Even for the doubly symmetric Ibeams under linear distributed moment, the design formulas given by codes of different countries are quite different. This paper will derive a dimensionless analytical solution via linear stability theory and propose a new design formula of the critical moment of the lateraltorsional buckling (LTB) of the simply supported Ibeams under linear distributed moment. Firstly, the assumptions of linear stability theory are reviewed, the dispute concerning the LTB energy equation is introduced, and then the thinking of PlateBeam Theory, which can be used to fully resolve the challenge presented by Ojalvo, is presented briefly; secondly, by introducing the new dimensionless coefficient of lateral deflection, the new dimensionless critical moment and Wagner’s coefficient are derived naturally from the total potential energy. With these independent parameters, the new dimensionless analytical buckling equation is obtained; thirdly, the convergence performance of the dimensionless analytical solution is discussed by numerical solutions and its correctness is verified by the numerical results given by ANSYS; finally, a new trilinear mathematical model is proposed as the benchmark of formulating the design formula and, with the help of 1stOpt software, the four coefficients used in the proposed dimensionless design formula are determined.
Structural stability has always been a key point in the design of steel structures [
Discrepancies of EUMF for a simply supported beam under linear distributed moment.
Due to the complexity of the LTB phenomenon of Ibeams under nonuniform distributed moment, so far only approximate analytical solutions or numerical solutions have been published.
Some approximate analytical solutions can be found in the classical text books such as Chajes [
The numerical solutions can be obtained from finite difference method [
It is noteworthy that, with the popularization and widespread use of finite element software, since the 2000s, some researchers have tried to use the generalpurpose finite element software to simulate the buckling behavior or to assess the related design formula of the steel Ibeams. For example, Mohri et al. [
In theory, FEM can be used not only to do research works but also to formulate design formulas. However, when you really try to formulate the design formulas through a lot of FEM analysis, you will have to deal with some real trouble. Firstly, the finite element analysis must be a dimensional analysis [
On the contrary, not only does the dimensionless analytic solution have the advantages of small input data amount and fast calculation speed, but also the formula formulated from these dimensionless results is more general than those obtained from FEM analysis. To the author’s knowledge, research on the dimensionless analytical solution is scare and began in 1970s with the Anderson and Trahair’s works [
According to the theory and discussion in [
As a continuation of the work of the author, this study shall present a dimensionless analytical solution via linear stability theory and design formula of the critical moment of the LTB of the simply supported Ibeams under linear distributed moment. In Section
As is known, for the slender (
For the sake of completeness, a brief review and comments of the basis of linear stability theory are presented hereafter.
Rigid section hypothesis: that is, the cross section is very rigid that its original shape is retained during buckling. This is the wellknown Vlasov’s rigid section hypothesis, which means that local and distortional buckling are excluded in the LTB analysis.
EulerBernoulli’s hypothesis: that is, there is no shear deformation in the middle surface of the cross section. This assumption is used to describe the inplane deformation of a shell/plate.
Kirchhoff’s hypothesis: that is, the shear deformation in the planes normal to the middle surface of the cross section is small and can be neglected. This assumption is used to describe the outofplane deformation of a shell/plate.
The material is an ideal isotropic material and follows Hooke’s law.
Displacements and twist angle are assumed to be small enough.
The first two hypotheses are first explicitly proposed by Vlasov [
Invoking assumption
Invoking assumption
In Vlasov’s monograph, (
In short, even though Vlasov’s theory is such invaluable that almost all the literature on the theory of thinwalled bars will refer to his work, thus making him an outstanding figure among the pioneers in the field [
The first correct LTB energy equation is proposed by Bleich [
In addition, both Vlasov and Bleich’s theories do not apply to the combined cross sections (e.g., openclosed/solidopen) and composite cross sections (e.g., steelconcrete).
Because the LTB energy equation is the foundation of the modern mechanics analysis, such as finite element method and approximate analytic solution, extensive research has been carried out since the 1960s. The related theoretical derivation on the LTB energy equation for Ibeams has been detailed in [
The most fatal challenge comes from the question of the validity of the Wagner hypothesis, which is a part of the theoretical basis for the traditional LTB theory. Since 1981, this problem has been continuously questioned and challenged by Ojalvo [
From the appearance point of view, the problem of the Wagner coefficient is only reflected in process of deriving the load potential; hence some attempts have been made to improve the Vlasov’s displacement filed in order to obtain the correct Wagner coefficient. For example, based upon a more complex geometric analysis, a longitudinal displacement with two additional nonlinear terms was proposed by Trahair (see [
In addition, even the concept of the rotation matrix, which was suitable for the nonlinear analysis with large rotation [
Obviously, incorrect linear strain energy will be obtained if the above displacements of Trahair and Pi et al. were used. Therefore, the above improvements are cumbersome and unnecessary.
In fact, The LTB problem is similar to the vibration problem of the prestressed concrete beam, and hence this problem belongs to the category of linear eigenvalue problem in essence. In modern mechanics terms, this problem is a small displacement finite strain problem, namely, in such analysis, assumptions
In the author’s opinion, so far the problem of how to derive the linear strain energy correctly based only upon the commonly used engineering mechanics theory has not been resolved in both the traditional theory and Ojalvo’s new theory [
In fact, there is a common theoretical basis in both Bleich’s theory and Ojalvo’s new theory; that is, they all tried to use engineering theory (beam or rod) to derive their LTB energy equation. For example, Navier’s plane section hypothesis is used in Bleich’s derivation of the linear strain energy. But these thoughts were rarely mentioned in the published LTB literature.
This study considers a simply supported steel Ibeam under linear distributed moment as shown in Figure
Calculation diagram of simply supported beam under linear distributed moment.
In this new theory, Vlasov’s hypotheses are preserved, but the warping function and the Vlasov’s longitudinal displacement are discarded. The longitudinal displacement, geometrical equation, strain energy, and initial stress potential energy of the inplane bending and the outplane bending and torsion of each flat plate are described by the Euler beam theory and KirchhoffPlate theory, respectively (Figure
Illustration of section deformation assumption and plate deformation analogy.
In short, the PlateBeam Theory provides a new way to resolve the dispute concerning the Wagner hypothesis completely, in which the rational elements of Bleich’s, Vlasov’s, and Ojalvo’s thoughts have been inherited and developed, while the warping function of Vlasov was abandoned. Furthermore, the correctness of the traditional LTB energy equation, such as (
For the simply supported Ibeam shown in Figure
For the case of the simply supported Ibeams, the following modal trial functions are proposed for the lateral deflection
From the perspective of structural dynamics,
Obviously, the above trial functions are orthogonal and satisfy the following geometric boundary conditions of the simply supported beams; that is,
Moreover, according to the theory and discussions given by literature [
Substituting the moment function (
Without loss of generality, the following new dimensionless parameters [
In the following derivation, the energy equation of (
The total potential energy of this problem is the sum of the above equations; that is,
According to the principle of energy variation method, the following stationary conditions that minimize the total potential energy are required:
From the condition
For each fixed
With the knowledge of the internal data structure of the above infinite series equation and using the following the notations listed as follows:
Similarly, from the condition
With the notations in (
If (
Equation (
Obviously, from the mechanical point of view, the matrix at the lefthand side of the equation represents the linear stiffness matrix of the steel girder and is independent of the load, while the matrix on the left side of the equation depends on the load pattern, which in the present case can be considered as a geometric stiffness matrix due to the linear distribution of moments.
It is noted that the dimensionless analytical solution of the buckling equation presented in the paper not only has a clear data structure and but also is easy to program and solve. In addition, unlike the FEM solutions published in literature, this solution can easily be verified by anyone.
There are different ways to define the dimensionless critical moment, in which the wellknown definition is the one proposed by Kitipornchai et al. [
Corresponding to this definition, the following dimensionless Wagner’s coefficient, the load position parameter, and the beam parameter are defined by Kitipornchai:
Obviously, all these parameters lack clear physical and geometric meanings. More importantly, since there is a common factor
Instead, the following dimensionless critical moment is used in this paper:
This definition is first proposed by Professor Zhang [
It is the first finding in this paper that this definition can be derived naturally from the total potential energy based upon the new definition of displacement function, that is, (
In addition, in the case of the LTB of a doubly symmetric beam under pure bending, the definition of (
That is, the dimensionless critical moment is the dimensionless distance between the fixed axis of rotation and the shear center for a cross section. It is clear that, for the doubly symmetric beams, the fixed axis of rotation always lies beyond the tension flange due to
Finally, and most importantly, the other dimensionless parameters defined in (
If
Under the conditions that the coefficient determinant is zero, then the firstorder approximation of the dimensionless critical moment of the Ibeams under linear distributed moment can be written in a compact form:
Obviously, this 1storder approximation only applies to cases where
If
It is easy to find that this expression is identical to the exact analytical solution in the textbook [
If
Equation (
It can be proved that the above result is exactly the same as that obtained by using trigonometric series with six terms as the modal function. This shows that the infinite series solution, that is, the exact analytical solution derived in previous section, is correct.
Our research results [
In order to obtain a numerical solution to the dimensionless analytical solution, we shall take the number of trigonometric series in (
If the commonused symbols used in traditional finite element method [
Mathematically, the problem of solving
Obviously, the final form of the dimensionless buckling equation (see (
First, the convergence performance of the trigonometric series is presented. Typical graphs of the relationship between the dimensionless buckling moment and the number of terms of trigonometric series used in the modal trial functions are shown in Figure
Convergence performance.
When
Next, the numerical algorithm is discussed. In fact, any method used to solve the generalized eigenvalue problem is applicable to LTB problem of this paper, because it is different from the largescale finite element analysis program, and the number of degrees of freedom involved in this paper is not large.
It is noted that the
In order to automatically obtain a numerical solution of any desired accuracy, it is necessary to perform an iterative process in the developed MATLAB program in which the following convergence criteria are used:
In order to facilitate the dimensionless parameter analysis and to perform the comparative study between the numerical solution from the exact dimensionless analytical solution and the ANSYS finite element analysis, a MATLAB program is developed according to the aforementioned numerical algorithm.
In this section, the generalpurpose finite element software ANSYS is used to verify the correctness of the exact solution and 6thorder approximations given previously.
It is well known that the BEAM189 in ANSYS software cannot correctly simulate the critical moment of LTB for the singly symmetric beams. Therefore, the SHELL63 element is used in this paper to simulate the buckling problem of the steel beams.
The second reason is that since a typical steel Ibeam is composed of three thin plates, it is theoretically correct using the SHELL elements to simulate the buckling problem of the Ibeams. This is in accordance with Vlasov’s work [
It is shown that only in this way all kinds of buckling phenomena such as local buckling, distortion buckling, and lateraltorsional buckling (LTB) can be simulated and captured. However, only the LTB phenomena is concerned in this investigation. Consequently, when the SHELL63 element is used to simulate the desired lateraltorsional buckling of a steel beam, there is a problem; that is, for beams simulated by SHELL63 element, the Vlasov’s rigid section hypothesis cannot be automatically satisfied as the BEAM elements. Therefore, special measures have to be taken in the FEM simulation to ensure that the overall buckling; that is, LTB rather than the local buckling or the distortional buckling appears first.
Some approaches have been proposed, such as the method of adding stiffeners or the method of treating the stiffeners as membrane elements [
After exploration and modeling practice, we found that, in the ANSYS finite element model, the “CRIG” command is such a very effective command that you can be more confident to simulate the rigid section features of the buckled beams, without worrying about the failure of FEM numerical simulation. Figure
Simulation of rigid section hypothesis by “CERIG” command.
Simulation of boundary conditions (fork supports).
Simulation of end moments.
The material properties are taken as follows: Young’s modulus
Although the analytical and numerical solutions given in the previous section are dimensionless, the finite element analysis must be a dimensional analysis in which the section size and span (or slenderness) of the steel Ibeams must be specified in advance.
Considering that the finite element analysis here is of a verification nature, only three typical cross sections are selected for the study purpose. Section A is a doubly symmetric section, and both Section B and Section C are singly symmetric sections, in which Section B has a larger tension flange, whereas Section C has a larger compression flange. All pictures of the selected sections along with their geometrical properties are shown in Table
Section dimensions and geometric properties for the selected beams.
Section A  Section B  Section C 






















The beam spans are chosen such that the spantodepth ratios lie between 15 and 30. Table
Critical moments for the selected beams under pure bending.
Section 








Diff.1/% 

Diff.2/% 

A  1.063  6  1  1  0  0.687  0.687  0.680  0.91  0.689  −0.35 
0.797  8  1  1  0  0.802  0.802  0.801  0.22  0.808  −0.66  
0.531  12  1  1  0  1.066  1.066  1.068  −0.23  1.076  −0.96  


B  0.569  6  0.125  1  −0.322  0.391  0.391  0.383  1.85  0.641  61.35 
0.427  8  0.125  1  −0.322  0.541  0.541  0.535  1.15  0.809  46.51  
0.284  12  0.125  1  −0.322  0.871  0.871  0.867  0.52  1.162  30.43  


C  0.569  6  8  1  0.322  1.035  1.035  1.048  −1.28  0.641  −39.11 
0.427  8  8  1  0.322  1.185  1.185  1.204  −1.55  0.809  −33.11  
0.284  12  8  1  0.322  1.515  1.515  1.539  −1.53  1.162  −25.02 
The abovementioned FEM model is used to calculate the critical moment of the Ibeams. Since, for a steel Ibeam under pure bending, the critical moment has exact solution, that is, (
It is observed that (1) for all cases where doubly symmetric Ibeams are studied, the deviations from the FEM results are within 0.91%; (2) for all cases where singly symmetric Ibeams are studied, the deviations from the FEM results are higher than those of the doubly symmetric beams, but the largest deviation is not more than 1.85%; (3) if three decimal places are retained, the results of the 6thorder approximate analytical solution and the numerical solution with 30 terms are exactly the same. This proves the appropriateness of the ANSYS FEM model developed in this paper.
In addition, since no exact solution exits for the beams under linear distributed moment, the ANSYS FEM model is used to compare with the published data which is obtained from newly developed FEM model. Table
Critical moments for the beams of HEA200 under linear distributed moment.
Number 

SHELL/10^{3} N⋅m  B3Dw/10^{3} N⋅m  Diff.1/%  Theory/10^{3} N⋅m  Diff.2/%  BEAM/10^{3} N⋅m  Diff.3/% 

L1  1  82.064  82.69  0.76  81.872  −0.23  82.687  −0.99 
L2  0.75  93.574  94.79  1.30  93.358  −0.23  94.285  −0.98 
L3  0.5  108.090  108.94  0.79  107.853  −0.22  108.916  −0.98 
L4  0.25  126.419  127.45  0.82  126.175  −0.19  127.393  −0.96 
L5  0  149.142  150.45  0.88  148.935  −0.14  150.318  −0.92 
L6  −0.25  175.909  177.61  0.97  175.823  −0.05  177.353  −0.86 
L7  −0.5  204.151  206.32  1.06  204.317  0.08  205.944  −0.79 
L8  −0.75  225.915  228.38  1.09  226.436  0.23  228.096  −0.73 
L9  −1  219.686  221.97  1.04  220.378  0.31  221.905  −0.69 


Section and its properties 














It can be observed that (1) all the results of B3DW are higher than those of the ANSYS model developed in this paper, and the largest deviation is less than 1.3%; (2) all the results given by the numerical solutions (referenced “theory”) are the lowest and closer to the those of ANSYS with the maximum difference within 0.31%. This provides an additional evidence of the appropriateness of the ANSYS FEM model, and the analytical and numerical solutions developed in the current work are verified preliminarily.
It should be pointed out that the BEAM189 in ANSYS software can only be used to predict the critical moment for a doubly symmetric beam but will yield erroneous results for a singly symmetric beam.
In summary, all the results show that the “CRIG” command in ANSYS can be used to simulate the Vlasov’s rigid section hypothesis, and its simulation effect is more effective and hence better than the method of adding stiffeners.
This paragraph will further to verify the accuracy of the analytical and numerical solutions developed in this paper.
For the steel Ibeams with the section listed in Table
Critical moments for the selected beams under linear distributed moment.
Section 








Diff.1/% 

A  1.063  6  1  0.5  0  0.906  0.906  0.897  0.99 
1.063  6  1  0.1  0  1.179  1.179  1.164  1.30  
1.063  6  1  0  0  1.265  1.265  1.247  1.45  
1.063  6  1  −0.1  0  1.357  1.357  1.336  1.63  
1.063  6  1  −0.5  0  1.766  1.766  1.720  2.69  
1.063  6  1  −1  0  1.872  1.872  1.809  3.48  


B  0.569  6  0.125  0.5  −0.322  0.512  0.512  0.502  1.92 
0.569  6  0.125  0.1  −0.322  0.649  0.649  0.635  2.18  
0.569  6  0.125  0  −0.322  0.689  0.689  0.673  2.29  
0.569  6  0.125  −0.1  −0.322  0.730  0.730  0.713  2.41  
0.569  6  0.125  −0.5  −0.322  0.908  0.908  0.883  2.88  
0.569  6  0.125  −1  −0.322  1.075  1.075  1.088  1.19  


C  0.569  6  8  0.5  0.322  1.365  1.365  1.385  −1.50 
0.569  6  8  0.1  0.322  1.767  1.767  1.784  −0.93  
0.569  6  8  0  0.322  1.889  1.889  1.900  −0.56  
0.569  6  8  −0.1  0.322  2.017  2.017  2.029  −0.60  
0.569  6  8  −0.5  0.322  2.263  2.262  2.228  1.55  
0.569  6  8  −1  0.322  1.126  1.126  1.090  3.29 
Buckling mode graph of the beams with doubly symmetric section (Section A).
Buckling mode graph of the beams with singly symmetric section (Section C).
It can be found that (1) if three decimal places are retained, the results of the 6thorder approximate analytical solution and the numerical solution with 30 terms are exactly the same as shown in Table
It is noted that, for all Ibeams studied in this paper, the results given by the 6thorder approximation are of high accuracy. Our other research results also support this conclusion [
Finally, it is noted that, for the case of Section A and Section B, SHELL solutions of ANSYS will provide lower critical moment predictions than the exact solution as expected, but, for Section C, the result is just the opposite. Therefore, for the case of the beam under linear distributed moment, FEM cannot be applied to formulate the design formula, because it lacks precision consistency in the analysis.
Nowadays, the concept of equivalent uniform moment coefficient (EUMF) is widely accepted in design codes in many countries. However, it is found that, even for the case of the doubly symmetric Ibeams under linear distributed moment, the design formulas given by the codes/specifications of different countries are quite different. The corresponding comparison between the design formulas [
This paper attempts to use a new mathematical model as the benchmark to solve this longstanding engineering problem with the abovementioned dimensionless analytical solutions.
Since the concept of equivalent uniform moment factor (EUMF) proposed by Salvadori [
In order to analyze the cause of failure, through the parametric analysis, we first study its change law of the data. Figures
Distribution pattern of critical moments with varied
Critical moments versus gradient moment factor with varied
Critical moments versus gradient moment factor with varied
With the aid of the idea of using a piecewise linear approximation to simulate an actual curve, this paper proposes a new trilinear mathematical model used as the benchmark of formulating the design formula. This new idea is represented by the graphics shown in Figure
Proposed trilinear model as the benchmark of design formula.
As we all know, there are three key issues in regression technology, namely, the choice of target formula, how to generate data, and how to regress the unknown coefficient in the target formula.
Firstly, according to the results of the previous parametric analysis, after a number of analyses, observations, and tries, we finally selected the following target formula:
In addition, the study also found that these undetermined regression coefficients for the cross sections with lager top flange are completely different from those of other cross sections in the range (
Next, the MATLAB program formulated in the paper can be employed to generate the abovementioned dimensionless data. In the following data analysis, the end moment ratio
Thirdly, due to the highly nonlinear nature of the target formula, we chose the 1stOpt [
Taking a cross section having a larger top flange and a negative moment (
Parameter  Best estimate 


2.220165826 

−0.458694891 

7.259270297 

1.482366765 

2.735838541 
The resulting relationship between the target value and the calculated value is depicted in Figure
Relationship of the target value and calculated value.
Using a similar approach, we can get all the undetermined coefficients and the relevant results are listed in Table
Formula and parameters for Ibeams under linear distributed moment.







0.234 (2.220)  1.387 (1.095)  5.983  1 

4.674 (−0.459)  0.822 (0.922)  0.287  1 

72.762 (7.259)  0.615 (10.860)  11.468  1 

0.861 (1.482)  0.587 (1.629)  11.713  1 

0.566 (2.736)  0.273 (1.154)  10.446  1 



 
 
 
 

It must be pointed out that the whole regression process takes only a few minutes, of which the calculation process of nearly 500 sets of data only takes a few seconds. If you intend to use the FEM to obtain such a large number of data sets, it may take at least a month. This fully demonstrates that the dimensionless solution has an unparalleled advantage over the finite element method in terms of efficiency and quality in obtaining a large number of regression data.
Finally, the proposed dimensionless design formula is used to calculate the critical moments of the Ibeams with the sections listed in Table
Comparison of the results of design formula with those of theory for Ibeams under linear distributed moment.
Section 








Diff./% 

A  1.063  6  1  1  0  0.687  0.687  0.687  0.00 
1.063  6  1  0  0  1.265  1.265  1.255  −0.72  
1.063  6  1  −0.5  0  1.766  1.766  1.737  −1.68  
1.063  6  1  −1  0  1.872  1.872  1.919  2.54  


B  0.569  6  0.125  1  −0.322  0.391  0.390  0.391  0.00 
0.569  6  0.125  0  −0.322  0.689  0.689  0.664  0.51  
0.569  6  0.125  −0.5  −0.322  0.908  0.908  0.915  5.43  
0.569  6  0.125  −1  −0.322  1.075  1.075  1.194  11.08  


C  0.569  6  8  1  0.322  1.035  1.035  1.035  0.00 
0.569  6  8  0  0.322  1.961  1.961  1.958  −0.19  
0.569  6  8  −0.5  0.322  2.263  2.262  2.286  0.88  
0.569  6  8  −1  0.322  1.126  1.126  1.150  7.00 
Comparison between theory and the proposed formula.
The results show that the proposed formula is of high accuracy and is applicable to both the doubly symmetric section and the singly symmetric section under positive and negative gradient moment.
It must be noted that the trilinear mathematical model can also be converted to a cubic curve model or by adding more points to obtain a quadratic or higherorder curve model. Due to the limited space, these will be discussed elsewhere.
Because there is no reliable analytical solution available, this study is initially intended to develop an exact analytical solution to study the applicability of the design formulas published in literature. However, we found that results given by the existing design formulas vary widely, especially in the range
(1) There is a challenge to part of the theory of monosymmetric beams, called the Wagner hypothesis, was presented by Ojalvo (1981). This challenge can be fully resolved by the PlateBeam Theory put forward by the author, in which only the commonly used plate and beam theory was used and the Vlasov’s warping function was discarded.
(2) For the case of Section A and Section B, SHELL solutions of ANSYS will provide lower critical moment predictions than the exact solution as expected, but, for Section C, the result is just the opposite. Therefore, for the case of the beam under linear distributed moment, FEM cannot be applied to formulate the design formula, because it lacks precision consistency in the analysis.
(3) By introducing a new dimensionless coefficient of lateral deflection, new dimensionless critical moment and dimensionless Wagner’s coefficient are derived naturally from the total potential energy. These new dimensionless parameters are independent of each other and thus can be used as independent variables in the process of regression formula.
(4) The results show that, in the case of the simplesupported Ibeams under linearly distributed moments, the dimensionless analytical solution presented in current work is exact in nature and has an unparalleled advantage over the finite element method in terms of efficiency and quality in obtaining a large number of regression data. Therefore, it provides a more convenient and effective method for the formulation of a more accurate design formula than finite element method.
(5) It is found that, for all Ibeams studied in this paper, the results given by the 6thorder approximation are of high accuracy. Our other research results also support this conclusion [
(6) The dimensionless parameters defined by Kitipornchai lack clear physical and geometric meanings. More importantly, none of these parameters is independent; thus it is difficult to use them to regress a rational design formula.
(7) The dimensionless analytical buckling equation is expressed as the generalized eigenvalue problem as that used in the traditional finite element method. This not only facilitates the solution in MATLAB programming but also makes the analytical solution have a clear physical meaning, more easy to understand and apply.
(8) To satisfy Vlasov’s rigid section hypothesis, the “CRIG” command is used in the FEM model of ANSYS. The results show that its simulation effect is more effective and hence better than the method of adding stiffeners.
(9) BEAM189 in ANSYS software can only be used to predict the critical moment for a doubly symmetric beam but will yield erroneous results for a singly symmetric beam.
(10) It is found that the concept of equivalent uniform moment factor (EUMF) is not applicable for the simplesupported Ibeams with unequal flanges subjected to unequal end moments; that is, a single EUMF does not exist in this case.
(11) This paper proposes a new trilinear mathematical model as the benchmark of formulating the design formula. The results show that the proposed design formula has a high accuracy and is applicable to both the doubly symmetric section and the singly symmetric section under positive and negative gradient moment. This model provides a new idea for the formulation of more reasonable design formulas, so it is suggested that the designers of the design specifications can formulate the corresponding design formulas with reference to this model.
The authors declare that they have no conflicts of interest.
This work is supported by the National Natural Science Foundation of China (51578120 and 51178087) and the Scientific Research Foundation of Nanjing Institute of Technology (YKJ201617).