Journal of Engineering Mechanics

Computer analysis of structures has traditionally been carried out using the displacement method combined with an incremental iterative scheme for nonlinear problems. In this paper, a Lagrangian approach is developed, which is a mixed method, where besides displacements, the stress resultants and other variables of state are primary unknowns. The method can potentially be used for the analysis of collapse of structures subjected to severe vibrations resulting from shocks or dynamic loads. The evolution of the structural state in time is provided a weak formulation using Hamilton’s principle. It is shown that a certain class of structures, known as reciprocal structures, has a mixed Lagrangian formulation in terms of displacements and internal forces. The form of the Lagrangian is invariant under finite displacements and can be used in geometric nonlinear analysis. For numerical solution, a discrete variational integrator is derived starting from the weak formulation. This integrator inherits the energy and momentum conservation characteristics for conservative systems and the contractivity of dissipative systems. The integration of each step is a constrained minimization problem and it is solved using an augmented Lagrangian algorithm. In contrast to the traditional displacement-based method, the Lagrangian method provides a generalized formulation which clearly separates the modeling of components from the numerical solution. Phenomenological models of components, essential to simulate collapse, can be incorporated without having to implement model-specific incremental state determination algorithms. The state variables are determined at the global level by the optimization method.

A theoretical formulation of the linear elastic in-plane and torsional behavior of corrugated web I-girders under in-plane loads is presented. A typical corrugated web steel I-girder consists of two steel flanges welded to a corrugated steel web. Under a set of simplifying assumptions, the equilibrium of an infinitesimal length of a corrugated web I-girder is studied, and the cross-sectional stresses and stress resultants due to primary bending moment and shear are deduced. The analysis shows that a corrugated web I-girder will twist out-of-plane simultaneously as it deflects in-plane under the action of in-plane loads. In the paper, the in-plane bending behavior is analyzed using conventional beam theory, whereas the out-of-plane torsional behavior is analyzed as a flange transverse bending problem. The results for a simply supported span subjected to a uniformly distributed load are presented. Finally, finite element analysis results are presented and compared to the theoretical results for validation.

The effect of size and the combined effect of the size and moisture sorption of test pieces on the long term creep behavior of wood composites were studied. Small-, wide-, and semisize test pieces from each of three commercial wood composites, particleboard (PB), oriented strand board (OSB), and medium density fiberboard (MDF) were tested. Three exposure regimes, constant 20°C/65%, a single change from 20°C/65% to 20°C/85%, and cyclic changes between 20°C/30% and 20°C/90% relative humidity (RH), were used. It was found that the width of test pieces had no effect while the length had a significant effect on long term behavior of wood composites, but the effect is in contrast to that of short term modulus of rupture (MOR) which ranged from 0.06 to 0.13 for the shape parameter and from 0.09 to 0.26 for the length parameter depending on the types of wood composites. The average ratio of the relative creep $\left(k_c\right)$ of small-:wide-:semisize was 1.14:1.13:1.00 for PB, 1.26:1.21:1.00 for MDF, and 1.24:1.24:1.00 for OSB, with the shape parameter ranging from 0.04 to 0.19 and the length parameter from 0.13 to 0.26. Change in RH significantly aggravated the size effect on $k_c$ with the most significant under cyclic RH, for which the ratio of $k_c$ small- to semisize was 1.45 for PB and 1.27 for OSB after 3 months’ exposure. Edge sealing on small test pieces efficiently prevented the effect of moisture sorption but the size effect on $k_c$ with a reduction of about 30 and 20%, respectively, for edge sealed PB and OSB in weekly changing climate between 20°C/30% and 20°C/90% RH compared to unsealed small-size test pieces. The findings elucidate the importance to take into account the size effects on short term strength, compensated size effect on long term creep, and the combined effects of size and moisture on long term behavior when predicting the long term load carrying capacity of wood composites in construction.

A method for analyzing the free vibration of complex structural systems, consisting of a simple oscillator attached to a beam with an internal hinge, is presented. A mathematical model possessing important features of singularity functions with their higher order derivatives is proposed to account for the effect of internal hinge and spring force interacting between the oscillator and supporting structure. The particular integral approach together with Laplace transformation is proved to be an efficient alternative to solve the generalized differential equation for the normal modes of dynamically combined systems. Exact vibration frequencies for clamped–pinned and pinned–pinned boundaries are determined. The results are extended to the cases in which the oscillator or internal hinge is removed from the structure. It is shown that the presence of internal hinge does not alter the generalized orthogonality relation for the combined system. The search for the optimal location of the internal hinge, which maximizes the desired natural frequency, is discussed. It is concluded that a combined system with an optimally positioned hinge vibrates with the same natural frequency as an equivalent system without a hinge.

Using the interval finite-element method, the vibration control problem of structures with interval parameters is discussed, which is approximated by a deterministic one. Based on the first-order Taylor expansion, a method to solve the interval dynamic response of the closed-loop system is presented. The expressions of the interval stiffness and interval mass matrix are developed directly with the interval parameters. With matrix perturbation and interval extension theory, the algorithm for estimating the upper and lower bounds of dynamic responses is developed. The results are derived in terms of eigenvalues and left and right eigenvectors of the second-order systems. The present method is applied to a vibration system to illustrate the application. The effect of the different levels of uncertainties of interval parameters on responses is discussed. The comparison of the present method with the classical random perturbation is given, and the numerical results show that the present method is valid when the parameter uncertainties are small compared with the corresponding mean values.

A nondestructive evaluation technique established on the basis of a global minimization method is presented for the system identification of laminated composite plates partially restrained by elastic edge supports. Six natural frequencies extracted from the vibration data of the flexibly restrained plate are used to identify the system parameters of the plate. In the identification process, the trial system parameters are used in the Rayleigh–Ritz method to predict the theoretical natural frequencies of the plate, an error function is established to measure the sum of the differences between the experimental and theoretical predictions of the natural frequencies, and the global minimization method is used to search for the best estimates of the parameters by making the error function a global minimum. The accuracy and efficiency of the proposed technique in identifying the parameters of several flexibly supported plates made of different composite materials are studied via both theoretical and experimental approaches. The excellent results obtained in this study have validated the applicability of the proposed technique.

Under pulse-type ground motions modal analysis is not quite efficient for estimating the elastic response of multi-degree-of-freedom systems, in particular when the effects of higher modes are significant. This paper first shows that the assumption of nondispersive damped waves for shear beams leads to inconsistent response estimation. Subsequently, a closed form time domain dispersive damped wave solution to the partial differential equation of motion is presented and it is verified with frequency domain solutions. Finally, using the solutions to the differential equation of motion, the response of frame structures with energy dissipating devices is studied.

A theoretical study of the steady-state response of a saturated poroelastic soil column during compressional and rotational harmonic vibrations is presented. Hydraulic damping due to Biot flow is evaluated for top-drained and double-drained boundary conditions and for compressional and rotational motions using the theory of a damped single-degree-of-freedom system. For compressional motions, the dynamic response of gravels and sands is highly influenced by the compressibility of the pore fluid. More hydraulic damping occurs as soil hydraulic conductivity increases and as the column boundary conditions change from top drained to double drained. On the other hand, hydraulic damping for rotational motions is significantly less than that for compressional motions and is dependent on a dimensionless hydraulic conductivity parameter $K_s$ . For $K_s$ within the range of ${10}^{-3}–100$ , hydraulic damping may have an important contribution to total soil damping, especially at small strain levels.

Based on Hamilton’s principle, a three-dimensional semianalytical method for the analysis of free vibration of a structure composed of piezoelectric and elastic materials is developed. Computations of dynamical behavior including natural frequencies, and mode shapes are of interest. In this approach, the mechanical displacement and electric potential functions, in each region, are expressed as products of a three-dimensional (3D) base function and a 3D polynomial with unknown coefficients. The base functions over every domain are constructed with respect to the kinematical boundary conditions, geometry of the structure, and the geometry of that domain. These base functions satisfy the necessary continuities in the displacement field and electric potential at the interfaces, and at the same time accounting for possible discontinuities in their derivatives at the interfaces. The mode shapes will be decomposed in accordance to the presence of any symmetry plane. The robustness of the proposed approach is demonstrated through comparison with the finite element method as applied to the following problems: (1) A perforated thick plate made of piezoelectric and elastic layers and (2) an elastic plate containing a hole, whose exact solution is available. For the latter problem, the result of the present study is in good agreement with the exact solution. Also, an example of a thick PZT plate containing an elastic inclusion with a complex interface, and various types of boundary conditions is considered.

A micromechanical approach for an accurate estimation of the effective moduli of composites reinforced with nondilute distribution of coated particles is developed. The theory is based on the extension of the equivalent inclusion method to interacting multiinhomogeneities. In this treatment the short and long range interactions of the reinforcing particles are appropriately incorporated through the homogenizing eigenstrain field. The periodicity of the microstructure of the medium suggests representing the eigenstrain field in terms of the Fourier series, accounting for the matrix–particle, interparticle, and intraparticle interactions with high precision. The proposed theory is valid for estimating the overall properties of solids containing thickly and/or functionally graded coated reinforcements. For demonstration, the problems of composites with body centered cubic (BCC) distribution of coated spherical particles as well as coated ellipsoidal particles are solved. The effect of volume fraction, coating thickness, coating stiffness, and shape and orientation of the coated ellipsoidal reinforcements on overall behavior of composites are thoroughly examined.

The paper deals with the fully coupled response characteristics of a multilayered composite plate with piezoelectric layers. The response quantities of the plate are coupled by the mechanical field and the electric field. Based on the three-dimensional linear piezoelectricity and the first-order shear deformation theory, the fundamental unknowns, such as the displacements and the electric potential, are assumed to be expandable through the plate thickness coordinate. The governing equations of motion of the plate are presented in terms of the unknown displacement and electrical potential coefficients. When the boundary conditions and electromechanical inputs are specified, the double Fourier series is used to obtain the response of the simply supported multilayered plates. Numerical results for the static and dynamic response of the laminated composite plates with different lamination schemes and having a PIC-151 piezoelectric material layer are obtained. The effects of the plate thinness ratio, plate aspect ratio, lamination scheme, fiber orientations, and piezoelectric coupling on the static and dynamic response are presented.

Initial imperfections of columns are often assumed to have the shape of the first buckling mode. In this technical note, the imperfection has the shape of the second mode. An elastica analysis is performed, and numerical results are obtained for two cases with the use of a shooting method. For the example of a pinned column, bifurcation occurs at a load slightly higher than the critical load for the perfect system. With further increase in load, the deflection changes smoothly from an antisymmetric shape with one node to a shape with no nodes. For the cantilevered-column example, a limit point occurs just beyond the critical load for the perfect system, and the equilibrium shape jumps from one state to another. As the load is increased further, the deflection passes smoothly to the other side of the column and loses its inflection point.

In this technical note, the critical buckling of simply supported functionally graded skew plate subjected to mechanical compressive loads is evaluated using first-order shear deformation theory in conjunction with the finite element approach. The material properties are assumed to vary in the thickness direction according to the power-law distribution in terms of volume fractions of the constituents. The effective material properties are estimated from the volume fractions and the properties of the constituents using the Mori–Tanaka homogenization method. The effects of aspect ratio, material gradient index, and skew angle on the critical buckling loads of functionally graded material plates are highlighted.