Journal of Engineering Mechanics

An application of the finite element method to the theory of thin‐walled bars of variable, open cross section is presented. A thin‐walled bar is considered as a special case of the membrane shell with internal constraints (Vlasov's and Wagner's assumptions). The bar is divided into elements along its longitudinal axis, then the shell midsurface of the element is approximated by arbitrary triangular subelements. Displacements of the element are represented by polynomials of the third degree, and both an equivalent stiffness matrix and a consistent mass matrix are obtained. An eigenvalue problem is analyzed using standard and generalized forms. Convergence of the method is discussed and numerical examples are presented for I‐beams of constant and variable cross sections.

Two methods for the solution of beams on variable two‐parameter elastic foundation are presented. The first is based on using the exact shape functions for beams on variable two‐parameter foundation, in order to obtain the stiffness, consistent mass, and geometric stiffness matrices. The second is based on using the cubic shape functions of a regular beam element and adding the contribution of the foundation as element foundation stiffness matrices. The two solutions are compared in an example that includes static, buckling, and vibration frequency analyses.

A simple procedure is outlined that inverts the specific compliance function for a linear viscoelastic material, similar to concrete, to its specific relaxation modulus by using Laplace transform and age shift function concept.

In second‐order reliability problems the limit state surface is (locally) replaced by a quadratic surface. This leads to a study of quadratic forms in the basic uncertainty variables. The present paper exploits a simple, closed‐form expression for the distribution function of a special class of quadratic forms in zero‐mean Gaussian variables to obtain bounds on the distribution function of general quadratic forms in such variables. It is shown that these bounds can be used to obtain useful estimates of the failure probability for such models.

A previous material model for the increase of creep of concrete caused by simultaneous drying is extended to describe the effect of both drying and wetting, as well as the increase of creep caused by temperature changes, both heating and cooling. By theoretical arguments and comparisons with numerous existing test data, it is shown that the creep increase due to temperature changes, sometimes called the transitional thermal creep, is physically the same phenomenon as the increase of creep due to humidity changes, known as the Pickett effect (or the drying creep effect). In accord with the previous model for drying creep alone, the present extended model explains the increase of creep due to humidity or temperature changes as a consequence of principally two effects: (1) Stress‐induced shrinkage (or swelling) or stress‐induced thermal expansion (or contraction); and (2) the distributed tensile cracking (or strain‐softening) of concrete. The former effect is explained by the previously advanced hypothesis that the creep viscosity depends on the pore humidity rate. The latter effect reduces measured overall deformations, and thus results in the true thermal expansion or shrinkage of uncracked material being considerably larger than observed on load‐free companion specimens when significant nonuniformly distributed self‐equilibrated stresses are produced by the temperature or humidity change. The proposed material model is suitable for finite‐element programs.

In the usual local finite element analysis, strain softening causes spurious mesh sensitivity and incorrect convergence when the element is refined to vanishing size. In a previous continuum formulation, these incorrect features were overcome by the imbricate nonlocal continuum, which, however, introduced some unnecessary computational complications due to the fact that all response was treated as nonlocal. The key idea of the present nonlocal damage theory is to subject to nonlocal treatment only those variables that control strain softening, and to treat the elastic part of the strain as local. The continuum damage mechanics formulation, convenient for separating the nonlocal treatment of damage from the local treatment of elastic behavior, is adopted in the present work. The only required modification is to replace the usual local damage energy release rate with its spatial average over the representative volume of the material whose size is a characteristic of the material. Avoidance of spurious mesh sensitivity and proper convergence are demonstrated by numerical examples, including static strain softening in a bar, longitudinal wave propagation in strain‐softening material, and static layered finite element analysis of a beam. In the last case, the size of the representative volume serving in one dimension as the averaging length for damage must not be less than the beam depth, due to the hypothesis of plane cross sections. It is also shown that averaging of the fracturing strain leads to an equivalent formulation, which could be extended to anisotropic damage due to highly oriented cracking.

This paper develops a novel solution scheme for the analysis of the transfer of an axially symmetric torque from a nonuniform elastic bar into a layered elastic halfspace. The torsional deformation of the bar is specified by a set of kinematically admissible functions that are indeterminate to within a set of arbitrary constants. The embedded bar‐elastic halfspace system is represented by extended halfspace and a fictitious bar. The fictitious traction distribution on the extended half‐space due to a specified displacement field is determined by a discretization procedure. The results of the analysis is used to develop the total potential energy functional for the bar‐elastic halfspace loading system. The minimization of the functional determines explicitly the torsional deformation field. The numerical results presented in the paper illustrate the manner in which the torsional stiffness of the bar is influenced by its nonuniformity and its relative rigidity.

Even after more than 20 years of research in the field of fracture mechanics applied to concrete, the influence of notch length on specimen behavior and fracture toughness, $K_{\mathrm{Iq}}\text{,}$ based on the experimental evidence, is not clear. In an effort to study this influence, we have employed a notched cylinder specimen geometry. Observed failure modes of portland cement concrete cylinders are reported and discussed, and $K_{\mathrm{Iq}}$ as calculated from both the maximum load and from the load at the end of the linear part of the load versus crack mouth opening displacement (CMOD) plots are compared. Values of $K_{\mathrm{Iq}}$ have been found to increase with increasing initial notch length. It is concluded that this is a consequence of the more ductile behavior due to increased microcracking of specimens with longer notches relative to those with shorter ones. Maximum linear load and initial notch length yield $K_{\mathrm{Iq}}$ values less dependent on initial notch length, thus suggesting that the notched cylinder may provide a good standard fracture mechanics specimen to measure the fracture toughness associated with the onset of microcracking ahead of a macrocrack.

The response of a rigid cylinder embedded in a homogeneous, isotropic half space to a step function incident stress wave is investigated. The boundary integral equation method is used directly in the time domain where two types of interfacial stresses are considered: In the first case, the interfacial stresses are not permitted to exceed a prescribed constant maximum value (elastic‐perfectly plastic interface behaviour); in the second case, a friction interface is considered where the limiting stresses vary along the interface according to the normal interfacial pressure. The angular distribution of shear stress at different times are examined to illustrate the redistribution of interfacial stresses. The relative displacements between the cylinder and half space are also presented.

The natural vibration frequencies of thin‐walled bars of open cross section are studied, with particular emphasis on the influence of asymmetry and destabilizing forces on the interaction between various modes. It is concluded that interactive vibrations are induced by both asymmetry and destabilizing forces. The validity of the solutions obtained is confirmed by the fact that the lowest natural frequencies reduce to zero when the applied forces are equal to the static critical forces for instability in the corresponding modes.

A theoretical model was developed based on a fluid mechanical force balance to describe the behavior of particles in pulsing airflow. Several attributes distinguished this flow regime from those that have been widely studied: (1) The fluid was a gas; (2) the particles were much larger than aerosol; (3) the airflow was pulsed, as opposed to the more commonly studied oscillatory flow; and (4) the flow was vertical, requiring the inclusion of a body force in the expression. Much effort in the theoretical formulation and calibration was devoted to the acceleration effect, due to its importance in time‐varying flow. Computer models were developed and presented that were used in laboratory calibration of the theoretical expression and in modeling particle separation in rising, pulsing airflow. Theory showed separations based upon density that were not possible in steady rising airstreams. Experimentation with air classifier designs suggested by the model showed greatly increased separation efficiency.

Experimental results related to the form drag coefficient of two-dimensional wall-mounted rectangular and round-nosed rectangular bodies are obtained and expressed as a function of the Reynolds number and the radius of the upstream top corner. The form drag coefficient is obtained by graphical integration of the pressure distribution on the front and rear faces of the body. The wall bodies tested bear a close resemblance to the form of rectangular broad-crested weirs which are widely used to measure open channel flows.