Advances in Civil Engineering

Volume 2019, Article ID 1602565, 11 pages

https://doi.org/10.1155/2019/1602565

## Developed Mathematical Model for Indeterminate Elements with Variable Inertia and Curved Elements with Constant Cross-Section

^{1}Structural Engineering Department, Faculty of Engineering, Mansoura University, Mansoura, Dakahlia 35516, Egypt^{2}Civil Engineering Department, College of Engineering and Islamic Architecture, Umm-Alqura University, Makkah, Saudi Arabia

Correspondence should be addressed to Mohamed A. El Zareef; moc.liamg@67feerazm

Received 8 December 2018; Accepted 24 March 2019; Published 7 April 2019

Academic Editor: Daniele Baraldi

Copyright © 2019 Mohamed A. El Zareef et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Issues such as analysis of indeterminate structural elements that have variable inertia as well as a curved shape still have no closed form solution and are considered one of the major problems faced by design engineers. One method to cope with these issues is by using suitable the finite element (FE) software for analyzing these types of elements. Although it saves time, utilization of FE programs still needs professional users and not all engineers are familiar with it. This paper has two main objectives; first, to develop simple mathematical models for analyzing indeterminate structural elements with variable inertia and that have a curved shape with constant cross section, this model is much easier to be used by engineers compared to the FE model. For simplicity and saving time, a MATLAB program is developed based on investigated mathematical models. The force method combined with numerical integration technique is used to develop these models. The developed mathematical models are verified using the suitable FE software; good agreement was observed between the mathematical and the FE model. The second objective is to introduce a mathematical formula to determine the accurate number of divisions that would be used in the mathematical models. The study proves that the accuracy of analysis depends on the number of divisions used in the numerical integration. The optimum number of divisions is obtained by comparing the output results for both FE and developed mathematical models. The developed mathematical models show a good agreement with FE results with faster processing time and easier usage.

#### 1. Introduction

Beams with variable sections have a lot of practical applications in concrete and in steel structures. Tapered beams made of welded plate are an example of variable sections beams that are used in steel structures; these types of beams are used mainly in rafter and columns of frames or as continuous beams, and the main advantage of these type of section is they are economical where the beams have variable heights along their length so that the beam capacity equals the acting moment [1, 2]. In concrete structures, beams with variable sections are used in many structural applications and in many precast concrete buildings [3].

Designing and analysis of variable section beams are considered one of the problems that engineers face. Due to lack of knowledge, using the mean value of stiffness or ignoring variation of stiffness along element length are used in the design process of such elements, which is not accurate and can lead to progressive collapse of the structure. In some special structures that have elements with variable stiffness, it is essential to find an accurate analysis method for these elements. One of the main issues in designing a beam with variable inertia is that the section which is subjected to maximum stress need not necessarily be at the midspan of the beam due the variation in beam depth, which leads to changing the inertia along the beam length [4].

Many researchers were motivated to study the behaviour of beams with variable sections under different loading scenarios. Boiangiu et al. [5]; Ece et al. [6] and Felsoufi and Azrar [7] studied the dynamic behaviour and the free body vibration of beams with variable cross section. Yang et al. [8] and Dumitrache [9] studied the shear stress distribution and analysis for variable sections beam. Yang et al. [8] found that design codes do not accurately calculate the stress distribution in both the cracked and elastic stage using the effective shear force method.

Koo Lee et al. [10], Rojas and Ramirez [11], and Sapountzakis and Tsiatas [12] investigated the elastic linear behaviour of beams with variable sections using the classical beam theory, and others studied the nonlinear behaviour of such beams [13, 14]. However, most of the proposed design equations are still complicated to be studied by the engineers.

Another structural element that has a beautiful and attractive architecture view is arches. Arches were previously used by the Persian, Egyptian, Babylonian, Greek, and Islamic civilizations for different types of structures, but the ancient Romans were the first to use them as bridges. Arches have been used in some bridges in China and have been used widely in Islamic mosques, castles, and palaces.

Nowadays, arches are used in different structural applications as railway and roadway bridges, pedestrian bridges, or as part of a building, hangers with a large open space without intermediate columns. The basic structural theory for arch is that it converts all stress into compressive and eliminates tensile stresses. This is useful especially for materials which carry less tensile stress.

Similar to beams with variable sections, engineers face difficulties in analyzing arches and determining the critical cross section and also calculating straining action along the arch length. Due to the importance of arches in our structures, many researches study the behaviour of arches under different loading conditions with different shape profiles. Sonavane [15] applied basic principles of the flexibility method in the analysis and design of symmetrical circular arches under different loading conditions. The results obtained from the analysis were compared with the finite element (FE) model results. MATLAB program was used to determine the internal forces in the circular arch design.

Ghannam and Najm [16] investigated the buckling behaviour of Islamic arches under different loading conditions. King and Brown [17] conducted the manufacturing and design of curved steel members using the British code (BS 5950–1 : 2000). King and Brown [17] included worked examples to clarify the design of arched steel used in different applications.

Designers tend to use FE software as one of the most effective methods that can be used in the analysis of arches or straight beams with variable cross sections. However, there is still no exact mathematical model for solving such problems.

Based on the above literature, it can be concluded that the main problem that faces engineers in designing structural elements that have variable inertia as well as that have a curved shape is the complication of the design methods that are available for these types of structures. Besides, not all engineers are familiar with using FE programs. Therefore, the main aim of this paper is to cover that gap in this area and to provide simple mathematical analytical models that would simplify the analysis and design of structural elements that have variable inertia and a curved shape. In this paper, the flexibility and numerical integration methods are used to develop simplified mathematical analytical models. These investigated models are used to evaluate the straining actions for straight beams with variable cross section and curved beams that have constant moment of inertia. To ease and save time of the solution process, simple MATLAB programs are established based on the investigated mathematical models. The output results from these models are verified using the suitable finite element software [18]. It was found that there is a good agreement between the developed program and the result of the FE program.

By comparing the output results from both the FE software and developed mathematical models, it is observed that the number of divisions used in the numerical integration of the mathematical models play a main role to obtain accurate results.

Therefore, the second objective of this paper is to present mathematical formulas for curved and straight beams with variable inertia to determine the optimum number of divisions used in the analytical model of each. A numerical analysis of the output results is applied to investigate these formulas. By adjusting the number of divisions used in the developed models, the output results show a good agreement with FE results.

#### 2. Flexibility Method (Force Method)

The flexibility or force method was originally developed by James Clerk Maxwell in 1874. This method is very common and a powerful method for the analysis of statically indeterminate structures without limitations as in other structural analysis methods.

The method is based on transforming the structure into a statically determinate system and calculating the value of redundant forces required to restore the geometric boundary conditions of the original structure. The force method (also called the flexibility method or method of consistent deformation) is used to calculate reactions and internal forces in statically indeterminate structures. The method can be used for either plane or space structures subjected to loads or forced deformations. It is well known that the analysis of beams with variable stiffness along length in a closed form solution cannot be obtained. Therefore, it is important to modify the existing method (flexibility method) for elements with constant stiffness to be suitable for beams with variable stiffness. Additionally, the exact analysis method for the curved beams with constant section is studied in this paper to fill the gap in the literature.

The flexibility and the stiffness for an element at a certain point and direction are reverse. The inverse of stiffness is flexibility. The relationship between flexibility and stiffness can be expressed as follows:where is the stiffness matrix and is the flexibility matrix of an element. Flexibility matrix for an element is generally determined by the following equation:

Equation (2) can be simplified according to the type of structure. For beams, equation (2) can be simplified into the following equation, where the bending moment is the governing staining action for the beam cross section:where () is the corresponding bending moment diagrams for force at ends (, ), respectively, () is the modulus of elasticity, and () is the cross section moment of inertia. The solution of the numerical integration can be obtained using trapezoidal rules presented in the following equation:

The equations of the force method can be written in the matrix form as follows:where and are the corresponding member force at the ends and , respectively, and are the member rotation at the ends and , respectively, and is the deflection in the direction of the redundant due to a unit load in the direction of the redundant.

#### 3. Proposed Analytical Model and Verifications

##### 3.1. Application 1: Beam with Variable Inertia

###### 3.1.1. Analytical Model for Beams with Variable Inertia

This section describes the proposed analytical model through an application in which the internal moments at supports are determined for the beam due to clockwise unit rotation separately at both ends of the beam as shown in Figure 1.