Elastic Euler Beam Research Articles

Nonlinear analysis of Euler beams resting on a tensionless soil with arbitrary configurations

BackgroundThe nonlinear interaction between an elastic Euler beam and a tensionless soil foundation is studied. The exact analytical solutions of the nonlinear problem are rather complicated. The main difficulty is imposing compatibility conditions at lift-off points. These points are determined as a part of the solution, although being needed to get the solution itself. In the current work, semi-analytical solutions are derived using the Rayleigh–Ritz method. The principle of vanishing variation of potential energy is adopted. The solution is approximated using a set of suitable trial functions. Accurate high-order approximate analytical solutions are obtained using MAXIMA symbolic manipulator. Lift-off points are identified through an iterative procedure and compatibility conditions are satisfied automatically. The methodology is designed to accommodate arbitrary configurations for the load distribution and the beam properties.ResultsExact solutions are revised briefly to verify the semi-analytical solutions in terms of deflection, bending moment, and shear. Semi-analytical solutions for constant beam properties including various support conditions and load distributions are verified. Convergence of high-order semi-analytical solutions is illustrated for cases including one and two contact points. A parametric study is provided to illustrate the effect of soil stiffness on the contact length. The case of a finite beam with free ends is considered. The semi-analytical solutions for variable beam moment of inertia are provided and verified.ConclusionsHighly accurate semi-analytical solutions can be obtained for the problem considered using the Rayleigh–Ritz method along with a symbolic manipulator. Arbitrary load and support configurations can be modeled, and the locations of lift-off points are well predicted. The semi-analytical solutions are extremely valuable for cases of variable moment inertia since exact solutions are rather rare.

Torsional response of a bisymmetric structure induced by bending–torsion interaction in vertical members

AbstractThe biaxially symmetric column induces torsional moment owing to the bending–torsion interaction when the top of the column is given horizontal displacement in the two directions of the principal axes of the cross section and the rotation of the column ends around the axes is constrained. Hence, horizontal ground motions in two directions can cause non‐negligible torsional vibration in a high‐rise building owing to autoparametric resonance. To investigate this phenomenon, in this study, we focus on a single‐story structure with an aggregated column, which is a model for columns or core structures in tall buildings and is assumed to be an elastic Euler beam; subsequently we derive the equations of motion under horizontal ground motions in two directions. The resonance condition in the torsional direction is also investigated. Time‐series response analyses are performed for input ground motions of harmonic waves, white noise, acceleration records of prior earthquakes, and a long‐period design wave, and the relationship between the ratio of two translational natural frequencies and the increase in maximum responses is explained. The results demonstrate that the absolute acceleration response to the long‐period design wave could increase by more than 50% in the worst‐case scenario.

ANALYTICAL SOLUTION FOR NONLINEAR INTERACTION OF EULER BEAM RESTING ON A TENSIONLESS SOIL

The nonlinear interaction between an elastic Euler beam and a tensionless soil foundation is studied. Exact analytical solutions of the challenging problem are rather complicated. The basic obstacle is imposing compatibility conditions at lift-off points. These points are determined as a part of the solution although being needed to get the solution itself. In the current work, solutions are derived using the approximate Rayleigh-Ritz method. The principal of vanishing variation of potential energy is adopted. The solution is approximated using a set of suitable trial functions. Lift-off points are identified through an iterative procedure and compatibility conditions are satisfied implicitly. Results are presented for various cases, including clamped support and free end condition. Various distributed loading conditions are analyzed. Exact solutions are revised briefly. Accurate high order approximate analytical solutions are obtained using MAXIMA symbolic manipulator. The convergence of approximate solutions towards the exact solutions is verified. For each case detailed results of deflection, bending moment and shear are presented.

Experiment of torsional response induced by the Q–delta resonance

SummaryBecause of geometric nonlinearities, vibrations in two horizontal directions can generate a torsional moment that induces a torsional response, even in a structure with no eccentricity; we term this phenomenon Q–Δ effect. In this study, the torsional response caused by the Q–Δ resonance was investigated by performing a shaking table test that involves a single‐layer symmetric specimen. The specimen was designed and fabricated by focusing on one of two resonance conditions. Its moment of inertia was adjustable by changing locations of weights, and the natural frequency in the torsional mode could be modified. We developed a numerical model of the specimen, in which the columns connected to the slab were integrated into an elastic Euler beam. It was confirmed that the torsional response increased near the predicted Q–Δ resonance point. In addition, the acceleration at the corner of the slab was significantly increased. The formulated equations of motion provided a better prediction of the actual phenomena. Because the specimen corresponds to a stiff and slender high‐rise building when converted into a full scale, the result suggests a need to be aware of the risk of torsional response increase due to the Q–Δ resonance.