Solution method of residual deformation of mid-damping flexible beam component under blast load

Abstract

The present application relates to a kind of solving method of residual deformation of middle damping flexible beam component under explosion load, belong to the technical field of blast-resistant design, specifically including middle damping flexible beam component refers to under the action of explosion, flexible beam component damping parameter x 2 It is equal to plastic hardening coefficient α ;And when the length of explosion load action t i Range, flexible beam component vibration does not reach positive elastic vibration maximum value, after explosion load unloading, rely on inertia force reaches positive elastic displacement maximum value t e At the moment y e After explosion load unloading, flexible beam component continues to vibrate to a certain time t m , reached flexible beam component total elastic-plastic displacement maximum value y m , and according to the whole process of explosion, the process is divided into elastic stage forced vibration, elastic stage free vibration and plastic stage free vibration, elastic rebound stage, plastic rebound stage, elastic vibration six stages, and then determine the residual deformation of middle damping flexible beam component under explosion load.

Description

Technical Field

[0001] This invention relates to a method for solving the residual deformation of a moderately damped flexible beam member under explosive loading, belonging to the field of blast-resistant design technology. Background Technology

[0002] The equivalent single-degree-of-freedom (SDOF) method, as a computationally efficient dynamic analysis method, is widely used in engineering blast-resistant codes of various countries and is also frequently used as a verification tool for displacement analysis in blast load tests of beam members. Building blast-resistant codes and most experimental designs allow beam members to have a certain degree of plastic displacement, indicating that the stiffness of the SDOF system should include both elastic and plastic components. When characterizing plastic stiffness, codes and most researchers often use an ideal elastoplastic resistance model that neglects plastic resistance, without in-depth research on the displacement influence of the resistance-enhancing effect in the plastic stage of beam members under blast loading. The bi-segmented resistance model is a good and accurate constitutive relation describing the resistance-enhancing effect in the plastic stage. Applying it to the vibration displacement analysis of beam members under blast loading will yield more accurate elastoplastic displacement analysis results. This invention uses the bi-segmented resistance model to establish a refined SDOF vibration equation. Based on the relationship between the damping ratio and the plastic resistance-enhancing coefficient, analytical solutions for the displacement in the forward and rebound stages under various conditions are obtained. Combined with typical working conditions, a method for solving the residual deformation of the member is derived. Summary of the Invention

[0003] To address the problems existing in the prior art, this invention provides a method for solving the residual deformation of a moderately damped flexible beam member under explosive loading.

[0004] To achieve the above objectives, the technical solution adopted in this invention is a method for solving the residual deformation of a moderately damped flexible beam member under explosive loading. The moderately damped flexible beam member refers to a flexible beam member whose damping parameter ξ is [value missing] under explosive loading. 2 It is equal to the plasticity strengthening coefficient α; and the duration of the explosive load is t. i Within the specified range, the vibration of the flexible beam component did not reach the maximum value of positive elastic vibration. After the explosive load was unloaded, it relied on inertial force to maintain its position at t. e The maximum positive elastic displacement y is reached at time . e After the explosive load is unloaded, the flexible beam component continues to vibrate until a certain moment t. m The maximum total elasto-plastic displacement y of the flexible beam member was reached. m Based on the entire process of the explosion, the process is divided into six stages: elastic forced vibration, elastic free vibration, plastic free vibration, elastic rebound stage, plastic rebound stage, and elastic vibration.

[0005] Currently, the equivalent single degree of freedom (SDOF) method is widely used in domestic and international standards and academic research. The resistance of flexible beam members considering positive vibration and springback vibration with enhanced resistance during the plastic stage is discussed in [reference needed]. Figure 1 The bilinear resistance model shown.

[0006] The specific expressions for the resistance of flexible beam members considering positive vibration and rebound vibration with enhanced resistance during the plastic stage are as follows:

[0007] (1)

[0008] a. Forced vibration in the elastic stage

[0009] In the elastic phase and within the load duration range of 0 <t<t i The vibration equation of the dynamic system is:

[0010] (2)

[0011] Where t is the time parameter under the explosive action of the flexible beam member, t i M represents the duration of the explosive load. e For the mass of the equivalent flexible beam member in the elastic stage, C e For the damping of the equivalent flexible beam member in the elastic stage, K e For the stiffness of the equivalent flexible beam member in the elastic stage, For the equivalent system vibration acceleration of the flexible beam member, Let y be the vibration velocity of the equivalent system of the flexible beam member, y be the vibration displacement of the equivalent system of the flexible beam member, and ΔP be the vibration velocity of the equivalent system of the flexible beam member. e (t) represents the explosive dynamic load borne by the flexible beam member, which varies with time t. The formulas for calculating the equivalent flexible beam member coefficients are as follows:

[0012] (3)

[0013] Where m is the mass per meter of the actual flexible beam member, l is the span of the actual flexible beam member, ξ is the damping ratio of the actual flexible beam member, K is the stiffness of the actual flexible beam member, and k M k is the mass transformation coefficient for the elastic stage. L This represents the load transformation coefficient for the elastic stage. Since the duration of the explosive impact load is extremely short, it is simplified to a linear load with equal impulse. The explosive load recommended by Chinese protective engineering standards is:

[0014] (4)

[0015] Among them, t i Δp represents the duration of the explosive load. mFor the peak overpressure of the explosive load, the initial displacement and initial velocity of the flexible beam member before bearing the explosive load are both 0. After solving the differential equation, the expressions for displacement and velocity in this stage are determined as follows:

[0016] (5)

[0017] (6)

[0018] Among them, the undamped natural frequency ω and the damped natural frequency ω d Peak overpressure Δp of explosive load m The static displacement y corresponding to static load st The parameters are calculated as follows:

[0019] (7)

[0020] At the time of unloading after the explosion load ends, t i At time t, the corresponding displacement and velocity are

[0021] (8)

[0022] (9)

[0023] b. Free vibration in the elastic stage

[0024] Because the flexible beam component designed is an explosion-proof flexible beam component, when the explosion load is removed, the flexible beam component remains in an elastic state. At this time, the flexible beam component is without external load and its displacement is y. i and speed v i For the initial condition of the damped elastic stage free vibration, that is, when t i <t<t e At that time, the vibration equation of the dynamic system is:

[0025] (10)

[0026] t e For a flexible beam member that has completed elastic vibration and is about to enter the critical moment of plastic vibration, after solving the equation, the displacement and velocity are as follows:

[0027] (11)

[0028] (12)

[0029] Substituting formulas (8) and (9) into formulas (11) and (12), and letting...

[0030] (13)

[0031] Then at t e At time t, the flexible beam member reaches its maximum displacement due to elastic vibration. At this time, the displacement and velocity are respectively:

[0032] (14)

[0033] (15)

[0034] c. Free vibration during the plastic stage

[0035] When the structural vibration time is greater than t e At any given time, there is no external load, and the value is y. e and v e Free vibration in the damped plastic stage under initial conditions, at t m At time t, the structural vibration reaches its maximum displacement, that is, when t e <t<t m At that time, the vibration equation of the dynamic system is:

[0036] (16)

[0037] In the formula, the parameters for the plastic stage are: m e For equivalent quality, c e The equivalent damping force is calculated using the following formula:

[0038] (17)

[0039] α is the ratio of the equivalent stiffness of the plastic stage to that of the elastic stage in a flexible beam member, and is called the plastic hardening coefficient; k m k l Let be the mass and load transformation coefficients during the plastic stage, respectively; the displacement and velocity solutions of equation (16) are:

[0040] (18)

[0041] (19)

[0042] And we can solve for C1 and C2 as follows:

[0043] (20)

[0044] Set equation (19) to 0, and calculate the maximum elastoplastic displacement y of the flexible beam member when it reaches positive vibration. m The total duration corresponding to the time is:

[0045] (twenty one)

[0046] d. Elastic rebound stage

[0047] The flexible beam member vibrates in the positive direction to the peak value of the elastoplastic displacement y m At that time, the vibration velocity v m When the value is zero, the resistance of the flexible beam member also reaches the maximum value of the elastoplastic resistance R. m It begins to vibrate elastically in the opposite direction. The vibration equation of the dynamic system is:

[0048] (twenty two)

[0049] After solving the equations, the displacement and velocity for this stage are obtained as follows:

[0050] (twenty three)

[0051] (twenty four)

[0052] y m v m Substituting into equations (23) and (24), we can solve for C3 and C4 as follows:

[0053] (25)

[0054] If the flexible beam member vibrates without plastic rebound, setting equation (24) to 0, we obtain the maximum rebound displacement y' of the flexible beam member. m The corresponding time t' m If the flexible beam member exhibits plastic rebound during vibration, let formula (23) y=y m -2y e The corresponding time is the total elastic rebound time t. n , will t n Substituting into equations (23) and (24), we obtain the maximum elastic displacement y of the flexible beam member during its first rebound. n Speed ​​v n .

[0055] e. Plastic springback stage

[0056] If the elastic rebound displacement of the flexible beam member is from the beginning to y m -2y e Since the vibration velocities are all non-zero, the flexible beam component will enter a state of plastic springback. The vibration equation of the dynamic system is:

[0057] (26)

[0058] Solving equation (26) yields:

[0059] (27)

[0060] (28)

[0061] Initial condition y n v n Substituting into equations (27) and (28), we can solve for C5 and C6 as follows:

[0062] (29)

[0063] Let t be the t corresponding to a velocity of 0. m , this t' m The corresponding displacement is the maximum elastoplastic displacement y' of the flexible beam member during rebound vibration. m .

[0064] f. Elastic vibration

[0065] Under the influence of damping and resistance, after reaching the first maximum elastic-plastic displacement, the flexible beam member continues to undergo periodic elastic rebound in the opposite direction. The vibration equation of the dynamic system is:

[0066] (30)

[0067] After solving, the displacement and velocity solutions for this stage are obtained as follows:

[0068] (31)

[0069] (32)

[0070] Initial condition y' m v' m Substituting into equations (31) and (32), we obtain C7 and C8 as follows:

[0071] (33)

[0072] When formula (32) is 0, the flexible beam member finally stops vibrating, and the corresponding displacement is the final residual deformation of the medium-damped flexible beam member. We also have a ready-made formula (34) for solving the residual deformation of the member, which can be obtained by substituting the formulas from the previous expressions.

[0073] (34).

[0074] Compared with existing technologies, this invention has the following technical advantages: Based on actual conditions, this invention fully considers the type of flexible beam component, component damping parameters, and the effect of plastic strengthening resistance on the final plastic residual deformation of the component under explosive loads. Furthermore, this method enables precise design of actual components and lays the foundation for blast-resistant design. Attached Figure Description

[0075] Figure 1 This is a diagram of the bilinear resistance model in this invention.

[0076] Figure 2 This is the displacement-time history curve after dimensionless processing of the strengthening coefficient 0.04 in this invention. Detailed Implementation

[0077] To make the technical problems, solutions, and beneficial effects of this invention clearer, the invention will be further described in detail below with reference to embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention.

[0078] The following example illustrates how to solve for the residual deformation of a moderately damped flexible beam under explosive loading, combined with practical blast-resistant design.

[0079] Take a typical flexible beam member (ωt) i =0.2) Damping ratio ξ=0.2, strengthening coefficient 0.04. The dimensionless displacement-time history curve is as follows: Figure 2 As shown: All points marked in the figure are rounded to four decimal places. The original data is:

[0080] y r / y st =(y m -y e +y' m -y n )*(1-α) / y st

[0081] = (0.125545263 - 0.013826345 + (0.097612053 - 0.097898671)) * 0.96

[0082] = 0.106975008

[0083] 0.106975008, rounded to approximately 0.1070

[0084] That is, the plastic displacement of the flexible beam member during the positive motion phase is 0.1073 y. st During the rebound phase, the plastic displacement of the flexible beam member is 0.0003 y. st Ultimately, the residual deformation of this flexible beam member is 0.1070 y. st。

[0085] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions and improvements made within the spirit and principles of the present invention should be included within the scope of the present invention.

Claims

1. A method for solving the residual deformation of a moderately damped flexible beam member under explosive loading, characterized in that: The aforementioned moderately damped flexible beam member refers to the flexible beam member with damping parameter ξ under explosive action. 2 It is equal to the plasticity strengthening coefficient α; and the duration of the explosive load is t. i Within the specified range, the vibration of the flexible beam component did not reach the maximum value of positive elastic vibration. After the explosive load was unloaded, it relied on inertial force to maintain its position at t. e The maximum positive elastic displacement y is reached at time . e After the explosive load is unloaded, the flexible beam component continues to vibrate until a certain moment t. m The maximum total elasto-plastic displacement y of the flexible beam member was reached. m ; Based on the entire process of the explosion, the process is divided into six stages: elastic forced vibration, elastic free vibration, plastic free vibration, elastic rebound stage, plastic rebound stage, and elastic vibration. The specific expressions for the resistance of flexible beam members subjected to positive vibration and rebound vibration during the plastic stage, determined by the equivalent single-degree-of-freedom method, are as follows: （1） a. Forced vibration in the elastic stage In the elastic phase and within the load duration range of 0 <t<t i The vibration equation of the dynamic system is: (2) Where t is the time parameter under the explosive action of the flexible beam member, t i M represents the duration of the explosive load. e For the mass of the equivalent flexible beam member in the elastic stage, C e For the damping of the equivalent flexible beam member in the elastic stage, K e For the stiffness of the equivalent flexible beam member in the elastic stage, For the equivalent system vibration acceleration of the flexible beam component, Let y be the vibration velocity of the equivalent system of the flexible beam member, y be the vibration displacement of the equivalent system of the flexible beam member, and ΔP be the vibration velocity of the equivalent system of the flexible beam member. e (t) represents the explosive dynamic load borne by the flexible beam member, which varies with time t. The formulas for calculating the equivalent flexible beam member coefficients are as follows: （3） Where m is the mass per meter of the actual flexible beam member, l is the span of the actual flexible beam member, ξ is the damping ratio of the actual flexible beam member, K is the stiffness of the actual flexible beam member, and k M k is the mass transformation coefficient for the elastic stage. L This represents the load transformation coefficient for the elastic stage. Since the duration of the explosive impact load is extremely short, it is simplified to a linear load with equal impulse. According to the protective engineering code, the explosive load used is: （4） Among them, t i Δp represents the duration of the explosive load. m For the peak overpressure of the explosive load, the initial displacement and initial velocity of the flexible beam member before bearing the explosive load are both 0. After solving the differential equation, the expressions for displacement and velocity in this stage are determined as follows: （5） （6） Among them, the undamped natural frequency ω and the damped natural frequency ω d Peak overpressure Δp of explosive load m The static displacement y corresponding to static load st The parameters are calculated as follows: （7） At the time of unloading after the explosion load ends, t i At time t, the corresponding displacement and velocity are （8） （9） b. Free vibration in the elastic stage Since the designed component type is a flexible beam with blast resistance, when the blast load is removed, the flexible beam remains in an elastic state. At this time, the flexible beam is without external load and its displacement is y. i and speed v i For the initial condition of the damped elastic stage free vibration, that is, when t i <t<t e At that time, the vibration equation of the dynamic system is: （10） t e For a flexible beam member that has completed elastic vibration and is about to enter the critical moment of plastic vibration, after solving the equation, the displacement and velocity are as follows: （11） （12） Substituting formulas (8) and (9) into formulas (11) and (12), and letting... （13） Then at t e At time t, the flexible beam member reaches its maximum displacement due to elastic vibration. At this time, the displacement and velocity are respectively: （14） （15） c. Free vibration during the plastic stage When the structural vibration time is greater than t e At any given time, there is no external load, and the value is y. e and v e Free vibration in the damped plastic stage under initial conditions, at t m At time t, the structural vibration reaches its maximum displacement, that is, when t e <t<t m At that time, the vibration equation of the dynamic system is: （16） In the formula, the parameters for the plastic stage are: m e For equivalent quality, c e The equivalent damping force is calculated using the following formula: （17） α is the ratio of the equivalent stiffness of the plastic stage to that of the elastic stage in a flexible beam member, and is called the plastic hardening coefficient; k m k l Let be the mass and load transformation coefficients during the plastic stage, respectively; the displacement and velocity solutions of equation (16) are: （18） （19） And we can solve for C1 and C2 as follows: （20） Set equation (19) to 0, and calculate the maximum elastoplastic displacement y of the flexible beam member when it reaches positive vibration. m The total duration corresponding to the time is: （21） d. Elastic rebound stage The flexible beam member vibrates in the positive direction to the peak value of the elastoplastic displacement y m At that time, the vibration velocity v m When the value is zero, the resistance of the flexible beam member also reaches the maximum value of the elastoplastic resistance R. m It begins to vibrate elastically in the opposite direction. The vibration equation of the dynamic system is: （22） After solving the equations, the displacement and velocity for this stage are obtained as follows: （23） （24） y m v m Substituting into equations (23) and (24), we can solve for C3 and C4 as follows: （25） If the flexible beam member vibrates without plastic rebound, setting equation (24) to 0, we obtain the maximum rebound displacement y' of the flexible beam member. m The corresponding time t' m If the flexible beam member vibrates and exhibits plastic rebound, let formula (23) y=y m -2y e The corresponding time is the total elastic rebound time t. n , will t n Substituting into equations (23) and (24), we obtain the maximum elastic displacement y of the flexible beam member during its first rebound. n Speed ​​v n ; e. Plastic springback stage If the elastic rebound displacement of the flexible beam member is from the beginning to y m -2y e Since the vibration velocities are all non-zero, the flexible beam component will enter a state of plastic springback. The vibration equation of the dynamic system is: （26） Solving equation (26) yields: （27） （28） Initial condition y n v n Substituting into equations (27) and (28), we can solve for C5 and C6 as follows: （29） Let t be the t corresponding to a velocity of 0. m , this t' m The corresponding displacement is the maximum elastoplastic displacement y' of the flexible beam member during rebound vibration. m ; f. Elastic vibration Under the influence of damping and resistance, after reaching the first maximum elastic-plastic displacement, the flexible beam member continues to undergo periodic elastic rebound in the opposite direction. The vibration equation of the dynamic system is: （30） After solving, the displacement and velocity solutions for this stage are obtained as follows: （31） （32） Initial condition y' m v' m Substituting into equations (31) and (32), we obtain C7 and C8 as follows: （33） When formula (32) is 0, the flexible beam member finally stops vibrating, and the corresponding displacement is the final residual deformation of the medium-damped flexible beam member. Thus, a ready-made formula (34) for solving the residual deformation of the flexible beam member is obtained. It can be solved by simply calculating and substituting the expressions mentioned above. （34）。

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