Solution method of residual deformation of high-damping rigid beam under blast loading

Abstract

The present application relates to a kind of solving method of high damping rigid beam component residual deformation under explosion load, belong to the technical field of blast-resistant design, specifically including high damping rigid beam component refers to under the action of explosion, rigid beam component damping parameter x 2 Greater than plastic hardening coefficient α ;And when the length of explosion load action t i Range, rigid beam component vibration does not reach the maximum value of positive elastic vibration, after explosion load unloading, rely on inertia force reaches the maximum value of positive elastic displacement t e Time y e , after explosion load unloading, rigid beam component continues to vibrate to a certain time t m , reached the maximum value of rigid beam component total elastic-plastic displacement 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 high damping rigid component under explosion load.

Description

Technical Field

[0001] This invention relates to a method for solving the final plastic residual deformation of a high-damping rigid beam member under explosive load, 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 resistance codes of various countries and is also often used as a verification tool for displacement analysis in blast load tests of beam members. Building blast resistance codes and most experimental designs allow beam members to have a certain degree of plastic displacement, which indicates 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 constitutive relation that can accurately describe the resistance-enhancing effect in the plastic stage. Applying it to the vibration displacement analysis of beam members under blast loading will bring 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 high-damping rigid beam member under explosive loading.

[0004] To achieve the above objectives, the technical solution adopted in this invention is a method for solving the final plastic deformation of a high-damped rigid beam member under explosive load. The high-damped rigid beam member refers to a rigid beam member whose damping parameter ξ is [value missing] under explosive loading. 2 The plasticity strengthening coefficient α is greater than that of the rigid beam member, and the rigid beam member completes the maximum elastic vibration y. e The critical moment t corresponding to the imminent entry into plastic vibration. e Less than the duration t of the explosive load i Numerical value: After the explosive load is unloaded, the rigid beam member continues to vibrate until a certain moment t. m The total elasto-plastic displacement y of the rigid beam member was reached. m Based on the entire process of the explosion, the process is divided into six stages: elastic stage forced vibration, plastic stage forced vibration and plastic stage 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 scholarly research. The resistance of rigid beam members considering positive vibration and springback vibration with enhanced resistance during the plastic stage is discussed in [reference needed].Figure 1 As shown.

[0006] The specific expressions for the resistance of rigid beam members considering the positive vibration and rebound vibration that strengthen the 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 rigid beam member, t i M represents the duration of the explosive load. e For the mass of the equivalent rigid beam member in the elastic stage, C e For the damping of the equivalent rigid beam member in the elastic stage, K e For the stiffness of the equivalent rigid beam member in the elastic stage, For the equivalent system vibration acceleration of the rigid beam member, Let y be the vibration velocity of the equivalent system of the rigid beam member, y be the vibration displacement of the equivalent system of the rigid beam member, and ΔP be the vibration velocity of the equivalent system of the rigid beam member. e (t) represents the explosive dynamic load borne by the rigid beam member, which varies with time t. The formulas for calculating the equivalent rigid beam member coefficients are as follows:

[0012] (3)

[0013] Where m is the mass per meter of the real rigid beam member, l is the span of the real rigid beam member, ξ is the damping ratio of the real rigid beam member, K is the stiffness of the real rigid 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 can be 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. m For the peak overpressure of the explosive load, the initial displacement and initial velocity of the rigid beam member before bearing the explosive load are both 0. After solving the differential equation, the expressions for displacement and velocity at this stage can be 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 e At time t, the corresponding displacement and velocity are:

[0021] (8)

[0022] (9)

[0023] b. Forced vibration during the plastic stage

[0024] When a rigid beam member just enters plastic vibration, the explosive load has not yet disappeared, that is, when t e <t<t i At that time, the vibration equation of the dynamic system is:

[0025] (10)

[0026] 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:

[0027] (11)

[0028] α is the ratio of the equivalent stiffness of a rigid beam member in the plastic stage to that in the elastic stage, 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 (10) are:

[0029] (12)

[0030] (13)

[0031] Initial condition y e v e Substituting into equations (12) and (13), we obtain C1 and C2 as follows:

[0032] (14)

[0033] in

[0034] Let t=t i Substituting these values ​​into the above expressions, we can obtain the values ​​for each case y corresponding to the end of the explosive load. i v i .

[0035] c. Free vibration during the plastic stage

[0036] After the explosive load is applied, the rigid beam member is y i v i Free vibration during the plastic stage under initial conditions, i.e., t i <t<t m At that time, the vibration equation of the dynamic system is:

[0037] (15)

[0038] Solving equation (15) yields:

[0039] (16)

[0040] (17)

[0041] Furthermore, C3 and C4 can be solved as follows:

[0042] (18)

[0043] Setting equation (17) to 0, we can obtain the maximum displacement y of the rigid beam member in the positive vibration direction. m The corresponding total duration is:

[0044] (19)

[0045] Let t=t m Substituting each into equation (16), we can obtain the corresponding y values ​​at the end of the positive vibration plastic stage. m value.

[0046] d. Elastic rebound stage

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

[0048] (20)

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

[0050] (twenty one)

[0051] (twenty two)

[0052] y m v m Substituting into equations (21) and (22), we can solve for C5 and C6 as follows:

[0053] (twenty three)

[0054] If the rigid beam member vibrates without plastic rebound, setting equation (22) to 0, we can obtain the maximum rebound displacement y' of the rigid beam member. m The corresponding time t' m If a rigid beam member exhibits plastic rebound during vibration, let formula (21) y=y m -2y e The corresponding time is the total elastic rebound time t. n , will t n Substituting into equations (21) and (22), the maximum elastic displacement y of the rigid beam member during its first rebound can be obtained. n Speed ​​v n .

[0055] e. Plastic springback stage

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

[0057] (twenty four)

[0058] Solving equation (24) yields: (67)

[0060] (25)

[0061] (26)

[0062] Initial condition y n v n Substituting into equations (25) and (26), we can solve for C7 and C8 as follows: (27)

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

[0064] f. Elastic vibration

[0065] Influenced by damping and resistance, after reaching the first maximum elastic-plastic displacement, the rigid beam member continues to undergo periodic elastic rebound in the opposite direction. For a similar situation, this paper presents the equation and solution for the second elastic rebound; other cases will not be elaborated upon. The vibration equation of the dynamic system is:

[0066] (28)

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

[0068] (29)

[0069] (30)

[0070] Initial condition y' m v' m Substituting into equations (29) and (30), we obtain C9 and C. 10 for:

[0071] (31)

[0072] When formula (30) is 0, the rigid beam member eventually stops vibrating, and the corresponding displacement is the final residual deformation of the high-damped rigid beam member. We can also obtain a ready-made formula (32) for solving the residual deformation of the rigid beam member, which can be solved by substituting the formulas in the previous text one by one.

[0073] (32)

[0074] Compared with existing technologies, the present invention has the following technical advantages: Based on actual conditions, the present invention fully considers the type of rigid beam component, the damping parameters of the rigid beam component, and the effect of plastic strengthening resistance on the final plastic residual deformation of the rigid beam component under explosive loads. Furthermore, this method enables precise design of actual rigid beam 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 0.08 in this invention. Detailed Implementation

[0077] To make the technical problems, technical 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 the solution method for the residual deformation of a high-damping rigid beam member under explosive loading, combined with practical blast-resistant design.

[0079] Take a typical rigid beam member (ωt) i =0.2) Damping ratio ξ=0.3, strengthening coefficient 0.08. 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] = (1.340269682 - 0.139375009 + (1.005202038 - 1.061579424)) * 0.92

[0082] = 1.052955904

[0083] 1.052955904, rounded to approximately 1.0530

[0084] That is, the plastic displacement of the rigid beam member during the positive motion phase is 1.1049 y. st During the rebound phase, the plastic displacement of the rigid beam member is 0.0519 y. st Ultimately, the residual deformation of this rigid beam member is 1.0530 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 high-damping rigid beam member under explosive loading, characterized in that: The aforementioned high-damping rigid beam member refers to the rigid beam member whose damping parameter ξ under explosive action. 2 The plasticity strengthening coefficient α is greater than that of the rigid beam member, and the rigid beam member completes the maximum elastic vibration y. e The critical moment t corresponding to the imminent entry into plastic vibration. e Less than the duration t of the explosive load i Numerical value: After the explosive load is unloaded, the rigid beam member continues to vibrate until a certain moment t. m The total elasto-plastic displacement y of the rigid beam member was reached. m ; Based on the entire process of the explosion, the process is divided into six stages: elastic stage forced vibration, plastic stage forced vibration and plastic stage free vibration, elastic rebound stage, plastic rebound stage, and elastic vibration. The specific expressions for the resistance of rigid 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 rigid beam member, t i M represents the duration of the explosive load. e For the mass of the equivalent rigid beam member in the elastic stage, C e For the damping of the equivalent rigid beam member in the elastic stage, K e For the stiffness of the equivalent rigid beam member in the elastic stage, For the equivalent system vibration acceleration of the rigid beam member, Let y be the vibration velocity of the equivalent system of the rigid beam member, y be the vibration displacement of the equivalent system of the rigid beam member, and ΔP be the vibration velocity of the equivalent system of the rigid beam member. e (t) represents the explosive dynamic load borne by the rigid beam member, which varies with time t. The formulas for calculating the equivalent rigid beam member coefficients are as follows: （3） Where m is the mass per meter of the real rigid beam member, l is the span of the real rigid beam member, ξ is the damping ratio of the real rigid beam member, K is the stiffness of the real rigid 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 is the duration of the explosive load. m For the peak overpressure of the explosive load, the initial displacement and initial velocity of the rigid 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 e At time t, the corresponding displacement and velocity are: （8） （9） b. Forced vibration during the plastic stage When a rigid beam member just enters plastic vibration, the explosive load has not yet disappeared, that is, when t e <t<t i At that time, the vibration equation of the dynamic system is: （10） 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: （11） α is the ratio of the equivalent stiffness of a rigid beam member in the plastic stage to that in the elastic stage, 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 (10) are: （12） （13） Initial condition y e v e Substituting into equations (12) and (13), we obtain C1 and C2 as follows: （14） in , Let t=t i Substituting these values ​​into the above expressions, we obtain the values ​​for each case y corresponding to the end of the explosive load. i v i ; c. Free vibration during the plastic stage After the explosive load is applied, the rigid beam member is y i v i Free vibration during the plastic stage under initial conditions, i.e., t i <t<t m At that time, the vibration equation of the dynamic system is: （15） Solving equation (15) yields: （16） （17） And we can solve for C3 and C4 as follows: （18） Setting equation (17) to 0, we obtain the maximum displacement y of the rigid beam member in the positive vibration direction. m The corresponding total duration is: （19） Let t=t m Substituting these values ​​into equation (16), we obtain the values ​​corresponding to the end of the positive vibration plastic stage. m value; d. Elastic rebound stage Rigid beam members vibrate in the positive direction to the peak value of elastoplastic displacement y m At that time, the vibration velocity v m When the value is zero, the resistance of the rigid beam member also reaches the maximum value of the elastoplastic resistance, R. m It begins to exhibit elastic rebound vibration in the opposite direction. The vibration equation of the dynamic system is: （20） After solving the equations, the displacement and velocity for this stage are obtained as follows: （21） （22） y m v m Substituting into equations (21) and (22), we can solve for C5 and C6 as follows: （23） If the rigid beam member vibrates without plastic rebound, setting equation (22) to 0, we obtain the maximum rebound displacement y' of the rigid beam member. m The corresponding time t' m ; If a rigid beam member vibrates and exhibits plastic rebound, let formula (21) y=y m -2y e The corresponding time is the total elastic rebound time t. n , will t n Substituting into equations (21) and (22), we obtain the maximum elastic displacement y of the rigid beam member during its first rebound. n Speed ​​v n ; e. Plastic rebound stage If the elastic rebound displacement of the rigid beam member is from the beginning to y m -2y e Since the vibration velocities are all non-zero, the rigid beam member will enter a state of plastic springback. The vibration equation of the dynamic system is: （24） Solving equation (24) yields: （67） （25） （26） Initial condition y n v n Substituting into equations (25) and (26), we can solve for C7 and C8 as follows: （27） 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 rigid 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 rigid beam member continues to undergo periodic elastic rebound in the opposite direction. The equation and solution for the second elastic rebound are also given. The vibration equation of this dynamic system is: （28） After solving, the displacement and velocity solutions for this stage are obtained as follows: （29） （30） Initial condition y' m v' m Substituting into equations (29) and (30), we obtain C9 and C. 10 for: （31） When formula (30) is 0, the rigid beam member eventually stops vibrating. The corresponding displacement is the final residual deformation of the high-damped rigid beam member, and a formula (32) for solving the residual deformation of the member is obtained. The solution can be obtained by substituting the above expression. （32）。

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