CN116304475A - A Bending Moment Calculation Method of Reinforced Concrete Members Considering the Change of Neutral Axis Position - Google Patents

A Bending Moment Calculation Method of Reinforced Concrete Members Considering the Change of Neutral Axis Position Download PDF

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CN116304475A
CN116304475A CN202310181099.9A CN202310181099A CN116304475A CN 116304475 A CN116304475 A CN 116304475A CN 202310181099 A CN202310181099 A CN 202310181099A CN 116304475 A CN116304475 A CN 116304475A
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reinforced concrete
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郑文杰
李戈
白雪冬
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Xian University of Architecture and Technology
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Abstract

The invention discloses a reinforced concrete member bending moment calculation method considering neutral axis position change, which comprises the following steps of firstly, judging whether the section of a reinforced concrete member is cracked or not; 2. when the section is not cracked, executing III; when cracking occurs, performing six; 3. calculating curvature; 4. acquiring the position of a central shaft; 5. calculating bending moment; 6. calculating the effective depth of the crack; 7. calculating an effective moment of inertia and curvature; 8. assuming the central axis position of the reinforced concrete member with the cracked section, so that the section is stressed in balance, and calculating the true moment of inertia; 9. acquiring a real central axis position of a reinforced concrete member with a cracked section; 10. the bending moment is calculated. The method has the advantages of simple steps, reasonable design and convenient realization, can be effectively applied to the calculation of the bending moment of the reinforced concrete member, has more accurate calculation result, considers the change of the neutral axis position, reflects the real situation of the bending rigidity of the cracking section, has obvious effect and is convenient to popularize.

Description

一种考虑中性轴位置变化的钢筋混凝土构件弯矩计算方法A Bending Moment Calculation Method of Reinforced Concrete Members Considering the Change of Neutral Axis Position

技术领域technical field

本发明属于岩土工程技术领域，具体涉及一种考虑中性轴位置变化的钢筋混凝土构件弯矩计算方法。The 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 considering the position change of a neutral axis.

背景技术Background technique

在平面弯曲和斜弯曲情形下，横截面与应力平面的交线上各点的正应力值均为零，这条交线称为中性轴。一般来说，当受弯钢筋混凝土梁的拉应力达到混凝土强度时，中性轴会向上移动，但此时钢筋不会屈服。然而，普通钢筋混凝土理论忽略从中性轴以下的混凝土抗拉强度，当开裂已经在钢筋混凝土构件部分发展，计算的抗弯性能没有考虑开裂后的抗拉强度，会导致对构件截面承载能力的错误估计。因此，隧道衬砌过度设计或过度使用加固是普遍现象，为了克服设计上的不足，急需建立一个考虑混凝土受拉开裂引起的截面非线性和真实有效惯性矩的混凝土构件弯曲行为的解析模型。In the case of plane bending and oblique bending, the normal stress value of each point on the intersection line of the cross section and the stress plane is zero, and this intersection line is called the neutral axis. Generally speaking, when the tensile stress of a flexural reinforced concrete beam reaches the concrete strength, the neutral axis will move upward, but the reinforcement will not yield at this time. However, ordinary reinforced concrete theory ignores the tensile strength of concrete from below the neutral axis. When cracking has developed in the reinforced concrete member part, the calculated flexural performance does not consider the tensile strength after cracking, which will lead to an error in the bearing capacity of the member section. estimate. Therefore, over-design or over-use of reinforcement in tunnel lining is a common phenomenon. In order to overcome the design deficiencies, it is urgent to establish an analytical model for the bending behavior of concrete members considering the section nonlinearity and true effective moment of inertia caused by concrete tensile cracking.

发明内容Contents of the invention

本发明所要解决的技术问题在于针对上述现有技术中的不足，提供一种考虑中性轴位置变化的钢筋混凝土构件弯矩计算方法，其方法步骤简单，设计合理，实现方便，能够有效应用在钢筋混凝土构件弯矩计算中，计算结果更加精确，考虑了中性轴位置的变化，反映了开裂截面弯曲刚度的真实情况，效果显著，便于推广。The technical problem to be solved by the present invention is to provide a method for calculating the bending moment of reinforced concrete members considering the position change of the neutral axis in view of the deficiencies in the above-mentioned prior art. The method has simple steps, reasonable design, convenient implementation, and can be effectively applied in In the calculation of the bending moment of reinforced concrete members, the calculation results are more accurate, considering the change of the neutral axis position, reflecting the real situation of the bending stiffness of the cracked section, the effect is remarkable, and it is easy to popularize.

为解决上述技术问题，本发明采用的技术方案是：一种考虑中性轴位置变化的钢筋混凝土构件弯矩计算方法，包括以下步骤：In order to solve the above-mentioned technical problems, the technical solution adopted in the present invention is: a method for calculating the bending moment of reinforced concrete members considering the position change of the neutral axis, comprising the following steps:

步骤一、判断钢筋混凝土构件的截面是否开裂；Step 1, judging whether the section of the reinforced concrete member is cracked;

步骤二、当所述钢筋混凝土构件的截面未开裂时，执行步骤三；当所述钢筋混凝土构件的截面发生开裂时，执行步骤六；Step 2. When the section of the reinforced concrete member is not cracked, perform step 3; when the section of the reinforced concrete member is cracked, perform step 6;

步骤三、计算截面未开裂的钢筋混凝土构件的曲率；Step 3, calculating the curvature of the uncracked reinforced concrete member of the section;

步骤四、获取截面未开裂的钢筋混凝土构件的中心轴位置；Step 4, obtaining the central axis position of the uncracked reinforced concrete member;

步骤五、计算截面未开裂的钢筋混凝土构件的弯矩；Step 5, calculating the bending moment of the uncracked reinforced concrete member of the section;

步骤六、计算截面开裂的钢筋混凝土构件的裂缝有效深度；Step 6, calculate the crack effective depth of the reinforced concrete member of section crack;

步骤七、计算截面开裂的钢筋混凝土构件的有效惯性矩和曲率；Step 7, calculating the effective moment of inertia and curvature of the reinforced concrete member with cracked section;

步骤八、根据所述裂缝有效深度和有效惯性矩，假设截面开裂的钢筋混凝土构件的中心轴位置，使得截面受力平衡，并计算真实惯性矩；Step 8, according to the effective depth of the crack and the effective moment of inertia, assuming the position of the central axis of the reinforced concrete member with cracked section, so that the section is stressed, and calculates the true moment of inertia;

步骤九、根据所述有效惯性矩和真实惯性矩，获取截面开裂的钢筋混凝土构件的真实的中心轴位置；Step 9, according to the effective moment of inertia and the real moment of inertia, obtain the real central axis position of the reinforced concrete member with cracked section;

步骤十、根据所述真实的中心轴位置，计算截面开裂的钢筋混凝土构件的弯矩。Step 10: Calculate the bending moment of the reinforced concrete member with cracked section according to the real central axis position.

上述的一种考虑中性轴位置变化的钢筋混凝土构件弯矩计算方法，步骤一中所述判断钢筋混凝土构件的截面是否开裂的具体过程包括：In the above-mentioned method for calculating the bending moment of a reinforced concrete member considering the change of the position of the neutral axis, the specific process of judging whether the section of the reinforced concrete member is cracked as described in step 1 includes:

步骤101、计算钢筋混凝土构件的载重弯矩；Step 101, calculating the load-bearing bending moment of the reinforced concrete member;

Figure BDA0004102357460000021
Figure BDA0004102357460000021

式中，M为载重弯矩，P为加载荷重，z为被测断面至支架的水平距离；In the formula, M is the load bending moment, P is the load weight, and z is the horizontal distance from the measured section to the support;

步骤102、计算钢筋混凝土构件的开裂弯矩；Step 102, calculating the cracking moment of the reinforced concrete member;

Figure BDA0004102357460000022
Figure BDA0004102357460000022

式中，Mcr为开裂弯矩，fr为断裂模量，Ig为截面惯性矩，yt为梁未开裂时拉力侧外缘至中性轴距离；where M cr is the cracking moment, f r is the modulus of rupture, I g is the section moment of inertia, and y t is the distance from the outer edge of the tension side to the neutral axis when the beam is not cracked;

步骤103、比较载重弯矩M和开裂弯矩Mcr，当M≤Mcr时，钢筋混凝土构件的截面未开裂；当M＞Mcr时，钢筋混凝土构件的截面发生开裂。Step 103: Comparing the load bending moment M and the cracking moment M cr , when M≤M cr , the section of the reinforced concrete member is not cracked; when M>M cr , the section of the reinforced concrete member is cracked.

上述的一种考虑中性轴位置变化的钢筋混凝土构件弯矩计算方法，步骤三中所述计算截面未开裂的钢筋混凝土构件的曲率的具体过程包括：In the above method for calculating the bending moment of a reinforced concrete member considering the change of the position of the neutral axis, the specific process of calculating the curvature of the uncracked reinforced concrete member described in step 3 includes:

Figure BDA0004102357460000031
Figure BDA0004102357460000031

式中，

Figure BDA0004102357460000032
为截面未开裂的钢筋混凝土构件的曲率，M为载重弯矩，Ec为弹性模量，Ig为截面惯性矩。In the formula,
Figure BDA0004102357460000032
is the curvature of the uncracked reinforced concrete member, M is the load bending moment, E c is the modulus of elasticity, and I g is the moment of inertia of the section.

上述的一种考虑中性轴位置变化的钢筋混凝土构件弯矩计算方法，步骤四中所述获取截面未开裂的钢筋混凝土构件的中心轴位置的具体过程包括：In the above method for calculating the bending moment of a reinforced concrete member considering the change of the position of the neutral axis, the specific process of obtaining the position of the central axis of the uncracked reinforced concrete member in step 4 includes:

步骤401、假设截面未开裂的钢筋混凝土构件的中性轴位置，使截面受力平衡；Step 401, assuming the position of the neutral axis of the uncracked reinforced concrete member, so that the force on the section is balanced;

步骤402、建立截面受力平衡方程；Step 402, establishing a cross-sectional force balance equation;

c＝Fap-Frp+Fsp-Fat+Frt-Fst c＝F ap -F rp +F sp -F at +F rt -F st

式中，c为平衡计算值，Fap为混凝土总压力，Frp为重复计算混凝土压力，Fsp为钢筋压力，Fat为混凝土总拉力，Frt为重复计算混凝土拉力，Fst为钢筋拉力；In the formula, c is the balance calculation value, F ap is the total concrete pressure, F rp is the repeated calculation concrete pressure, F sp is the steel bar pressure, F at is the total concrete tension, F rt is the repeated calculation concrete tension, F st is the steel bar tension ;

步骤403、当平衡计算值大于设定值时，移动中性轴位置，重复步骤402，直至平衡计算值小于设定值，得到截面未开裂的钢筋混凝土构件的中心轴位置。Step 403 , when the calculated balance value is greater than the set value, move the position of the neutral axis, repeat step 402 until the calculated balance value is less than the set value, and obtain the central axis position of the uncracked reinforced concrete member.

上述的一种考虑中性轴位置变化的钢筋混凝土构件弯矩计算方法，步骤五中所述计算截面未开裂的钢筋混凝土构件的弯矩的具体过程包括：In the aforementioned method for calculating the bending moment of a reinforced concrete member considering the change of the position of the neutral axis, the specific process of calculating the bending moment of the uncracked reinforced concrete member described in step 5 includes:

Mext＝Mst+Mpt+Mat+Map-Mrt-Mrp M ext =M st +M pt +M at +M ap -M rt -M rp

式中，Mext为截面未开裂的钢筋混凝土构件的弯矩计算值，Mst为拉力钢筋弯矩，Mpt为压力钢筋弯矩，Mat为混凝土拉力侧总弯矩，Map为混凝土压力侧总弯矩，Mrt为重复计算混凝土拉力弯矩，Mrp为重复计算混凝土压力弯矩。In the formula, M ext is the calculated value of bending moment of reinforced concrete members with uncracked section, M st is the bending moment of tension steel bar, M pt is the bending moment of pressure steel bar, M at is the total bending moment of concrete tension side, and M ap is the concrete pressure The total lateral bending moment, M rt is the repeated calculation of the concrete tension bending moment, and M rp is the repeated calculation of the concrete pressure bending moment.

上述的一种考虑中性轴位置变化的钢筋混凝土构件弯矩计算方法，步骤六中所述计算截面开裂的钢筋混凝土构件的裂缝有效深度的具体过程包括：In the above-mentioned method for calculating the bending moment of reinforced concrete members considering the change of neutral axis position, the specific process of calculating the effective depth of cracks of reinforced concrete members with cracked cross-sections described in step 6 includes:

Figure BDA0004102357460000041
Figure BDA0004102357460000041

式中，X为裂缝有效深度，fr为断裂模量，Ec为弹性模量，

Figure BDA0004102357460000042
为截面未开裂的钢筋混凝土构件的曲率。where X is the effective depth of the crack, f r is the modulus of rupture, E c is the modulus of elasticity,
Figure BDA0004102357460000042
is the curvature of the uncracked reinforced concrete member.

上述的一种考虑中性轴位置变化的钢筋混凝土构件弯矩计算方法，步骤七中所述计算截面开裂的钢筋混凝土构件的有效惯性矩的具体过程包括：In the above method for calculating the bending moment of a reinforced concrete member considering the change of the position of the neutral axis, the specific process of calculating the effective moment of inertia of the reinforced concrete member with cracked cross-section described in step 7 includes:

Figure BDA0004102357460000043
Figure BDA0004102357460000043

式中，Ie为有效惯性矩，Mcr为开裂弯矩，M为载重弯矩，Ig为截面惯性矩，Icr为开裂的截面惯性矩。In the formula, I e is the effective moment of inertia, Mc cr is the cracking moment, M is the load bending moment, I g is the section moment of inertia, and I cr is the section moment of inertia of cracking.

上述的一种考虑中性轴位置变化的钢筋混凝土构件弯矩计算方法，步骤七中所述计算截面开裂的钢筋混凝土构件的曲率的具体过程包括：In the above method for calculating the bending moment of reinforced concrete members considering the change of the position of the neutral axis, the specific process of calculating the curvature of the reinforced concrete member with cracked cross-section described in step 7 includes:

Figure BDA0004102357460000044
Figure BDA0004102357460000044

式中，

Figure BDA0004102357460000045
为截面开裂的钢筋混凝土构件的曲率，M为载重弯矩，Ec为弹性模量，Ie为有效惯性矩。In the formula,
Figure BDA0004102357460000045
is the curvature of the reinforced concrete member with cracked section, M is the load bending moment, E c is the modulus of elasticity, and I e is the effective moment of inertia.

上述的一种考虑中性轴位置变化的钢筋混凝土构件弯矩计算方法，步骤八中所述根据裂缝有效深度和有效惯性矩，假设截面开裂的钢筋混凝土构件的中心轴位置，使得截面受力平衡的具体过程包括：In the above method for calculating the bending moment of reinforced concrete members considering the change of the position of the neutral axis, in step 8, according to the effective depth of the crack and the effective moment of inertia, the position of the central axis of the reinforced concrete member with a cracked section is assumed, so that the force on the section is balanced The specific process includes:

步骤801、假设截面开裂的钢筋混凝土构件的中性轴位置，使截面受力平衡；Step 801, assuming the position of the neutral axis of the reinforced concrete member with a cracked section, so that the force on the section is balanced;

步骤802、建立关于平衡计算值的截面受力平衡方程；Step 802, establishing a cross-sectional force balance equation about the balance calculation value;

步骤803、当平衡计算值大于设定值时，移动中性轴位置，重复步骤802，直至平衡计算值小于设定值，得到截面开裂的钢筋混凝土构件的中心轴位置。Step 803 , when the calculated balance value is greater than the set value, move the position of the neutral axis, repeat step 802 until the calculated balance value is less than the set value, and obtain the position of the central axis of the reinforced concrete member with a cracked section.

上述的一种考虑中性轴位置变化的钢筋混凝土构件弯矩计算方法，步骤九中所述根据有效惯性矩和真实惯性矩，获取截面开裂的钢筋混凝土构件的真实的中心轴位置的具体过程包括：In the above method for calculating the bending moment of a reinforced concrete member considering the change of the neutral axis position, the specific process of obtaining the real central axis position of the reinforced concrete member with a cracked section according to the effective moment of inertia and the real moment of inertia described in step 9 includes :

令

Figure BDA0004102357460000051
当a＞0.00006时，将此时得到的真实惯性矩Iexact的值赋给此时的有效惯性矩Ie；然后重复步骤七～步骤九，直到a≤0.00006，将此时假设的截面开裂的钢筋混凝土构件的中心轴位置作为真实的中性轴位置。make
Figure BDA0004102357460000051
When a > 0.00006, assign the value of the real moment of inertia I exact obtained at this time to the effective moment of inertia I e at this time; then repeat steps 7 to 9 until a ≤ 0.00006, and assign the value of the assumed section cracked at this time The position of the central axis of the reinforced concrete member is taken as the true neutral axis position.

本发明与现有技术相比具有以下优点：Compared with the prior art, the present invention has the following advantages:

1、本发明方法步骤简单，设计合理，实现方便。1. The method of the present invention has simple steps, reasonable design and convenient implementation.

2、本发明在受弯钢筋混凝土构件中，考虑了中性轴位置的变化，采用真实的惯性矩去计算钢筋混凝土构件的弯矩。2. In the flexural reinforced concrete member, the present invention considers the change of the position of the neutral axis, and uses the real moment of inertia to calculate the bending moment of the reinforced concrete member.

3、本发明在钢筋混凝土构件开裂时，计算的抗弯性能考虑了开裂后的混凝土抗拉强度。3. When the reinforced concrete member cracks in the present invention, the calculated flexural performance takes into account the concrete tensile strength after cracking.

4、本发明在钢筋混凝土构件抗弯性能分析中引入了考虑混凝土抗拉强度和实际有效惯性矩的解析模型，考虑了截面开裂对抗弯刚度及承载能力降低的影响，反映了开裂截面弯曲刚度的真实情况。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 members, considers the impact of section cracking on the flexural stiffness and the reduction in bearing capacity, and reflects the flexural stiffness of the cracked section the real situation.

5、本发明能够有效应用在钢筋混凝土构件弯矩计算中，计算结果更加精确，效果显著，便于推广。5. The present invention can be effectively applied to the calculation of the bending moment of reinforced concrete members, the calculation result is more accurate, the effect is remarkable, and it is easy to popularize.

综上所述，本发明方法步骤简单，设计合理，实现方便，能够有效应用在钢筋混凝土构件弯矩计算中，计算结果更加精确，考虑了中性轴位置的变化，反映了开裂截面弯曲刚度的真实情况，效果显著，便于推广。In summary, the method of the present invention has simple steps, reasonable design, and convenient implementation, and can be effectively applied in the calculation of the bending moment of reinforced concrete members. The real situation, the effect is remarkable, and it is easy to promote.

下面通过附图和实施例，对本发明的技术方案做进一步的详细描述。The technical solutions of the present invention will be described in further detail below with reference to the accompanying drawings and embodiments.

附图说明Description of drawings

图1为本发明的方法流程图；Fig. 1 is method flowchart of the present invention;

图2为跨中荷载-弯矩关系与理论解对比图。Figure 2 is a comparison diagram between the mid-span load-bending moment relationship and the theoretical solution.

具体实施方式Detailed ways

如图1所示，本发明的考虑中性轴位置变化的钢筋混凝土构件弯矩计算方法，包括以下步骤：As shown in Figure 1, the calculation method of the reinforced concrete member bending moment considering the neutral axis position variation of the present invention comprises the following steps:

步骤一、判断钢筋混凝土构件的截面是否开裂；Step 1, judging whether the section of the reinforced concrete member is cracked;

步骤二、当所述钢筋混凝土构件的截面未开裂时，执行步骤三；当所述钢筋混凝土构件的截面发生开裂时，执行步骤六；Step 2. When the section of the reinforced concrete member is not cracked, perform step 3; when the section of the reinforced concrete member is cracked, perform step 6;

步骤三、计算截面未开裂的钢筋混凝土构件的曲率；Step 3, calculating the curvature of the uncracked reinforced concrete member of the section;

步骤四、获取截面未开裂的钢筋混凝土构件的中心轴位置；Step 4, obtaining the central axis position of the uncracked reinforced concrete member;

步骤五、计算截面未开裂的钢筋混凝土构件的弯矩；Step 5, calculating the bending moment of the uncracked reinforced concrete member of the section;

步骤六、计算截面开裂的钢筋混凝土构件的裂缝有效深度；Step 6, calculate the crack effective depth of the reinforced concrete member of section crack;

步骤七、计算截面开裂的钢筋混凝土构件的有效惯性矩和曲率；Step 7, calculating the effective moment of inertia and curvature of the reinforced concrete member with cracked section;

步骤八、根据所述裂缝有效深度和有效惯性矩，假设截面开裂的钢筋混凝土构件的中心轴位置，使得截面受力平衡，并计算真实惯性矩；Step 8, according to the effective depth of the crack and the effective moment of inertia, assuming the position of the central axis of the reinforced concrete member with cracked section, so that the section is stressed, and calculates the true moment of inertia;

步骤九、根据所述有效惯性矩和真实惯性矩，获取截面开裂的钢筋混凝土构件的真实的中心轴位置；Step 9, according to the effective moment of inertia and the real moment of inertia, obtain the real central axis position of the reinforced concrete member with cracked section;

步骤十、根据所述真实的中心轴位置，计算截面开裂的钢筋混凝土构件的弯矩。Step 10: Calculate the bending moment of the reinforced concrete member with cracked section according to the real central axis position.

本实施例中，步骤一中所述判断钢筋混凝土构件的截面是否开裂的具体过程包括：In this embodiment, the specific process of judging whether the section of the reinforced concrete member described in step 1 is cracked includes:

步骤101、计算钢筋混凝土构件的载重弯矩；Step 101, calculating the load-bearing bending moment of the reinforced concrete member;

Figure BDA0004102357460000061
Figure BDA0004102357460000061

式中，M为载重弯矩，P为加载荷重，z为被测断面至支架的水平距离；In the formula, M is the load bending moment, P is the load weight, and z is the horizontal distance from the measured section to the support;

步骤102、计算钢筋混凝土构件的开裂弯矩；Step 102, calculating the cracking moment of the reinforced concrete member;

Figure BDA0004102357460000062
Figure BDA0004102357460000062

式中，Mcr为开裂弯矩，fr为断裂模量，Ig为截面惯性矩，yt为梁未开裂时拉力侧外缘至中性轴距离；where M cr is the cracking moment, f r is the modulus of rupture, I g is the section moment of inertia, and y t is the distance from the outer edge of the tension side to the neutral axis when the beam is not cracked;

步骤103、比较载重弯矩M和开裂弯矩Mcr，当M≤Mcr时，钢筋混凝土构件的截面未开裂；当M＞Mcr时，钢筋混凝土构件的截面发生开裂。Step 103: Comparing the load bending moment M and the cracking moment M cr , when M≤M cr , the section of the reinforced concrete member is not cracked; when M>M cr , the section of the reinforced concrete member is cracked.

具体实施时，当钢筋混凝土构件的截面处于弹性阶段未开裂时，截面分析中采用载重弯矩M；当钢筋混凝土构件的截面拉应力达到混凝土屈服强度时，中性轴向受压侧发生偏移，然而，钢筋还没有达到屈服状态，此时，截面分析中采用开裂弯矩Mcr进行分析。In practice, when the section of the reinforced concrete member is in the elastic stage without cracking, the load-bearing bending moment M is used in the section analysis; when the tensile stress of the section of the reinforced concrete member reaches the yield strength of the concrete, the neutral axial compression side shifts , however, the steel bar has not yet reached the yield state, at this time, the cracking moment M cr is used in the section analysis for analysis.

本实施例中，步骤三中所述计算截面未开裂的钢筋混凝土构件的曲率的具体过程包括：In this embodiment, the specific process of calculating the curvature of the uncracked reinforced concrete member described in step 3 includes:

Figure BDA0004102357460000071
Figure BDA0004102357460000071

式中，

Figure BDA0004102357460000072
为截面未开裂的钢筋混凝土构件的曲率，M为载重弯矩，Ec为弹性模量，Ig为截面惯性矩。In the formula,
Figure BDA0004102357460000072
is the curvature of the uncracked reinforced concrete member, M is the load bending moment, E c is the modulus of elasticity, and I g is the moment of inertia of the section.

本实施例中，步骤四中所述获取截面未开裂的钢筋混凝土构件的中心轴位置的具体过程包括：In this embodiment, the specific process of obtaining the central axis position of the uncracked reinforced concrete member described in step 4 includes:

步骤401、假设截面未开裂的钢筋混凝土构件的中性轴位置，使截面受力平衡；Step 401, assuming the position of the neutral axis of the uncracked reinforced concrete member, so that the force on the section is balanced;

步骤402、建立截面受力平衡方程；Step 402, establishing a cross-sectional force balance equation;

c＝Fap-Frp+Fsp-Fat+Frt-Fst c＝F ap -F rp +F sp -F at +F rt -F st

式中，c为平衡计算值，Fap为混凝土总压力，Frp为重复计算混凝土压力，Fsp为钢筋压力，Fat为混凝土总拉力，Frt为重复计算混凝土拉力，Fst为钢筋拉力；In the formula, c is the balance calculation value, F ap is the total concrete pressure, F rp is the repeated calculation concrete pressure, F sp is the steel bar pressure, F at is the total concrete tension, F rt is the repeated calculation concrete tension, F st is the steel bar tension ;

具体实施时，混凝土总压力Fap、重复计算混凝土压力Frp、钢筋压力Fsp、混凝土总拉力Fat、重复计算混凝土拉力Frt和钢筋拉力Fst的计算均与中性轴位置有关。In specific implementation, the calculations of total concrete pressure F ap , recalculated concrete pressure F rp , steel bar pressure F sp , total concrete tensile force F at , recalculated concrete tensile force F rt , and steel bar tensile force F st are all related to the position of the neutral axis.

步骤403、当平衡计算值大于设定值时，移动中性轴位置，重复步骤402，直至平衡计算值小于设定值，得到截面未开裂的钢筋混凝土构件的中心轴位置。Step 403 , when the calculated balance value is greater than the set value, move the position of the neutral axis, repeat step 402 until the calculated balance value is less than the set value, and obtain the central axis position of the uncracked reinforced concrete member.

具体实施时，设定值为0.0005，当c＞0.0005时，令yf＝yl-0.00001，其中，yf为假设的中性轴位置，yl为前次假设的中性轴位置，当c＜0.0005时，此时假设的中性轴位置作为截面未开裂的钢筋混凝土构件的中心轴位置。During specific implementation, the set value is 0.0005. When c>0.0005, let y f =y l -0.00001, wherein, y f is the assumed neutral axis position, y l is the assumed neutral axis position last time, when When c<0.0005, the assumed neutral axis position at this time is taken as the central axis position of the uncracked reinforced concrete member.

本实施例中，步骤五中所述计算截面未开裂的钢筋混凝土构件的弯矩的具体过程包括：In this embodiment, the specific process of calculating the bending moment of the uncracked reinforced concrete member described in step 5 includes:

Mext＝Mst+Mpt+Mat+Map-Mrt-Mrp M ext =M st +M pt +M at +M ap -M rt -M rp

式中，Mext为截面未开裂的钢筋混凝土构件的弯矩计算值，Mst为拉力钢筋弯矩，Mpt为压力钢筋弯矩，Mat为混凝土拉力侧总弯矩，Map为混凝土压力侧总弯矩，Mrt为重复计算混凝土拉力弯矩，Mrp为重复计算混凝土压力弯矩。In the formula, M ext is the calculated value of bending moment of reinforced concrete members with uncracked section, M st is the bending moment of tension steel bar, M pt is the bending moment of pressure steel bar, M at is the total bending moment of concrete tension side, and M ap is the concrete pressure The total lateral bending moment, M rt is the repeated calculation of the concrete tension bending moment, and M rp is the repeated calculation of the concrete pressure bending moment.

本实施例中，步骤六中所述计算截面开裂的钢筋混凝土构件的裂缝有效深度的具体过程包括：In this embodiment, the specific process of calculating the crack effective depth of the reinforced concrete member with section cracking described in step 6 includes:

Figure BDA0004102357460000081
Figure BDA0004102357460000081

式中，X为裂缝有效深度，fr为断裂模量，Ec为弹性模量，

Figure BDA0004102357460000082
为截面未开裂的钢筋混凝土构件的曲率。where X is the effective depth of the crack, f r is the modulus of rupture, E c is the modulus of elasticity,
Figure BDA0004102357460000082
is the curvature of the uncracked reinforced concrete member.

本实施例中，步骤七中所述计算截面开裂的钢筋混凝土构件的有效惯性矩的具体过程包括：In this embodiment, the specific process of calculating the effective moment of inertia of the reinforced concrete member with section cracking described in step 7 includes:

Figure BDA0004102357460000083
Figure BDA0004102357460000083

式中，Ie为有效惯性矩，Mcr为开裂弯矩，M为载重弯矩，Ig为截面惯性矩，Icr为开裂的截面惯性矩。In the formula, I e is the effective moment of inertia, Mc cr is the cracking moment, M is the load bending moment, I g is the section moment of inertia, and I cr is the section moment of inertia of cracking.

本实施例中，步骤七中所述计算截面开裂的钢筋混凝土构件的曲率的具体过程包括：In this embodiment, the specific process of calculating the curvature of the reinforced concrete member with section cracking described in step 7 includes:

Figure BDA0004102357460000084
Figure BDA0004102357460000084

式中，

Figure BDA0004102357460000085
为截面开裂的钢筋混凝土构件的曲率，M为载重弯矩，Ec为弹性模量，Ie为有效惯性矩。In the formula,
Figure BDA0004102357460000085
is the curvature of the reinforced concrete member with cracked section, M is the load bending moment, E c is the modulus of elasticity, and I e is the effective moment of inertia.

本实施例中，步骤八中所述根据裂缝有效深度和有效惯性矩，假设截面开裂的钢筋混凝土构件的中心轴位置，使得截面受力平衡的具体过程包括：In this embodiment, according to the effective depth of the crack and the effective moment of inertia described in step 8, the specific process of assuming the position of the central axis of the reinforced concrete member with a cracked section so that the section is stressed includes:

步骤801、假设截面开裂的钢筋混凝土构件的中性轴位置，使截面受力平衡；Step 801, assuming the position of the neutral axis of the reinforced concrete member with a cracked section, so that the force on the section is balanced;

步骤802、建立关于平衡计算值的截面受力平衡方程；Step 802, establishing a cross-sectional force balance equation about the balance calculation value;

具体实施时，平衡计算值的计算与中性轴位置有关。During specific implementation, the calculation of the balance calculation value is related to the position of the neutral axis.

步骤803、当平衡计算值大于设定值时，移动中性轴位置，重复步骤802，直至平衡计算值小于设定值，得到截面开裂的钢筋混凝土构件的中心轴位置。Step 803 , when the calculated balance value is greater than the set value, move the position of the neutral axis, repeat step 802 until the calculated balance value is less than the set value, and obtain the position of the central axis of the reinforced concrete member with a cracked section.

具体实施时，当裂缝有效深度尚未超过混凝土保护层厚度时，在截面受力平衡分析时，应该从产生的拉力中减去混凝土在受拉钢筋位置处的计算拉力，以避免重复计算。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 tensioned reinforcement should be subtracted from the generated tensile force in the force balance analysis of the section to avoid repeated calculations.

本实施例中，步骤九中所述根据有效惯性矩和真实惯性矩，获取截面开裂的钢筋混凝土构件的真实的中心轴位置的具体过程包括：In this embodiment, according to the effective moment of inertia and the real moment of inertia described in step 9, the specific process of obtaining the real central axis position of the reinforced concrete member with section cracking includes:

令

Figure BDA0004102357460000091
当a＞0.00006时，将此时得到的真实惯性矩Iexact的值赋给此时的有效惯性矩Ie；然后重复步骤七～步骤九，直到a≤0.00006，将此时假设的截面开裂的钢筋混凝土构件的中心轴位置作为真实的中性轴位置。make
Figure BDA0004102357460000091
When a > 0.00006, assign the value of the real moment of inertia I exact obtained at this time to the effective moment of inertia I e at this time; then repeat steps 7 to 9 until a ≤ 0.00006, and assign the value of the assumed section cracked at this time The position of the central axis of the reinforced concrete member is taken as the true neutral axis position.

为了验证所提的计算方法，对跨中荷载-弯矩关系与理论解进行比较比较结果如图2所示，从图2可以得到在同时考虑混凝土抗拉强度和实际惯性矩时，其荷载-弯矩关系与荷载范围的理论解较为接近，从而验证了本发明计算方法的适用性。In order to verify the proposed calculation method, the comparison between the mid-span load-bending moment relationship and the theoretical solution is shown in Figure 2. From Figure 2, it can be obtained that when the concrete tensile strength and actual moment of inertia are considered at the same time, the load- The bending moment relationship is relatively close to the theoretical solution of the load range, thus verifying the applicability of the calculation method of the present invention.

以上所述，仅是本发明的较佳实施例，并非对本发明作任何限制，凡是根据本发明技术实质对以上实施例所作的任何简单修改、变更以及等效结构变化，均仍属于本发明技术方案的保护范围内。The above are only preferred embodiments of the present invention, and do not limit the present invention in any way. All simple modifications, changes and equivalent structural changes made to the above embodiments according to the technical essence of the present invention still belong to the technical aspects of the present invention. within the scope of protection of the scheme.

Claims (10)

1. The reinforced concrete member bending moment calculation method taking the neutral axis position change into consideration is characterized by comprising the following steps of:
step one, judging whether the section of the reinforced concrete member is cracked or not;
step two, executing a step three when the section of the reinforced concrete member is not cracked; executing the step six when the section of the reinforced concrete member is cracked;
step three, calculating the curvature of the reinforced concrete member with the section not cracked;
step four, obtaining the central axis position of the reinforced concrete member with the cross section not cracked;
step five, calculating bending moment of the reinforced concrete member with the section not cracked;
step six, calculating the effective depth of cracks of the reinforced concrete member with the cracked cross section;
step seven, calculating the effective moment of inertia and curvature of the reinforced concrete member with the cracked cross section;
step eight, according to the effective depth and the effective moment of inertia of the crack, assuming the central axis position of the reinforced concrete member with the cracked section, so that the stress of the section is balanced, and calculating the actual moment of inertia;
step nine, acquiring a real central axis position of the reinforced concrete member with the cracked section according to the effective moment of inertia and the real moment of inertia;
and step ten, calculating the bending moment of the reinforced concrete member with the cracked section according to the real central shaft position.
2. The method for calculating bending moment of reinforced concrete member in consideration of position change of neutral axis according to claim 1, wherein the specific process for judging whether the section of the reinforced concrete member is cracked in the first step comprises:
step 101, calculating the load bending moment of the reinforced concrete member;
Figure FDA0004102357450000011
wherein M is a load bending moment, P is a loading load, and z is the horizontal distance from the section to be measured to the bracket;
102, calculating a cracking bending moment of the reinforced concrete member;
Figure FDA0004102357450000012
wherein M is cr F is a cracking bending moment r For modulus of rupture, I g Is the section moment of inertia, y t The distance from the outer edge of the tension side to the neutral axis is the distance from the outer edge of the tension side to the neutral axis when the beam is not cracked;
step 103, comparing the load bending moment M with the cracking bending moment M cr When M is less than or equal to M cr When the reinforced concrete member is in use, the section of the reinforced concrete member is not cracked; when M > M cr When the reinforced concrete member is broken in cross section.
3. A method for calculating bending moment of reinforced concrete member in consideration of position change of neutral axis according to claim 1, wherein the specific process for calculating the curvature of reinforced concrete member having a non-cracked cross section in step three comprises:
Figure FDA0004102357450000021
in the method, in the process of the invention,
Figure FDA0004102357450000022
the curvature of the reinforced concrete member with the cross section not cracked is M is a load bending moment E c Modulus of elasticity, I g Is the moment of inertia of the section.
4. The method for calculating bending moment of reinforced concrete member in consideration of position change of neutral axis according to claim 1, wherein the specific process for obtaining the center axis position of the reinforced concrete member with the uncracked cross section in the fourth step comprises:
step 401, assuming neutral axis positions of reinforced concrete members with uncracked sections, and balancing the section stress;
step 402, establishing a section stress balance equation;
c＝F ap -F rp +F sp -F at +F rt -F st
wherein c is a balance calculation value, F ap Is the total pressure of the concrete, F rp To repeatedly calculate the concrete pressure, F sp For the pressure of the reinforcing steel bar, F at F is the total tension of the concrete rt To repeatedly calculate the concrete tension, F st Is the tension of the steel bar;
and 403, when the balance calculated value is greater than the set value, moving the neutral axis position, and repeating the step 402 until the balance calculated value is less than the set value, so as to obtain the central axis position of the reinforced concrete member with the cross section not cracked.
5. A method for calculating a bending moment of a reinforced concrete member in consideration of a change in a neutral axis position according to claim 1, wherein the concrete process for calculating the bending moment of a reinforced concrete member having a non-cracked cross section in the fifth step comprises:
M ext ＝M st +M pt +M at +M ap -M rt -M rp
wherein M is ext Calculated bending moment for reinforced concrete member with uncracked cross section, M st Is the bending moment of the tensile steel bar, M pt Is a bending moment of a pressure steel bar, M at Is the total bending moment of the tension side of the concrete, M ap Is the total bending moment of the concrete pressure side, M rt To repeatedly calculate the tensile bending moment of the concrete, M rp To repeatedly calculate the concrete pressure bending moment.
6. A method for calculating bending moment of reinforced concrete member in consideration of position change of neutral axis according to claim 1, wherein the specific process for calculating effective depth of crack of reinforced concrete member with cracked cross section in step six comprises:
Figure FDA0004102357450000031
wherein X is the effective depth of the crack, f r For modulus of rupture, E c Is the modulus of elasticity of the material,
Figure FDA0004102357450000035
is the curvature of a reinforced concrete member with an uncracked cross section.
7. A method for calculating bending moment of reinforced concrete member in consideration of position change of neutral axis according to claim 1, wherein the specific process for calculating effective moment of inertia of the reinforced concrete member having cracked cross section in step seven comprises:
Figure FDA0004102357450000032
wherein I is e To be effective moment of inertia, M cr Is a cracking bending moment, M is a loading bending moment, I g Is the section moment of inertia, I cr Is the cross-sectional moment of inertia of the crack.
8. A method for calculating a bending moment of a reinforced concrete member in consideration of a change in a neutral axis position according to claim 7, wherein the step seven of calculating the curvature of the reinforced concrete member having a cracked cross section comprises:
Figure FDA0004102357450000033
in the method, in the process of the invention,
Figure FDA0004102357450000034
the curvature of the reinforced concrete member with cracked cross section, M is a load bending moment and E c Modulus of elasticity, I e Is the effective moment of inertia.
9. A method for calculating bending moment of reinforced concrete member in consideration of position change of neutral axis according to claim 8, wherein the specific process of balancing the section stress based on the effective depth of the crack and the effective moment of inertia assuming the center axis position of the reinforced concrete member with the section cracked in the eighth step comprises:
step 801, assuming neutral axis positions of reinforced concrete members with cracked sections, balancing the section stress;
step 802, establishing a section stress balance equation about a balance calculation value;
and 803, when the balance calculated value is larger than the set value, moving the neutral axis position, and repeating the step 802 until the balance calculated value is smaller than the set value, so as to obtain the central axis position of the reinforced concrete member with the cracked cross section.
10. A method of calculating a bending moment of a reinforced concrete member in consideration of a change in a neutral axis position according to claim 9, wherein the specific process of obtaining a true center axis position of the reinforced concrete member having a cracked cross section based on the effective moment of inertia and the true moment of inertia in step nine comprises:
order the
Figure FDA0004102357450000041
When a > 0.00006, the true moment of inertia I obtained at this time exact Is assigned to the effective moment of inertia I at that time e The method comprises the steps of carrying out a first treatment on the surface of the And repeating the steps seven to nine until a is less than or equal to 0.00006, and taking the central axis position of the reinforced concrete member with the cross section cracked supposed at the moment as the actual neutral axis position.
CN202310181099.9A 2023-02-28 2023-02-28 A Bending Moment Calculation Method of Reinforced Concrete Members Considering the Change of Neutral Axis Position Pending CN116304475A (en)

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Cited By (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN117371089A (en) * 2023-10-04 2024-01-09 四川大学 Complex degree calculating method, device, computer equipment and medium

Cited By (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN117371089A (en) * 2023-10-04 2024-01-09 四川大学 Complex degree calculating method, device, computer equipment and medium

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