Static and Dynamic Stability Analysis of a Functionally Graded Timoshenko Beam

Abstract

This article presents the evaluation of static and dynamic behavior of functionally graded ordinary (FGO) beam and functionally graded sandwich (FGSW) beam for pined–pined end condition. The variation of material properties along the thickness is assumed to follow exponential and power law. A finite element method is used assuming first order shear deformation theory for the analysis. The element chosen is different from the conventional elements as the shape functions of the element are obtained from the exact solution of the static part of governing differential equation derived according to Hamilton's principle. Moreover, the shape functions depend on length, cross-section and material properties which ensure better accuracy of the solution. The effect of power law index on critical buckling load and natural frequencies of FGO beam is investigated. The critical buckling load of FGO beam with steel-rich bottom increases with power law index whereas the trend reverses for beam with Al-rich bottom. The first three natural frequencies of FGO beam are found to decrease to a minimum value and then increase as the power law index increases from one. The dynamic stability of FGO beam with steel-rich bottom is found to be more than that of beam with Al-rich bottom. An FGSW beam with alumina as bottom skin, steel top skin, and mixture of alumina and steel as core is chosen for analysis. The critical buckling load of FGSW beam increases with the increase of core thickness for variation of material properties in core as per power law with index more than one, whereas it decreases with the increase of core thickness for variation of material properties in core as per exponential law. The first three natural frequencies of the FGSW beam increase with the increase of FGM core for both the types of property distributions. The dynamic stability of the FGSW beam is enhanced as the thickness of the FGM core is increased.

Dynamic stability of moderately thick beams and frames with the use of harmonic balance and perturbation methods

“…The harmonic balance method using the Fourier series with period 2T (12) The first three areas of dynamic instability are determined based on equations (18), (20) and (24).…”

“…Various aspects for Timoshenko beam theory have been investigated, for example, beams with asymmetric cross-section [11], beams excited by a sequence of moving masses [12], twisted beams [13,14], beams with damping [14] and beams under tangential follower forces [15][16][17]. The Timoshenko beam theory can also be used to study dynamic stability of sandwich beams [18][19][20]. With the resonance areas identified, dynamic parameters of the system can be determined, these parameters being very sensitive to damage in structures [21][22][23].…”

Finite Element Modelling for Static and Free Vibration Response of Functionally Graded Beam

“…Najeeb and Alam (2012) presented a one dimensional finite element model using an efficient layerwise (zigzag) theory for the dynamic analysis of laminated beams integrated with piezoelectric sensors and actuators. Mohanty et al (2012) presented the evaluation of static and dynamic behavior of functionally graded ordinary (FGO) beam and functionally graded sandwich (FGSW) beam for pined-pined end condition. The variation of material properties along the thickness is assumed to follow exponential and power law.…”

Strength of the Three Layer Beam with Two Binding Layers

“…They solved the dynamic problem in the frequency domain of soft core sandwich beams. Mohanty et al [14] presented the evaluation of the static and the dynamic behavior of functionally graded Timoshenko beams. Smith et al [20] and Szyniszewski et al [22], [23] characterized mechanical properties of hollow sphere steel foam.…”