CN107748833A - A kind of method that aseismic joint width is determined based on earthquake pounding vulnerability - Google Patents
A kind of method that aseismic joint width is determined based on earthquake pounding vulnerability Download PDFInfo
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Abstract
The invention discloses a kind of method that aseismic joint width is determined based on earthquake pounding vulnerability, the time-varying failure probability of single-degree-of-freedom and system with several degrees of freedom in the case where bearing rest tuned mass damper is calculated using classical Vanmarcke, amendment Vanmarcke and Poisson approximation method, obtains the modified computing formulae of white Gaussian noise excitation mean collisional probability next year；Critical gap value corresponding to specific collision probability is calculated.The inventive method can be overcome at present on the limitation for the aseismic joint width for calculating adjacent structure respective objects collision probability value, it is a kind of aseismic joint computational methods with important theory significance and engineering application value, there is very big reference value in the vibration control field to structure.
Description
Technical Field
The invention belongs to the field of vibration control of engineering structures, and particularly relates to a method for determining the width of a shockproof joint based on seismic collision vulnerability.
Background
The collision of adjacent structures can bring large economic loss and immeasurable social influence. From the viewpoint of measures for preventing or alleviating collision of adjacent structures, the most direct and effective method is to increase the width of the shock-proof seam of the adjacent structures. Aiming at the research of the minimum shockproof seam width for avoiding collision of adjacent structures, the current representative research methods mainly comprise a response combination method, an analytical method and a numerical simulation method. Lopez-Garcia and Soong et al (2009) examined the accuracy of the DDC method (also referred to simply as CQC rule) in predicting the width of the shock slot of adjacent linear single degree of freedom structures. Based on the Monte-Carlo method, it is verified that the accuracy of the DDC method depends not only on the ratio of the natural periods of the neighboring structures, but also on the relationship between the structures' natural periods and the period values related to the dominant frequency of the excitation. The Wang Tong Ning Yongbo of scholars in China and the like (2009) take the relative displacement of the bridge end of a shock-insulation continuous beam as a research object, and deduce a maximum value reaction spectrum calculation method of the relative displacement of a beam body under the action of an earthquake by using a bridge structure splitting method and applying a vibration mode decomposition method; and (3) adopting an equivalent bilinear lead core rubber support model, calculating the relative displacement of the beam end through iteration, and analyzing the applicability of the SRSS and CQC vibration mode combination method. Some scholars study on the calculation of the width of the shockproof gap of the adjacent structure based on theories of random vibration, reliability and the like. Lin (1997) adopts a linear elastic multi-degree of freedom (MDOF) structure model to simulate adjacent structures, theoretical analysis is carried out on the critical distance between the adjacent structures based on a random vibration theory, and the precision of the method is verified through comparison with a numerical simulation result. The Bipin (2013) analytically gives the anti-vibration seam width of the adjacent structure for avoiding collision, and the anti-vibration seam width of the adjacent linear and nonlinear SDOF structure calculated by the method is compared with the SRSS method and the DDC method; further proposed is the effect of different normalized anti-shock slot widths on the collision response of adjacent linear and non-linear SDOF structures. In addition to the two methods, some students also discuss the width of the shockproof joint of the adjacent structure by adopting a numerical simulation method, hao and Shen (2001) discuss the width of the shockproof joint for avoiding collision of the asymmetric adjacent building structure under seismic excitation, consider the torsional coupling lateral response of the structure, and study the influence of the self-vibration frequency, the torsional rigidity and the eccentricity of the adjacent asymmetric structure on the width of the shockproof joint through a large amount of parametric analysis. The Lumingqi and Yangqingshan (2012) study the maximum relative displacement of adjacent structures under the action of earthquake, define the maximum relative displacement coefficient, select 100 actual earthquake records corresponding to hard soil, medium hard soil and soft soil field conditions, and discuss the influence of the adjacent structure period ratio, the height ratio, the displacement ductility coefficient ratio and the field conditions on the maximum relative displacement coefficient.
In summary, in the research of calculating the width of the anti-vibration gap of the adjacent structure by adopting the response combination method, the analytic method and the numerical simulation method, the collision risk of the adjacent structure is not evaluated from the perspective of probability, so that the probability of collision of the calculated width value of the anti-vibration gap within the structural design service life is unknown (unknown safety level).
Disclosure of Invention
The invention aims to solve the technical problem of providing a method for determining the width of a shockproof joint based on earthquake collision vulnerability, and carrying out earthquake vulnerability analysis on adjacent structures based on random vibration and reliability theory. And (3) solving a collision probability curve of adjacent structures based on the seismic vulnerability, and obtaining a critical interval value corresponding to a specific collision probability through an error control iterative program.
The technical scheme adopted by the invention for solving the technical problems is as follows: the method comprises the following steps of firstly, evaluating and calculating the earthquake collision probability risk of adjacent structures; calculating the non-geometric spectral characteristics of random response of the linear system under non-stationary excitation, then calculating the response characteristics of the linear multi-degree-of-freedom system under modulation Gaussian white noise excitation, further calculating the time-varying bandwidth parameters of the linear elastic system, calculating the failure probability of the system under the first transfinite reliability, and calculating the earthquake collision vulnerability of the adjacent structure; and step three, solving the annual average overrun probability based on the earthquake collision vulnerability of the shockproof joint width of the adjacent linear structure, then solving the collision probability, solving a collision probability curve, and then obtaining a critical distance value corresponding to the determined collision probability through an error control iterative program.
According to the technical scheme, the first step specifically comprises the annual average exceeding probability v of collision of adjacent structures p Can be given by:
the collision probability under the seismic motion intensity, namely the collision vulnerability, is as follows:
probability of conditional failure P P|IM That is, the calculation of the seismic collision vulnerability of adjacent structures can be expressed as solving the single-barrier first-time over-limit reliability problem:
P p|IM ＝P[U rel，max (t)≥d|IM＝im] (3)
the annual average overrun probability of a collision in (1) can be expressed as:
adjacent structure in design service life t L Inner collision probability:
according to the above technical solution, in the second step, the calculating the non-geometric spectral feature of the random response of the linear system under the non-stationary excitation is specifically that the spectral feature required by the reliability application is as follows:
all quantities in equation (11) can be calculated from the spectral features of the following complex-valued non-stationary process:
according to the technical scheme, the calculating of the response characteristic of the linear multi-degree-of-freedom system under the excitation of the modulated white gaussian noise specifically includes that after algebraic operation, the spectral characteristics in equation (12) are obtained as follows:
according to the above technical solution, in the second step, the further calculating the time-varying bandwidth parameter of the linear elastic system specifically includes defining a bandwidth parameter q (t) of a complex-valued non-stationary random process:
according to the technical scheme, in the second step, the calculating of the failure probability of the system under the first overrun reliability specifically includes the following formulas:
according to the above technical solution, in the second step, the calculation of the earthquake collision vulnerability of the adjacent structure is specifically that after mathematical derivation, the covariance of the relative displacement of the adjacent structure system is as follows:
cov(u rel ，v rel )＝cov(u A -u B ，v A -v B )
＝cov(u A ，v A )+cov(u A ，-v B )+cov(-u B ，v A )+cov(-u B ，-v B )
＝cov(u A ，v A )+cov(-u B ，-ν B )
＝cov(u A ，v A )+cov(u B ，v B ) (20)
cov(u rel ,Y rel )＝cov(u A ,Y A )+cov(u B ,Y B ) (21)
speed, hilbert transformation Process Y of neighboring architecture rel The covariance of (c) can also be derived.
The collision probability P of the corresponding targets P The calculation of the width of the shockproof joint of the lower adjacent structure is converted into the problem of inverse reliability: finding the width d of the shockproof gap * Make it possible toAnd (3) determining the inverse reliability problem of the critical interval, namely solving the corresponding critical interval by knowing the collision probability, wherein the flow is shown in the figure 2.
In order to find the critical spacing ξ * The patent equates the inverse reliability problem represented by the flow chart to the problem of making change, namely:
ξ * ＝Zero[f(ξ)] (22)
the error control method is as shown in fig. 3.
According to the technical scheme, the solving of the annual average transcendental probability in the third step is specifically that the annual average transcendental probability of the maximum relative displacement is surpassed:
seismic vulnerability function:
with respect to vulnerability function parameter m R And beta R The method is used for solving the collision probability of the structure by adopting a method of fitting the collision probability of the structure on the basis of the known collision vulnerability curve of the adjacent structure. The structure system is a linear elastic adjacent single degree of freedom System (SDOF) (T) A ＝1.0s，T B ＝0.5s，) And adjacent multiple degree of freedom systems (MDOF) (T) A ＝0.915s，T B ＝0.562s，) The change interval of the adjacent spacing is assumed to be (0-0.5 m), and the seismic oscillation duration is 20s. The results of fitting the two structures at different adjacent pitches are shown in FIGS. 4-11.
The analytic expression of the annual average transcendental probability is as follows:
in considering the intrinsic uncertainty, the displacement-based seismic vulnerability function:
the annual average failure probability is:
a simplified modified form of equation (26):
according to the technical scheme, in the third step, the solved 50-year collision probability is,
in the present invention, the proposed width of the anti-seismic seam has a known probability of collision.
The invention has the following beneficial effects: the method for calculating the width of the shockproof gap of the adjacent structure evaluates the collision risk of the adjacent structure from the angle of probability, calculates the collision probability of the shockproof gap width value within the structural design service life, can overcome the limitation of the shockproof gap width related to the calculation of the corresponding target collision probability value of the adjacent structure at present, is a shockproof gap calculation method with important theoretical significance and engineering application value, and has great reference value to the field of vibration control of the structure.
Drawings
The invention will be further described with reference to the following drawings and examples, in which:
FIG. 1 is a schematic diagram of a neighboring structure in which collisions may occur;
FIG. 2 is a flow chart of an inverse reliability problem for determining critical spacing;
FIG. 3 is a flow chart of an error control iteration method;
FIG. 4 is a comparison of vulnerability equation at ξ =0-0.06m and the fitting results of a known curve;
FIG. 5 is a comparison of the vulnerability equation at ξ =0.07-0.10m versus the results of a fit of a known curve;
FIG. 6 is a comparison of the vulnerability equation at ξ =0.11-0.26m versus the results of a fit of a known curve;
FIG. 7 is a comparison of the vulnerability equation at ξ =0.27-0.50m versus the results of a fit of a known curve;
FIG. 8 is a comparison of the vulnerability equation at ξ =0-0.05m versus the fitting results of a known curve (MDOF);
FIG. 9 is a comparison of the results of fitting a vulnerability equation to a known curve when ξ =0.06-0.09M (MDOF);
FIG. 10 is a comparison of the vulnerability equation at ξ =0.10-0.23m versus the results of a fit of a known curve (MDOF);
FIG. 11 is a comparison of the vulnerability equation at ξ =0.24-0.50m versus the results of a fit of a known curve (MDOF);
FIG. 12 is a collision vulnerability curve when adjacent single degree of freedom structures have different critical spacings when ξ =0.2 m;
FIG. 13 is a collision vulnerability curve when adjacent single degree of freedom structures have different critical spacings when ξ =0.5 m;
FIG. 14 is a graph of collision vulnerability at different critical spacings when ξ =0.2 m;
FIG. 15 is a graph of collision vulnerability at different critical spacings when ξ =0.5 m;
FIG. 16 is a 50-year collision probability for two adjacent architectures and the results of calculations based on the Monte Carlo (MC) method.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.
The embodiment of the invention provides a method for determining the width of a shockproof joint based on earthquake collision vulnerability, which comprises the following steps of firstly, carrying out evaluation calculation on the probability risk of earthquake collision of adjacent structures; calculating the non-geometric spectral characteristics of the random response of the linear system under the non-stationary excitation, then calculating the response characteristics of the linear multi-degree-of-freedom system under the excitation of the modulated Gaussian white noise, further calculating the time-varying bandwidth parameters of the linear elastic system, calculating the failure probability of the system under the first overrun reliability, and calculating the earthquake collision vulnerability of the adjacent structure; and step three, solving the annual average overrun probability based on the earthquake collision vulnerability of the shockproof joint width of the adjacent linear structure, then solving the collision probability, solving a collision probability curve, and then obtaining a critical distance value corresponding to the determined collision probability through an error control iterative program.
Further, the step one specifically comprises the average annual exceeding probability v of collision of adjacent structures p Can be given by:
the collision probability under the seismic motion intensity, namely the collision vulnerability, is as follows:
probability of conditional failure P P|IM That is, the calculation of the seismic collision vulnerability of adjacent structures can be expressed as solving the single barrier first-time over-limit reliability problem:
P p|IM ＝P[U rel，max (t)≥d|IM＝im] (3)
the annual average overrun probability of a collision in (1) can be expressed as:
adjacent structure at design life time t L Inner collision probability:
further, in the second step, the calculating of the non-geometric spectral feature of the random response of the linear system under the non-stationary excitation is specifically that the spectral feature required by the reliability application is as follows:
all quantities in equation (11) can be calculated from the spectral features of the following complex-valued non-stationary process:
further, the calculating of the response characteristic of the linear multiple degree of freedom system under the excitation of the modulated white gaussian noise specifically includes that after algebraic operation, the spectral feature in equation (12) is obtained as follows:
further, in the second step, the further calculating the time-varying bandwidth parameter of the linear elastic system specifically includes defining a bandwidth parameter q (t) of a complex-valued non-stationary random process:
further, in the second step, the calculating of the failure probability of the system under the first overrun reliability is specifically that a classical Vanmarcke and a modified Vanmarcke formula are respectively:
further, in the second step, the calculating of the seismic collision vulnerability of the adjacent structure is specifically that after the covariance of the relative displacement of the adjacent structure system is mathematically derived, the following formula is shown:
cov(u rel ，v rel )＝cov(u A -u B ，v A -v B )
＝cov(u A ，v A )+cov(u A ，-v B )+cov(-u B ，v A )+cov(-u B ，-v B )
＝cov(u A ，v A )+cov(-u B ，-v B )
＝cov(u A ，v A )+cov(u B ，v B ) (20)
cov(u rel ,Y rel )＝cov(u A ,Y A )+cov(u B ,Y B ) (21)
speed, hilbert transformation Process Y of neighboring architecture rel The covariance of (c) may also be derived.
The collision probability P of the corresponding targets P The calculation of the width of the shockproof joint of the lower adjacent structure is converted into the problem of inverse reliability: finding the width d of the shockproof gap * Make it possible toAnd determining an inverse reliability problem of the critical distance, namely solving the corresponding critical distance by knowing the collision probability, wherein the flow is shown in the figure 2.
In order to find the critical spacing ξ * The patent equates the inverse reliability problem represented by the flow chart to the problem of making change, namely:
ξ * ＝Zero[f(ξ)] (22)
the error control method is shown in fig. 3.
Further, the solving of the annual average transcendence probability in the third step is specifically that the annual average transcendence probability of the transcendental maximum relative displacement is:
seismic vulnerability function:
with respect to vulnerability function parameter m R And beta R The method is used for solving the collision probability of the structure by adopting a method of fitting the collision probability of the structure on the basis of the known collision vulnerability curve of the adjacent structure. The structure system is a linear elastic adjacent single degree of freedom System (SDOF) (T) A ＝1.0s，T B ＝0.5s，) And adjacent multiple degree of freedom systems (MDOF) (T) A ＝0.915s，T B ＝0.562s，) The change interval of the adjacent spacing is assumed to be (0-0.5 m), and the seismic oscillation duration is 20s. The results of fitting the two structures at different adjacent pitches are shown in FIGS. 4-11.
The analytic expression of the annual average transcendental probability is as follows:
in considering the intrinsic uncertainty, the displacement-based seismic vulnerability function:
the annual average failure probability is:
a simplified modified form of equation (26):
furthermore, in the third step, the solved 50-year collision probability is,
in the present invention, the proposed width of the anti-seismic seam has a known probability of collision.
According to the method, a time-varying failure probability of a single-degree-of-freedom and multi-degree-of-freedom system under initial static non-stationary seismic excitation is calculated by utilizing a classical Vanmarke method, a modified Vanmarke method and a Poisson approximation method, a modified calculation formula of an annual average collision probability under Gaussian white noise excitation is obtained, and a critical distance value corresponding to a specific collision probability is calculated. The performance-based anti-seismic design theory is applied to the anti-seismic performance evaluation of the adjacent structure, and a method for calculating the width of the anti-seismic joint of the adjacent structure based on collision vulnerability is provided through research methods such as theoretical analysis and numerical simulation, so that the limitation of calculating the width of the anti-seismic joint of the corresponding target collision probability value (for example, the probability of exceeding collision within 50 years is 10% or 2%) of the adjacent structure can be overcome. Fig. 1 is a schematic diagram of an adjacent structure in which a collision may occur. FIGS. 4-7 are linear elastic adjacent single degree of freedom Systems (SDOF) (T) A ＝1.0s,T B ＝0.5s,) The results of the fitting at different adjacent spacings are shown as "+" signs, and the solid line represents the known impact vulnerability curve of the adjacent structure. FIGS. 8-11 are neighboring multiple degree of freedom systems (MDOF) (T) A ＝0.915s,T B ＝0.562s,) The results of the fitting at different adjacent spacings are shown as "+" signs, and the solid line represents the known impact vulnerability curve of the adjacent structure.
In the preferred embodiment of the present invention, the following numerical example is provided.
1) Analyzing the collision vulnerability of the linear elastic single-degree-of-freedom adjacent structure:
supposing two adjacent linear elastic single-degree-of-freedom structures A and B with the periods of T respectively A And T B On the top floor of the lower building, the respective corresponding displacement is u A And u B Then if the relative displacement u rel ＝u A -u B And xi, wherein xi is a given critical spacing value, the structure A and the structure B collide, and the collision event is a single potential barrier problem. Assuming the damping ratio of the two structuresCollision vulnerability through approximate analytic risk function h VM (ξ,t)，h mVM (xi, t) and h p (xi, t) is calculated, and the adjacent space of the two structures takes two conditions into consideration, namely xi 1 ＝0.2，ξ 2 =0.5; get T A ＝1.0s，T B ＝0.5s。
As can be seen from FIGS. 12-13, the coincidence degree of the collision vulnerability curves obtained by the two approximation methods of classical Vanmarke (cVM) and modified Vanmarke (mVM) is very high, and is very close to the calculation result of Monte Carlo (MC), which proves the correctness of programming; meanwhile, as can be seen from fig. 12-13, the poisson approximation (P) relatively underestimates the probability of collision of adjacent single-degree-of-freedom structures. In addition, when the critical spacing is increased from 0.2m to 0.5m, the difference in collision probability obtained by the three approximation methods is reduced.
2) Analyzing the collision vulnerability of the linear elastic multi-degree-of-freedom adjacent structure:
assuming that two adjacent linear elastic multi-degree-of-freedom structures A and B are both shear type anti-bending steel frames, wherein A is an eight-storey building and the interlayer rigidity K A =628801kN/m (same for each storey), storey mass m A =454.545t (same for each storey), B is a four-storey building, the storey stiffness K is B =470840kN/m (same for each storey), floor mass m B =454.545t (same for each layer), damping ratio of two structuresThe basic natural vibration periods of the two buildings are respectively T through calculation A ＝0.915s,T B =0.562s. Taking a critical spacing xi 1 ＝0.2m，ξ 2 =0.5m, and in order to truly reflect the characteristics of the actual building structure, the linear displacement assumption is adopted here, i.e. the displacement of the multiple degree of freedom system at the collision position is equivalent to the displacement of the corresponding generalized single degree of freedom system. Resolving a risk function h by approximation VM (ξ,t)，h mVM (xi, t) and h p The collision vulnerability curves calculated for (ξ, t) are shown in FIGS. 14-15.
As shown in fig. 14 to 15, the analysis result of the multi-degree-of-freedom system is similar to that of the single-degree-of-freedom system, the collision probability is reduced with the increase of the critical distance, and the attenuation amplitude is more obvious than that of the single-degree-of-freedom structure system; the cVM approximation method is very close to the result obtained by the mVM method in a smaller critical distance and is also very close to the calculation result of Monte Carlo (MC), and the programming accuracy is proved; meanwhile, as can be seen from fig. 14 to 15, when the anti-vibration gap width is small, the poisson approximation (P) relatively underestimates the probability of collision of adjacent structures (as in fig. 14); when the width of the anti-vibration gap is large, the probability of collision of adjacent structures is relatively overestimated by the poisson approximation (P) (as shown in fig. 15), so that the subsequent calculation of the width of the anti-vibration gap with the known collision probability by using the poisson approximation (P) analytic method is not recommended. The patent adopts an mVM method.
3) Shockproof gap width values with consistent target collision probability:
by adopting the hybrid algorithm (HYB) of analysis and numerical simulation of the width of the shockproof seam of the adjacent structure with known collision probability based on the earthquake collision vulnerability, the adjacent line elastic multi-degree-of-freedom system mentioned above is taken as an example (the period of the multi-degree-of-freedom system is determined as T) A ＝0.915s，T B =0.562s, damping ratio of the two structures) Fig. 16 shows the 50-year collision probability of two adjacent architectures.
As can be seen from FIG. 16, the results obtained by the method proposed in this patent and Monte Carlo Method (MC) fit well, demonstrating the correctness of the programming. Meanwhile, it can be seen that the collision probability of the structural system gradually decreases with the increase of the adjacent distance, when the distance is smaller (< 0.1 m), the collision probability approaches to 1 infinitely, and for the structural safety, the adjacent distance should be set to be at least greater than 100mm.
Given the probability of collision, the corresponding critical spacing can be derived as: the minimum anti-shock seam width values required for adjacent multiple degree of freedom structural systems are 0.1213, 0.1884, and 0.2599, respectively, when the 50-year collision probability is 50%, 10%, and 2%, respectively.
It will be appreciated that modifications and variations are possible to those skilled in the art in light of the above teachings, and it is intended to cover all such modifications and variations as fall within the scope of the appended claims.
Claims (9)
1. A method for determining the width of a shockproof joint based on earthquake collision vulnerability is characterized by comprising the following steps of firstly, evaluating and calculating the earthquake collision probability risk of an adjacent structure; calculating the non-geometric spectral characteristics of random response of the linear system under non-stationary excitation, then calculating the response characteristics of the linear multi-degree-of-freedom system under modulation Gaussian white noise excitation, further calculating the time-varying bandwidth parameters of the linear elastic system, calculating the failure probability of the system under the first transfinite reliability, and calculating the earthquake collision vulnerability of the adjacent structure; and step three, solving the annual average overrun probability based on the earthquake collision vulnerability of the shockproof joint width of the adjacent linear structure, then solving the collision probability, solving a collision probability curve, and then obtaining a critical interval value corresponding to the determined collision probability through an error control iterative program.
2. The method for determining seismic joint width based on seismic collision vulnerability of claim 1, wherein the step one specifically comprises an average annual transcendental probability v of adjacent structure collisions p Can be represented by the following formulaThe following are given:
the collision probability under the earthquake motion intensity, namely the collision vulnerability is as follows:
probability of conditional failure P P|IM That is, the calculation of the seismic collision vulnerability of adjacent structures can be expressed as solving the single barrier first-time over-limit reliability problem:
P p|IM ＝P[U rel，max (t)≥d|IM＝im] (3)
the annual average overrun probability of a collision in (1) can be expressed as:
adjacent structure in design service life t L Inner collision probability:
3. the method for determining the anti-seismic joint width based on the seismic collision vulnerability of claim 2, wherein in the second step, the non-geometric spectral feature for calculating the random response of the linear system under the non-stationary excitation is specifically that the spectral features required by the reliability application are as follows:
all quantities in equation (11) can be calculated from the spectral features of the following complex-valued non-stationary process:
4. the method for determining the width of the anti-seismic joint based on the seismic collision vulnerability of claim 3, wherein in the second step, the response characteristic of the linear multi-degree-of-freedom system under the excitation of the modulated Gaussian white noise is calculated, and after algebraic operation, the spectral characteristics in equation (12) are obtained as follows:
5. the method for determining the width of the anti-seismic joint based on the seismic collision vulnerability of claim 4, wherein in the second step, the time-varying bandwidth parameter of the linear elastic system is further calculated, specifically, the bandwidth parameter q (t) defining a complex-valued non-stationary random process:
6. the method for determining the width of the anti-seismic joint based on the seismic collision vulnerability of claim 5, wherein in the second step, the calculation of the failure probability of the system under the first overrun reliability is specifically that a classical Vanmarke formula and a modified Vanmarke formula are respectively:
7. the method for determining the seismic joint width based on the seismic impact vulnerability of claim 6, wherein in the second step, the calculation of the seismic impact vulnerability of the adjacent structures is specifically that the covariance of the relative displacement of the adjacent structures is mathematically derived as shown in the following formula:
cov(u rel ，v rel )＝cov(u A -u B ，v A -v B )
＝cov(u A ，v A )+cov(u A ，-v B )+cov(-u B ，v A )+cov(-u B ，-v B )
＝cov(u A ，v A )+cov(u B ，v B )
＝cov(u A ，v A )+cov(u B ，v B ) (20)
cov(u rel ,Y rel )＝cov(u A ,Y A )+cov(u B ,Y B ) (21)
speed, hilbert transformation Process Y of neighboring architecture rel The covariance of (c) can also be derived.
8. The method for determining the width of the anti-seismic joint based on the seismic collision vulnerability according to claim 7, wherein the solving of the annual average exceeding probability in the third step is specifically the annual average exceeding probability exceeding the maximum relative displacement:
seismic vulnerability function:
with respect to vulnerability function parameter m R And beta R On the basis of the known collision vulnerability curve of the adjacent structure, the method of fitting the collision probability of the structure is adopted to solve, the structural system is a linear elastic adjacent single-degree-of-freedom system and an adjacent multi-degree-of-freedom system, the change interval of the adjacent distance is assumed to be (0-0.5 m), the earthquake motion duration is 20s,
the analytic expression of the annual average transcendental probability is as follows:
when considering intrinsic uncertainties, the displacement-based seismic vulnerability function:
the annual average failure probability is:
a simplified modified form of equation (26):
9. the method for determining the width of the anti-seismic joint based on the seismic collision vulnerability of claim 8, wherein in the third step, the 50-year collision probability is solved as,
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| CN110333041A (en) * | 2019-04-29 | 2019-10-15 | 武汉工程大学 | The method for studying adjacent non-resilient SDOF structures collision based on dimensional analysis |
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