Calculation method for local equivalent stiffness of beam structure equivalent to deflection or rotation angle

Abstract

The present invention discloses a calculation method for the local equivalent stiffness of a beam structure with equivalent deflection or rotation angle, and the steps are as follows: Set an appropriate length ε of the local damaged beam segment according to the length d of the damaged area of the beam structure; Calculate the equivalent stiffness coefficient c eq ; Calculate the equivalent stiffness of the local damaged beam segment, and the equivalent stiffness = c eq × the section flexural stiffness of the undamaged beam; The equivalent stiffness coefficient c eq can adopt an approximate calculation method or an accurate calculation method. The approximate calculation method calculates c eq for the local damaged beam segments in the ranges from the hinged end to L / 4, from the cantilever end to L / 4, and other positions of the beam structure respectively, where L is the span of the beam. The accurate calculation method calculates c eq respectively according to the types of beam structures, such as simply supported beams, cantilever beams, statically indeterminate beams, and prismatic beams with damage at the middle position of the local damaged beam segments. The present invention proposes a concise calculation method for the equivalent stiffness between local damage measurement points, providing a theoretical basis for the calculation of the actual damage degree when conducting quantitative tests on the damage degree based on static force indicators.

Description

Technical Field

[0001] The present invention belongs to the field of structural health monitoring, and relates to a method for calculating the theoretical damage degree of a beam structure, specifically to a method for calculating the local equivalent stiffness of a beam structure equivalent in deflection or rotation angle. Background Art

[0002] In recent years, there have been more and more old bridges in our country, and the problems emerging have become increasingly prominent. The damage of a beam structure usually shows local damages such as cracking and concrete crushing. When identifying damages, the measuring point spacing is usually fixed. When a damage is found in the structure, it is very likely that a local damage occurs between two measuring points. At this time, what is the equivalent damage degree between the two measuring points? This problem is the key to reasonably interpreting the results of damage quantitative indicators. Due to the great difficulty in quantifying the damage degree, there are few literature reports on experimental verification at present. According to the deflection or rotation angle equivalence of the beam structure, the present method proposes a simple method for calculating the equivalent stiffness between local damage measuring points, providing a theoretical basis for calculating the actual damage degree when carrying out damage degree quantitative tests based on static indicators. Summary of the Invention

[0003] Aiming at the problem of calculating the static equivalent stiffness of local damage of a beam structure, the present invention proposes a method for calculating the local equivalent stiffness of a beam structure equivalent in deflection or rotation angle.

[0004] The method for calculating the local equivalent stiffness of a beam structure equivalent in deflection or rotation angle according to the present invention is as follows:

[0005] (1) According to the length d of the damage area of the beam structure, set an appropriate length ε of the local damage beam segment;

[0006] (2) Calculate the equivalent stiffness coefficient c eq ;

[0007] (3) Calculate the equivalent stiffness of the local damage beam segment, equivalent stiffness = c eq × flexural stiffness of the cross-section of the beam without damage;

[0008] Specifically, in step (2), the equivalent stiffness coefficient c eq can adopt an approximate calculation method or an accurate calculation method:

[0009] a) Approximate calculation method

[0010] ① For the local damage beam segment within the range from the simply supported end of the beam structure to L / 4 and from the cantilever end to L / 4, the calculation method of the equivalent stiffness coefficient is:

[0011]

[0012] where L is the span of the beam, c eqis the equivalent stiffness coefficient, with the origin at the hinged end or the cantilever end of the beam, the x-axis along the length of the beam, pointing to the other end of the beam, a is the distance of the locally damaged beam segment from the origin, ε is the length of the locally damaged beam segment, EI u (x) is the flexural stiffness of the cross-section of the undamaged beam at position x, EI d (x) is the flexural stiffness of the cross-section of the locally damaged beam segment at position x within the range of [a, a + ε];

[0013] When it is a beam with a constant cross-section, EI u (x) is a constant EI, and the equivalent stiffness coefficient c eq can be simplified and calculated by the following formula:

[0014]

[0015] ② For other positions of the beam structure, the calculation method of the equivalent stiffness coefficient is:

[0016]

[0017] When it is a beam with a constant cross-section, the equivalent stiffness EI eq can be simplified and calculated by the following formula:

[0018]

[0019] Among them, EI di is the flexural stiffness of the cross-section of the i-th segment of the locally damaged beam segment with a length of ε divided into m equal parts, and m is a positive integer;

[0020] b) Exact calculation method

[0021] According to the specific type of the structure, it is calculated by one of the following methods:

[0022] 1) Simply supported beam

[0023] The calculation method of the equivalent stiffness coefficient is as follows:

[0024] ① When uniformly loaded:

[0025]

[0026]

[0027]

[0028] Among them, c eql 、c eqr are the equivalent stiffness coefficients when the deflections or rotations on the left and right sides of the locally damaged beam segment of the simply supported beam are equal respectively;

[0029] ② When a concentrated load acts at the mid-span:

[0030]

[0031]

[0032]

[0033] 2) Cantilever beam

[0034] The calculation method of the equivalent stiffness coefficient is as follows:

[0035]

[0036] Among them, when the uniform load acts and the deflection is equivalent, n = 3; when the uniform load acts and the rotation angle is equivalent, or when the concentrated load acts and the deflection is equivalent, n = 2; when the concentrated load acts and the rotation angle is equivalent, n = 1;

[0037] 3) Hyperstatic structure

[0038] When calculating the equivalent stiffness of the locally damaged beam segment within half a span from the hinged support end, the basic structure is simplified to a simply supported beam; when calculating the equivalent stiffness of the locally damaged beam segment within half a span from the fixed support end, the basic structure is simplified to a cantilever beam; for other positions, the equivalent stiffness can be calculated according to the basic structures of simply supported beam or cantilever beam;

[0039] 4) Uniform cross-section beam with damage at the middle position of the locally damaged beam segment

[0040] ① Uniform damage at the middle position of the locally damaged beam segment, the first locally damaged beam segment adjacent to the edge hinged support end or the cantilever end:

[0041]

[0042] Among them, b is the length of the undamaged area on one side in the locally damaged beam segment, d is the length of the damaged area at the middle position of the locally damaged beam segment, 2b + d = ε, EI d is the flexural stiffness of the cross-section of the damaged beam segment, EI d <EI;

[0043] Other locally damaged beam segments:

[0044]

[0045] ② Crack damage at the middle position of the locally damaged beam segment, the first locally damaged beam segment adjacent to the edge hinged support end or the cantilever end:

[0046]

[0047] Other locally damaged beam segments:

[0048]

[0049] Among them, K ris the additional spring stiffness of the crack. For a single-sided crack in a rectangular cross-section beam, it is calculated according to the following formula:

[0050]

[0051]

[0052]

[0053] where h is the height of the rectangular cross-section beam, and h cr is the crack height, and h cr <h.

[0054] Specifically, in step (1), the locally damaged beam segments are generally divided into equal lengths, and the length ε of the locally damaged beam segment is not greater than L / 4.

[0055] Specifically, in step (2), the integration can be calculated using mathematical software such as matlab.

[0056] The present invention takes the equivalent stiffness of local damage of the beam structure as the research object. Through the derivation of simply supported beams, cantilever beams, and hinged-fixed beams, a calculation method for the local equivalent stiffness of the beam structure based on deflection or rotation equivalence is proposed. Through examples of simply supported beams, cantilever beams, hinged-fixed beams, and three-span continuous beams, the effectiveness of the calculation method for the local equivalent stiffness of the beam structure with deflection or rotation equivalence is verified, providing a theoretical basis for the calculation of the actual damage degree when carrying out quantitative tests on the damage degree based on static force indexes. Description of the Drawings

[0057] Figure 1 is a schematic diagram of the stiffness equivalence of the locally damaged beam segment of the simply supported beam of the present invention.

[0058] Figure 2 is the bending moment diagram of the uniformly distributed load acting on the locally damaged simply supported beam of the present invention.

[0059] Figure 3 is the bending moment diagram of the unit force acting on the left side of the locally damaged beam segment of the locally damaged simply supported beam of the present invention.

[0060] Figure 4 is the bending moment diagram of the unit force acting on the right side of the locally damaged beam segment of the locally damaged simply supported beam of the present invention.

[0061] Figure 5 is the bending moment diagram of the uniformly distributed load acting on the equivalent stiffness simply supported beam of the present invention.

[0062] Figure 6 is the bending moment diagram of the unit force acting on the left side of the locally damaged beam segment of the equivalent stiffness simply supported beam of the present invention.

[0063] Figure 7It is the bending moment diagram of the simply supported beam with equivalent stiffness of the present invention when a unit force acts on the right side of the locally damaged beam segment.

[0064] Figure 8 It is the bending moment diagram of the simply supported beam with local damage of the present invention under the action of a concentrated load at the mid-span.

[0065] Figure 9 It is the bending moment diagram of the simply supported beam with equivalent stiffness of the present invention under the action of a concentrated load at the mid-span.

[0066] Figure 10 It is the bending moment diagram of the simply supported beam with local damage of the present invention when a unit bending moment acts on the left side of the locally damaged beam segment.

[0067] Figure 11 It is the bending moment diagram of the cantilever beam of the present invention under the action of a uniformly distributed load of damage.

[0068] Figure 12 It is the bending moment diagram of the cantilever beam of the present invention when a unit force acts on the cantilever end.

[0069] Figure 13 It is the bending moment diagram of the cantilever beam of the present invention when a unit bending moment acts at the z position (on the left side of the damage).

[0070] Figure 14 It is the bending moment diagram of the cantilever beam of the present invention under the action of a concentrated load of damage.

[0071] Figure 15 It is the model diagram of the hinged-fixed beam of the present invention.

[0072] Figure 16 It is the bending moment diagram of the external load under the basic structure of the hinged-fixed beam and cantilever beam of the present invention.

[0073] Figure 17 It is the bending moment diagram of the redundant force under the basic structure of the hinged-fixed beam and cantilever beam of the present invention.

[0074] Figure 18 It is the bending moment diagram of the redundant unit force under the basic structure of the hinged-fixed beam and cantilever beam of the present invention.

[0075] Figure 19 It is the bending moment diagram of the unit force (z is on the right side of the damage) under the basic structure of the hinged-fixed beam and cantilever beam of the present invention.

[0076] Figure 20 It is the bending moment diagram of the unit force (z is on the left side of the damage) under the basic structure of the hinged-fixed beam and cantilever beam of the present invention.

[0077] Figure 21 It is the schematic diagram of local damage at the middle position of the hinged-fixed beam of the present invention.

[0078] Figure 22 It is the bending moment diagram of the redundant force under the basic structure of the hinged-fixed beam and simply supported beam of the present invention.

[0079] Figure 23 It is the redundant unit force moment diagram under the basic structure of the simply supported beam of the hinged-fixed beam of the present invention.

[0080] Figure 24 It is a schematic diagram of local damage at the fixed end of the hinged-fixed beam of the present invention.

[0081] Figure 25 It is a schematic diagram of crack damage at the midpoint of the pure bending section of the simply supported beam of the present invention.

[0082] Figure 26 It is a model diagram of the simply supported beam in the first embodiment of the present invention.

[0083] Figure 27 It is the equivalent stiffness coefficient of the left 1 / 3 damaged section of the simply supported beam in the first embodiment of the present invention (under the action of q).

[0084] Figure 28 It is the equivalent stiffness coefficient of the left 1 / 3 damaged section of the simply supported beam in the first embodiment of the present invention (under the action of P).

[0085] Figure 29 It is the comparison of deflection errors of the left 1 / 3 damaged section of the simply supported beam segment 5 in the first embodiment of the present invention (under the action of P).

[0086] Figure 30 It is the equivalent stiffness coefficient of the middle 1 / 3 damaged section of the simply supported beam in the first embodiment of the present invention (under the action of q).

[0087] Figure 31 It is the equivalent stiffness coefficient of the middle 1 / 3 damaged section of the simply supported beam in the first embodiment of the present invention (under the action of P).

[0088] Figure 32 It is the equivalent stiffness coefficient of the right 1 / 3 damaged section of the simply supported beam in the first embodiment of the present invention (under the action of q).

[0089] Figure 33 It is the equivalent stiffness coefficient of the right 1 / 3 damaged section of the simply supported beam in the first embodiment of the present invention (under the action of P).

[0090] Figure 34 It is the equivalent stiffness coefficient of the left 1 / 3 damaged section of the locally damaged beam segment of the cantilever beam in the second embodiment of the present invention.

[0091] Figure 35 It is the equivalent stiffness coefficient of the middle 1 / 3 damaged section of the locally damaged beam segment of the cantilever beam in the second embodiment of the present invention.

[0092] Figure 36 It is the equivalent stiffness coefficient of the right 1 / 3 damaged section of the locally damaged beam segment of the cantilever beam in the second embodiment of the present invention.

[0093] Figure 37It is the comparison of the deflection error of the left 1 / 3 of the beam segment 5 of the three-hinged-fixed beam in the third embodiment of the present invention (under the action of P).

[0094] Figure 38 It is the deflection (under the action of P) of the right 1 / 3 of the beam segment 20 of the three-hinged-fixed beam in the third embodiment of the present invention.

[0095] Figure 39 It is the comparison of the deflection difference of the right 1 / 3 of the beam segment 20 of the three-hinged-fixed beam in the third embodiment of the present invention (under the action of P).

[0096] Figure 40 It is the model diagram of the three-span continuous beam in the fourth embodiment of the present invention

[0097] Figure 41 It is the comparison of the deflection error of the middle 1 / 3 of the beam segment 1 of the three-span continuous beam in the fourth embodiment of the present invention (under the action of P).

[0098] Figure 42 It is the comparison of the deflection difference of the middle 1 / 3 of the beam segment 20 of the three-span continuous beam in the fourth embodiment of the present invention (under the action of P).

[0099] Figure 43 It is the comparison of the deflection error of the left 1 / 3 of the beam segment 20 of the three-span continuous beam in the fourth embodiment of the present invention (under the action of P). Specific embodiments

[0100] The present invention will be further described below in conjunction with the accompanying drawings and embodiments. When the following description involves the accompanying drawings, unless otherwise indicated, the same numbers in different drawings represent the same or similar elements.

[0101] Figure 1 It is the schematic diagram of the stiffness equivalence of the locally damaged beam segment of the simply supported beam of the present invention. In the figure, L is the beam length, the origin is at the left end of the beam, the x-axis is along the beam length direction, pointing to the right end of the beam, z is the action position of the concentrated load P, a is the distance of the locally damaged beam segment from the origin, ε is the length of the locally damaged beam segment, d is the length of the actual damaged area, b is the distance of the actual damaged beam segment from the left measuring point, EI d is the stiffness of the actual damaged beam segment, EI eq is the equivalent stiffness of the beam segment between the measuring points, w d is the deflection in the damaged state, w eq is the deflection of the equivalent model.

[0102] The calculation method of the local equivalent stiffness of the beam structure with equivalent deflection or rotation angle according to the present invention is as follows:

[0103] Step 1: Set a suitable length ε of the locally damaged beam segment according to the length d of the damaged area of the beam structure;

[0104] Step 2: Calculate the equivalent stiffness coefficient c eq ;

[0105] Step 3: Calculate the equivalent stiffness of the locally damaged beam segment. The equivalent stiffness = c eq × the flexural stiffness of the cross-section of the undamaged beam;

[0106] Specifically, in Step 2, the equivalent stiffness coefficient c eq can be calculated using an approximate calculation method or an exact calculation method:

[0107] a) Approximate calculation method

[0108] For the locally damaged beam segment within the range from the hinged end to L / 4 or from the cantilever end to L / 4 of the beam structure, the equivalent stiffness calculation method is as follows:

[0109]

[0110] where L is the span of the beam, c eq is the equivalent stiffness coefficient, c eq ·EI u (x) is the equivalent stiffness, with the origin at the hinged end or the cantilever end of the beam, the x-axis along the beam length direction, pointing to the other end of the beam, a is the distance of the locally damaged beam segment from the origin, ε is the length of the locally damaged beam segment, EI u (x) is the flexural stiffness of the cross-section of the undamaged beam at position x, EI d (x) is the flexural stiffness of the cross-section of the locally damaged beam segment at position x within the range of [a, a + ε];

[0111] When it is a prismatic beam, EI u (x) is a constant EI, and the equivalent stiffness EI eq can be simplified to the following formula for calculation:

[0112]

[0113] For other positions of the beam structure, the equivalent stiffness calculation method is as follows:

[0114]

[0115] When it is a prismatic beam, the equivalent stiffness EI eq can be simplified to the following formula for calculation (series spring model):

[0116]

[0117] where EI di is the flexural stiffness of the cross-section of the i-th segment of the locally damaged beam segment with length ε divided into m equal parts, and m is a positive integer;

[0118] b) Exact calculation method

[0119] To further obtain a more accurate equivalent stiffness EI eq, the equivalent stiffness coefficient can be calculated separately according to the types of beam structures, such as simply supported beams, cantilever beams, and statically indeterminate beams;

[0120] For a beam with a constant cross-section, when the damage occurs at the middle position of the locally damaged beam segment:

[0121] ① When the damage at the middle position of the locally damaged beam segment is uniform, except for the first locally damaged beam segment adjacent to the simply supported end and the cantilever end, the flexural stiffness of the cross-section of other locally damaged beam segments can be calculated by the following method (series spring model):

[0122]

[0123] where d is the length of the damaged area at the middle position of the locally damaged beam segment, and EI d is the flexural stiffness of the cross-section of the damaged beam segment, and EI d <EI;

[0124] ② When the damage at the middle position of the locally damaged beam segment is a crack, for the first locally damaged beam segment adjacent to the simply supported end and the cantilever end:

[0125]

[0126] For other locally damaged beam segments (series spring model):

[0127]

[0128] where K r is the additional spring stiffness of the crack. For a rectangular cross-section beam, it is calculated according to the following formula:

[0129]

[0130]

[0131]

[0132] where h is the beam height of the rectangular cross-section beam, and h cr is the crack height, and h cr <h.

[0133] In step 2, for a simply supported beam, the schematic diagram of the calculation principle of the local equivalent stiffness is as Figure 1 , the length of the equivalent beam segment is ε, and the distance from the origin is a. Since the simply supported beam is symmetric about the left and right, only the case where the local damage is located in the left half-span is analyzed. For the action of a uniformly distributed load, the bending moment diagram of the locally damaged beam is as Figure 2 , and the bending moment diagrams when a unit force acts on the left and right sides of the damage are respectively as Figure 3 , Figure 4 , and the bending moment diagrams of the equivalent stiffness beam are respectively as Figures 5 to 7 , when equivalent according to the deflection, if it is required that the deflections on the left side of the damage are the same, fromFigure 2 , Figure 3 The deflection of the locally damaged beam can be obtained by graphical multiplication as follows:

[0134]

[0135] From Figure 5 , Figure 6 The deflection of the equivalent stiffness beam can be obtained by graphical multiplication as follows:

[0136]

[0137] Comparing the two equations, when w zd = w zeq , we have:

[0138]

[0139]

[0140] Similarly, from the equivalence of the deflections on the right side of the damage, we can obtain:

[0141]

[0142]

[0143] The equivalent stiffness coefficient takes the average of the two, which is:

[0144]

[0145] where c eql and c eqr are the equivalent stiffness coefficients when the deflections or rotations on the left and right sides of the locally damaged beam section of the simply supported beam are equal, respectively.

[0146] Similarly, when the simply supported beam is subjected to a concentrated load at the mid-span, by performing graphical multiplication with Figure 8 , Figure 9 and the corresponding unit force moment diagram, we can obtain:

[0147]

[0148]

[0149]

[0150] From Figure 10 it can be seen that compared with its moment diagram and Figure 3 , there is only a constant difference, and the shapes of the moment diagrams are the same. Therefore, the analysis results of rotation equivalence and deflection equivalence are exactly the same, and the above results are also applicable to the calculation of rotation equivalence of simply supported beams.

[0151] In step 2, for the cantilever beam, the moment diagram under the uniformly distributed load is asFigure 11 When equivalent to the deflection at the cantilever end, the moment diagram of the unit force is as shown in Figure 12 . By multiplying the diagrams, we get:

[0152]

[0153]

[0154] When equivalent to the rotation at the z position, from Figure 11 and Figure 13 , by multiplying the moment diagrams, we get:

[0155]

[0156] The moment diagram under the concentrated load at the cantilever end is as shown in Figure 14 . When equivalent to the deflection at the cantilever end, from Figure 12 and Figure 14 , by multiplying the moment diagrams, we get:

[0157]

[0158] When equivalent to the rotation at the z position, from Figure 13 and Figure 14 , by multiplying the moment diagrams, we get:

[0159]

[0160] In step 2, for the statically indeterminate beam, taking the simply supported - fixed beam under the concentrated load at the mid - span as an example for analysis, the model is as shown in Figure 15 . Taking the cantilever beam as the basic structure, the moment diagrams under the external load, redundant force, redundant unit force, unit force at the right side of the damage, and unit force at the left z position are respectively as shown in Figures 16 to 20 . Assume that the redundant force under the external load of the locally damaged beam is X, and the redundant force of the equivalent stiffness beam is X′.

[0161] 1) Analyze the deflection at the right side of the damage. The damage position is as shown in Figure 21 . From Figure 19 respectively multiplied by the moment diagrams of Figure 16 and Figure 17 , we get:

[0162]

[0163]

[0164] From w zd = w zeq , it can be known that X = X′.

[0165] The redundant force is calculated by multiplying Figure 18 respectively by the moment diagrams of Figure 16 and Figure 17 :

[0166]

[0167] From X = X′, we get:

[0168]

[0169]

[0170] The equivalent stiffness coefficient c eqr has the same result as that of the cantilever beam (see Equation (24)).

[0171] 2) Analyze the deflection on the damaged left side. First, find the redundant force X of the locally damaged beam. Since the calculation is relatively complex, a numerical solution is taken as an example for illustration. Assume the beam length L = 10 m, the beam segment length ε = 0.5 m, the position of the locally damaged beam segment a = 2 m, and the redundant force X is calculated by Figure 18 respectively multiplied by Figure 16 , Figure 17 through the moment diagram multiplication:

[0172]

[0173]

[0174]

[0175] Similarly, the redundant force X′ of the equivalent stiffness beam can be obtained:

[0176]

[0177]

[0178] There are two unknowns in Equation (35): X′ and c eql , and one more equation is needed to solve. Assume the deflections at z = 1 m are equivalent. From Figure 20 respectively multiplied by Figure 16 , Figure 17 through the moment diagram multiplication, we get:

[0179]

[0180]

[0181] From w zd = w zeq we can obtain:

[0182]

[0183]

[0184]

[0185] By simultaneously solving equations (35) and (39), we can obtain:

[0186]

[0187] If we make an equivalent deflection at z = 2m, we can obtain the formula:

[0188]

[0189] By simultaneously solving equations (35) and (41), we can obtain:

[0190]

[0191] It can be seen that the two sets of solutions are basically the same, and the values of X' and X are also very close. Can we consider that the redundant forces before and after damage are equal? By directly solving equations (39) and (41) for c eql , and their values are respectively:

[0192]

[0193]

[0194] The difference between the two is relatively large, indicating that the change in redundant forces needs to be considered. Therefore, for statically indeterminate structures, directly calculating the equivalent stiffness according to the equivalent deflection is a relatively complex process, and the results are prone to errors.

[0195] From equation (30), we know that:

[0196]

[0197] Therefore, the equivalent stiffness coefficient is

[0198]

[0199] 3) Analyze whether it affects the calculation results for different basic structures. Take the basic structure of a simply supported beam to analyze the equivalent deflection of the right side of the damage. The redundant force and the bending moment diagram of the unit redundant force are as Figure 22 , Figure 23 , and for other bending moment diagrams, refer to the simply supported beam analysis section.

[0200] The redundant force X is obtained by multiplying the diagrams of Figure 23 respectively with Figure 22 , Figure 8 :

[0201]

[0202]

[0203] Similarly, the redundant force X' of the equivalent stiffness beam can be obtained:

[0204]

[0205]

[0206] Assume that the deflections at z = 5m are equivalent. From Figure 4 Multiply and calculate respectively with Figure 22 and Figure 8 of the bending moment diagrams, we get:

[0207]

[0208]

[0209]

[0210] By solving equations (50) and (53) simultaneously, we can obtain:

[0211]

[0212] Comparing with equation (45), the results of the two are exactly the same, indicating that the choice of the basic structure does not affect the calculation result of the equivalent stiffness. However, when taking the simply supported beam as the basic structure, the calculation difficulty of c eqr is significantly greater than that of the cantilever beam basic structure.

[0213] 4) If the cantilever beam basic structure is adopted, by multiplying the bending moment diagram of a single unit force with the redundant force or the bending moment diagram of the external load, and only analyzing the damaged position, because it can only be calculated equivalently by :

[0214]

[0215]

[0216] When the simply supported beam basic structure is adopted, by multiplying the bending moment diagram of a single unit force with the redundant force or the bending moment diagram of the external load, and only analyzing the damaged position, it is calculated equivalently by . When the deflection on the right side of the damaged beam segment is equivalent, the result is the same as that of the above-mentioned cantilever beam:

[0217]

[0218] When the deflection on the left side of the damaged beam segment is equivalent:

[0219]

[0220]

[0221] equivalent and For concentrated loads, the calculation results of the two are the same.

[0222] According to The calculation result by equivalent calculation is the same as that in Equation (56).

[0223] According to the basic structure of a simply supported beam, from The comparison of the equivalent stiffness coefficient calculation results with the theoretical values by equivalent calculation is shown in Table 1. It can be seen that for the hinged support end, the calculation results by using the basic structure of a simply supported beam are basically the same as the theoretical values.

[0224] Table 1 Comparison of equivalent stiffness coefficients

[0225]

[0226] 5) For the case of local damage at the fixed support end, such as Figure 24 , take the basic structure of a cantilever beam to analyze the equivalent mid-span deflection.

[0227] The redundant force X is obtained by Figure 18 Multiplication of the moment diagrams of Figure 16 , Figure 17 respectively:

[0228]

[0229]

[0230] Similarly, the redundant force X' of the equivalent stiffness beam can be obtained:

[0231]

[0232]

[0233] From the equivalent mid-span deflection (at z = 5m), we get:

[0234]

[0235]

[0236] By solving Equations (63) and (65) simultaneously, we can obtain:

[0237]

[0238] 6) If the basic structure of a cantilever beam is adopted, by multiplying the moment diagram of a single unit force with the redundant force or the moment diagram of the external load, and only analyzing the damage position, the calculation is carried out by equivalent calculation:

[0239]

[0240]

[0241] Calculated equivalently by :

[0242]

[0243]

[0244] Calculated equivalently by :

[0245]

[0246]

[0247] Adopt the basic structure of a simply supported beam, multiply the moment diagram of a single unit force by the redundant force or the moment diagram of the external load, and only analyze the damaged position. Calculated equivalently by :

[0248]

[0249]

[0250] Calculated equivalently by :

[0251]

[0252]

[0253] Calculated equivalently by When calculating equivalently, due to the M X diagrams of the two basic structures being the same, the results are also the same.

[0254] The comparison between the results of the equivalent stiffness coefficients calculated by each equivalent method and the theoretical values is shown in Table 2. It can be seen that for the fixed end, the result calculated by adopting the basic structure of a cantilever beam is the closest to the theoretical value, and the result of the basic structure of a simply supported beam has a larger error.

[0255] Therefore, for a statically indeterminate structure, for the side hinged support end, it can be simplified to a simply supported beam for calculating the equivalent stiffness, and for the fixed end, it can be simplified to a cantilever beam for calculation.

[0256] Table 2 Equivalent stiffness coefficient c eq Comparison

[0257]

[0258] In step 2, for a beam with a constant cross-section, when the middle position of the locally damaged beam segment is damaged, such as the crack damage in Figure 25 , h cris the crack height. For a rectangular cross-section beam with a beam height of h, the additional spring stiffness K of the crack r can be obtained by integral calculation from the crack stress manual, as shown in Equation (8).

[0259] When the deflection in the pure bending region on the right side of the crack is equivalent (series spring model):

[0260]

[0261]

[0262]

[0263]

[0264] When the deflection in the pure bending region on the left side of the crack is equivalent (series spring model):

[0265]

[0266]

[0267]

[0268]

[0269] It can be seen that c eqr = c eql , so it is only necessary to analyze the deflection on the damaged side. The following analyzes the case under a concentrated load at the mid-span. When the deflection on the right side of the crack is equivalent:

[0270]

[0271]

[0272] When it is the damage of the side beam segment, a = 0, the above formula is simplified to:

[0273]

[0274]

[0275] When it is not the side beam segment, a ≥ 1, at this time:

[0276]

[0277] Therefore, (series spring model):

[0278]

[0279] When the deflection on the left side of the crack is equivalent (the crack is located in the left half-span) (series spring model):

[0280]

[0281]

[0282]

[0283]

[0284] Therefore, the influence of the load form on the results is very small. Similarly, the cases of a cantilever beam and the uniform damage at the middle position of a locally damaged beam segment can be analyzed.

[0285] The results of the first locally damaged beam segment adjacent to the simply supported end and the cantilever end are as shown in Equation (95).

[0286]

[0287] Among them, b is the length of the undamaged area on one side in the locally damaged beam segment, d is the length of the damaged area at the middle position of the locally damaged beam segment, 2b + d = ε. When d << ε, the above formula can be reduced to Equation (96), which is consistent with the formula (88) of crack damage in form.

[0288]

[0289] The flexural stiffness of the cross-section of other locally damaged beam segments can be calculated by the following method (series spring model):

[0290]

[0291] In Step 1, the locally damaged beam segments are generally divided into equal lengths, and the number of locally damaged beam segments within the range of L / 4 is not less than 1, that is, each span is at least divided into 4 beam segments.

[0292] Example 1: A simply supported beam with a span of 10 m, divided into one beam segment every 0.5 m, the cross-sectional size is b×h = 30 cm×50 cm, C40 concrete, elastic modulus E = 3.25×10 4 MPa, Poisson's ratio is 0.2, and the density is 25 kN / m 3 . The locally damaged beam segment of 0.5 m is equally divided into three segments, and the equivalent stiffness when the stiffness of each small segment area drops to EI d = 0.6EI is analyzed respectively. The model is as Figure 26 .

[0293] (1) When the stiffness of the left 1 / 3 area of the locally damaged beam segment uniformly drops to 0.6EI:

[0294] According to Step 1, the length of the damaged area of the structure d = 0.5 / 3 m, and the length of the locally damaged beam segment ε = 0.5 m.

[0295] Calculate the equivalent stiffness coefficients c of beam segments 1 - 5 and 16 - 20 by the approximate method in step 2 eq(1) :

[0296]

[0297] Calculate the equivalent stiffness coefficients c of beam segments 6 - 15 by the approximate method in step 2 eq(2) :

[0298]

[0299] c eq(1) and c eq(2) The equivalent stiffness coefficients of each beam segment calculated jointly are denoted as c eq(1,2) .

[0300] According to step 2, the exact solution of the equivalent stiffness coefficient of the simply supported beam can be calculated. The results of equivalent calculation from the left deflection (or rotation angle) of the locally damaged beam segment are c eql , and c eqr for the right - hand side equivalent. The average value is c eq . The results under the action of uniform load q and concentrated load P are as shown in Figure 27 and Figure 28 respectively. It can be seen that the results of uniform load and concentrated load are almost equal, and the difference between c eql and c eqr is not significant. The results of c eq(1,2) and c eq are close, and the method has good effect.

[0301] When beam segment 5 is damaged, the comparison of the deflection error between the equivalent stiffness beam and the locally damaged beam under the action of the mid - span concentrated load P is as shown in Figure 29 . It can be seen that c eql satisfies that the left deflection of beam segment 5 is 0, c eqr satisfies that the right deflection of beam segment 5 is 0, and the comprehensive deflection error of c eq is the smallest. The deflection error of c eq(2) calculated by the method of step 2 increases slightly, and c eq(1) = c eqr .

[0302] Step 3: Equivalent stiffness = c eq EI.

[0303] (2) When the stiffness in the middle 1 / 3 region of the locally damaged beam segment uniformly drops to 0.6EI:

[0304] The results under the action of uniform load q and concentrated load P are as shown in Figure 30 and Figure 31 respectively. It can be seen that except for the first beam segment, the different equivalent stiffness coefficients of each beam segment are basically the same, and the method has good effect.

[0305] From Step 2, for the equal cross-section beam, the middle position of the locally damaged beam segment has uniform damage. Except for the first locally damaged beam segment adjacent to the simply supported end and the cantilever end, the flexural stiffness of the cross-section of other locally damaged beam segments can be calculated as follows:

[0306]

[0307] c eq Compared with c eq(1,2) Only the first beam segment has more differences, so the middle beam segments can all adopt c eq(2) .

[0308] (3) When the stiffness in the right 1 / 3 region of the locally damaged beam segment uniformly drops to 0.6EI:

[0309] The results under the action of uniform load q and concentrated load P are respectively as Figure 32 , Figure 33 . The results are similar to those of the damage in the left 1 / 3 region. The equivalent stiffness coefficients of each method are close to the theoretical values. The difference is that the equivalent stiffness coefficients near both ends are less than those in the middle position.

[0310] Example 2: A 10m-span cantilever beam with the fixed end at the left end and other parameters the same as those of the simply supported beam.

[0311] When the stiffness in the left, middle, and right 1 / 3 regions of the locally damaged beam segment uniformly drops to 0.6EI, the equivalent stiffness coefficients calculated by each method are respectively as Figures 34 to 36 (The results of the equivalent of rotation under the action of uniform load and the equivalent of deflection under the action of concentrated load are the same). As can be seen from the figure, c eq(1,2) is relatively close to the equivalent stiffness coefficients calculated by other methods. Therefore, when the accuracy requirement is not very high, c eq(1,2) can be used to calculate the equivalent stiffness coefficient c eq of the cantilever beam.

[0312] Example 3: A 10m-span simply supported - fixed beam with the simply supported end at the right end and the fixed end at the left end, and other parameters the same as those of the simply supported beam.

[0313] When the stiffness in the left 1 / 3 region of beam segment 5 uniformly drops to 0.6EI, the specific analysis results of the equivalent stiffness coefficient are shown in Table 1 of the previous part. The c eq calculated according to the simply supported beam is basically the same as the theoretical value. Since beam segment 5 is exactly at the L / 4 position, the equivalent stiffness coefficient is calculated by the approximate method in Step 2. The calculated c eq(1) = c eqr . The calculated c eq(2) = 0.8182, and c eq(1) is closer to the theoretical value. The deflection errors calculated by each stiffness coefficient are as Figure 37, It can be seen that the deflection errors are relatively close and are all small.

[0314] Table 1 Comparison of equivalent stiffness coefficients

[0315]

[0316] When the stiffness of the 20th right 1 / 3 region of the beam segment uniformly drops to 0.6EI, the deflection under the concentrated load at the mid-span is as Figure 38 , The specific analysis results of the equivalent stiffness coefficients are shown in Table 2 in the previous part. Calculated as a cantilever beam, c eq(悬臂梁) = 0.80782394 is the closest to the theoretical value c eq(理论值) = 0.79995804. Calculated by the approximate method in step 2, c eq(2) = 0.8182. The deflection errors calculated by each stiffness coefficient are as Figure 39 , It can be seen that c eq(理论值) can make the deflection difference of the 11 nodes be 0. The deflection differences of c eq(悬臂梁) and c eq(2) slightly increase. The equivalent stiffness coefficients of each method are relatively close.

[0317] Table 2 Equivalent stiffness coefficient c eq Comparison

[0318]

[0319] Example 4: A 3×10m three-span continuous beam, divided into one beam segment every 0.5m, with other parameters the same as those of the simply supported beam. The model is as Figure 40 .

[0320] When the stiffness of the middle 1 / 3 region of the beam segment 1 uniformly drops to 0.6EI, the three-span continuous beam is simplified to the basic structure of a simply supported beam. Under the concentrated load at the mid-span of the first span, from step 2, c eq(1) = c eqr = 0.8523, c eq(2) = 0.8182. The deflection errors are as Figure 41 , It can be seen that when c eq(1) = c eqr = 0.8523, the deflection error is 0. When c eq(2) = 0.8182, the deflection error is also very small.

[0321] When the stiffness of the right 1 / 3 region of the beam segment 1 uniformly drops to 0.6EI, from step 1, c eq(1) = 0.6807, the deflection error is 0, and c eq(1) is the theoretical equivalent stiffness value.

[0322] When the stiffness of the middle 1 / 3 region of the beam segment 20 uniformly drops to 0.6EI, under the concentrated load at the mid-span of the second span, from step 2, c eq(2)= 0.8182. Since the three-span continuous beam is a two-time statically indeterminate structure, it will be very complex to directly calculate the theoretical equivalent stiffness by the force method according to the deflection equivalence. Here, the method of finite element model correction is used for iterative solution. The objective functions are the mid-span deflection differences of the first and second spans. By minimizing the objective function, the theoretical equivalent stiffness coefficient c can be obtained from the mid-span deflection equivalence of the first span eq(理论值1跨中w) = 0.8184, and the theoretical equivalent stiffness coefficient c can be obtained from the mid-span deflection equivalence of the second span eq(理论值2跨中w) = 0.81821. The deflection differences are as shown in Figure 42 . It can be seen that when c eq(理论值1跨中w) , the mid-span deflection error of the first span is 0, and when c eq(理论值2跨中w) , the deflection differences of the second span are all 0. c eq(2) is basically the same as c eq(理论值2跨中w) , and c eq(2) can be used to calculate the value of the theoretical equivalent stiffness coefficient

[0323] When the stiffness of the right 1 / 3 area of beam segment 20 uniformly drops to 0.6EI, under the action of the concentrated load at the mid-span of the second span, according to step 2, c eq(2) = 0.8182, and the theoretical equivalent stiffness coefficient c eq(理论值1跨中w) = 0.7983 can be obtained from the mid-span deflection equivalence of the first span, and the theoretical equivalent stiffness coefficient c eq(理论值2跨中w) = 0.8131 can be obtained from the mid-span deflection equivalence of the second span. The deflection differences are as shown in Figure 43 . It can be seen that when c eq(理论值1跨中w) , the mid-span deflection error of the first span is 0, and when c eq(理论值2跨中w) , the deflection differences of the second span are all 0. c eq(2) is basically the same as c eq(理论值2跨中w) , and c eq(2) can be used to calculate the value of the theoretical equivalent stiffness coefficient

[0324] The above are only 4 embodiments of the present invention. All equivalent changes and modifications made according to the scope of the patent application of the present invention shall fall within the scope covered by the present invention

Claims

1. A calculation method for the local equivalent stiffness of a beam structure with equivalent deflection or rotation angle, characterized in that It includes the following steps: (1) Set an appropriate length ε of the locally damaged beam segment according to the length d of the damaged area of the beam structure; (2) Calculate the equivalent stiffness coefficient c eq ; (3) Calculate the equivalent stiffness of the locally damaged beam segment. The equivalent stiffness = c eq × the flexural stiffness of the cross-section of the undamaged beam; Specifically, in step (2), the equivalent stiffness coefficient c eq adopts an approximate calculation method or an exact calculation method: a) Approximate calculation method ① For the locally damaged beam segment within the range from the hinged end to L / 4 and from the cantilever end to L / 4 of the beam structure, the calculation method of the equivalent stiffness coefficient is as follows: Among them, L is the span of the beam, c eq is the equivalent stiffness coefficient, the origin is at the hinged end or the cantilever end of the beam, the x-axis is along the length direction of the beam, pointing to the other end of the beam, a is the distance of the locally damaged beam segment from the origin, ε is the length of the locally damaged beam segment, EI u (x) is the flexural rigidity of the cross-section of the undamaged beam at the position x, EI d (x) is the flexural rigidity of the cross-section of the locally damaged beam segment at the position x within the range of [a, a + ε]; When it is a beam with a constant cross-section, EI u (x) is the constant EI, and the equivalent stiffness coefficient c eq is simplified to the following formula for calculation: ② For other positions of the beam structure, the calculation method of the equivalent stiffness coefficient is as follows: When it is a beam with a constant cross-section, the equivalent stiffness EI eq is simplified to be calculated by the following formula: Among them, EI di is the sectional flexural rigidity of the i-th beam segment when the locally damaged beam segment with a length of ε is equally divided into m segments, and m is a positive integer; b) Exact calculation method Calculate according to one of the following methods according to the specific type of the structure: 1) Simply supported beam The calculation method of the equivalent stiffness coefficient is as follows: ① When under uniformly distributed load: Among them, c eql and c eqr are respectively the equivalent stiffness coefficients when the deflections or rotations on the left and right sides of the locally damaged beam segment of the simply supported beam are equal; ② When under a concentrated load at the mid-span: 2) Cantilever beam The calculation method of the equivalent stiffness coefficient is as follows: Among them, when under uniformly distributed load and deflection equivalence, n = 3; when under uniformly distributed load and rotation equivalence or under concentrated load and deflection equivalence, n = 2; when under concentrated load and rotation equivalence, n = 1; 3) Hyperstatic structure When calculating the equivalent stiffness of the locally damaged beam segment within the range of half-span from the hinged end, the basic structure is simplified to a simply supported beam; when calculating the equivalent stiffness of the locally damaged beam segment within the range of half-span from the fixed end, the basic structure is simplified to a cantilever beam; for other positions, calculate according to the basic structure of a simply supported beam or a cantilever beam; 4) Uniform cross-section beam with damage at the middle position of the locally damaged beam segment ① When the middle position of the locally damaged beam segment is uniformly damaged, for the first locally damaged beam segment adjacent to the side hinged end or the cantilever end: Among them, b is the length of the undamaged area on one side in the locally damaged beam segment, d is the length of the damaged area at the middle position of the locally damaged beam segment, 2b + d = ε, and EI d is the flexural stiffness of the cross-section of the damaged beam segment, EI d <EI; For other locally damaged beam segments: ② When the middle position of the locally damaged beam segment has a crack damage, for the first locally damaged beam segment adjacent to the side hinged end or the cantilever end: For other locally damaged beam segments: Among them, K r is the additional spring stiffness of the crack. For a single-edge crack in a rectangular cross-section beam, it is calculated according to the following formula: where h is the height of the rectangular cross-section beam, and h cr is the crack height, and h cr <h.

2. The method for calculating the local equivalent stiffness of a beam structure equivalent in deflection or rotation angle according to claim 1, characterized in that: In step (1), the length ε of the locally damaged beam segment is not greater than L / 4.

3. The method for calculating the local equivalent stiffness of a beam structure equivalent to deflection or rotation angle according to claim 1, characterized in that: In step (2), the integration is calculated using the matlab mathematical software.

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