JP4892214B2 - Analysis method of vertical earthquake response - Google Patents

Description

  The present invention relates to an analysis method, a design method, and a structure of a vertical earthquake response of a structure having group piles.

  Conventionally, when analyzing the seismic response of a structure having group piles, it is common to replace the group pile-ground system with equivalent springs and dashpots. However, the piles constituting the group pile influence each other during vibration, and the values of the spring and the dashpot change. For this reason, in order to perform an earthquake response analysis using a mass system model of a structure having group piles, it is necessary to determine the spring constant and the damping coefficient in consideration of the influence of each pile.

As a method for evaluating the dynamic impedance (ie, spring constant and damping coefficient) of such a group pile with high accuracy and simplicity, for example, Non-Patent Document 1 includes all other piles for each pile constituting the group pile. The method of calculating the dynamic impedance of a group pile is described by making the influence coefficient between them into a matrix and applying this influence coefficient to the dynamic impedance of a single pile.
Katsuichiro Hijikata, 2 others, “Simple evaluation method for dynamic impedance of group piles”, Architectural Institute of Japan, Proc. 455, Architectural Institute of Japan, January 1994, p73-82

  However, in this dynamic impedance evaluation method, in order to calculate the influence from all other piles for each pile, a matrix calculation must be performed. For this reason, in order to calculate a group pile coefficient, the calculation of the number of times proportional to the square of the number of piles is needed, and calculation will take time and effort.

In addition, the relationship between the friction force around the pile and the deformation becomes non-linear when the ground deformation increases, but in this dynamic impedance evaluation method, the relationship between the friction force around the pile and the deformation is treated as a linear. For this reason, there has been a problem that the accuracy is lowered in the calculation of the response history that causes nonlinear deformation.
The present invention has been made in view of the above problems, and an object thereof is to provide an earthquake response analysis method for a structure having a simple group pile in consideration of nonlinearity of pile peripheral friction force. It is.

The vertical earthquake response analysis method of the present invention is a method for analyzing the vertical seismic response of a structure having a group pile, and the vertical pile circumferential friction force per unit length of the group pile Modeled into a mass system model having a mass point representing and a spring connected to the mass point, and the spring constant of the spring is calculated based on the number of piles, the pile spacing, and the pile diameter, and the group pile coefficient and the number of piles , Calculated by multiplying the spring constant of the spring representing the vertical friction force per unit length of a single pile, and using the calculated spring constant to analyze the seismic response of the mass system model And

The vertical earthquake response analysis method of the present invention is a method for analyzing the vertical earthquake response of a structure having a group pile,
The vertical pile perimeter friction force per unit length of the group pile is modeled into a mass system model having a mass point representing the group pile and a spring connected to the mass point,
The spring constant of the spring is k _fg , the group pile coefficient of the real part is α _r , the number of piles is N _P , the pile spacing is X, the pile diameter is D, and the vertical pile circumference friction spring per unit length of a single pile The constant is k _f , the ground shear modulus is G, the Poisson's ratio of the ground is ν, the pile length is L, the pile diameter is D, and the parameter related to the number of piles is a, and the variable r _m = 2.5 (1 -Ν) When L,
The spring constant k _fg is calculated by the following formulas (1) to (3),
The seismic response of the mass system model is analyzed using the calculated spring constant.

In the above vertical earthquake response analysis method, the mass system model has a dashpot related to the vertical pile perimeter friction force per unit length of the group pile,
The damping coefficient of the dashpot is C _fg , the group pile coefficient of the imaginary part is α _i , the pile circumferential friction spring damping coefficient in the vertical direction per unit length of the single pile is C _f , and the group pile coefficient of the real part is applied When the parameter is b, the ground natural frequency is ω _gv , and the ground damping constant is h _f ,
The method for analyzing a vertical earthquake response, wherein the attenuation coefficient C _fg is calculated by the following equations (4) to (6).
α _i = b × α _r (4)
C _f = k _f tan (2 sin ⁻¹ h _f ) / ω _gv (5)
_{C fg = α i N P C} f ... (6)
However, when α _i ≧ 1.0, α _i = 1.0

  According to the above vertical earthquake response analysis method, the matrix calculation required in the conventional analysis method is unnecessary, and the calculation can be performed easily.

The vertical earthquake response analysis method of the present invention is a method for analyzing the vertical earthquake response of a structure having a group pile,
The vertical pile perimeter friction force per unit length of the group pile is modeled into a mass system model having a mass point representing the group pile and a spring connected to the mass point,
The spring constant of the spring is k _fg , the group pile coefficient of the real part is α _r , the number of piles is N _P , the pile spacing is X, the pile diameter is D, the vertical pile circumference friction force per unit length of a single pile The spring constant of the spring representing k _f , the ground shear modulus G, the Poisson's ratio of the ground ν, the pile length L, the pile diameter D, the pile circumferential friction force τ, the ultimate circumferential friction force τ _S , When the parameter relating to the number of piles is a and the variable r _m = 2.5 (1-ν) L in the formula,
The spring constant k _fg is calculated by the following formulas (7) to (11),
The seismic response of the mass system model is analyzed using the calculated spring constant.

In addition, the mass system model has a dashpot related to the pile peripheral friction force in the vertical direction per unit length of the group pile,
The damping coefficient of the dashpot is C _fg , the group pile coefficient of the imaginary part is α _i , the pile circumferential friction spring damping coefficient in the vertical direction per unit length of the single pile is C _f , and the group pile coefficient of the real part is applied When the parameter is b, the ground natural frequency is ω _gv , and the ground damping constant is h _f ,
The attenuation coefficient C _fg may be calculated from the following formulas (1 2 ) to (1 4 ).
α _i = b × α _r (12)
C _f = k _f tan (2 sin ⁻¹ h _f ) / ω _gv (13)
_{C fg = α i N P C} f ... (14)
However, when α _i ≧ 1.0 or τ> τ _s / 6 , α _i = 1.0

Further, in the above vertical earthquake response analysis method, the parameter a relating to the number of piles may be determined so as to satisfy the following equation (15).
−0.1 ≧ a ≧ −0.9 (15)

You may determine the parameter b concerning the group pile coefficient of the said real part so that the following formula | equation (16) may be satisfy | filled.
1.0 ≦ b ≦ 3.0 (16)

  According to the vertical earthquake response analysis method of the present invention, the matrix calculation for evaluating the influence of each single pile constituting the group pile, which was necessary in the conventional analysis method, is not required, so the calculation is simpler. It can be performed. Furthermore, since the analysis taking into account the non-linearity of the ground deformation can be performed, the calculation accuracy is improved.

Hereinafter, an embodiment of a method for analyzing a vertical earthquake response according to the present invention will be described with reference to the drawings.
FIG. 1 is a diagram showing a mass system model 10 assumed to analyze a vibration response of a structure having a group pile in the vertical earthquake response analysis method of the present embodiment. As shown in the figure, in the vertical earthquake response analysis method of the present embodiment, the group pile and the structure are represented by using a plurality of discrete mass points 11 and 12, and the pile peripheral friction force acting on the group pile is expressed as follows. It is modeled by a pile circumferential friction spring 15 and a dashpot 16 connected between a mass point 13 representing free ground and a mass point 11 representing a group pile. The pile tip support force of the ground is modeled using a pile tip ground spring 17 and a dash pot 18 connected to a mass point 14 corresponding to the pile tip.

  In order to analyze the vertical earthquake response, it is necessary to set the spring constant of the pile circumferential friction spring 15 and the damping characteristic of the dashpot 16. However, since the piles that constitute the group piles affect each other, adjacent piles have a group pile effect in which the spring constant and the value of the dashpot change as compared to the single pile. Therefore, the inventors propose the following method as a method for simply evaluating the group pile effect and analyzing the vertical earthquake response.

FIG. 2 is a flowchart showing the flow of the earthquake response analysis method of the present embodiment. As shown in the figure, first, the spring constant of the pile peripheral friction spring of the single pile is calculated (S101). The spring constant k _f of the single pile pile friction spring is based on the elasticity theory described in, for example, “Analysis of deformation of vertically loaded piles” Journal of the geotechnical engineering division GT12, MFRandolph, CPWroth, p.1465-1488. According to the method, it can be calculated by the following equation (17). Incidentally, the soil shear modulus G, the Poisson's ratio of the ground [nu, the pile length L, and pile diameter is D, a variable _{r m = 2.5 (1-ν} ) L in the formula.

Next, a group pile coefficient is calculated (S102). Inventors propose the following relational expression as a method of calculating a group pile coefficient more simply. First, in order to calculate the spring constant of the pile circumference friction spring 15 shown in FIG. 1, the group pile coefficient α _r of the real part is calculated by the following equation (18). However, X in the formula is a pile distance, D is the pile diameter, N _P represents a pile number. In addition, the constant a in the equation (18) is determined so as to satisfy the equation (19).
α _r = ((X / D) / 2) × N _P ^a (18)
−0.1 ≧ a ≧ −0.9 (19)
However, when α _r ≧ 1.0, α _r = 1.0.

Moreover, in order to calculate the damping coefficient of the dashpot 16 related to the pile peripheral friction force in FIG. 1, the group pile coefficient α _i of the imaginary part is calculated by the equation (20). In addition, the constant b in Formula (20) shall be defined so that Formula (21) may be satisfy | filled.
α _i = b × α _r (20)
1.0 ≦ b ≦ 3.0 (21)
However, when α _i ≧ 1.0, α _i = 1.0.

  Note that the constant a has a width as shown in the equation (19), but is more preferably −0.5, which is the average of the maximum value and the minimum value. Similarly, the constant b is given a width as shown in Expression (21), but is more preferably 2.0, which is the average of the maximum value and the minimum value.

Hereinafter, the investigations conducted by the inventors to derive the above formulas (18) to (21) will be described.
In order to examine the group pile coefficient of the real part and the imaginary part applied to the group pile from the experiment and numerical analysis, the inventors obtained the group pile coefficient of the real part and the imaginary part using the experiment and the numerical analysis. The relationship between the number of piles, the pile diameter ratio between piles and the group pile coefficient was obtained. The experiment was performed by a model experiment in a centrifugal field when the number of piles was 9 and the pile interval was 2.5D, and the number of piles was 4 and the pile interval was 2.5D.

  In addition, the numerical analysis shows that the pile interval is 2.5D, the pile numbers are 4, 9, 16 and the ground deformation is linear, the pile number is 9, the pile interval is 2.5D, The numerical analysis was performed using the thin layer element method (hereinafter referred to as the “detailed analysis method”) that can take into account the decrease in rigidity of the ground adjacent to the pile when the ground deformation is set to 5.0D and 7.5D, and the ground deformation is linear and nonlinear, respectively. .

FIG. 3 (A) is a graph comparing the pile group coefficient α _r of the real part at each frequency when the pile interval is 2.5D and different pile numbers, and FIG. 3 (B) is a group of imaginary parts. It is a graph which compares pile coefficient (alpha) _i . As shown in FIG. 5A, in the range of 5 Hz or more, which is near the ground-coupled natural frequency of vertical motion, and 7.5 Hz or less, which is an upper limit value for obtaining a stable calculation result, a group of real parts is obtained. It was confirmed that the pile effect increased and the group pile coefficient α _r of the real part tended to decrease as the number of piles increased. Similarly, it was confirmed that the group pile coefficient α _i of the imaginary part tends to decrease as the number of piles increases. Further, the group pile coefficient α _i of the imaginary part shows a tendency to increase as the frequency is high, and is 1.0 or more on the high frequency side from the vicinity of the natural frequency.

FIG. 4A is a graph comparing the group pile coefficient α _r of the real part at each frequency when the number of piles is nine and different pile intervals, and FIG. 4B is the group pile of the imaginary part. It is a graph which compares coefficient (alpha) _i . As shown in the figure, it can be confirmed that when the pile interval is increased, the group pile effect is considerably reduced (that is, α _r and α _i are close to 1.0) including when the ground is nonlinear. Moreover, the comparison regarding the group pile coefficient of the imaginary part in 2.5D confirmed the tendency for the group pile coefficient near the natural frequency to become larger when the ground is nonlinear.
Therefore, the inventors assumed the following equations (22) and (23).
α _r = ((X / D) / 2) × N _P ^a (22)
α _i = b × α _r (23)

FIG. 5 is a graph showing the value of the index value a in the equation (22) obtained by back calculation based on the group pile coefficient α _r obtained by the analysis. As shown in the figure, in the range of 5 Hz or more, which is near the natural frequency of vertical motion, and 7.5 Hz or less, which is the upper limit value for obtaining a stable calculation result, the range of the exponent value a is -0.1 to 0.1. -0.9 (average -0.5). Therefore, by setting the constants a to satisfy equation (19) can calculate the value of exactly alpha _r.

In addition, the inventors consider that a proportional relationship is established between the group pile coefficient of the real part and the group pile coefficient of the imaginary part following the stiffness proportional type damping. The ratio b of the group pile coefficient α _i of the part and the group pile coefficient α _r of the real part was calculated. FIG. 6 is a graph showing the ratio b between the group pile coefficient α _i of the imaginary part and the group pile coefficient α _r of the real part. As shown in the figure, when the imaginary part / real part ratio b of the ground spring group pile coefficient is 5 to 7.5 Hz, the range of b is 1.0 to 3.0 (average 2.0). Therefore, the value of α _i can be accurately calculated by setting the constant b so as to satisfy the equation (21).
Note that when a = −0.5 and b = 2.0, the group pile coefficient calculated by the analysis method of this embodiment when X / D = 2 is substantially consistent with the results of the experiment and numerical analysis. In the formula (22), X / D is divided by 2.

On the other hand, if α _r and α _i are too large, the spring constant and damping coefficient representing the peripheral surface friction of the county piles become large, so that the obtained seismic response becomes too small and the structural design may be difficult. Therefore, the maximum value of α _r and α _i is set to 1.0, and when α _r and α _i calculated by the equations (18) and (20) exceed 1.0, 1.0 is assumed. did.

Further, as described above, since the group pile coefficient α _i of the imaginary part is larger near the natural frequency and is always 1.0 or more, the stiffness of the pile circumferential friction spring is reduced due to the non-linearity of the ground described later. In this case (τ> τ _S / 6), α _i = 1.0. As described above, even if α _r and α _i are estimated to be small, the obtained seismic response is a vibration larger than the actual vibration, so that the design is on the safe side.

After calculating the group pile coefficient as described above, the spring constant of the pile peripheral friction spring of the group pile is calculated (S103 in FIG. 2). Since the spring constant of a single pile is multiplied by the number of piles and the group pile effect is taken into consideration, the following formula (24) is obtained when the spring coefficient of the group pile is k _fg .
_{k fg = α r N P k} f ... (24)

Next, the damping coefficient C _f related to the pile peripheral friction force of the single pile is obtained by the following formula (25) (S104). ω _gv indicates the natural frequency of the ground, and is obtained by ω _gv = 2π (V _P / 4H). In the equation, _hf represents the ground attenuation constant, _VP represents the ground P wave velocity, and H represents the surface ground thickness. The equation (25) is equivalent attenuation constant in frequency in the vicinity of ground natural frequency of the pile friction spring single piles, and about the same as the ground damping constant h _f, the spring constant of the equivalent damping constant and ground spring Derived from the relationship.
C _f = k _f tan (2 sin ⁻¹ (h _f )) / ω _gv (25)

Next, the damping coefficient C _fg of the dashpot 16 related to the pile circumferential friction force of the group pile is obtained by the following formula, similarly to the case where the spring constant of the pile circumference friction spring 15 of the group pile is obtained (S105).
_{C fg = α i N P C} f ... (26)

Then, to calculate the damping coefficient _{C eg} spring constant _{k eg} the pile tip dashpot 18 of pile tip vertical ground spring 17 (S106, S107). The spring constant k _eg and the damping coefficient C _eg are described in, for example, “Architectural Structure University 1 Earthquake Engineering”, Shokokusha, p. What is necessary is just to calculate by the following formula | equation (27) and (28) derived based on the elasticity theory described in 83-85. In the formula, Γ ₁ is 1.21 (ν = 1/3) to 1.26 (ν = 1/2), ν is the Poisson's ratio of the pile tip ground, ρ is the ground density, and V _s. Indicates S wave velocity.
k _eg = 4 Gr / (1−ν) × α _r N _P (27)
_{C eg = 8Γ 1 / (π} 2 (1-ν)) ρV s π (D / 2) 2 × α i N P ... (28)

Here, since the relationship between the actual pile circumference friction force and displacement is non-linear, it is desirable to calculate the seismic response in consideration of this non-linearity. Therefore, the inventors use the spring constant k _fg calculated from the equation (24) when the pile circumference friction force τ is less than 1/6 of the limit circumferential surface friction force τ _S , and the pile circumference friction force τ is When the limit peripheral frictional force τ _S is 1/6 or more, the spring constant k _fg ′ calculated by the equation (29) is used (S108).
When τ ≧ τ _S / 6, k _fg ′ = k _fg / 2 (29)

Note that the limit peripheral frictional force τ _S is calculated according to “Guidelines for Architectural Foundation Structural Design” Architectural Institute of Japan, p. As described in 205, τ _S = 3.3N for sandy soil and τ _S = Qu / 2 for viscous soil may be used. N represents the N value of the ground, and Qu represents the uniaxial compressive strength.

  The above-mentioned simple evaluation method for non-linearity of pile circumference friction force proposed by the inventors (referred to as simple method) is derived by regressing the relationship between pile circumference friction force and displacement obtained from the above-described experiment in a straight line. It is a thing. 7 (A) and (B) are graphs showing the pile circumference frictional force obtained by the above experiment and the pile circumference frictional force obtained by the simple method proposed by the inventors. In the case of 9 piles, (B) shows the case of 4 piles. As shown in the figure, it can be confirmed that the expression (26) proposed by the inventors accurately represents the relationship between the pile peripheral friction force and the displacement obtained by the experiment.

  Next, an equation of motion is derived (S109). Pile circumference friction spring, pile tip ground spring, stiffness matrix (K matrix) composed of each spring constant of the ground, pile, and structure, and damping matrix (C matrix) composed of the damping coefficient of each dashpot, The equation of motion of the mass point system model 10 is obtained from the mass matrix (M matrix) composed of the masses of the respective mass points.

  The vibration response of each part of the vibration model is analyzed by inputting acceleration to the material point of the material point system model 10 (S110). In this case, the non-linearity of the pile circumferential friction spring modeled by the above simple method is taken into account according to the pile circumferential surface friction force. Thereby, the earthquake response of each part is computable.

  Next, each part of the structure is evaluated based on the obtained earthquake response (S111). For example, it is possible to design on the safe side by preventing the vibration response of each member obtained by the above analysis from exceeding the maximum deformation of each member.

According to the vertical ground motion analysis method for a structure having a group pile according to this embodiment, the following effects can be obtained.
In the conventional seismic motion analysis method, matrix calculation is necessary to evaluate the influence between each pile of the group pile, and it took time and effort to calculate, but according to the analysis method of this embodiment, matrix calculation is omitted. Therefore, labor can be reduced and workability can be improved.

  Moreover, in the analysis method of this embodiment, since the non-linearity of the ground deformation can be reflected in the vibration analysis, a more accurate earthquake response analysis can be performed.

In the analysis method of the vertical earthquake response of this embodiment, it is an example of the mass system model assumed in order to analyze the vibration response of the structure which has a group pile. It is a flowchart which shows the flow of the analysis method of the earthquake response of this embodiment. (A) is a graph comparing the pile pile coefficient α _i of the imaginary part with a pile interval of 2.5D and comparing the pile pile coefficient α _r of the real part at each frequency in the case of different pile numbers. It is a graph to compare. (A) is a graph comparing the group pile coefficient α _r of the real part at each frequency when the number of piles is 9, and (B) is the group pile coefficient α _i of the imaginary part. It is a graph to compare. It is a graph which shows the value of the index value a calculated | required by calculating backward based on the group pile coefficient (alpha) _r obtained by analysis. It is a graph which shows ratio b of the group pile coefficient of an imaginary part, and the group pile coefficient of a real part. It is a graph which shows the pile circumference frictional force obtained by experiment, and the pile circumference frictional force computed by the simple method which the inventors propose, (A) shows the case of nine piles, and (B) shows four piles. This case is shown.

Explanation of symbols

10 Mass point spring model 11 Mass point representing group pile 12 Mass point representing structure 13 Mass point representing free ground 14 Mass point equivalent to pile tip 15 Pile circumference friction spring 16 Pile circumference dash pot 17 Pile tip ground spring 18 Pile tip dash pot

Claims (7)

A method for analyzing the vertical seismic response of a structure having group piles,
The vertical pile perimeter friction force per unit length of the group pile is modeled into a mass system model having a mass point representing the group pile and a spring connected to the mass point,
The spring constant of the spring,
Group pile coefficient calculated based on the number of piles, pile interval, and pile diameter, and the number of piles,
Calculated by multiplying the spring constant of the spring representing the vertical friction force of the pile per unit length of a single pile,
An earthquake response analysis method, comprising: analyzing the earthquake response of the mass system model using the calculated spring constant. A method for analyzing the vertical seismic response of a structure having group piles,
The vertical pile perimeter friction force per unit length of the group pile is modeled into a mass system model having a mass point representing the group pile and a spring connected to the mass point,
The spring constant of the spring is k _fg , the group pile coefficient of the real part is α _r , the number of piles is N _P , the pile spacing is X, the pile diameter is D, and the vertical pile circumference friction spring per unit length of a single pile The constant is k _f , the ground shear modulus is G, the Poisson's ratio of the ground is ν, the pile length is L, the pile diameter is D, and the parameter related to the number of piles is a, and the variable r _m = 2.5 (1 -Ν) When L,
The spring constant k _fg is calculated by the following formulas (1) to (3),
An analysis method of a vertical earthquake response, wherein the earthquake response of the mass system model is analyzed using the calculated spring constant.

A method of analyzing a vertical earthquake response according to claim 2,
The mass system model has a dashpot related to the vertical pile perimeter friction force per unit length of the group pile,
The damping coefficient of the dashpot is C _fg , the group pile coefficient of the imaginary part is α _i , the pile circumferential friction spring damping coefficient in the vertical direction per unit length of the single pile is C _f , and the group pile coefficient of the real part is applied When the parameter is b, the ground natural frequency is ω _gv , and the ground damping constant is h _f ,
The method for analyzing a vertical earthquake response, wherein the attenuation coefficient C _fg is calculated by the following equations (4) to (6).
α _i = b × α _r (4)
C _f = k _f tan (2 sin ⁻¹ h _f ) / ω _gv (5)
_{C fg = α i N P C} f ... (6)
However, when α _i ≧ 1.0, α _i = 1.0 A method for analyzing the vertical seismic response of a structure having group piles,
The vertical pile perimeter friction force per unit length of the group pile is modeled into a mass system model having a mass point representing the group pile and a spring connected to the mass point,
The spring constant of the spring is k _fg , the group pile coefficient of the real part is α _r , the number of piles is N _P , the pile spacing is X, the pile diameter is D, the vertical pile circumference friction force per unit length of a single pile The spring constant of the spring representing k _f , the ground shear modulus G, the Poisson's ratio of the ground ν, the pile length L, the pile diameter D, the pile circumferential friction force τ, the ultimate circumferential friction force τ _S , When the parameter relating to the number of piles is a and the variable r _m = 2.5 (1-ν) L in the formula,
The spring constant k _fg is calculated by the following formulas (7) to (11),
An analysis method of a vertical earthquake response, wherein the earthquake response of the mass system model is analyzed using the calculated spring constant.

A method of analyzing a vertical earthquake response according to claim 4,
The mass system model has a dashpot related to the pile peripheral friction force in the vertical direction per unit length of the group pile,
The damping coefficient of the dashpot is C _fg , the group pile coefficient of the imaginary part is α _i , the pile circumferential friction spring damping coefficient in the vertical direction per unit length of the single pile is C _f , and the group pile coefficient of the real part is applied When the parameter is b, the ground natural frequency is ω _gv , and the ground damping constant is h _f ,
A method of analyzing a vertical earthquake response, wherein the attenuation coefficient C _fg is calculated from the following equations (1 2 ) to (1 4 ):
α _i = b × α _r (1 2 )
C _f = k _f tan (2 sin ⁻¹ h _f ) / ω _gv (1 3 )
_{C fg = α i N P C} f ... (1 4)
However, αi = 1.0 when α _i ≧ 1.0 or τ> τ _s / 6 Analysis method of vertical seismic response of any one of claims 2 5, characterized in that defining the parameter a according to the pile number so as to satisfy the following equation (1 5).
−0.1 ≧ a ≧ −0.9 (1 5 ) The vertical earthquake response analysis method according to claim 3 or 5, wherein the parameter b relating to the group pile coefficient of the real part is determined so as to satisfy the following formula (1 6 ).
1.0 ≦ b ≦ 3.0 (1 6 )

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