A novel one-dimensional dual-material periodic foundation optimization design method

Abstract

The present invention provides a novel one-dimensional dual-material periodic foundation optimization design method, and studies the factors that affect the vibration isolation effect of the one-dimensional dual-material periodic foundation on a crushed stone base, namely, the coupling effect of the upper structure, the number of units and the band gap. Based on the band gap approximate analytical formula, by ensuring that the compressive strength FRS of the upper structure is quite mild at any frequency under consideration, each series resonance zone RZ is covered by the corresponding band gap BG, and for all potential designs that meet the above requirements, the maximum energy function PF (h1, h2) is satisfied, thereby achieving a good design of the one-dimensional reinforced concrete periodic foundation. The superiority of the present invention is verified by comparing the vibration reduction performance with that of the traditional design method. For the cases of S waves and P waves, the present invention is significantly better than the concrete foundation and the benchmark foundation. In addition, the vibration attenuation performance of the optimal design foundation for S waves is very outstanding. If the configuration can strictly follow the results obtained by the proposed method, it is expected that the attenuation effect of the optimal design foundation will be better.

Description

Technical Field

[0001] The invention belongs to the technical field of structural mechanics, and in particular relates to a novel one-dimensional dual-material periodic foundation optimization design method. Background Art

[0002] The configuration of the one-dimensional periodic structure of the gravel foundation can be found in Figure 1 In the design of 1-D RC periodic foundation and superstructure with series resonance zone (RZS), the frequency band of ground motion attenuation is FB = (ω FB-start ,ω FB-end ), the bracket represents an open inertia quantity; the RZ set of S waves is denoted by RZ S , the RZ of the qth layer unit is The RZ set of P waves is denoted as RZ P , the RZ of the r-th layer unit is The band gaps of the p-th layer under S-wave and P-wave are denoted as BG p-S =(ω p-start-S ,ω p-end-S ) and BG p-P =(ω p-start-P ,ω p-end-P ); the energy function PF(h1,h2) is used to measure the quality of foundation design (i.e. PF(h1,h2) = n. Maximizing the number of unit cells will effectively increase the attenuation effect).

[0003] A well-designed 1-D reinforced concrete periodic foundation should ensure that the compressive strength FRS of the superstructure is fairly moderate at any frequency under consideration, which requires that each RZ is covered by a corresponding BG. In addition, for all potential designs that meet the above requirements, the optimal design should also maximize the energy function PF(h1,h2). Therefore, the optimization problem becomes finding h1 and h2 and maximizing PF(h1,h2) under the following conditions: 1. Any RZ within the FB q-S Should be included in BG p-S 2. Any RZ in FB q-P Should be included in BG p-P 3. Configuration constraint CC(h1,h2,n) satisfies CC(h1,h2,n)=n(h1+h2)<H max , where H max is the permissible foundation height. Summary of the invention

[0004] The technical problem to be solved by the present invention is to provide a novel one-dimensional dual-material periodic foundation optimization design method for optimizing the vibration reduction performance of the foundation.

[0005] The technical solution adopted by the present invention to solve the above technical problems is: a novel one-dimensional dual-material periodic foundation optimization design method, comprising the following steps:

[0006] S1: Determine the seismic resistance level of the foundation according to the seismic requirements of the foundation;

[0007] S2: Frequency response FR of the superstructure according to the foundation S The series resonance region RZ is determined by the bandwidth FB S ;

[0008] S3: Let the RZ corresponding to the P wave be RZ r-P , each RZ r-P Mapped to the RZ corresponding to the S wave, denoted as RZ r-S , and add to RZ S middle;

[0009] S4: Fusion RZ S Overlapping elements in RZ q-S With RZ r-S Get RZ i-S ;

[0010] S5: Enumerate each RZ S Elements in RZ i-S , the specific steps are:

[0011] S51: Iteration 1≤p≤p max For each p in , the specific steps are:

[0012] S511: Set RZ i -s upper bandwidth UB and BG p-S The upper bandwidth UB is equal to that of the trajectory BI S-trail Enumeration Calculation BI i-p-S-trail , where the subscript trail means the case where the resonance region is within the band gap, i represents enumeration, p is the interval number, which is an integer from 1 to 100, and S represents s-wave; the subscript temp means that not all cases may satisfy the case where the resonance region is within the band gap;

[0013] S512: Check BI by testing whether there are UBs in BGs cutting an element in RZs i-p-S-trail effectiveness;

[0014] S513: If BI i-p-S-trail If valid, then set E k is Young's modulus, v k is Poisson's ratio, ρ k is the density, calculate the rubber layer tracking thickness h 1-trail :

[0015]

[0016] S514: Set the concrete layer tracking thickness h 2-trail = 0, enumerate each RZ in RZs j -s Calculate the concrete layer tracking thickness h 2-trail , the specific steps are:

[0017] S5141: Set RZ j -s is equal to the lower limit bandwidth LB of the band gap; the endpoints of BGs form an arithmetic sequence, and the increment of the arithmetic sequence at the junction is defined as the band interval BI; let C k is the corresponding velocity constant, D is the correction wave coefficient, is the starting position frequency of the s-wave of the u-th layer unit, is the starting position frequency of the s-wave of the s-th layer unit; calculate h 2-temp :

[0018]

[0019] S5142: h 2-temp Insert into h 2-temp middle;

[0020] S5143: By selecting h 2-temp The largest element in gets h 2-trail :

[0021] h 2-trail =max(h 2-temp );

[0022] S515: Calculate the tracking performance function PF trail =PF(h 1-trail ,h 2-trail ) and insert it into PF trail ;

[0023] S6: Let h 1-opt and h 2-opt are the optimal thicknesses of the rubber layer and the concrete layer, respectively, and search for PF trail The maximum value in h 1-opt and h 2-opt :

[0024] PF max =PF(h 1-opt ,h 2-opt )=max(PF trail );

[0025] S7: When the configuration constraint CC(h is satisfied 1-opt ,h 2-opt ,n), calculate the maximum number of units n.

[0026] According to the above scheme, in step S2, the specific steps are: determining the relationship between the frequency range of the series resonance zone RZ of the upper structure of the foundation and the bandwidth FB:

[0027] If the frequency range of the series resonance region RZ is included in the bandwidth FB, the series resonance region RZ is accepted;

[0028] If the frequency range of the series resonance region RZ is not completely included in the bandwidth FB, the series resonance region RZ is not accepted;

[0029] If the frequency range of the series resonant region RZ is partially included in the bandwidth FB, the frequency range of the series resonant region RZ that is included in the bandwidth FB is accepted, and the frequency range of the series resonant region RZ is updated.

[0030] According to the above scheme, in step S3, the specific steps are: MR is mapping ratio, , is the starting position frequency of the p-wave of the r-th layer unit, is the frequency of the p-wave end position of the r-th layer unit, according to the S-wave frequency ω l-S and P wave frequency ω l-P The conversion relationship formula ω l-S =MRω l-P Get the start and end frequencies of RZr-s:

[0031]

[0032] According to the above scheme, in step S4, the specific steps are:

[0033] R i-S Also includes RZ q-S With RZ r-S , from RZ S Delete RZ q-S With RZ r-S , and RZ i-S Insert RZ S middle.

[0034] Furthermore, in step S511, is the frequency of the end position of the s-wave of the t-th layer unit, then the trajectory BI S-trail for:

[0035]

[0036] Furthermore, in step S512, the specific steps are:

[0037] Let brackets [] represent the operator for obtaining the integer part, is the starting position frequency of the S wave of the s-th layer unit, is the frequency of the end position of the s-wave of the s-th layer unit, if Corresponding to all RZ S RZ s-S , then BI i-p-S-trail efficient.

[0038] Furthermore, in step S513, the specific steps are:

[0039] According to the formula Calculate h 1-trail .

[0040] Furthermore, in step S5141, the specific steps are:

[0041] According to the formula Calculate h 2-temp .

[0042] A computer storage medium stores a computer program executable by a computer processor, wherein the computer program executes a novel one-dimensional dual-material periodic basis optimization design method.

[0043] The beneficial effects of the present invention are:

[0044] 1. A novel one-dimensional dual-material periodic foundation optimization design method of the present invention studies the factors affecting the vibration isolation effect of the crushed stone-based one-dimensional dual-material periodic foundation, namely, the superstructure coupling effect, the number of units and the band gap. Based on the band gap approximate analytical formula, by ensuring that the compressive strength FRS of the superstructure is quite mild at any frequency under consideration, each series resonance zone RZ is covered by the corresponding band gap BG, and for all potential designs that meet the above requirements, the maximum energy function PF(h1,h2) is satisfied, thereby achieving a good design of the one-dimensional reinforced concrete periodic foundation.

[0045] 2. The superiority of the present invention is verified by comparing the vibration reduction performance with the traditional design method. For the S-wave and P-wave cases, the optimized design foundation is significantly better than the concrete foundation and the reference foundation. In addition, the vibration attenuation performance of the optimal design foundation for S-waves is very outstanding, and if the configuration can strictly follow the results obtained by the proposed method, the attenuation effect of the optimal design foundation is expected to be better. BRIEF DESCRIPTION OF THE DRAWINGS

[0046] Figure 1 is a configuration diagram of a one-dimensional periodic structure according to an embodiment of the present invention.

[0047] Figure 2 is a flow chart of an embodiment of the present invention.

[0048] Figure 3It is a schematic diagram of the best performance of an embodiment of the present invention.

[0049] Figure 4 Graph showing the frequency response and resonance region of the upper structure of an embodiment of the present invention.

[0050] Figure 5 1 is a diagram of the band gap and resonance region of the best design of an embodiment of the present invention.

[0051] Figure 6 is a frequency response graph of exemplary superstructures with different foundations according to an embodiment of the present invention. DETAILED DESCRIPTION

[0052] The present invention will be further described in detail below in conjunction with the accompanying drawings and specific embodiments.

[0053] To meet the requirements include:

[0054] 1. Any RZ in FB q-S Should be included in BG p-S Inside;

[0055] 2. Any RZ in FB q-P Should be included in BG p-P Inside;

[0056] 3. Configuration constraint CC(h1,h2,n) satisfies CC(h1,h2,n)=n(h1+h2)<H max , where H max is the permissible foundation height;

[0057] Under the three conditions of , we can find h1 and h2 and maximize the energy function PF(h1,h2), where the energy function PF(h1,h2) is a monotonically decreasing function, because this assumption is consistent with many different PF choices, that is, PF = n, or PF represents the minimum number of material costs. Figure 2 As shown, a novel one-dimensional dual-material periodic foundation optimization design method according to an embodiment of the present invention comprises the following steps:

[0058] S1: Determine the seismic resistance level according to the seismic requirements;

[0059] S2: Determine the RZs according to the FRs and FB of the superstructure. For example, the superstructure has three RZs, namely (2.5, 4.5) Hz, (10.1, 20.1) Hz and (40.4, 60.6) Hz. If FB = (5.0 50.0) Hz, then RZ = (2.5, 4.5) Hz is unacceptable because it is not within the FB, RZ = (10.1, 20.1) Hz is acceptable because it is included in the FB, and RZ = (40.4, 60.6) Hz is changed to (40.4, 50.0) Hz and then accepted;

[0060] S3: Each RZ r-P Mapped to the RZ corresponding to the S wave, denoted as RZ r-S , according to the formula ω l-S =MRω l-P and add it to RZ S Right now

[0061] S4: Merge overlapping elements into RZ S , that is, RZ q-S =(3.3,5.5), RZ r-S =(4.4,6.6), then from RZ S Delete both RZ q-S and RZ r-S , then RZ t-S =(3.3,6.6) is inserted into it;

[0062] S5: List each RZ as in step 6 S Chinese element RZ t-S ;

[0063] S6: for 1≤p≤p max Each p in performs steps S7 to S9 and step S15;

[0064] S7: Assume RZ t-S UB and BG in p-S UB in is equal and the trajectory BI is calculated S-trail ,Right now

[0065] S8: Check BI S-trail The effectiveness of the RZ was tested by testing whether there were no UBs within the BGs. S An element in BI S-trail is effective if Corresponding to all RZ S RZ s-S , where the brackets represent operators that obtain the integer part;

[0066] S9: For an effective BI S-trail , execute steps S10 to S11 to obtain the corresponding tracking thickness of the rubber layer and the concrete layer, which are represented as h 1-trail and h 2-trail ;

[0067] S10: According to the formula Calculate h 1-trail ,Right now

[0068] S11: Through RZS Each RZ u-S Execute steps S12 to S14 to obtain h 2-trail ;

[0069] S12: Assume RZ u-S The LB in is equivalent to the band gap of LB, according to the formula Calculate, that is

[0070] S13: h 2-temp Insert into h 2-temp middle;

[0071] S14: By selecting h 2-temp The largest element in gets h 2-trail , that is, h 2-trail =max(h 2-temp );

[0072] S15: Calculate the tracking performance function PF trail =PF(h 1-trail ,h 2-trail )Insert into PF trail ;

[0073] S16: Search PF trail The maximum value among them, namely PF max =PF(h 1-opt ,h 2-opt )=max(PF trail ), h 1-opt and h 2-opt are the optimal thicknesses of the rubber layer and the concrete layer, respectively;

[0074] S17: When the configuration constraint CC(h is satisfied 1-opt ,h 2-opt ,n), calculate the maximum number of units.

[0075] Theoretical explanation

[0076] First, since the BGs and RZS of the P wave can be mapped one-to-one to the corresponding quantities of the S wave, it is sufficient to consider only the case of the S wave. S = {RZ t-S , 1≤t≤t max} and a group of BG S = {BG p-S , 1≤p≤p max} represents BG p-S =(ω p-start-S ,ω p-end-S ), each RZ t-S Are embedded in the corresponding BGp-S According to the formula ω l-p =(p-Δo p )BI→ω l-q =(q-Δo q )BI, RZ can be mapped to the first BG to form a new set consisting of RZ MP1 = {RZ i-MP1 , 1≤i≤m}. MP1 The lower limit bandwidth LB and upper limit bandwidth UB are respectively given by ω start-MP1 and ω end-MP1 express.

[0077] like Figure 3 As shown, from 0 to ω start-MP1 ,ω 1-start-S to start-MP1 , 0 to ω end-MP1 ,ω end-MP1 to 1-end-S The distances are represented by Δ1, Δ2, Δ3 and Δ4, where Δ2 ≥ 0 and Δ4 ≥ 0.

[0078]

[0079]

[0080] Calculate

[0081]

[0082]

[0083] Because PF(h1,h2) is assumed to be a monotonically decreasing function of h1+h2, the optimal design is characterized by min[h1+h2]. When Δ2=0, min[h1+h2] can be achieved, resulting in Similar if Therefore, when Δ2=Δ4=0, h1+h2 reaches the minimum value, which means that the optimization design must ensure that the BG of the qth layer is equal to the LB of the sth layer RZ, and the UB of the pth layer BG is equal to the UB of the tth layer RZ.

[0084] Therefore, the optimal design can be obtained by exhausting all possible combinations of p, q, r and s. The proposed design method is actually a traversal algorithm for all possible combinations, which ensures that the final output of the design is optimal in a global sense. Specifically, steps S5 to S7 are assumed to be In addition, BI should group RZs without cutting any RZs, so step S8 checks whether there is a positioning ω in RZ. p-end-SAfter determining the valid BI, the next task is to find the corresponding combination of h1 and h2, where h1 is directly obtained through the formula in step S10 Then, by assuming You can use the formula Calculate the trajectory of h2. From the formula It can be seen that p is a monotonically increasing function of h2. This means that the enumeration The maximum trajectory h2 after each possible combination of is the effective value h2 of the current BI, which is the meaning of steps S11 to S14. The best combination of h1 and h2 is the combination that maximizes the performance function. Therefore, by selecting the maximum value of the performance function of all combinations of h1 and h2, h 1-opt and h 2-opt , which is performed by steps S15 to S16. Finally, the number of unit cells can be calculated by step S17.

[0085] Design Examples

[0086] To further demonstrate its practicality and performance, the proposed method was applied to design an optimal 1-D RC periodic foundation for a steel frame described in Table 2 for the range of ground motions performed on the slab. The designed periodic foundation is a 10 m × 10 m2 foundation with a maximum depth equal to 4 m, where the materials used are rubber and concrete as described in Table 1. During the design process, the FB was determined to be between 2.5 and 60.0 Hz.

[0087] First, the frequency response functions of the steel frame are calculated using “SteadyStateDirectStep” in ABAQUS 6.14, where the frequency response functions of S-waves and P-waves are obtained by specifying unit displacements for the four bottom nodes in the x-direction and y-direction, respectively. Then, the RZ is determined by checking whether the FRFs in the FB are greater than the corresponding thresholds. S and RZ P , where the thresholds for S and P waves are 0.0 and 3.0, respectively. Consider RZ P The threshold is greater than RZ S The threshold is because horizontal ground motion is generally more dangerous. Figure 4 As shown, (a) is the S wave diagram, (b) is the P wave diagram, RZ S It is determined to be a set of the following 10 frequency intervals: 3.1048~3.4789, 6.6254~7.0567, 12.1200~14.4425, 23.5164~23.8595, 29.2337~29.5127, 30.8233~31.8888, 34.8555~35.5267, 36.8418~37.3217, 42.4463~43.0980; RZP The optimal thicknesses of the rubber layer and concrete layer were determined to be 0.3779m and 0.6555m, respectively. The results of designing AZs and BGs using the predicted optimal configuration are shown in Figure 2. Figure 5 As shown, (a) is the S wave diagram and (b) is the P wave diagram. It can be seen from the figure that all AZs are evenly distributed in the BGs with minimal waste. Finally, after considering the actual manufacturing, the thickness of the rubber layer and the concrete layer are determined to be 0.36m and 0.64m respectively, and the number of units is 4.

[0088] Numerical verification

[0089] To further verify the optimality of the proposed design method, the performance of the optimal design foundation, concrete foundation, and benchmark foundation was numerically compared. The finite element analysis of the optimal design foundation and concrete foundation was similar to that of the benchmark foundation, where the thickness of the rubber layer and concrete layer were 0.36m and 0.64m for the optimal design foundation, respectively. The material of the first layer was concrete for the concrete foundation. Figure 6 The simulation results of the FRF of each layer of the steel frame are shown, corresponding to the three different foundations. It can be seen that the optimally designed foundation is significantly better than the concrete foundation and the benchmark foundation for both S-wave and P-wave cases. In addition, the vibration attenuation performance of the optimally designed foundation for S-wave is very outstanding, with almost all FRFs in the FB being less than 0 and most even below -50, while the benchmark foundation fails to attenuate the AZS outside the BGS. In addition, if the configuration can strictly follow the results obtained by the proposed method, the attenuation effect of the optimally designed foundation is expected to be even better. However, due to the RZ P The FRF threshold is high and the attenuation performance of P waves is relatively ordinary, which is acceptable because the P-wave condition is not the main design goal.

[0090] The above embodiments are only used to illustrate the design ideas and features of the present invention, and their purpose is to enable those skilled in the art to understand the content of the present invention and implement it accordingly. The protection scope of the present invention is not limited to the above embodiments. Therefore, any equivalent changes or modifications made based on the principles and design ideas disclosed by the present invention are within the protection scope of the present invention.

Claims

1. A novel one-dimensional dual-material periodic foundation optimization design method, characterized by: The following steps are involved: S1: Determine the seismic resistance level of the foundation according to the seismic requirements of the foundation; S2: Frequency response FR of the superstructure according to the foundation S The series resonance region RZ is determined by the bandwidth FB S ; S3: Let the RZ corresponding to the P wave be RZ r-P , each RZ r-P Mapped to the RZ corresponding to the S wave, denoted as RZ r-S , and add to RZ S middle; S4: Fusion RZ S Overlapping elements in RZ q-S With RZ r-S Get RZ i-S ; S5: Enumerate each RZ S Elements in RZ i-S , the specific steps are: S51: Iteration 1≤p≤p max For each p in , the specific steps are: S511: Set RZ i -s upper bandwidth UB and BG p-S The upper bandwidth UB is equal to that of the trajectory BI S-trail Enumeration Calculation BI i-p-S-trail , where the subscript trail means the case where the resonance region is within the band gap, i represents enumeration, p is the interval number, which is an integer from 1 to 100, and S represents s-wave; the subscript temp means that not all cases may satisfy the case where the resonance region is within the band gap; S512: Check BI by testing whether there are UBs in BGs cutting an element in RZs i-p-S-trail effectiveness; S513: If BI i-p-S-trail If valid, then set E k is Young's modulus, v k is Poisson's ratio, ρ k is the density, calculate the rubber layer tracking thickness h 1-trail : S514: Set the concrete layer tracking thickness h 2-trail = 0, enumerate each RZ in RZs j -s Calculate the concrete layer tracking thickness h 2-trail , the specific steps are: S5141: Set RZ j -s is equal to the lower bandwidth LB of the band gap; the endpoints of BGs form an arithmetic sequence, and the increment of the arithmetic sequence at the junction is defined as the band interval BI; let c k is the corresponding velocity constant, D is the correction wave coefficient, is the starting position frequency of the s-wave of the u-th layer unit, is the starting position frequency of the s-wave of the s-th layer unit; calculate h 2-temp : S5142: h 2-temp Insert into h 2-temp middle; S5143: By selecting h 2-temp The largest element in gets h 2-trail : h 2-trail =max(h 2-temp ); S515: Calculate the tracking performance function PF trail =PF(h 1-trail ,h 2-trail ) and insert it into PF trail ; S6: Let h 1-opt and h 2-opt are the optimal thicknesses of the rubber layer and the concrete layer, respectively, and search for PF trail The maximum value in h 1-opt and h 2-opt : PF max =PF(h 1-opt ,h 2-opt )=max(PF trail ); S7: When the configuration constraint CC(h is satisfied 1-opt ,h 2-opt ,n), calculate the maximum number of units n.

2. A novel one-dimensional dual-material periodic foundation optimization design method according to claim 1, characterized in that: In the step S2, the specific steps are: determining the relationship between the frequency range of the series resonance zone RZ of the upper structure of the foundation and the bandwidth FB: If the frequency range of the series resonance region RZ is included in the bandwidth FB, the series resonance region RZ is accepted; If the frequency range of the series resonance region RZ is not completely included in the bandwidth FB, the series resonance region RZ is not accepted; If the frequency range of the series resonant region RZ is partially included in the bandwidth FB, the frequency range of the series resonant region RZ that is included in the bandwidth FB is accepted, and the frequency range of the series resonant region RZ is updated.

3. A novel one-dimensional dual-material periodic foundation optimization design method according to claim 1, characterized in that: In the step S3, the specific steps are: MR is mapping ratio, is the starting position frequency of the p-wave of the r-th layer unit, is the frequency of the p-wave end position of the r-th layer unit, according to the S-wave frequency ω l-S and P wave frequency ω l-P The conversion relationship formula ω l-S =MRω l-P Get the start and end frequencies of RZr-s:

4. A novel one-dimensional dual-material periodic foundation optimization design method according to claim 1, characterized in that: In the step S4, the specific steps are: R i-S Also includes RZ q-S With RZ r-S , from RZ S Delete RZ q-S With RZ r-S , and RZ i-S Insert RZ S middle.

5. A novel one-dimensional dual-material periodic foundation optimization design method according to claim 3, characterized in that: In the step S511, is the frequency of the end position of the s-wave of the t-th layer unit, then the trajectory BI S-trail for:

6. A novel one-dimensional dual-material periodic foundation optimization design method according to claim 5, characterized in that: In the step S512, the specific steps are: Let brackets [] represent the operator for obtaining the integer part, is the starting position frequency of the S wave of the s-th layer unit, is the frequency of the end position of the s-wave of the s-th layer unit, if Corresponding to all RZ S RZ s-S , then BI i-p-S-trail efficient.

7. A novel one-dimensional dual-material periodic foundation optimization design method according to claim 6, characterized in that: In the step S513, the specific steps are: According to the formula Calculate h 1-trail .

8. A novel one-dimensional dual-material periodic foundation optimization design method according to claim 7, characterized in that: In the step S5141, the specific steps are: According to the formula Calculate h 2-temp .

9. A computer storage medium, characterized in that: A computer program executable by a computer processor is stored therein, and the computer program executes a novel one-dimensional dual-material periodic basis optimization design method as described in any one of claims 1 to 8.

Images