Method for evaluating the progressive collapse resistance of locally prestressed concrete frames

Abstract

The application provides a method for evaluating the continuous collapse resistance of a locally prestressed concrete frame. Since the locally prestressed structure is generally asymmetric, the existing theoretical method is generally difficult to obtain a good prediction effect. The method for evaluating the continuous collapse resistance of the locally prestressed concrete frame is based on the basic parameters of the structure and the material, and is based on the assumed three-fold line limit state model. The displacement coordination and stress balance conditions of the substructure are utilized, and the influence of the plastic hinge area is considered. The method can better solve the difficulties caused by the geometric and physical asymmetry of the locally prestressed structure, and can obtain the reliable limit displacement and limit resistance of the locally prestressed concrete frame substructure. Further, the method can be used for evaluating the continuous collapse resistance of the structure.

Description

TECHNICAL FIELD

[0001] The application discloses a theoretical calculation method for predicting the limit state of resistance to progressive collapse of a local prestressed reinforced concrete frame beam-column substructure under a middle column failure state based on a backup load path method, which can be used for evaluating the resistance to progressive collapse of the local prestressed concrete frame and belongs to the field of disaster prevention and mitigation of concrete frame structures and relates to theoretical calculation and program compilation. BACKGROUND

[0002] The local prestressed structure refers to a structure in which prestress is applied in partial areas and no prestress is applied in other areas. The local prestress is a new type of structure form, which has been applied more and more in many civil buildings such as exhibition centers, office buildings and stadiums in recent years due to the characteristics of flexibility, variability, simple process and adaptability to various use requirements. The most widely used form is the combination of prestressed beams and non-prestressed beams, and the application proposes the corresponding theoretical prediction method for this working condition.

[0003] The progressive collapse refers to the extension from a local damage to members, and finally leads to partial structure collapse or overall structure collapse. Since the 1960s, the progressive collapse problem has gradually been paid attention to by the academic circles and engineering personnel in the field of civil engineering. Although the occurrence probability of the progressive collapse of a structure is low, once the progressive collapse occurs, the consequences are very serious, and often brings huge life and property losses. Therefore, it is necessary to establish a reliable evaluation method for the resistance to progressive collapse of a building structure. At present, the widely recognized and applied method for the research on the resistance to progressive collapse of a structure is the backup load path method (also referred to as the removal member method). The method assumes that a specific column is failed under external actions such as explosion, impact and earthquake, so as to examine the ability of the remaining structure to cross the local damage and not to have the progressive collapse.

[0004] The resistance to progressive collapse of the local prestressed concrete frame structure mainly includes the following three stages: (1) the compression arch stage: the axial force of the beam as a whole is in compression, and an arch-shaped force transmission path is formed in the beam under the action of the vertical load; (2) the transition stage: the prestressed beam as a whole is in tension, and the non-prestressed beam as a whole is in compression, and this stage is a conversion stage between the compression arch stage and the catenary stage; (3) the catenary stage: with the further development of the vertical displacement, the axial force of the non-prestressed beam is completely converted into tension, and the prestressed beam and the non-prestressed beam are both in tension, at this time, the resistance of the structure is mainly provided by the steel bars and the prestressed steel strands, and finally the limit state of the structure is reached in this stage.

[0005] In the practical engineering application of local prestress, the prestressed span usually has the characteristics of large span and large section, and the span and section of the non-prestressed span are relatively small, which means that the structure has significant geometric and physical asymmetry. And this asymmetry also leads to the asymmetric continuous collapse failure mode of the local prestress structure. Therefore, the theory must consider the influence of the prestressed span and the non-prestressed span, and ensure the mutual coordination and joint action of the two. SUMMARY

[0006] The purpose of the present application is to provide a method for evaluating the progressive collapse resistance of a local prestressed concrete frame. Since the local prestressed concrete frame usually has strong asymmetry, if the existing ordinary reinforced concrete frame theoretical formula is used, there is a big difference in the failure mode assumption, and the result often has a big error. The method for evaluating the progressive collapse resistance of the local prestressed concrete frame proposed in the present application, based on the assumed three-fold line model, uses the displacement coordination and force balance of the substructure, considers the influence of the plastic hinge region, can obtain the limit displacement and limit resistance of the local prestressed concrete frame substructure, and further be used to evaluate the progressive collapse resistance of the structure.

[0007] The present application adopts the following technical scheme:

[0008] The method for evaluating the progressive collapse resistance of a local prestressed concrete frame; the specific process of the method for evaluating the progressive collapse resistance of the local prestressed concrete frame beam-column substructure under the condition of middle column failure based on the alternate load path method includes the following steps:

[0009] (1), preparation work:

[0010] 1a), determine the geometric size, reinforcement scheme, material parameters and other information of the structure, and establish prestressed beam, non-prestressed beam, zero beam segment, first beam segment and second beam segment;

[0011] 1b), using the information in step 1a), establish a double-fold line stress-strain model of the steel bar and a three-fold line limit state model of the substructure;

[0012] (2), establish displacement coordination equation:

[0013] 2a), calculate the vertical displacement of the zero beam segment: through the rotation angle of the end of the zero beam segment, the vertical displacement at the key section 1 is obtained according to the straight line assumption;

[0014] 2b), calculate the length of each plastic hinge region by using Mattock formula; the plastic hinge regions are respectively the key section 1, the plastic hinge at the joint of the middle column and the non-prestressed beam and the key section 2;

[0015] 2c), according to the strain distribution of the top and bottom of the first beam segment and the second beam segment and the plane section assumption, the strain distribution at the central axis of the beam segment is obtained; the elongation of the first beam segment and the second beam segment is obtained by calculation;

[0016] 2d), according to the displacement coordination condition that the prestressed beam and the non-prestressed beam should satisfy at the middle column, the vertical displacement of the two at the middle column should be equal, and a nonlinear equation related to the limit displacement at the middle column is established;

[0017] (3), the angle correction of the plastic hinge curvature is considered:

[0018] 3a), the plastic hinge curvature at the key section 1 is calculated by using the limit curvature calculation formula;

[0019] 3b), the angle after correction is obtained by solving the geometric relationship in combination with the rotation angle of the first beam segment relative to the initial horizontal position obtained in the foregoing step;

[0020] (4), the force balance equation is established:

[0021] 4a), according to the plane section assumption, the strain of the steel bars at the key section 1 and the key section 2 is determined, and the stress state of each steel bar and the prestressed steel strand is obtained;

[0022] 4b), according to the vertical force balance, the limit load is equal to the total vertical component of the tensile force provided by the steel bars and the prestressed steel strand in the tensile area of the structure at this time; the concrete has been seriously cracked and is out of work.

[0023] (5), the limit displacement and the limit resistance of the local prestressed concrete structure against continuous collapse are obtained by solving the displacement coordination equation, the plastic hinge curvature relationship and the force balance equation established in (2), (3) and (4).

[0024] The continuous collapse resistance evaluation method of the local prestressed concrete frame provided by the application, the geometric size, the reinforcement scheme and the material parameter information of the structure determined in the step (1) include: the span length L PC of the prestressed beam, RC the span length L s of the non-prestressed beam, the sectional size of the prestressed beam and the non-prestressed beam, the length L0, L1 and L2 of the zero, first and second beam segments, the column width b, the elastic modulus E y of the steel bar, the yield strength f u , the limit strength f y , the yield strain ε u of the steel bar, and the limit strain ε s1,top .

[0025] The specific reinforcement of each section includes the top reinforcement A s1,w and the waist reinforcement A s1,w .

[0026] Top reinforcement A at key section 2 s2,top and waist reinforcement A s2,w ,

[0027] Concrete ultimate compressive strain ε hu , relative limit compressive zone height ξ p , prestressed steel strand yield strength f P , prestressed steel strand area A P ;

[0028] The junction of the zero beam segment and the first beam segment is the key section 1, and the end of the non-prestressed beam away from the column is the key section 2;

[0029] In step 1b), the steel reinforcement yield strength f y and the ultimate strength f u , the steel reinforcement yield strain ε y and the ultimate strain ε u can be used to establish a bilinear stress-strain model of the steel reinforcement, and the steel reinforcement strain hardening rate b in the model is obtained according to the following formula h :

[0030]

[0031] Wherein, E p and E s are the plastic deformation modulus and the elastic deformation modulus of the steel reinforcement in the model.

[0032] In the method for evaluating the progressive collapse resistance of the locally prestressed concrete frame, in step 2a), it is assumed that the zero beam segment rotates as a rigid body, and the rotation angle is γ, and the vertical displacement of the beam end at the key section 1 caused by the rotation is δ0; it is assumed that the top steel reinforcement of the zero beam segment just yields, and the elongation of the top steel reinforcement at the boundary of the zero beam segment is ε y L0; then according to the geometric relationship, the vertical displacement δ0 of the beam end of the zero beam segment is obtained according to the following formula:

[0033] δ0=L0 tanγ

[0034]

[0035] Wherein, h0 is the effective height of the section;

[0036] In step 2b), the length of the plastic hinge at the junction of the column and the non-prestressed beam at the key section 1 and the length of the key section 2 are calculated by using the Mattock formula:

[0037] l p =0.5h0+0.05Z

[0038] Wherein, h0 is the effective height of the section, and Z is the distance from the maximum bending section to the inflection point.

[0039] The method for evaluating the progressive collapse resistance of the locally prestressed concrete frame according to the present application, in step 2a), it is assumed that the zero beam segment rotates as a rigid body, and the rotation angle is γ, and the vertical displacement of the beam end at the key section 1 caused by the rotation is δ0; it is assumed that the top reinforcement of the zero beam segment just yields, and the elongation of the top reinforcement at the boundary of the zero beam segment is ε y L0; then according to the geometric relationship, the vertical displacement δ0 of the beam end of the zero beam segment is obtained as follows:

[0040] δ0 = L0 tan γ

[0041]

[0042] wherein h0 is the effective height of the section;

[0043] In step 2b), the lengths of the plastic hinges at the joint of the middle column and the non-prestressed beam at the key section 1 and the key section 2 are calculated by using the Mattock formula:

[0044] l p = 0.5h0 + 0.05Z

[0045] wherein h0 is the effective height of the section, and Z is the distance from the maximum bending section to the inflection point;

[0046] In step 2c), after obtaining the strains at the top and bottom of the beam segment according to the extracted strain distribution, the strain distribution at the central axis of the beam segment can be obtained according to the plane section assumption, and the elongation at the central axis of each beam segment after excluding the plastic hinge area can be obtained by integrating the following formula respectively:

[0047]

[0048]

[0049] wherein L 10 and L 20 are the lengths of the first beam segment and the second beam segment respectively after excluding the plastic hinge area, ε 10 (x) and ε 20 (x) are the strain distribution functions along the longitudinal direction of the beam at the central axis, and x is the length variable for determining the longitudinal position of the beam; in step 2d), the following trigonometric function relationship is used:

[0050]

[0051]

[0052]

[0053] The nonlinear equation related to the limit displacement at the middle column is established as follows:

[0054] δ = δ0 + δ1 = δ2

[0055] δ1 = (L1 + △L1) sinθ1

[0056] δ2 = (L2 + △L2) sinθ2

[0057] That is,

[0058] δ0 + (L1 + △L1) sinθ1 = (L2 + △L2) sinθ2

[0059]

[0060] The method for evaluating the progressive collapse resistance of the partially prestressed concrete frame, in step 3a), the connection between the zero beam segment and the first beam segment is approximately considered according to the plastic hinge equivalent limit curvature at the key section 1, and the limit curvature is calculated by the following formula:

[0061]

[0062] wherein, is the plastic hinge limit curvature, ε hu is the limit compressive strain of the concrete, ξ p is the relative compression zone height when the section is damaged, and h0 is the effective section height.

[0063] In step 3b), the relationship between the modified included angle α and the included angle of the first beam segment and the horizontal line is obtained by the following formula:

[0064]

[0065] Wherein, β is the central angle corresponding to the plastic hinge region.

[0066] The method for evaluating the progressive collapse resistance of the partially prestressed concrete frame, in step 4a), the strain distribution of each plastic hinge region is obtained through step 2c), according to the plane section assumption, the strain of each steel bar is calculated, and the stress state of each steel bar and prestressed steel strand is obtained; the stress and strain state of the waist steel bar can be considered at the central axis, as shown in the following formula:

[0067]

[0068] f w = f y +b h E s (ε w -ε y )​

[0069] where ε w and f w are the strain and stress of the waist reinforcement, respectively.

[0070] Step 4b) the top reinforcement, waist reinforcement and prestressed steel strands at the critical section 1 and the top reinforcement and bottom reinforcement at the critical section 2 under the action of vertical load, the force mechanism is shown as follows:

[0071] P = F1 sin a + F2 sin 0 2

[0072] where P is the ultimate load, F1 and F2 are the axial tension of the components that play a role when the structure reaches the ultimate state, respectively.

[0073] The expressions of F1 and F2 in the above formula are as follows, respectively:

[0074] F1 = T1 + T P

[0075] T1 = T top1 + T w1 = f u A s1,top + f w1 A s1,w

[0076] T P = f P A P

[0077] F2 = T2

[0078] T2 = T top2 + T w2 = f u A s2,top + f w2 A s2,w

[0079] where T1 and T P are the tension provided by the reinforcement and prestressed steel strands at the critical section 1, and T2 is the tension provided by the reinforcement at the critical section 2; the ultimate load is equal to the sum of the vertical components of the tension provided by the reinforcement and prestressed steel strands in the tension zone of the structure at this time, which is expressed by the following formula:

[0080] P = (T1 + T P ) sin a + T2 sin 0 2

[0081] That is: P = (f u A s1,top + f w1 A s1,w + f P A P)sinα+(f u A s2,top +f w2 A s2,w )sinθ2。

[0082] Beneficial effects:

[0083] The method for evaluating the progressive collapse resistance of the local prestressed concrete frame structure of the application is based on displacement coordination, force balance and angle correction considering the curvature of the plastic hinge area, a theoretical method for predicting the progressive collapse resistance limit state of the local prestressed structure is proposed, and the progressive collapse resistance of the local prestressed structure can be evaluated. The method has the following advantages:

[0084] (1) According to the establishment of the limit state model according to the structure failure mode, and based on the geometric and physical relationship, a theoretical formula is proposed, the idea is clear and easy to understand, and has high precision, stability and reliability, and can be used for the prediction of the progressive collapse resistance limit state of the local prestressed concrete frame substructure. Through sample library verification, the structure generally has good prediction effect for the assumed failure mode, and can be considered to be universal for local prestressed frame structures.

[0085] (2) The parameters used in the formula are taken from the structure design information and material performance, which can be determined or adjusted according to the actual situation, and are practical and widely applicable. In the design stage of the structure, the theoretical method proposed by the application can not only accurately predict the progressive collapse resistance of the structure, but also facilitate parameterized analysis, providing convenience for the design and optimization of the structure.

[0086] (3) The plastic hinge is a very important stress mechanism in disaster prevention and mitigation of structures, especially in the process of resisting progressive collapse. In addition to displacement coordination and force balance, the theory proposed by the application also considers the influence of the plastic hinge area, which makes the established theory more in line with the actual situation, avoids the problem of decreased prediction accuracy due to excessive simplification, and can truly achieve the dual goals of simplicity and high efficiency, and has sufficient use value.

[0087] (4) The theoretical method proposed by the application is complete, simple and efficient, the analysis process does not involve complex transcendental equation solving, and the solving idea is a one-way process, which is convenient for programming related programs and realizes the "input-output" solving mode. Since the theoretical method does not contain selection and loop processes, the program algorithm is significantly simplified, the amount of calculation is reduced, and the expected goal can be achieved without complex and lengthy iteration methods, while improving the calculation efficiency with good accuracy. BRIEF DESCRIPTION OF DRAWINGS

[0088] Figure 1 The flowchart of the method for evaluating the progressive collapse resistance of the local prestressed concrete frame structure.

[0089] Figure 2 Figure for double-fold line stress-strain model of steel bar.

[0090] Figure 3 Figure for three-fold line ultimate state model of substructure.

[0091] Figure 4 Figure for deformation of zero beam segment.

[0092] Figure 5 Figure for strain distribution of non-plastic hinge area of first and second beam segments.

[0093] Figure 6 Figure for strain distribution of plastic hinge area.

[0094] Figure 7 Figure for schematic diagram of ultimate curvature of plastic hinge area.

[0095] Figure 8 Figure for schematic diagram of stress mechanism of structure.

[0096] Figure 9 Figure for distribution of relative error.

[0097] Figure 10 Figure for comparison of ultimate displacement.

[0098] Figure 11 Figure for comparison of ultimate resistance.

[0099] Figure 12 Figure for parameter information of local prestressed beam designed according to design software PKPM based on national standard. DETAILED DESCRIPTION

[0100] In order to make the purpose and technical solutions of the embodiments of the present application clearer, the technical solutions of the embodiments of the present application will be described clearly and completely below in combination with the drawings of the embodiments of the present application. Obviously, the described embodiments are part of the embodiments of the present application, rather than all the embodiments of the present application. Based on the described embodiments of the present application, all other embodiments obtained by those of ordinary skill in the art without any creative effort belong to the scope of protection of the present application.

[0101] The specific process of the method for evaluating the progressive collapse resistance of the local prestressed concrete frame beam-column substructure under the condition of failure of the middle column based on the alternate load path method of the present application comprises the following steps:

[0102] (1) Preparation work:

[0103] 1a) input prestressed beam span length L PC , non-prestressed beam span length L RC , cross-sectional size of two beams (including cross-sectional height h1 and h2, cross-sectional effective height h10 and h 20 The cross-sectional widths b1 and b2), the lengths L0, L1, and L2 of beam segments 0, 1, and 2, the column width b, and the elastic modulus E of the steel reinforcement. s The yield strength f of the steel reinforcement y and ultimate strength f u ε yield strain of steel reinforcement y and ultimate strain ε u Specific reinforcement details for each section (including the top reinforcement A at critical section 1) s1,top and waist reinforcement A s1,w Top reinforcement A at two key sections s2,top and waist reinforcement A s2,w ), ultimate compressive strain ε of concrete hu The relative height of the boundary pressure zone ξ p Yield strength f of prestressed steel strand P The area A of the prestressed steel strand P .

[0104] 1b) Establish the bilinear stress-strain model of the reinforcing steel and the trilinear limit state model of the substructure, as shown below. Figure 2 , 3 As shown. Utilizing the yield strength f of the reinforcing steel. y and ultimate strength f u ε yield strain of steel bars y and ultimate strain ε u A bilinear stress-strain model of the reinforcing steel can be established, and the strain hardening rate b of the reinforcing steel in the model can be obtained by the following formula. h :

[0105]

[0106] Among them, E p and E s These are the plastic deformation modulus and elastic deformation modulus of the steel bars in the model, respectively.

[0107] (2) Establish the displacement compatibility equation:

[0108] 2a) Assume that beam segment zero undergoes rigid body rotation as a whole, and the deformation diagram is as follows. Figure 4 As shown, let the rotation angle be γ, and the vertical displacement of the beam end at the critical section 1 due to rotation be δ0. Assuming the top reinforcement of beam segment zero has just yielded, the elongation of the top reinforcement at the boundary of beam segment zero is ε. y L0. Based on geometric relationships, the vertical displacement δ0 at the end of beam segment zero can be obtained using the following formula:

[0109] δ0=L0 tanγ

[0110]

[0111] where h0 is the effective height of the cross section.

[0112] 2b) Calculate the length of each plastic hinge region using Mattock formula:

[0113] l p = 0.5h0+0.05Z

[0114] where h0 is the effective height of the cross section and Z is the distance from the maximum moment section to the inflection point.

[0115] There are three plastic hinge regions to be calculated, which are key section 1, plastic hinge at the joint of middle column and non-prestressed beam and key section 2.

[0116] 2c) The simplified strain distribution is as follows: in the first beam segment, the strain of the top reinforcement is triangularly distributed, with the maximum being the yield strain ε y , and the strain of the bottom reinforcement is zero along the whole segment. In the second beam segment, the strain of the top reinforcement is distributed in two triangles along the whole length, from the compression yield at the section close to the middle column to the strain of 0 at the midspan, and then into the tensile state to the tensile yield. At the same time, the strain state of the bottom reinforcement is just the opposite, from the tensile yield at the section close to the middle column to the strain of 0 at the midspan, and then into the compression state to the compression yield. The strain distribution is shown in the attached figure Figure 5 . According to this strain distribution assumption, the extracted strain distribution change is approximately considered as the top and bottom strain distribution of the first and second beam segments as shown in Figure 5 . After obtaining the top and bottom strain of the beam segment, the strain distribution at the central axis of the beam segment can be obtained according to the plane section assumption, and the elongation of each beam segment after removing the plastic hinge region at the central axis can be obtained by integrating the following formula respectively:

[0117]

[0118]

[0119] where L 10 and L 20 are the lengths of the first and second beam segments respectively after removing the plastic hinge region,

[0120] ε 10 (x) and ε 20 (x) are the strain distribution functions along the longitudinal direction of the central axis of the first and second beam segments respectively, and x is the length variable that determines the longitudinal position of the beam.

[0121] For the elongation calculation of the plastic hinge region, a different strain distribution needs to be used, which is shown in Figure 6 , approximately considering that the top reinforcement of the key section reaches the ultimate tensile strain ε u , and the bottom reinforcement reaches the compression yield strain εy , and the top and bottom reinforcements of the end of the plastic hinge zone are in tension and compression, respectively. According to the plane section assumption, the ultimate tensile strain ε w at the mid-axis can be obtained, and the force state of the waist reinforcement can be calculated by using the strain here. According to the Mattock formula in the second b) step, the lengths of the three plastic hinge zones l p1 , l p2 , and l p3 are obtained, respectively.

[0122]

[0123]

[0124] △l p3 =△l p2

[0125] wherein ε w1 and ε w2 are the strains at the horizontal midline of the key sections 1 and 2, respectively.

[0126] The total elongations of the first and second beam segments are obtained by adding the elongations of the plastic hinge zones to the elongations of the non-plastic hinge zones:

[0127]

[0128]

[0129] 2d) According to the displacement compatibility condition that the vertical displacements of the pre-stressed beam and the non-pre-stressed beam at the mid-column should be equal, i.e. the vertical displacement difference δ0 of the zero beam segment plus the vertical displacement difference δ1 of the first beam segment is equal to the vertical displacement difference δ2 of the second beam segment, as shown in Figure 3 The trigonometric function relationship used in the process can be calculated according to the following formula:

[0130]

[0131]

[0132]

[0133] Using the above trigonometric function relationship, a nonlinear equation related to the ultimate displacement at the mid-column can be established:

[0134] δ = δ0 + δ1 = δ2

[0135] δ1 = (L1 +△L1) sin θ1

[0136] δ2 = (L2 + AL2) sin θ2

[0137] That is,

[0138] δ0 + (L1 + AL1) sin θ1 = (L2 + AL2) sin θ2

[0139]

[0140] (3) The angle correction considering the plastic hinge curvature:

[0141] 3a) In the actual situation, the connection between the zero beam segment and the first beam segment is not a broken line, but should be a curve transition. Therefore, the equivalent limit curvature of the plastic hinge at key section 1 needs to be calculated to approximately consider the connection between the zero beam segment and the first beam segment, and to be applied in the subsequent stress analysis. The limit curvature calculation formula used is as follows:

[0142]

[0143] wherein, is the plastic hinge limit curvature, ε hu is the limit compressive strain of concrete, ξ p is the relative compression zone height when the section is destroyed, and h0 is the effective section height.

[0144] 3b) The geometric relationship of the plastic hinge area is shown in Figure 7 . Through this geometric relationship, the corrected angle α and the angle between the first beam segment and the horizontal line exist the following relationship:

[0145]

[0146] wherein, β is the central angle corresponding to the plastic hinge area.

[0147] (4) Establish the stress balance equation:

[0148] 4a) In the second c) step, we have obtained the strain distribution of each plastic hinge area as shown in Figure 6 . According to the plane section assumption, the strain of each steel bar can be calculated, and the stress state of each steel bar and the prestressed steel strand can be obtained. The stress and strain state of the web steel bar can be considered at the central axis, as shown in the following formula:

[0149]

[0150] f w = f y +b h E s (ε w -ε y )

[0151] where ε w and f w are the strain and stress of the waist reinforcement, respectively.

[0152] 4b) Under the action of vertical load, the top reinforcement, waist reinforcement and prestressed steel strand at the key section 1 and the top reinforcement and bottom reinforcement at the key section 2 mainly play a resistance role, and the force mechanism diagram is shown in Figure 8 . (The concrete has been seriously cracked and has withdrawn from work). Then:

[0153] P = F1 sin a + F2 sin 0 2

[0154] where P is the ultimate load, and F1 and F2 are the axial tension of the components that play a role when the structure reaches the limit state, respectively.

[0155] In the key section 1, the components that play a role are the top reinforcement, waist reinforcement and prestressed steel strand, and in the key section 2, the components that play a role are the top reinforcement and waist reinforcement. Then the expressions of F1 and F2 are respectively:

[0156] F1 = T1 + T P

[0157] T1 = T top1 + T w1 = f u A s1,top + f w1 A s1,w

[0158] T P = f P A P

[0159] F2 = T2

[0160] T2 = T top2 + T w2 = f u A s2,top + f w2 A s2,w

[0161] where T1 and T P are the tension provided by the reinforcement and prestressed steel strand at the key section 1, respectively, and T2 is the tension provided by the reinforcement at the key section 2, A s1,top , A s1,w and A P are the cross-sectional areas of the top reinforcement, waist reinforcement and prestressed steel strand at the key section 1, respectively, and A s2,top and A s2,wT and T are the sectional areas of the top reinforcement and the waist reinforcement at the critical section 2, respectively top1 w1 P top2 w2 f and f are the tensile forces provided by the reinforcement or the steel strand, respectively u w1 w2 f and f are the ultimate tensile strengths of the reinforcement, the reinforcement stresses corresponding to the strains at the midlines of the critical sections 1 and 2, respectively P f is the yield strength of the steel strand.

[0162] According to the vertical force balance, the ultimate load is equal to the sum of the vertical components of the tensile forces provided by the reinforcement and the steel strand in the tensile region of the structure at this time, and thus there is:

[0163] P = (T1 + T P ) sin a + T2 sin q2

[0164] That is:

[0165] P = (f u A s1,top +f w1 A s1,w +f P A P ) sin a + (f u A s2,top +f w2 A s2,w ) sin q2

[0166] (5) The equation obtained in the above steps is solved by substituting the known parameters, and the ultimate displacement and the ultimate resistance of the local pre-stressed concrete structure against progressive collapse are obtained.

[0167] Example verification:

[0168] A series of frame samples conforming to the national standards of China are designed by using the design software PKPM, and a substructure is selected from the samples as a verification example of the theory proposed in the present application. The parameter information of the sample structure is shown in Table 1. A model is established according to the selected structure by using the finite element software OpenSEES, and finite element analysis is performed. The results obtained are used as a control of the calculation of the theoretical method, and the comparison results are shown in Tables 2, 3 and 4. It can be considered that the present theoretical method has high precision and good prediction effect, and can be used to evaluate the progressive collapse resistance of local pre-stressed concrete frames. Figure 12 Figure 9 10

[0169] ​​​​​​​​​The above merely describes preferred specific embodiments of the present application, but the protection scope of the present application is not limited thereto, any changes or replacements within the technical scope disclosed by the present application, which can be easily thought by those skilled in the art, should be covered within the protection scope of the present application. Therefore, the protection scope of the present application should be subject to the protection scope of the claims.

Claims

1. A method for evaluating the progressive collapse resistance of a locally prestressed concrete frame, characterized in that: The method for evaluating the progressive collapse resistance of the local prestressed concrete frame beam-column substructure based on the alternate load path method and considering the failure of the middle column comprises the following steps: (1) Preparation work: 1a) Determine the geometric size, reinforcement scheme, and material parameter information of the structure, and establish prestressed beams, non-prestressed beams, zero beam segments, first beam segments, and second beam segments; 1b) Use the information in step 1a) to establish a bilinear stress-strain model of the steel bars and a three-fold line ultimate state model of the substructure; (2) Establish the displacement coordination equation: 2a) Calculate the vertical displacement of the zero beam segment: According to the straight line assumption, the vertical displacement at the key section 1 is obtained through the rotation angle at the end of the zero beam segment; 2b) Calculate the length of each plastic hinge region using the Mattock formula; the plastic hinge regions are the key section 1, the plastic hinge at the joint of the middle column and the non-prestressed beam, and the key section 2; 2c) Obtain the strain distribution at the central axis of the beam segment according to the strain distribution at the top and bottom of the first beam segment and the second beam segment and the plane section assumption; the elongation of the first beam segment and the second beam segment is obtained by calculation; 2d) According to the displacement coordination condition that the prestressed beam and the non-prestressed beam should satisfy at the middle column, the vertical displacements of the two at the middle column should be equal, and a nonlinear equation related to the ultimate displacement of the middle column is established; (3) Consider the angle correction of the plastic hinge curvature: 3a) Calculate the plastic hinge curvature at the key section 1 using the ultimate curvature calculation formula; 3b) Obtain the corrected angle by solving the geometric relationship combined with the rotation angle of the first beam segment relative to the initial horizontal position obtained in the foregoing steps; (4) Establish the force balance equation: 4a) Determine the strain of the steel bars at the key section 1 and the key section 2 according to the plane section assumption, and obtain the stress state of each steel bar and prestressed steel strand; 4b) According to the vertical force balance, the ultimate load is equal to the total vertical component of the tensile force provided by the tensile zone steel bars and prestressed steel strands of the structure at this time; (5) Solve the displacement coordination equation, plastic hinge curvature relationship, and force balance equation established in (2), (3), and (4) to obtain the ultimate displacement and ultimate resistance of the local prestressed concrete structure against progressive collapse.

2. The method for evaluating the progressive collapse resistance of a locally pre-stressed concrete frame according to claim 1, wherein: The step (1) includes determining the geometric size, the reinforcement scheme, and the material parameter information of the structure, which includes the span length L of the prestressed beam PC , the span length L of the non-prestressed beam RC , the cross-sectional size of the prestressed beam and the non-prestressed beam, the length L0, L1, L2 of the zero, first, and second beam segments, the column width b, the elastic modulus E of the steel bar s , the yield strength f of the steel bar y , the ultimate strength f u , the yield strain ε of the steel bar y , and the ultimate strain ε u . The specific reinforcement of each section includes the top reinforcement A at the key section 1 s1,top , the waist reinforcement A s1,w , Top reinforcement A at key section 2 s2,top and waist reinforcement A s2,w , concrete ultimate compressive strain ε hu , relative limit compressed zone height ξ p , prestressed steel strand yield strength f P , prestressed steel strand area A P ; The junction of the zero beam segment and the first beam segment is the key section 1, and the end of the non-prestressed beam away from the middle column is the key section 2; In step 1b) the steel reinforcement yield strength f y and ultimate strength f u , the steel reinforcement yield strain ε y and ultimate strain ε u A bilinear stress-strain model of the steel reinforcement can be established and the steel reinforcement strain hardening rate b h in the model is obtained as follows: ； where E p and E s are the plastic and elastic deformation modulus of the reinforcement in the model, respectively.

3. The method for evaluating the progressive collapse resistance of a locally pre-stressed concrete frame according to claim 1, wherein: In step 2a), it is assumed that the No. 0 beam segment rotates as a rigid body, and the rotation angle is γ. The vertical displacement of the beam end at the key section 1 due to the rotation is δ0. It is assumed that the top reinforcement of the No. 0 beam segment just yields, and the elongation of the top reinforcement at the boundary of the No. 0 beam segment is ε y L0; then according to the geometric relationship, the vertical displacement δ0 of the beam end of the No. 0 beam segment is obtained as follows: ； where h0 is the effective height of the section; In step 2b), the lengths of the plastic hinge at the joint of the key section 1, the middle column and the non-prestressed beam, and the key section 2 are calculated using the Mattock formula: ； where h0 is the effective height of the section, and Z is the distance from the maximum bending section to the inflection point; In step 2c), after obtaining the top and bottom strains of the beam segment according to the extracted strain distribution, the strain distribution at the central axis of the beam segment can be obtained according to the plane section assumption. The elongation of each beam segment at the central axis except the plastic hinge region can be obtained by integrating the following formula: ； ； wherein L 10 and L 20 are the lengths of the first and second beam segments, respectively, minus the plastic hinge region, ε 10 (x) and ε 20 (x) are the strain distribution functions along the longitudinal direction of the beam at the axis, x is the length variable that determines the longitudinal position of the beam; and step 2d) is performed by the following trigonometric relationship: ； ; The nonlinear equation related to the ultimate displacement at the middle column is established as follows: ； ； ； That is: ； 。 4. The method for evaluating the progressive collapse resistance of a partially prestressed concrete frame according to claim 1, wherein: In step 3a), the connection between the zero beam segment and the first beam segment is approximately considered according to the equivalent ultimate curvature of the plastic hinge at the key section 1. The ultimate curvature is calculated using the following formula: ； where φ u is the plastic-hinge limit curvature, ε hu is the limit compressive strain of the concrete, ξ p is the relative compressive zone height at failure of the section, and h0 is the effective section height; The relationship between the corrected angle a and the angle θ1 between the first beam segment and the horizontal line is obtained in step 3b) by: ； Where β is the central angle corresponding to the plastic hinge region.

5. The method for evaluating the progressive collapse resistance of a partially prestressed concrete frame according to claim 1, wherein: In step 4a), the strain distribution of each plastic hinge region is obtained in step 2c), and according to the plane section assumption, the strain of each steel bar is calculated, and the stress state of each steel bar and the prestressed steel strand is obtained; the stress and strain state of the web steel bar can be considered at the central axis, as shown in the following formula: ； where ε w and f w are the strain and stress of the waist reinforcement, respectively; In step 4b), under the action of vertical load, the stress mechanism of the top steel bar, the web steel bar and the prestressed steel strand at the key section 1, and the top steel bar and the bottom steel bar at the key section 2 is as follows: ； Where P is the ultimate load, F1 and F2 are the axial tension of the components acting when the structure reaches the ultimate state, respectively; The expressions of F1 and F2 in the above formula are as follows: ； ； Wherein, T1 and T P T1 and T2 are the tensile forces provided by the steel bars and the prestressed steel strands at the key section 1 and the key section 2, respectively. A s1,top , A s1,w and A P are the top reinforcement, the web reinforcement and the prestressing strand cross-sectional area at the critical section 1, respectively. A s2,top and A s2,w A and A are the cross-sectional areas of the top reinforcement and the web reinforcement, respectively, at the critical section 2. T top1 , T w1 , T P , T top2 and T w2 are the tensile forces provided by the respective steel bars or steel strands; f u , f w1 and f w2 are the ultimate tensile strength of the reinforcement, the stress in the reinforcement at the midline of the critical sections 1 and 2, respectively, f P is the yield strength of the steel strand; The ultimate load is equal to the sum of the vertical components of the tension provided by the steel bars and the prestressed steel strands in the tension zone of the structure at this time, which is expressed by the following formula: ； That is: 。

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