A method for calculating bending moments of reinforced concrete members considering the change of neutral axis position

Abstract

The present invention discloses a method for calculating the bending moment of reinforced concrete components taking into account the change in the position of the neutral axis, including the steps of: 1. judging whether the cross section of the reinforced concrete component is cracked; 2. executing step 3 when the cross section is not cracked; 3. executing step 6 when cracking occurs; 4. calculating the curvature; 5. obtaining the position of the central axis; 6. calculating the effective depth of the crack; 7. calculating the effective moment of inertia and the curvature; 8. assuming the position of the central axis of the reinforced concrete component with a cracked cross section so that the cross section is subjected to force balance and calculating the true moment of inertia; 9. obtaining the true position of the central axis of the reinforced concrete component with a cracked cross section; 10. calculating the bending moment. The method of the present invention has simple steps, reasonable design, and convenient implementation. It can be effectively applied to the calculation of the bending moment of reinforced concrete components. The calculation result is more accurate, takes into account the change in the position of the neutral axis, reflects the true situation of the bending stiffness of the cracked cross section, has a significant effect, and is easy to promote.

Description

Technical Field

[0001] The present invention belongs to the technical field of geotechnical engineering, and in particular relates to a method for calculating the bending moment of a reinforced concrete member taking into account the change in the position of the neutral axis. Background Art

[0002] In plane bending and oblique bending, the normal stress value at each point on the intersection of the cross section and the stress plane is zero. This intersection is called the neutral axis. Generally speaking, when the tensile stress in a reinforced concrete beam subjected to bending reaches the concrete strength, the neutral axis will move upward, but the steel bars will not yield at this time. However, ordinary reinforced concrete theory ignores the tensile strength of the concrete below the neutral axis. When cracking has already developed in the reinforced concrete component, the calculated bending performance does not take into account the tensile strength after cracking, which will lead to an incorrect estimation of the bearing capacity of the component section. Therefore, over-design of tunnel linings or excessive use of reinforcement is a common phenomenon. To overcome these design deficiencies, it is urgent to establish an analytical model for the bending behavior of concrete components that considers the cross-sectional nonlinearity and true effective moment of inertia caused by tensile cracking of concrete. Summary of the Invention

[0003] The technical problem to be solved by the present invention is to address the deficiencies in the above-mentioned prior art and provide a method for calculating the bending moment of reinforced concrete components taking into account the change in the position of the neutral axis. The method has simple steps, reasonable design, and easy implementation. It can be effectively applied to the calculation of the bending moment of reinforced concrete components. The calculation results are more accurate, taking into account the change in the position of the neutral axis, and reflecting the actual situation of the bending stiffness of the cracked section. The method has significant effects and is easy to promote.

[0004] To solve the above technical problems, the present invention adopts a technical solution: a method for calculating the bending moment of reinforced concrete members taking into account the change in the position of the neutral axis, comprising the following steps:

[0005] Step 1: Determine whether the cross section of the reinforced concrete component is cracked;

[0006] Step 2: When the cross section of the reinforced concrete component is not cracked, execute step 3; when the cross section of the reinforced concrete component is cracked, execute step 6;

[0007] Step 3: Calculate the curvature of the reinforced concrete member with an uncracked cross section;

[0008] Step 4: Obtain the neutral axis position of the reinforced concrete member with an uncracked cross section;

[0009] Step 5: Calculate the bending moment of the reinforced concrete member with uncracked cross section;

[0010] Step 6: Calculate the effective depth of cracks in reinforced concrete components with cracked sections;

[0011] Step 7: Calculate the effective moment of inertia and curvature of the reinforced concrete member with cracked cross section;

[0012] Step 8: Based on the effective crack depth and effective moment of inertia, assume the neutral axis position of the reinforced concrete member with a cracked cross section so that the cross section is under force equilibrium, and calculate the true moment of inertia;

[0013] Step 9: Obtain the true neutral axis position of the reinforced concrete member with a cracked cross section based on the effective moment of inertia and the true moment of inertia;

[0014] Step 10: Calculate the bending moment of the reinforced concrete member with cracked cross section based on the true neutral axis position.

[0015] In the above-mentioned method for calculating the bending moment of reinforced concrete members considering the change in the neutral axis position, the specific process of determining whether the cross section of the reinforced concrete member is cracked in step 1 includes:

[0016] Step 101, calculating the load bending moment of the reinforced concrete member;

[0017]

[0018] Where M is the load bending moment, P is the applied load, and z is the horizontal distance from the measured section to the support;

[0019] Step 102: Calculate the cracking bending moment of the reinforced concrete member;

[0020]

[0021] Where M cr is the cracking moment, f r is the modulus of rupture, I g is the section moment of inertia, y t It is the distance from the outer edge of the tension side to the neutral axis when the beam is not cracked;

[0022] Step 103: Compare the load bending moment M and the cracking bending moment M cr , when M≤M cr When M>M cr When the load is too high, the cross section of the reinforced concrete component will crack.

[0023] In the above-mentioned method for calculating the bending moment of reinforced concrete members considering the change in the position of the neutral axis, the specific process of calculating the curvature of the reinforced concrete member with an uncracked cross section in step 3 includes:

[0024]

[0025] Where, is the curvature of the reinforced concrete member with no cracks in the cross section, M is the load bending moment, and E c is the elastic modulus, I g is the section moment of inertia.

[0026] In the above-mentioned method for calculating the bending moment of reinforced concrete members considering the change in the neutral axis position, the specific process of obtaining the neutral axis position of the reinforced concrete member with an uncracked cross section in step 4 includes:

[0027] Step 401: Assume the neutral axis position of the reinforced concrete member with an uncracked cross section so that the cross section is subjected to force equilibrium;

[0028] Step 402: Establish a cross-section force equilibrium equation;

[0029] c=F ap -F rp +F sp -F at +F rt -F st

[0030] Where c is the equilibrium calculated value, F ap is the total concrete pressure, F rp To repeat the calculation of concrete pressure, F sp is the steel bar pressure, F at is the total tensile force of concrete, F rt To repeatedly calculate the concrete tension, F st is the steel bar tension;

[0031] Step 403: When the equilibrium calculation value is greater than the set value, the neutral axis position is moved and step 402 is repeated until the equilibrium calculation value is less than the set value, thereby obtaining the neutral axis position of the reinforced concrete member with an uncracked cross section.

[0032] In the above-mentioned method for calculating the bending moment of reinforced concrete members considering the change in the position of the neutral axis, the specific process of calculating the bending moment of the reinforced concrete member with an uncracked cross section in step 5 includes:

[0033] M ext =M st +M pt +M at +M ap -M rt -M rp

[0034] Where M ext is the calculated value of the bending moment of the reinforced concrete member with no cracks in the cross section, M st is the bending moment of the tensile reinforcement, M pt is the bending moment of the compression reinforcement, M at is the total bending moment on the tensile side of concrete, Map is the total bending moment on the concrete pressure side, M rt To repeatedly calculate the concrete tensile and bending moments, M rp To repeatedly calculate the concrete compression and bending moment.

[0035] In the above-mentioned method for calculating the bending moment of reinforced concrete members considering the change in the position of the neutral axis, the specific process of calculating the effective crack depth of the reinforced concrete member with a cracked cross section in step 6 includes:

[0036]

[0037] Where X is the effective depth of the crack, f r is the modulus of rupture, E c is the elastic modulus, is the curvature of the reinforced concrete member with uncracked cross section.

[0038] In the above-mentioned method for calculating the bending moment of reinforced concrete members considering the change in the position of the neutral axis, the specific process of calculating the effective moment of inertia of the reinforced concrete member with a cracked cross section in step 7 includes:

[0039]

[0040] Where, I e is the effective moment of inertia, M cr is the cracking bending moment, M is the load bending moment, I g is the section moment of inertia, I cr is the moment of inertia of the cracked section.

[0041] In the above-mentioned method for calculating the bending moment of reinforced concrete members considering the change in the position of the neutral axis, the specific process of calculating the curvature of the reinforced concrete member with a cracked cross section in step 7 includes:

[0042]

[0043] Where, is the curvature of the reinforced concrete member with cracked section, M is the load bending moment, E c is the elastic modulus, I e is the effective moment of inertia.

[0044] In the above-mentioned method for calculating bending moments of reinforced concrete members considering the change in the neutral axis position, the specific process of assuming the neutral axis position of the reinforced concrete member with a cracked cross section based on the effective crack depth and the effective moment of inertia to achieve cross-sectional force equilibrium as described in step 8 includes:

[0045] Step 801: Assume the neutral axis position of the reinforced concrete member with a cracked cross section to balance the cross section forces;

[0046] Step 802: Establish a cross-section force equilibrium equation based on the equilibrium calculation value.

[0047] Step 803: When the equilibrium calculation value is greater than the set value, the neutral axis position is moved and step 802 is repeated until the equilibrium calculation value is less than the set value, thereby obtaining the neutral axis position of the reinforced concrete member with cracked cross section.

[0048] In the above-mentioned method for calculating bending moments of reinforced concrete members considering changes in the neutral axis position, the specific process of obtaining the true neutral axis position of a reinforced concrete member with a cracked cross section based on the effective moment of inertia and the true moment of inertia in step nine includes:

[0049] make When a>0.00006, the actual moment of inertia I exact The value of the effective moment of inertia I is assigned to the e Then repeat steps 7 to 9 until a≤0.00006, and take the neutral axis position of the reinforced concrete member with the assumed cracked cross section as the true neutral axis position.

[0050] Compared with the prior art, the present invention has the following advantages:

[0051] 1. The method of the present invention has simple steps, reasonable design and easy implementation.

[0052] 2. The present invention takes into account the change in the position of the neutral axis in the bending reinforced concrete member and uses the real moment of inertia to calculate the bending moment of the reinforced concrete member.

[0053] 3. When a reinforced concrete member cracks, the present invention calculates the bending performance taking into account the tensile strength of the concrete after cracking.

[0054] 4. The present invention introduces an analytical model that considers the tensile strength of concrete and the actual effective moment of inertia in the analysis of the flexural performance of reinforced concrete components, takes into account the impact of section cracking on the reduction of flexural stiffness and bearing capacity, and reflects the actual situation of the flexural stiffness of the cracked section.

[0055] 5. The present invention can be effectively applied to the calculation of bending moments of reinforced concrete members, with more accurate calculation results, significant effects, and easy promotion.

[0056] In summary, the method of the present invention has simple steps, reasonable design, and easy implementation. It can be effectively applied to the bending moment calculation of reinforced concrete components. The calculation results are more accurate, taking into account the changes in the neutral axis position, and reflecting the actual bending stiffness of the cracked section. The effect is significant and easy to promote.

[0057] The technical solution of the present invention is further described in detail below through the accompanying drawings and embodiments. BRIEF DESCRIPTION OF THE DRAWINGS

[0058] Figure 1 is a flow chart of the method of the present invention;

[0059] Figure 2 Comparison diagram of the load-moment relationship at mid-span and the theoretical solution. DETAILED DESCRIPTION

[0060] like Figure 1 As shown, the method for calculating the bending moment of reinforced concrete members considering the change of the neutral axis position of the present invention includes the following steps:

[0061] Step 1: Determine whether the cross section of the reinforced concrete component is cracked;

[0062] Step 2: When the cross section of the reinforced concrete component is not cracked, execute step 3; when the cross section of the reinforced concrete component is cracked, execute step 6;

[0063] Step 3: Calculate the curvature of the reinforced concrete member with an uncracked cross section;

[0064] Step 4: Obtain the neutral axis position of the reinforced concrete member with an uncracked cross section;

[0065] Step 5: Calculate the bending moment of the reinforced concrete member with uncracked cross section;

[0066] Step 6: Calculate the effective depth of cracks in reinforced concrete components with cracked sections;

[0067] Step 7: Calculate the effective moment of inertia and curvature of the reinforced concrete member with cracked cross section;

[0068] Step 8: Based on the effective crack depth and effective moment of inertia, assume the neutral axis position of the reinforced concrete member with a cracked cross section so that the cross section is under force equilibrium, and calculate the true moment of inertia;

[0069] Step 9: Obtain the true neutral axis position of the reinforced concrete member with a cracked cross section based on the effective moment of inertia and the true moment of inertia;

[0070] Step 10: Calculate the bending moment of the reinforced concrete member with cracked cross section based on the true neutral axis position.

[0071] In this embodiment, the specific process of determining whether the cross section of the reinforced concrete component is cracked in step 1 includes:

[0072] Step 101, calculating the load bending moment of the reinforced concrete member;

[0073]

[0074] Where M is the load bending moment, P is the applied load, and z is the horizontal distance from the measured section to the support;

[0075] Step 102: Calculate the cracking bending moment of the reinforced concrete member;

[0076]

[0077] Where M cr is the cracking moment, f r is the modulus of rupture, I g is the section moment of inertia, y t It is the distance from the outer edge of the tension side to the neutral axis when the beam is not cracked;

[0078] Step 103: Compare the load bending moment M and the cracking bending moment M cr , when M≤M cr When M>M cr When the load is too high, the cross section of the reinforced concrete component will crack.

[0079] In specific implementation, when the cross section of the reinforced concrete component is in the elastic stage and has not cracked, the load bending moment M is used in the cross section analysis; when the cross section tensile stress of the reinforced concrete component reaches the yield strength of the concrete, the neutral axial compression side deviates, but the steel bar has not yet reached the yield state. At this time, the cracking bending moment M is used in the cross section analysis. cr Conduct analysis.

[0080] In this embodiment, the specific process of calculating the curvature of the reinforced concrete member with an uncracked cross section in step 3 includes:

[0081]

[0082] Where, is the curvature of the reinforced concrete member with no cracks in the cross section, M is the load bending moment, and E c is the elastic modulus, I g is the section moment of inertia.

[0083] In this embodiment, the specific process of obtaining the neutral axis position of the reinforced concrete member with an uncracked cross section in step 4 includes:

[0084] Step 401: Assume the neutral axis position of the reinforced concrete member with an uncracked cross section so that the cross section is subjected to force equilibrium;

[0085] Step 402: Establish a cross-section force equilibrium equation;

[0086] c=F ap -F rp +F sp -F at +F rt-F st

[0087] Where c is the equilibrium calculated value, F ap is the total concrete pressure, F rp To repeat the calculation of concrete pressure, F sp is the steel bar pressure, F at is the total tensile force of concrete, F rt To repeatedly calculate the concrete tension, F st is the steel bar tension;

[0088] When it is implemented, the total concrete pressure F ap , Repeat calculation of concrete pressure F rp , steel bar pressure F sp , total concrete tension F at , Repeated calculation of concrete tension F rt and steel bar tension F st The calculations are all related to the neutral axis position.

[0089] Step 403: When the equilibrium calculation value is greater than the set value, the neutral axis position is moved and step 402 is repeated until the equilibrium calculation value is less than the set value, thereby obtaining the neutral axis position of the reinforced concrete member with an uncracked cross section.

[0090] In specific implementation, the setting value is 0.0005. When c>0.0005, let y f =y l -0.00001, where y f is the assumed neutral axis position, y l is the neutral axis position assumed previously. When c < 0.0005, the neutral axis position assumed at this time is used as the neutral axis position of the reinforced concrete member with uncracked section.

[0091] In this embodiment, the specific process of calculating the bending moment of the reinforced concrete member with an uncracked cross section in step 5 includes:

[0092] M ext =M st +M pt +M at +M ap -M rt -M rp

[0093] Where M ext is the calculated value of the bending moment of the reinforced concrete member with no cracks in the cross section, M st is the bending moment of the tensile reinforcement, M pt is the bending moment of the compression reinforcement, M at is the total bending moment on the tensile side of concrete, M ap is the total bending moment on the concrete pressure side, Mrt To repeatedly calculate the concrete tensile and bending moments, M rp To repeatedly calculate the concrete compression and bending moment.

[0094] In this embodiment, the specific process of calculating the effective crack depth of the reinforced concrete member with a cracked cross section in step 6 includes:

[0095]

[0096] Where X is the effective depth of the crack, f r is the modulus of rupture, E c is the elastic modulus, is the curvature of the reinforced concrete member with uncracked cross section.

[0097] In this embodiment, the specific process of calculating the effective moment of inertia of the reinforced concrete member with a cracked cross section in step 7 includes:

[0098]

[0099] Where, I e is the effective moment of inertia, M cr is the cracking bending moment, M is the load bending moment, I g is the section moment of inertia, I cr is the moment of inertia of the cracked section.

[0100] In this embodiment, the specific process of calculating the curvature of the reinforced concrete member with a cracked cross section in step seven includes:

[0101]

[0102] Where, is the curvature of the reinforced concrete member with cracked section, M is the load bending moment, E c is the elastic modulus, I e is the effective moment of inertia.

[0103] In this embodiment, the specific process of assuming the neutral axis position of the reinforced concrete member with a cracked cross section based on the effective crack depth and the effective moment of inertia in step eight to achieve cross-sectional force balance includes:

[0104] Step 801: Assume the neutral axis position of the reinforced concrete member with a cracked cross section to balance the cross section forces;

[0105] Step 802: Establish a cross-section force equilibrium equation based on the equilibrium calculation value.

[0106] During specific implementation, the calculation of the equilibrium calculation value is related to the position of the neutral axis.

[0107] Step 803: When the equilibrium calculation value is greater than the set value, the neutral axis position is moved and step 802 is repeated until the equilibrium calculation value is less than the set value, thereby obtaining the neutral axis position of the reinforced concrete member with cracked cross section.

[0108] In specific implementation, when the effective depth of the crack has not exceeded the thickness of the concrete cover, the calculated tensile force of the concrete at the position of the tensile steel bar should be subtracted from the generated tensile force during the cross-section force equilibrium analysis to avoid repeated calculations.

[0109] In this embodiment, the specific process of obtaining the true neutral axis position of the reinforced concrete member with a cracked cross section based on the effective moment of inertia and the true moment of inertia in step nine includes:

[0110] make When a>0.00006, the actual moment of inertia I exact The value of the effective moment of inertia I is assigned to the e Then repeat steps 7 to 9 until a≤0.00006, and take the neutral axis position of the reinforced concrete member with the assumed cracked cross section as the true neutral axis position.

[0111] In order to verify the proposed calculation method, the mid-span load-moment relationship is compared with the theoretical solution. The results are shown in Figure 2. Figure 2 As shown, from Figure 2 It can be found that when the tensile strength and actual moment of inertia of concrete are considered simultaneously, the load-moment relationship is relatively close to the theoretical solution of the load range, thereby verifying the applicability of the calculation method of the present invention.

[0112] The above description is only a preferred embodiment of the present invention and does not limit the present invention in any way. Any simple modification, change and equivalent structural change made to the above embodiment based on the technical essence of the present invention shall still fall within the scope of protection of the technical solution of the present invention.

Claims

1. A method for calculating the bending moment of reinforced concrete members considering the change of the neutral axis position, characterized in that: The following steps are involved: Step 1: Determine whether the cross section of the reinforced concrete component is cracked; Step 2: When the cross section of the reinforced concrete component is not cracked, execute step 3; when the cross section of the reinforced concrete component is cracked, execute step 6; Step 3: Calculate the curvature of the reinforced concrete member with an uncracked cross section; Step 4: Obtain the neutral axis position of the reinforced concrete member with an uncracked cross section; Step 5: Calculate the bending moment of the reinforced concrete member with uncracked cross section; Step 6: Calculate the effective depth of cracks in reinforced concrete components with cracked sections; Step 7: Calculate the effective moment of inertia and curvature of the reinforced concrete member with cracked cross section; The specific process of calculating the effective moment of inertia of the reinforced concrete member with cracked cross section includes: Where, I e is the effective moment of inertia, M cr is the cracking bending moment, M is the load bending moment, I g is the section moment of inertia, I cr is the moment of inertia of the cracked section; Step 8: Based on the effective crack depth and the effective moment of inertia, the neutral axis position of the reinforced concrete member with a cracked cross section is assumed to be balanced, and the true moment of inertia is calculated. The specific process of assuming the neutral axis position of the reinforced concrete member with a cracked cross section to be balanced according to the effective crack depth and the effective moment of inertia includes: Step 801: Assume the neutral axis position of the reinforced concrete member with a cracked cross section to balance the cross section forces; Step 802: Establish a cross-section force equilibrium equation based on the equilibrium calculation value. Step 803: When the equilibrium calculation value is greater than the set value, the neutral axis position is moved, and step 802 is repeated until the equilibrium calculation value is less than the set value, thereby obtaining the neutral axis position of the reinforced concrete member with a cracked cross section. Step 9: Obtaining the true neutral axis position of the reinforced concrete member with a cracked cross section based on the effective moment of inertia and the true moment of inertia. The specific process of obtaining the true neutral axis position of the reinforced concrete member with a cracked cross section based on the effective moment of inertia and the true moment of inertia includes: make When a>0.00006, the actual moment of inertia I exact The value of the effective moment of inertia I is assigned to the e Then repeat steps 7 to 9 until a≤0.00006, and take the neutral axis position of the reinforced concrete member with the assumed cracked cross section as the true neutral axis position; Step 10: Calculate the bending moment of the reinforced concrete member with cracked cross section based on the true neutral axis position.

2. A method for calculating bending moments of reinforced concrete members taking into account changes in the position of the neutral axis according to claim 1, characterized in that: The specific process of determining whether the cross section of the reinforced concrete component is cracked in step 1 includes: Step 101, calculating the load bending moment of the reinforced concrete member; Where M is the load bending moment, P is the applied load, and z is the horizontal distance from the measured section to the support; Step 102: Calculate the cracking bending moment of the reinforced concrete member; Where M cr is the cracking moment, f r is the modulus of rupture, I g is the section moment of inertia, y t It is the distance from the outer edge of the tension side to the neutral axis when the beam is not cracked; Step 103: Compare the load bending moment M and the cracking bending moment M cr , when M≤M cr When M>M cr When the load is too high, the cross section of the reinforced concrete component will crack.

3. A method for calculating bending moments of reinforced concrete members taking into account changes in the position of the neutral axis according to claim 1, characterized in that: The specific process of calculating the curvature of the reinforced concrete member with an uncracked cross section described in step 3 includes: Where, is the curvature of the reinforced concrete member with no cracks in the cross section, M is the load bending moment, and E c is the elastic modulus, I g is the section moment of inertia.

4. A method for calculating bending moments of reinforced concrete members taking into account changes in the position of the neutral axis according to claim 1, characterized in that: The specific process of obtaining the neutral axis position of the reinforced concrete member with an uncracked cross section in step 4 includes: Step 401: Assume the neutral axis position of the reinforced concrete member with an uncracked cross section so that the cross section is subjected to force equilibrium; Step 402: Establish a cross-section force equilibrium equation; c=F ap -F rp +F sp -F at +F rt -F st Where c is the equilibrium calculated value, F ap is the total concrete pressure, F rp To repeat the calculation of concrete pressure, F sp is the steel bar pressure, F at is the total tensile force of concrete, F rt To repeatedly calculate the concrete tension, F st is the steel bar tension; Step 403: When the equilibrium calculation value is greater than the set value, the neutral axis position is moved and step 402 is repeated until the equilibrium calculation value is less than the set value, thereby obtaining the neutral axis position of the reinforced concrete member with an uncracked cross section.

5. A method for calculating bending moments of reinforced concrete members taking into account changes in the position of the neutral axis according to claim 1, characterized in that: The specific process of calculating the bending moment of the reinforced concrete member with an uncracked cross section described in step 5 includes: M ext =M st +M pt +M at +M ap -M rt -M rp Where M ext is the calculated value of the bending moment of the reinforced concrete member with no cracks in the cross section, M st is the bending moment of the tensile reinforcement, M pt is the bending moment of the compression reinforcement, M at is the total bending moment on the tensile side of concrete, M ap is the total bending moment on the concrete pressure side, M rt To repeatedly calculate the concrete tensile and bending moments, M rp To repeatedly calculate the concrete compression and bending moment.

6. A method for calculating bending moments of reinforced concrete members taking into account changes in the position of the neutral axis according to claim 1, characterized in that: The specific process of calculating the effective crack depth of the reinforced concrete member with cracked cross section described in step 6 includes: Where X is the effective depth of the crack, f r is the modulus of rupture, E c is the elastic modulus, is the curvature of the reinforced concrete member with uncracked cross section.

7. A method for calculating bending moments of reinforced concrete members taking into account changes in the position of the neutral axis according to claim 1, characterized in that: The specific process of calculating the curvature of the reinforced concrete member with cracked cross section described in step 7 includes: Where, is the curvature of the reinforced concrete member with cracked section, M is the load bending moment, E c is the elastic modulus, I e is the effective moment of inertia.

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