Gurmail S. Benipal

Bilinear Dynamics of SDOF Concrete Structures under Sinusoidal Loading

Concrete structures are cracked in tension at working loads. The opening and closing of the existing cracks during load variations renders these structures nonlinear elastic. Under proportional load variations and for single loads, these structures exhibit bilinear mechanical response in the form of discontinuity at origin in their load displacement relations. The typical values of bilinearity ratio, i. e., the ratio of the two values of stiffness, of common reinforced concrete beams range up to 10. In this paper, the dynamic response of concrete structures under working loads has been studied by modeling them as damped single-degree-of-freedom bilinear systems under the action of sinusoidal forcing function. Using the available techniques of bilinear dynamic analysis of moored buoys developed by Thompson and coworkers, concrete structures has been shown to exhibit sub-harmonic resonance response and extreme sensitivity to initial conditions leading to chaos. The limitations of the current practice of dynamic analysis have been exposed and the practical relevance of the work done has been discussed.

Stability of concrete beam-columns under follower forces

An axial follower force acting on the free end of a beam-column is known to remain tangential to its elastica at that point. Elastic beam-columns exhibit infinitely high buckling resistance to static compressive follower load. Loss of their dynamic stability is known to occur at critical follower loads, by flutter characterized by vanishing lateral displacement and infinitely high natural frequency. Classical theory deals with physically linear nonconservative beam-columns. Physical nonlinearity exhibited by concrete beam-columns under service loads is caused by the closing and reopening of the extant transverse cracks. In this Paper, analytical expressions for the lateral displacement and lateral stiffness of such concrete beam-columns are derived. Using these expressions, the stability of physically nonlinear elastic flanged concrete beam-columns under the action of a follower compressive axial force and a lateral force is investigated. The significance of the analytical approach and the theoretical predictions is discussed.

CONSTITUTIVE MODEL FOR BIMODULAR ELASTIC DAMAGE OF CONCRETE

An elastic damage model for concrete has been proposed considering damage-induced bimodularity. A scalar damage parameter has been chosen to quantify the damage. Expressions for the material compliance tensor components have been derived from the assumed strain and complementary energy functions stated in terms of the principal stresses and strains. Incremental constitutive equations have been derived incorporating the elastic behavior due to stress increments as well as stiffness degradation. Within the current damage surface, the stiffness of the material with constant damage varies with applied stress variations. During loading beyond the current damage surface, the material experiences stiffness degradation due to increase in extent of damage suffered by it. Using the proposed elastic damage model, the material response has been predicted for different load histories.

Elastic Stability of Concrete Beam-Columns

Following the empirical-computational methodology, the contemporary investigations deal with inelastic stability and dynamics of concrete beam-columns. Even under service loads, the concrete structures exhibit physical nonlinearity due to presence of axio-flexural cracks. The objective of the present paper is to analyze the static and dynamic stability of conservative physically nonlinear fully cracked flanged concrete beam–columns. In this paper, using proper reference frames, analytical expressions are developed for the lateral displacement and stiffness of a flanged concrete cantilever under axial compressive and lateral forces. Two critical values of both the axial and lateral loads are identified. For constant lateral force smaller than its first critical value, the concrete beam–columns exhibit brittle buckling mode. Higher lateral forces lesser than the second critical value introduce alternate stable and unstable domains with increase in axial force. The lateral stiffness is predicted to vanish when the axial loads reach the critical values and when the limiting displacement is reached for axial load exceeding its second critical value. The load-space is partitioned into stable and unstable regions. Accessibility of these equilibrium states in the load space has been investigated. Such distinguishing aspects of the predicted behavior of elastic concrete beam–columns are discussed. Their dynamic stability is investigated in second part of the paper.

Seismic Analysis of Weightless Sagging Elasto-flexible Cables

There exists considerable literature which deals with the dynamic response of cables with distributed self-weight and some lumped masses, if any. Seismic response of single weightless cable structures has not yet been sufficiently investigated. In this Paper, seismic response of a single weightless planer elasto-flexible sagging cable with lumped nodal masses is studied. This investigation is informed by the appreciation that weightless flexible cables lack unique natural state. Rate-type constitutive equation and third order differential equations of motion have been derived earlier. Using these equations, the dynamic response of such cables subjected to harmonic excitation has also been studied by the Authors. Configurational response is distinguished from the elastic response. The scope of the present Paper is limited to prediction of vibration response of a weightless sagging planer two-node cable structure with lumped masses and sustained gravity loads subjected to horizontal and vertical seismic excitations in the presence of sustained gravity loads. The horizontal and vertical seismic excitations are predicted to cause predominantly configurational and elastic displacements from the equilibrium state. Also, the tensile forces in the inclined and horizontal segments are caused predominantly by these excitations respectively. Cross effects due to mode coupling are predicted. No empirical validation of the theory is attempted. The theoretical predictions are validated by comparing with the seismic response of heavy cable nets predicted by other researchers. The theoretical significance of the approach followed here is critically evaluated.

Parametric resonance in concrete beam-columns

A dynamic instability, called parametric resonance, is exhibited by undampedelastic beam-columns when under the action of pulsating axial force. The scope of the existing theory of parametric resonance is restricted to physically linear beam-columns undergoing finite lateral displacements. In this Paper, the dynamic behaviour of physically nonlinear elastic cracked concrete beam-columns under pulsating axial force and constant lateral force is investigated. The constitutive equations derived earlier by Authors in the form of force-displacement relations are employed here to formulate equations of motion of the SDOF cantilever with mass lumped at its free end. The expected phenomenon of parametric resonance is exhibited in the form of regular subharmonic resonance at about the frequency ratio of two. Resonance peaks broaden with increase in pulsating force. Like damping, physical nonlinearity is also predicted to stabilize the dynamic response at resonance frequencies. In some particular statically unstable conditions, the loss of dynamic stability is shown to occur by divergence. Unexpectedly, similar phenomenon of parametric resonance is exhibited by these physically nonlinear beam-columns undergoing even small lateral displacements. The contribution made to the theory of parametric resonance and the potential relevance of the proposed theory to design of concrete beam-columns is discussed.

Linearized Dynamic Analysis of Weightless Sagging Planar Elastic Cables

Nonlinear dynamic analysis of elastic structures is known to be much more complex than their linear analysis. There are many sources of nonlinearity of the structural response of elastic cables, viz., physical nonlinearity due to nonlinear tension–extension relations, geometric nonlinearity associated with finite elastic displacements and nonlinearity of nodal load–displacement relations due to the presence of self-weight. Incremental second-order differential equations of motion are used to predict the vibration amplitudes relative to the equilibrium state caused by additional dynamic forces. Generally, the tangent stiffness matrices are determined by adding the tangent elastic and geometric stiffness matrices. Many a time, an approximate linearized dynamic analysis is attempted. In this paper, the initial tangent stiffness matrix corresponding to the equilibrium state is used in the second-order linear differential equation of motion. The dynamic response relative to the equilibrium state of the structure subjected to additional dynamic loads is predicted. The predictions of linearized dynamic analysis are generally considered acceptable for small elastic displacements from the equilibrium state. The validity of such linearized dynamic analysis for elasto-flexible cables obeying third-order differential equation of motion is explored.

First order homogeneous dynamical systems 1: theoretical formulation

Of late, the attention of the dynamicists has increasingly been focused on the multi-degree of freedom (MDOF) nonlinear dynamical systems. In the present paper, a new class of conservative two-DOF nonlinear dynamical systems - first order homogeneous dynamical (FOHD) systems - has been proposed. This investigation is motivated by two-DOF cracked concrete beams undergoing small deformations. For these mechanical systems, the nodal forces are functions homogeneous of order one of the nodal displacements and vice-versa. Under assumptions of lumped nodal masses and classical damping, the equations of motion have been derived in the paper. The nodal displacement space has been partitioned into four elastically-distinct regions. Within the two nonlinear elastic regions, the stiffness and damping coefficients as well as the modal frequencies have been shown to vary continuously but remain constant within the two linear regions. Peculiar characteristics distinguishing the FOHD systems from other known MDOF nonlinear dynamical systems have been identified. Theoretical significance of the proposed FOHD systems in the general nonlinear dynamical systems theory has been brought out. The issues such as empirical validation of the predicted dynamical response and the practical relevance of the work done for the concrete beams under working loads have also been discussed.

First order homogeneous dynamical systems 2: application to cracked concrete beams

In this two-part paper, a new class - first order homogeneous dynamical (FOHD) systems - of two-DOF conservative nonlinear dynamical systems has been explored. Theoretical formulation and significance of proposed theory have been presented in part-I. Using the proposed theory, the dynamical behaviour of the two-DOF cracked concrete beam has been predicted here. A new type of phase plot for MDOF dynamical systems has been proposed. Depending upon the loading details and system parameters, the vibration response of these essentially nonlinear systems can be linear, bilinear or nonlinear. Forced vibrations about the passive state have been predicted to resemble linear vibrations in some respects. Like other nonlinear dynamical systems, concrete beam response has also been found to be quite sensitive to initial conditions and system parameters, and to exhibit sub-harmonics and combination sub-harmonics. Feasibility of a nonlinear tuned mass vibration absorber has also been explored. Empirical validation and practical relevance of the proposed theory have been discussed.

Stability of Highly Damped Concrete Beam-Columns

In an earlier paper, authors have investigated the static and dynamic stability of cracked concrete flanged conservative beam-columns. The cantilever column with a lumped mass at its free end is subjected to an axial compressive force and a lateral force. Two critical values each of the axial and lateral loads are defined. Loss of static stability is predicted to occur by excessive displacements or brittle buckling while dynamic instability called divergence is shown to occur by vanishing natural frequency. In this paper, the dynamic stability of highly damped concrete beams-columns is investigated. Two measures—damping ratio and damping coefficient—of structural damping have been employed. Critical loads and displacements are not affected by level of structural damping. Application of inadmissible set of loads results in dynamic instability by divergence at all levels of damping. In particular cases, higher damping has been predicted to destabilize even these conservative structures. Effects of initial conditions and higher damping on the inelastic stability, passive stability control and creep buckling of concrete beam-columns have been delineated. Theoretical significance and practical relevance of the paper are also discussed.