JP2002146718A - Earthquake-resistant design method for structure - Google Patents

Abstract

PROBLEM TO BE SOLVED: To efficiently design a structure which can be constructed most economically. SOLUTION: In an earthquake-resistant design method, the components of a structure are abstracted in the first step; for each component, structural specifications required for non-linear dynamic response analysis is determined for each component, and an analytic model is set. In the second step, among the structural specifications, the design variables are set in such a manner that the characteristic values showing the non-linear behavior can be contained. In the third step, the design variable is set as a parameter, and an objective function expressing the construction cost is set for each component. In the fourth step, a plurality of levels are set for the design variables in response to the size of earthquake scale. In the fifth step, by using the orthogonal table of experimental planning method, non-linear dynamic response analysis treatment in the combination of a plurality of levels of design variables is performed, in the sixth step, by using the results of the analytic treatment, an estimating equation of dynamic behavior to be considered in the restricting conditions is prepared based on the experimental planning method, and the restricting conditions for the dynamic behavior is set. Thereafter, in the seventh step, the optimization problem is turned into formulation and the optimization value is determined based on the mathematical programming method.

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a seismic design method for civil engineering and architectural structures such as bridges, and more particularly to a seismic design method for structures that can be constructed more economically while satisfying seismic standards. Things.

[0002]

2. Description of the Related Art When designing a structure such as a bridge, a design procedure which is generally performed by a designer first includes, based on the experience of the designer himself, an appropriate structure such as a size of a member. After determining the specifications (design variables), a prototype experiment and simulation (structural analysis) are performed based on these structural specifications.
Calculate the proof stress and the behavior of the structure, etc., check whether these are within the allowable values of the conditions that should be satisfied in the design, and if they do not meet the allowable values, With a slight modification, the above procedure is repeated to determine structural specifications satisfying the tolerance.

In such a design method, an experienced designer can carry out the design in several repetitions, but a less experienced designer can use a general-purpose software, for example, to design. Even so, a large amount of repetitive calculations and labor are often required.

[0004] Incidentally, the basic procedure in the seismic design method of a structure is the same as the above-mentioned method. For example, the design procedure for the seismic design of a road bridge is indicated in the road bridge specification book. I have.

[0005] However, such a conventional seismic design method has the following problems.

[0006]

That is, the seismic design method described in the current specification for road bridges may involve complicated analysis including highly nonlinear, for example, dynamic analysis, etc. Takes a very long time.

Further, in the seismic design method shown in the current specification for road bridges, many constraints (permissible values) for securing the seismic safety are set, and these constraints are satisfied. The relationship between the value of the design variable and the construction cost is not taken into account, and therefore, there is a problem that uneconomical design is apt to be made in order to ensure earthquake-resistant safety.

The present invention has been made in view of such conventional problems, and an object of the present invention is to provide a structure which can be constructed most economically by repeatedly calculating a structure for securing earthquake resistance. An object of the present invention is to provide a seismic design method that can be designed efficiently with a small number of times.

[0009]

In order to achieve the above object, the present invention provides a method for seismic design of civil engineering and architectural structures, such as bridges, by extracting components of the structures, and for each of the components. A first step of determining the structural parameters required for the nonlinear dynamic response analysis and setting a nonlinear dynamic response analysis model; A second step of setting a design variable so as to be included, a third step of setting an objective function representing a construction cost for each of the components using the design variable as a parameter, and an earthquake magnitude for the nonlinear behavior. A fourth step of setting a plurality of levels for the design variable in accordance with the magnitude of the design variable; a fifth step of performing the non-linear dynamic response analysis processing on the plurality of level combinations of the design variables; Using the results of response analysis process, sixth prepared in accordance with the estimated equation of the dynamic behavior should be considered in constraints experimental design, sets the constraints of the dynamic behavior
And a seventh step of formulating an optimization problem and obtaining an improved solution of the design variables based on mathematical programming, and repeating the steps of the fourth to seventh steps as necessary. Was decided. According to the seismic design method for a structure thus configured, in the third step, an objective function representing a construction cost for each component is set using design variables as parameters, and the objective function is included in a form including the objective function. Since the optimization problem is formulated and the optimum value is obtained based on the experimental design method and the mathematical programming method, the obtained optimum value reflects the construction cost. In the sixth step, using the results of the non-linear dynamic response analysis processing, an expression for estimating the dynamic behavior to be considered in the constraint condition is created based on the experimental design method, and in the seventh step, the optimization problem is solved. Since the optimal value is obtained based on the mathematical programming based on the formulation, the response surface of the design variable can be approximated by using a small nonlinear dynamic response analysis result, so that the optimal solution can be determined extremely efficiently. Furthermore, in the formulation of the optimization problem, the objective function and constraints are formulated by not considering the dimensions of each structural element as design variables, but considering the nonlinear history characteristics of each structural element as design variables. Optimization can be performed efficiently. In the fourth step, the levels are set to three levels, and a combination of each level of the design variable can be determined by an orthogonal table of an experimental design. The structure may be a bridge having a seismic isolation structure, and the constituent elements may be a superstructure, a seismic isolation bearing of each pier, a pier, and a foundation. The design variable is a characteristic value indicating the nonlinear behavior of the seismic isolation bearing, a shear spring constant is selected, the yield load is set to the design variable of the seismic isolation bearing, and the characteristic indicating the nonlinear behavior of the pier As a value, a bending stiffness can be selected and its yield bending moment can be set to the design variable of the pier. The seventh step is a step of solving the optimization problem by a dual method using a Lagrangian function to obtain an improved solution of the design variable; and a construction cost of the structure and the design variable of the design variable. Determining that the value has converged to a constant value;
Determining whether the obtained improved solution is within the standard.

[0010]

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS Preferred embodiments of the present invention will be described below in detail with reference to the accompanying drawings. 1 to 5 show an embodiment of a method for designing a structure for earthquake resistance according to the present invention.

The seismic design method shown in these figures exemplifies a case where the present invention is applied to a base-isolated bridge 10 as a structure. The seismic design method of the present embodiment is executed according to the procedure shown in FIG.

In the design procedure shown in FIG. 1, for example, FIG.
In the first step, components of the seismic isolation bridge (structure) 10 are extracted, and the components required for the nonlinear dynamic response analysis are extracted for each component in a first step. Determine the structural specifications and set up a nonlinear dynamic response analysis model.

In this case, the seismic isolation bridge 10 shown in FIG.
An upper structure 12 such as a bridge girder, a seismic isolation bearing 14 in which laminated rubber is sandwiched between steel plates, and six (P1
To P6) and the foundation 18 of each pier 16.

The foundation 18 has a plurality of piles cast in the ground, and the seismic isolation bearings 14 are interposed between the superstructure 12 and the piers 16 respectively. In the seismic isolation bridge 10 having such a structure, the superstructure 12, the seismic isolation bearing 14, the pier 1
6, four parts of the foundation 18 are extracted as components.

Incidentally, in the base-isolated bridge 10 having such a structure, when a large-scale earthquake occurs, the base-isolated bearing 14 and the pier 16 are allowed to undergo plastic deformation, and the foundation 18 is kept within the elastic limit. Design under optimal conditions.

In the first step, a nonlinear dynamic response analysis model is set. As the structural specifications required for the nonlinear dynamic response analysis of the base-isolated bridge 10, the upper structure 12 includes a plurality of bridges. Considering the axial masses ms1 to ms6 and the damping constant hs.

Further, as for the seismic isolation bearing 14 of each pier 16, shear spring constants kb1 to kb6 exhibiting non-linear behavior,
For each pier 16, a plurality of bridge axis direction masses mp11 to
Considering mp61, mp12 to mp62, mp13 to mp63, nonlinear bending stiffness kp1 to kp6, and damping constant hp.

Further, regarding the foundation 18 of each pier 16, the mass in the bridge axis direction and the rotational masses mf 1 to mf 6, the horizontal spring constants kh 1 to kh 6, the rotational spring constants kθ 1 to kθ 6, and the damping constant hf for each are taken into consideration. 3
A non-linear dynamic response analysis model of the base-isolated bridge 10 as shown in FIG.

In the next second step, the above-described structural parameters are set so as to include characteristic values indicating the non-linear behavior, and these are considered as design variables.

In general, in an optimal design problem as an object of the present invention, structural parameters for analysis are often set as design variables. In the case of the present embodiment, however, characteristic values indicating nonlinear behavior are set. As shown in FIG. 4, the shear spring constants kb1 to kb6 of the seismic isolation bearing 14 and the bending stiffnesses Kp1 to Kp6 of the pier 16 as shown in FIG. 5 are selected, and these characteristic values are designed. Adopted as a variable.

More specifically, as for the seismic isolation bearing 14, the yield load Qd at the point A shown in FIG.
About the yield bending moment M at the point B shown in FIG.
y is a design variable.

As for the foundation 18, the pile arrangement is assumed such that the horizontal spring constant Kh and the rotation spring constant Kθ are dependent on each other, and the horizontal spring constant Kh is adopted as a design variable.

In the next third step, the design variables set in the second step are used as parameters, and the purpose is to express the construction cost of each of the components, that is, the seismic isolation bearing 14, the pier 16 and the foundation 18 of the seismic isolation bridge 10. Set the function W.

In this case, the objective function W is
Each construction cost of the pier 16 and the foundation 18 is expressed as a function W (Qd, My, Kh) of the design variable determined in the second step. In this case, the value of the coefficient of the objective function W for each design variable (Qd, My, Kh) is
y, Kh) is a coefficient that gives the minimum construction cost.

In the next fourth step, a plurality of levels are set as design variables in response to the magnitude of the earthquake scale with respect to the above-described nonlinear behavior. In the case of this embodiment, the level is 3
The levels (the first level is the lower limit, the second level is the intermediate value, the third level is the upper limit), and the design variables (Qd, My, K
The combination with h) was determined by an orthogonal table of the design of experiment.

That is, in the fourth step, the non-linear dynamic behavior of the analytical model set in the first step with respect to large-scale earthquake motion is converted into the design variables (Qd,
(My, Kh).

In this case, each design variable (Qd, My, K
The combination of the three levels of h) is determined according to the orthogonal table of the experimental design described later, and the number of analysis required at this time is 9 when the number of design variables is four.
In the case of 13 times, it becomes 27 times.

By the way, if the basic model of the analysis is determined as shown in FIG. 3, only the value of the design variable of interest at the time of the analysis is changed, so that the analysis in a short time is required. Will be possible. The design variables (Qd, My,
When the level value of Kh) is set, the value of the first level (the lower limit value of the design variable) is given, for example, at the seismic intensity method level where the magnitude of the earthquake is relatively small. Will be possible.

This is because in the practical design, the cross-sectional data is less determined by the seismic intensity method, which is a conventional design technique, and the cross-sectional data is determined by the seismic holding horizontal strength method assuming a larger earthquake motion. It can be in line with the fact that the source is increasingly determined.

In the next fifth step, the design variables (Qd,
My, Kh) is subjected to nonlinear dynamic response analysis processing in a combination of a plurality of levels. In this step, the nonlinear dynamic behavior of the analytical model set in the first step with respect to the large-scale earthquake motion is converted into the design variable (Q
(d, My, Kh) is analyzed for the combination of the three levels of the design variables set in (d, My, Kh).

In this case, as the constraint condition of the nonlinear dynamic behavior of interest, the maximum response horizontal displacement (δ
b), maximum response horizontal displacement (δp) of the pier 16, foundation 18
The maximum response horizontal displacement (δf) is considered.

In the next sixth step, using the result of the nonlinear dynamic response analysis processing in the fifth step, an expression for estimating the dynamic behavior to be considered in the constraint condition is created based on the experimental design method. Set constraints on behavior.

That is, in this step, using the nonlinear dynamic analysis result of the fifth step, the dynamic behavior to be considered in the constraint condition, that is, the Chebyshev orthogonal polynomial,
Equations for estimating the maximum response horizontal displacement (δb) of the base-isolated bearing 14, the maximum response horizontal displacement (δp) of the pier 16, and the maximum response horizontal displacement (δf) of the foundation 18 as functions of the design variables (Qd, My, Kh). Set.

In the following seventh step, the optimization problem is formulated by the above steps, and based on mathematical programming,
The optimum value is obtained by repeating the process of obtaining the improved solution of the design variable as needed.

In the seventh step, the optimization problem is solved by a dual method using a Lagrange function to obtain improved solutions of the design variables (Qd, My, Kh). The method includes a step 7c for determining whether or not the value converges to a value and a step 7d for determining whether or not the obtained improved solution is within a standard.

In step 7a, the design variable (Qd, Qd) is calculated using the objective function W (Qd, My, Kh) set in step 3 and the constraint equations δb, δp, δf obtained in steps 5 and 6. , My, Kh) is formulated as an optimization problem.

In step 7b, the design variables (Qd, M
(y, Kh) are determined as improved solutions Qd ', My', Kh '. This is because the optimization problem set in the step 7a is approximated to a convex approximation design problem of a variable separation type, which is solved by a dual method using a Lagrangian function to improve the design variable Q
Determine the improved solutions of d ′, My ′, Kh ′ and the objective function W. In this case, in order to maintain the accuracy of the obtained improved solution, pharmaceutical conditions regarding the move limit of the design variable are also considered.

In step 7c, the convergence of the design variables and the improved value of the objective function obtained in step 7b is determined.
If the convergence condition is satisfied, the process proceeds to the next step 7d. If the convergence condition is not satisfied, the process returns to step 7b to determine another improved solution.

In the step 7d, the convergence solution obtained in the step 7c is set to 3 of the design variables set in the step 4.
It is determined whether the value is within the range of the level value. When the convergence solution (improved solution) is within the set level region, the optimal solution has been obtained within the region of the dynamic behavior estimation formula, and the procedure ends.

On the other hand, if the convergence solution (improved solution) is out of the set level region, the process returns to the fourth step, and three levels of new design variables are set with the improved solution as the median value. By repeating the steps, an optimal solution can be determined.

The seismic isolation bridge 10 constructed as described above is now described.
According to the seismic design method described above, in the third step, the objective function W representing the construction cost of the seismic isolation bearing 14, the pier 16, and the foundation 18 of each component, that is, the seismic isolation bridge 10, is set using the design variables as parameters. The optimization problem is formulated in a form including the objective function W, and the optimal value is calculated based on the experimental design method and the mathematical programming method. It will be reflected.

Further, in the sixth step, based on the experimental design method, it is possible to estimate the response surface relating to the design variable of the nonlinear behavior of the bridge using a small amount of analysis results.
In the step, the optimization problem is formulated and the optimal value is obtained based on the mathematical programming, so that the optimal solution can be determined very efficiently.

Further, in formulating the optimization problem, the objective function and the constraint conditions are formulated by considering the non-linear hysteresis characteristics of each structural element as design variables without considering the dimensions of each structural element as design variables. Therefore, optimization can be performed efficiently.

Next, a more specific design method of the present invention will be described. This specific example is a case where the present invention is applied to the seismic design of the base-isolated bridge 10 having the structure shown in FIG. Setting of analysis model (corresponding to the first step in FIG. 1) In this specific example, the base-isolated bridge 10 has a bridge length of 220 m (40 m + 40 m).
m + 60m + 40m + 40m), 5 with a total width of 12.0m
It is a continuous span steel girder bridge with six reinforced concrete piers 16 (P1 to P6) or (i = 1, 2,..., 6) and a cast-in-place pile foundation 18 (i = 1, 2,..., 6). It is assumed that it is supported.

The ground conditions of the foundation 18, the materials used for the lower structure such as piles, and the vertical load from the upper structure 12 were set as shown in FIG. 6 and Table 1, respectively.

[0046]

[Table 1]

In the seismic isolation pier 10 having such a configuration,
As described in the first step, when the modeling is performed by focusing on the ground motion in the bridge axis direction, an analysis model shown in FIG. 3 is obtained.

Here, the codes shown in FIG.
ij and mfi are the mass in the bridge axis direction of the upper structure 12, the mass in the bridge axis direction of the pier 16 (i), the mass in the bridge axis direction of the footing, and the rotating mass, respectively.

Also, kbi, kpi, khi and kθi
Are the nonlinear shear spring constant of the seismic isolation bearing 14 of the pier 16 (i), the nonlinear bending stiffness of the pier 16 (i), the horizontal spring constant of the foundation ground of the pier 16 (i), and the rotating spring constant, respectively.

Further, hs, hp, and hf are a damping constant for the horizontal vibration of the upper structure 12, a damping constant for the horizontal vibration of the pier 16, and a damping constant for the horizontal and rotational vibrations of the foundation structure, respectively.

In this specific example, the damping constant of the upper structure 12 is set to hs = 0.02 in consideration of the damping in the elastic region, and the damping constant of the pier 16 is set to hp = 0.02 since the nonlinear hysteresis model is used. The damping constant of the foundation 18 was assumed to be hf = 0.1 in consideration of the dissipation damping.

(2) Setting of design variables (corresponding to the second step in FIG. 1) Setting of non-linear hysteresis characteristic of seismic isolation bearing 14 and design variable Qd FIG. It shows an approximate relationship between the horizontal load H and the horizontal displacement U, which is generally used in nonlinear time history response analysis.

When the dynamic characteristics of the seismic isolation bearing 14 are to be defined by one typical parameter, various characteristic values can be selected. The parameters governing the amount of energy absorption include the yield load. Qd and the secondary rigidity k _b2 are important.

In a laminated rubber bearing (LRB) containing lead plugs used in an actual seismic isolation bridge, the primary rigidity k _b1 of the seismic isolation bearing is determined by the magnitude of the dead load reaction force (normal force) acting thereon. Is determined irrespective of the magnitude of the horizontal yield load Qd, and the ratio between the primary stiffness k _b1 and the secondary stiffness k _b2 is 6.5;

Further, in consideration of the fact that if Qd is given, a corresponding production cost is required, the yield load Qd of the seismic isolation bearing 14 is used as a design variable in this specific example. Therefore, in the subsequent nonlinear time history response analysis, if the value of Qd is given, the corresponding bilinear model in FIG. 7 is necessarily defined.

The nonlinear history characteristic of the pier 16 and the design variable M
Setting of y The non-linear hysteresis characteristic of the reinforced concrete pier 16 is assumed to represent the non-linear hysteresis characteristic in the plastic hinge section, assuming a case where a main plastic hinge occurs at the base.

As a nonlinear hysteresis characteristic of the reinforced concrete bridge pier 16, a Takeda model of a reduced rigidity type as shown in FIG. 8, which is generally used in a nonlinear time history response analysis, is used. When the dynamic characteristics of the pier 16 are to be defined by one typical parameter, the yield bending moment My is an important factor as a parameter that governs the amount of energy absorption as in the case of the seismic isolation bearing 14.

Here, My and its hysteresis behavior are described by the pier 1
6, the cross-sectional shape is given as a constant value from the viewpoint of workability and economy, and My and the non-linear hysteretic behavior are governed by changes in the reinforcing bar amount. It was taken.

When the cross-sectional shape is constant, the yield bending moment My and the rotation angle θy at the time of yield change depending on the amount of rebar, but My and θy have a one-to-one correspondence. Can be defined, and the corresponding construction cost can be clearly expressed as a function of My.

For this reason, in this specific example, the yielding bending moment My is used as a pier design variable as a symbolized value that defines the nonlinear behavior of the pier 16 and the objective function value.

The dynamic spring constant of the foundation-ground system and the design variable kh The dynamic spring constant of the foundation-ground is determined by the ground conditions of the bridge and the shape of the foundation. As a model of the nonlinear time history response analysis, it is possible to directly use the nonlinear history characteristics of the foundation body and the nonlinear history characteristics of the ground, but in this specific example, it is assumed that plasticization does not occur in the foundation structure. Then, it was decided to express it with a linear spring model that considered only the plasticization of the ground.

The linear spring constant is a spring constant in consideration of the dynamic behavior, and is given as a horizontal spring and a rotary spring that give equivalent displacement on the underside of the footing. Here, the horizontal and rotation spring constants vary greatly depending on the arrangement of the piles and the shape of the piles, but in this specific example, the number of pile rows in the direction perpendicular to the bridge is constant, and the number of pile rows in the bridge axis direction is set at the minimum interval. It is assumed that the above two spring constants have a one-to-one correspondence by changing.

Under such conditions, once the horizontal spring constant is determined, the rotational spring constant can be uniquely determined as its dependent function, and the pile with the minimum construction cost for a certain horizontal spring constant Kh is obtained. It is only necessary to determine the diameter.
For this reason, in this specific example, the horizontal spring constant Kh is adopted as a design variable of the pile foundation as a symbolized value that defines the dynamic behavior and the objective function value of the pile foundation.

(3) Setting of the objective function W (corresponding to the third step in FIG. 1) Formulation of the objective function W In the optimal design problem in the present invention, the seismic isolation bearing 1
4, the sum of the construction costs of the pier 16 and the foundation 18 was defined as the objective function W. In this case, as described above, the seismic isolation bearing 1 of the pier 16 is used.
4 is a function of Qd,
The construction cost Wpi of No. 6 is expressed as a function of My, and the construction cost Wf of the foundation 18 of the pier 16 is expressed as a function of kh. The objective function W can be expressed by the following equation.

[0065]

(Equation 1)

Here, np: the radix of the pier 16 (= 6), Wbi: the production cost of the seismic isolation bearing 14 of the pier 16, Wpi: the construction cost of the pier 16, Wfi: the pier i
The construction cost of the foundation 18.

Next, a relational expression between Wbi, Wpi, Wfi and a design variable will be described. a. Relational expression between yield load Qdi of seismic isolation bearing 14 and manufacturing cost Wbi In this specific example, the weight of superstructure 12 is a constant value.
The vertical load acting on 6 has a constant value. FIG. 9 shows dead load reaction force Rdi = 4500 kN (i = 1, 6) and 5490 kN.
The graph which plotted the relationship between the yield load Qd of the base-isolated bearing 14 and the production cost Wbi in the case of (i = 2, 5) and 7150 kN (i = 3, 4) is shown.

As apparent from FIG. 9, the yield load Q
The relational expression between d and the production cost Wbi can be formulated by a simple linear expression, and the production cost increases as the dead load reaction force Rdi increases. This relational expression is expressed as follows. Wbi = 1.02 Qdi + 9000 (thousand yen) (i = 1, 6) Wbi = 1.84 Qdi + 9100 (thousand yen) (i = 2.5) Wbi = 4.08 Qdi + 9250 (thousand yen) (i = 3, 4) (4) )

B. Yield bending moment Myi of pier 16
In general, when the dead load reaction force from the superstructure 12 and the materials of concrete and reinforcing steel to be used are known, the yield bending moment Myi of the pier 16 is known.
And the construction cost Wpi vary depending on the cross-sectional dimension of the pier 16 and the amount of rebar, but in this specific example, it is assumed that the cross-sectional dimension is given as a constant value.

FIG. 10 shows the yield bending moment Myi and the construction obtained by changing the amount of reinforcing steel by setting the column width in the bridge axis direction to a constant value of 2.5 m and the column width in the direction perpendicular to the bridge axis to a constant value of 5.0 m. The relationship with the cost Wpi is shown.

In general, the relationship between the yield bending moment Myi and the construction cost Wpi differs depending on the magnitude of the vertical force (dead load reaction force) acting on the pier 16.
In the seismic isolation bridge 10 shown in FIG. 2, the influence was slight, so the relationship between the yield bending moment Myi and the construction cost Wpi shown in FIG. 10 was adopted for each pier 16. The relational expression in this case is shown below. Wpi = 0.056 Myi + 5250 (thousand yen) (5)

C. Relational expression between the horizontal spring constant Khi of the foundation and the ground system and the construction cost Wfi In general, when the ground conditions and the material used for the pile foundation are known, it is dominant in calculating the horizontal spring constant Khi of the foundation-ground system of the pier 16. Elements are
The shape of the pile and the arrangement of the pile.

In this specific example, since the focus is on the seismic duty in the bridge shelf direction, the pile arrangement of all the pile foundations in the direction perpendicular to the bridge axis is as follows.
With three rows, the relational expression between the pile diameter and the pile arrangement in the bridge axis direction and the construction cost Wfi was obtained.

FIG. 11 shows a pile diameter 100 generally used.
The graph which plotted the relationship between the horizontal spring constant Khi and the construction cost Wfi obtained by changing the number of pile rows in the bridge axis direction in the case of 0 mm, 1200 m, and 1500 mm to 2, 3, and 4 rows. I have.

As can be seen from FIG. 11, a simple linear expression can be used to formulate a relational expression for any pile diameter, and the seismic isolation pier 10 shown in FIG.
It was found that the relationship between the horizontal spring constant Khi and the construction cost Wfi in the case of m can be formulated by almost the same linear equation, and that the construction can be more economical than a pile foundation having a pile diameter of 1500 mm.

In this specific example, considering that the relationship between the horizontal spring constant Khi and the construction cost Wfi is the most discrete among the three variables, it is formulated as a relational expression covering the case of a pile diameter of 1000 mm and a pile diameter of 1200 mm. , Decided to adopt this. Horizontal spring constant Khi of foundation-ground system and construction cost Wfi
Is shown below. Wfi = 0.0063Khi (thousand yen) (i = 1, ‥, 6) (6)

(4) Method of Nonlinear Dynamic Response Analysis (corresponding to the fifth step in FIG. 1) In this specific example, nonlinear time history response analysis was performed by the Newmark-β method of the direct integration method. Here, β = 1/4,
Integration time interval is 0.01 seconds, convergence error is 0.000
1 or less. Note that the attenuation matrix is Rayle
In this specific example, the main mode is 1
Next mode and third mode were selected.

(5) Setting of constraints for nonlinear dynamic behavior (corresponding to the sixth step in FIG. 1) Formulation of constraints for nonlinear dynamic behavior This specific example is based on seismic isolation obtained by nonlinear time history response analysis. Focusing on the maximum response horizontal displacement of the pier 16 of the bearing 14 and the foundation 18, the constraints of the optimal design problem were formulated. The maximum response horizontal displacement of the seismic isolation bearing 14 of the pier 16 when receiving a large-scale earthquake (the maximum horizontal deformation of the seismic isolation bearing 14) is δbi,
Assuming that the maximum response horizontal displacement of the pier 16 (the maximum bending deformation in the horizontal direction of the pier 16) is δpi and the maximum response horizontal displacement of the foundation 18 of the pier 16 (the maximum horizontal displacement of the footing top) is δfi, the nonlinear dynamic The constraint equation for the dynamic behavior is given by the following equation. δbi−δbai ≦ 0 δpi−δpai ≦ 0 δfi−δfai ≦ 0 (i = 1, 2,..., 6) (7)

Here, δbai: seismic isolation bearing 1 of pier 16
4, the allowable horizontal displacement of the pier 16 and δfai: the allowable horizontal displacement of the foundation 18 of the pier 16. The method of setting the allowable horizontal displacement of the nonlinear dynamic behavior in the above equation is not fully understood at present, but here, the allowable displacement by the seismic holding capacity method during earthquakes shown in the specification for road bridges is described. Δba
The setting method of i, δbai, δfai will be described.

Setting of Allowable Horizontal Displacement δbai of Seismic Isolation Bearing 14 The allowable horizontal displacement δbai of the seismic isolation bearing 14 of the pier 16 is determined by setting the allowable shear strain 250% to the rubber thickness in accordance with the specification for road bridges.
Multiplied by. Here, the rubber thickness of the seismic isolation bearing 14 is
Although it depends on the value of the design variable Qdi, in this specific example,
From the previous design example, the dead load reaction force
The allowable horizontal displacement δbai was set using the rubber thickness as a guide when% is defined as the horizontal yield load Qdi of the seismic isolation bearing 14.

Setting of Allowable Horizontal Displacement δpai of Pier 16 The allowable horizontal displacement δbai of the pier 16 is calculated by the following equation according to the road specification. δbai = μai × δyi (8) where μai is the allowable plasticity of the pier 16 and δy
i is the yield displacement of the pier 16. The allowable plastic modulus μai of the pier 16 in Expression (8) is expressed by the following expression.

Μai = 1 + (δui−δyi) / αδyi (9) where δui is the ultimate displacement of the pier 16 and α is
It is a safety factor. The safety coefficient α is twice as large as that of a normal bridge in the case of the base-isolated bridge 10 as shown in the road bridge specification book.

Here, in this specific example, μai and δyi change as the design variable Myi changes, but in the base-isolated bridge 10 shown in FIG. 2, the allowable horizontal displacement μai × δyi
Is substantially constant, the allowable horizontal displacement δ of the pier 16 is
pai was set as a constant value regardless of the value of the design variable Myi.

Setting of Allowable Horizontal Displacement δfai of Foundation 18 The allowable horizontal displacement δfai of the foundation 18 of the pier 16 is set so as not to cause major damage or major non-linearity to the foundation structure, and also takes into consideration the safety of the entire bridge. Set.

In this example, the sum of the deformations at the position of the superstructure 12 is considered and the sum of the deformations of the seismic isolation bearing 14, the pier 16, and the foundation 18 is considered in consideration of the problem of girder collision during an earthquake. Is set to be about 50.0 cm or less.

(6) Introduction of an equation for estimating the maximum response horizontal displacement by the experimental design method (corresponding to the fourth and sixth steps in FIG. 1) Outline of the experimental design method It is a method to analyze the measured value statistically when it is necessary to clarify it, and to simulate the experimental value accurately using the result. Now, for example, assuming that a characteristic value z obtained by an experiment is affected by the factors of x1, x2,..., Xn, z = f (x1, x2,.
n).

In this case, each factor is changed to three levels (three different values), the characteristic value z in the combination of the levels of each factor is obtained by experiment, and an estimation formula of z is determined using these values. This method is called an n-factor three-level experiment.

In this example, the target of the seismic design is the seismic isolation bridge 1.
0, which is symmetrical in the bridge axis direction, the yield loads Qdl to Qd3 of the seismic isolation bearing 14, the yield bending moments My1 to My3 of the pier 16, and the spring constant Kh1 of the pile of the foundation 18
To Kh3, the maximum response horizontal displacements δbi, δp of the base-isolated bearing 14, pier 16, and foundation 18 obtained by the nonlinear time-history response analysis method.
i, δfi (hereinafter referred to as δmax = [δbi, δp
i, δfi] T), and an estimation formula of δmax was introduced using the results of an n-factor three-level experiment.

In this case, since the total number of factors is nine,
From known literature (Genichi Taguchi, Design of Experiments, published by Maruzen Co., Ltd.), 3 of each factor in the orthogonal table shown in Table 2 below
Analysis (experiment) was performed for 27 combinations of the two level levels (1, 2, 3), δmax was determined, and an estimation formula was introduced using the Chebyshev orthogonal polynomial shown in the following formula (10). y = b0 + b1 (A- A ') + b2 {(A-A') 2 - (a 2 -1) h 2/12} + ...... (10) here, A is variable, A 'is Average level, a
Is a level number, h is a level interval, and b0, b1, and b2 are coefficients of each order term.

[0090]

[Table 2]

The total number of combinations of all discrete levels of the nine factors is 19683. By using the above-described statistical method, the analysis value of only 27 discrete levels is used. , Can be accurately estimated.

In the experiment design method, when the number of design variables is four, the estimation formula can be introduced by nine kinds of analysis in the orthogonal table. Can be introduced by 27 kinds of analysis as in this specific example.

Setting of Initial Level Values Each design variable Qdl- of the analysis model used in this example is
First to third of Qd3, My1 to My3, Kh1 to Kh3
The initial values of the levels were set as shown in Table 3 below, and nonlinear time history analysis was performed on the combinations in Table 2 using the respective level values.

[0094]

[Table 3]

The initial level values shown in this table can theoretically be set at random so that the difference between the level values is constant, but in this specific example, the initial level value is The following settings were made with reference to the existing design examples.

A. Level value of the yield load Qdi of the seismic isolation bearing 14 The yield load Qdi of the seismic isolation bearing 14 was set within a range where the increase in the horizontal displacement at all times could be expected and the seismic isolation effect could be expected. In this specific example, attention is paid to the piers 16 at both ends having the smallest weight of the superstructure to be supported, and the 10% of the weight of the pier 16 is defined as the first level, and the 20% is defined as the second level. The one having a yield load of 30% was defined as a third level.

B. Yield bending moment Myi of pier 16
The level value of the yield bending moment Myi of the pier 16 was set within a range that is safe against seismic intensity level ground motion. In this specific example, the column width in the bridge axis direction is fixed at 2.5 m, the yield bending moment when two steps of reinforcing bars having a diameter of 25 mm are arranged is the first level, and two steps of reinforcing bars having a diameter of 29 mm are arranged. The yielding bending moment when the bar was reinforced was set to the second level, and the yielding bending moment when the reinforcing bar having a diameter of 32 mm was arranged in two steps was set to the third level.

C. The level value of the horizontal spring constant Khi of the foundation-soil system The level value of the horizontal spring constant Khi of the foundation-soil system is such that the number of piles in the direction perpendicular to the bridge axis is constant in three rows, and a pile of 1000 mm in diameter is a two-row bridge axis. The horizontal spring constant when arranged in the same direction was set to the first level, the horizontal spring constant when arranged in the same three rows was set to the second level, and the horizontal spring constant when arranged in the same four rows was set to the third level.

Introducing an equation for estimating the maximum response horizontal displacement The equation for estimating the maximum response horizontal displacement of the base-isolated bearing 14, the pier 16, and the foundation 18 was introduced by the method described above. As an example, a formula for estimating the maximum response displacement of the seismic isolation bearing 14 of the pier 16 (i = 1... 6) is shown below.

[0100]

(Equation 3)

Table 4 below shows a comparison between the analysis values and the estimated values obtained from the estimation formulas for the 27 combinations shown in Table 2. The error between the analytical value and the estimated value is
In each case, it was 5% or less, and it was confirmed that the response horizontal displacement could be accurately estimated.

[0102]

[Table 4]

(7) Formulation of optimization design problem and optimization method (corresponding to the seventh step in FIG. 1) Constraints on nonlinear dynamic behavior are formulated using the above-mentioned estimation formula, and objective function W is minimized. Design variables Qdi,
The optimization problem for determining Myi and Khi is formulated as follows.

[0104]

(Equation 4)

In this example, the dual method using the Lagrange function is employed as a method for solving the optimal design problem represented by the equation (12). Is known, for example,
"Tadaji Okubo, Kazuhiro Taniwaki, Dual Theory and Sub of Materials
Optimizing Design of Truss Structures by Optimization ", Transactions of the Japan Society of Civil Engineers, October 1984.

(8) Example of Optimal Design Large-Scale Earthquake Ground Motion Used for Design The optimal seismic design of the base-isolated bridge was performed by the method described above. The ground motion used in the design example is a standard acceleration waveform (type II ground motion-II
Seed ground-3, Hyogoken-Nanbu Earthquake, Osaka gas roofing supply station ground).

Allowable Horizontal Displacement As the allowable horizontal displacement of each structural element, the allowable horizontal displacement specified by the horizontal strength holding method at the time of earthquake was adopted. The allowable horizontal displacement δbai of the seismic isolation bearing 14 was calculated by multiplying the rubber thickness when the yield load of the seismic isolation bearing 14 by 10% of the dead load reaction force acting on each pier 16 by 250%. In the present design example, since the rubber thickness of each pier 16 was about 13.5 cm, the allowable horizontal displacement δbai was set to 34.0 cm.

The allowable horizontal displacement δpai of the pier 16 was calculated by multiplying the yield displacement of each pier 16 by the allowable plasticity factor. In the present design example, the allowable horizontal displacement δpai of each pier 16 was set to 14.0 cm, since it was confirmed that the influence of the difference in dead load reaction force was small and the value was almost constant.

The allowable horizontal displacement δfai of the foundation 18 is such that the sum of the deformations of the foundation 18, the pier 16 and the seismic isolation bearing 14 is 50.0.
cm or less. In this design example, δ
Considering the values of bai and δpai, the allowable horizontal displacement δfai of the foundation 18 of each pier 16 was set to 3.0 cm. This value is
This corresponds to twice the allowable horizontal displacement of 1.5 cm according to the seismic intensity method.

In the present design example, in order to see the change of the optimum value when the allowable horizontal displacement changes, the optimum design when the allowable horizontal displacement δpai of the pier 16 is set to 12.0 cm and 16.0 cm is also performed. The optimal solutions for each design condition were compared.

Comparison of optimal values of design variables Allowable horizontal displacement δbai of base-isolated bearing 14 is 34.0 cm,
The allowable horizontal displacement δfai of the foundation 18 is 3.0 cm,
And the allowable horizontal displacement δpai of the pier 16 is 16.0 cm,
Table 5 shows a comparison between the optimum values of the design variables Qdi, Myi, and Khi when the values are changed to 14.0 cm and 12.0 cm.

In this case, the initial value of each design variable is
The level limit was used and the move limit for a single improvement of the design variables was 20% in each case. In order to confirm whether the same optimal solution can be obtained when the initial values of the respective design variables are changed, the allowable horizontal displacement δbai of the base-isolated bearing 14 is 34.0 cm, the allowable horizontal displacement δpai of the pier 16 is 16 cm, Allowable horizontal displacement δfai of foundation 18
Is set to 3.0 cm, and the optimal solution when the first level value and the third level value in Table 3 are used as the initial values of the design variables is shown in Table 6.
Is shown in

[0113]

[Table 5]

[0114]

[Table 6]

As shown in the ITE column of Tables 5 and 6, the optimum values of any of the design conditions are obtained by repetitive improvement of up to 10 times. It became clear that was required. Here, in the optimal solution of each constraint, δb, δp
And δf are all active.

The optimal solutions obtained by changing the initial values of the design variables shown in Table 6 have exactly the same values.
It can be determined that the overall optimal solution is obtained by the method described in the present invention.

Note that the maximum response horizontal displacement is the level 1, 2,
Since it was determined by the experimental design method using the value of 3,
In the process of improving the design variables, the improved value becomes the first level or the third level.
If it is outside the level, it may be possible to improve in a different direction because the exact constraints are not obtained. Therefore, when the design variables are improved beyond the range of the first or third level, it is necessary to reapply the maximum response displacement surface in the vicinity thereof.

The yield seismic intensity at the optimum value shown in Table 5, that is, the value obtained by dividing the yield strength of each structural element by the weight of the element above the element is 0.1% for the seismic isolation bearing 14.
In the range of 1 to 0.31, for the pier 16, 0.29 to
The range is 0.42.

The maximum response plasticity at the optimum value, that is, the value obtained by dividing the maximum response displacement of each structural element by the yield displacement, is in the range of 10.8 to 24.1 for the seismic isolation bearing 14, and for the pier 16 , 2.61 to 3.86.

The yield load Qd of the seismic isolation bearing 14 of each pier 16
Comparing the optimal values of
The construction cost of d is larger than that of the other piers 16, and the yield strength of the seismic isolation bearing 14 of the pier 16 (P3, P4) is smaller than that of the other piers 16.

Further, regarding the optimum value of the pier 16, the yield seismic intensity of the pier 16 (P3, P4) is small.
These facts make the seismic isolation bearing 14 and the pier 16 of the pier 16 having a large dead load reaction force and a large construction unit cost more easily plastically deform than the other piers, and cause the other piers 16 having a large rigidity to bear more seismic force. This is more effective from the viewpoint of minimizing the construction cost of the seismic isolation bridge 10.

This tendency is caused by the allowable horizontal displacement δp of the pier 16.
It was found that as ai became larger, it became more prominent. Confirmation of the reliability of the optimal value To confirm the reliability of the optimal value obtained in Table 5, a nonlinear time history response analysis was performed using the optimal value of the design variable, and the maximum response horizontal displacement calculated by the estimation formula was calculated. Was confirmed.

Table 7 shows a comparison table when the allowable horizontal displacement δpai of the pier 16 is 14.0 cm.

[0124]

[Table 7]

From the results shown in Table 7, the error of the estimated value of the maximum response horizontal displacement and the error of the maximum response horizontal displacement based on the nonlinear time history response analysis are all within 10%, and the average error is 4.5.
%, And it was confirmed that the maximum response horizontal displacement can be accurately estimated by the method of experimental design.

In Table 7, the error of the pier 16 is slightly larger than the errors of the other structural elements. This is because the horizontal displacement of the top of the pier shows a large change even by a slight change of the rotation angle of the pier base. It seems to be receiving.

In the above embodiments and the specific examples, the case where the present invention is applied to a seismic isolation bridge is illustrated. However, the present invention is not limited to this, and a bridge without a seismic isolation structure is used. It can also be applied to the seismic design of non-seismic structures and non-seismic structures other than bridges.

[0128]

As described in detail above, according to the seismic design method for a structure according to the present invention, a structure which can be constructed most economically can be constructed with a small number of repetitive calculations for securing the earthquake resistance. It can be designed efficiently.

Further, since the optimization process is performed by a computer using a mathematical programming method, the values of the design variables that make the structure most economical can be theoretically accurately and automatically calculated by the computer. Can be determined in a short time.

In this case, according to the present invention, by combining the experimental design method and the dual method (mathematical programming method) using the Lagrange function, the structure exhibiting a complicated nonlinear dynamic behavior can be obtained.
The global and most economical optimal values of the non-linear dynamic properties of a plurality of components can be efficiently designed with very little iterative improvement.

Further, in formulating an optimal design problem for a bridge exhibiting non-linear dynamic behavior in response to a large-scale earthquake, a representative value of the non-linear hysteresis characteristic of each structural element is treated as a symbolic continuous design variable. Optimal design problems dealing with complex nonlinear dynamic behavior can be simplified.

Further, by using the experimental design method, the nonlinear time-history response displacement of a multi-span bridge system subjected to a large-scale earthquake is calculated as 9 when the number of design variables is four.
Nonlinear analysis, when the number of design variables is up to 13, 2
By performing the nonlinear analysis seven times and using the obtained analysis results, it is possible to introduce an estimation formula with high accuracy.

[Brief description of the drawings]

FIG. 1 is a flowchart showing the design procedure of a method of designing a structure for earthquake resistance according to the present invention.

FIG. 2 is an explanatory view of a base-isolated bridge to which the design method of FIG. 1 is applied.

FIG. 3 is an explanatory diagram of a nonlinear time history response analysis model of the base-isolated bridge shown in FIG. 2;

FIG. 4 is a non-linear hysteresis characteristic diagram of a base-isolated bearing of the base-isolated bridge shown in FIG. 2;

FIG. 5 is a non-linear hysteresis characteristic diagram of the pier of the base-isolated pier shown in FIG.

FIG. 6 is an explanatory diagram of ground conditions of a base-isolated bridge assumed in a specific example of the present invention.

FIG. 7 is a characteristic diagram of a bilinear model of a seismic isolation bearing assumed in a specific example of the present invention.

FIG. 8 is a characteristic diagram of a bilinear model of a pier assumed in a specific example of the present invention.

FIG. 9 is a graph showing a relationship between a yield load of a base-isolated bearing and a manufacturing cost assumed in a specific example of the present invention.

FIG. 10 is a graph showing a relationship between a bending moment of a pier and a construction cost assumed in a specific example of the present invention.

FIG. 11 is a graph showing a relationship between a horizontal spring constant of a pile foundation and a construction cost assumed in a specific example of the present invention.

[Explanation of symbols]

 DESCRIPTION OF SYMBOLS 10 Seismic isolation bridge 12 Superstructure 14 Seismic isolation bearing 16 Pier 18 Foundation

Claims (5)

[Claims] 1. A seismic design method for civil engineering and architectural structures such as bridges, the method comprising: extracting structural elements of the structure; and determining structural parameters required for nonlinear dynamic response analysis for each of the structural elements. A first step of setting a non-linear dynamic response analysis model; a second step of setting a design variable so as to include a characteristic value indicating the non-linear behavior of the structural parameters for each of the constituent elements; A third step of setting an objective function representing a construction cost for each of the constituent elements using a variable as a parameter; A fourth step of setting; a fifth step of performing the non-linear dynamic response analysis processing at the plurality of level combinations of the design variables; and considering a constraint condition using a result of the non-linear dynamic response analysis processing. A sixth step of creating an estimation formula of exponential dynamic behavior based on an experimental design and setting the constraints of the dynamic behavior; and formulating an optimization problem, and formulating the design variable based on a mathematical programming method. And a seismic design method for a structure that determines an optimal solution by repeating the processes of the fourth to seventh steps as necessary. 2. The seismic design method for a structure according to claim 1, wherein in the fourth step, the level is set to three levels, and a combination with the design variable is determined by an orthogonal table of an experimental design method. . 3. The structure according to claim 1, wherein the structure is a bridge having a seismic isolation structure.
3. The method of claim 1, wherein the pier and the foundation are used. 4. The design variable selects a shear spring constant as a characteristic value indicating the non-linear behavior of the seismic isolation bearing, sets a yield load thereof to the design variable of the seismic isolation bearing, and sets the design variable of the pier. The method according to claim 3, wherein bending stiffness is selected as the characteristic value indicating the non-linear behavior, and the yield bending moment is set as the design variable of the pier. 5. The seventh step is to solve the optimization problem by a dual method using a Lagrangian function to obtain an improved solution of the design variable; and the construction cost of the structure and the value of the design variable 5. The method according to claim 1, further comprising: judging that has converged to a constant value; and judging whether or not the obtained improved solution is within a standard.
Seismic design method for structures described in the section.