Exact Solution of Thin walled Open Section Beam using a Coupled Field - PDF document

18 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS Exact Solution of Thin walled Open Section Beam using a Coupled Field Formulation Srinivasan Ramaprasad 1* , Darsi Nagendra Kumar 2 1 Center of Excellence Aerospace and Defense, Mahindra Satyam Computer Services Ltd., Bangalore, 2 Lead Engineer - Aircraft Systems, Cassidian Air Systems, EADS DS India Pvt. Ltd., Bangalore * Corresponding author (srinivasan_ramaprasad @ mahindrasatyam.com) Keywords : Composite Beams, Warping, Flexural, Torsional, Coupling, Anisotropic 1 Abstract response to applied shear, bending and twisting The exact static solutions of shear flexible thin loads. Also the end restraints cause a non-uniform walled laminated I-beams are derived using a out-of-plane warping and estimates of torsional coupled field formulation. The formulation stiffness based on Saint-Venant theory are accommodates the effect of elastic couplings due to inaccurate. The effect is even more significant in material anisotropy, shell wall thickness, warping open sections and Vlasov theory is normally adopted shear, transverse shear deformation and constrained to incorporate restraint warping effect. warping effects. The governing equations are first derived in terms of forces or stress resultants. The A 1-D mathematical model is usually used to spatial distribution of beam forces and the analyze a TWCB. The kinematics of the beam is displacements along the length of the beam are derived by expressing the local displacements in the derived in closed form. Examples of isotropic and thin-walled shells in terms of generalized beam laminated composite I-beam subjected to bending displacements which include extension, bending in and torsion forces are studied and compared two directions, shear in two directions, and the twist. favorably with available numerical results. The twisting includes the component of both St. Venant torsional moment and a bi-moment which arises due to restrained warping effect. Jung et.al [1] 2 General Introduction Composites are being extensively used as a material has compiled an extensive survey of existing of choice in the aircraft industry since they offer a numerical and analytical composite beam theories. high strength to weight ratio, increased fatigue life Chandra and Chopra [2] included the extension and improved damage tolerance performance. Thin bending - coupling stiffness, transverse shear effects walled structures are an integral component of a and generalized the theory to accommodate material typical aero structure. Structures like rotor blades, coupling due to unsymmetric laminate stacking wing spars can be modeled as one dimensional beam sequence. Jung et.al [3] developed a mixed method since their cross sectional dimensions are much applicable to coupled composite beams, with small compared to its length. Additionally they are arbitrary cross section. Numerical simulations were being increasingly used in aircraft structures as carried out and showed good accuracy with stiffeners whose primary objective is to improve experimental results. panel stability. Thin walled composite beams (TWCB ) demonstrate very complex behavior under Most of the numerical simulations for TWCB up to the application of bending and twisting loads. now have been carried out using finite element (FE) Several non-classical effects like material coupling, method due to its versatility. A displacement based transverse shear and restrained torsional warping 1-D FE model for flexural torsional buckling of must be included in developing an analytical model composite I-beams was developed by Lee and Kim for TWCB. [4]. Jaehong Lee [5] presented a shear deformable beam theory and applied it for the flexural analysis Composites have a very low shear modulus to of TWCB using a FE analysis. He introduced several extensional modulus and hence transverse shear non-classical effects displayed by them like deformation has a significant influence on their transverse shear, warping shear, material coupling,

restrained warping effects etc. Mira Mitra et.al [6] enables any aircraft designer to explore the influence presented a new super convergent TWCB element of different design parameters on the fully coupled for box beam analysis. They used the shape response of the TWCB. The solutions can be used to functions which satisfied the governing equations develop closed form expressions for any arbitrary cross section provided the corresponding stiffness’s and demonstrated superior convergence in FE results. are correctly derived from the geometry of the cross section. The developed solutions are validated by studying the bending and torsional response of Analytical solutions for static analysis of composite isotropic and laminated I-beams with NASTRAN beams have also been made by researchers. Jung and results. Lee [7] derived a closed form solution for the static response of both symmetric and anti-symmetric lay- 3 Theoretical Formulation up I-beam with transverse shear coupling and A Thin walled beam is characterized as a flexible included most of the non-classical effects in their body whose length is much larger than its cross formulation. Most of the analytical solutions have sectional dimensions. The kinematics of the thin been achieved for simplified laminate stacking walled beam is quite complex and is developed sequence. Obtaining the exact solution for the static based on some simplifying assumptions [10] as behavior or arbitrary laminated TWCB and mentioned below. including all the non-classical effects is very 1. The contour of a cross section does not deform difficult due to the complexities arising from the in its own plane. coupling effects of extensional, flexural and 2. A general plate segment of the beam is torsional deformations. D.K. Shin et.al [8] presented modeled as a thin plate but with shear the development of exact stiffness matrix for TWCB deformation. Hence, transverse shear strains with arbitrary lamination from the solution of and warping shear are introduced and assumed spatially coupled ordinary differential equations that uniform over the cross section. arise in the solution process. The exact stiffness 3. The tangential stress (“s”) is negligible and matrix was used to derive closed form expressions to symmetric laminated TWCB with various boundary stiffness of the plate is derived based on plane stress reduction from the 3D Constitutive conditions. From the foregoing literature it is evident equations. that closed form analytical solutions for thin walled 4. Each plate element in a cross section is composite beams are very few. governed by the First Order shear Deformation theory. The objective of the present study is to 5. The Shear strain in the Cross sectional plane is systematically develop the exact solutions to TWCB assumed to be zero. incorporating arbitrary lamination sequence and all The beam coordinate system is shown in fig. 1 non-classical effects. The solution is based on a new method called coupled field formulation (CFE) introduced by S. Ramaprasad [9] for the analysis of planar laminated curved beams. The CFE method [9] is particularly attractive for deriving exact solutions for 1-D problems since it not only satisfies the governing equations of motion, but also satisfies all the boundary conditions of the problem. In this work, the TWCB problem is formulated in terms of 8 generalized beam forces and six rigid body Fig.1. Beam Coordinate System The shell displacements are denoted by “ v ”, “ w ” and displacements for a total of 14 field variables. “ u ” respectively along the “s”,” n” and “x” Explicit simple closed form expressions for the displacements and forces are derived which are directions. The beam displacements in the element applicable even for a highly coupled lamination Coordinate system are denoted by U, V, and W sequence. The simplicity of the solutions developed along the beam axis and transverse to the beam axis