Follower forces: Leipholz's early researches in elastic stability

1990 ◽  
Vol 17 (3) ◽  
pp. 277-286 ◽  
Author(s):  
G. M. L. Gladwell
Keyword(s):  
Critical Load ◽  
Key Words ◽  
Lower Bounds ◽  
Critical Loads ◽  
Elastic Stability ◽  
Follower Force ◽  
Historical Account ◽  
Nonconservative Systems ◽  
Flutter Instability ◽  
Force Systems

This paper provides an historical account of Leipholz's research into elastic stability. Emphasis is placed on divergence and flutter instability of follower force systems, the derivation of lower bounds for the critical load for divergence, and estimates for critical loads for flutter. Key words: elastic stability, divergence, flutter, lower bounds, nonconservative systems, symmetrisable matrix.

Elastic Stability of Two-Span Continuous Beams under Moving Loads

2005 ◽  
Vol 8 (2) ◽  
pp. 157-172 ◽  
Author(s):  
Lei Zhang ◽  
Gengshu Tong
Keyword(s):  
Critical Load ◽  
Linear Approximation ◽  
Critical Loads ◽  
Elastic Stability ◽  
The Other ◽  
Moving Loads ◽  
Absolute Moment ◽  
Load Combination ◽  
Continuous Beams ◽  
Shear Centre

The elastic stability of two span continuous beams has been studied using FEA methods. Two formulae for estimating the critical loads are proposed, one is suitable for two-span beams with one span loaded, while the other is suitable for two-span beams with both spans equally loaded. Two identical concentrated loads symmetrically located about the mid-span of each loaded span were considered in the derivation of both formulae, and the effect of the height of loaded points for doubly symmetric beams was included. The formulae presented are also accurate enough in calculating the critical loads for two-span continuous beams with the mono-symmetric sections used in practice if the point of load application is at or above the shear centre. A linear approximation is suggested for the interaction of two spans when the two spans of the beam are not equally loaded. For a two-span continuous runway girder supporting moving cranes, the minimum critical load and the maximum absolute moment were investigated, some possible combination of wheel forces on beams considered, and approaches to calculating the critical load for each load combination are suggested when the girder has either one or two cranes moving along it.

The effect of finite deformations in the elastic stability of plane frames

1962 ◽  
Vol 266 (1324) ◽  
pp. 47-67 ◽  
Keyword(s):  
Critical Load ◽  
Failure Load ◽  
Elastic Stability ◽  
Load Level ◽  
Load Factor ◽  
Proportional Loading ◽  
Arbitrary Load ◽  
Plane Frames ◽  
The Difference ◽  
Rigid Joints

The circumstances are discussed under which orthogonal relations exist between the elastic critical modes of plane frames subjected to proportional loading. Orthogonal relations may be obtained provided the loading does not produce any components of deformation associated with any of the critical modes at arbitrary levels of the load factor, and provided no part of the structure remains statically indeterminate due to bar forces when all rigid joints are replaced by pin joints. When at arbitrary load factors, the structure deforms with components associated with any of the buckling modes, the elastic failure load is not identical with the lowest elastic critical load, although for many frames the two loads may be very close. A general expression is obtained which reveals the relation between the deformations at an arbitrary load level and the deflexions given by linear analysis. The difference between the elastic failure load and the elastic critical load is discussed, and an approximate treatment applicable to certain types of frame and associated loading is developed.

scholarly journals Estimates of Exceedances of Critical Loads for Acidifying Deposition in Alberta and Saskatchewan

2018 ◽  
Author(s):  
Paul A. Makar ◽  
Ayodeji Akingunola ◽  
Julian Aherne ◽  
Amanda S. Cole ◽  
Yayne-abeba Aklilu ◽  
...  
Keyword(s):  
Critical Load ◽  
Critical Loads ◽  
Oil Sands ◽  
Aquatic Ecosystems ◽  
Reasonable Agreement ◽  
Base Cation ◽  
Spatial Extent ◽  
Fugitive Dust ◽  
Sulphur Deposition ◽  
Total Deposition

Abstract. Estimates of potential harmful effects to ecosystems in the Canadian provinces of Alberta and Saskatchewan due to acidifying deposition were calculated, using a one year simulation of a high resolution implementation of the Global Environmental Multiscale – Modelling Air-quality and Chemistry (GEM-MACH) model, and estimates of aquatic and terrestrial ecosystem critical loads. The model simulation was evaluated against two different sources of deposition data; total deposition in precipitation and total deposition to snowpack in the vicinity of the Athabasca oil sands. The model captured much of the variability of observed ions in wet deposition in precipitation (observed versus model sulphur, nitrogen and base cation R2 values of 0.90, 0.76 and 0.72, respectively), while being biased high for sulphur deposition, and low for nitrogen and base cations (slopes 2.2, 0.89 and 0.40, respectively). Aircraft-observation-based estimates of fugitive dust emissions, shown to be a factor of ten higher than reported values (Zhang et al., 2017), were used to estimate the impact of increased levels of fugitive dust on model results. Model comparisons to open snowpack observations were shown to be biased high, but in reasonable agreement for sulphur deposition when observations were corrected to account for throughfall in needleleaf forests. The model-observation relationships for precipitation deposition data, along with the expected effects of increased (unreported) base cation emissions, were used to provide a simple observation-based correction to model deposition fields. Base cation deposition was estimated using published observations of base cation fractions in surface collected particles (Wang et al., 2015). Both original and observation-corrected model estimates of sulphur, nitrogen and base cation deposition were used in conjunction with critical load data created using the NEG-ECP (2001) and CLRTAP (2004, 2016, 2017) protocols for critical loads, using variations on the Simple Mass Balance model for forest and terrestrial ecosystems, and the Steady State Water Chemistry and the First-order Acidity Balance models for aquatic ecosystems. Potential ecosystem damage at 2013/14 emissions and deposition levels was predicted for regions within each of the ecosystem critical load datasets examined here. The spatial extent of the regions in exceedance of critical loads varied between 1 × 104 and 3.3 × 105 km2, for the more conservative observation-corrected estimates of deposition, with the variation dependant on the ecosystem and critical load protocol. The larger estimates (for aquatic ecosystems) represent a substantial fraction of the area of the provinces examined. Base cation deposition was shown to have a neutralizing effect on acidifying deposition, and the use of the aircraft and precipitation observation-based corrections to base cation deposition resulted in reasonable agreement with snowpack data collected in the oil sands area. However, critical load exceedances calculated using both observations and observation-corrected deposition suggest that the neutralization effect is limited in spatial extent, decreasing rapidly with distance from emissions sources, due to the rapid deposition of emitted primary particles dust particles as a function of their size.

scholarly journals Critical loads of sulphur and nitrogen for freshwaters in Great Britain and assessment of deposition reduction requirements with the First-order Acidity Balance (FAB) model

2000 ◽  
Vol 4 (1) ◽  
pp. 125-140 ◽  
Author(s):  
C. Curtis ◽  
T. Allott ◽  
J. Hall ◽  
R. Harriman ◽  
R. Helliwell ◽  
...  
Keyword(s):  
Great Britain ◽  
Water Chemistry ◽  
Critical Load ◽  
Acid Deposition ◽  
Critical Loads ◽  
Freshwater Ecosystems ◽  
Acid Neutralizing Capacity ◽  
First Order ◽  
Fab Model ◽  
S Deposition

Abstract. The critical loads approach is widely used within Europe to assess the impacts of acid deposition on terrestrial and freshwater ecosystems. Recent work in Great Britain has focused on the national application of the First-order Acidity Balance (FAB) model to a freshwaters dataset of 1470 lake and stream water chemistry samples from sites across Britain which were selected to represent the most sensitive water bodies in their corresponding 10 km grid square. A ``Critical Load Function" generated for each site is compared with the deposition load of S and N at the time of water chemistry sampling. The model predicts that when catchment processes reach steady-state with these deposition levels, increases in nitrate leaching will depress acid neutralizing capacity (ANC) below the critical threshold of 0 μeql-1 at more than a quarter of the sites sampled, i.e. the critical load of acid deposition is exceeded at these sites. The critical load exceedances are generally found in upland regions of high deposition where acidification has been previously recognised, but critical loads in large areas of western Scotland are also exceeded where little biological evidence of acidification has yet been found. There is a regional variation in the deposition reduction requirements for protection of the sampled sites. The FAB model indicates that in Scotland, most of the sampled sites could be protected by sufficiently large reductions in S deposition alone. In the English and Welsh uplands, both S and N deposition must be reduced to protect the sites. Current international commitments to reduce S deposition throughout Europe will therefore be insufficient to protect the most sensitive freshwaters in England and Wales. Keywords: critical loads; acidification; nitrate; FAB model; acid deposition

scholarly journals Critical load of steel trusses - graphical methods

2018 ◽  
Vol 219 ◽  
pp. 02015
Author(s):  
Piątkowski Michał
Keyword(s):  
Numerical Analysis ◽  
Critical Load ◽  
Critical Loads ◽  
Experimental Tests ◽  
Experimental Results ◽  
Graphical Methods ◽  
Critical Moment ◽  
Beam Elements ◽  
Steel Truss ◽  
Steel Trusses

The article presents graphical methods for determine critical loads of column and beam elements based on experimental results. The author presents the principles of using each method and the results of author's experimental tests on the instability phenomenon of planar steel truss with imperfections. The discussed methods were used to determine critical load of the tested truss, next compared with the results of numerical analysis. The validity of methods for determining the critical moment in the truss analysis has been confirmed.

The estimation of the critical load of a braced framework

1955 ◽  
Vol 231 (1184) ◽  
pp. 25-36 ◽  
Keyword(s):  
Critical Load ◽  
Critical Loads ◽  
The Other ◽  
Simple Criterion ◽  
Relaxation Methods ◽  
The Stability

With present methods of estimating the critical loads of triangulated frameworks by relaxation methods it is difficult to decide near the critical load whether the process is converging and the structure is stable, or whether the process is diverging and the structure unstable. This difficulty does not arise in the method presented here. Each triangle of the framework in turn is replaced by a hypothetical member until finally only one member of the truss remains, and this member has been modified in such a way as to take into account the stiffnesses of all the other members of the truss. A simple criterion for the stability of this final equivalent member is established and an example of the application of the method given.

CRITICAL LOADS OF SQUARE PLATES UNDER DIFFERENT INPLANE LOAD CONFIGURATIONS ON OPPOSITE EDGES

2001 ◽  
Vol 01 (02) ◽  
pp. 283-291 ◽  
Author(s):  
S. G. LEE ◽  
S. C. KIM ◽  
J. G. SONG
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Stability Analysis ◽  
Critical Load ◽  
Critical Loads ◽  
Patch Load ◽  
Fixity Factor ◽  
The Stability ◽  
Opposite Tendency ◽  
Width Factor

The elastic critical load coefficients of square plates, under different inplane load configurations on opposite plate edges, are determined and the results compared. The stability analysis was performed by a finite element method that was developed by the authors. The parameters considered in the analysis are the Kinney's fixity factor, and the width factor of the patch load. It was found that the coefficients of the critical loads increase with increasing values of fixity and width factors. The opposite tendency is that a plate under a patch loaded towards the two corners of an edge is more stable than a plate loaded concentrically at the center of the edge.

Sur les opérateurs différentiels symétrisants relatifs aux systèmes non-conservatifs

1998 ◽  
Vol 76 (5) ◽  
pp. 403-420
Author(s):  
R El Abdi ◽  
G Gambart
Keyword(s):  
Exact Solutions ◽  
Critical Load ◽  
General Expression ◽  
Critical Frequency ◽  
Differential Operators ◽  
Adjoint Problem ◽  
New Method ◽  
Numerical Comparison ◽  
Nonconservative Systems

For nonconservative systems, which we call systems with follower loads, a study is proposed concerning the differential operators which lead to a self-adjoint problem for a generalization of the Rayleigh quotient.In the case of punctual loads, we give the general expression for the identification of these operators. For some systems under follower loads, a new method is developed for the identification of the eigenvalues (critical load and critical frequency) when these operators do not exist. A numerical comparison is presented when the exact solutions do exist. PACS Nos. : 02.00 et 03.00

Modeling of effects of adhesive interlayers on contact-induced radial cracking in brittle coatings on substrates

2003 ◽  
Vol 18 (6) ◽  
pp. 1481-1486 ◽  
Author(s):  
Chun-Hway Hsueh ◽  
Jong Ho Kim ◽  
Do Kyung Kim
Keyword(s):  
Finite Element ◽  
Analytical Solution ◽  
Critical Load ◽  
Elastic Foundation ◽  
Critical Loads ◽  
Glass Substrates ◽  
Adhesive Interlayer ◽  
Semiempirical Equation ◽  
Radial Cracks ◽  
Fitting Parameters

The effects of soft adhesive interlayers on contact-induced radial cracking in brittle coatings on supporting substrates were investigated recently. A semiempirical equation for the critical load to initiate radial cracking was derived, which used three fitting parameters obtained by comparison with the finite element results. An analytical model is derived in the present study to illustrate the effects of adhesive interlayers. This is achieved by adopting the analogy between the coating/substrate system and a plate on an elastic foundation. In the presence of an adhesive interlayer, the interlayer/substrate bilayer is treated as the effective elastic foundation. The effective modulus of foundation for the bilayer is derived, and the solution for coating/interlayer/substrate systems can be obtained from the existing solution for coating/substrate systems by replacing the modulus of foundation. Specific results are calculated for critical loads to initiate radial cracks in silicon coatings bonded by adhesive interlayers of different materials and thicknesses to glass substrates. The present analytical solution provides an alternative other than the existing semiempirical equation in predicting the effects of adhesive interlayers.

Sign in / Sign up

Export Citation Format

Share Document