Determination of the Critical Buckling Load of Shear Deformable Unified Beam

Abstract This paper presents a finite element formulation for the determination of the critical buckling load of unified beam element that is free from shear locking using the energy method. The formulated element is used for the determination of the critical buckling load of beams with different boundary conditions. The developed formulae for the determination of critical buckling load are based on the effect of shear deformation. Numerical results for the critical buckling load of the classical Euler-Bernoulli beam and Timoshenko beam are presented and compared with the exact solutions. It is shown that the proposed technique provides a unified approach for the stability analysis of beams with any end conditions. It is concluded that for design purposes, shear deformation may be safely ignored for beam span-to-depth (l/d) ratio greater than 10, while for l/d ratio of 5 and less, shear deformation effect is significant and should be accounted for.

  The coefficients 1 a , 2 a , 3 a and 4 a are determined from the boundary conditions of the beam element: The coefficients 1 b , 2 b and 3 b are determined from the boundary conditions:

Bending-Shear Interaction Factor
To ensure continuous interaction between the bending and shear components as a function and to avoid the use of partial derivatives, an expression for the total cross sectional rotation  is proposed as: where ) (x  is the total cross-sectional rotation of the beam is the cross-sectional rotation of the Euler-Bernoulli beam ) (x s  is the cross-sectional rotation of the shear beam  is the bending-shear interaction factor and is expressed as the ratio of bending strain energy to total strain energy of a simply-supported beam under load. That is: where  = ( 1 1 ) where E is the elastic modulus of the beam material. I = moment of inertia of the beam section. Consider a simply supported beam with a point load P at midspan. The bending moment at a section, distance x from a support, is given by: Since the maximum bending moment occurs at midspan (x=L/2),


( 1 3 ) The shear force at any section, distance x from a support, is: The integral expressions for shear strain energy is given by the familiar expression dx kAG Substituting for ) (x Q in Equation (15) gives the shear strain energy as: where E= Young's modulus G= shear modulus A= cross-sectional area k = shear coefficient depending on the shape of cross-section. Edem [8] proposed that the bending-shear interaction factor,  , be based on the value of  for midspan point load, i.e. Equation (17).

Beam Element Stiffness Matrix
The bending strain energy of the Euler-Bernoulli beam is given Equation (11): Using the expression for the total cross-sectional rotation (Equation 9), the total energy in the unified beam element under a distributed normal load q is expressed as: From Equations (3) and (7): for the shear beam From Castigliano's first theorem, the stiffness coefficient K ij is given by The assembled unified beam element stiffness matrix is K is

Beam Element Stability Matrix
The stability matrix is formulated using the kinetic energy principle. Consider a long compression member shown in  If an axial load P is applied and increased slowly, it will ultimately reach a value P cr that will cause buckling of the beam or column. P cr is called the critical buckling load of the beam or column. The total kinetic energy for the rotation of the cross-section due to bending is given by Following the finite element method philosophy, the element displacement field is interpolated for flexure and shear rotation by shape functions, Equations (3) and (7): The rotation of the cross-section is given by Equation (9): where the dx The stability coefficient is given by The assembled unified beam element stability matrix is The strain energy due to bending is: The force P will move through a distance given by Thus, the total potential energy is given by

  
(32) For equilibrium, the total potential energy is minimum: i.e.
, , 0 For stability analysis and determination of the magnitude of the static compressive axial load that will cause the beam to buckle, the lateral load q=0. Thus, the critical buckling load P cr that satisfies Equation (33) is: But from Equations (22) and (28): Substituting into Equation (34) and taking u=w:

Presentation And Discussion Of Results
Consider the propped cantilever beam loaded axially with P as shown in Fig. 5:   i.e.  3 Substituting for  in Equation (39) and L/d=100, the critical buckling load of the beam is: In general, the Euler critical buckling load for any support condition is given by: Similar finite element analysis using a 2-element mesh is performed for other support conditions of the beam and the results are presented in Table 1:  3 d/L The effect of span-to-depth (l/d) ratio of beam on the critical buckling load for different support conditions is presented in Table 2. The parameters assumed for the beam section are, [11]: (i) Shear correction factor, k=5/6 (ii) Poisson's ratio, v=0.25 Length of beam =1.0 m The beam was discretised into five finite elements in each case. The results in Table 1 demonstrate the advantage of the shear locking-free unified beam elements. The results show that it requires just a 2-element mesh to produce excellent results, in contrast to the 16-element mesh used by Reddy [11] for the same problem. The unified beam element thus models shear deformation extremely closely, [12]. The results also show that the shear deformation parameter  influences the critical buckling load. As the beam changes from stocky to slender, the shear deformation parameter reduces from 1 to 0, and the critical buckling analysis reduces to classical solution. This is also applicable to the effective length factor. The results in Table 2 show that shear deformation contributes less than 5% to the critical buckling load for L/d ratio greater than 10. Also, the critical buckling load increases as the L/d ratio decreases The results in Table 3 demonstrate the accuracy of the unified beam element model for both deep and slender beams in comparison with the analytical and Timoshenko beam models.

Conclusion
In this paper, a unified beam element model of the Euler-Bernoulli and Timoshenko beam theories is developed. Bending stiffness and stability coefficients of the unified beam element are derived by employing a bendingshear interaction factor. Explicit formulae for the critical buckling loads and effective lengths of Timoshenko beams based on the proposed unified element have been developed for different support conditions. The numerical examples presented demonstrate the validity and accuracy of the proposed unified beam element model in the evaluation of the critical buckling load of beams. The results suggest that for design purposes, shear deformation may be safely ignored for beam span-to-depth (l/d) ratio greater than 10, while for l/d ratio of 5 and less, shear deformation effect is significant and should be accounted for.