scholarly journals Determining the range of allowable axial force for the third-order Beam Constraint Model

2018 ◽  
Vol 9 (1) ◽  
pp. 71-79 ◽  
Author(s):  
Fulei Ma ◽  
Guimin Chen ◽  
Guangbo Hao
Keyword(s):  
Axial Force ◽  
Strain Energy ◽  
Beam Length ◽  
Positive Definite Quadratic Form ◽  
Third Order ◽  
Intermediate Range ◽  
Modeling Errors ◽  
Dimensional Boundary ◽  
Constraint Model ◽  
Nonlinear Behaviors

Abstract. The Beam Constraint Model (BCM) was developed for the purpose of accurately and analytically modeling nonlinear behaviors of a planar beam flexure over an intermediate range of transverse deflections (10 % of the beam length). The BCM is expressed in the form of Taylor's expansion associated with the axial force. It has been found that the BCM may yield large predicting errors (>  5 %) when the applied axial force goes beyond a certain boundary, even the deflection is still in the intermediate range. However, this boundary has not been clearly identified so far. In this work, we mathematically determine the non-dimensional boundary of the axial force by the condition that the strain energy expression of the BCM is a positive definite quadratic form, and by the buckling condition relate to compressing axial force. Several examples are analyzed to demonstrate the effects of the axial force on the modeling errors of the BCM. When using the BCM for modeling, it is always suggested to check if the axial force is within this boundary to avoid large modeling errors. If the axial force is beyond the boundary, the Chained Beam Constraint Model (CBCM) can be used instead.

Bifurcation of rotating circular cylindrical elastic membranes

1980 ◽  
Vol 87 (2) ◽  
pp. 357-376 ◽  
Author(s):  
D. M. Haughton ◽  
R. W. Ogden
Keyword(s):  
Axial Force ◽  
Strain Energy ◽  
Elastic Strain Energy ◽  
Strain Energy Function ◽  
Angular Speed ◽  
Elastic Membrane ◽  
Axial Extension ◽  
End Conditions ◽  
Elastic Membranes ◽  
Realistic Choice

SummaryBifurcation from a finitely deformed circular cylindrical configuration of a rotating circular cylindrical elastic membrane is examined. It is found (for a physically realistic choice of elastic strain-energy function) that the angular speed attains a maximum followed by a minimum relative to the increasing radius of the cylinder for either a fixed axial extension or fixed axial force.At fixed axial extension (a) a prismatic mode of bifurcation (in which the cross-section of the cylinder becomes uniformly non-circular) may occur at a maximum of the angular speed provided the end conditions on the cylinder allow this; (b) axisyim-metric modes may occur before, at or after the angular speed maximum depending on the length of the cylinder and the magnitude of the axial extension; (c) an asymmetric or ‘wobble’ mode is always possible before either (a) or (b) as the angular speed increases from zero for any length of cylinder or axial extension. Moreover, ‘wobble’ occurs at lower angular speeds for longer cylinders.At fixed axial force the results are similar to (a), (b) and (c) except that an axisym-metric mode necessarily occurs between the turning points of the angular speed.

Analytical and Experimental Study of Beam Torsional Stiffness With Large Axial Elongation

1988 ◽  
Vol 55 (1) ◽  
pp. 171-178 ◽  
Author(s):  
M. Degener ◽  
D. H. Hodges ◽  
D. Petersen
Keyword(s):  
Experimental Data ◽  
Axial Force ◽  
Energy Function ◽  
Strain Energy ◽  
Strain Energy Function ◽  
Torsional Stiffness ◽  
Deformation Tensor ◽  
Axial Elongation ◽  
Green Strain ◽  
Strain Components

The axial force and effective torsional stiffness versus axial elongation are investigated analytically and experimentally for a beam of circular cross section and made of an incompressible material that can sustain large elastic deformation. An approach based on a strain energy function identical to that used in linear elasticity, except with its strain components replaced by those of some finite-deformation tensor, would be expected to provide only limited predictive capability for this large-strain problem. Indeed, such an approach based on Green strain components (commonly referred to as the geometrically nonlinear theory of elasticity) incorrectly predicts a change in volume and predicts the wrong trend regarding the experimentally determined axial force and effective torsional stiffness. On the other hand, use of the same strain energy function, only with the Hencky logarithmic strain components, correctly predicts constant volume and provides excellent agreement with experimental data for lateral contraction, tensile force, and torsional stiffness—even when the axial elongation is large. For strain measures other than Hencky, the strain energy function must be modified to consistently account for large strains. For comparison, theoretical curves derived from a modified Green strain energy function are added. This approach provides results identical to those of the Neo-Hookean formulation for incompressible materials yielding fair agreement with the experimental results for coupled tension and torsion. An alternative approach, proposed in the present paper and based on a modified Almansi strain energy function, provides very good agreement with experimental data and is somewhat easier to manage than the Hencky strain energy approach.

A Framework for Energy-Based Kinetostatic Modeling of Compliant Mechanisms

2017 ◽  
Author(s):  
Guimin Chen ◽  
Fulei Ma ◽  
Ruiyu Bai ◽  
Spencer P. Magleby ◽  
Larry L. Howell
Keyword(s):  
Strain Energy ◽  
Compliant Mechanisms ◽  
Modeling Framework ◽  
Geometric Nonlinearities ◽  
Complementary Strain ◽  
Minimum Strain Energy ◽  
Constraint Model ◽  
Future Work

Although energy-based methods have advantages over the Newtonian methods for kinetostatic modeling, the geometric nonlinearities inherent in deflections of compliant mechanisms preclude most of the energy-based theorems. Castigliano’s first theorem and the Crotti-Engesser theorem, which don’t require the problem being solved to be linear, are selected to construct the energy-based kinetostatic modeling framework for compliant mechanisms in this work. Utilization of these two theorems requires explicitly formulating the strain energy in terms of deflections and the complementary strain energy in terms of loads, which are derived based on the beam constraint model. The kinetostatic modeling of two compliant mechanisms are provided to demonstrate the effectiveness of using Castigliano’s first theorem and the Crotti-Engesser theorem with the explicit formulations in this framework. Future work will be focused on incorporating use of the principle of minimum strain energy and the principle of minimum complementary strain energy.

scholarly journals An Energy-Based Framework for Nonlinear Kinetostatic Modeling of Compliant Mechanisms Utilizing Beam Flexures

2021 ◽  
pp. 1-18
Author(s):  
Guimin Chen ◽  
Fulei Ma ◽  
Ruiyu Bai ◽  
Weidong Zhu ◽  
Spencer P Magleby ◽  
...  
Keyword(s):  
Strain Energy ◽  
Compliant Mechanisms ◽  
Modeling Framework ◽  
Geometric Nonlinearities ◽  
Complementary Strain ◽  
Constraint Model

Abstract Although energy-based methods have advantages over the Newtonian methods for kinetostatic modeling, the geometric nonlinearities inherent in deflections of compliant mechanisms preclude most of the energy-based theorems. Castigliano's first theorem and the Crotti-Engesser theorem, which don't require the problem being solved to be linear, are selected to construct the energy-based kinetostatic modeling framework for compliant mechanisms in this work. Utilization of these two theorems requires explicitly formulating the strain energy in terms of deflections and the complementary strain energy in terms of loads, which are derived based on the beam constraint model. The kinetostatic modeling of two compliant mechanisms are provided to demonstrate the effectiveness of the explicit formulations in this framework derived from Castigliano's first theorem and the Crotti-Engesser theorem.

scholarly journals Vibration frequency of prestress slender beams resting on Winkler elastic foundation

2006 ◽  
Vol 28 (4) ◽  
pp. 241-251
Author(s):  
Nguyen Dinh Kien
Keyword(s):  
Axial Force ◽  
Vibration Frequency ◽  
Strain Energy ◽  
Natural Frequencies ◽  
Winkler Foundation ◽  
The Finite Element Method ◽  
Various Boundary Conditions ◽  
Winkler Elastic Foundation ◽  
Mass Matrices ◽  
Slender Beams

The present paper investigates the vibration frequency of slender beams prestressing by axial force and resting on an elastic Winkler foundation by the finite element method. A beam element taking the effects of both the prestress and foundation support into account is formulated using the expression of strain energy. Using the developed element, the natural frequencies of beams having various boundary conditions are computed for different values of the axial force and foundation stiffness. The influence of the axial force and the foundation stiffness on the frequency of the beams is investigated. The effect of partial support by the foundation and the type of mass matrices on the vibration frequency of the beam is also studied and highlighted.

Post-Buckling Analysis of a Uniform Self-Weight Beam with Application to Catenary Riser

2019 ◽  
Vol 19 (04) ◽  
pp. 1950047 ◽  
Author(s):  
Ong-Art Punjarat ◽  
Somchai Chucheepsakul
Keyword(s):  
Axial Force ◽  
Strain Energy ◽  
Equilibrium Configuration ◽  
Virtual Work ◽  
Beam Theory ◽  
Energy Functional ◽  
Arc Length ◽  
Bending Strain ◽  
Highly Nonlinear ◽  
Post Buckling

This paper focused on a simply supported beam under uniform self-weight, subjected to an axial force at the roller end. The principle of virtual work-energy was used to formulate the equation for the nonlinear deformation of the beam, which involves the bending strain energy, the virtual work due to self-weight, and the virtual work of the axial force applied at the free-sliding roller end. The work–energy functional was expressed in terms of the arc-length coordinate. The functional vanished, yielding the static equilibrium configuration of the beam — a highly nonlinear problem. Finite element and Newton–Raphson iterative methods were used to solve the problem. The beam theory was extended to large sag analysis of a catenary riser. With this, some interesting features of the various configurations of the catenary riser under various end forces were evaluated.

Nonlinear Strain Energy Formulation to Capture the Constraint Characteristics of a Spatial Symmetric Beam

2011 ◽  
Author(s):  
Shiladitya Sen ◽  
Shorya Awtar
Keyword(s):  
Cross Sections ◽  
Strain Energy ◽  
Virtual Work ◽  
Nonlinear Strain ◽  
The Past ◽  
Transverse Bending ◽  
Spatial Beams ◽  
Cross Axis ◽  
Constraint Model ◽  
Energy Formulation

In the past, a beam constraint model (BCM) that captures pertinent geometric nonlinearities associated with large displacements has been proposed for slender spatial beams with uniform and symmetric cross-sections. By providing closed-form parametric relations between the end-loads and end-displacements of the beam, the BCM quantifies the constraint characteristics of the beam in terms of stiffness variations, parasitic error motions, and the cross-axis coupling. This paper presents a nonlinear strain and strain energy formulation for the spatial symmetric beam, based on assumptions that are consistent with the BCM. This strain energy derivation, employing the Principle of Virtual Work, provides a simpler mathematical approach for the analysis of flexure mechanisms with multiple spatial beams. Using this formulation, we obtain the stiffness relations in the transverse bending directions, the constraint relations in the axial and torsional directions, and the overall strain energy expression in terms of the beam end-loads and end-displacements. These expressions, collectively the BCM, are in form that is suitable for the analysis of multi-beam flexure mechanisms.

Higher-Order Beam Theories for Mode II Fracture of Unidirectional Composites

2003 ◽  
Vol 70 (6) ◽  
pp. 840-852 ◽  
Author(s):  
D. V. T. G. Pavan Kumar ◽  
B. K. Raghu Prasad
Keyword(s):  
Finite Element ◽  
Energy Release Rate ◽  
Release Rate ◽  
Strain Energy ◽  
Strain Energy Release Rate ◽  
Beam Theory ◽  
Strain Energy Release ◽  
Mode Ii ◽  
Third Order ◽  
Beam Theories

Mathematical models, for the stress analyses of unidirectional end notch flexure and end notch cantilever specimens using classical beam theory, first, second, and third-order shear deformation beam theories, have been developed to determine the interlaminar fracture toughness of unidirectional composites in mode II. In the present study, appropriate matching conditions, in terms of generalized displacements and stress resultants, have been derived and applied at the crack tip by enforcing the displacement continuity at the crack tip in conjunction with the variational equation. Strain energy release rate has been calculated using compliance approach. The compliance and strain energy release rate obtained from present formulations have been compared with the existing experimental, analytical, and finite element results and found that results from third-order shear deformation beam theory are in close agreement with the existing experimental and finite element results.

Numerical and experimental investigation of modal-energy-based damage localization for offshore wind turbine structures

2018 ◽  
Vol 21 (10) ◽  
pp. 1510-1525 ◽  
Author(s):  
Yingchao Li ◽  
Min Zhang ◽  
Wenlong Yang
Keyword(s):  
Finite Element ◽  
Wind Turbine ◽  
Strain Energy ◽  
Offshore Wind ◽  
Mode Shapes ◽  
Offshore Wind Turbine ◽  
Energy Index ◽  
Modal Strain Energy ◽  
Modeling Errors ◽  
Energy Indices

Offshore wind turbine structures are prone to deterioration and damage during their service life in harsh marine environment. To explore highly efficient and robust damage detection methods for offshore wind turbine structures, three well-known modal strain energy indices are reviewed first and then a new index named total modal energy method is proposed. The innovation of the new index is the simultaneous use of modal strain energy and modal kinetic energy. To investigate the feasibility and robustness of the four modal-energy-based methods, numerical and experimental studies are conducted on a tripod-type offshore wind turbine structure with simulated and measured data. It is indicated that all the four modal-energy-based methods work well with limited incomplete modal data, especially for the single-damage cases. While for the cases of multiple damage locations, the new total modal energy index significantly outperforms the traditional modal strain energy indices. Moreover, high robustness is shown for the indices, when the measured mode shapes of undamaged and damaged structures are polluted with the same noise level. However, when their noise levels have some difference, two of the modal strain energy indices turn invalid, but the new total modal energy index still shows stronger robustness. As frequencies are also used in the total modal energy index, its robustness to the noise in modal frequencies is also studied. It is shown that the results are slightly affected by the measurement noise in modal frequencies. Besides, the influence of finite element modeling errors is also investigated with both simulated and experimental data. Results show that all the four modal-energy-based methods are all very stable and insensitive to certain modeling errors. So, finite element model updating is not necessary in the test structure herein.

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