A Refined Zigzag Beam Theory for Composite and Sandwich Beams

Abstract

A new refined theory for laminated composite and sandwich beams that contains the kinematics of the Timoshenko Beam Theory as a proper baseline subset is presented. This variationally consistent theory is derived from the virtual work principle and employs a novel piecewise linear zigzag function that provides a more realistic representation of the deformation states of transverse-shear flexible beams than other similar theories. This new zigzag function is unique in that it vanishes at the top and bottom bounding surfaces of a beam. The formulation does not enforce continuity of the transverse shear stress across the beam's cross-section, yet is robust. Two major shortcomings that are inherent in the previous zigzag theories, shear-force inconsistency and difficulties in simulating clamped boundary conditions, and that have greatly limited the utility of these previous theories are discussed in detail. An approach that has successfully resolved these shortcomings is presented herein. Exact solutions for simply supported and cantilevered beams subjected to static loads are derived and the improved modelling capability of the new "zigzag" beam theory is demonstrated. In https://ntrs.nasa.gov/search.jsp?R=20090020418 2019-04-30T01:36:45+00:00Z 2 particular, extensive results for thick beams with highly heterogeneous material lay-ups are discussed and compared with corresponding results obtained from elasticity solutions, two other "zigzag" theories, and high-fidelity finite element analyses. Comparisons with the baseline Timoshenko Beam Theory are also presented. The comparisons clearly show the improved accuracy of the new, refined "zigzag" theory presented herein over similar existing theories. This new theory can be readily extended to plate and shell structures, and should be useful for obtaining relatively low-cost, accurate estimates of structural response needed to design an important class of high-performance aerospace structures. KEYWORDS

Two-noded zigzag beam element accounting for shear effects based on an extended Euler Bernoulli theory

“…The beam segment is defined in the ( , ) x z Cartesian system, where x is the beam longitudinal axis and z is the thickness coordinate. The superscript k denotes quantities within the th k layer with Using the same notation as for the RZT [18,19,21,23], we propose the following displacement field for the th k layer…”

“…Different works [18][19][20] have shown that beam models requiring 1 C continuity for the deflection field, such as that presented in Eq. (1), have theoretical difficulties to represent correctly the shear strain and the shear stress distributions in a clamped support where the following kinematic conditions are required…”

“…Zigzag theories typically have as baseline the kinematics assumptions of the classical EulerBernoulli [14,15] and Timoshenko [16][17][18][19] beam theories (EBT and TBT). Some researchers consider that the 1 C continuity requirement for the deflection field of the EBT is a drawback versus the simpler 0 C requirement of the TBT [18,20].…”

“…Some researchers consider that the 1 C continuity requirement for the deflection field of the EBT is a drawback versus the simpler 0 C requirement of the TBT [18,20]. The main reason for this consideration is the theoretical difficulty for the 1 C approximation to satisfy the equilibrium of forces at clamped supports.…”

“…Tessler et al [18,19] have recently developed a Refined Zigzag Theory (RZT) for beams and plates based in the TBT kinematic assumptions. In this theory the zigzag functions have the property of vanishing on the top and bottom surfaces of the laminate.…”