Abstract
Behaviour of beam depends on its depth. A beam is considered as deep, if the depth span ratio is 0.5 or more. In the available beam theories, we have to apply correction in case of deep beams. In the present work, method of initial functions (MIF) is used to study the effect of depth on the behaviour of concrete beam. The MIF is an analytical method of elasticity theory. It gives exact solutions of different types of problems without the use of assumptions about the character of stress and strain. In this method, no correction factor is required for beams having larger depth. Results are obtained for three different cases of depth span ratios and compared with available theory and finite element methodbased software ANSYS. It is observed that deep beam action starts at depth span ratio equal to 0.25.
Introduction
A beam is considered as a deep beam when the ratio of effective span to overall depth is <2.0 for simply supported members. The beam theories which are based on assumptions are useful in case of those problems, where thickness of beams is moderate. Available beam theories which are based on assumptions produces two types of errors. The first is the error in the stress and the second error is in the strains, i.e. in deflections. So we need a theory to analyse the beams having higher depth span ratio. In this paper, we have used method of initial functions (MIF) for the analysis of concrete beams of different depth span ratios. It gives exact solutions of different types of problems without the use of assumptions about the character of stress and strain. In comparison to Timoshenko beam theory which is used for analysis of deep beam, this method requires no assumption regarding position of neutral axis of beams and no shear correction factor is required.
A method was suggested for solving problems of theory of elasticity for the analysis of thick plates as well as shells and was known as the MIF. In this method, unknowns of the problem were expanded in Maclaurin’s series in the thickness coordinate and hence the solutions were obtained in terms of unknown initial functions on the reference plane (Vlasov 1957). Twodimensional elasticity equations were used in this method (Timoshenko and Goodier 1951).
Method of initial functions was used for the analysis of beams under symmetric central loading and uniform loading for different end conditions (Iyengar et al. 1974). It was used for the analysis of free vibration of rectangular beams of arbitrary depth. The frequency values were calculated for different values of Poisson’s ratio (Iyengar and Raman 1979). MIF had been applied for deriving theories for laminated composite thick rectangular plates. The governing equations had been obtained for perfectly and imperfectly bonded plates subjected to normal loads (Iyengar and Pandya 1986). Governing equations were developed for composite laminated deep beams by using MIF and results were compared with the available theory (Dubey 2000).
Method of initial functions has been applied for the composite beams having two layers of orthotropic material (Patel et al. 2012). MIF is successfully applied for the analysis of brickfilled reinforced concrete beams (Patel et al. 2013).
In deep beams, the bending stress distribution across any transverse section deviates appreciably from straight line distribution as assumed in the elementary theory of beam. Consequently, a transverse section which is plane before bending does not remain approximately plane after bending and the neutral axis does not usually lie at the middepth (Krishna Raju 2005).
There are so many other theories which are used in the place of prevailing theories for the analysis of beams. hyperbolic shear deformation theory was developed for transverse shear deformation effects. It was used for the static flexure analysis of thick isotropic beams. The results of the present theory are compared with those of other refined shear deformation theories of beams (Ghugal and Sharma 2011). A layerwise trigonometric shear deformation theory was used for the analysis of twolayered crossply laminated simply supported and fixed beams subjected to sinusoidal load. Virtual work principle was employed to obtain governing equations and boundary conditions (Ghugal and Shinde 2013). Keeping in view the limitations of theories in practice and advantages of MIF, it is clear that we can use this theory effectively for beams of any depth span ratio. Significance of the research is that the available theory like bending theory is not useful for the beam sections having more depth.
MIF formulation
The equations of equilibrium for solids ignoring the body forces for twodimensional case are:
The stress–strain relations for isotropic material are:
The values of the coefficients C′_{11}–C′_{33} for isotropic materials are given in the “Appendix”.
The strain displacement relations for small displacements are:
Eliminating σ_{ x } between Eqs. (1) and (2) the following equations are obtained, which can be written in matrix form as
where
Equation (9) can be expressed as:
The solution of Eq. (10) is
where {S_{0}} is the vector of initial functions, being the value of the state vector {S} on the initial plane.
If u_{0}, v_{0}, Y_{0} and X_{0} are values of u, v, Y and X, respectively, on the initial plane, then
Expanding Eq. (13) in the form of a series
Consider a simply supported beam of isotropic material having length l, depth, d and loaded with uniformly distributed load P in the y direction.
The bottom plane of the beam is taken as the initial plane. Due to loading at the top plane of the beam one has X_{0} = Y_{0} = 0.
On the plane, y = d, the conditions are X = 0, Y = −P.
Y = −P on y = d, after simplification yields the governing partial differential equation:
Initial functions are obtained by substituting the value of Φ:
From the value of initial functions the value of displacements and stresses are obtained.
Analysis of concrete beams
The following values of concrete beam dimensions are chosen for the particular problem,
d = 400, 750 and 1,500 mm, l = 3,000 mm
The following material properties are taken, E = 22,360 N/mm^{2}, G = 10,164 N/mm^{2}, µ = 0.10 (Fig. 1).
The boundary conditions of the simply supported edges are:
The boundary conditions are exactly satisfied by the auxiliary function Φ = A_{1}sin (πx/l). A uniformly distributed load P = 20.0 N/mm is applied, on the top surface of the beam. The value of auxiliary function Φ is obtained from Eq. 15. Using this value of auxiliary function, the values of initial functions u_{0} and v_{0} are obtained from Eq. 16. These are substituted in Eq. 11 for obtaining the values of displacements and stresses.
Results and discussion
Numerical results have been given in Tables 1, 2 and 3 for uniformly distributed load. The values of displacements and stresses are obtained using MIF for different depth span ratios. The results obtained by MIF are compared with bending theory and FEMbased software ANSYS. Twodimensional analyses are performed taking PLANE 183 element. Material properties E and µ are required for concrete. Mapped meshing is used for the modelling. The displacements and stresses across the thickness in the particular problem of concrete beams are presented in Figs. 2, 3, 4, 5, 6, 7, 8 and 9.
Conclusions
It is observed that the deep beam action is seen at d/l = 0.25 which is less than the specified value d/l = 0.5. It is necessary to apply an appropriate method of analysis for beams having large depth span ratio. Deep beam effect is not seen in bending theory based on assumptions that transverse sections which are plane before bending remain plane after bending. MIF gives correct result for both shallow and deep beams. In this method we need not apply corrections. Also in this method it is not necessary to assume the position of neutral axis; it incorporates the position of neutral axis by itself. So we can conclude that analysis done by MIF provides more realistic behaviour of beam sections of any depth.
Abbreviations
 L :

Effective span of beam
 d :

Total thickness of beam
 E :

Young’s modulus of elasticity
 G :

Shear modulus of elasticity
 µ :

Poisson’s ratio
 ε :

Strain
 σ _{ x } :

Bending stress
 σ _{ y } :

Normal stress
 τ _{ xy } :

Shear stress
 u :

Displacements in x direction
 v :

Displacements in y direction
 α :

$\frac{\mathit{\partial}}{\mathit{\partial}x}$
References
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Patel, R., Dubey, S.K. & Pathak, K.K. Effect of depth span ratio on the behaviour of beams. Int J Adv Struct Eng 6, 3 (2014). https://doi.org/10.1007/s4009101400563
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Keywords
 Method of initial functions
 Deep beam
 Displacement
 Stress
 Depth span ratio