A calculation method for the fire resistance of T-shaped composite beams under fire

Abstract

A method for calculating the fire resistance limit of a T-shaped composite beam under fire, which relates to the technical field of safety risk assessment, includes steps of thermal-mechanical coupling analysis; predicting and judging the width and height of cracks; simulating the influence of different parameters on the fire resistance limit of a T-shaped composite beam; using SPSS data analysis software to fit the fire resistance limit formula of a T-shaped composite beam under different working conditions, and verifying the accuracy through test values ​​and simulation values. The present invention provides a method for calculating the fire resistance limit of a T-shaped composite beam under fire, the purpose of which is to determine the influence of different factors on the fire resistance limit of a T-shaped composite beam under fire, including the thickness of the concrete cover C, the load holding level L, the load carrying capacity ... r , overlap parameter D h and friction coefficient F h The fire resistance limit of the specimen is determined by comprehensively considering various factors.

Description

Technical Field

[0001] The present invention relates to the technical field of safety risk assessment, and in particular to a method for calculating the fire resistance limit of a T-shaped composite beam under fire. Background Art

[0002] The fire resistance limit calculation method for different concrete cover thickness and load ratios proposed in the fire resistance test of reinforced concrete simply supported beams under fire is obtained through regression of a large number of existing tests. The formula is as follows: f =η(β c / β M / Mu ), β M / Mu =1962.8(M / M u ) 2 -17.351(M / M u )+240.05,β c =136124.9C+3094.4, η=12.418. This method has two drawbacks. First, it only considers the load ratio and concrete cover thickness, but not the composite parameter, namely the precast slab thickness. When the composite parameter is small, the integrity of the composite beam is reduced, its fire resistance performance is weakened, and the risk of structural collapse under fire is increased. Second, it does not consider the influence of the friction coefficient. Different friction coefficients at the composite surface also affect the temperature distribution of the T-shaped composite beam under fire, thereby reducing the accuracy of the fire resistance limit prediction. In actual engineering, composite beams with different composite parameters and friction coefficients have different fire resistance performance under fire. Therefore, a fire resistance limit calculation method for T-shaped composite beams under different working conditions should be proposed. Summary of the Invention

[0003] The present invention provides a method for calculating the fire resistance limit of a T-shaped composite beam under fire, the purpose of which is to determine the influence of different factors on the fire resistance limit of a T-shaped composite beam under fire, including the thickness of the concrete cover C, the load holding level L, and the influence of the concrete cover thickness C, the load holding level L, and the fire resistance limit of the T-shaped composite beam under fire. r , overlap parameter D h and friction coefficient F h The fire resistance limit of the specimen is determined by comprehensively considering various factors.

[0004] To achieve the above object, the technical solution of the present invention is as follows:

[0005] A method for calculating the fire resistance limit of a T-shaped composite beam under fire includes the following steps:

[0006] (1) The finite element software ABAQUS was used to perform sequential thermal-mechanical coupling simulation of the T-shaped composite beam under fire test. The temperature field distribution of the T-shaped composite beam section was obtained through heat transfer analysis. The temperature field distribution data was then imported into the static model as the initial condition for thermal-mechanical coupling analysis.

[0007] (2) With reference to the crack locations on the beam surface in the field test, the extended finite element method (XFEM) was used to preset cracks in the finite element model. The crack width and height were predicted and determined by comparing with the test results.

[0008] (3) The Coulomb-cohesion model was used to simulate the bond-slip characteristics between the precast concrete slab and the cast-in-place concrete slab at the superimposed surface of the T-shaped composite beam under fire, and the results were compared with the experimental values. Based on the verified finite element model, the effects of different parameters on the fire resistance of the T-shaped composite beam were simulated. The parameters included the thickness of the concrete cover C, the load holding level L of the composite beam, and the load-bearing capacity of the composite beam. r , overlap parameter D h And the friction coefficient F of the overlapping surface h ;

[0009] (4) Based on the simulation results, SPSS data analysis software was used to fit the fire resistance limit formula of T-shaped composite beams under different working conditions, and the accuracy was verified by experimental values ​​and simulation values.

[0010] Preferably, the step (1) includes the following specific steps: designing and manufacturing a number of prefabricated assembled concrete T-section composite beams, considering the fire test conditions, the actual situation of composite beam use and the requirements of the specification, determining the total height, beam length, flange width, beam rib width, and total thickness of the flange part of the test composite beam; and setting the thickness of the precast concrete slab and the cast-in-place concrete slab at the composite beam superposition surface; setting the protective layer thickness of the composite beam according to the requirements of the concrete structure design specification for the protective layer thickness; wherein, in the finite element model of the composite beam, the concrete adopts an 8-node linear heat transfer unit DC3D8, and the steel adopts a two-node linear heat transfer unit DC1D2. In order to ensure heat transfer between the steel and concrete, Tie constraints are used to connect the concrete and the steel skeleton. The five sides of the beam body are exposed to fire, namely the bottom surface of the beam, the side surfaces of the two webs, and the bottom surfaces of the two flanges. Subsequently, the temperature field distribution data is imported into the static model as the initial condition for thermal-mechanical coupling analysis.

[0011] Preferably, the step (2) includes the following specific steps:

[0012] (21) Temperature field simulation with cracks, verifying the accuracy of the composite beam finite element model by comparing the test values ​​with the simulation values;

[0013] (22) XFEM extended finite element simulation: The finite element model is pre-set based on the actual crack positions of the composite beam after the fire to determine the crack development trend, crack width and height. The results are then compared with the actual crack width and height to verify the accuracy of the model.

[0014] (23) Crack data simulation value: During the crack modeling process, the width and depth of the crack are calculated based on the grid division length.

[0015] Preferably, the step (3) includes the following specific steps:

[0016] (31) Considering the thickness of the concrete cover C and the load-bearing level L of the composite beam r , overlap parameter D h And the friction coefficient F of the overlapping surface h The fire resistance of composite beams is analyzed by taking different values ​​of the thickness of the protective layer, the load holding level of the composite beam, the composite parameters, and the friction coefficient.

[0017] (32) In the overlap parameter D h Under the same conditions as the concrete cover thickness C, the fire resistance limit and load holding level L of the composite beam are analyzed. r the changing relationship;

[0018] (33) At the load holding level L r Under the same conditions as the concrete cover thickness C, the fire resistance limit and composite parameter D of the composite beam are analyzed. h the changing relationship;

[0019] (34) To ensure the load holding level L r and the overlap parameter D h Under the same conditions, the relationship between the fire resistance limit and friction coefficient of the composite beam is analyzed.

[0020] (35) At the load holding level L r and the friction coefficient F of the overlapping surfaces h Under the same conditions, the relationship between the thickness of the concrete cover C and the fire resistance limit of the composite beam is analyzed.

[0021] Preferably, the step (4) includes the following specific steps:

[0022] Comprehensively consider the thickness of the concrete cover C, the load-bearing level of the composite beam L r , overlap parameter D h and the interface friction coefficient F h The influence on the fire resistance limit of composite beams is studied by using the statistical analysis software SPSS regression to obtain the relationship between the fire resistance limit and various influencing factors based on the data obtained in step (3). The relationship is given according to different protective layer thicknesses. Among them, the relationship (1) when the concrete protective layer thickness C is 20 mm is:

[0023]

[0024] In formula (1): R T1—Fire resistance limit of composite beam, in min; F h —friction coefficient of the overlapping surfaces; D h —Superposition parameter; L r —Load holding level; 0.4≤F h ≤0.8, 0.4≤D h ≤0.6, 0.4≤L r ≤0.6;

[0025] In formula (1), the residual sum of squares RSS is 30.96, the regression sum of squares ESS is 231054.04, and R 2 =0.998;

[0026] Among them, the relationship equations (2) and (3) for the protective layer thickness of 30mm and 40mm are as follows:

[0027]

[0028]

[0029] In formula (2), the residual sum of squares RSS is 59.30, and the regression sum of squares ESS is 257085.70; in formula (3), the residual sum of squares RSS is 76.32, and the regression sum of squares ESS is 278018.68. The R of formulas (2) and (3) is 2 Both are 0.996.

[0030] The beneficial effects of the method for calculating the fire resistance limit of a T-shaped composite beam under fire of the present invention are:

[0031] The present invention comprehensively considers the thickness of the concrete protective layer C, the load-bearing level L of the composite beam, and the r , overlap parameter D h and friction coefficient F h The fire resistance limit of T-shaped composite beams under fire is calculated by taking into account the influence of many factors such as fire hazard, fire hazard and fire hazard. This calculation method can accurately calculate the fire resistance limit of composite beams under fire and accurately estimate the fire resistance performance of components under fire, thereby reducing the risk of structural collapse caused by serious fire, protecting people’s lives and property, and facilitating fire fighting and post-disaster reinforcement and repair. BRIEF DESCRIPTION OF THE DRAWINGS

[0032] Figure 1 : Composite beam dimensions and thermocouple arrangement diagram;

[0033] Figure 2 : Composite beam temperature field model diagram;

[0034] Figure 3 , mid-span section temperature cloud map;

[0035] Figure 4, simulated and measured values ​​of temperature at different measuring points of composite beam;

[0036] Figure 5 , comparison of cracks between PT2 specimen and numerical model;

[0037] Figure 6 , the change of fire resistance of composite beams with the load level;

[0038] Figure 7 , the fire resistance of composite beams changes with composite parameters;

[0039] Figure 8 , the variation of fire resistance of composite beams with friction coefficient;

[0040] Figure 9 , the fire resistance limit of composite beams changes with the thickness of the protective layer. DETAILED DESCRIPTION

[0041] The following describes in detail the implementation methods of the present invention in a step-by-step manner. This description is only a preferred embodiment of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions and improvements made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

[0042] In the description of the present invention, it should be noted that the terms "up", "down", "left", "right", "top", "bottom", "inside", "outside", etc., indicating the orientation or position relationship, are based on the orientation or position relationship shown in the accompanying drawings. They are only for describing the present invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, and a specific orientation structure and operation. Therefore, they cannot be understood as limiting the present invention.

[0043] Please refer to Figure 1-9 As shown:

[0044] A specific implementation method of a method for calculating the fire resistance limit of a T-shaped composite beam under fire is as follows:

[0045] Step (1): The finite element software ABAQUS is used to perform sequential thermal-mechanical coupling simulation on the T-shaped composite beam under the fire test. The temperature field distribution of the T-shaped composite beam section is obtained through heat transfer analysis. The temperature field distribution data is then imported into the static model as the initial condition for thermal-mechanical coupling analysis.

[0046] (11) Fire test simulation:

[0047] Eight prefabricated assembled concrete T-section composite beams were designed and manufactured. Considering the fire furnace test conditions, actual conditions and regulatory requirements, the total height of the test composite beams was 300mm, the beam length was 3000mm, the flange width was 450mm, and the beam rib width was 150mm. The total thickness of the flange part was 100mm, and the thickness of the prefabricated concrete slab and the cast-in-place concrete slab at the composite beam composite surface was set. The thickness of the prefabricated slab was 40mm and 60mm respectively, and the corresponding cast-in-place slab thickness was 60mm and 40mm. According to the requirements of the concrete structure design code for the thickness of the protective layer, the protective layer thickness of the composite beams was 20mm and 30mm respectively. The test grouping is shown in Table 1, and the reinforcement diagram of the test beam is shown in Figure 1 .

[0048] (12) Simulation of temperature field without cracks:

[0049] In the finite element model of the T-shaped composite beam, the concrete adopts the 8-node linear heat transfer element DC3D8, and the steel adopts the two-node linear heat transfer element DC1D2. To ensure the heat transfer between the steel and concrete, the Tie constraint is used to connect the concrete and steel skeleton. The beam is exposed to fire on five sides: the bottom surface of the beam, the side surfaces of the two webs, and the bottom surfaces of the two flanges.

[0050] Some temperature field results are as follows Figure 2 、 3 As shown, a fire test simulation was carried out on this basis.

[0051] Step (2): referring to the crack positions on the beam surface in the field test, the extended finite element method (XFEM) is used to preset cracks in the finite element model, and compared with the test results, the crack width and height are predicted and determined;

[0052] (21) Temperature field simulation with cracks:

[0053] The accuracy of the composite beam finite element model was verified by comparing the test values ​​with the simulation values. Taking specimen PT2 as an example, the temperature at the same position as the test measuring point was extracted on the model, that is, the temperature-fire time curves of measuring point 3 at the mid-span concrete position and measuring point 4 at the steel bar position were plotted on the same graph. The results were compared. Figure 4 (a) and (b) show the accuracy of the composite beam finite element model by comparing the test values ​​with the simulation values. Taking specimen PT2 as an example, the temperature at the same position as the test measuring point is extracted on the model, that is, the temperature-fire time curves of measuring point 3 at the mid-span concrete position and measuring point 4 at the steel bar position are plotted on the same figure. The results are compared. Figure 4 As shown in (a) and (b).

[0054] (22) XFEM extended finite element simulation:

[0055] According to the actual crack position of the composite beam after the fire, the finite element model is preset to determine the development trend of the cracks and the width and height of the cracks. Finally, the width and height of the cracks are compared with the actual crack width and height to verify the accuracy of the model. The crack situation of the composite beam PT2 specimen after reaching the fire resistance limit is compared with the crack situation of the thermal-mechanical coupling model. Figure 5 shown.

[0056] (23) Crack data simulation value:

[0057] In the crack modeling process, the width and depth of the crack can be calculated according to the grid division length. The width and height of the crack of the detailed model are shown in Table 2.

[0058] Step (3): Use the Coulomb-cohesion model to simulate the bond-slip characteristics between the precast concrete slab and the cast-in-place concrete slab at the superimposed surface of the T-shaped composite beam under fire, and compare and verify with the experimental values;

[0059] (31) Considering the thickness of the concrete cover C and the load-bearing level L of the composite beam r , overlap parameter D h and the friction coefficient F of the overlapping surfaces h The fire resistance of composite beams under different conditions is shown in Table 3.

[0060] (32) The internal force of the composite beam is related to the external load acting on the beam. h Under the same conditions as the concrete cover thickness C, the fire resistance of the composite beam is plotted against the load level. Figure 6 The figure clearly shows that when the holding level of the composite beam increases from 0.4 to 0.5, the fire resistance of the composite beam decreases sharply by approximately 35 minutes. When the holding level increases from 0.5 to 0.6, the fire resistance of the composite beam also decreases by approximately 25 minutes. The higher the holding level of the composite beam, the greater the deformation of the composite beam and the lower its fire resistance at high temperatures. As the holding level increases, the fire resistance of the composite beam components decreases rapidly, and the influence of the holding level on the fire resistance of the composite beam is very significant.

[0061] (33) Under the same load level and concrete cover thickness, the fire resistance limit of the composite beam is changed with the composite parameter D h The changes in Figure 7 From the figure, we can see that the composite parameter D of the composite beam hThe influence on fire resistance limit is small. When the overlap parameter increases from 0.4 to 0.5, the fire resistance limit is extended by about 3 minutes. h The larger the value, the higher the fire resistance limit. The main reason is that the thicker the precast slab is, the greater the temperature loss is in the process of transferring from the precast slab to the cast-in-place slab, the slower the concrete deterioration in the compression zone, the slower the deflection growth, and the higher the fire resistance limit.

[0062] (34) When analyzing the influence of friction coefficient, under the condition of ensuring the same load level and superposition parameters, the change of fire resistance limit friction coefficient of composite beam is plotted on the Figure 8 As can be seen from the figure, the friction coefficient F of the composite beam composite interface h The overall effect on the fire resistance of composite beams is that the greater the friction coefficient, the greater the fire resistance. The friction coefficient of the composite interface represents the overall degree of bonding between the precast panel and the cast-in-place panel. The greater the friction coefficient, the stronger the integrity between the two, the greater the bonding strength, the better the integrity, and the longer the fire exposure time required for deformation and failure. When the friction coefficient increases from 0.4 to 0.6, the fire resistance is extended by about 6 minutes; the load holding level L r When the load increases, the significance of the effect of the increase in friction coefficient on the fire resistance limit decreases. The reason is that after the load level increases, the deformation of the composite beam increases, the internal stress increases, and the impact on the composite surface is greater, reducing the growth of the fire resistance limit of the composite beam.

[0063] (35) The relationship between the thickness of the concrete cover C and the fire resistance limit of the composite beam is positively correlated, such as Figure 9 The concrete cover prevents heat transfer during fire. Due to the thermal inertia of concrete itself, the thicker the cover, the better the insulation effect. The slower the temperature rise of the tensile reinforcement inside the composite beam, the less its strength decreases. r and the friction coefficient F of the overlapping surfaces h Under the same conditions, the greater the thickness of the concrete protective layer C, the greater the fire resistance limit of the composite beam, and the fire resistance limit of the composite beam and the thickness of the protective layer change almost linearly; after the load level is greater than 0.5, the effect of the increase in the protective layer thickness on the fire resistance limit decreases significantly. This is because the deformation of the composite beam increases after the load increases, the cracks at the bottom of the composite beam develop more seriously, and the temperature inside the structure rises faster, resulting in accelerated damage, so the fire resistance limit increases less.

[0064] Step (4): Based on the simulation results, SPSS data analysis software is used to fit the fire resistance formula of T-shaped composite beams under different working conditions, and the accuracy is verified by test values ​​and simulation values:

[0065] Comprehensively consider the thickness of the concrete cover C, the load-bearing level of the composite beam L r , overlap parameter D h and the interface friction coefficient Fh The influence of four factors on the fire resistance of composite beams. The above parameter analysis shows that, under the same other conditions, the load holding level has the most significant impact on the fire resistance of composite beams. Secondly, the fire resistance of composite beams tends to increase linearly with the increase of protective layer thickness and composite surface friction coefficient. When the load ratio is too large, the influence of the two factors decreases significantly. Compared with the other three influencing factors, the effect of the increase in composite parameters on the fire resistance is not obvious. The data in Table 3 were sorted and analyzed, and the statistical analysis software SPSS regression was used to give the relationship between the fire resistance and each influencing factor. The relationship is given according to different protective layer thicknesses. Among them, the commonly used relationship when the concrete protective layer thickness C is 20mm is:

[0066]

[0067] Where: R T1 —Fire resistance limit of composite beam, in min; F h —friction coefficient of the overlapping surfaces; D h —Superposition parameter; L r —Load holding level;

[0068] In order to more reasonably calculate the fire resistance time of composite beams, the above formula needs to be used within a certain range: 0.4≤F h ≤0.8, 0.4≤D h ≤0.6, 0.4≤L r ≤0.6. For the regression equation, the residual sum of squares RSS is 30.96, the regression sum of squares ESS is 231054.04, and R 2 =0.998. The relationship formulas for protective layer thickness of 30mm and 40mm are shown in (2) and (3):

[0069]

[0070]

[0071] For the regression equation (2), the residual sum of squares RSS is 59.30, and the regression sum of squares ESS is 257085.70; for the equation (3), the residual sum of squares RSS is 76.32, and the regression sum of squares ESS is 278018.68. 2 Both are 0.996.

[0072] Appendix: The test data table is as follows:

[0073] Table 1. Grouping of T-shaped composite beam specimens

[0074]

[0075] Table 2. Crack width and height

[0076]

[0077] Table 3 Fire resistance limit of composite beams under different working conditions (min)

[0078]

[0079] Table 4. Fire resistance test values, simulation values ​​and calculated values

[0080]

[0081]

[0082] Note: R T T is the test value, R T S is the simulation value, R T F is the formula value. w1 is R T T With R T F The error of w2 is R T S With R T F error.

Claims

1. A method for calculating the fire resistance of a T-shaped composite beam under fire, characterized by: The steps include: (1) The finite element software ABAQUS was used to perform sequential thermal-mechanical coupling simulations on the T-shaped composite beam under fire test. The temperature field distribution of the T-shaped composite beam section was obtained through heat transfer analysis. The temperature field distribution data was then imported into the static model as the initial condition for thermal-mechanical coupling analysis. (2) With reference to the crack positions on the beam surface in the field test, the extended finite element method (XFEM) was used to preset cracks in the finite element model, and the crack width and height were predicted and determined by comparing with the test results. (3) The Coulomb-cohesion model was used to simulate the bond-slip characteristics between the precast concrete slab and the cast-in-place concrete slab at the superimposed surface of the T-shaped composite beam under fire, and the results were compared with the experimental values. Based on the verified finite element model, the effects of different parameters on the fire resistance of the T-shaped composite beam were simulated. The parameters included the thickness of the concrete cover C, the load holding level L of the composite beam, and the load-bearing capacity of the composite beam. r , overlap parameter D h and the friction coefficient F of the overlapping surfaces h ; (4) Based on the simulation results, SPSS data analysis software was used to fit the fire resistance formula of T-shaped composite beams under different working conditions, and the accuracy was verified by experimental values ​​and simulation values; The step (3) includes the following specific steps: (31) Considering the influence of the thickness of the concrete cover C, the load holding level Lr of the composite beam, the composite parameter Dh and the composite surface friction coefficient Fh on the fire resistance limit of the composite beam, the fire resistance limit of the composite beam under different values ​​of the cover thickness, the load holding level Lr of the composite beam, the composite parameter Dh and the friction coefficient Fh are analyzed; (32) Under the same conditions of composite parameter Dh and concrete cover thickness C, the relationship between the fire resistance limit and load holding level of composite beams is analyzed; (33) Under the same conditions of load level Lr and concrete cover thickness C, the relationship between the fire resistance limit of the composite beam and the composite parameter Dh is analyzed; (34) Under the condition of ensuring the same load level Lr and composite parameter Dh, the relationship between the fire resistance limit of the composite beam and the friction coefficient Fh is analyzed; (35) When the load level Lr and the friction coefficient Fh of the composite surface are the same, the relationship between the thickness of the concrete cover C and the fire resistance limit of the composite beam is analyzed.

2. The method for calculating the fire resistance of a T-shaped composite beam under fire according to claim 1, wherein: The step (1) includes the following specific steps: designing and manufacturing a plurality of prefabricated assembled concrete T-section composite beams, considering the fire test conditions, the actual use of composite beams and the requirements of the specification, and determining the total height, beam length, flange width, beam rib width, and total thickness of the flange portion of the test composite beam; The thickness of the precast concrete slab and the cast-in-place concrete slab at the superimposed surface of the composite beam is set; the protective layer thickness of the composite beam is set according to the requirements of the concrete structure design code for the protective layer thickness; in the finite element model of the composite beam, the concrete adopts the 8-node linear heat transfer unit DC3D8, and the steel bar adopts the two-node linear heat transfer unit DC1D2. In order to ensure the heat transfer between the steel bar and the concrete, the Tie constraint is used to connect the concrete and the steel bar skeleton. The five sides of the beam are exposed to fire, namely the bottom surface of the beam, the side surfaces of the two webs and the bottom surfaces of the two flanges. The temperature field distribution data is then imported into the static model as the initial condition for thermal-mechanical coupling analysis.

3. The method for calculating the fire resistance limit of a T-shaped composite beam under fire according to claim 2, wherein the step (2) comprises the following specific steps: (21) Temperature field simulation with cracks, verifying the accuracy of the composite beam finite element model by comparing the experimental values ​​with the simulation values; (22) XFEM extended finite element simulation: The finite element model is pre-set based on the actual crack positions of the composite beams after the fire to determine the crack development trend, crack width and height. The results are then compared with the actual crack width and height to verify the accuracy of the model. (23) Crack data simulation value: During the crack modeling process, the width and depth of the crack are calculated based on the grid division length.

4. The method for calculating the fire resistance of a T-shaped composite beam under fire according to claim 3, wherein: The step (4) includes the following specific steps: Comprehensively consider the thickness of the concrete cover C, the load-bearing level of the composite beam L r , overlap parameter D h and the interface friction coefficient F h The influence on the fire resistance limit of composite beams is studied by using the statistical analysis software SPSS regression to obtain the relationship between the fire resistance limit and various influencing factors based on the data obtained in step (3). The relationship is given according to different protective layer thicknesses. Among them, the relationship (1) when the concrete protective layer thickness C is 20 mm is: ； In formula (1): R T1 — Fire resistance limit of composite beam, in min; F h — Friction coefficient of the overlapping surfaces; D h —Superposition parameter; L r — Load holding level; 0.4 ≤ F h ≤ 0.8, 0.4 ≤ D h ≤ 0.6, 0.4 ≤ L r ≤ 0.6; In formula (1), the residual sum of squares RSS is 30.96, the regression sum of squares ESS is 231054.04, and R 2 =0.998; Among them, the relationship equations (2) and (3) for the protective layer thickness of 30 mm and 40 mm are as follows: ； In formula (2), the residual sum of squares RSS is 59.30, and the regression sum of squares ESS is 257085.70; in formula (3), the residual sum of squares RSS is 76.32, and the regression sum of squares ESS is 278018.

68. The R of formulas (2) and (3) is 2 Both are 0.996.

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