# Introduction beam may be considered as one of the most commonly used structural elements in engineering applications. A beam is said to be a deep beam when the depth is comparable to its span. Design of deep beams based on classical Euler bending theory can be seriously erroneous, since the simple theory of flexure takes no account of the effect of normal pressures on the top and bottom edges of the beam caused by the loads and reactions (Chow, Conway and Winter, 1952). The effect of normal pressures on the stress distribution in deep beams is such that the distribution of bending stresses on vertical sections is not linear and the distribution of shear stresses is not parabolic. Consequently, a plane transverse section does not remain plane after bending, and the neutral axis does not lie at the mid-depth, which eventually causes the basis of classical theory to be violated. In an attempt to make up the limitation, different theories as well as methods of solution have been reported in the literature (Conway, Chow and Morgan, 1951;Conway and Ithaca, 1953;Murty, 1984;Suzuki, 1986). However, each solution possesses certain limitations, and eventually none of the solutions are found to conform to all the physical characteristics of the problem for deep beam appropriately. Even, photoelastic studies (Uddin, 1966), finite element analysis (Hardy and Pipelzadeh, 1991) and finite difference solutions (Ahmed, Idris and Uddin, 1966; Ahmed, Khan and Uddin, 1998; Ahmed, Idris and Uddin, 1999; Ahmed, Hossain and Uddin 2005; Akanda, Ahmed and Uddin, 2002) have also been carried out for deep beams on two supports, mainly because all the physical conditions imposed on the beam could not be fully taken into account in the analytical methods of solution. Among the existing mathematical models of elasticity for the plane boundary-value problems, the stress function approach and the displacement formulation are noticeable. The stress function approach accepts boundary conditions in terms of loading only; boundary restraints cannot be satisfactorily imposed on it. On the other hand, the displacement formulation involves extreme difficulty especially when the boundary conditions are a mixture of restraints and stresses. As a consequence, serious attempts had hardly been made in the past for stress analysis using this formulation. As such, neither of the existing formulations is suitable for solving problems of mixed boundary conditions. Further, the use of standard structures, like beams, columns, etc. with guides on part or full of their bounding surfaces is receiving increased importance in order to satisfy precise and strict design criteria in many of the engineering applications. Guided boundaries usually help in reducing the level of deformation in the structural elements, which eventually resist the change of the original shape of the bounding surfaces under loading. But structures with guided boundaries always remain away from the scope of analytical solutions, because the physical conditions of guided boundaries need to be mathematically modelled in terms of a mixed mode of boundary conditions. Since the exact analytical solution of mixedboundary-value elastic problems, is beyond the scope of existing mathematical models of elasticity, the use of a new mathematical formulation will be investigated to analyze the elastic behavior of a guided deep beam under three point bending loading and support arrangements. It would be worth mentioning that, as far as the reporting in the literature is concerned, the author has not come across any reliable study of the present problem. Therefore, the analytical solution for a guided deep beam under three point bending is yet to be developed. # II. # Boundary Conditions The physical conditions at different boundaries of the beam are expressed mathematically as follows: ? u x = 0 at the edge of x = 0 ? u x = 0 at the edge of x = L ? ? xy (0,y) = 0 at the edge of x = 0 ? ? xy (L,y) = 0 at the edge of x = L ? ? xy (0,y) = 0 at the edge of y = 0 ? ? xy (0,y) = 0 at the edge of y = D ? The lateral stress at the edge of y = D is related to the applied load for the three point bending. Since the point load is actually acting over a certain area of the beam, for instance it can be considered for the length of x=0.45L to 0.55L. Again it is considered that the load intensity is ? o . Therefore, the magnitude of point load, P= 0.1L? o . Then for x=0.45L to 0.55L ? Similarly, the lateral stress at the edge of y = 0 is related to the reactions at the support. In this case ? yy (x,0) = ? o /2 for x=0 to 0.1L and 0.9L to L. # Analytical Solution The equation of equilibrium for isotropic material is as follows (Timoshenko and Goodier, 1970): 0 2 4 4 2 2 4 4 4 = ? ? + ? ? ? + ? ? y y x x ? ? ? (1) The expressions of displacement and stress components in terms of function ?(x, y) are as follows (Nath, Ahmed and Afsar, 2006): ( ) y x y x u x ? ? ? = ? 2 , (2a) ( ) ( ) ? ? ? ? ? ? ? ? ? + ? ? + ? = 2 2 2 2 1 2 1 1 , y x y x u y ? µ ? µ (2b) ( ) ( ) ? ? ? ? ? ? ? ? ? ? ? ? + ? = 2 2 2 3 2 1 , y y x E y x xx ? µ ? µ ? (2c) ( ) ( ) ( ) ? ? ? ? ? ? ? ? + ? ? ? + + ? = 3 3 2 3 2 2 1 , y y x E y x yy ? ? µ µ ? (2d) ( ) ( ) ? ? ? ? ? ? ? ? ? ? ? ? + ? = 2 3 3 3 2 1 , y x x E y x xy ? µ ? µ ? (2e) The potential function ?(x, y) is first assumed in a way so that the physical conditions of the two opposing guided ends are automatically satisfied. At the same time solution has to satisfy the 4 th order partial differential equation. After a long trial and error process, the solution of the governing equation ( 1) is thus approximated as follows: cos ) ( ) , ( y K x y Y y x m m + = ? ? = ? ?(3) where, ) ( y f Y m = , ) L / m ( ? = ? , K is an arbitrary constant and m= 1, 2, 3, ??.. ? . 3) with respect to x and y are substituted in Eq. ( 1) and following equation is obtained: # Derivatives of equation ( Y m //// ? 2? 2 Y m // + ? 4 Y m = 0 (4) The solution of the above 4 th order ordinary differential equation with constant coefficients [Eq. ( 4)] can normally be approximated as follows: ( ) Now, the reactions on the bottom boundary (y = 0) are acting over the two supports. It is considered that the supports are located at x=0 to 0.1L and x=0.9L to L respectively. The total length for reaction is 20 percent of beam length. Now the compressive load exerted at the mid-span on the edge D y = of the beam may be considered as acting over at least some length of the beam, for instance x= 0.45L to 0.55L. As a result the intensity of reaction is half of the load intensity. Therefore, the reactions over the beam at the supports can be taken as Fourier series in the following manner: y x y x u x ? ? ? = ? 2 , ( ) ( ) [ ] x e y D e C e y B e A m y m y m y m y m ? ? ? ? ? ? ? ? ? ? sin 1 1 1 ? ? = ? ? ? ? ? + + ? = (7a) ( ) y x u y , ( ) ( ) ? ? ? ? ? ? ? ? ? + ? ? + ? = 2 2 2 2 1 2 1 1 y x ? µ ? µ ( ) ( ) ( ) ( ) ( ) ( ) ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? + + ? + ? ? ? + + ? + ? = ? ? = ? ? 1 2 2 1 6 cos 2 2( ) ( ) ? ? ? ? ? ? ? ? ? ? ? ? + = 3 3 2 3 2 1 , y y x E y x xx ? µ ? µ ? ( ) ( ) ( ) ( ) ( ) ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? ? + + ? ? + + ? + + + + + + ? = ? ? = ? ? K x e y y D e C e y y B e A E m y m y m y m y m µ ? ? µ µ? ? µ ? µ µ? ? µ ? µ ? ? ? ? 6 cos 1 3 1 1 3 1 1 2 1 2 (7c) ( ) ( ) ( ) ? ? ? ? ? ? ? ? + ? ? ? + + ? = 3 3 2 3 2 2 1 , y y x E y x yy ? ? µ µ ? ( ) ( ) ( ) ( ) ( ) ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? ? + ? + + + + + ? ? ? + ? ? + ? = ? ? = ? ? K x e y y D e C e y y B e A E m y m y m y m y m 6 cos 1 1 1 1 1 2 1 2 ? ? µ µ? ? µ ? µ µ? ? µ ? µ ? ? ? ? (7d) ( ) ( ) ? ? ? ? ? ? ? ? ? ? ? ? + ? = 2 3 3 3 2 1 , y x x E y x xy ? µ ? µ ? ( ) ( ) ( ) ( ) ( ) ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + + + + + + + + + ? = ? ? = ? ? x e? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? = 20 9 sin 20 11 sin 2 0 L L L ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? = 20 9 sin 20 11 sin 2 0 ? ? ? ? m m m (8b) The reaction load at the support on the edge 0 = y can also be given by a Fourier series as follows: and using Eq. (9a) and Eq. ( 12) the arbitrary constant K can be obtained as follows: L L L m xdx xdx L E ? ? ? ? ( ) ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? = 10 9 sin sin 0 10 sin 0 L L L L ? ? ? ? ? ( ) ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? =100 = y ( ) ( ) ( ) ( ) { } ( ) ( ) ? ? ? ? = ? = ? = + = ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? ? + ? + + + + ? + + + ? + ? 1 1 2 3 2 3 2 1 2 cos 6 cos 3 3 cos 2 1 m o m m m m m m m m m m m E x E K x D C B A x D C B A E ? ? ? ? ? ? ? ? ? ? µ µ(12) Therefore, ( ) ( ) ( ) ( )( ) [ ] m m m m m E D C B A E ? + ? + + + µ ? µ µ ? µ µ ? 1 1 1 1 1 ( ) 10 6 1 0 0 2 ? µ = = + ? E K E or, ( ) E K 60 1 2 0 µ ? + ? = (15) The simultaneous equations ( 10), ( 11), ( 13) and ( 14 # ( ) 2 2 11 1 µ ? + ? = E Z Therefore, stress and displacement components at various points of the beam can be obtained using equations of (7). IV. # Result Analysis The analytical solutions of displacement and stress components are obtained for various aspect ratios (L/D) of the beam. The material of the beam is mild steel whose modulus of elasticity is E=209 X 10^9 and poison's ratio ?=0.3. The result of a guided isotropic beam having aspect ratio two and the uniform loading parameter 0 ? = 40 N/mm is presented in sequence of axial displacement (u x ), lateral displacement (u y ), bending stress (? xx ), normal stress (? yy ) and shearing stress (? xy ). Axial displacements (u x ) are found to be zero at the mid-section of span and at the lateral guided boundaries. Zero value of u x at the guided ends verifies the boundary condition of those edges of the beam. Axial displacements distribution is found skewsymmetric about the mid-span of the beam. The values of u x for sections 0