Moment Equilibrium Equations Research Articles

THE FOUR‐NODE C0 SHELL ELEMENT REFORMULATED

A reformulated four-node shell element, based on the analysis of the moment redistribution mechanism development by C0 plate bending and shell elements, is presented. The moment redistribution mechanism of a finite shell element model is shown to be predominantly activated by the membrane flexural action of the shell. This action is triggered through the membrane strain components which participate in the moment equilibrium equations of the finite element assembly system. An equivalent elastic foundation action, along with the activation of the in-plane twisting stiffness of the shell, may also contribute to the moment redistribution mechanism of the finite shell element model. The proposed shell element formulation aims at retaining the non-spurious contribution of the transverse shear/membrane strain energy to the flexural behaviour of the shell, through the activation of the moment redistribution mechanism. Yet, any potentially spurious, whether locking or kinematic, mechanism is rejected. In warped configurations, the element activates appropriate coupling mechanisms of the bending terms to nodal translations. The so-obtained reformulated four-node shell element exhibits an excellent behaviour without experiencing any locking phenomena or zero-energy modes, while its formulation is kept simple, based on physical considerations. The proposed formulation performs equally well in flat as well as in warped shell element applications.

Comparison of muscle forces and joint load from an optimization and EMG assisted lumbar spine model: Towards development of a hybrid approach

The purpose of this study was to determine whether the same estimates of individual muscle and L4 L5 lumbar joint compressive forces result from an optimization (OPT) compared to an electromyography (EMG) assisted approach for solving the inderminate moment equilibrium equations in the same anatomical model. Four male subjects performed near maximum, isometric, ramp efforts in trunck flexion, extension and lateral bending in a testing apparatus. The EMG approach was sensitive to subject and trial differences in the magnitudes of individual muscle forces needed to produce the same reaction moment. In contrast, the OPT method converged on a similar estimate of muscle forces for all subjects and trials producing the same moment. The OPT method predicted lower L4 L5 joint compression values, on average, by 32, 43 and 23% in trunk extension, flexion and lateral bending, respectively, because, unlike the EMG method, it could not predict co-contraction of anatomically antagonistic muscles. We incorporated the OPT method's advantage of forcing an equilibrium in the reaction moments into the EMG method in a new approach we have called ‘EMG assisted optimization’ (EMGAO). Muscle force estimates from the EMG and EMGAO methods differed from those from the OPT method, on average, by 123% (RMS) for flexion and extension and by 218% for lateral bends. Data from the two approaches result in different conclusions about spine mechanics. We have more confidence in the EMG assisted methods because they respond to variation in muscle synergy and co-contraction patterns commonly observed in different trials and subjects for the same reaction moments.

EMG assisted optimization: A hybrid approach for estimating muscle forces in an indeterminate biomechanical model

There are two basic approaches to estimate individual muscle forces acting on a joint, given the indeterminacy of moment balance equations: optimization and electromyography (EMG) assisted. Each approach is characterized by unique advantages and liabilities. With this in mind, a new hybrid method which combines the advantages of both of these traditional approaches, termed ‘EMG assisted optimization’ (EMGAO), was described. In this method, minimal adjustments are applied to the individual muscle forces estimated from EMG, so that all moment equilibrium equations are satisfied in three dimensions. The result is the best possible match between physiologically observed muscle activation patterns and the predicted forces, while satisfying the moment constraints about all three joint axes. Several forms of the objective function are discussed and their effect on individual muscle adjustments is illustrated in a simple two-dimensional example.

A nonlinear transverse shear deformable plate theory allowing a multiple choice of trial displacement and stress distributions

This paper deals with a refined formulation of a general, geometrically nonlinear, shear deformable, static plate theory which allows a multiple choice of trial displacement and, consequently, stress distributions through the plate thickness. This is achieved with the incorporation, into a basic trial displacement approximation, of a general function of the transverse coordinate. An appropriate choice of the derivative of that function is, essentially, adequate for an a posteriori control of the transverse shear strain and stress distributions. However, a particular choice of that general function is not provided, leaving, therefore, the theory in its most general form. The development of the theory is based on the concept of small strains and moderate rotations. Its force and moment equilibrium equations are derived variationally, on the basis of the principle of the total potential energy. Under certain conditions, the present nonlinear theory becomes analogous to several linear and nonlinear plate theories available in the literature, in the sense that it can easily adopt and make use of their main characteristics.

A three-dimensional static torso model for the six human lumbar joints

A simple three-dimensional static torso model for the prediction of the force distributions at the six human lumbar levels in different activities was developed. There are two procedures involved in the model calculations, the intersegmental resultant joint forces and moments of the presumed 26 joints and the internal muscoloskeletal force distributions of the 6 intervertebral disc joints. In formulation of the joint force distribution problem, 6 muscle forces and 3 moment equilibrium equations at one of the six disc joints (e.g., L3/L4) with muscle stress (muscle force divided by physiological cross sectional area - PCSA) upper limits were used. The disc compressive force was minimized as the cost function of the linear programming to solve the force distribution problem. This solution of disc force distribution was further substituted into the next level. Such sequential substitutions of joint level's force equilibrium equations were used to solve for the disc forces at all 6 joint levels.

多指ハンドによる3次元つりあい把持のための指先位置計算アルゴリズム

This paper proposes an algorithm for computing the fingertip positions of an equilibrium grasp for a polyhedral object. The authors has already developed such an algorithm for a planar grasp where the problem can be reduced to a linear programming problem using the static equivalence: the effect of changing a fingertip position along an edge is equivalent to the effect of changing the fingertip forces of two virtual fingers, one fixed at each edge endpoint. This equivalance makes the moment equilibrium equation in the planar grasp linear. In a spatial grasp, introducing such virtual fingers to the faces of the object decomposes the moment equilibrium equation into linear and nonlinear constraints. The nonlinear constraints still remain, but exhibit the same properties as the integer requirement in an integer programming problem. This paper proposes an algorithm based on the branch-and-bound method that decomposes each finger region until the nonlinear constraint associated with the finger is satisfied. Numerical examples show that the algorithm effectively solves problems of practical size. >

Generalized Slice Method for Slope Stability Analysis

ABSTRACT Many slice methods already proposed for slope stability analysis have two weak points, assumption on interslice forces and complicated calculation technique. In this paper, an extended wedge method in which equilibrium equations of forces acting on slices are directly presented and safety factors on interslice planes are defined, is proposed. Followings are clarified after mathematical consideration. If the problem to obtain safety factor along main slip surface is considered as optimization problem with constraints, the solution can be obtained even if the number of variables exceeds the number of equations. This means that the safety factor can be optimized not only by geometry of slip surface but also by assumption on interslice forces. The equilibrium equations of forces are sufficient to obtain and minimize safety factor without equilibrium equations of moment. Safety factor, Fsmin, which assumes no tangential force on interslice planes, and safety factor Fsmax, which assumes that interslice planes are just failing, are minimum and maximum limit of safety factors, respectively. Together with Fsmed which assumes that safety factor along main slip surface is equal to ones along interslice planes, Fsmin and Fsmax are the standard for evaluation of other slice methods from the state of forces acting on interslice planes. Then, some comparisons suggest that Fellenius method and simplified Janbu method give the lowest safety factors which are nealy equal to Fsmin and that Morgenstern-Price method gives the safety factor higher than Fsmed.

A planar model of the knee joint to characterize the knee extensor mechanism

A simple planar static model of the knee joint was developed to calculate effective moment arms for the quadriceps muscle. A pathway for the instantaneous center of rotation was chosen that gives realstic orientations of the femur relative to the tibia. Using the model, nonlinear force and moment equilibrium equations were solved at one degree increments for knee flexion angles from 0 (full extension) to 90°, yielding patellar orientation, patellofemoral contact force and patellar ligament force and direction with respect to both the tibial insertion point and the tibiofemoral contact point. The computer-derived results from this two-dimensional model agree with results from more complex models developed previously from experimentally obtained data. Due to our model's simplicity, however, the operation of the patellar mechanism as a lever as well as a spacer is clearly illustrated. Specifically, the thickness of the patella was found to increase the effective moment arm significantly only at flexions below 35° even though the actual moment arm exhibited an increase throughout the flexion range. Lengthening either the patella or the patellar ligament altered the force transmitted from the quadriceps to the patellar ligament, significantly increasing the effective moment arm at flexions greater than 25°. We conclude that the levering action of the patella is an essential mechanism of knee joint operation at moderate to high flexion angles.

Convergence of transverse shear stresses in the finite element analysis of plates

AbstractIf bending moments converge at a sufficiently fast rate, it is proved that transverse shear stress resultants calculated from the moment equilibrium equation also converge. The result holds independent of the plate theory or type of element employed.

Model analysis of factors influencing the prediction of muscle forces at the knee.

A three-dimensional stochastic mathematical muscle model of the knee joint has been developed and applied to a study in which the influence of both mechanical and physiological factors were examined in relation to the prediction of muscular forces about the joint. The model includes a representation of the proximal portion of the tibia and distal portion of the femur along with a mathematical expression of the patellar mechanism and 13 muscles crossing the knee joint. The model accounts for the rolling and gliding movement of the tibial-femoral articulation. The computational technique involves equilibrating three components of external moments at the knee joint to the internal moments generated by muscular forces and soft tissue. The variables contained in the moment equilibrium equation are randomly chosen based on the choice of the tibial-femoral contact point. The randomness of the variables, reflected in the final solution, defines a stochastic process in the context of the present model. Studies with the model indicated that a very important mechanical aspect of the model was the capability to simulate the moving contact point between the tibia and femur. The moving contact point increased the mechanical advantage of the quadriceps muscles by 50%, which corresponded to in vivo EMG measurements. Muscle force predictions during normal gait have shown the capability of the model to determine the presence of synergistic and antagonistic muscle action.

Dynamic Plastic Deformation of Clamped Disk Subjected to Electrohydraulic Impulsive Pressure

Equilibrium equations of tension and moment for radial and circumferential directions are induced on the basis of the finite deformation theory. On the assumption of a deflection consisting of the plastic part which does not vibrate and the vibration part, the dynamic deformation process is explained using the equilibrium equations. In connection with the former, the dynamic deformation process is explained using Lunge-Kutta method. In connection with the latter, dynamic process is explained using Galerkin method from the viewpoint of vibration stability condition. Results of the calculation are compared with the experimental results obtained by an electrohydraulic forming machine for aluminum alloy. As a result, under the assumption of the waveform which is not affected by vibration, the deformation develops with a repeatedly applied pressure. On the other hand, under the assumption of vibrating waveform, it is revealed that deformation is developed in parallel with vibration.

General helicoidal shells undergoing large, one-dimensional strains or large inextensional deformations

An earlied paper showed that general helicoidal shells (which include shells of revolution and general cylindrical shells as limiting cases) admit arbitrarily large, one-dimensional strain fields. In the present paper, the associated rotation and stress function fields are found. Introduction of the Euler parameters from rigid body dynamics reduces the determination of the rotation field to a linear problem. The equilibrium and compatibility equations are shown to reduce to six coupled scalar equations involving three rotation functions, three stress functions, extensional strains, stress couples, and four constants, two measuring the gross axial displacement and twist of the shell, and two measuring the net axial force and torque. One field equation is a first integral of the compatibility equations; another, a first integral of the moment equilibrium equations. Reissner's equations for the pure bending of curved tubes and Wan's equations for the gross twisting and extension of right helicoidal shells fall out as special cases. Determination of the displacement field in large inextensional bending reduces to quadratures, generalizing Reissner's result for a slit shell of revolution.

Balancing a force on the fingertip of a two-dimensional finger model without intrinsic muscles

A slightly flexed human middle finger can balance an external force on the fingertip. Internal stabilization is also possible, which means that the externally unloaded finger can be kept stiff. We want to analyse whether in these situations the intrinsic hand muscles are needed. Distances from tendons to flexion axes are taken from the literature and are substituted in the moment equilibrium equations of a two-dimensional finger model. Diagrams illustrate the statically indeterminate problem of solving tendon forces. The possibilities for equilibrium without intrinsics appear to depend mainly on four tendon-to-joint distances. These distances determine to which of two groups a finger belongs: (1) one in which intrinsics are not necessary for internal stabilization nor for balancing a force on the fingertip in any direction in the sagittal plane; (2) one in which, without intrinsics, internal stabilization is impossible and only dorso-distally directed forces on the fingertip can be balanced.

A finite element model for plane elasticity problems using the complementary energy theorem

AbstractA finite element stress analysis capability for plane elasticity problems, employing the principle of stationary complementary energy, is developed. Two models are investigated. The first is a 24 d.o.f. rectangular finite element. The second model consists of an 18 d.o.f. triangular element. In order to allow for self‐equilibrating stresses which are continuous within the element, the well‐known Airy stress function ø is used. The function ø is represented by means of quintic Hermitian polynomials within the finite element. The values of the ø function and its derivatives up to order two are used as nodal parameters. For matching the stress function with the prescribed boundary tractions, additional equations are developed considering the force and moment equilibrium equations on the boundary consistent with the assumed stress function. These additional boundary equations are incorporated into the system equations using the Lagrangian multiplier technique. Excellent results are obtained for linear elastic problems even with coarse finite element discretization. Some examples of plane elasticity problems are solved and results compared.

Simple methods to analyze buckling of rock slopes

Simple Methods to Analyze Buckling of Rock Slopes Buckling modes of slope failure are a possibility whenever a throughgoing discontinuity, approximately parallel to the slope, separates a thin slab of rock. Failure by buckling may be initiated by forces external to the slab, by groundwater pressures, by applied forces or by the weight of the slab itself, especially if the slab is curved convex upward. Simple formulae are developed for three possible cases of buckling: flexural buckling of plane slopes, three hinge buckling of plane slopes and three hinge buckling of curved slopes. For flexural buckling, Euler's formula is applied after making several simplifying assumptions regarding the geometry of a buckling slab. Three hinge buckling may occur if a slab is cut by cross-joints perpendicular to the slope. Stability of each of the two blocks is determined using equations of force and moment equilibrium, resulting in a series of equations describing the interblock forces required for stability. The applicability of the three hinge model was tested by applying the equations to a documented failure on an excavated slope. The method gave reasonable results.

Stability and vibration of thin rectangular plates by simplified mixed finite elements

Rectangular finite elements based on a Reissner type variational statement for plate bending are applied to stability and free vibration of rectangular plates. The finite elements are constructed from the weak statements of the two normal moments-displacement relations and the moment equilibrium equation in terms of the two normal moments. These finite elements are algebraically simple and yield better accuracies for the critical loads and natural frequencies when compared to conventional finite elements. Linear and quadratic rectangular finite elements are used to calculate frequencies and buckling loads of rectangular plates with various edge conditions.

In-plane free vibrations of tapered oval rings

A set of differential force and moment equilibrium equations for the in-plane dynamic strains in an oval ring of variable thickness is determined. This set is reduced to algebraic form by substituting Fourier series for the stresses and deformations. The resulting matrix is then truncated and evaluated in order to determine the first five normal modes and the first eight natural frequencies for an elliptical ring of uniform wall thickness. Excellent agreement is shown between theoretical and experimental data for several aluminum rings of pseudoelliptical shape.

The linear and non-linear equilibrium equations for thin elastic shells according to the Kirchhoff-Love hypotheses

The currently accepted bending theory of thin elastic shells contains equilibrium equations derived on the quite general premise that the shell can withstand an unspecified transverse shear load. The transverse shear stress resultants are then eliminated between the force-equilibrium and moment-equilibrium equations. This procedure is, of course, acceptable. But it is shown to be inconsistent with the universal practice of relating these equilibrium equations to the displacement components and their derivatives by means of strain-displacement relations formulated according to the Kirchhoff-Love hypothesis that normals to the shell middle surface before deformation remain normal after deformation, which prescribes zero transverse shears, both strains and stresses. Furthermore, it is revealed that in actually relating the stress resultants and couples to the strains, the accepted practice includes inconsistencies in the stress couple terms which are of the same order of magnitude as (and in some cases identical to) the terms retained. A new formulation of the thin-shell equilibrium equations, under the Kirchhoff-Love hypotheses, is presented so that a complete self-consistent theory of thin elastic shells is produced. This new shell theory is used elsewhere to analyse the buckling of thin cylindrical shells under axial compression. The analysis is a linear infinitesimal deformation solution for the actual buckling load of an ideal shell, not the post-buckling minimum load capacity, and is in good agreement with experiments, both from the viewpoint of the buckling stress and the deformation pattern. The buckling stress is found to be one-half of the classical value. The inconsistencies in the currently accepted thin-shell theories are thus seen to be too important to overlook.

A non-linear theory of elastic directed curves

A non-linear theory of elastic directed curves is developed on the basis of a postulated principle of virtual work. A strain energy function is assumed such that in addition to the classical stress, there exists double stress with and without moment. The basic equilibrium equations, boundary conditions and constitutive relations are derived. In addition to the classical equations of force and moment equilibrium, there are found to exist six equations which govern the double stress without moment. In the case of a special class of deformations in which the directors do not deform, the equilibrium equations reduce to the classical set. A simple example is considered which illustrates the possibility of the theory leading to permanent deformations.

On the errors of and possible corrections to classical linear shell theory

The currently accepted bending theory of thin elastic shells contains equilibrium equations derived on the quite general premise that the shell can withstand an unspecified transverse shear load. The transverse shear stress resultants are then eliminated between the force-equilibrium and moment-equilibrium equations. This procedure is, of course, acceptable. But it is shown to be inconsistent with the universal practice of relating these equilibrium equations to the displacement components and their derivatives by means of strain-displacement relations formulated according to the Kirchhoff-Love hypothesis that normals to the shell middle surface before deformation remain normal after deformation, which prescribes zero transverse shears, both strains and stresses. Furthermore, it is revealed that in actually relating the stress resultants and couples to the strains, the accepted practice includes inconsistencies in the stress couple terms which are of the same order of magnitude as (and in some cases identical to) the terms retained. A new formulation of the thin-shell equilibrium equations, under the Kirchhoff-Love hypotheses, is presented so that a complete self-consistent theory of thin elastic shells is produced. This new shell theory is used elsewhere to analyse the buckling of thin cylindrical shells under axial compression. The analysis is a linear infinitesimal deformation solution for the actual buckling load of an ideal shell, not the post-buckling minimum load capacity, and is in good agreement with experiments, both from the viewpoint of the buckling stress and the deformation pattern. The buckling stress is found to be one-half of the classical value. The inconsistencies in the currently accepted thin-shell theories are thus seen to be too important to overlook.