A method for calculating shear deformation of beam-column joints

Abstract

In view of the shortcomings of the traditional node shear deformation calculation method, the present invention uses the "similarity transformation" of the matrix to describe the "parallelogram assumption" and the "diagonal non-rotation assumption" in the above-mentioned traditional calculation formula, avoiding the introduction of the small deformation assumption and the assumption that the diagonal length changes are approximately equal in the mathematical derivation process, and can obtain an accurate solution that satisfies the "parallelogram assumption" and the "diagonal non-rotation assumption", overcoming the shortcoming of the traditional node shear deformation calculation formula that the calculation result has a gradually increasing error in the later stage of force, and has high practical application significance.

Description

Technical Field

[0001] The invention relates to a calculation method for shear deformation of a beam-column node and belongs to the technical field of structural engineering. Background Art

[0002] For reinforced concrete frames, steel frames, rectangular steel tube concrete columns-reinforced concrete beam frames, rectangular steel tube concrete columns-steel beam frames and other structures, nodes are key parts connecting beams and columns. However, under reciprocating loads such as earthquakes, beam-column nodes may produce large inelastic shear deformations, which have a significant impact on the overall structure. The shear deformation of the node is an important parameter that reflects the inelastic deformation of the beam-column node and evaluates the seismic performance of the frame structure and beam-column assembly specimens.

[0003] The stress in the beam-column joint area is complex, and the deformation mode of the node is not as simple as the pure shear deformation of linear elastic materials. First, the different bearing capacities of the positive and negative moments at the beam ends will lead to different shear deformations of the beam-column joint in the clockwise and counterclockwise directions. Therefore, in general, the beam-column joint after shear deformation is no longer a completely idealized parallelogram. In addition, the concrete in the core area of ​​the beam-column joint of the reinforced concrete frame in the later loading period expands under pressure, and the strain of the stirrups increases significantly, causing the original right-angled sides of the rectangular joint to bulge. In addition, the development of concrete cracks in the node area and the partial shedding of the protective layer concrete affect the edge straight line of the initial rectangular area of ​​the beam-column joint. After shear deformation, it generally becomes a curve with an arc. In short, the actual deformation characteristics of the node are relatively complex.

[0004] Due to the limitation of measuring instruments, except for a very small number of scholars who use image recognition to directly measure the shear strain of nodes, the traditional calculation formula for node shear deformation is generally proposed based on the "parallelogram assumption" and "diagonal non-rotation assumption". However, the traditional calculation formula for node shear deformation adopts some approximate assumptions in the derivation process, such as the small deformation assumption, which causes the calculation error of the traditional calculation formula for node shear deformation to increase with the increase of node shear deformation. Since the calculation results of the large inelastic deformation stage of the node in the later stage of force are the key data for evaluating the inelastic deformation of the node, the seismic performance of the frame structure and the beam-column assembly specimens, it is particularly important to reduce the calculation error of the shear deformation of the beam-column node in the later stage of force.

[0005] At present, the traditional method for calculating the shear deformation of a node is to arrange displacement meters at the two diagonals of the node rectangular area and measure the length change of the two diagonals δ 1 , δ 2 , and define the average diagonal deformation of the node core area The shear strain at the beam end is

[0006] The shear strain at the column end is:

[0007] The total shear strain at the node is:

[0008] In the above calculation method, the traditional node shear deformation calculation formula has two sources of error. First, it assumes that the length of the two diagonal lines is approximately equal, which is inconsistent with the actual situation. Second, it assumes that the deformation of the node satisfies the small deformation assumption. Obviously, in the later stage of force bearing, as the node stirrups yield, the width of the node concrete crack continues to develop, and the inelastic shear deformation of the beam-column node will gradually increase. Therefore, the use of the traditional node shear deformation calculation formula will definitely cause obvious errors.

[0009] In view of this, the present invention aims at the shortcomings of the traditional node shear deformation calculation method, and uses the "similarity transformation" of the matrix to describe the "parallelogram assumption" and the "diagonal non-rotation assumption" in the above-mentioned traditional calculation formula, avoiding the introduction of the small deformation assumption and the assumption that the diagonal length changes are approximately equal in the mathematical derivation process, and can obtain an accurate solution that satisfies the "parallelogram assumption" and the "diagonal non-rotation assumption", overcoming the shortcoming of the traditional node shear deformation calculation formula that the calculation result has a gradually increasing error in the later stage of force, and has high practical application significance. Summary of the invention

[0010] The present invention aims at the shortcoming that the calculation error of the traditional node shear deformation calculation formula increases with the increase of node deformation, and proposes an improved calculation method for the shear deformation of the beam-column node on the premise of satisfying the two basic assumptions of "parallelogram assumption" and "diagonal non-rotation assumption". The method has a strict mathematical derivation and geometric physics background, and the measuring device of the node shear deformation is the same as the traditional method. Therefore, while satisfying the simplicity of the measuring device, the node shear deformation calculation formula proposed by the present invention improves the accuracy of the node shear deformation calculation results in the entire force process.

[0011] To achieve the above object, the present invention proposes a method for calculating the shear deformation of a beam-column node, comprising the following steps (only a specific deformation state is used as an example for description):

[0012] Step 1: Arrange wire displacement meters at the two diagonals ① and ② of the node rectangular area respectively, and record the initial lengths of the two diagonals of the node rectangular area before the test loading, which are recorded as m and n respectively.

[0013] Step 2: At a specific moment during the test, record the lengths of the two diagonals after they change, and record them as m' and n' respectively. According to the m and n recorded in step 1, calculate the lengths of the two diagonals according to the formula and Calculate the expansion multiples λ of the two diagonals separately 1 and λ 2 .

[0014] Step 3: Get the vector with the same direction as the two diagonals. This vector is the eigenvector. For example, the vector with the same direction as the diagonal ① is (h c ,h b ) T , this vector is the eigenvalue λ 1 The corresponding eigenvector; the vector with the same direction as the diagonal line ② is (h c ,-h b ) T , this vector is the eigenvalue λ 2 The corresponding eigenvector. c is the width of the node rectangular area, h b is the height of the node rectangular area.

[0015] Step 4: Assume that the lower left corner of the node rectangle before deformation is the origin, and the two basis vectors of the right-angled sides are i and j, which become i' and j' after the node is deformed. The eigenvalues ​​and eigenvectors obtained in step 3 can be transformed by similarity of matrix theory to obtain the basis vectors i' and j' of the node rectangle at a specific moment.

[0016] Step 5: The i, j, i', j' obtained in step 4 can be used to calculate the beam end shear strain γ at each moment during the test. b and column end shear strain γ c .

[0017] As a further technical solution of the present invention, the present invention sets the diagonal expansion factor λ of the node rectangular area to 1 and λ 2 They are used as transformation parameters respectively, assuming that the node rectangle is deformed into a parallelogram and the diagonal is kept unchanged.

[0018] As a further technical solution of the present invention, the basis vector i' and basis vector j' of the node rectangle in step 4 are determined as follows:

[0019]

[0020] As a further technical solution of the present invention, the beam end shear strain γ in step 5 b and column end shear strain γ c Determine by the following formula:

[0021]

[0022]

[0023] In the above formula, ||i||, ||i'||, ||j||, ||j'|| represent the modulus lengths of vectors i, i', j, j' respectively. The total shear strain of the node γ=γ b+γ c .

[0024] Compared with the traditional calculation formula, the node shear deformation calculation formula proposed by the present invention has the following technical effects: based on matrix similarity transformation, it satisfies the theoretical formula of "parallelogram assumption" and "diagonal non-rotation assumption", and has strict mathematical derivation and geometric physics background; the node area measurement device of the present invention is the same as the traditional method. Therefore, while satisfying the simplicity of the measurement device, the node shear deformation calculation formula proposed by the present invention improves the accuracy of the calculation results of the shear deformation of the beam-column node. BRIEF DESCRIPTION OF THE DRAWINGS

[0025] Figure 1 It is a schematic diagram of the traditional node shear deformation calculation formula;

[0026] Figure 2 This is a schematic diagram of the derivation of the traditional node shear deformation calculation formula;

[0027] Figure 3 It is a schematic diagram of the calculation formula for node shear deformation provided by the present invention;

[0028] Figure 4 It is the geometric dimensions and reinforcement diagram of a reinforced concrete beam-column node;

[0029] Figure 5 It is a graph showing the variation of shear stress τ with shear deformation γ. DETAILED DESCRIPTION

[0030] In order to further illustrate the technical solution disclosed in the present invention, the following is a detailed description in conjunction with the accompanying drawings and specific embodiments. Those skilled in the art should know that the optimization and improvements made without violating the spirit of the present invention fall within the scope of protection of the present invention, and the conventional means and customary techniques in the field are not recorded and described in detail in this specific embodiment.

[0031] like Figure 3 The figure shows a schematic diagram of a calculation method for shear deformation of beam-column joints based on matrix similarity transformation. Taking a reinforced concrete beam-column mid-joint specimen as an example, the cross-sectional size of the beam is b b ×h b =300mm×500mm, the cross-sectional dimension of the column is b c ×h c=400mm×400mm, the longitudinal reinforcement of beams and columns are both made of HRB500 steel bars, with a diameter of 20mm, and the measured yield strength value is 433.7MPa; the stirrups of beams and columns are both made of HRB335 steel bars, with a diameter of 10mm, and the measured yield strength value is 244MPa; the stirrup spacing in the non-densified area of ​​the column is 90mm, and the stirrup spacing in the densified area is 60mm. The measured yield strength of the node stirrups is 456MPa, and the node stirrup characteristic value is 0.376. The measured value of the concrete cube compressive strength of the specimen is 29.8MPa, the axial compression ratio is 0.25, and the shear compression ratio is 0.26. During the test loading process, the vertical axial force is first applied and kept constant throughout the test process, and then the horizontal repeated load is applied to the top of the column. The geometric dimensions and reinforcement of the specimen are as follows Figure 4 shown.

[0032] The change of node shear stress τ with shear deformation γ is calculated as follows: Figure 5 As shown by Figure 5 It can be seen that the calculated stress-deformation curve is basically consistent with the skeleton curve of the hysteresis curve measured by the test loading, which shows that the accuracy of the algorithm is relatively high. Therefore, the present invention can be used to calculate the shear deformation of the beam-column node.

[0033] The above embodiments are only preferred embodiments of the present invention, and the protection scope of the present invention is not limited to the above embodiments. For those skilled in the art, improvements and changes obtained without departing from the technical concept of the present invention should also be regarded as the protection scope of the present invention.

Claims

1. A method for calculating node shear deformation based on matrix similarity transformation, characterized in that: The following steps are involved: Step 1: Arrange the wire displacement meter at the two diagonals ① and ② of the node rectangular area respectively, and record the initial lengths of the two diagonals of the node rectangular area before the test loading, which are recorded as m and n respectively; Step 2: At a specific moment during the test, record the lengths of the two diagonals after they change, and record them as m' and n' respectively. and The expansion and contraction multiples λ1 and λ2 of the two diagonals are calculated respectively; Step 3: Get a set of vectors with the same direction as the two diagonals. This set of vectors is the eigenvector. For example, the vector with the same direction as the diagonal ① is (h c ,h b ) T , this vector is the eigenvector corresponding to the eigenvalue λ1; the vector with the same direction as the diagonal line ② is (h c ,-h b ) T , which is the eigenvector corresponding to the eigenvalue λ2; where h c is the width of the node rectangular area, h b is the height of the node rectangular area; Step 4: Assume that the lower left corner of the node rectangle before deformation is the origin, and the two basis vectors of the right-angled sides are i and j respectively, which become i' and j' respectively after the node is deformed; the eigenvalues ​​and eigenvectors obtained in step 3 can be transformed through similarity transformation of matrix theory to obtain the basis vectors i' and j' of the node rectangle at a specific time; Step 5: The i, j, i', j' obtained in step 4 can be used to calculate the beam end shear strain γ at each moment during the test. b and column end shear strain γ c .

2. The node shear deformation calculation method based on matrix similarity transformation according to claim 1 is characterized in that: The basis vector i' and basis vector j' of the node rectangle in step 4 are determined as follows: Wherein: A is a matrix composed of basis vectors i' and j' as column vectors.

3. The node shear deformation calculation method based on matrix similarity transformation according to claim 1 is characterized in that: Shear strain at the beam end in step 5 γ b and column end shear strain γ c Determine by the following formula: Where: ‖i‖, ||i'||, ||j||, ‖j'‖ represent the modulus lengths of vectors i, i', j, j' respectively; the total shear strain of the node γ = γ b +γ c .

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