Method for calculating economic reinforcement ratio of concrete beam and analyzing cost

Abstract

The invention relates to a CFRP concrete beam economic reinforcement ratio calculation and cost analysis method. The method comprises the following steps that 1, an internal force balance equation is written into a balance equation containing reinforcement ratio variables according to the stress-strain sketch diagram relation, the strain coordination condition and the reinforcement ratio of a concrete beam; 2, solving the height x of the compression area according to a variable balance equation containing the reinforcement ratio; 3, according to the variable balance equation containing the reinforcement ratio, the flexural capacity of the concrete beam is written into a flexural capacity expression containing the reinforcement ratio; 4, constructing a target function by considering the prices of the steel bars and the concrete and the strength grades of the steel bars and the concrete; 5, determining the constraint condition of the flexural capacity of the normal section of the concrete beam, the constraint condition of the minimum reinforcement ratio and the constraint condition of the maximum reinforcement ratio; 6, a Lagrange function expression is written according to the concrete beam normal section flexural capacity constraint condition and the target function, analysis and solution are carried out, and a reinforcement ratio expression is given; 7, on the basis of the deduced economic reinforcement ratio expression, on the premise that the minimum reinforcement ratio constraint condition and the maximum reinforcement ratio constraint condition are met, the corresponding range of the ratio of the steel bar price to the concrete price and the relation between the steel bar price and the concrete price are given on the premise that the economic reinforcement ratio exists; 8, giving a relation equation of the concrete beam manufacturing cost and the reinforcement ratio. The method has certain innovativeness and practicability.

Description

Technical Field

[0001] The invention relates to the field of civil engineering structure design and economic technology, and in particular to an analysis method for calculating the optimal reinforcement ratio and the most economical construction cost of a concrete rectangular cross-section beam. Background Art

[0002] Reinforced concrete beams and other bending members are widely used in structural engineering and account for a large proportion of engineering structures. In actual engineering, the reinforcement ratio is selected as long as it satisfies ρ min ≤ρ≤ρ max That is, such reinforcement may cause unnecessary waste. In order to avoid waste and save investment, it is particularly important to determine the optimal reinforcement ratio of concrete beams to minimize the cost. For reinforced concrete beams, the "Concrete Structure Design Code" has given its relative limit compression zone height ξ b The calculation formula and ρ min However, there is currently no effective calculation method for determining the optimal reinforcement ratio, which has a great impact on the engineering design of concrete beams. Summary of the invention

[0003] The purpose of the present invention is to determine the optimal reinforcement ratio of a reinforced concrete beam so as to provide a calculation method which minimizes the cost of the reinforced concrete beam.

[0004] Reinforcement ratio and cost analysis method of reinforced concrete beams. This method follows the following basic assumptions:

[0005] 1. The reinforced concrete beam complies with the plane section assumption;

[0006] 2. Without considering the tensile strength of concrete, all tensile forces are borne by the longitudinal tensile reinforcement;

[0007] 3. When concrete is under compression, stress and strain are proportional, and when steel bars are under tension, stress and strain are proportional;

[0008] The method of the present invention specifically comprises the following steps:

[0009] Step 1: According to the stress-strain diagram relationship, strain coordination conditions and reinforcement ratio of the reinforced concrete beam, the internal force equilibrium equation is written as an equilibrium equation containing the reinforcement ratio variable.

[0010] α 1 βf c bx=f y A s ,

[0011] make

[0012] ∵f c =Ec ε c ,f y =E s ε s ,

[0013] ∴α 1 βbxE c ε c =E s ε s A s ,Right now After sorting, α 1 βx 2 +a E ρh 0 x E ρh 0 2 =0

[0014] In the formula, ε c is the compressive strain of concrete at the edge of the compression zone, ε s is the tensile strain of the reinforcement in the tension zone, f c is the design value of concrete axial compressive strength, f y is the design value of tensile strength of ordinary steel bars, A s is the longitudinal reinforcement area in the tension zone, b is the width of the rectangular section, h is 0 is the effective height of the section, α 1 is the stress of the rectangular stress diagram and the design value of the axial compressive strength f c ratio, β is the ratio of the calculated pressure height to the actual pressure height, E s is the elastic modulus of the steel bar, E c is the elastic modulus of concrete, ρ is the tensile reinforcement ratio;

[0015] Step 2: Determine the height x of the compression zone based on the equilibrium equation containing the reinforcement ratio variable. Specifically: To simplify, we can

[0016]

[0017] In the formula, a E is the ratio of the elastic modulus of steel to the elastic modulus of concrete;

[0018] Step 3: Based on the determined compression zone height x, the flexural bearing capacity of reinforced concrete beam is written as the flexural bearing capacity expression including the reinforcement ratio. Specifically:

[0019]

[0020] because but

[0021] Step 4: Calculate the cross-sectional area A of the tensile reinforcement according to step S1 s , the objective function is constructed by considering the price of steel bars and concrete and the strength grade of steel bars and concrete, and the corresponding measures fee and formwork fee are not considered. If they are considered, they can be combined into the corresponding comprehensive fee. Specifically:

[0022]

[0023] P E =P c A c +P s A s =P c (bh-A s )+P s A s =P c bh+(P s -P c )A s =P c bh 0 +P c ba+(P s -P c )A s

[0024] ∴

[0025]

[0026] Where P s —Unit price of steel bars per ton, × 7.85 yuan / m 3 ;

[0027] P c —Unit price of concrete, yuan / m 3 ;

[0028] Step 5: Determine the constraints of the bending bearing capacity of the reinforced concrete beam cross section, the minimum reinforcement ratio, and the maximum reinforcement ratio. Specifically:

[0029] Constraints on the bending bearing capacity of the normal section:

[0030]

[0031] Right now Constraints for minimum reinforcement ratio:

[0032] Y 2 =ρ-ρ min ≥0, that is Constraints on the maximum reinforcement ratio:

[0033] Y 3 = ρ - ρ max ≤ 0, that is

[0034] Among them,

[0035] In the formula, f t is the design value of the axial tensile strength of concrete, ε cu is the ultimate compressive strain of concrete, ξ b is the boundary relative compression zone height;

[0036] Step 6, according to the objective function in Step S4 and the flexural bearing capacity constraint condition of the reinforced concrete beam in Step S5, write the Lagrangian function expression and analyze and solve it. The reinforcement ratio expressions are given as follows:

[0037] The Lagrangian function expression is written as:

[0038]

[0039] The reinforcement ratio expressions obtained by analyzing and solving the Lagrangian function are:

[0040]

[0041] Step 7, based on the optimized reinforcement ratio expression derived in Step S6, under the constraint conditions of meeting the minimum reinforcement ratio and the maximum reinforcement ratio in Step S5, the corresponding relationship between the steel price and the concrete price is given on the premise of the existence of the optimized reinforcement ratio as:

[0042]

[0043] In the formula, ρ min is the minimum reinforcement ratio of the tensile zone of the flexural member.

[0044] Step 8, give the equation of the relationship between the cost of the reinforced concrete beam and the reinforcement ratio as:

[0045]

[0046] The method of the present invention takes reinforced concrete single-bar rectangular cross-section beams as the research object, and takes the lowest cost per unit length as the objective function on the basis of considering the material price. A calculation method based on strain coordination conditions and the bearing capacity of the positive section of reinforced concrete beams is adopted, and a reasonable expression of the economic reinforcement ratio is derived using the Lagrangian function. And on the basis of the derived expression of the economic reinforcement ratio, a method is given to determine the range of the ratio of the corresponding steel bar price to the concrete price under the premise of the existence of the economic reinforcement ratio, by satisfying the constraints of the maximum and minimum reinforcement ratio ranges corresponding to the corresponding steel bar grade and concrete strength grade. This method is practical and helps to avoid unnecessary waste in engineering design. BRIEF DESCRIPTION OF THE DRAWINGS

[0047] Figure 1 The stress and strain diagram of reinforced concrete rectangular cross-section beam

[0048] Figure 2 The relationship between the reinforcement ratio and cost of HPB300 and C20

[0049] Figure 3 The relationship between the reinforcement ratio and cost of HPB300 and C25

[0050] Figure 4 The relationship between the reinforcement ratio and cost of HPB300 and C30

[0051] Figure 5 The relationship between the reinforcement ratio and cost of HPB300 and C35

[0052] Figure 6 Relationship between reinforcement ratio and cost of HPB300 and C40

[0053] Figure 7 Relationship between reinforcement ratio and cost of HPB300 and C45

[0054] Figure 8 The relationship between the reinforcement ratio and cost of HPB300 and C50

[0055] Fig. 9 Relationship between reinforcement ratio and cost of HRB335 and C20

[0056] Fig.10 Relationship between reinforcement ratio and cost of HRB335 and C25

[0057] Fig.11 Relationship between reinforcement ratio and cost of HRB335 and C30

[0058] Fig.12 Relationship between reinforcement ratio and cost of HRB335 and C35

[0059] Fig.13 Reinforcement Ratio and Cost Relationship for the Matching of HRB335 and C40

[0060] Fig.14 Reinforcement Ratio and Cost Relationship for the Matching of HRB335 and C45

[0061] Fig.15 Reinforcement Ratio and Cost Relationship for the Matching of HRB335 and C50

[0062] Fig.16 Reinforcement Ratio and Cost Relationship for the Matching of HRB400 and C25

[0063] Fig.17 Reinforcement Ratio and Cost Relationship for the Matching of HRB400 and C30

[0064] Fig.18 Reinforcement Ratio and Cost Relationship for the Matching of HRB400 and C35

[0065] Fig.19 Reinforcement Ratio and Cost Relationship for the Matching of HRB400 and C40

[0066] Fig. 20 Reinforcement Ratio and Cost Relationship for the Matching of HRB400 and C45

[0067] Fig.21 Reinforcement Ratio and Cost Relationship for the Matching of HRB400 and C50

[0068] Fig. 22 Reinforcement Ratio and Cost Relationship for the Matching of HRB500 and C35

[0069] Fig.23 Reinforcement Ratio and Cost Relationship for the Matching of HRB500 and C40

[0070] Fig.24 Reinforcement Ratio and Cost Relationship for the Matching of HRB500 and C45

[0071] Fig.25 Reinforcement Ratio and Cost Relationship for the Matching of HRB500 and C50 Specific Embodiment

[0072] The present invention will be described in detail below with reference to the accompanying drawings and specific embodiments. Obviously, the described embodiments are part of the embodiments of the present invention, rather than all embodiments. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts, such as the calculation of the economic reinforcement ratio of other similar cross-sectional reinforced concrete beams, should fall within the scope of protection of the present invention.

[0073] According to step 7, the P values ​​corresponding to the reinforced concrete beams with different grades of steel bars and different concrete strength grades (≤C50) calculated by the formula are s / P c The value range is shown in Table 1. s / P c When the value range exceeds the value range shown in Table 1, there is no economic reinforcement ratio.

[0074] Table 1 Ps / Pc calculation table

[0075]

[0076] Example 1

[0077] Assume a reinforced concrete rectangular cross-section simply supported beam, M = 220kN·m, b = 250mm. According to a local market price survey, the steel grade HRB335 has a market price of 3500 yuan / t; the concrete strength grade C30 has a unit price of 370 yuan / m 3 α 1 =1.0,β=0.8,α s =35mm, determine its economic reinforcement ratio and minimum cost.

[0078]

[0079] Where P s / P c The range does not exceed that shown in Table 1, and there is an economic reinforcement ratio.

[0080] According to the derived economic reinforcement ratio formula, we can get

[0081]

[0082] Example 2

[0083] Assume a reinforced concrete rectangular cross-section simply supported beam, M = 220kN·m, b = 250mm. According to a local market price survey, the market price of steel bars fluctuates between (3000-4000) yuan / t, and the steel bar grades are HPB300, HRB335, HRB400, and HRB500. The market price fluctuation range of concrete units of different strength grades is shown in Table 2. 1 =1.0,β=0.8,α s =35mm, determine its economic reinforcement ratio.

[0084] First calculate P through step 7 or find it in Table 1 s / P c There is a range of economic reinforcement ratio. If P s / P cIf the value is within the range of economic reinforcement ratio, the results of the economic reinforcement ratio range calculated according to step 6 are shown in Table 2. In this example, P is given for each combination. s(max) / P c(max) , P s(max) / P c(min) , P s(mid) / P c(mid) , P s(min) / P c(max) , P s(min) / P c(min) In several cases, the objective function P E(ρ) The law of change with ρ is shown in Figure 2 to Figure 25 .

[0085] Table 2 Calculation table of economic reinforcement ratio of reinforced concrete beams

[0086]

[0087]

Claims

1. A method for analyzing the economic reinforcement ratio and cost of concrete beams, characterized in that: The following steps are involved: S1. The stress-strain diagram relationship, strain coordination conditions and reinforcement ratio of the root concrete beam. The internal force equilibrium equation is written as an equilibrium equation containing the reinforcement ratio variable: α1βx 2 +a E ρh0x-a E ρh0 2 =0 Where h0 is the effective height of the section, α1 is the stress of the rectangular stress diagram and the design value of the axial compressive strength f c ratio, β is the ratio of the calculated pressure height to the actual pressure height, a E is the ratio of the elastic modulus of steel bar to the elastic modulus of concrete, ρ is the reinforcement ratio of tensile steel bar; S2. According to the balance equation containing the distribution (reinforcement) rate variable in step S1, the height x of the compression zone is solved as follows: x=f(ρ)·h0 S3. According to the height x of the compression zone solved in step S2, the bending bearing capacity of the concrete beam is expressed as the bending bearing capacity expression including the reinforcement ratio: Where M is the design value of the bending moment, f c is the design value of the axial compressive strength of concrete, b is the width of the rectangular section, h0 is the effective height of the section, α1 is the stress of the rectangular stress diagram and the design value of the axial compressive strength f c Ratio, β is the ratio of the calculated pressure height to the actual pressure height; S4, steel and concrete prices and steel and concrete strength grades are used to construct the objective function, ignoring the corresponding measures fees and formwork costs. If these are considered, they can be combined into the corresponding comprehensive costs: Where P s is the price of steel bar per ton, P c is the unit price of concrete; S5. The constraints for determining the bending bearing capacity of the positive section of the concrete beam, the minimum reinforcement ratio constraint conditions, and the maximum reinforcement ratio constraint conditions are: ⑴Constraints on the bending bearing capacity of the normal section: Right now (2) Constraints for minimum reinforcement ratio: ⑶Constraints of maximum reinforcement ratio: Y3=r-r max ≤0, even in, Where ξ b is the relative limit compression zone height; S6. Write the Lagrangian function expression according to the objective function of step S4 and the bending bearing capacity constraint condition of the concrete beam cross section in step S5 and analyze and solve it, and give the reinforcement ratio expressions respectively: Written as Lagrangian function expression: The reinforcement ratio expression obtained by analyzing and solving the Lagrangian function is: S7. Based on the economic reinforcement ratio expression derived in step S6, the range of the ratio of steel bar price to concrete price and the relationship between steel bar price and concrete price under the premise of economic reinforcement ratio are given by satisfying the minimum reinforcement ratio constraint and the maximum reinforcement ratio constraint: In the formula, ρ min It is the minimum reinforcement ratio in the tension zone of a flexural member. S8. The relationship between the cost of concrete beams and the reinforcement ratio is given as:

2. A method for analyzing economic reinforcement ratio and cost of concrete beams according to claim 1, characterized in that: The specific content of step S1 is: After sorting, α1βx 2 +a E ρh0x-a E ρh0 2 =0 In the formula, ε c is the compressive strain of concrete at the edge of the compression zone, ε s is the tensile strain of the reinforcement in the tension zone, f c is the design value of concrete axial compressive strength, f y is the design value of tensile strength of ordinary steel bars, A s is the longitudinal reinforcement area in the tension zone, b is the width of the rectangular section, h0 is the effective height of the section, α1 is the stress of the rectangular stress diagram and the design value of the axial compressive strength f c ratio, β is the ratio of the calculated pressure height to the actual pressure height, E s is the elastic modulus of the steel bar, E c is the elastic modulus of concrete, and ρ is the tensile reinforcement ratio.

3. A method for analyzing economic reinforcement ratio and cost of concrete beams according to claim 1, characterized in that: The specific content of step S2 is: Among them Then x=f(ρ)·h0 In the formula, a E It is the ratio of the elastic modulus of steel to the elastic modulus of concrete.

4. A method for analyzing economic reinforcement ratio and cost of concrete beams according to claim 1, characterized in that: The specific content of step S3 is: because but 5. A method for analyzing economic reinforcement ratio and cost of concrete beams according to claim 1, characterized in that: The specific content of step S4 is: P E =P c A c +P s A s =P c (bh-A s )+P s A s =P c bh+(P s -P c )A s =P c bh0+P c ba+(P s -P c )A s 6. A method for analyzing economic reinforcement ratio and cost of concrete beams according to claim 1, characterized in that: The specific content of step S6 is: The constraint condition (1) of step S4 and the objective function of step S5 are expressed as Lagrangian functions: Right now The minimum value of the above Lagrangian function is: From (2), we can get Bundle On behalf of others (1), After sorting, Solved From (3), we can get 7. A method for analyzing economic reinforcement ratio and cost of concrete beams according to claim 1, characterized in that: The specific content of step S7 is: Combining constraints (2) and (3) in step S5, we get: Generally speaking, P s >P c , after calculation and analysis, we can get:

8. A method for analyzing economic reinforcement ratio and cost of concrete beams according to claim 1, characterized in that: The specific content of step S8 is: Right now When ρ=ρ e When P E( ρ ) Minimum. For the objective function P E As for the expression, it is relatively complicated. Through analysis, it can be known that within the range of appropriate reinforcement ratio and So the first term increases gradually with the decrease of ρ; while for the second term (P s -P c ) / P c ·f c / 2f y One term gradually increases with the increase of ρ. So (P s -P c ) / P c ·f c / 2f y The smaller the value, the higher the steel grade and the lower the concrete strength grade. s -P c ) / P c The smaller the value, the smaller the target function P E The change with the increase of ρ is not very sensitive, while P E It is relatively sensitive to changes as ρ decreases; (P s -P c ) / P c ·f c / 2f y The larger the value, the lower the steel grade and the higher the concrete strength grade. s -P c ) / P c The larger the value, the higher the objective function P E It is not very sensitive to the decrease of ρ, but P E It is relatively sensitive to changes as ρ increases.

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