a three-dimensional finite element model will be described below. A three-dimensional finite element model will be described below by taking a model of two main composite girders as an example. Figures 27 and 28 are diagrams showing an example of a target structure referring to two main composite girders. Figure 27 shows a cross section, and Figure 28 shows a cross section of a girder. The hatched parts in Figures 27 and 28 are made of concrete, and the other parts are made of steel, and are modeled as being integrated on the upper surface of the upper flange, and reinforcing bars, shear stoppers, cross beams, etc. are not modeled. Both steel and concrete are isotropic linear elastic bodies. Figure 29 is a diagram showing an example of material constants of steel and concrete in a model of two main composite girders. The cross section is modeled with 1770 elements, the axial element length is 100 mm, and a total of 1,239,000 elements of first-order hexahedral elements. The total number of nodes is 1,610,898.

[0063] As in the planar problem above, a two-span continuous beam of 30,000 mm and 40,000 mm is used, with the end support on the 30,000 mm span side taken as the origin, the axis from the origin to the end support on the opposite side as the x-axis, the vertical downward direction with the neutral axis of bending of the cross section as the origin as the z-axis, and the horizontal direction perpendicular to the axis with the center of the deck as the origin as the y-axis. At the support, the underside of the lower flange is linearly supported in the z-direction, and restraints in other directions are the minimum restraints that restrain rigid body motion. Figure 30 is a cross-sectional view of the vicinity of the point of interest in the model of two main composite girders. The point of interest is the point indicated by a circle in Figure 30 at x=10,000 mm. Figure 31 is a diagram showing the overall diagram of the analysis model and the position of the point of interest. Taking into account the three-dimensional constitutive relationship, a load P in the x-direction is applied to each node adjacent to the point of interest in the x-direction.x A load P in the y direction is applied to each adjacent node in the y direction. y In addition, since the point of interest is the end point in the z direction, the load P in the z direction is applied to the node adjacent to the point of interest in the z direction. z These loads were applied in opposite directions.

number

number

number

number

[0064] FIG. 32 is an influence line contour diagram of the x-direction vertical stress at a point of interest on the top surface of the deck. FIG. 32 also shows the positions of the main girders and supports. To confirm the validity of this result, three points A, B, and C shown in FIG. 32 are selected as reference points, and a conventional unit load analysis is performed to compare with the stress at the point of interest. FIG. 33 is a diagram summarizing the coordinates of the reference points, the stress at the point of interest when a unit load is applied to each reference point, and the values ​​of the influence lines at the reference points using the method according to this embodiment. As shown in FIG. 33, the method according to this embodiment is 10 times better than the solution using the conventional solution method for loading a unit load. -1 The agreement is within about % accuracy.

[0065] As described above, the method for calculating a stress influence line according to the first aspect of this embodiment includes a load setting step of setting loads corresponding to any stress at a point of interest of the finite element model at multiple positions based on the reciprocity theorem, and an influence line calculation step of calculating any stress influence line at the point of interest for a unit load by performing finite element analysis on the finite element model to which the loads have been set.

[0066] According to this configuration, by utilizing the reciprocity theorem in finite element analysis, it is possible to calculate the influence line by performing only one analysis for one point of interest. In addition, the load applied to the model for the analysis of the stress influence line can be easily determined from the element length and material constant, which makes it easy to implement in a general-purpose finite element analysis code.

[0067] In the method for calculating a stress influence line according to the second aspect of the present embodiment, in the first aspect, a finite element model of a steel deck is used as the finite element model.

[0068] According to this configuration, when calculating stress influence lines for a finite element model of a steel deck, it is sufficient to perform a single finite element analysis for one point of interest, and there is no need to modify the finite element model.

[0069] In the method for calculating a stress influence line according to the third aspect of this embodiment, in the first aspect, a plane stress model is used as the finite element model.

[0070] According to this configuration, when calculating a stress influence line for a plane stress model, it is sufficient to perform a single finite element analysis for one point of interest, and correction of the finite element model is not required.

[0071] In the method for calculating a stress influence line according to a fourth aspect of this embodiment, in the first aspect, a three-dimensional finite element model is used as the finite element model.

[0072] According to this configuration, when calculating a stress influence line for a three-dimensional finite element model, it is sufficient to perform a single finite element analysis for one point of interest, and correction of the finite element model is not required.

[0073] A method for evaluating fatigue of a steel deck according to a fifth aspect of this embodiment includes the steps of generating a finite element model of the steel deck, setting a point of interest in the generated finite element model, calculating a stress influence line of corrected structural hotspot stress at the point of interest for a unit load for the finite element model in which the point of interest has been set using the above-mentioned stress influence line calculation method, generating a stress waveform occurring at the point of interest in the finite element model based on the calculated stress influence line, and calculating fatigue at the point of interest based on the generated stress waveform.

[0074] According to this configuration, by utilizing the reciprocity theorem in finite element analysis, it is possible to calculate the influence line of the corrected structural hotspot stress by performing only one analysis for one point of interest. In addition, since there are no restrictions on the element division of the load acting surface, it is possible to obtain the response of the point of interest due to any distributed load. Therefore, there is no need to modify or add to the finite element model when the load model, etc. is changed. In addition, there is no need for post-processing to relate the response of the point of interest to the deck surface. In addition, the load to be applied to the model for the analysis of the stress influence surface can be easily determined from the element length and material constants, making it easy to implement in a general-purpose finite element analysis code.

[0075] In the sixth aspect of the present embodiment, in the fifth aspect of the fatigue evaluation method for a steel deck, in the step of generating a stress waveform, a vehicle model, vehicle weight, and vehicle running position in a finite element model are set for a vehicle running on the steel deck, and a stress waveform is generated based on the settings.

[0076] According to this configuration, the stress influence surface for a unit load can be calculated regardless of the load model, so that the response for any load model set for a vehicle traveling on a steel deck can be easily obtained.

[0077] In the seventh aspect of the present embodiment, in the fifth aspect of the fatigue evaluation method for a steel deck, in the step of calculating the fatigue of a point of interest, a stress range frequency distribution for the point of interest is calculated based on the generated stress waveform, and a cumulative fatigue damage ratio of the point of interest is calculated based on the calculated stress range frequency distribution.

[0078] According to this configuration, the cumulative fatigue damage ratio of the point of interest can be efficiently calculated.

[0079] A method for evaluating fatigue of a steel deck according to an eighth aspect of this embodiment is, in the fifth aspect, the steel deck comprises a deck plate, a horizontal plate joined to the underside of the deck plate and extending in a first direction, a vertical plate extending in a second direction perpendicular to the first direction and joined to the underside of the deck plate so as to penetrate the horizontal plate in the second direction, and a weld joining the horizontal plate and the vertical plate, and the focus point is a point at the toe of the weld.

[0080] According to this configuration, it is possible to easily and efficiently calculate the influence line when the point at the toe of the weld is set as the point of interest.

[0081] The technical scope of the present invention is not limited to the above-described embodiment, and appropriate modifications can be made without departing from the spirit of the present invention.

[0082] The method described in the above embodiment can be used without any restrictions on the type of element (solid, shell, etc.) used in the finite element analysis or the stress components.

[0083] The method described in the above embodiment can calculate a stress influence surface for a unit load without relying on a load model, and therefore can obtain a response for an arbitrary load model.

[0084] The method described in the above embodiment can be used regardless of the type of bridge (new bridge, existing bridge, reconstructed bridge (deck replacement, etc.), road bridge, railway bridge). [Explanation of symbols]

[0085] 10 Deck plate 20 Main digit 30 Horizontal ribs 40 Vertical ribs 50 Welding 100 Finite Element Models

Claims

1. a load setting step of setting loads corresponding to any stress at a point of interest of the finite element model at a plurality of positions corresponding to the point of interest of the finite element model based on the reciprocity theorem; an influence line calculation step of calculating a stress influence line of an arbitrary stress at the point of interest for a unit load by performing a finite element analysis on the finite element model in which the load is set; A method for calculating stress influence lines, including:

2. The finite element model is a finite element model of a steel deck. The method for calculating a stress influence line according to claim 1.

3. The finite element model uses a plane stress model. The method for calculating a stress influence line according to claim 1.

4. The finite element model is a three-dimensional finite element model. The method for calculating a stress influence line according to claim 1.

5. generating the finite element model of a steel deck; Setting a focus point of the generated finite element model; Calculating a stress influence line of a corrected structural hot spot stress of the point of interest for a unit load by the stress influence line calculation method according to claim 1 for the finite element model in which the point of interest is set; generating a stress waveform occurring at the point of interest of the finite element model based on the calculated stress influence line; Calculating fatigue at the point of interest based on the generated stress waveform; A method for fatigue assessment of steel decks, including:

6. In the step of generating the stress waveform, a vehicle model, a vehicle weight, and a vehicle running position in the finite element model are set for a vehicle running on the steel deck, and the stress waveform is generated based on the settings. The method for evaluating fatigue of a steel deck according to claim 5.

7. In the step of calculating the fatigue of the target point, a stress range frequency distribution for the target point is calculated based on the generated stress waveform, and a cumulative fatigue damage ratio of the target point is calculated based on the calculated stress range frequency distribution. The method for evaluating fatigue of a steel deck according to claim 5.

8. The steel deck is Deck plate and A horizontal plate joined to the lower surface of the deck plate and extending in a first direction; A vertical plate extending in a second direction perpendicular to the first direction and joined to the underside of the deck plate so as to penetrate the horizontal plate in the second direction; a welded portion that joins the horizontal plate and the vertical plate; Equipped with The focus point is a point at the toe of the weld. The method for evaluating fatigue of a steel deck according to claim 5.

Images