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\title{Investigation of Stresses in a Beam with Fixed Connection using Finite Difference Technique}
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             \author[1]{Muhammad Ziaur  Rahman}

             \author[2]{Tousif  Ahmed}

             \author[3]{Raihan Md.  Imtiaz}

             \affil[1]{  Bangladesh University of Engineering and Technology}

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\date{\small \em Received: 9 December 2012 Accepted: 1 January 2013 Published: 15 January 2013}

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\begin{abstract}
        


In this study a simple rectangular beam has been analyzed. For simplicity two dimensional plane strain problem is considered. The beam is fixed at the left and right edges of the bottom surface. Uniformly distributed load is applied along the length of the beam. A new formulation, known as displacement potential function formulation has been used here. Stresses and displacements at different sections along the length and depth of the beam have been plotted and discussed. The curves are found to be in complete conformity with the available literature and physical condition of the beam. The present literature reveals that the fixed connections of the beam are extremely critical regions and have the highest probability to fail. 

\end{abstract}


\keywords{stresses, beam with fixed connection, finite-difference, displacement potential function.}

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\let\tabcellsep& 	 	 		 
\section[{Notations}]{Notations}\par
Introduction ixed connections are very common. Steel structures of many sizes are composed of elements which are welded together. A cast-inplace concrete structure is automatically monolithic and it becomes a series of rigid connections with the proper placement of the reinforcing steel. Fixed connections demand greater attention during construction and are often the source of building failures. This paper is an attempt to find out what actually happens in a beam with fixed connection at the bottom when it is loaded with uniformly distributed force. In this regard two dimensional elastic problem has been used. It enabled to manage the mixed mode of the boundary conditions as well as the zone of their transition.\par
In modern age, stress related problems are classical topics. But still there are some shortcomings in analyzing the problems. Specially at the boundary, where stress and deformation co-exist, the problem turns into a complex situation.\par
The formulation of two dimensional elastic problems used here was first introduced by Uddin \hyperref[b12]{[1]}, later Idris et. al. \hyperref[b3]{[5]}\hyperref[b4]{[6]}\hyperref[b5]{[7]} used it for obtaining analytical solutions of a number of mixed boundary value elastic problems and Ahmed \hyperref[b0]{[2,}\hyperref[b6]{[8]}\hyperref[b7]{[9]} extended its use where he obtained finite difference solution of a number of mixed boundary value problems of simple rectangular bodies. Later, Akanda developed a numerical scheme \hyperref[b8]{[10]}\hyperref[b9]{[11]}\hyperref[b10]{[12]} by which he solved irregular shaped elastic bodies under mixed mode of loading. This study focuses on the solution of the problem of rectangular beam with fixed connection.\par
In this paper stresses and deformations at different longitudinal and transverse sections of the beam have been plotted and discussed elaborately. For the convenience of analysis the body is divided into 30 meshes in y direction and 10 meshes in x direction. 
\section[{III.}]{III.} 
\section[{Formulation of the Problem}]{Formulation of the Problem}\par
Analysis of stresses in an elastic body is generally a three-dimensional problem. But in the cases of plane stress or plane strain, the stress analysis of three-dimensional body can easily be resolved into twodimensional problem. The problem studied here has been considered to be a plane strain problem. In the case of the absence of any body forces, the equations governing the three stress components ? x , ? y , ? xy under the state of plane stress or plain strain are as follows: \par
(\par
(3)\par
Replacement of the stress components in Eqs. (1-2) by their relations with the displacement components u and v makes Eq.(  {\ref 3}) surplus and Eq.( 1) and (2) becomes : (4) \hyperref[b3]{(5)} This two equations can be used for the solution. But it is still difficult to solve for two functions simultaneously. To overcome the difficulty, the two equations are transformed into a single equation with a single function. So a new function called displacement potential function (?) is defined as a function of displacement components to reduce the no. of governing differential equations into a single equation like following : \hyperref[b4]{(6)} When the displacement components in Eqs.(  {\ref 4}) and (  {\ref 5}) are replaced by ? (x,y) Eq.(  {\ref 4}) is automatically satisfied and the only condition that ? (x,y) has to satisfy becomes \hyperref[b12]{[1]}: \hyperref[b5]{(7)} So the problem is now reduced in such a way that a single function ? (x,y) has to be evaluated from the bi-harmonic Eq. (  {\ref 7}), satisfying the boundary conditions that are specified at the boundary. 
\section[{IV.}]{IV.} 
\section[{Boundary Conditions}]{Boundary Conditions}\par
Fig. \hyperref[fig_3]{1} shows the simple rectangular beam subjected to uniformly distributed load. Surface AB is under compression and points C, D are fixed. On the other hand AC and BD surfaces are free from any external load.\par
The normal and tangential stress components for the top surface AB are given by The normal and tangential stress components for the surfaces AC and BD are given by The normal and tangential stress components of the surface CD except the points C, D are given by The normal and tangential displacement components for the points C and D are given by In order to solve the problem using the biharmonic equation \hyperref[b5]{(7)} it is necessary to express the known boundary functions interms of the single potential function ? like following(8) (9) (10) (11) (12)\par
So it is clear that all the boundary conditions can be discretized in terms of displacement potential function ? with the help of finite difference technique.\par
V. 
\section[{Solution Procedure a) Method of Solution}]{Solution Procedure a) Method of Solution}\par
The finite-difference technique, frequently used to transform differential equations into corresponding algebraic equations has been used here to generate the algebraic equations of the bi harmonic Eq.(  {\ref 7}) and partial differential Eq. \hyperref[b6]{(8)}\hyperref[b7]{(9)}\hyperref[b8]{(10)}\hyperref[b9]{(11)}\hyperref[b10]{(12)} which are associated with boundary conditions. The discrete values of the potential function ? (x,y) at the mesh points of the domain concerned (Fig. \hyperref[fig_4]{2}) are obtained from a system of linear algebraic equations resulting from the discretization of the governing equation and the prescribed boundary conditions.\par
The problem is discretized in a desired number of mesh points and the value of ? (x,y) is evaluated only at these points. A false boundary, exterior to the physical boundary is introduced to keep the order of error of the difference equations to a minimum. The discretization of the domain concerned is shown in Fig. \hyperref[fig_4]{2}. The division into mesh points can be done in any regular and irregular shape. But due to the shape of the problem rectangular meshing is performed. The governing biharmonic equation which is used to evaluate the function only at the internal mesh points is expressed in its corresponding difference equation using central difference operators. Replacing the derivatives of Eq. (  {\ref 7}) with central difference formulae, the finite difference equation stands like following v(x,y)=0 for x/a=1 and y/b= 0, 1/30, 29/30, 1\par
Let O (i,j) is an internal mesh point as shown in Fig. \hyperref[b0]{(2)}. From Eq.(  {\ref 13}) it is evident that O has 13 mesh points including itself.   
\section[{b) Management of Boundary Conditions}]{b) Management of Boundary Conditions}\par
Boundary conditions are specified either in deformation parameter or stress parameter of combination of two. There are four possible combination of boundary condition. They are (1)normal stress and shear stress (2) normal stress and tangential displacement (3) shear stress and normal displacement (4) normal displacement and tangential displacement. Each mesh point on the physical boundary entertains two conditions at a time out of four. The computer program is organized in such a fashion that out of these two conditions one is used for evaluation of ? (x,y) at the concerned boundary point and the other point for the corresponding point on the exterior false boundary. Thus when the boundary conditions are expressed by their appropriate difference equations, every mesh point of the domain will have a single linear algebraic equation. Table \hyperref[tab_0]{1} is the demonstration of boundary conditions for all the surfaces namely AB, CD, AC and BD.  Where R=k/h R 4 \{?(i-2,j)+?(i+2, j)\}-4R 2 (1+R 2 )\{?(i-1,j) + ?(i+1,j)\}-4(1+R 2 )\{ ?(i,j+1)+ ?(i,j-1)\}+(6R 4 +8R 2 +6) ?(i,j)+2R 2 \{?(i-1,j-1) + ?(i-1,j+1) + ?(i+1,j-1) + ?(i+1,j+1)\}+ ?(i,j-2)+ ?(i,j+2)=0Right BD ? y , ? xy ? y /(i,n) ? xy /(i,n-1) Point C,D u,v u/(m-1,2), (m- 1\textbf{,}\par
(13)(14)\textbf{(15)}\par
In Fig.  {\ref 3}(a) and (b) the discretization process used for the two equations above has been demonstrated. 
\section[{Fig. 3(a)}]{Fig. 3(a)}\par
Referring to Fig. 4, assuming B as the corner mesh point, it is evident that B is common for both the edges AB and BD. So point B has four boundary conditions, two from each edges. In the present solution, three of the four conditions have been used and one is considered to be redundant. Thus the values of ? at the point B and 1 have been evaluated from the boundary conditions coming from the edge AB and point 2 has been evaluated from the boundary BD. Management of the boundary conditions at the corner mesh points has been demonstrated in table 2 for the problem shown in Fig. \hyperref[fig_3]{1}.   the system. But it has some short comings like it has very slow convergence rate and it fails to produce any solution for other complex boundary conditions. Considering all these, a triangular decomposition method has been used here ensuring better reliability and better accuracy of solution in a shorter period of time. The matrix decomposition method used here solves the present system of equations directly.  The finite difference equation corresponding to corner mesh point B of Fig. 4 is like following c) Solution of the system of algebraic equations A system of equations having a numerous number of unknowns needs to be solved in the present problem. An iterative method could be chosen in solving b/a, u increases. That means at the transverse sections nearer to mid transverse section, u is higher and it is highest at the mid section. 
\section[{b) Distribution of v}]{b) Distribution of v}\par
Distribution of displacement along y axis, v is shown in the Fig.  {\ref 7}. The distribution is counter symmetric about the transverse mid section of the beam. At the mid-section, v is zero. All these is in complete conformity with the physical condition and loading of the problem.\par
Fig. 8 shows the distribution of v along the depth of the beam. It is zero at the left most transverse section where value of x/a is 1. The effect of b/a ratio is also evident in the Fig. 8. With increasing b/a ratio, v decreases. That means at the inner sections the displacement along y direction dies down and at the mid-section there is no v value. c) Distribution of ? x Fig.  {\ref 9} shows the distribution of normal stress component ? x with respect to x at various longitudinal sections of the beam.  
\section[{Results And Discussion}]{Results And Discussion}\par
Distribution of the deformation parameters and stress parameters has been plotted along the length and depth of the beam. All the solutions of interest obtained through the ? formulation conform to the symmetric and anti-symmetric characteristics of the problem. Again the famous Saint-Venant's principle that the effects of sharp variation of parameter on the boundary die down and become uniform with the increase of distance of points in the body from the boundary is also supported by all the solutions obtained.\par
The elastic problem that has been studied here is considered to be made of ordinary steel having poison ratio, ?=0.3 and modulus of elasticity E=200 GPa. Uniformly distributed load is applied at the top surface of the beam. The stresses and deformation are symmetrically distributed about the transverse midsection of the beam. a) Distribution of u Fig. 5 shows the distribution of u, displacement along x axis, along the length of the beam. The distribution is symmetric about the mid-section of the beam. At the mid-section, value of u is maximum. Again u is zero at points C and D. All are in complete conformity with the physical condition of the beam. Change of the distribution of u with the increase of distance from the top fiber is also evident in the figure. The distribution is completely symmetric about the transverse mid-section of the beam. The distribution is negative at the top fiber and positive at the bottom fiber. At the mid longitudinal section, the value of ? y is negligible. All these are in complete conformity with the physical condition of the beam. The most critical points as far as ? y is concerned, are found at the fixed region and they are at around y/b= 0.03 and 0.97 respectively. Again it conforms to the available literature \hyperref[b7]{[9]} From Fig. \hyperref[fig_4]{12} the effect of b/a ratio on ? y can be studied. It is zero at almost all transverse sections except at b/a=0.1. At the section b/a=0.1, there are at least two critical points between 0.8 ? x/a ? 1.0 e) Distribution of ? xy Distribution of shear stress ? xy at various longitudinal sections(Fig. \hyperref[fig_3]{13}) of a beam with fixed connection reveals that shearing stress is almost zero at all the longitudinal sections. It conforms to the physical and loading condition of the beam under consideration. But again there are atleast two critical points found at the fixed connection region which again reveals that fixed connection is an extremely critical region where the probability of failure is maximum.\par
Finally Fig. 14 is a demonstration of shear stress at different transverse sections of the beam. Shear stress is almost zero at different sections except at the fixed region, which again conforms to the physical condition of the beam. 
\section[{VII.}]{VII.} 
\section[{Conclusions}]{Conclusions}\par
In the elementary formulas of strength of materials, the boundary conditions are satisfied in an approximate way. As a result it often fails to give the exact distribution of stresses at the restrained boundaries. The present displacement potential function approach is free from this type of limitations. So it is capable of representing the actual distribution of normal and shear stresses at critical regions. For example, from the elementary solution it is observed that the magnitude of the shearing stress is maximum at the mid-section of the beam. It is not supported by the numerical solution presented here,  x/a \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-2.png}
\caption{\label{fig_0}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-3.png}
\caption{\label{fig_1}F}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-4.png}
\caption{\label{fig_2}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{1}\includegraphics[]{image-5.png}
\caption{\label{fig_3}Figure 1 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{2}\includegraphics[]{image-6.png}
\caption{\label{fig_4}Figure 2 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-7.png}
\caption{\label{fig_5}}\end{figure}
          \begin{figure}[htbp]
\noindent\textbf{1} \par 
\begin{longtable}{P{0.23500000000000001\textwidth}P{0.11499999999999999\textwidth}P{0.40499999999999997\textwidth}P{0.095\textwidth}}
\tabcellsep Given\tabcellsep \multicolumn{2}{l}{Correspondence between mesh}\\
Boundary\tabcellsep boundary\tabcellsep \multicolumn{2}{l}{points and given boundary}\\
\tabcellsep conditions\tabcellsep \multicolumn{2}{l}{conditions}\\
\tabcellsep \tabcellsep condition/mesh\tabcellsep condition/mesh\\
\tabcellsep \tabcellsep point\tabcellsep point\\
\multicolumn{2}{l}{Top AB ? Bottom ?}\tabcellsep \\
CD\tabcellsep \tabcellsep \\
(except\tabcellsep \tabcellsep \\
point C\tabcellsep \tabcellsep \\
and D)\tabcellsep \tabcellsep \end{longtable} \par
  {\small\itshape [Note: x , ? xy ? x /(2,j) ? xy /(1,j) x , ? xy ?x /(m,j) ? xy /(m-1,j) Left AC ? y , ? xy ? y /(i,1) ? xy /(i,2)]} 
\caption{\label{tab_0}Table 1 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{2} \par 
\begin{longtable}{P{0.85\textwidth}}
corner mesh points\end{longtable} \par
  {\small\itshape [Note: [u,v, ? y ] (m-1,2),(m,2),(m-1,1) Point D [? y, ? xy ] on AC [u,v] on CD [u,v, ? y ] (m-1,n-1),(m,n-1),(m -1,n)Treating the transaction Point B]} 
\caption{\label{tab_1}Table 2 :}\end{figure}
 			\footnote{© 2013 Global Journals Inc. (US)} 		 		\backmatter  			 
\subsection[{Global}]{Global}\par
From the figure it is observed that the distribution is symmetric about the transverse midsection of the beam. It is also evident that the fixed region is the most critical region as far as the normal stress is concerned. It is supported by the available literature \hyperref[b7]{[9]}.\par
The effect of b/a ratio on the distribution of ? x is shown in Fig.  {\ref 10}. The value of ? x decreases with an increasing b/a ratio. The most critical points are found at the region of fixed connection which is conforming to the available literature \hyperref[b7]{[9]}. So it can concluded from the distributions shown in Fig.  {\ref 10} and Fig.  {\ref 11} that the most critical points with respect to ? x are around x/a =1, y/b=0 and x/a =1, y/b=1. x/a ? x /E because the approach used here reveals that shearing stress as well as normal stresses are maximum at the restrained boundaries. So, a more reliable and accurate study on the distribution of normal and shear stresses has been provided in the present literature. Besides, all the results obtained here are compared with the available literature and are in complete conformity with it and with the boundary and loading conditions. Such both the qualitative and quantitative results obtained through ?-formulation of a beam with fixed connection at the bottom surface and subjected under uniformly distributed load establish the soundness and acceptance of the approach adopted here.  
\subsection[{Global}]{Global}			 			  				\begin{bibitemlist}{1}
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\end{document}
