A shear center demonstration model using 3D-printing

Locating the shear, or flexural, center of non-symmetric cross-sectional beams is a key element in the teaching of structural mechanics. That is, establishing the point on the plane of the cross-section where an applied load, generating a bending moment about a principal axis, results in uni-directional deflection, and no twisting. For example, in aerospace structures it is particularly important to assess the propensity of an airfoil section profile to resist bending and torsion under the action of aerodynamic forces. Cross-sections made of thin-walls, whether of open or closed form are of special practical importance and form the basis of the material in this paper. The advent of 3D-printing allows the development of tactile demonstration models based on non-trivial geometry and direct observation.

The Comparative Study of Biaxial Bending Analysis of Steel Sections Using AISC and Eurocode Approaches

ABSTRACT This study investigates the capacity of the steel section using both AISC and Eurocode approaches by Robot Structure Analysis software. Three types of steel sections were subject to biaxial bending by applying loads to both main axes and examined by both approaches. The concept of Fisher was also adopted as an approach. The findings suggested that the Eurocode approach is more conservative in the design of steel sections subject to biaxial bending as it takes into account the level at which the load is applied, the type of the section whether rolled or welded and its height-to-width ratio (lateral buckling effect), the effects which are not considered in AISC approach. The AISC approach considers the shear center of the section as the level at which the loads are applied. The conservatism of the results was more pronounced when the section is close to H-section. Fisher`s concept of structural design of biaxial bending of structural steel is more conservative than both AISC and Eurocode approaches of analysis.

Three-dimensional nonlinear displacement-based beam element for members with asymmetric thin-walled sections

Asymmetric thin-walled sections such as steel angles and tees are widely used in truss structures and transmission towers. To address extreme limit states that these structures encounter due to extreme events such as hurricanes and earthquakes, it is important to capture their response due to large deformations caused by static or dynamic loading. In the nonlinear large deformation regime, these members have coupled axial-flexural-torsional deformation due to the so-called Wagner effect and the noncoincident shear center and centroid. A three-dimensional corotational total Lagrangian beam element is formulated and implemented in the OpenSees corotational framework to account for these coupling effects by invoking Green-Lagrange strains referenced to a basic system. In the basic system, shear forces and torque are defined with respect to the shear center, axial force is referred to the centroid, and flexure is defined around the section principle axes but in the planes containing the shear center. The element tangent stiffness matrix is derived through linearization of the governing equation obtained from the principle of virtual work. Cubic Hermitian functions for the transverse displacements and a linear shape function for the axial and torsional deformation are adopted in the development. Before conducting the corotational transformation, all element end forces and displacements are transformed to act about the shear center. In order to remedy membrane locking in the inextensional bending mode, the high order bending terms in the axial strain are replaced by a constant effective strain. Cyclic material nonlinearity is considered by discretizing the cross section into a grid of fibers, tracking the steel uniaxial stress-strain constitutive at each fiber, and performing numerical integration over the cross section to obtain the section stiffness matrix. The formulation is compared against a set of experimental and numerical results to validate that the element can model geometric and material nonlinearities accurately and efficiently.

A novel stress analysis method for composite Z-stiffeners under mechanical and thermal loads

A closed-form analytical solution is developed for analyzing laminated composite beam with asymmetric Z cross-section. The explicit expressions for evaluating sectional properties such as centroid, shear center, equivalent bending/torsional stiffness and warping stiffness are formulated based upon modified lamination theory and taken into consideration of the structural deformation characteristics of beam with narrow section. The ply stresses of flanges and web laminates are computed for composite Z-stiffener under axial, bending, and torsional loads. The present results give excellent agreement with the results from ANSYS™. A parametric study of their centroid and shear center with various layup sequences was performed by using the developed solution. It is found that the sectional properties are not only dependent of structural configuration but also the laminate property. Moreover, these properties are only dependent of structural configuration if the entire Z-stiffener is made of the same family laminates regardless their ply orientation and stacking sequence. It is concluded that the present approach is a viable and efficient method for designing composite Z-stiffener.

Effect of mass and shear center offset on the dynamic response of a rotating blade

In this work, a model that accounts for the extensional, chordwise, flapwise and torsional vibrations of a flexible rotating blade was developed. The model also takes into consideration the offset between the elastic and inertial axes of the blade. In order to account for the centrifugal stiffening effect, expression for the strain energy was obtained based on an ordering scheme that retains terms up to 2nd order. Hence, a set of four nonlinear coupled partial differential equations governing the deformations of the blade was derived. The linearized equations were non-dimensionalized and then spatially discretized by the FEM (Finite Element Method). State space techniques were used to obtain the blade's natural modes and response to initial excitation. Effect of the mass and shear center offset on the coupling between the modes and veering regions at different rotor speeds were investigated.