JP2013030138A - Structure analysis method - Google Patents

Abstract

PROBLEM TO BE SOLVED: To easily analyze elastic-plastic action of a steel beam during a fire with high accuracy and to greatly reduce an analysis time, a computer load and work labor.SOLUTION: A quaternary simultaneous equation having an equation showing respective balances of axial force and a moment at an axis of member orthogonal cross section of a material end of a steel beam, an equation showing a displacement conformity condition in an axis of member direction at the material end of the steel beam, and an equation showing a yield condition at the axis of member orthogonal cross section of the steel beam made plastic by a temperature rise is created. An unknown solution of the quaternary simultaneous equation is calculated by solving the quaternary simultaneous equation. Axial force N, a plastic area length l, and temperature histories of compressive strain εand tensile strain εare calculated by integrating the solution of the quaternary simultaneous equality by temperature T.

Description

  The present invention relates to a structural analysis method for analyzing elasto-plastic behavior of a steel beam installed between steel columns of a steel structure building at the time of a fire.

  Conventionally, a structural analysis method using a finite element method has been used as means for analyzing nonlinear elastoplastic behavior of a static structure such as a steel structure building during a fire (for example, Patent Document 1). reference). The finite element method is a method of obtaining a behavior of a structure by discretizing a structure into a plurality of finite elements, obtaining a governing equation of the finite element, and solving the governing equation.

  The governing equation obtained at this time is a non-linear equation, and an iterative method such as Newton-Raphson method is preferably used for obtaining a solution of the governing equation. The iterative method is a method of obtaining an approximate solution of the governing equation by setting an appropriate initial value for the unknown of the governing equation and repeating the operation using the governing equation starting from the initial value. In this iterative method, if the initial value is not properly set, it does not converge to an approximate solution by iterative calculation. For this reason, in the iterative method, the convergence judgment that determines whether or not the solution after the iteration operation has converged to the approximate solution is performed after an appropriate number of iteration operations, and if it has not converged, the initial value is corrected and the correction is made. It is necessary to repeat the iterative operation starting from a later value.

JP 2005-141645 A

  However, when analyzing the behavior of an indefinite structure using the finite element method as described above, it is necessary to perform an iterative operation with convergence judgment to find the solution of the governing equation of the finite element. Therefore, the analysis time is prolonged and the calculation load of the computer is increased. In this case, since it is necessary to perform an input operation for dividing the statically indeterminate structure into a plurality of finite elements, the analysis time is prolonged and the burden of human labor is increased accordingly. It was.

  Therefore, the present invention has been devised in view of the above-mentioned problems, and the object of the present invention is to easily and accurately analyze the elasto-plastic behavior of a steel beam at the time of a fire, and to analyze time, An object of the present invention is to provide a structural analysis method capable of greatly reducing the computer load and work labor.

  In order to solve the above-mentioned problems, the present inventors have invented the following structural analysis method after intensive studies.

A structural analysis method according to a first aspect of the present invention is a structural analysis method for analyzing elastoplastic behavior of a steel beam installed between steel columns of a steel structure building at the time of a fire, wherein the axial force of the steel beam is N, the plastic region length l _p from the end of the steel beam and the center of the span, the compressive strain ε _c of the lower cross section in the cross section perpendicular to the axis of the material end of the steel beam, and the upper cross section With respect to the tensile strain ε _t , when the steel beams are heated from the temperature T by the increment dT, the increments dN, dl _p , dε _c , dε _t are the increments dN, increment dlp, increment dε, respectively. _{an equation} including any one or more of _c and increment dεt as an unknown and including the increment dT, and an equation indicating the balance of axial force and moment in the material axis orthogonal section of the material end of the steel beam; , An equation indicating the displacement matching condition in the material axis direction at the material end of the steel beam and A quaternary simultaneous equation having a yield condition in a cross section perpendicular to the axis of the steel beam plasticized by temperature rise is created, and the solution of each unknown is calculated by solving the quaternary simultaneous equation. The temperature history of the axial force N, the plastic region length l _p , the compressive strain ε _c and the tensile strain ε _t is calculated by integrating the solution of the quaternary simultaneous equations at the temperature T.

ここで、式（１）におけるＥ_t（ε，Ｔ）は歪ε、温度Ｔとしたときの鋼製梁の応力−歪関係の接線剛性であり、Ｅ₀は、鋼製梁の応力−歪関係において歪硬化が始まる点での歪をε_sとしたときに、下記式（２）により表される。

The structural analysis method according to a second aspect of the present invention is the structural analysis method according to the first aspect, wherein an increase in deformation amount in the material axis direction of the plastic region when the temperature of the steel beam is increased from the temperature T by an increment dT is expressed by the following formula (1 The expression evaluated using γ expressed by (2) is used as an expression indicating the displacement adaptation condition.

Here, E _t (ε, T) in equation (1) is the tangential stiffness of the stress-strain relationship of the steel beam when the strain ε and the temperature T are set, and E ₀ is the stress-strain of the steel beam. When the strain at the point where strain hardening starts in the relationship is ε _s , it is expressed by the following equation (2).

The structural analysis method according to a third aspect of the present invention is the first or second aspect of the present invention, in the first aspect or the second aspect, based on an expression indicating the yield condition in the material axis orthogonal section of the material end and span center of the steel beam. By calculating the elastic limit temperature T ₀ when the material axis and the cross section of the material axis at the center of the span are fully plasticized, and integrating the solution of the quaternary simultaneous equations at the temperature T in the range of the temperature T ₀ or more, The temperature history of the axial force N, the plastic region length l _p , the compressive strain ε _c and the tensile strain ε _t is calculated.

  A structural analysis program according to a fourth invention causes a computer to execute the structural analysis method according to any one of the first to third inventions.

  A recording medium according to a fifth aspect is characterized in that the structural analysis program according to the fourth aspect is recorded in a computer-readable manner.

  According to the first to fifth inventions, when analyzing the elasto-plastic behavior of a steel beam at the time of a fire, it is not necessary to perform an iterative operation with a convergence determination. In addition, it is possible to reduce the computational load of the computer. In addition, when analyzing the elasto-plastic behavior of steel beams in the event of a fire, input work to model steel beams to be a continuum consisting of multiple elements is no longer necessary, reducing input work time. The analysis time can be shortened and the burden of human labor can be reduced.

  In particular, according to the second invention, it is possible to accurately analyze the elastic-plastic behavior of a steel beam having a stress-strain relationship other than the bilinear type, that is, a stress-strain relationship in accordance with the material characteristics of an actual steel material. Is possible.

(A) is a side view which shows typically the structure of a steel structure building, (b) is a side view which shows typically the state after the deformation | transformation at the time of the fire. It is a graph showing the temperature history of the axial force ratio N / N _y of steel beams. (A) is a side view which shows the model of the steel structure building which has the steel beam used as the analysis object in the structural analysis method based on this invention, (b) shows the model of the steel beam used as the analysis object It is a side view. It is front sectional drawing which shows the model of the steel beam used as the analysis object in the structural analysis method based on this invention. (A) is a graph which shows a bilinear type stress-strain relationship, (b) is a graph which shows a round house type stress-strain relationship. It is a graph which shows the temperature history of the compressive strain (epsilon) _c in the material end of steel beams. (A) is a graph which shows the temperature history of the axial force ratio of the steel beam which has a stress-strain relationship of a round house type, (b) is a graph which shows the temperature history of the compressive strain (epsilon) _c of the steel beam. is there. It is a block diagram showing roughly the composition of the structure analysis device concerning a 1st embodiment. It is a flowchart which shows the process sequence of the structural analysis program which the processor of the structural analysis apparatus which concerns on 1st Embodiment performs.

  Hereinafter, embodiments for carrying out a structural analysis method to which the present invention is applied will be described in detail with reference to the drawings.

  First, the general elasto-plastic behavior of steel beams during a fire will be described.

  When a fire occurs in the steel structure building 1, as shown in FIG. 1 (a), the heated steel beam 2 extends due to a temperature rise, and the extension is a steel pillar 3, other steel. By being restrained by a peripheral member such as a beam, an axial compression force due to thermal stress is generated on the steel beam 2. At this time, in addition to the thermal stress, a bending moment due to a long-term load loaded on the steel beam 2 acts on the steel beam 2.

In the temperature range S1 in which the steel beam 2 exhibits only elastic behavior, as shown in FIG. 2, the thermal stress increases linearly as the temperature rises unless an unstable phenomenon such as buckling occurs in the steel beam 2. It will be. When the temperature exceeds a certain temperature T ₀ , the material axis orthogonal cross section where the stress of the steel beam 2 is maximized becomes fully plastic. Hereinafter, the temperature T ₀ will be described as the elastic limit temperature T ₀ . In the example shown in FIG. 1B, the case where the material ends 21 and the span center 22 on both sides of the steel beam 2 are made completely plastic is shown.

After the elastic limit temperature T ₀ , the linear increase of the thermal stress according to the temperature rise stops. As the temperature rises, as shown in FIG. 1B, the plastic deformation in which the entire steel beam 2 bends downward, the cross section of the plastic region contracts in the material axis direction, or a part of the elastic region Plastic compressive deformation that becomes a plastic region proceeds. At the time of plastic deformation of the entire beam, the cross section of the all plastic state behaves as a plastic hinge.

Thereafter, until the temperature T ₁ at which the proof stress of the steel material starts to decrease, the thermal stress increases with strain hardening due to the temperature increase. In the example shown in FIG. 2, such behavior is shown in the temperature range S2.

After the temperature T ₁ is exceeded, the proof stress of the steel material begins to increase, and the thermal stress begins to attenuate due to the progress of the above-mentioned plastic compression deformation as the temperature rises. The amount of deformation due to plastic compressive deformation accompanying this temperature rise increases as the temperature rises, and along with this, the thermal stress continues to attenuate, and finally the thermal stress becomes zero. In the example shown in FIG. 2, such behavior is shown in the temperature range S3. This state shows a state in which the bending moment at the material end 21 due to the long-term load loaded on the steel beam 2 and the bending strength of the material axis orthogonal section at the material end 21 are balanced. The final state of the steel beam 2 is shown.

In order to analyze the elasto-plastic behavior of the steel beam 2 as described above, in the structural analysis method according to the present invention, an equation indicating the balance of the axial force and moment of the steel beam 2 and the material axis at the end of the material are shown. A quaternary simultaneous equation having an equation indicating the displacement conforming condition in the direction and an equation indicating the yield condition in the cross section perpendicular to the material axis is created, and the elasto-plastic behavior of the steel beam 2 is expressed based on the quaternary simultaneous equation. The temperature history of parameters such as the axial force N and the plastic region length l _p is calculated. The details will be further described below.

  First, a model used in the structural analysis method according to the present invention will be described.

  In this model, as shown in FIG. 3, a line due to a temperature rise of the steel beam 2 is caused by peripheral members such as a steel column 3 and other steel beams 2 arranged around the material end 21 of the steel beam 2. It is assumed that the expansion is constrained. At this time, it is assumed that the peripheral members such as the steel column 3 and the other steel beams 2 remain at room temperature (20 ° C.) regardless of the temperature rise of the steel beam 2 to be analyzed.

  Further, in this model, as shown in FIG. 3B, the length l of the half span from the span center 22 to the material end 21 of the steel beam 2 is set as the analysis target. The length l for this half span is assumed to be 3.5 (m) here. The steel beam 2 is assumed to have an H-shaped cross section as shown in FIG. The long-term load acting on the steel beam 2 is represented by a constant concentrated load P (N) acting on one point of the span center 22.

The boundary condition is assumed for the material end 21 of the steel beam 2 on the assumption that the bending deformation is constrained by the peripheral members and the rotational degree of freedom is constrained. This is for simplifying the later-described theoretical calculation by making the bending moment distribution in the material axis direction reversely symmetric with the span center 22 as the center of symmetry. Further, a boundary condition is assumed for the material end 21 of the steel beam 2 on the assumption that the horizontal deformation due to the linear expansion of the steel beam 2 is constrained by the elastic spring 41 connected to the material end 21. The elastic spring 41 functions as a representation of the constraining effect when the constrained portion by the peripheral member of the material end 21 of the steel beam 2 is considered to have elasticity with respect to the horizontal displacement of the steel beam 2. The spring stiffness K (N / m) of the elastic spring 41 is equivalent to the stiffness of the peripheral member that restrains the linear expansion of the steel beam 2, and is made of steel disposed above and below the material end of the steel beam 2. If the material end 31 of the column 3 is fixed, it can be evaluated by the following equation (3).

Next, consider a case where the temperature rises by a minute temperature increment dT from an arbitrary temperature T equal to or higher than the elastic limit temperature T ₀ using the above model.

First, the balance of the axial force and moment in the cross section perpendicular to the material axis of the material end 21 of the steel beam 2 will be considered. In this cross section perpendicular to the material axis, the tensile stress σ _t uniformly increases by the increment dσ _t in the upper cross section 25 a above the neutral shaft 25 due to the temperature increase of the increment dT, and the lower cross section 25 b below the neutral shaft 25. Suppose that the compressive stress σ _c increases uniformly by an increment dσ _c . In this case, since the increment dN of the axial force N in the cross section perpendicular to the material axis is balanced with the increase in the total stress in the cross section perpendicular to the material axis, (A0) will be derived. In this case, the following formula (B0) is derived as a formula indicating the balance of moments in the cross section perpendicular to the material axis. In the following formula (A0), A is the cross-sectional area (mm ² ) of the cross section perpendicular to the material axis of the steel beam 2.

Here, assuming that the stress-strain relationship of the steel beam 2 at the temperature T is expressed by the function f described in the following equation (4), the following is obtained by partially differentiating both sides of the equation (4). Equation (5) is derived.

Then, the equation (A0) indicating the balance of the force in the material axis direction in the material axis orthogonal section of the material end 21 is expressed by the following equation (A1) by combining the equations (B0) and (5). Will be. In the following formula (A1), ε _c is a compressive strain (−) in the lower cross section 25 _b below the neutral shaft 25 in the cross section perpendicular to the material axis of the material end 21 of the steel beam 2.

Similarly, the formula (B0) indicating the balance of moments in the cross section perpendicular to the material axis of the material end 21 is expressed by the following formula (B1) by combining the formula (5). In the following equation (B1), ε _t is a tensile strain (−) in the upper cross section 25a above the neutral shaft 25 in the cross section perpendicular to the material axis of the material end 21 of the steel beam 2.

Next, the displacement matching conditions in the material axis direction at the material end 21 of the steel beam 2 will be considered. As a factor affecting the displacement of the material end 21 of the steel beam 2 in the material axis direction when the temperature rises by the increment dT, the material in the elastic region 24 of the steel beam 2 as shown in FIG. Increment dδ _e in the axial deformation amount, Increment dδ _p in the axial direction in the plastic region 23 of the steel beam 2, Increment in deformation amount of the elastic spring 41 connected to the material end 21 of the steel beam 2 And an increase in deformation amount due to linear expansion of the entire steel beam 2. Since the increment of the deformation amount of the elastic spring 41 can be expressed by dN / K, and the increment of the deformation amount due to the linear expansion of the entire steel beam 2 can be expressed by α · l · dT. The following expression (C0) is derived as an expression indicating the condition. Α is a coefficient of linear expansion (1 / ° C.) of the steel material.

Here, the increment dδ _e of the deformation amount in the material axis direction in the elastic region 24 of the steel beam 2 can be evaluated by the following equation (6) from the Hooke's law. In addition, E (T) in the following formula (6) is an elastic coefficient (N / m ² ) of the steel beam 2 at the temperature T. Further, in the following formula (6), the increment dT of the temperature T and the increment dl _{p of} the plastic region length l _p are not taken into consideration, but even in this case, the deformation amount increment dδ _e of the elastic region 24 with respect to the temperature change. There since sufficiently smaller than the deformation amount increment d? _p plastic region 23, if sufficiently small increment dT during integral calculation will be described later, even ignoring the calculation error becomes very small.

Further, the increment dδ _p of the deformation amount in the material axis direction in the plastic region 23 of the steel beam 2 is the plasticity when the stress-strain relationship of the steel beam 2 is a bilinear type as shown in FIG. When the mean compressive strain in wood axis orthogonal cross section in the region 23 and epsilon _0, an increment of deformation of the steel beams 2 of wood end 21 side and the mid span 22 side each plastic region 23 (ε ₀ · l _p) Since it becomes what was added, it can evaluate by following formula (7).

At this time, the average strain increment of the material axis orthogonal cross section at the material end 21 of the steel beam 2 is the increment dε _t of the tensile strain ε _t of the upper cross section 25a and the increment dε of the compressive strain ε _c of the lower cross section 25b. mean value of _c can be expressed by _{(dε t + dε c) /} 2. Further, the average compressive strain increment dε ₀ over the entire length of the plastic region 23 in the material axis direction is the average of the material axis orthogonal cross section at the material end 21 when the stress-strain relationship of the steel beam 2 is bilinear. Since the value is 1/2 of the strain increment, the following equation (8) is derived.

Then, the following equation (9) is derived from the equations (7) and (8).

Then, the equation (C0) indicating the displacement matching condition in the material axis direction at the material end 21 of the steel beam 2 is expressed by the following equation (C1) by combining the equations (6) and (9). Will be. The following formula (C1) is based on the premise that the stress-strain relationship is bilinear.

Next, let us consider the yield condition in the cross-section perpendicular to the material axis that is plasticized just by the temperature increase of the increment dT from the temperature T. The material axis orthogonal cross section to be plasticized here is a cross section of the boundary 26 as shown in FIG. 3B, that is, the plastic region length l _p + dl _p from the beam end 21 or the span center 22. It is the cross section in the position. This yield condition can be expressed by the following equation (D1) from the axial force and bending moment acting on the cross section perpendicular to the material axis, the yield strength N _y (T), and the total plastic moment M _p (T). The yield strength N _y (T) represents the yield strength (N / m ² ) at the temperature T, and the total plastic moment M _p (T) represents the total plastic moment (N · m) at the temperature T.

Here, the above-mentioned formulas (A1), (B1), (C1), and (D1) are respectively the increment dN of the axial force N, the increment dl _{p of} the plastic region length l _p , and the material axis of the material end 21 It includes at least one of an increase dε _c of the compressive strain ε _c of the lower cross section 25b and an increase dε _t of the tensile strain ε _t of the upper cross section, and an increase dT of the temperature T. It is out. Therefore, if these four increments dN, dl _p , dε _c , and dε _t are unknown numbers, the same number of equations are established for the unknown numbers, so the above-described equations (A1) and (B1 ), (C1), and (D1) are solved as a set to solve the unknowns.

The solutions of the quaternary linear simultaneous equations thus obtained are expressed so as to include the temperature increment dT from the temperature T. From this, the integral range is set to the elastic limit temperature T ₀ or more, and the obtained solution is integrated with the temperature T, whereby the axial force N, the plastic region length l _p in the temperature range of the elastic limit temperature T ₀ or more, The temperature history of the compressive strain ε _c and the tensile strain ε _t is obtained. At this time, the initial value of the integral operation uses l _p as zero, and N, ε _c , and ε _t use elastic solutions at the elastic limit temperature T ₀ .

FIG. 2 shows the temperature history expressed by the axial force ratio N / N _y for the axial force N obtained by finding the solution of the above-mentioned quaternary linear simultaneous equations and then integrating the solution with the temperature T. FIG. 6 is a graph showing the temperature history of the obtained compression strain ε _c . 2 and 6, a plurality of steel beams having different strain hardening coefficient ratios e when the ratio of the strain hardening coefficient and the elastic coefficient is the strain hardening coefficient ratio e (= strain hardening coefficient / elastic coefficient). 2 shows a temperature history.

  2 and 6, the elastic-plastic behavior of the steel beam in the event of a fire under the same material conditions as the elastic modulus, yield strength, total plastic moment, etc., and the computer that performs the analysis process. The result when analyzing by the finite element method (FEM) is also shown. Thus, the analysis result by the structural analysis method according to the present invention and the analysis result by the finite element method are almost the same, and it can be understood that the elastic-plastic behavior of the steel beam 2 can be analyzed with high accuracy.

  In addition, as mentioned above, Formula (C1) presupposes the case where the stress-strain relationship of the steel beam 2 is a bilinear type. In this case, since the rigidity in the material axis direction in the plastic region 23 is constant, the above equation (8) is established. On the other hand, as shown in FIG. 5B, when the stress-strain relationship of the steel beam is a round house type, the rigidity in the material axis direction in the plastic region 23 changes. In the above equation (8), the average strain of the cross section perpendicular to the material axis at the material end 21 of the steel beam 2 is underestimated.

Therefore, in order to solve such a problem, in the present invention, the increment dδ _p of the deformation amount in the material axis direction in the plastic region 23 of the steel beam 2 is used by using γ represented by the following formula (E1). The following evaluated expression (C2) is used as an expression indicating the displacement matching condition. Note that E _t (ε, T) in the following formula (E1) is the tangential stiffness of the stress-strain relationship of the steel beam 2 at the time of strain ε and temperature T, and E _t (ε _c , T) is This is expressed as tangential rigidity at the time of strain ε _c as shown in FIG. Further, E ₀ is an average strain hardening coefficient. As shown in FIG. 5B, when the strain at the point where strain hardening starts in the stress-strain relationship of the steel beam 2 is ε _s , It is represented by the formula (E2). Formula (E1) is, as to represent the average tangent stiffness in the plastic region 23 at E _0, the E ₀ and tangent stiffness E _t in wood axis perpendicular cross-section of the steel beam 2 of wood end 21 (epsilon _c , T).

FIG. 7A shows a quaternary primary with the above-described formulas (A1), (B1), (C2), and (D1) as a set for the steel beam 2 having a round house type stress-strain relationship. FIG. 4 is a graph showing a temperature history expressed by an axial force ratio N / Ny for an axial force N obtained by integrating the solution with a temperature T after obtaining a solution of simultaneous equations, and (b) is obtained. 6 is a graph showing the temperature history of the compression strain ε _c . In FIG. 7, the elasto-plastic behavior of the steel beam 2 in the event of a fire is finite under the same material conditions such as elastic modulus, yield strength, total plastic moment, and the computer that performs the analysis process. The results when analyzed by the element method (FEM) are also shown.

  Thus, even when the elastic-plastic behavior of a steel beam having a round house type stress-strain relationship is analyzed by the structural analysis method according to the present invention, the analysis result and the analysis result by the finite element method are almost identical. Therefore, it can be understood that the elasto-plastic behavior of the steel beam 2 can be analyzed with high accuracy by using the equation evaluated using γ represented by the above equation (E1).

Further, regarding the method of obtaining the elastic limit temperature T ₀ when the material axis 21 of the steel beam 2 and the material axis orthogonal cross section of the span center 22 are fully plasticized, this is the material end 21 of the steel beam 2. And it is calculated | required from the yield conditions when the material axis orthogonal cross section of the span center 22 yields. Specifically, since the yield condition is expressed by the following equation (10), the elastic limit temperature T ₀ can be obtained by solving the equation (10).

Explaining in detail the basis from which the above equation (10) is derived, originally, the yield condition when the cross section perpendicular to the axis of the bar subjected to the compression axial force and bending moment yields is the following equation (11) ) Is known. Here, M is a bending moment acting on the yielding material axis orthogonal cross section, and is represented by 0.5 × P × l under the conditions shown in FIG.

Here, when the hardness of the elastic spring 41 is infinite, the axial force N caused by the thermal stress acting on the material end 21 and the span center 22 of the steel beam 2 is expressed by the following formula (12). Works. On the other hand, when the hardness of the elastic spring 41 is finite, the axial force N is smaller than that expressed by the equation (12), and the axial force N is calculated by the following equation (13). It is known to be represented. By combining this equation (13) and equation (11), the above equation (10) is derived.

  Next, a structural analysis apparatus suitable for executing the structural analysis method according to the first embodiment of the present invention will be described.

  As shown in FIG. 8, the structural analysis device 5 stores an input unit 51 for an operator to input various information, a processor 52 for controlling the operation of the entire structural analysis device 5, and various information. The storage unit 53, an output unit 54 for outputting various types of information, and an external storage unit 55 for storing various types of information in the recording medium 56 are configured as a computer. The components such as the input unit 51 and the processor 52 of the structural analysis device 5 are connected to each other by a data bus 57.

  The input unit 51 includes a keyboard, a mouse, and the like. The processor 52 includes a CPU and the like. The storage unit 53 includes a ROM, a RAM, a hard disk, and the like. The output unit 54 includes a display, a printer, and the like.

  The external storage device 55 is configured to be loaded with a recording medium 56. The external storage device 55 is configured to be able to read and write information with respect to the recording medium 56 when the recording medium 56 is loaded. The recording medium 56 is composed of a CD-ROM, MO, DVD or the like.

  In the first embodiment, it is assumed that a structure analysis program 58 for realizing the above-described structure analysis method is recorded on a recording medium 56 as a CD-ROM. At the time of structural analysis, after the structural analysis program 58 is read from the recording medium 56, the structural analysis program 58 is executed by the processor 52.

  FIG. 9 is a flowchart showing the processing procedure of the structural analysis program executed by the processor 52.

First, in step S1, initial parameters necessary for executing the above structural analysis method are set. As the initial parameters, for example, the cross-sectional area A, the linear expansion coefficient α, the elastic coefficient E (T), the yield strength N _y (T), the total plastic moment M _p (T), the stress − Examples include material conditions such as strain relation f (ε, T), boundary conditions such as rigidity K of elastic spring 41, long-term load P, secondary moment of inertia Ic of steel column 3, floor height h, total number n, and the like. . This material condition is set according to the material of the steel beam to be analyzed, and a known value or a value obtained by an experiment is set. The boundary condition is set as appropriate according to the steel structure building 1 to be analyzed. As the initial parameters, those stored in advance in the storage unit 53 or those input by an operator operating the input unit 51 are used.

Next, in step S2, the elastic limit temperature T ₀ of the steel beam 2 is calculated. The elastic limit temperature T ₀ is calculated based on, for example, the equation (8).

  Next, in step S3, the above-described quaternary simultaneous equations are created. The quaternary simultaneous equations, for example, use the equations (A1), (B1), (C1), and (D1) to analyze the elastoplastic behavior of the steel beam 2 having a bilinear stress-strain relationship. In analyzing the elastoplastic behavior of the steel beam 2 having a house-type stress-strain relationship, the equation (C2) is used instead of the equation (C1). The parameters set in step S1 are used for parameters other than the unknown number of the quaternary simultaneous equations and the increment dT.

Next, in step S4, by solving the quaternary simultaneous equation created in the previous step S3, an unknown solution of the quaternary simultaneous equation is calculated. The unknown here means the increment dN of the axial force N, the increment dl _{p of} the plastic region length l _p , and the compressive strain ε of the lower cross section 25b in the cross section perpendicular to the material axis of the material end 21 of the steel beam 2. _It means the increment dε _{c of c} and the increment dε _t of the tensile strain ε _t of the upper cross section 25a.

Next, in step S5, the solution of the quaternary simultaneous equations calculated in the previous step S4, that is, the increment dN, the increment dl _p , the increment dε _c , and the increment dε _t are set to a temperature _equal to or greater than T _0. By integrating with T, the temperature history of the axial force N, the plastic region length l _p , the compressive strain ε _c , and the tensile strain ε _t is calculated. The temperature history of these parameters is stored in the storage unit 53 and the recording medium 56. Moreover, the temperature history of these parameters is output by the output unit 54 in the next step S6 as necessary. Output by the output unit 54 is executed by display on a display, printing by a printer, or the like.

  According to the above, it is possible to shorten the analysis time by shortening the calculation time because it is not necessary to perform an iterative calculation with convergence judgment in analyzing the elastic-plastic behavior of the steel beam 2 at the time of fire. In addition, it is possible to reduce the computational load of the computer. In addition, when analyzing the elasto-plastic behavior of the steel beam 2 in the event of a fire, the input work of modeling the steel beam 2 so as to be a continuum consisting of a plurality of elements becomes unnecessary. By shortening, it becomes possible to shorten the analysis time and to reduce the burden of human labor.

  The results of studying how much the analysis time is shortened by applying the present invention will be described. A steel beam having the same material conditions and boundary conditions when the conventional finite element method is used as a means for analyzing the elastoplastic behavior of the steel beam 2 and when the structural analysis means according to the present invention is used. The time required to analyze the elastoplastic behavior of 2 was measured. As a result, after the input work such as setting initial parameters is completed, the time required until the analysis result is actually calculated is about 20 seconds in the conventional structural analysis by the finite element method. On the other hand, in the structural analysis method according to the present invention, the analysis result was calculated in about 1.3 seconds. In this example, by applying the structural analysis method according to the present invention, the analysis time can be reduced by about 15 times compared to the case of using the conventional finite element method.

  As mentioned above, although the example of embodiment of this invention was demonstrated in detail, all the embodiment mentioned above showed only the example of actualization in implementing this invention, and these are the technical aspects of this invention. The range should not be construed as limiting.

DESCRIPTION OF SYMBOLS 1 Steel structure building 2 Steel beam 3 Steel pillar 5 Structural analysis apparatus 21 Material end 22 Span center 23 Plastic area 24 Elastic area 25 Neutral shaft 26 Boundary 31 Material end 41 Elastic spring 51 Input part 52 Processor 53 Memory | storage part 54 Output 55 External storage device 56 Recording medium 57 Data bus 58 Structure analysis program

Claims (5)

In the structural analysis method to analyze the elasto-plastic behavior of steel beams erected between steel columns of steel structure buildings at the time of fire,
The axial force N of the steel beam, the plastic region length l _p from the end of the steel beam and the center of the span, and the compressive strain of the lower cross section of the steel beam at the material axis orthogonal cross section With respect to ε _c and the tensile strain ε _t of the upper cross section, when the steel beams are heated from the temperature T by the increment dT, the increments are dN, dl _p , dε _c , dε _t , respectively. an axial force at a cross-section perpendicular to the axis of the steel beam at the end of the steel beam as an equation including any one or more of dN, increment dl _p , increment dε _c, and increment dε _t as an unknown and the increment dT The equation showing the balance of each moment, the equation showing the displacement matching condition in the material axis direction at the material end of the steel beam, and the yield condition in the cross section perpendicular to the material axis of the steel beam plasticized by the temperature rise. Create a quaternary simultaneous equation with
Calculating the solutions for each of the unknowns by solving the quaternary simultaneous equations,
Integrating the solution of the quaternary simultaneous equations at the temperature T, the temperature history of the axial force N, the plastic region length l _p , the compressive strain ε _c and the tensile strain ε _t is calculated. Method. For the increment of the deformation amount in the material axis direction of the plastic region when the temperature of the steel beam is increased by an increment dT from the temperature T, an expression evaluated using γ expressed by the following expression (1) The structural analysis method according to claim 1, wherein the structural analysis method is used as an expression indicating the displacement matching condition.

Here, E _t (ε, T) in equation (1) is the tangential stiffness of the stress-strain relationship of the steel beam at strain ε and temperature T, and E ₀ is the stress-strain relationship of the steel beam. Is expressed by the following formula (2), where ε _s is the strain at the point where strain hardening starts.

Elastic limit when the material end of the steel beam and the material axis orthogonal cross section at the center of the span are fully plasticized based on the expression indicating the yield condition at the material axis orthogonal cross section of the steel beam Calculate the temperature T ₀ ,
The temperature history of the axial force N, the plastic region length l _p , the compressive strain ε _c and the tensile strain ε _t is calculated by integrating the solution of the quaternary simultaneous equations at the temperature T in the range of the temperature T ₀ or higher. The structural analysis method according to claim 1 or 2, wherein: A structural analysis program that causes a computer to execute the structural analysis method according to claim 1. A recording medium in which the structural analysis program according to claim 4 is recorded in a computer-readable manner.

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