Analysis of Beam-Column Designs by Varying Axial Load with Internal Forces and Bending Rigidity Using a New Soft Computing Technique

School of Urban Construction, Wuhan University of Science and Technology, Wuhan 430074, China Fujian University Engineering Research Center of Disaster Prevention and Reduction of Southeast Coastal Engineering Structure (JDGC03), Putian 351100, China Putian College Institute of Civil Engineering, Putian 351100, China Department of Mathematics, Abdul Wali Khan University Mardan, Mardan 23200, Khyber Pakhtunkhwa, Pakistan


Introduction
Most of the problems arising in numerical science and engineering such as physics, biology, economics, mathematics, and astronomy are modeled as differential equations with initial or boundary values.
e modeling enables us to analyze and understand the concept of a particular event in a precise way.
us, finding an exact and analytical solution for such models gained immense importance in recent years. Generally, finding exact solutions to the physical problems modeled as differential equations is a difficult task and, in most cases, the analytic solution does not exist. erefore, it becomes necessary to study numerical methods for finding solutions to the differential equations representing physical phenomenon.
Among various proposed methods for solving ODE's, to solve the two-point Boundary value problem (BVP), a finite difference method [1] was proposed. For solving fourth-order and sixth-order BVP's, the homotopy perturbation method [2,3] is proposed. Multipoints' BVP's are solved by optimal homotopy ssymptotic proposed by Ali [4]. Also, for the solution of two-points and multipoints' BVP's of higher order, a domain decomposition method [5][6][7] was developed. Jacobi polynomials were used by Doha [8] to propose a spectral Galerkin algorithm for two-points' BVP's of third and fifth order. For the solution of even order DEs, Chebyshev polynomials are used by Doha [9] to develop spectral Galerkin algorithm. Shannon wavelet method [10], Haar wavelets method [11], Saadatmandi and Dehghan technique [12], a fifthfourth continuous block implicit hybrid method [13], and spectral Bernstein residual method [14] were proposed for the solution of higher-order differential equations. To solve higherorder linear and nonlinear differential equations with two boundary values, variational methods [15][16][17] were introduced. In recent years, many techniques have been developed to solve nonlinear stiff models, i.e., blood flow of Ree Eyring fluid [18] and numerical study of the DNA dynamics arising in oscillator chain of the Peyrard-Bishop model [19]. Besides the advantages that these methods give a good approximation to the solution, it is required to have an initial guess near a required solution and the approximate solution is differentiable and continuous in the domain, and hence, it seriously affects the stability of these numerical techniques.
Artificial neural networks are one of the intelligent techniques in solving different types of differential equations. e results obtained by the approximation through neural networks are close enough to the analytic or exact solution. In general, neural networks are widely used to solve fractional differential equations, integro differential equations (IDEs), partial differential equations (PDEs), and ordinary differential equations (ODEs). Feedforward artificial neural networks are used to solve two-point boundary value problems (BVP's) of fourth order. Higher-order ODE's with boundary values are solved by the radial basis activation function in neural networks as presented by Mai-Duy [20].
According to no free lunch theorem, researchers are compelled to hybridize the strengths of two or more techniques so that high-quality solutions are calculated with fewer efforts. In recent times, metaheuristic techniques have gained the attention of researchers in various fields of science and engineering. Impacts of different crossover operators are investigated for handling multiobjective problems [21]. A plant propagation algorithm (PPA) and its modified version were developed to solve design engineering problems [22]. In electrical engineering, several methodologies are used to solve complex optimization problems [23]. Optimal design and temperature distribution of heat fin is solved by using hybridization of artificial neural networks and metaheuristic algorithms [24,25] and, furthermore, oscillatory behavior of heart beat [26]. Most of the real-application problems are highly nonlinear ODEs and provide less information about the continuity and differentiability of resulting solutions in solution space. e authors of this paper are addressing this issue and hence developed a novel solver that combines weighted Legendre polynomials and an efficient single path following optimizer the Nelder-Mead (NM) algorithm. e outcome of our experiments dictates that the LeNN-NM algorithm can handle ODEs and provide quality solutions.
Salient features of this research are summarized as follows: (i) We have derived a mathematical model that represents different structures of beam-columns. By varying axial load with or without internal forces including bending rigidity, we have studied different cases of the problem. (ii) We have also developed a novel solver, the LeNN-NM algorithm, which consists of weighted Legendre polynomials and a single path following optimizer the Nelder-Mead (NM) algorithm. (iii) A graphical overview of the novel LeNN-NM algorithm is presented in Figure 1. (iv) ree versions of the structural designs of beamcolumns are analyzed with the help of LeNN-NM algorithm, see Problems 1-3.
(v) To examine the quality of solutions calculated by LeNN-NM algorithm, we have calculated values of performance indicators mean absolute deviation (MAD), eil's inequality coefficient (TIC), Nash Sutcliffe efficiency (NSE), and error in Nash Sutcliffe efficiency (ENSE). (vi) Series solutions are presented in equations (30), (31), and (32), which may be used by researchers for further investigation. (vii) Our solutions are in strong agreement with reference solutions and absolute errors are less than the reference errors, see Tables 1-3. (viii) Graphical illustrations for performance indicators are presented in Figure 2. Lower values of MAD, TIC, and ENSE show that our solutions are of good quality. (ix) We have repeated our simulations for hundred times and overall performance is depicted in terms of fitness values, MAD, TIC, and ENSE, for 100 runs, see Figure 3. (x) e success rate of the LeNN-NM algorithm is shown by plotting the frequency graphs of fitness values, MAD, TIC, and ENSE, with normal distribution fittings. All these analyses dictate that the LeNN-NM algorithm is stable, efficient, and compatible for solving design engineering problems.

Mathematical Model Representing the Design of Beam-Columns
Most of the problems related to the physical phenomenon are modeled as differential equations. Higher-order differential equations appear in mathematical models of the elastic stability theory. Hence, most of the physical problems are posed with nonlinear and complex ODEs. To understand and analyze the design of beam-column, a differential relation should be developed between the various cross-sections of the physical problem. Cross-section analysis of a beam-column problem is shown in Figure 1. An element at a distance dx, as shown in the figure, is taken from the cross-section of beam-column and is subject to both spread load q perpendicular to the axis and the axial load P, see Figure 1. Internal forces may arise in the element, and mathematically, it can be written as Equilibrium equation in y-direction can be written as an ordinary differential equation given in [29][30][31]: e sum of forces acting on each surface of the crosssection element will be the same because of the equilibrium state. Mathematically, it can be written as 2 Complexity         Complexity where M is the bending moment that tries to bend the crosssection element and V is the shear force acting on the surface of the element. If we assume that rotations are small and if we neglect the terms of second order in dx, then equation (3) becomes Since the rotations are assumed to be small and d 2 /dx 2 � − M/EI, then equation (4) can be written as where bending rigidity is denoted by EL. Taking derivative on both sides of equation (5) with respect to x, then we get differential equation of fourth order for the elastic curve and is given as

Approximate Solutions and Weighted Legendre Polynomials
Higher-order boundary value problems occur in several areas of applied mathematics, fluid mechanics, elasticity, and quantum mechanics, as well as other branches of science and engineering. Consider a general form of fourth-order ODE as where α 1 , α 2 ,α 3 , and α 4 are constants, t is the independent variable, y is the dependent variable, and a and b are bounds on the independent variable. Prior to the discussion of the approximate solution in equation (1) for the differential equation, the weighted Legendre polynomials are discussed first. Ln(t) denotes the Legendre polynomials. Here, n denotes order of Legendre polynomials. ese polynomials are orthogonal and thus form a set of orthogonal polynomials on the basis of [− 1, 1]. In Table 4, the first ten Legendre polynomials are given. e recursive formula given below generates higherorder Legendre polynomials: Approximate series solution for equation (7) is considered as where ζ n , ψ n , and θ n are unknown parameters. Since, nth order continuous derivatives of equation (9) exist, so first, second, third, and fourth derivative of equation (9) is given by where ζ n , ψ n , and θ n are real-valued unknown parameters. Plugging equations (10)- (13) in equation (7), where a ≤ t ≤ b. Equation (7) is now converted into an equivalent algebraic system of equations. Equation (14) can be solved for unknown parameters ζ n , ψ n , and θ n using an efficient solver such as the LeNN-NM algorithm. All the symbols and abbreviations used in the paper are defined in Table 5.

Nelder-Mead Algorithm
Nelder-Mead (NM) algorithm is a single-path following optimizer and is used in this paper to optimize the fitness function developed for each problem. To reduce a function, NM algorithm sets up the simplex by using n + 1 points. Such points are defined as the vector directions on an n-dimensional search space. In recent times, NM algorithm is used to find numerical solution of the dynamical model of Li-ion batteries for electric vehicle [32], nonlinear Muskingum models [33], economic load dispatch problem with valve point loading effect [34], optimization of TIG welding parameters [35], optimization of noisy CNLS problems [36], and parameter identification of chaotic systems [37]. Implementation of NM algorithm is based on four basic operators [38]. Below, we present details of these operators:

Reflection. Reflection point is determined by
where X is called centroid and is defined as Reflection coefficient is denoted by α. "X r " is accepted, and iterations are terminated if f(X 1 ) ≤ f(X r ) < f(X n ).

Expansion.
To calculate expansion point, equation (17) is used: Equation (17) will be evaluated if value of function at X r is less then X 1 . Expansion coefficient is denoted by β. If f(X e ) ≤ f(X r ), then we accept "X e ," and the process of iteration is terminated. X r will be accepted if the above condition is not satisfied.

4.3.
Contraction. If f(X r ) ≥ f(X n ), then the process of contraction occurs.
(1) Outside contraction is performed by equation (18) if f(X r ) < f(X n+1 ): where "c" is coefficient of contraction. We accept "X c " if f(X c ) ≤ f(X r ). Otherwise, NM algorithm will move to the shrinking step.
Otherwise, NM algorithm will move to the shrink step.

Our Proposed LeNN-NM Algorithm
Steps for the proposed technique are summarized as follows: Step 1: initialize a random population of weights and define a fitness function. Approximate solution is constructed by using activation function given by equation (9).
Step 2: solution along with higher derivatives are approximated at arbitrary generated search points.
Step 3: put approximate solution along with higher derivatives in a given differential equation.
Step 4: an equivalent set of system of algebraic equations corresponding to given differential equation will be generated by Step 3.
Step 5: optimize the system unknown parameters ζ n , ψ n , and θ n using LeNN.
Step 6: Nelder-Mead algorithm will start the process of optimizing the system by taking ζ n , ψ n , and θ n as its initial guess.
Step 7: Nelder-Mead algorithm evaluates the fitness function, and the results will be displayed when the stopping criteria are achieved.
Step 8: best values for the weights ζ n , ψ n , and θ n obtained will be plugged in approximate solution equation (9).
Step 9: it will be the solution.
Working strategy of proposed algorithm is also shown in Figure 4. Parameters' setting for LeNN and NM algorithm is given in Table 6.

Performance Indicators
To study the performance of LeNN-NM technique for solving Beam equations, the performance indications such as mean absolute deviation (MAD), eil's inequality coefficient (TIC), and error in Nash Sutcliffe efficiency (ENSE) are implemented. e formulation of these performance indices are given by [38] MAD � 1 n n m�1 y(t) − y approx (t) , where y(t) � 1 n n m�1 y(t), where n represents the number the mesh points.

Numerical Applications
Problem 1. Consider the homogenous beam equation of the fourth order with bending rigidity [27], EL � 1, P � − 1, and q(t) � 0: with boundary conditions

Start Start
Approximate solution is considered using by using eq (9) At randomly generated search points approximate solution and its derivatives are calculated.
Values of approximate solution along with their derivatives is substituted into given DE which results in a set of algebraic system of equations

Legendre neural networks
Nealder-mead algorithm The algebraic system is solved to find the unknown parameters involved in the equations.
The trained neurons are taken as starting point by NM algorithm.
NM algorithm calculate the fitness value and displays result when minimum value is obtained.
The best value for the unknown parameter when attained will be plugged into eq (9) . .
Then, eq (9) will be called the approximate solution for the given differential equation   Complexity e exact solution for the physical problem modeled as equation (22) is given by e result obtain by LeNN-NM algorithm along with absolute errors is given in Table 1 and graphically illustrated through Figures 5(a) and 6(a), respectively. e unknown parameters achieved by the proposed method are revealed in Table 7. e result of MAD, TIC, and ESNE for equation (22) is shown in Figures 2 and 3. Normal provability plots for fitness evaluation, MAD, TIC, and ENSE, are shown in Figures 7-10, respectively. Hence, the graphs and tables show the dominance of LeNN-NM algorithm in solving higher-order ordinary differential  Complexity equations subjecting to multiple initial or boundary conditions.

Problem 2.
Consider the nonhomogenous linear beam equation of the fourth order with bending rigidity [28], EL � 1, P � − 2, and q(t) � − 8e t , for with boundary conditions e exact solution for equation (25) is y(t) � t(1 − t)e t . e result obtained by LeNN-NM algorithm is compared with the spline method [28] and given in Table 2. Figures 5(b) and 6(b) show that the solution obtained by proposed algorithm overlaps the exact solution with minimum absolute errors. Unknown neurons in LeNN structure for optimization of Problem 2 are shown in Figure 11. e result of MAD, TIC, and ESNE for equation (24) is shown in Figures 2 and 3. Normal provability plots for fitness evaluation, MAD, TIC, and ENSE, are shown in Figures 7-10, respectively. e statistical analysis shows the dominance of the proposed method.
Problem 3. Consider the homogenous linear beam equation of the fourth order with bending rigidity [28], EL � 1, P � 0, and q(t) � 0: with boundary conditions 10 Complexity e exact solution for equation (27) is given by e result obtained by LeNN-NM algorithm along with absolute errors are given in Table 3 and graphically presented through Figures 5(c) and 6(c), respectively. e unknown parameters achieved by the proposed method are revealed in Table 7. Convergence of the fitness value during 100 independent runs of the proposed algorithm is shown in Figure 12. e result of MAD, TIC, and ESNE for equation (27) is shown in Figures 2 and 3. Normal provability plots for fitness evaluation and performance measures are shown through Figures 7-10 Approximate series solution for Problem 3 is given as 16 Complexity

Conclusion
In this paper, we have modeled physical problems arising in the beam-column theory as the fourth ordinary differential equation. Moreover, the paper presents a technique which is named as LeNN-NM algorithm for solving an ordinary differential equation of higher order subjecting to single or multi-initial boundary conditions. e different cases of beam equations are studied and have been solved by the proposed technique. e results given in Tables 1-3 show that LeNN-NM algorithm converges rapidly and dominates the existing algorithm in literature for finding solutions to higher-order differential equations such as beam equations. e analysis reveals that the proposed algorithm looks like a promising methodology to be exploited as an alternate, accurate, reliable, and robust computing framework for solving a variety of the problems arising in astrophysics, atomic physics, plasma physics, nonlinear optic, electric machines, nanotechnology, fuel ignition model, fluid dynamics, bioinformatics, and financial mathematics. e proposed method gives the direction of using the proposed algorithm for solving fractional differential equations.

Data Availability
e data used to support the findings of the study are available within the article.