Lateral–Torsional Buckling of Cantilever Steel Beams under 2 Types of Complex Loads
Lateral–Torsional Buckling of Cantilever Steel Beams under 2 Types of Complex Loads
Cai, Yong;Ling, Angyang;Lv, Xiaoyong
2023-05-09 00:00:00
applied sciences Article Lateral–Torsional Buckling of Cantilever Steel Beams under 2 Types of Complex Loads 1 1 2 , Yong Cai , Angyang Ling and Xiaoyong Lv * School of Civil Engineering, Central South University, Changsha 410075, China; caiyong@csu.edu.cn (Y.C.); lingangyang1225@163.com (A.L.) School of Civil Engineering, Central South University of Forestry and Technology, Changsha 410004, China * Correspondence: lvxiaoyong10@163.com; Tel.: +86-15773179538 Abstract: Cantilever steel beams are an essential structural element in civil engineering fields such as bridges and buildings. However, there is very little research on the critical moment (M ) of cantilever cr beams subjected to a concentrated load (CL) or a combination of concentrated load and uniformly distributed load (CUDL) when the concentrated load is not limited to the free end. Therefore, the focus of the current paper is the calculation of M for cantilever steel beams under CL and CUDL. cr This paper proposes a program and a simple closed-form solution for M that are applicable to the cr elastic buckling analysis of cantilever I-beams under CL and CUDL. Based on the Rayleigh–Ritz method, a matrix equation and the corresponding procedure about M under CL and CUDL are cr derived by using infinite trigonometric series for the buckling deformation functions. The value of M and the corresponding mode of buckling can be obtained efficiently by considering the symmetry cr of the section, the ratio of two load values and the load action position. Experimental results and finite element calculations validate the numerical solutions of the procedure. A closed-form solution for M is derived according to the assumption of a small torsion angle and the specific values of cr each coefficient in the closed-form solution of M are calculated by the proposed procedure. The cr results show that the procedure and closed-form solution for M presented in this paper have a high cr degree of accuracy in calculating the M of the cantilever beam under CL and CUDL. The deviations cr between the results calculated by the proposed procedure and data from existing literature are less than 8%. These conclusions are capable of solving the calculation problem of M for cantilever cr beams under CL or CUDL, which are both significant load cases in engineering. The study provides Citation: Cai, Y.; Ling, A.; Lv, X. a reference for the design of cantilever steel beams. Lateral–Torsional Buckling of Cantilever Steel Beams under 2 Types Keywords: cantilever steel beam; lateral–torsional buckling; total potential energy equation; of Complex Loads. Appl. Sci. 2023, 13, 5830. https://doi.org/10.3390/ combined load; critical moment app13105830 Academic Editor: Rodrigo Gonçalves 1. Introduction Received: 23 March 2023 Lateral–torsional buckling (LTB) is a common global instability phenomenon for slen- Revised: 6 May 2023 der structures, where structures that are bent in the plane of greatest flexural rigidity may Accepted: 6 May 2023 buckle laterally and torsionally as the external load reaches the critical value. Therefore, Published: 9 May 2023 it is necessary to consider LTB in the design of the beam when the flexural rigidity of the beam in the plane of bending is greater than the lateral flexural rigidity. Cantilevers are a common slender structure in civil engineering. Therefore, it is necessary to con- sider the limiting load of LTB in addition to deformation and stress analysis during the Copyright: © 2023 by the authors. engineering design process. The American National Standard Institute/American Insti- Licensee MDPI, Basel, Switzerland. tute of Steel Construction (ANSI/AISC 360-16) proposed the lateral–torsional buckling This article is an open access article modification factor, C , when calculating the LTB of simply supported beams. C had distributed under the terms and b b an explicit formula but it was only recommended directly taking the value of 1.0 when conditions of the Creative Commons calculating the C of cantilever beams [1]; EN 1993-1-1:2005 also had a specific formulas for Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ lateral–torsional buckling of simply supported beams, but it did not mention the instability 4.0/). calculation of cantilever [2]. Appl. Sci. 2023, 13, 5830. https://doi.org/10.3390/app13105830 https://www.mdpi.com/journal/applsci Appl. Sci. 2023, 13, 5830 2 of 27 As early as 1960, Clark proposed a formula for calculating the critical moment (M ) cr using C , C and C factors [3]. Since then, the calculation method of C , C and C has been 1 2 3 1 2 3 increasingly refined. Many scholars have studied C , C and C factors of lateral–torsional 1 2 3 buckling of simply supported beams. Zhang et al. [4] studied the general formula of M cr under linear distributed moment. The study used dimensionless parameters and infinite series to calculate M and fitted the specific values of C , C and C . It was revealed that cr 1 2 3 the more terms the buckling deformation function takes, the more accurate the calculation results will be. However, due to the complexity of the external load, the solution to M also cr needs to depend on other methods. Bresser et al. [5] further studied the expression of C of simply supported beams under support moments, point loads, uniformly distributed loads and a combination thereof. Kucukler et al. [6] presented a theory of stiffness reduction coefficient for the LTB design of welded web-tapered steel beams. Rossi et al. [7] conducted elastic LTB analysis on supported beams using ABAQUS and GBTul. Sahraei et al. [8] assessed the LTB mode of plane frames by finite element theory, which greatly reduced the amount of modeling and calculation. So far, numerous problems about the elastic LTB of simply supported beams are being gradually solved. For cantilever steel beams, buckling deformation is more complex due to the char- acteristics of the boundary condition. Unlike simply supported beams, the maximum displacement and torsion angle of cantilever are located at the free end rather than near the mid-span. Therefore, the coefficients of C , C and C are no longer fixed values. 1 2 3 Timoshenko used infinite series to study the singly symmetric cross-section cantilever beam and obtained the critical load of the lateral–torsional buckling [9]. Anderson and Attard [10,11] carried out LTB experiments on four types of I-shaped cantilever steel beams with different cross-sections. Andrade et al. [12] extended the domain of application of C , C and C factors to cantilevers. In addition, Andrade evaluated the performance 1 2 3 of a 1D model of elastic LTB behavior [13]. Since then, many scholars have calculated M ’s formula of cantilever beams with different sections under a large number of different cr loads. Zhang et al. [14] proposed a set of formulae to calculate M under concentrated cr load at the free end or uniformly distributed load. Ozbasaran et al. [15,16] proposed a closed-form expression for calculating elastic critical LTB loads and a design procedure for cantilevers. The design procedure considered elastic buckling and inelastic buck- ling and the calculation results of the procedure were highly consistent with Eurocode 3-2005 or AISC360-10. Because the boundary conditions of cantilever beams are more com- plex than those of simply supported beams, the analytical solutions of M of the cantilever cr are highly complex. Therefore, the solution to M requires more approaches, such as the cr finite difference [17], finite element [18–20], energy method [21,22], finite integral [10] or the lateral–torsional buckling modification factor [1]. Among them, the energy method is one of the most common and basic methods. Therefore, the correct derivation or selection of total potential energy is the key to obtaining accurate calculation results. Additionally, Trahair et al. [23,24] studied the inelastic LTB of cantilevers and continuous steel beams in order to provide better design methods than current specifications. Similarly, Demirhan et al. [25] carried out inelastic buckling experiments on biaxial symmetric cantilever steel beams and proposed a numerical simulation method to accurately predict the displacement and inelastic critical buckling load of cantilever steel beams. In addition to the study of ordinary simply supported beams and cantilever beams, many scholars also considered other factors, such as pre-stressed beams, initial imperfec- tions, flange–web interaction and material properties. Lorkowski et al. [26] studied the M of LTB of pre-stressed two-chord columns by experiment and numerical analysis. Kim cr et al. [27,28] presented an LTB theory of pre-stressed steel H-beams. This theory considered the initial rotation angle and it was applicable to simply supported beams and cantilever beams. Zhang used the energy method to obtain the buckling load of pre-stressed steel I-beams [29]. Lebastard et al. [30] proposed analytical formulations for supported beams whose support was restrained from warping, and some scholars [31–33] have studied the LTB of special sections such as built-up sections. Saoula et al. [34] proposed a new theory Appl. Sci. 2023, 13, 5830 3 of 27 to predict the shape and amplitude of unique global and local initial imperfections for the LTB of doubly symmetric I-section cantilevers. Erkmen [35] and Kimura [36] studied the elastic LTB mode considering web-distortion or flange–web interaction. Jáger et al. [37–39] studied the elastic LTB mode of corrugated web beams. In addition, an increasing number of scholars have considered the influence of material properties on the LTB performance of beams [40–42]. For example, Khalaj et al. [43,44] studied the effect of chemical compo- sition of materials on their strength by using neural networks and artificial intelligence technology. Virgin et al. [45] studied the LTB performance of cantilever beams using 3D printing technology. It could be seen from those studies on the lateral–torsional buckling of cantilever steel beams that most of their transverse loads were mainly a single load. The position of the load acting on the cross-section was restricted to the top flange (TF), bottom flange (BF) and the shear center (SC). The position of the load acting in the beam length direction was restricted to the free end. There was less research on lateral–torsional buckling of singly symmetric I-beams under CL or CUDL. In engineering practice, cantilever steel beams are usually affected by the concentrated load or the combination of concentrated load and uniformly distributed load, and the position of the concentrated load is not limited to the free end. For example, some cantilever beams will be subject to the concentrated load from the secondary beams and uniformly distributed load from the floor slabs. The position of the concentrated load may be in the middle of the span or other positions. In view of this, CL and CUDL are both very important load cases in the engineering field. Therefore, it is necessary to study this situation. The innovation of this paper is the numerical method and the closed-form solution about M , which aims to efficiently obtain the M and corresponding mode of buckling of cr cr cantilever steel beams under CL and CUDL. A matrix equation with M is developed by cr using infinite trigonometric series for the buckling deformation functions and Rayleigh– Ritz method. Based on the matrix equation, a MATLAB program was developed for LTB analysis of cantilever I-beams. The MATLAB program considered the ratio of two loads, the symmetry of the section, the action position of the concentrated load in the direction of the beam length and the vertical position of two loads. It is worth noting that the matrix equation and MATLAB program can efficiently obtain the M and buckling mode cr of the cantilever beam under any single load except constant moment. Therefore, it has a wide range of applications. Moreover, it was able to solve M and the buckling mode of cr cantilever beams under CL or CUDL, which are both common load cases in engineering applications. Then M calculated by the MATLAB program was compared with results cr reported in existing references [10,11,16,18] to verify the correctness of the theory. Last, a simple closed-form solution for M under CL and CUDL is proposed by analyzing the cr factors affecting the LTB modes of cantilever beams, which enables computing M with cr relative ease. 2. Calculation of Total Potential Energy Equation under CL and CUDL 2.1. Introduction and Comparison of Total Potential Energy Equation Figures 1 and 2 illustrate the LTB of a singly symmetric cantilever I-beam. As shown in Figures 1 and 2, L is cantilever length. P is concentrated load and q is uniformly y y distributed load, their coordinates are respectively (0, y ) and (0, y ), the shear center ’s y q y coordinates are (0, y ), where a = y y , a = y y . l is the ratio of distance s P y P y s q y q y s between the action point of P and the fixed end to the cantilever ’s length. Zhang and Tong [14,46] thought that the LTB theory of steel beams should consider linear strain energy, nonlinear longitudinal strain energy, nonlinear shear strain energy and nonlinear transverse strain energy at the same time. Compared with the experimental results in reference [10] and reference [11], the total potential energy proposed by Zhang is accurate in calculating the M of the cantilever beam. At the same time, it is also confirmed cr that the total potential energy proposed by Zhang and Tong is applicable to the boundary conditions of cantilever beam through the discussion in reference [14]. Appl. Sci. 2023, 13, x FOR PEER REVIEW 4 of 28 Appl. Sci. 2023, 13, x FOR PEER REVIEW 4 of 28 is accurate in calculating the Mcr of the cantilever beam. At the same time, it is also con- firmed that the total potential energy proposed by Zhang and Tong is applicable to the boundary conditions of cantilever beam through the discussion in reference [14]. is accurate in calculating the Mcr of the cantilever beam. At the same time, it is also con- For the load case shown in Figures 1 and 2, the total potential energy of Zhang and firmed that the total potential energy proposed by Zhang and Tong is applicable to the Tong can be expressed as Equation (1). boundary conditions of cantilever beam through the discussion in reference [14]. For the load case shown in Figures 1 and 2, the total potential energy of Zhang and 1 1 22 2 2 2 2 ′′ Π+ = (EI u'' EI ϕ ''+ GIϕϕ '− 2M u' '+2β Mϕ '+ 2β Mϕϕ '− 2M u'ϕ− q (β− a )ϕ )dz− (β− a )Pϕ (λl ) (1) y ω t x xx xx x y x q x p y Tong can be expressed as Equation (1). yy 2 2 1 1 22 2 2 2 2 ′′ Π+ = (EI u'' EI where ϕ ''+ GIEϕϕ 'is Y− 2o Mung’s mod u' '+2β Mϕ u'lus, + 2βGM is ϕ the shear ϕ '− 2M u'ϕmod− q (uβ lus, − a I)ωϕ is )dtzh−e w (βarping − a )Pconstϕ (λl )a nt, It is the (1) y ω t x xx xx x y x q x p y yy 2 2 torsional constant, Iy is the moment of inertia about the y-axis, βx is Wagner’s coefficient, u where E is Young’s modulus, G is the shear modulus, Iω is the warping constant, It is the and φ are the lateral deflection and twist angle of section, respectively, φ(λl) is the twist torsional constant, Iy is the moment of inertia about the y-axis, βx is Wagner’s coefficient, u angle at the point where the transverse concentrated load acts during the lateral–torsional and φ are the lateral deflection and twist angle of section, respectively, φ(λl) is the twist buckling of the beam, and Mx is the moment around the x-axis. Appl. Sci. 2023, 13, 5830 4 of 27 angle at the point where the transverse concentrated load acts during the lateral–torsional buckling of the beam, and Mx is the moment around the x-axis. y a Py S(0,ys) P Py q S(0,ys) y P a y qy y y qy λl λl y y y y Figure 1. LTB about singly symmetric I-section cantilevers under CUDL. Figure 1. LTB about singly symmetric I-section cantilevers under CUDL. Figure 1. LTB about singly symmetric I-section cantilevers under CUDL. y u S(0,y ) Py s P a y Py S(0,ys) P λl λl y y y y Figure 2. LTB about singly symmetric I-section cantilevers under CL. Figure 2. Figure L 2.TL B a TBbabout out sing singly ly sy symmetric mmetric I-s I-section ection cantilev cantilevers er under s under C CL. L. The ratio of the uniformly distributed load to concentrated load in Figure 1 is as- For the load case shown in Figures 1 and 2, the total potential energy of Zhang and sumed to be as follows: The ratio of the uniformly distributed load to concentrated load in Figure 1 is as- Tong can be expressed as Equation (1). ql sumed to be as follows: l α = 1 1 (2) 2 2 2 2 0 0 0 0 0 0 0 0 2 2 00 00 P = (E I u + E I j + G I j 2M u j +2b M j + 2b M jj 2M u j q (b a )]j )dz (b a )P j (ll) (1) y ! t x x x x x x P y x q x p y y y 2 2 ql α = (2) where E is Young’s modulus, G is the shear modulus, I is the warping constant, I is the The magnitude of the moment at any position of cant ! ilever (Mx) is expressed by Eq t ua- torsional constant, I is the moment of inertia about the y-axis, b is Wagner ’s coefficient, u y x tion (3). and j are the lateral deflection and twist angle of section, respectively, j(ll) is the twist The magnitude of the moment at any position of cantilever (Mx) is expressed by Equa- angle at the point where the transverse concentrated load acts during the lateral–torsional tion (3). buckling of the beam, and M is the moment around the x-axis. The ratio of the uniformly distributed load to concentrated load in Figure 1 is assumed to be as follows: q l a = (2) The magnitude of the moment at any position of cantilever (M ) is expressed by Equation (3). P a y 2 (l z) P (ll z) , 0 z ll 2 l M = (3) P a : y 2 (l z) ,ll z l 2 l Note that when l is equal to 0, M takes the second function in Equation (3). When l is equal to 1, M takes the first function in Equation (3). x Appl. Sci. 2023, 13, 5830 5 of 27 According to Equation (3), boundary conditions and the properties of the integral, the following equations can be obtained. Z Z Z l l l 0 0 0 l 0 0 0 2M u j dz = 2M u j + 2( M u ) jdz = 2( M u + M u )jdz (4) x x x x x 0 0 0 Z Z l l 0 0 0 2 l 0 0 2 2b M jj dz = M b j ( M b ) j dz (5) x x x x x x 0 0 M = q (6) x y If l = 1, then M (l) = P and j(0) = 0. The following equation can be obtained by Equations (5) and (6). Z Z l l 0 0 2 2 2b M jj dz = P b j (l) + q b j dz (7) x x y x y x 0 0 If l < 1, the following equation can be obtained by Equations (5) and (6): R R R l ll l 0 0 0 0 0 0 2b M jj dz = 2b M jj dz + 2b M jj dz x x x x x x 0 0 ll R R ll l 0 2 ll 0 0 2 0 2 l 0 0 2 = M b j ( M b ) j dz + M b j ( M b ) j dz x x x x x x x x 0 0 ll ll P a y ll 2 2 (8) = ( (l ll) + P )b j (ll) 0 + q b j dz+ y x y x l 0 P a P a y y l 2 2 2 ( (l l))b j (l) (l ll)b j (ll) + q b j dz x x y x l l ll 2 2 = P b j (ll) + q b j dz y x y x Equation (9) can be obtained by substituting Equations (4), (7) and (8) into Equation (1). Equation (9) is the expression of the traditional total potential energy in reference [47]. Therefore, two total potential energy equations are equivalent under the load case studied in this paper. For convenience of calculation, Equation (9) with the simpler form is selected as the basis for calculation. 1 1 2 2 2 2 0 0 2 2 00 00 00 P = E I u + E I j + G I j + 2M u j+ 2b M j + a q j )dz + a P j (ll) (9) y ! t x x x q y p y y y 2 2 2.2. The Buckling Deformation Functions When using the Rayleigh–Ritz method for stability analysis, the assumption of the buckling deformation functions is very critical, which directly affects the calculation ac- curacy. First, the buckling deformation functions should match the actual deformation form as much as possible and can be expressed by infinite series. Secondly, the buckling deformation functions must meet the displacement boundary conditions. If it can also meet the mechanical boundary conditions, the accuracy will be higher. Based on the above situation, the following trigonometric series are used to describe the lateral deflection u(z) and twist angle j(z) of cantilever steel beams [4,12]. (2n 1)z u(z) = h A 1 cos (10) 2l n=1 (2n 1)z j(z) = B 1 cos (11) å n 2l n=1 where h is the distance between the centroids of the top and bottom flange, and A and B are mutually independent generalized coordinates. From the perspective of structural dynamics, it can be seen that the dimensions of A and B are different. Therefore, h is n n introduced into the expression of u(z) to eliminate the dimension of A , so that A and B n n n Appl. Sci. 2023, 13, 5830 6 of 27 are both undetermined dimensionless coefficients. This idea is from reference [4] and it can make the following calculation easier. Equations (10) and (11) satisfy the displacement boundary condition, i.e., u(0) = u (0) = 0 0 0 0 and j(0) = j (0) = 0. For the free end, it satisfies u(l) 6= 0, u (l) 6= 0, j(l) 6= 0 and j (l) 6= 0. 2.3. Calculation of Total Potential Energy Equation The maximum absolute value of moment (j M j) is expressed as the following equation: P a P l(a + 2l) y y j M j = l + P ll = (12) 0 y 2 l 2 On substituting the moment function (3) and the buckling deformation functions (10) and (11) into the total potential energy Equation (9) and integrating over the beam length, one can obtain the integration results of the total potential energy. In order to make the results more simple, new dimensionless parameters in reference [4] are introduced below. 2 a j M j b I E I P x ! y y 0 1 f f f f M = ; b = ; h = ; K = ; a = ; a = (13) 0 x P q y y 2 2 2 h I h h ( E I /l )h G I l y 2 t where I is the moment of inertia of the top flange around the y-axis, and I is the moment 1 2 of inertia of the bottom flange around the y-axis. Some equations are derived from Equation (13) and defined thus: I h = h (14) (1 + h) 2 2 h h G I = E I (15) t y 2 2 K l (1 + h) The expression of the total potential energy equation is divided into four parts. Each 3 2 part is integrated and multiplied by l /h EI . Equations (13)–(15) are substituted to obtain the specific potential energy results. The specific results are as follows. h i l 2 1 l 00 2 00 2 0 P = (E I u + E I j + G I j )dz 1 y ! t 2 2 0 h E I (16) ¥ 4 ¥ 4 ¥ 2 4 2 4 2 4 2 ( 1+2n) A ( 1+2n) h B ( 1+2n) h B n n n = + + å å å 2 2 64(1+h) 16K (1+h) n=1 n=1 n=1 h i l 2 1 l P = 2b M j dz 2 x x 2 0 h E I 2 2 f f b M B 2 x 0 2 2 = (a + 3l )(1 2n) 6(1 + a)+ 6 cos[( 1 + 2n)l]g 24(a+2l) n=1 ¥ ¥ f f ( 1+2r)( 1+2n)b M B B (17) x 0 n r å å f( 1 + 2r)( 1 + 2n)(1 + a) 2 2 4(n r) ( 1+r+n) (a+2l) n=1 r = 1 r 6= n 2 2 ( 1 + r + n) cos((r n)l) + ( r + n) cos[( 1 + r + n)l] Appl. Sci. 2023, 13, 5830 7 of 27 1 l P = 2M u jdz 3 x 2 0 h E I M A B 2 0 n n 2 2 = f96a cos(n) ( 1 + 2n) 42(1 + a) + (1 2n) (a + 3l )+ 24( 1+2n)(a+2l) n=1 io ¥ ¥ M A B 1+2n 0 n r 48 cos( l) 6 cos(( 1 + 2n)l) + å å 2 2 4( r+n) ( 1+r+n) ( 1+2n)(a+2l) n=1 r = 1 (18) r 6= n 2 2 2 2 2 f 16a(r n) ( 1 + r + n) cos(n) + p( 1 + 2n)[ (1 2r) ( 1 2r + 2r + 6n 6n )(1 + a)+ 2 2 2 2 8(r n) ( 1 + r + n) cos( (1 2n)l) + (1 2n) ( ( 1 + r + n) cos((r n)l) (r n) cos(( 1 + r + n)l))]g 1 l P = a P j(l) p y 4 2 y h E I 2 2 ¥ f ¥ ¥ f g a M aB g a M aB B q 0 n q 0 n r 4 cos(r) 4 cos(n) y y = (3 6n 8 cos(n)) + (2 )+ å å å 2( 1+2n)(a+2l) 2(a+2l) (1 2r) (1 2n) n=1 n=1 r = 1 r 6= n (19) 2 2 2 ¥ f ¥ ¥ f ag M B ag M B B P 0 n P 0 n r y y 2n 1 2n 1 2r 1 å (1 cos( l)) + å å (1 cos( l))(1 cos( l)) a+2l 2 a+2l 2 2 n=1 n=1 r = 1 r 6= n The total potential energy equation can be written as the sum of Equations (16)–(19): P = P + P + P + P = P (20) 1 2 3 4 å i i=1 According to the principle of stationary total potential energy, the stability conditions of the system can be obtained as follows. ¶P = 0, n= 1, 2, 3 . . . . . . , ¥; ¶ A (21) ¶P = 0, n= 1, 2, 3 . . . . . . , ¥; ¶B The following expressions are made available by Equation (21). ¥ 4 ¥ ( 1+2n) A M B ¶P n n = + f 96a cos(n) + ( 1 + 2n)[ 42(1 + a)+ å å ¶ A 32 24( 1+2n)(a+2l) n=1 n=1 2 2 1+2n (1 2n) (a + 3l ) + 48 cos( l) 6 cos(( 1 + 2n)l)]g+ ¥ ¥ M B 2 2 0 r f 16a(r n) ( 1 + r + n) cos(n) + p( 1 + 2n) å å 2 2 4( r+n) ( 1+r+n) ( 1+2n)(a+2l) (22) n=1 r = 1 r 6= n 2 2 2 2 2 1 [ (1 2r) ( 1 2r + 2r + 6n 6n )(1 + a) + 8(r n) ( 1 + r + n) cos( (1 2n)l)+ 2 2 2 (1 2n) ( ( 1 + r + n) cos((r n)l) (r n) cos(( 1 + r + n)l))] Appl. Sci. 2023, 13, 5830 8 of 27 ¥ ¥ ¥ 4 4 2 4 2 fg ( 1+2n) B h ( 1+2n) B h b M B ¶P n n x n 2 0 2 2 = å + å + å (a + 3l )(1 2n) 6(1 + a)+ 6 cos[( 1 + 2n)l]g+ 2 2 ¶B 2 n 12(a+2l) 32(1+h) 8K (1+h) n=1 n=1 n=1 ¥ ¥ fg ( 1+2r)( 1+2n)b M B 2 x 0 r f( 1 + 2r)( 1 + 2n)(1 + a) ( 1 + r + n) cos((r n)l)+ å å 2 2 2(r n) ( 1+r+n) (a+2l) n=1 r = 1 r 6= n 2 M A 2 0 n 2 2 ( r + n) cos[( 1 + r + n)l]g + å f 96a cos(n) + ( 1 + 2n) 42(1 + a) + (1 2n) (a + 3l )+ 24( 1+2n)(a+2l) n=1 io ¥ ¥ M A 1+2n r 48 cos( l) 6 cos(( 1 + 2n)l) + å å 2 2 4( n+r) ( 1+r+n) ( 1+2r)(a+2l) n=1 r = 1 r 6= n (23) 2 2 2 2 2 2 2 f 16a(n r) ( 1 + r + n) cos(r) + p( 1 + 2r)[ (1 2n) ( 1 2n + 2n + 6r 6r )(1 + a) + 8(n r) ( 1 + r + n) 2 2 2 cos( (1 2r)l) + (1 2r) ( ( 1 + r + n) cos((r n)l) (n r) cos(( 1 + r + n)l))] ¥ g ¥ ¥ g agM aB ag M aB q 0 n q 0 r 4 cos(r) 4 cos(n) y y (3 6n 8 cos(n)) + (2 )+ å å å ( 1+2n)(a+2l) (a+2l) (1 2r) (1 2n) n=1 n=1 r = 1 r 6= n 2 2 ¥ g ¥ ¥ g 2ag M B 2ag M B n 2 q 0 r Py 0 y 2n 1 2n 1 2r 1 (1 cos( l)) + (1 cos( l))(1 cos( l)) å å å a+2l 2 (a+2l) 2 2 n=1 n=1 r = 1 r 6= n The condition for the linear equations to hold is that the coefficient determinant of the independent parameters A and B is zero. For convenience of writing, the following n n matrices can be defined. f Ag = A A A 1 2 3 (24) fBg = B B B 1 2 3 ( 1 + 2n) 0 0 R = ; R = 0, n 6= r (25) n,n n,r 0 0 S = T = 0 (26) n,r n,r 4 2 4 4 ( 1 + 2n) h ( 1 + 2n) h 0 0 Q = + ; Q = 0, n 6= r (27) n,n 2 2 n,r 32(1 + h) 8K (1 + h) R = 0 (28) n,r 1 2 2 S = f 96a cos(n) + ( 1 + 2n) 42(1 + a) + (1 2n) (a + 3l )+ n,n 24( 1+2n)(a+2l) io (29) 1+2n 48 cos( l) 6 cos(( 1 + 2n)l) 2 2 S = f 16a(r n) ( 1 + r + n) cos(n) + p( 1 + 2n) 2 2 n,r 4( r+n) ( 1+r+n) ( 1+2n)(a+2l) 2 2 2 2 2 (30) [ (1 2r) ( 1 2r + 2r + 6n 6n )(1 + a) + 8(r n) ( 1 + r + n) cos( (1 2n)l)+ 2 2 2 (1 2n) ( ( 1 + r + n) cos((r n)l) (r n) cos(( 1 + r + n)l))]g , n 6= r 1 1 T = S (31) n,r r,n b 2 1 2 2 Q = (a + 3l )(1 2n) 6(1 + a) + 6 cos[( 1 + 2n)l]g+ n,n 12(a+2l) (32) g a a 2ag q 2 y y 2n 1 (3 6n 8 cos(n)) (1 cos( l)) a+2l 2 ( 1+2n)(a+2l) Appl. Sci. 2023, 13, 5830 9 of 27 ( 1+2r)( 1+2n)b 2 Q = f( 1 + 2r)( 1 + 2n)(1 + a) ( 1 + r + n) cos((r n)l)+ 2 2 n,r 2(r n) ( 1+r+n) (a+2l) (33) ( r + n) cos[( 1 + r + n)l]g 2 2 a a 2ag 4 cos(r) 4 cos(n) P y y 2n 1 2r 1 (2 ) (1 cos( l))(1 cos( l)), n 6= r (a+2l) (1 2r) (1 2n) a+2l 2 2 Equation (21) can be written as the following matrix equation. " #( ) ( ) 0 1 0 1 f f R R M S S M A 0 0 0 = (34) 0 1 0 1 f f B 0 T T M Q Q M 0 0 Equation (34) is derived from Equation (1) through integration. In the integration process, the buckling deformation functions Equations (10) and (11) are substituted into Equation (1). Since Equations (1), (10) and (11) meet the boundary conditions of cantilever beams, Equation (34) is applicable to the calculation of the M of cantilever beams. cr For convenience of writing, the following matrices can be defined. 0 0 R S D = (35) 0 0 T Q 1 1 R S D = (36) 1 1 T Q It can be seen from Equation (34) that the solution of the dimensionless critical moment (M ) is the smallest positive real number of the generalized eigenvalues of the matrices D 0 0 and D . The matrix formed by the generalized coordinates is the generalized eigen- vector corresponding to the smallest positive real number of the generalized eigenvalues of the matrices D and D . Our MATLAB program can be compiled to solve M and . 0 1 cr The flowchart given in Figure 3 elaborates on the idea of the procedure. Appl. Sci. 2023, 13, x FOR PEER REVIEW 10 of 28 Appl. Sci. 2023, 13, 5830 10 of 27 Figure 3. Flowchart of the procedure. Figure 3. Flowchart of the procedure. 3. Verification and Analysis of Matrix Equation 3. Verification and Analysis of Matrix Equation 3.1. Verification of the Results and Discussion 3.1. Verification of the Results and Discussion The proposed matrix equation is verified according to the experimental data [10,11,18] and finite element analysis data [16] of cantilever beams in existing literature. The dimensions The proposed matrix equation is verified according to the experimental data [10,11,18] of specimens are shown in Figure 4 and Table 1. and finite element analysis data [16] of cantilever beams in existing literature. The dimen- sions of specimens are shown in Figure 4 and Table 1. Appl. Sci. 2023, 13, x FOR PEER REVIEW 11 of 28 Appl. Sci. 2023, 13, 5830 11 of 27 b1 tw b2 Figure Figure 4. 4. Symbolic Symbolic de description scription of of cr cross- oss-sectional sectional dim dimensions. ensions. Table 1. Dimensions of specimens. Table 1. Dimensions of specimens. b1 t1 b2 t2 h tw E G b t b t h t E G Reference Section 1 1 2 2 w 2 2 Reference Section 2 2 (mm) (mm) (mm) (mm) (mm) (mm) (N/mm) (N/mm ) (mm) (mm) (mm) (mm) (mm) (mm) (N/mm ) (N/mm ) 1 31.52 3.13 31.52 3.13 72.44 2.19 69,400 26,000 1 31.52 3.13 31.52 3.13 72.44 2.19 69,400 26,000 [10] 2 31.52 3.13 15.88 3.13 72.44 2.19 69,400 26,000 [10] 2 31.52 3.13 15.88 3.13 72.44 2.19 69,400 26,000 3 31.47 3.14 31.45 1.15 71.44 2.19 69,400 26,000 3 31.47 3.14 31.45 1.15 71.44 2.19 69,400 26,000 1 31.75 3.2 15.63 3.16 72.89 2.87 65,700 24,500 1 31.75 3.2 15.63 3.16 72.89 2.87 65,700 24,500 2 31.67 2.18 15.63 3.118 72.711 2.25 64,800 24,200 2 31.67 2.18 15.63 3.118 72.711 2.25 64,800 24,200 [11] [11] 3 31.68 3.17 16.06 3.19 72.86 2.90 65,400 24,400 3 31.68 3.17 16.06 3.19 72.86 2.90 65,400 24,400 4 31.75 3.21 15.75 3.19 72.75 2.89 65,400 24,400 4 31.75 3.21 15.75 3.19 72.75 2.89 65,400 24,400 [18] 31.4 3.1 31.4 3.1 72.60 2.2 69,400 26,000 [18] 31.4 3.1 31.4 3.1 72.60 2.2 69,400 26,000 Section I 82 7.4 82 7.4 152.6 5 200,000 76,923 Section I 82 7.4 82 7.4 152.6 5 200,000 76,923 Section II-reinforced Section II-reinforced top82 7.4 41 7.4 152.6 5 200,000 76,923 top flange [16] 82 7.4 41 7.4 152.6 5 200,000 76,923 [16] flange Section II-reinforced 41 7.4 82 7.4 152.6 5 200,000 76,923 Section II-reinforced bot- bottom flange 41 7.4 82 7.4 152.6 5 200,000 76,923 tom flange Because it is difficult to obtain the LTB load of a cantilever beam under the uniformly Because it is difficult to obtain the LTB load of a cantilever beam under the uniformly distributed load and other complex loads, most lateral–torsional buckling tests applied distributed load and other complex loads, most lateral–torsional buckling tests applied concentrated load at the free end of the cantilever beam. In order to verify the accuracy of concentrated load at the free end of the cantilever beam. In order to verify the accuracy of the method in this paper and the applicability of the total potential energy equation based the method in this paper and the applicability of the total potential energy equation based on the experimental data, it is necessary to make a = 0 and l = 1. At this time, the load on the experimental data, it is necessary to make α = 0 and λ = 1. At this time, the load case case of the cantilever beam became a concentrated load at the free end. The experimental of the cantilever beam became a concentrated load at the free end. The experimental (Mex) (M ) [10,11,18] and theoretically predicted (M ) critical moment and the M /M ratio ex ex th th [10,11,18] and theoretically predicted (Mth) critical moment and the Mth/Mex ratio are given are given in Table 2 for all the selected specimens. The specimens in reference [18] are not in Table 2 for all the selected specimens. The specimens in reference [18] are not numbered, numbered, so the vertical position of the concentrated load is used instead of the specimen so the vertical position of the concentrated load is used instead of the specimen numbers. numbers. At the same time, as the experimental data can only verify the load case of CL, it At the same time, as the experimental data can only verify the load case of CL, it is neces- is necessary to extract the data of finite element analysis from existing literature to verify sary to extract the data of finite element analysis from existing literature to verify the load the load case of CUDL. The critical moment calculated by finite element analysis (M ) FEM case of CUDL. The critical moment calculated by finite element analysis (MFEM) from ref- from reference [16] and theoretically predicted (M ) critical moment and the M /M th th FEM erence [16] and theoretically predicted (Mth) critical moment and the Mth/MFEM ratio are ratio are given in Tables 3–5 for all the selected specimens. given in Tables 3–5 for all the selected specimens. In Table 2, compared with the experimental results, the maximum value of the Mth/Mex ratio is 1.04, and the minimum value is 0.92. In Tables 3–5, there are four sets of t1 t2 Appl. Sci. 2023, 13, 5830 12 of 27 Table 2. Expermental (M ) and theoretically predicted (M ) critical moment. ex th Specimen M (Nm) M (Nm) M /M Specimen M (Nm) M (Nm) M /M ex th th ex ex th th ex Anderson et al. 1972 [10] Attard et al. 1990 [11] 1Aa65 438.44 405.1 0.92 1TA 253.45 250.11 0.99 1Ac65 533.91 528.61 0.99 1BA 292.30 293.51 1.00 1Aa50 515.21 504.45 0.98 1TB 305.25 295.09 0.97 1Ac50 758.13 751.23 0.99 1BB 408.85 404.95 0.99 2Aa65 277.42 264.08 0.95 2TA 227.70 225.59 0.99 2Ab65 421.27 413.60 0.98 2BA 272.25 275.70 1.01 2Ba65 240.73 227.94 0.95 2TB 282.15 263.23 0.93 2Bb65 276.69 279.43 1.01 2BB 417.45 415.85 1.00 2Aa50 320.66 321.08 1.00 3TA 326.25 316.58 0.97 2Ab50 594.47 608.05 1.02 3BA 384.25 386.32 1.01 2Ba50 287.92 287.86 1.00 3TB 382.80 368.65 0.96 2Bb50 365.82 373.09 1.02 3BB 558.25 567.67 1.02 3Aa65 261.45 259.75 0.99 4TA 380.00 361.91 0.95 3Ab65 433.75 438.04 1.01 4BA 447.50 452.43 1.01 3Ba65 281.08 274.65 0.98 4TB 428.75 416.80 0.97 3Bb65 350.80 352.71 1.01 4BB 690.00 708.46 1.03 3Aa50 347.76 336.10 0.97 Ings et al. 1987 [18] 3Ab50 630.03 656.93 1.04 TF 234.19 236.32 1.01 3Ba50 344.37 326.52 0.95 BF 567.39 561.30 0.99 3Bb50 514.87 507.76 0.99 Table 3. The critical moment predicted by finite element analysis(M ) and theoretically predicted FEM (M ) critical moment for Section I. th CL CUDL l (m) M (kNm) M (kNm) M /M M (kNm) M (kNm) M /M FEM th th FEM FEM th th FEM TF 38.31 41.18 1.07 46.37 49.27 1.06 SC 98.06 99.04 1.01 118.50 120.40 1.02 1.5 BF 143.09 141.38 0.99 181.10 179.36 0.99 TF 31.34 32.94 1.05 37.08 39.10 1.05 2 SC 65.28 64.04 0.98 77.88 77.40 0.99 BF 87.48 84.61 0.97 108.96 106.63 0.98 TF 23.22 23.90 1.03 27.00 28.01 1.04 3 SC 36.06 35.65 0.99 42.12 42.73 1.01 BF 43.98 43.15 0.98 54.27 53.66 0.99 TF 18.20 18.51 1.02 21.12 21.56 1.02 SC 24.32 24.13 0.99 28.56 28.76 1.01 BF 28.28 27.88 0.99 34.56 34.27 0.99 Table 4. The critical moment predicted by finite element analysis (M ) and theoretically predicted FEM (M ) critical moment for Section II—reinforced top flange. th CL CUDL l (m) M (kNm) M (kNm) M /M M (kNm) M (kNm) M /M FEM FEM FEM FEM th th th th TF 23.75 24.76 1.04 27.44 28.61 1.04 1.5 SC 98.06 27.84 0.28 32.06 32.36 1.01 BF 40.10 40.81 1.02 51.17 52.18 1.02 TF 18.76 19.26 1.03 21.48 22.11 1.03 SC 65.28 20.86 0.32 24.00 24.12 1.01 BF 27.50 27.84 1.01 34.50 34.96 1.01 TF 13.11 13.36 1.02 14.85 15.29 1.03 SC 36.06 13.99 0.39 16.07 16.12 1.00 BF 16.95 17.13 1.01 20.66 20.97 1.02 TF 10.12 10.26 1.01 11.52 11.74 1.02 SC 24.32 10.60 0.44 12.24 12.20 1.00 BF 12.36 12.45 1.01 14.88 15.01 1.01 Appl. Sci. 2023, 13, 5830 13 of 27 Table 5. The critical moment predicted by finite element analysis (M ) and theoretically predicted FEM (M ) critical moment for Section II—reinforced bottom flange. th CL CUDL l (m) M (kNm) M (kNm) M /M M (kNm) M (kNm) M /M FEM th th FEM FEM th th FEM TF 19.25 20.23 1.05 23.12 24.20 1.05 SC 83.94 83.77 1.00 103.07 103.81 1.01 1.5 BF 91.07 89.48 0.98 114.41 111.96 0.98 TF 17.34 18.86 1.09 20.58 22.30 1.08 SC 52.04 51.50 0.99 63.72 63.37 0.99 BF 55.64 54.38 0.98 68.70 67.51 0.98 TF 14.19 14.89 1.05 16.61 17.40 1.05 SC 27.54 27.25 0.99 33.48 33.14 0.99 BF 28.89 28.33 0.98 35.24 34.72 0.99 TF 11.64 11.95 1.03 13.44 13.89 1.03 SC 18.12 17.95 0.99 21.84 21.66 0.99 BF 18.72 18.50 0.99 22.80 22.45 0.98 In Table 2, compared with the experimental results, the maximum value of the M /M ratio is 1.04, and the minimum value is 0.92. In Tables 3–5, there are four sets of ex th data with significant differences. These data came from SC under CL in Table 4. In refer- ence [16], the distance between the SC and TF of the section II with a reinforced top flange is very small and the differences between their M should not be significant. However, the cr differences in Table 4 are particularly significant and it can be concluded from reference [16] that M is more accurate than M . Therefore, these four sets of data are not included in th FEM the analysis scope. In Tables 3–5, compared with the data of finite element analysis from reference [16], the maximum value of the M /M ratio is 1.07, and the minimum value FEM th is 0.97. The average of all the ratios is 0.98 with a standard deviation of 0.13. Therefore, critical moment can be accurately solved by the numerical method proposed in this paper. The predictions of the matrix equation and the data from existing literature are com- pared in Figure 5. The comparison method in Figure 5 uses the modified version of demerit point classification (DPC) [40]. DPC is used to evaluate the predictions of the matrix equation in terms of safety, accuracy and economic aspects. A specific explanation of the modified version of DPC can be found in Table 6. According to Table 6 and Figure 5, a penalty is assigned to each range of the M /M or M /M ratio. The predictions ex th th FEM based on Equation (34) are all safe in Figure 5. To sum up, the predictions of 100% of the Appl. Sci. 2023, 13, x FOR PEER REVIEW 14 of 28 specimens are within the appropriate safety range (total penalty = 0). Therefore, it can be concluded that Equation (34) resulted in a positive prediction in terms of safety, accuracy and economic aspects. 2.0 2.0 1.8 1.8 1.6 1.6 D = 0 D = 0 1.4 1.4 1.2 1.2 1.176 1.176 AS = 68 AS = 38 1.0 1.0 0.869 0.869 0.8 0.8 C = 0 C = 0 0.6 0.6 0.5 0.5 0.4 0.4 Total penalty = 0 Total penalty = 0 EC = 0 EC = 0 0.2 0.2 0.0 0.0 0 4 8 1216 20 242832 3640 0 102030405060 70 N N (a) (b) Figure 5. Prediction capability of Equation (34) based on modified DPC [40]: (a) comparison with Figure 5. Prediction capability of Equation (34) based on modified DPC [40]: (a) comparison with the the experimental data from reference [10,11,18]; (b) comparison with the data of finite element anal- experimental data from reference [10,11,18]; (b) comparison with the data of finite element analysis ysis from reference [16]. (D: dangerous; AS: appropriate safety; C: conservative; EC: extra conserva- from reference [16]. (D: dangerous; AS: appropriate safety; C: conservative; EC: extra conservative). tive). Table 6. Modified version of the Demerit Points Classification (DPC) criteria. Mth/Mex or Mth/MFEM Classification Penalty (PEN) >2 Extra dangerous 10 [1.176–2] Dangerous 5 [0.869–1.176] Appropriate safety 0 [0.5–0.869] Conservative 1 ≤0.5 Extra conservative 2 In addition, the impact of the number of generalized coordinates (n) on the accuracy of calculation results is studied. The specimens of section II in Table 1 are selected and their beam lengths are 4 m. The specific results are shown in Table 7 and Figure 6. Accord- ing to Table 7 and Figure 6, the maximum error between Mth and MFEM is only 3.61%, and calculation accuracy is high when n is increased to the convergence term. The maximum error between Mth and MFEM is only 3.96% when n is 10, which is within the allowable error range, and is only 0.35% higher than that corresponding to the convergence term. When n is equal to 1 or 2, the error between Mth and MFEM is greater than 17%. Table 7. Comparison of predicted critical moment corresponding to different n values and MFEM. Load Cases: CUDL, Section: II—Reinforced Top Flange n Mth/kN·m Error TF SC BF TF SC BF 1 30.47 35.29 116.13 164.52% 188.35% 680.47% 2 14.06 14.81 20.18 22.03% 20.97% 35.62% 5 11.89 12.35 15.20 3.22% 0.92% 2.16% 10 11.78 12.24 15.05 2.28% 0.02% 1.13% Convergence term 11.77 12.22 15.03 2.16% 0.14% 0.12% MFEM/kN·m 11.52 12.24 14.88 —— —— —— Load Cases: CUDL; Section: Section: II—Reinforced Bottom Flange n Mth/kN·m Error TF SC BF TF SC BF 1 22.55 74.21 85.95 67.81% 239.79% 276.98% 2 15.75 26.99 28.265 17.19% 23.58% 23.97% 5 14.19 22.29 23.13 5.58% 2.08% 1.44% 10 13.97 21.80 22.60 3.96% 0.19% 0.88% M /M th ex M /M th FEM Appl. Sci. 2023, 13, 5830 14 of 27 Table 6. Modified version of the Demerit Points Classification (DPC) criteria. M /M or M /M Classification Penalty (PEN) th ex th FEM >2 Extra dangerous 10 [1.176–2] Dangerous 5 [0.869–1.176] Appropriate safety 0 [0.5–0.869] Conservative 1 0.5 Extra conservative 2 In addition, the impact of the number of generalized coordinates (n) on the accuracy of calculation results is studied. The specimens of section II in Table 1 are selected and their beam lengths are 4 m. The specific results are shown in Table 7 and Figure 6. According to Table 7 and Figure 6, the maximum error between M and M is only 3.61%, and th FEM calculation accuracy is high when n is increased to the convergence term. The maximum error between M and M is only 3.96% when n is 10, which is within the allowable th FEM error range, and is only 0.35% higher than that corresponding to the convergence term. When n is equal to 1 or 2, the error between M and M is greater than 17%. th FEM Table 7. Comparison of predicted critical moment corresponding to different n values and M . FEM Load Cases: CUDL, Section: II—Reinforced Top Flange n M /kNm Error th TF SC BF TF SC BF 1 30.47 35.29 116.13 164.52% 188.35% 680.47% 2 14.06 14.81 20.18 22.03% 20.97% 35.62% 5 11.89 12.35 15.20 3.22% 0.92% 2.16% 10 11.78 12.24 15.05 2.28% 0.02% 1.13% Convergence 11.77 12.22 15.03 2.16% 0.14% 0.12% term M /kNm 11.52 12.24 14.88 —— —— —— FEM Load Cases: CUDL; Section: Section: II—Reinforced Bottom Flange M /kNm Error th TF SC BF TF SC BF 1 22.55 74.21 85.95 67.81% 239.79% 276.98% 2 15.75 26.99 28.265 17.19% 23.58% 23.97% 5 14.19 22.29 23.13 5.58% 2.08% 1.44% 10 13.97 21.80 22.60 3.96% 0.19% 0.88% Convergence 13.93 21.69 22.48 3.61% 0.71% 1.41% term M /kNm 13.44 21.84 22.80 —— —— —— FEM M M FE M th Note: Error = . Convergence term refers to the value of n when the M unchang as n increases. th FE M Through the comparative analysis of finite elements, it can be seen that the critical moment of cantilever beam under the load cases studied in this paper can be calculated within the allowable error range when n is greater than or equal to 10, and when n is equal to 1 or 2, the error is too large to calculate. Therefore, the program in Figure 3 needs to be modified. The value of n in this program can directly take any value greater than 10, such as 100, without the need for a loop statement. It can greatly improve the efficiency of the procedure in Figure 3. In addition, the procedure of this paper refences the principles of GBTUL software in reference [48]. The specific content of the code can be found in Appendix A. At the same time, a user graphical interface is developed for users. 3.2. Comparative Analysis between Matrix Equation and Equations Proposed in Other References In past studies, many scholars have also proposed the equation of the M of cantilever cr beam, such as references [12,17]. Appl. Sci. 2023, 13, 5830 15 of 27 Appl. Sci. 2023, 13, x FOR PEER REVIEW 15 of 28 The method of reference [12] is the Rayleigh–Ritz method. It used a set of non- dimensional parameters and deduced the governing equations of cantilever beams. Ref- Convergence term 13.93 21.69 22.48 3.61% 0.71% 1.41% erence [12] solved the governing equation by discretizing the beam model. However, MFEM/kN·m 13.44 21.84 22.80 —— —— —— reference [12] only describes its method and did not derive the explicit expression of the Note: Error = . Convergence term refers to the value of n when the Mth unchang as n in- discrete matrix equation. Because the discrete matrix had too many physical quantities and creases. was different from the expression in this paper, it will not be introduced in this paper. See Section 3 in reference [12] for details. 10 TF TF SC SC 4 BF 8 BF 4 2 01 23 456 78 9 10 11 012345 6789 10 11 (a) (b) Figure 6. Comparison of predicted critical moment corresponding to different n values and MFEM: Figure 6. Comparison of predicted critical moment corresponding to different n values and M : FEM (a) comparison of cantilevers with section II with reinforced top flange; (b) comparison of cantilevers (a) comparison of cantilevers with section II with reinforced top flange; (b) comparison of cantilevers with section II with reinforced bott om flange. with section II with reinforced bottom flange. Through t The method he com used parat in refer ive ence analy [17 sis o ] is fthe finite elements, it can be see finite difference approach. Itn established that the critical the governing equation of cantilever beams using the Galerkin method. The finite difference moment of cantilever beam under the load cases studied in this paper can be calculated method was used to discretize the beam model and solve M . cr within the allowable error range when n is greater than or equal to 10, and when n is equal In order to compare the advantages and disadvantages of the equations in reference [12,17] to 1 or 2, the error is too large to calculate. Therefore, the program in Figure 3 needs to be and this paper, section I in Table 1 is selected for calculating M . The lengths of the can- cr modified. The value of n in this program can directly take any value greater than 10, such tilever is 1.5, 2, 3 and 4 m, respectively, and the other properties of the beam are shown as 100, without the need for a loop statement. It can greatly improve the efficiency of the in Table 1. The concentrated load or uniformly distributed load is applied at the TF, SC procedure in Figure 3. In addition, the procedure of this paper refences the principles of and BF of the beam. The specific calculation results are shown in Figure 7. In Figure 7, the GBTUL software in reference [48]. The specific content of the code can be found in Appen- horizontal coordinates represent different load cases. When the numbers are 1, 2, 3 and 4, dix A. At the same time, a user graphical interface is developed for users. the position of the load is TF and the lengths of the beams are 1.5 m, 2 m, 3 m, and 4 m, respectively. When the numbers are 5, 6, 7 and 8, the position of the load is SC and the 3.2. Comparative Analysis between Matrix Equation and Equations Proposed in lengths of the beams are 1.5 m, 2 m, 3 m, and 4 m, respectively. When the numbers are 9, Other Refere 10, 11, and nc12, es the position of the load is BF and the lengths of the beams are 1.5 m, 2 m, 3 m, and 4 m, respectively. In past studies, many scholars have also proposed the equation of the Mcr of cantile- It can be seen from Figure 7 that the calculation results of Equation (34) are similar to ver beam, such as references [12,17]. those of reference [12], but are slightly different from those of reference [17]. Through com- The method of reference [12] is the Rayleigh–Ritz method. It used a set of non-dimen- parison and analysis, it can be concluded that Equation (34) in this paper has the following sional parameters and deduced the governing equations of cantilever beams. Reference advantages: (i) Equation (34) has higher accuracy than reference [17]; (ii) Equation (34) can [12] solved the governing equation by discretizing the beam model. However, reference calculate M under CUDL, CL or uniformly distributed load, while the equation proposed cr [12] only describes its method and did not derive the explicit expression of the discrete in reference [12] can only calculate M under a single load and the equation proposed in cr matrix equation. Because the discrete matrix had too many physical quantities and was reference [17] can only calculate M under CL and CUDL when the concentrated load is cr difflimited erent fro to m the the expression free end; (iii) Equation in this p (34) aper, it considers will the not ratio be of int urniformly oduced in distributed this paper. See load Sect to ion 3 in concentrated reference load, [1the 2] for symmetry details. of the section, the action position of the concentrated load in the direction of the beam length and the vertical position of two loads. It considers The method used in reference [17] is the finite difference approach. It established the very comprehensive factors and can calculate M in all cases. The specific comparison cr governing equation of cantilever beams using the Galerkin method. The finite difference results can be seen in Tables 8–11. In Tables 8–11, “ ” indicates that it can be calculated method was used to discretize the beam model and solve Mcr. and “” indicates that it cannot be calculated. In order to compare the advantages and disadvantages of the equations in reference [12,17] and this paper, section I in Table 1 is selected for calculating Mcr. The lengths of the cantilever is 1.5, 2, 3 and 4 m, respectively, and the other properties of the beam are shown in Table 1. The concentrated load or uniformly distributed load is applied at the TF, SC and BF of the beam. The specific calculation results are shown in Figure 7. In Figure 7, the horizontal coordinates represent different load cases. When the numbers are 1, 2, 3 and 4, M /M th FEM M /M th FEM Appl. Sci. 2023, 13, x FOR PEER REVIEW 16 of 28 the position of the load is TF and the lengths of the beams are 1.5 m, 2 m, 3 m, and 4 m, respectively. When the numbers are 5, 6, 7 and 8, the position of the load is SC and the lengths of the beams are 1.5 m, 2 m, 3 m, and 4 m, respectively. When the numbers are 9, 10, 11, and 12, the position of the load is BF and the lengths of the beams are 1.5 m, 2 m, 3 m, and 4 m, respectively. It can be seen from Figure 7 that the calculation results of Equation (34) are similar to those of reference [12], but are slightly different from those of reference [17]. Through comparison and analysis, it can be concluded that Equation (34) in this paper has the fol- lowing advantages: (i) Equation (34) has higher accuracy than reference [17]; (ii) Equation (34) can calculate Mcr under CUDL, CL or uniformly distributed load, while the equation proposed in reference [12] can only calculate Mcr under a single load and the equation proposed in reference [17] can only calculate Mcr under CL and CUDL when the concen- trated load is limited to the free end; (iii) Equation (34) considers the ratio of uniformly distributed load to concentrated load, the symmetry of the section, the action position of the concentrated load in the direction of the beam length and the vertical position of two loads. It considers very comprehensive factors and can calculate Mcr in all cases. The spe- cific comparison results can be seen in Tables 8–11. In Tables 8–11, “√” indicates that it can Appl. Sci. 2023, 13, 5830 be calculated and “×” indicates that it cannot be calculated. 16 of 27 200 Andrade+2007 Andrade+2007 Ozbasaran+2013 Ozbasaran+2013 Eq.(34) in this paper Eq.(34) in this paper 160 350 0 0 0 1234 56789 10 11 12 13 012 3456789 10 11 12 13 number number (a) (b) Figure 7. Comparison of calculation results of three equations(ʺAndrade+2007” is reference [12] and Figure 7. Comparison of calculation results of three equations("Andrade+2007” is reference [12] and “Ozbasaran+2013” is reference [17]): (a) calculation results under concentrated load at the free end; “Ozbasaran+2013” is reference [17]): (a) calculation results under concentrated load at the free end; (b) calculation results under uniformly distributed load. (b) calculation results under uniformly distributed load. Table 8. Comparison of the ratio of uniformly distributed load to concentrated load. Table 8. Comparison of the ratio of uniformly distributed load to concentrated load. Ratio of Uniformly Distributed Load Equation Reference [12] Reference [17] Ratio of Uniformly Distributed Load to Concentrated Load Equation (3(34) 4) Reference [12] Reference [17] to Concentrated Load α = 1 √ × √ p p a = 1 Α ≠ 1 √ × × A 6= 1 Table 9. Comparison of cross-sectional symmetry. Table 9. Comparison of cross-sectional symmetry. Equation TheThe S Symmetry ymmetry of of the Section the Section Equation (34) Reference Reference [1 [12]2] Reference Reference [1 [17] 7] (34) p p p doubly symmetric doubly symmetric √ √ √ p p singly symmetric singly symmetric √ √ × Table 10. Comparison of applicable load cases. Load Cases Equation (34) Reference [12] Reference [17] p p p CL (l = 1) CL (l < 1) p p CUDL (l = 1) CUDL (l < 1) Table 11. Comparison of applicable vertical position of all loads. The Vertical Equation (34) Reference [12] Reference [17] Position of All Loads p p TF p p p SC p p BF Other vertical position 4. Deduction of the Closed-Form Solution of M cr 4.1. Analysis of the Reason Why M Cannot Be Expressed Accurately by Analytic Expression cr When n is greater than 1, Equation (34) can be transformed into a 2n degree equa- tion about M , which cannot be easily solved by mathematical methods. The following derivation process reveals the fundamental reasons for determining the highest order of M in Equation (34). M /kNm cr M /kNm cr Appl. Sci. 2023, 13, 5830 17 of 27 The following buckling deformation functions are used. h i (2i 1)z u(z) = h A 1 cos 2l i=1 (37) h i (2j 1)z j(z) = B 1 cos 2l j=1 h i A A A . . . A A = 1 2 3 n f g (38) fBg = B B B . . . B 1 2 3 According to Equation (21), a matrix equation of n + r order similar to Equation (34) can still be obtained: " #( ) ( ) 0 1 0 1 f f R R M S S M A 0 0 0 = (39) Appl. Sci. 2023, 13, x FOR PEER REVIEW 18 of 28 0 1 0 1 f f B 0 T T M Q Q M 0 0 0 0 0 0 1 1 1 1 where the physical meaning and expression of R , S , T , Q , R , S , T and Q are the same Therefore, only when r is equal to 1 and n is greater than or equal to r can Equation as before. (40) be a quadratic polynomial approximating 𝑀 , and then 𝑀 can be accurately ex- According to Equations (26) and (28), the coefficient determinant of independent pressed by mathematical analytic expression. parameter A (n = 1, 2, 3 . . . ) and B (n = 1, 2, 3 . . . ) can be expressed concisely as follows: n n Figure 8 shows the twist angle of each point along the z-axis of the beam when λ = 1, 0 1 0 1 0 1 f f f aP = aq and the torsion R R aM l stiffness coe S S Mfficient K fo R r each is 0 S.5M and 1, 2.5, R respect0 ively. y y 0 0 0 = + (40) It can be seen f 0 rom 1 Figure 0 8 tha 1 t the twist b 1 uckling mode 1 of a c1antilever be 0 am is f f f f f T M Q T T M Q Q M T M Q M 0 0 0 0 0 related to the torsional stiffness coefficient K. However, it is difficult to find a twist angle function cont If n is ain gr ing only o eater than ne generalized coor or equal to r, accor dina ding te to uni to Equation formly descri (40) and be the twi the definition st buck- of ling mode of the cantilever beam according to the results of Figure 8 and the existing lit- determinant, that is, the determinant of order n is the algebraic sum of the products of erature n elements . Therefore, only w taken fromh dif en the num ferent rows ber of and terms columns, of the twist angl the operation e rfe usult nction i of Equation s greater (40) than or equal to 10 can the twist angle deformation of the cantilever beam be accurately is a 2r degree polynomial approximating M . If n is less than r, the operation result of described according to the conclusions in Table 7 and Figure 6. However, it has been Equation (40) is a n + r degree polynomial approximating M . proven that the analytical expression of 𝑀 can be obtained accurately only when r equals Therefore, only when r is equal to 1 and n is greater than or equal to r can Equation (40) 1, that is, the number of terms of twist angle function takes as 1. The contradiction formed be a quadratic polynomial approximating M , and then M can be accurately expressed by before and after means that the Mcr of the cantilev 0 er beam cannot be 0 accurately expressed mathematical analytic expression. by the analytical expression. Therefore, this paper must use the numerical calculation Figure 8 shows the twist angle of each point along the z-axis of the beam when l = 1, method to obtain the Mcr’s expression. a = a and the torsional stiffness coefficient K for each is 0.5 and 1, 2.5, respectively. P y q y 1.2 K=2.5 K=1.0 K=0.5 0.9 0.6 0.3 0.0 0.0 0.2 0.4 0.6 0.8 1.0 z/l Figure 8. The torsion angle j varies with the coefficient of K. Figure 8. The torsion angle φ varies with the coefficient of K. It can be seen from Figure 8 that the twist buckling mode of a cantilever beam is related 4.2. Derivation of Closed-Form Solution of Mcr Based on Twist Angle Function to the torsional stiffness coefficient K. However, it is difficult to find a twist angle function With the assumption that torsional rotation is small, the second derivative of u with containing only one generalized coordinate to uniformly describe the twist buckling mode respect to z can be writt en as given below from [9]. of the cantilever beam according to the results of Figure 8 and the existing literature. Therefore, only when the number of terms of the twist angle function is greater than or M ϕ ′′ u =− (41) EI The moment distribution function 𝜅 𝑧 , the twist angle function φ(z) and Mmax are defined as follows. κ() z = (42) ϕϕ ()zB = ()z (43) M = − M (44) max 0 where M0 is the moment at the fixed end of the cantilever beam. The value of M0 is less than 0. φ0(z) is the basic function of twist angle function φ(z) of the cantilever, and B is an undetermined coefficient. According to the conclusions obtained in Table 7 and Figure 6, φ0(z) could use the following formula. ϕ Appl. Sci. 2023, 13, 5830 18 of 27 equal to 10 can the twist angle deformation of the cantilever beam be accurately described according to the conclusions in Table 7 and Figure 6. However, it has been proven that the analytical expression of M can be obtained accurately only when r equals 1, that is, the number of terms of twist angle function takes as 1. The contradiction formed before and after means that the M of the cantilever beam cannot be accurately expressed by the cr analytical expression. Therefore, this paper must use the numerical calculation method to obtain the M ’s expression. cr 4.2. Derivation of Closed-Form Solution of M Based on Twist Angle Function cr With the assumption that torsional rotation is small, the second derivative of u with respect to z can be written as given below from [9]. M j u = (41) E I The moment distribution function k(z), the twist angle function j(z) and M are max defined as follows. k(z) = (42) j(z) = Bj (z) (43) M = M (44) max where M is the moment at the fixed end of the cantilever beam. The value of M is less 0 0 than 0. j (z) is the basic function of twist angle function j(z) of the cantilever, and B is an undetermined coefficient. According to the conclusions obtained in Table 7 and Figure 6, j (z) could use the following formula. (2n 1)z j (z) = B 1 cos (45) 0 n 2l n=1 Some physical quantities are defined as follows. 2 2 r = k (z)j (z)dz (46) 1 0 r = (j (z)) dz (47) 2 0 r = (j (z)) dz (48) 3 0 r = k(z)(j (z)) dz (49) 4 0 r = j (z)dz (50) 5 0 r = j (ll) (51) 6 0 Then Equation(1) can be written as follows. 2 2 M B 1 1 1 1 2 2 2 2 2 P = r + E I B r + G I B r + b B M r + a q B r + a P B r (52) 1 ! 2 t 3 x 0 4 q y 5 P y 6 2E I 2 2 2 2 y Appl. Sci. 2023, 13, 5830 19 of 27 ¶P M ’s expression can be obtained from = 0. cr ¶B q 2 E I y 2r 2r 2r 2r 5 6 5 6 M = E I (2r b a a ) + E I (2r b a a ) + 4r (E I r + G I r ) (53) cr y 4 x q P y 4 x q P 1 ! 2 t 3 y y 2 y 2 y 2r (a + 2l)l (a + 2l)l (a + 2l)l (a + 2l)l When the load case is CUDL, a = 1 and a = a are considered. Some physical y q y quantities are defined as follows. R = 2r (54) 1 1 2r 2r + 2r l 6 5 6 R = or R = (55) 2 2 (a + 2l)l (a + 2l)l R = 2r (56) 3 4 R = r (57) 4 2 R = r (58) 5 3 The expression of M can be written as follows. cr E I M = E I (R a + R b ) + 2R (E I R + G I R ) E I (R a + R b ) (59) cr y 2 3 x 1 ! 4 t 5 y 2 3 x 5. Practical Expressions for Design under CL and CUDL 5.1. Practical Calculation Formula of M cr For the load case of CL, this paper assumes that a = 0 and calculates the M ’s value cr for some common load cases when l is equal to 1/3, 1/2, 2/3 and 1, respectively. For the load case of CUDL, since the vertical position of the concentrated load (a ) P y and the uniform load (a ) on the beam are usually the same under actual load cases, this q y paper assumes that a = 1 and a = a and calculates the M ’s value for some common P y q y cr load cases when l is equal to 1/3, 1/2, 2/3 and 1, respectively. It can be seen from Figure 8 that the twist angle mode of the cantilever beam during lateral–torsional buckling is mainly related to the torsional stiffness coefficient K. Since most of the commonly used values of K are between 0.3 and 2.5, this paper takes 15 different values of K between 0.3 and 2.5 and calculates B by substituting relevant parameters into the MATLAB program compiled according to Equation (34) so as to obtain the specific expression of j (z) in Equation (45). Then the specific values of R ~R can be obtained 0 1 5 as shown in Tables 12 and 13. According to reference [49], Wagner ’s coefficient has only a slight impact on the buckling mode. Therefore, the conclusions in Tables 12 and 13 are applicable to both uniaxial symmetric and biaxial symmetric I-beams. Corresponding R ~R ’s values can be obtained by interpolation when K has other values. Therefore, the 1 5 M under CL or CUDL can be solved through Tables 12 and 13. cr 5.2. Example Verification Section I, Section II that reinforced bottom flange and their properties in Table 1 are selected for calculating M . The length of cantilever is 4 m. It is assumed that the material cr is elastic and the component is free from defects. M is obtained according to Equation (59) cr and the finite element analysis respectively when l takes different values. Abaqus software is used to generate finite element models of the cantilever beams. The Finite Element Method (FEM) is applied to this software. The models of cantilever beams were generated in the CAE module and calculations were carried out in the STANDARD module [50]. Cantilever beams are divided to 13,000 finite elements. S8R5 shell elements are used in this model and results are obtained by eigenvalue. In order to keep the original shape of the section and better conform to the rigid perimeter assumption, one stiffener Appl. Sci. 2023, 13, 5830 20 of 27 is set for each l/6 in the length direction and a total of 5 stiffeners are set. In order not to create unnecessary constraints on other deformations of the cantilever beams, the stiffener is only rigidly connected with the web, and only couples with the in-plane displacement of top and bottom flanges. Tables 14 and 15 show that the results obtained by Equation (59) and Abaqus are nearly identical at SC. For TF and BF, difference between Abaqus results and Equation (59) results increases as l decreases. Andrade and Camotim [13] explained this phenomenon. They verified that 1D frame element assumption loses its validity as the beam length decreases. And the situation of l decreases is similar to this situation. Table 12. Specific values of R , R , R , R and R under CL. 1 2 3 4 5 l = 1/3 l = 1/2 K R R R R R R R R R R 1 2 3 4 5 1 2 3 4 5 2.5 0.0062 0.2548 0.2257 1.7400 0.7071 0.0191 0.3306 0.2985 1.0582 0.6602 2.22 0.0075 0.2350 0.2106 1.3099 0.6024 0.0244 0.3218 0.2955 0.8523 0.6072 1.81 0.0087 0.1738 0.1594 0.6922 0.3828 0.0302 0.2486 0.2364 0.4852 0.4209 1.57 0.0096 0.1394 0.1304 0.4436 0.2752 0.0326 0.1925 0.1885 0.3079 0.3011 1.4 0.0103 0.1161 0.1105 0.3097 0.2101 0.0346 0.1553 0.1562 0.2133 0.2284 1.19 0.0115 0.0888 0.0869 0.1856 0.1433 0.0374 0.1133 0.1188 0.1265 0.1539 1 0.0127 0.0658 0.0665 0.1067 0.0958 0.0403 0.0796 0.0878 0.0718 0.1011 0.84 0.0140 0.0480 0.0504 0.0608 0.0644 0.0429 0.0550 0.0640 0.0403 0.0668 0.7 0.0152 0.0339 0.0372 0.0335 0.0428 0.0452 0.0366 0.0454 0.0218 0.0435 0.63 0.0159 0.0274 0.0310 0.0236 0.0338 0.0462 0.0286 0.0370 0.0151 0.0340 0.57 0.0165 0.0222 0.0259 0.0168 0.0269 0.0469 0.0225 0.0302 0.0107 0.0268 0.5 0.0172 0.0167 0.0204 0.0108 0.0200 0.0474 0.0162 0.0231 0.0067 0.0196 0.44 0.0177 0.0125 0.0159 0.0069 0.0149 0.0475 0.0116 0.0176 0.0042 0.0144 0.36 0.0182 0.0077 0.0107 0.0034 0.0093 0.0466 0.0068 0.0112 0.0020 0.0088 0.30 0.0183 0.0049 0.0073 0.0018 0.0060 0.0449 0.0041 0.0074 0.0010 0.0055 l = 2/3 l = 1 K R R R R R R R R R R 1 2 3 4 5 1 2 3 4 5 2.5 0.0417 0.3939 0.3601 0.7324 0.6235 0.1266 0.5289 0.4805 0.4365 0.5743 2.22 0.0520 0.3722 0.3474 0.5783 0.5686 0.1516 0.4780 0.4439 0.3313 0.5169 1.81 0.0778 0.3426 0.3343 0.4018 0.4888 0.2123 0.4055 0.3959 0.2160 0.4362 1.57 0.0826 0.2563 0.2598 0.2520 0.3480 0.2749 0.3647 0.3728 0.1667 0.3921 1.4 0.0861 0.2007 0.2107 0.1730 0.2624 0.3434 0.3367 0.3599 0.1386 0.3631 1.19 0.0907 0.1399 0.1553 0.1014 0.1747 0.3978 0.2511 0.2891 0.0913 0.2741 1 0.0949 0.0935 0.1109 0.0568 0.1129 0.3909 0.1522 0.1925 0.0496 0.1706 0.84 0.0980 0.0614 0.0783 0.0315 0.0732 0.3795 0.0906 0.1271 0.0267 0.1058 0.7 0.0998 0.0388 0.0537 0.0167 0.0467 0.3629 0.0521 0.0818 0.0137 0.0644 0.63 0.1001 0.0295 0.0429 0.0115 0.0360 0.3513 0.0377 0.0632 0.0093 0.0484 0.57 0.0998 0.0227 0.0345 0.0080 0.0281 0.3390 0.0277 0.0493 0.0064 0.0369 0.50 0.0985 0.0159 0.0257 0.0050 0.0203 0.3211 0.0185 0.0354 0.0038 0.0258 0.44 0.0963 0.0111 0.0191 0.0031 0.0147 0.3020 0.0125 0.0255 0.0023 0.0182 0.36 0.0911 0.0063 0.0119 0.0014 0.0087 0.2700 0.0067 0.0150 0.0010 0.0104 0.30 0.0849 0.0037 0.0076 0.0007 0.0054 0.2403 0.0038 0.0092 0.0005 0.0062 Table 13. Specific values of R , R , R , R and R under CUDL. 1 2 3 4 5 l = 1/3 l = 1/2 K R R R R R R R R R R 1 2 3 4 5 1 2 3 4 5 2.5 0.0168 0.2722 0.2426 0.6697 0.6114 0.0244 0.3336 0.3024 0.7721 0.6263 2.22 0.0208 0.2570 0.2333 0.5279 0.5562 0.0305 0.3168 0.2929 0.6132 0.5718 1.81 0.0282 0.2148 0.2028 0.3323 0.4314 0.0384 0.2453 0.2363 0.3581 0.4096 1.57 0.0298 0.1611 0.1573 0.2085 0.3053 0.0411 0.1852 0.1849 0.2263 0.2919 1.4 0.0310 0.1268 0.1275 0.1434 0.2289 0.0432 0.1465 0.1509 0.1564 0.2203 1.19 0.0326 0.0893 0.0941 0.0844 0.1509 0.0460 0.1038 0.1125 0.0926 0.1470 1 0.0340 0.0607 0.0676 0.0476 0.0965 0.0487 0.0709 0.0814 0.0525 0.0954 0.84 0.0352 0.0409 0.0481 0.0267 0.0619 0.0511 0.0478 0.0584 0.0295 0.0622 Appl. Sci. 2023, 13, 5830 21 of 27 Table 13. Cont. l = 1/3 l = 1/2 K R R R R R R R R R R 1 2 3 4 5 1 2 3 4 5 0.7 0.0360 0.0268 0.0335 0.0143 0.0391 0.0529 0.0312 0.0408 0.0159 0.0399 0.63 0.0363 0.0209 0.0270 0.0100 0.0301 0.0536 0.0241 0.0329 0.0110 0.0309 0.57 0.0363 0.0165 0.0219 0.0070 0.0234 0.0539 0.0189 0.0267 0.0078 0.0243 0.5 0.0361 0.0120 0.0166 0.0044 0.0168 0.0539 0.0135 0.0202 0.0049 0.0176 0.44 0.0357 0.0087 0.0126 0.0028 0.0122 0.0534 0.0097 0.0153 0.0031 0.0128 0.36 0.0343 0.0052 0.0080 0.0013 0.0073 0.0516 0.0056 0.0097 0.0015 0.0077 0.30 0.0326 0.0032 0.0052 0.0007 0.0045 0.0491 0.0034 0.0063 0.0007 0.0049 l = 2/3 l = 1 K R R R R R R R R R R 1 2 3 4 5 1 2 3 4 5 2.5 0.0391 0.3798 0.3473 0.6907 0.6172 0.0878 0.4635 0.4202 0.4627 0.5792 2.22 0.0486 0.3569 0.3333 0.5438 0.5620 0.1057 0.4214 0.3904 0.3532 0.5221 1.81 0.0715 0.3214 0.3141 0.3716 0.4762 0.1497 0.3624 0.3526 0.2332 0.4422 1.57 0.0757 0.2392 0.2431 0.2329 0.3390 0.1957 0.3299 0.3356 0.1817 0.3989 1.4 0.0788 0.1866 0.1965 0.1599 0.2555 0.2466 0.3082 0.3271 0.1523 0.3708 1.19 0.0828 0.1294 0.1442 0.0937 0.1698 0.2522 0.2043 0.2326 0.0884 0.2456 1 0.0864 0.0861 0.1026 0.0526 0.1095 0.2502 0.1263 0.1568 0.0484 0.1535 0.84 0.0890 0.0564 0.0723 0.0292 0.0708 0.2454 0.0769 0.1048 0.0262 0.0957 0.7 0.0905 0.0356 0.0495 0.0155 0.0451 0.2371 0.0453 0.0683 0.0136 0.0586 0.63 0.0906 0.0272 0.0395 0.0107 0.0347 0.2309 0.0333 0.0531 0.0092 0.0442 0.57 0.0902 0.0209 0.0317 0.0074 0.0271 0.2240 0.0248 0.0417 0.0063 0.0338 0.50 0.0889 0.0147 0.0237 0.0046 0.0195 0.2136 0.0168 0.0302 0.0039 0.0237 0.44 0.0868 0.0103 0.0176 0.0029 0.0141 0.2023 0.0115 0.0219 0.0023 0.0168 0.36 0.0820 0.0059 0.0109 0.0013 0.0084 0.1826 0.0063 0.0131 0.0011 0.0097 0.30 0.0764 0.0035 0.0070 0.0007 0.0052 0.1639 0.0037 0.0081 0.0005 0.0058 Table 14. Comparison of critical moment under CL. Section II-Reinforced Section I Bottom Flange M M M /M M M M /M cr1 cr2 cr1 cr2 cr1 cr2 cr1 cr2 TF 56.84 50.43 1.13 50.71 42.98 1.18 SC 123.07 122.87 1.00 103.08 101.99 1.01 1/3 BF 177.32 158.27 1.12 125.93 120.11 1.05 TF 37.25 33.73 1.10 32.84 28.37 1.16 1/2 SC 64.77 64.26 1.01 51.51 50.92 1.01 BF 85.37 77.01 1.11 61.10 58.77 1.04 TF 28.08 25.97 1.08 23.79 20.65 1.15 2/3 SC 42.26 42.16 1.00 32.60 32.45 1.00 BF 52.51 48.85 1.07 37.71 36.82 1.02 TF 18.49 17.49 1.06 14.81 13.46 1.10 SC 24.11 24.1 1.00 17.98 17.76 1.01 BF 27.86 26.52 1.05 20.02 19.81 1.01 Table 15. Comparison of critical moment under CUDL. Section II-Reinforced Section I Bottom Flange M M M /M M M M /M cr1 cr2 cr1 cr2 cr1 cr2 cr1 cr2 TF 40.35 34.91 1.16 24.46 20.78 1.18 1/3 SC 64.50 64.58 1.00 52.37 52.2 1.00 BF 91.93 80.14 1.15 56.20 53.18 1.06 Appl. Sci. 2023, 13, 5830 22 of 27 Table 15. Cont. Section II-Reinforced Section I Bottom Flange M M M /M M M M /M cr1 cr2 cr1 cr2 cr1 cr2 cr1 cr2 TF 34.75 30.86 1.13 21.14 18.09 1.17 SC 54.70 54.56 1.00 43.21 43.03 1.00 1/2 BF 72.37 64.79 1.12 45.67 43.79 1.04 TF 29.22 27.12 1.08 18.09 15.93 1.14 2/3 SC 43.30 43.1 1.00 33.53 33.36 1.01 BF 54.58 50.12 1.09 35.13 34.33 1.02 TF 21.53 20.11 1.07 13.93 13.78 1.01 1 SC 28.74 28.56 1.01 21.68 21.5 1.01 BF 34.26 32.56 1.05 22.48 22.12 1.02 Note: M is the calculated value by Equation (59). M is the calculated value by ABAQUS. cr1 cr2 6. Conclusions A novel numerical method and a new closed-form solution for the elastic LTB analysis of cantilever steel beams have been successfully developed and validated in this paper. The numerical method was developed based on postulated buckling deformation functions and the Rayleigh–Ritz method. The closed-form solution was developed based on the assumption that the torsion angle is small and based on the numerical method. The values of M predicted by the proposed numerical method are sufficiently comparable to the data cr from existing literature and the following findings can be drawn: (1) M and the generalized coordinates A and B in the buckling deformation functions cr n n of cantilever I-beams under CL and CUDL can be accurately calculated by the nu- merical method proposed in this paper. Compared with the experimental results and the finite element results of relevant reference, the maximum of the M /M and ex th M /M ratio is 1.07, and the minimum error is 0.92. The average of all the ratios is th FEM 0.98 with a standard deviation of 0.13. (2) The safety, accuracy and economic aspects of the prediction of the numerical method are assessed using DPC. The results show that the numerical method can effectively predict M of cantilever steel beams under CL and CUDL. cr (3) The number of generalized coordinates determines the accuracy of M . The number cr of generalized coordinates is defined as n. The obtained errors of M do not exceed cr 3.96% when n is greater than or equal to 10. The errors of M exceed 17% when n is cr equal to 1 or 2. (4) The degree of M in Equation (34) varies with the number of generalized coordinates cr of twist angle function. When the number of generalized coordinates of twist angle function is n, the degree of M in Equation (34) is 2n. cr (5) The torsional buckling mode of the cantilever steel beam is mainly determined by K, which is the torsional stiffness coefficient. However, in order to provide reference for the calculation of the M of cantilever cr steel beams matching the specification of ANSI/AISC 360-16, a large number of numerical calculations and statistical evaluations should be performed to determine the expression of C , which is the lateral–torsional buckling modification factor. In addition, further investigations are needed on the LTB performance of beams composed of other materials that are not the isotropic material, such as Glass Fiber Reinforced Polymer (GFRP), laminate composite material and pultruded fiber reinforced polymer (PFRP) material, which are increasingly used in beams. Author Contributions: Conceptualization, software, validation, writing—original draft preparation, A.L.; writing—review and editing, supervision, Y.C. and X.L. All authors have read and agreed to the published version of the manuscript. Appl. Sci. 2023, 13, 5830 23 of 27 Funding: This research was funded by National Natural Science Foundation of China (Grant no. 51078354). Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Acknowledgments: The author gratefully acknowledge the financial support of National Natural Science Foundation of China (51078354). Conflicts of Interest: The authors declare no conflict of interest. Nomenclature LTB lateral–torsional buckling M critical moment cr CL concentrated load CUDL combination of concentrated load and uniformly distributed load C lateral–torsional buckling modification factor C , C , C correction factor of critical moment 1 2 3 l cantilever length x, y, z Cartesian coordinates P , q concentrated load and uniformly distributed load y y a , a the difference between the ordinate of load and the ordinate of shear center P y q y l the ratio of distance between the action point of Py and the fixed end to the cantilever ’s length E Young’s modulus G shear modulus I , I , I warping constant, torsional constant and moment of inertia about y-axis ! t y b Wagner ’s coefficient u, j the lateral deflection and twist angle of section u(z), j(z), j (z) the buckling deformation functions j(ll) the twist angle at the point where the transverse concentrated load acts during the lateral–torsional buckling of the beam M the moment about the x-axis a the ratio of the product of q multiplied by l to P y y h the distance between the centroids of the top and bottom flanges A , B generalized coordinates n n n, r the numbers of terms of buckling deformation functions M , |M | the moment when z = 0, the maximum absolute value of moment 0 0 I , I the moment of inertia of the top flange around the y-axis, the moment of 1 2 inertia of the bottom flange around the y-axis h the ratio of I to I 1 2 M the ratio of |M | to the product of Euler critical load multiplied by h 0 0 b , af , af the ratio of b to h, the ratio of a to h, the ratio of a to h x P qy x P y q y K torsional stiffness coefficient 0 0 0 0 1 1 1 1 R , S , T , Q , R , S , T , Q n-dimensional vectors about total potential energy D ,D 2n-dimensional vectors about total potential energy 0 1 TF, SC, BF top flange, shear center and bottom flange M , M , M the experimental values, the model predicted values, the predicted ex th FEM values by ABAQUS in reference [16] DPC demerit points classification M , M the values calculated by Equation (59), the values calculated by ABAQUS cr1 cr2 r , r , r , r , r , r , R , R , R , R , R coefficients of M in Equation (53) and Equation (59) 1 2 3 4 5 6 1 2 3 4 5 cr Appendix A. Sample Coding of MATLAB clear;clc;%This program is compiled according to Equation (34) in the paper %The following are the parameters that the user needs to input length = 1.651;%the beam length h = 0.07243699; b = 0.0315214; t = 0.00312928; b = 0.015875; 2 Appl. Sci. 2023, 13, 5830 24 of 27 t = 0.00312674; t = 0.00219;%the definition of h, b , t , b , t and t can be seen in Figure 4 w w 1 1 2 2 E = 65,990.00010ˆ6;% Young’s modulus G = 25,650.00010ˆ6;% shear modulus ap = 0.01123573;%the difference between the ordinate of uniformly distributed load and the ordinate of shear center. It can be seen in Figure 1 aq = 0;%the difference between the ordinate of concentrated load and the ordinate of shear center. It can be seen in Figure 1. namuda = 1;%the ratio of distance between the action point of P and the fixed end to the cantilever ’s length, i.e., l in Equation (1) aerfa = 0;%the ratio of the product of q multiplied by l to P ,i.e., a in Equation (2) y y N = 100;%the number of generalized coordinates %The following is the calculation program of the system and does not need to be changed s = b t + b t + tw(h t /2 t /2);%Cross-sectional area 1 1 2 2 1 2 d1 = ((b t ˆ2/2 + t (h t /2 t /2)(t + (1/2)(h t /2 t /2)) + b t (h + 2 2 1 2 2 1 2 1 1 t /2)))/s;%Distance from centroid to the the lowest end of the bottom flange I = (1/12)t b ˆ3((1/12)b ˆ3t h/((1/12)b ˆ3t + (1/12)b ˆ3t ))ˆ2 + (1/12)t 1 1 2 2 2 2 1 1 2 b ˆ3(h (1/12)b ˆ3t h/((1/12)b ˆ3t + (1/12)b ˆ3t ))ˆ2;%warping constant 2 2 2 2 2 1 1 I = (1/12)(b ˆ3t + b ˆ3t + t ˆ3(h t /2 t /2));%moment of inertia y w 1 1 2 2 1 2 about y-axis I = (1/3)(b t ˆ3 + b t ˆ3 + t ˆ3(h t /2 t /2));%torsional constant t 1 1 2 2 w 1 2 I = (1/12)(b t ˆ3 + b t ˆ3 + t (h t /2 t /2)ˆ3) + b t (h + t /2 d1)ˆ2 + x 1 1 2 2 w 1 2 1 1 2 b t (d1 t /2)ˆ2 + t (h t /2 t /2)(d1 (h + t /2 + t /2)/2)ˆ2;%moment of inertia 2 2 2 w 1 2 1 2 about x-axis yita = (1/12)b ˆ3t /((1/12)b ˆ3t );%the ratio of I to I , i.e., h in Equation (13). I 1 1 2 2 1 2 1 is the moment of inertia of the top flange around the y-axis and I is the moment of inertia of the bottom flange around the y-axis. betaba = (1/(2I )( (h + t /2 d1)((1/12)b ˆ3t + b t (h + t /2 d1)ˆ2) + (d1 2 1 1 1 1 2 t /2)((1/12)b ˆ3t + b t2(d1 t /2)ˆ2) + 0.25t (((d1 t /2) t /2)ˆ4 ((h + t /2 2 2 2 2 2 w 2 2 2 d1) t /2)ˆ4)) ( (h + t /2 + t /2 d1 t /2 (1/12)b ˆ3t h/((1/12)b ˆ3t + 1 1 2 1 2 2 2 2 (1/12)b ˆ3t1))))/h;%the ratio of Wagner ’s coefficient to h, i.e., b in Equation (13) 1 x K = sqrt(piˆ2EI /(GI lengthˆ2));%torsional stiffness coefficient w t apba = ap/h; %i.e., a in Equation (13) aqba = aq/h; %i.e., a in Equation (13) R = zeros(N,N); for m = 1:N for r = 1:N if m == r R (m,r) = ( 1 + 2m)ˆ4piˆ4/32; end end end S = zeros(N,N); T = zeros(N,N); Q = zeros(N,N); for m = 1:N for r = 1:N if m = =r Q (m,r) = ( 1 + 2m)ˆ4piˆ4yita/(32(1 + yita)ˆ2) + ( 1 + 2m)ˆ2piˆ4yita/(8Kˆ2(1 + yita)ˆ2); end end Appl. Sci. 2023, 13, 5830 25 of 27 end R = zeros(N,N); S = zeros(N,N); for m = 1:N for r = 1:N if m = =r SS1 = pi( 96aerfacos(mpi) + ( 1 + 2m)pi( 42(1 + aerfa) + (1 2m)ˆ2piˆ2 (aerfa + 3namudaˆ2) + 48cos(0.5( 1 + 2m)pinamuda) 6cos(( 1 + 2m)pinamuda))) /(24( 1 + 2m)(aerfa + 2namuda)); SS2 = 0; S (m,r) = SS1 + SS2; else SS1 = pi( 16aerfa(r m)ˆ2( 1 + r + m)ˆ2cos(pim) + pi( 1 + 2m)( (1 2r)ˆ2 ( 1 + 2( 1 + r)r 6( 1 + m)m)(1 + aerfa) + 8(r m)ˆ2( 1 + r+m)ˆ2cos(0.5pi (1 2m)namuda) + (1 2m)ˆ2( ( 1 + r + m)ˆ2cos(pi(r m)namuda) (r m)ˆ2cos (pi( 1 + r + m)namuda))))/(4( r + m)ˆ2( 1 + r + m)ˆ2( 1 + 2m)(aerfa + 2namuda)); SS2 = 0; S (m,r) = SS1 + SS2; end end end T = zeros(N,N); for m = 1:N for r = 1:N T (m,r) = S1(r,m); end end Q = zeros(N,N); for m = 1:N for r = 1:N if m = =r QQ1 = betabapiˆ2((1 2m)ˆ2piˆ2(aerfa + 3namudaˆ2) 6(1 + aerfa) + 6cos(( 1 + 2m)pinamuda))/(12(aerfa + 2namuda)); QQ2 = aqbapiaerfa(3pi 6mpi 8cos(mpi))/(( 1 + 2m)(aerfa + 2namuda)) 2 apbapiˆ2(1 cos(0.5(2m 1)pinamuda))ˆ2/(aerfa + 2namuda); Q (m,r) = QQ1 + QQ2; else QQ1 = 2( 1 + 2r)( 1 + 2m)betabapiˆ2(( 1 + 2r)( 1 + 2m)(1 + aerfa) ( 1 + r + m)ˆ2cos(pi(r m)namuda) + ( r + m)ˆ2cos(pi( 1 + r + m)namuda))/(4(r m)ˆ2( 1 + r + m)ˆ2(aerfa + 2namuda)); QQ2 = 2aqbaaerfapiˆ2(2 4cos(pir)/(pi(1 2r)) 4cos(pim)/(pi(1 2m)))/ (2(aerfa + 2namuda)) 2apbapiˆ2(1 cos(0.5(2m 1)pinamuda))(1 cos(0.5(2r 1) pinamuda))/(aerfa + 2namuda); Q (m,r) = QQ1 + QQ2; end end end 0 0 0 0 D = [R ,S ;T ,Q ];%Equation (35) in the paper 1 1 1 1 D = [R ,S ;T ,Q ];%Equation (36) in the paper [V,D] = eig(D , D );%D is the diagonal matrix composed of generalized eigenvalues, 0 1 and V is the corresponding eigenvector D = eig(D , D ); 0 1 L = 0; for i = 1:2N Appl. Sci. 2023, 13, 5830 26 of 27 if isreal(D(i)) = =1 && D(i) > 0 L = L + 1; Eg(L) = D(i); end end P1 = min(Eg); [row,column] = find(D == P1); A = V(1:10,row);%Vector composed of generalized coordinates of the lateral deflection function u(z) B = V(N + 1:N + 10,row);%Vector composed of generalized coordinates of the twist angle function j(z) M = P1piˆ2hEI /(lengthˆ2);% M is the critical moment cr y cr References 1. ANSI/AISC 360-16; Specification for Structural Steel Buildings. AISC: Chicago, IL, USA, 2016; pp. 44–69. 2. EN 1993-1-1; Eurocode 3: Design of Steel Structures, Part 1-1: General Rules and Rules for Buildings. Comité Européen de Normalisation (CEN): Brussels, Belgium, 2005; pp. 60–64. 3. Clark, J.W.; Hill, H.N. Lateral buckling of beams. J. Struct. Div. 1960, 127, 180–201. [CrossRef] 4. Zhang, W.F.; Liu, Y.C.; Chen, K.S.; Deng, Y. Dimensionless Analytical Solution and New Design Formula for Lateral-Torsional Buckling of I-Beams under Linear Distributed Moment via Linear Stability Theory. Math. Probl. Eng. 2017, 2017, 4838613. [CrossRef] 5. Bresser, D.; Ravenshorst, G.J.P.; Hoogenboom, P.C.J. General formulation of equivalent moment factor for elastic lateral torsional buckling of slender rectangular sections and I-sections. Eng. Struct. 2020, 207, 110230. [CrossRef] 6. Kucukler, M.; Gardner, L. Design of web-tapered steel beams against lateral-torsional buckling through a stiffness reduction method. Eng. Struct. 2019, 190, 246–261. [CrossRef] 7. Rossi, A.; Martins, C.H.; Nicoletti, R.S. Reassesment of lateral torsional buckling in hot-holled I-beams. Structures 2020, 26, 524–536. [CrossRef] 8. Sahraei, A.; Mohareb, M. Lateral torsional buckling analysis of moment resisting plane frames. Thin-Walled Struct. 2019, 134, 233–254. [CrossRef] 9. Timoshenko, S.P.; Gere, J.M. Theory of Elastic Stability, 2nd ed.; McGraw-Hill: New York, NY, USA, 1961; pp. 251–278. 10. Anderson, J.M.; Trahair, N.S. Stability of monosymmetric beams and cantilevers. J. Struct. Div. 1972, 98, 269–286. [CrossRef] 11. Attard, M.M.; Bradford, M.A. Bifurcation experiments on monosymmetric cantilevers. In Proceedings of the 12th Australasian Conference on the Mechanics of Structures and Materials, Brisbane, Australia, 10 December 1990; pp. 207–213. 12. Andrade, A.; Camotim, D.; Providência e Costa, P. On the evaluation of elastic critical moments in doubly and singly symmetric I-section cantilevers. J. Constr. Steel Res. 2007, 63, 894–908. [CrossRef] 13. Andrade, A.; Camotim, D.; Borges Dinis, P. Lateral–torsional buckling of singly symmetric web-tapered thin-walled I-beams: 1D model vs. shell FEA. Comput. Struct. 2007, 85, 1343–1359. [CrossRef] 14. Zhang, L.; Tong, G.S. Elastic flexural-torsional buckling of thin-walled cantilevers. Thin-Walled Struct. 2008, 46, 27–37. [CrossRef] 15. Ozbasaran, H. A Parametric study on lateral torsional buckling of European IPN and IPE cantilevers. Int. J. Civil. Archit. Struct. Constr. Eng. 2014, 8, 739–744. 16. Ozbasarann, H.; Aydin, R.; Dogan, M. An alternative design procedure for lateral–torsional buckling of cantilever I-beams. Thin-Walled Struct. 2015, 90, 235–242. [CrossRef] 17. Ozbasaran, H. Finite differences approach for calculating elastic lateral torsional buckling moment of cantilever I sections. Anadolu Univ. J. Sci. Technol. A Appl. Sci. Technol. 2013, 14, 143–152. 18. Ings, N.L.; Trahair, N.S.; ASCE, M. Beam and column buckling under directed loading. J. Struct. Eng. 1987, 113, 1251–1263. [CrossRef] 19. Gonçalves, R. An assessment of the lateral-torsional buckling and post-buckling behaviourof steel I-section beams using a geometrically exact beam finite element. Thin-Walled Struct. 2019, 143, 106222. [CrossRef] 20. Arizou, R.; Mohareb, M. Finite element formulation for distortional lateral buckling of I-beams. Eng. Struct. 2022, 262, 114265. [CrossRef] 21. Roberts, T.M.; Burt, C.A. Instability of monosymmetric I-beams and cantilevers. Int. J. Mech. Sci. 1985, 27, 313–324. [CrossRef] 22. Asgarian, B.; Soltani, M.; Mohri, F. Lateral-torsional buckling of tapered thin-walled beams with arbitrary cross-sections. Thin-Walled Struct. 2013, 62, 96–108. [CrossRef] 23. Trahair, N.S. Inelastic lateral buckling of continuous steel beams. Eng. Struct. 2019, 190, 238–245. [CrossRef] 24. Trahair, N.S. Inelastic lateral buckling of steel cantilevers. Eng Struct. 2020, 208, 109918. [CrossRef] 25. Demirhan, A.L.; Eroglu, ˘ H.E.; Mutlu, E.O.; Yılmaz, T.; Anil, Ö. Experimental and numerical evaluation of inelastic lateral-torsional buckling of I-section cantilevers. J. Constr. Steel Res. 2020, 168, 105991. [CrossRef] Appl. Sci. 2023, 13, 5830 27 of 27 26. Lorkowski, P.; Gosowski, B. Experimental and numerical research of the lateral buckling problem for steel two-chord columns with a single lacing plane. Thin-Walled Struct. 2021, 165, 107897. [CrossRef] 27. Kim, M.; Hayat, U.; Mehdi, A.I. Lateral–torsional buckling of steel beams pre-stressed by straight tendons with a single deviator. Thin-Walled Struct. 2021, 163, 107642. [CrossRef] 28. Kim, M.; Hayat, U.; Kim, S.; Mehdi, A.I. Stabilizing effects of discrete deviators on LTB of mono-symmetric thin-walled beams pre-stressed by rectilinear tendon cables. Thin-Walled Struct. 2022, 176, 109329. [CrossRef] 29. Zhang, W.F. Symmetric and antisymmetric lateral–torsional buckling of prestressed steel I-beams. Thin-Walled Struct. 2018, 122, 463–479. [CrossRef] 30. Lebastard, M.; Couchaux, M.; Santana, M.V.B.; Bureau, A. Elastic lateral-torsional buckling of beams with warping restraints at supports. J. Constr. Steel. Res. 2022, 197, 107410. [CrossRef] 31. Pezeshky, P.; Sahraei, A.; Rong, F.; Sasibut, S.; Mohareb, M. Generalization of the Vlasov theory for lateral torsional buckling analysis of built-up monosymmetric assemblies. Eng. Struct. 2020, 221, 111055. [CrossRef] 32. Saoula, A.; Selim, M.M.; Meftah, S.A.; Benyamina, A.B.; Tounsi, A. Simplified analytical method for lateral torsional buckling assessment of RHS beams with web openings. Structures 2021, 34, 2848–2860. [CrossRef] 33. Rossi, A.; Ferreira, F.P.V.; Martins, C.H.; Júnior, E.C.M. Assessment of lateral distortional buckling resistance in welded I-beams. J. Constr. Steel Res. 2020, 166, 105924. [CrossRef] 34. Agüero, A.; Balaz, I.; Kolekova, Y. New method for metal beams sensitive to lateral torsional buckling with an equivalent geometrical UGLI imperfection. Structures 2021, 29, 1445–1462. [CrossRef] 35. Erkmen, R.E. Elastic buckling analysis of thin-walled beams including web-distortion. Thin-Walled Struct. 2022, 170, 108604. [CrossRef] 36. Kimura, Y.; Fujak, S.M.; Suzuki, A. Elastic local buckling strength of I-beam cantilevers subjected to bending moment and shear force based on flange–web interaction. Thin-Walled Struct. 2021, 162, 107633. [CrossRef] 37. Jager, B.; Dunai, L. Nonlinear imperfect analysis of corrugated web beams subjected to lateral-torsional buckling. Eng. Struct. 2021, 245, 112888. [CrossRef] 38. Jager, B.; Dunai, L.; Kovesdi, B. Lateral-torsional buckling strength of corrugated web girders—Experimental study. Structures 2022, 43, 1275–1290. [CrossRef] 39. Bärnkopf, E.; Jager, B.; Kovesdi, B. Lateral–torsional buckling resistance of corrugated web girders based on deterministic and stochastic nonlinear analysis. Thin-Walled Struct. 2022, 180, 109880. [CrossRef] 40. Wakjira, T.G.; Ebead, U. A shear design model for RC beams strengthened with fabric reinforced cementitious matrix. Eng. Struct. 2019, 200, 109698. [CrossRef] 41. Zeinali, E.; Nazari, A.; Showkati, H. Experimental-numerical study on lateral-torsional buckling of PFRP beams under pure bending. Compos. Struct. 2020, 237, 111925. [CrossRef] 42. Pham, P.V. An innovated theory and closed form solutions for the elastic lateral torsional buckling analysis of steel beams/columns strengthened with symmetrically balanced GFRP laminates. Eng. Struct. 2022, 256, 114046. [CrossRef] 43. Khalaj, G.; Pouraliakbar, H.; Mamaghani, K.R.; Khalaj, M. Modeling the correlation between heat treatment, chemical composition and bainite fraction of pipeline steels by means of artificial neural networks. Neural Netw. World 2013, 4, 351–367. [CrossRef] 44. Khalaj, G.; Khalaj, M. Modeling the correlation between yield strength, chemical composition and ultimate tensile strength of X70 pipeline steels by means of gene expression programming. Int. J. Mater. Res. 2013, 104, 697–702. [CrossRef] 45. Virgin, L.N.; Harvey, P.S. A lateral–torsional buckling demonstration model using 3D printing. Eng. Struct. 2023, 280, 115682. [CrossRef] 46. Tong, G.S.; Zhang, L. A Controversy and Its Settlement in the Calculation of Buckling Moments of Thin-walled Beams with Monosymmetrical I-sections Under Distributed Loads. J. Build. Struc. 2002, 23, 44–51. 47. Bleich, F. Buckling Strength of Metal Structures; McGraw Hill: New York, NY, USA, 1952; pp. 153–158. 48. Bebiano, R.; Camotim, D.; Gonçalves, R. GBTUL 2.0—A second-generation code for the GBT-based buckling and vibration analysis of thin-walled members. Thin-Walled Struct. 2018, 124, 235–257. [CrossRef] 49. Camotim, D.; Andrade, A.; Basaglia, C. Some thoughts on a surprising result concerning the lateral–torsional buckling of monosymmetric I-section beams. Thin-Walled Struct. 2012, 60, 216–221. [CrossRef] 50. ABAQUS. Abaqus/CAE User’s Guide; Dassault Systems Simulia Corporation: Providence, RI, USA, 2020. 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