Nonlinear Creep Deformations of Columns of Rectangular Cross Section

1959 ◽  
Vol 26 (4) ◽  
pp. 517-525
Author(s):  
H. H. Bleich ◽  
O. W. Dillon
Keyword(s):  
Cross Section ◽  
Rectangular Cross Section ◽  
Critical Time ◽  
Bending Moment ◽  
Qualitative Reasoning ◽  
Upper And Lower Bounds ◽  
Nonlinear Creep ◽  
Essential Point ◽  
History Of ◽  
Creep Deformations

Abstract Creep deformations of columns of rectangular cross section are studied for the case of materials following the nonlinear law ϵ̇ = (1/E) σ̇ + λσk. The essential point of the paper is the following: The time rate of the curvature κ̇ of an element of a bar loaded by a constant force P and an increasing bending moment M(t) has bounds, which depend on P and on the instantaneous values of M and Ṁ, but not on the history of M. In combination with the collocation method, this permits the formulation of ordinary differential equations for upper and lower bounds on the deformations. Closed solutions for the critical time are obtained for one bound, while the other requires numerical integration. The bounds which are a function of the initial eccentricity are reasonably close and are presented in tables and graphs. By qualitative reasoning it is further shown that the location of the actual critical time with respect to the two bounds is governed by the ratio of the column load P and the nominal Euler buckling load PE of the column if it were elastic.

The Effect of Shear and Normal Forces on the Fully Plastic Moment of a Beam of Rectangular Cross Section

1961 ◽  
Vol 28 (2) ◽  
pp. 269-274 ◽  
Author(s):  
B. G. Neal
Keyword(s):  
Cantilever Beam ◽  
Cross Section ◽  
Lower Bounds ◽  
Rectangular Cross Section ◽  
Bending Moment ◽  
Upper And Lower Bounds ◽  
Collapse Load ◽  
Shear Forces ◽  
Normal Forces ◽  
Interaction Relations

The value of the fully plastic moment of a beam is known to be reduced by both normal and shear forces, and their separate effects have been studied in some detail, but little attention has been paid to the reduction caused by normal and shear forces acting simultaneously. This problem is discussed with reference to a cantilever beam of rectangular cross section subjected to both shear and normal forces at the free end. Upper and lower bounds to the collapse load are determined, and the results are presented in the form of interaction relations between the shear and normal forces and the bending moment at the clamped end of the cantilever at collapse.

Bearing Capacity of Structures Made of Materials with Different Tensile and Compression Strengths

2019 ◽  
Vol 968 ◽  
pp. 200-208
Author(s):  
Mykola Soroka
Keyword(s):  
Cross Section ◽  
Rectangular Cross Section ◽  
Limit State ◽  
Analytical Calculation ◽  
Bending Moment ◽  
Maximum Load ◽  
Longitudinal Force ◽  
Plastic State ◽  
Limiting State ◽  
Tension And Compression

The paper considers the problem of the ultimate load finding for structures made of a material with different limits of tensile strength and compression. The modulus of elasticity under tension and compression is the same. It is assumed that upon reaching the ultimate strength, the material is deformed indefinitely. The calculations use a simplified material deformation diagram — Prandtl diagrams. The limiting state of a solid rectangular section under the action of a longitudinal force and a bending moment is considered. The dependences describing the boundary of the strength of a rectangular cross section are obtained. Formulas allowing the calculation of the values of the limit forces and under the action of which the cross section passes into the plastic state are derived. Examples of the analytical calculation of the maximum load for the frame and two-hinged arch are given. An algorithm is proposed and a program for calculating arbitrary flat rod systems according to the limit state using the finite element method is compiled. The proposed algorithm does not involve the use of iterative processes, which leads to an exact calculation of the maximum load within the accepted assumptions.

scholarly journals NUMERICAL INVESTIGATION OF THE ELASTIC-PLASTIC LINEAR HARDENING STRESS-STRAIN STATE OF THE FRAME ELEMENT CROSS-SECTION

2016 ◽  
Vol 8 (3) ◽  
pp. 94-100
Author(s):  
Andrius Grigusevičius ◽  
Gediminas Blaževičius
Keyword(s):  
Cross Section ◽  
Axial Force ◽  
Cross Sections ◽  
Rectangular Cross Section ◽  
Bending Moment ◽  
Software Environment ◽  
Stress Strain ◽  
Hardening Material ◽  
Frame Element ◽  
Nonlinear Stress

The aim of this paper is to present a solution algorithm for determining the frame element crosssection carrying capacity, defined by combined effect of bending moment and axial force. The distributions of stresses and strains inside a cross-section made of linearly hardening material are analysed. General nonlinear stress-strain dependencies are composed. All relations are formed for rectangular cross-section for all possible cases of combinations of axial force and bending moment. To this end, five different stress-strain states are investigated and four limit axial force values are defined in the present research. The nonlinear problem is solved in MATLAB mathematical software environment. Stress-strain states in the cross-sections are investigated in detail and graphically analysed for two numerical experiments.

scholarly journals Cross-Sectional Analysis of the Resistance of Rc Members Subjected to Bending With/without Axial Force

2021 ◽  
Author(s):  
Marek Lechman
Keyword(s):  
Cross Section ◽  
Axial Force ◽  
Rectangular Cross Section ◽  
Bending Moment ◽  
Equilibrium Equations ◽  
Cross Sectional ◽  
Cross Sectional Analysis ◽  
Nonlinear Stress ◽  
The Cross ◽  
Sectional Analysis

The paper presents section models for analysis of the resistance of RC members subjected to bending moment with or without axial force. To determine the section resistance the nonlinear stress-strain relationship for concrete in compression is assumed, taking into account the concrete softening. It adequately describes the behavior of RC members up to failure. For the reinforcing steel linear elastic-ideal plastic model is applied. For the ring cross-section subjected to bending with axial force the normalized resistances are derived in the analytical form by integrating the cross-sectional equilibrium equations. They are presented in the form of interaction diagrams and compared with the results obtained by testing conducted on RC columns under eccentric compression. Furthermore, the ultimate normalized bending moment has been derived for the rectangular cross-section subjected to bending without axial force. It was applied in the cross-sectional analysis of steel and concrete composite beams, named BH beams, consisting of the RC rectangular core placed inside a reversed TT welded profile. The comparisons made indicated good agreements between the proposed section models and experimental results.

scholarly journals STRESS-STRAIN STATE OF REINFORCED CONCRETE BEAMS STRENGTHENED WITH A NEW EXTERNAL STEEL STRUCTURE

2017 ◽  
Vol 1 (48) ◽  
pp. 90-99
Author(s):  
V. P. Zhyrakhivskyi ◽  
М. G. Chekanovych ◽  
О. М. Chekanovych
Keyword(s):  
Reinforced Concrete ◽  
Cross Section ◽  
Concrete Beams ◽  
Rectangular Cross Section ◽  
Bending Moment ◽  
Steel Structure ◽  
Research Data ◽  
Reinforced Concrete Beams ◽  
Reinforced Beams ◽  
Steel Bars

The study presents a new structure for strengthening of one-span reinforced concrete beams in rectangular cross-section using external steel bars. The specific feature of the proposed strengthening is the unloading of the compressed upper zone of a beam with simultaneous compression of its lower stretched zone. The article considers some variants of making the strengthening structure with rigid and flexible reinforcement elements for faster tension of external bars, and the variant including only flexible elements. It provides a design scheme and method for such reinforced beams. The study provides experimental research data on the series of beams with different parameters of the strengthening structure in the form of «bending moment – deflection» and «bending moment - deformation of concrete» dependencies.

Analysis of Helical Torsion Spring for Epicyclic Traction Drive

2017 ◽  
Vol 261 ◽  
pp. 408-415
Author(s):  
Geza Nemeth
Keyword(s):  
Cross Section ◽  
Rectangular Cross Section ◽  
Bending Moment ◽  
Initial Shape ◽  
Traction Drive ◽  
Torsion Spring ◽  
Spring Wire ◽  
Tangential Forces ◽  
Radial Forces ◽  
Proper Operation

Let us consider a simple epicyclic traction drive containing a sun wheel, an annular wheel, planetary wheels and a planet carrier. The annular wheel is substituted by a helical torsion spring with rectangular cross section. The spring has initial tensioning, tightened to the planetary wheels. When the number of coils is z and the number of planet wheels is N, there are zN piece of concentrated forces acting to the spring from inside towards outside. The main load of the spring is bending, it is computable along the spring wire. The bending moment is limited by the spring material and the cross section of spring. The radial forces acting to the spring are governed by the constraint of stress equality, the deflections of the contacting parts are determined by the radial (and slightly the tangential) forces. An initial shape of the spring that assures the proper operation of the drive after the assembly, is calculated by elementary mechanical calculation methods. The main goal is the developing of a traction drive in which the clamping force is proportional to the torque which should be transmitted, for the sake of the favourable life rating and the efficiency.

Shape Memory Elements in Bending: Influence of the Shape of their Cross-Section

2000 ◽  
Vol 11 (12) ◽  
pp. 977-984 ◽  
Author(s):  
Vratislav Kafka ◽  
David Vokoun
Keyword(s):  
Shape Memory ◽  
Cross Section ◽  
Elastic Limit ◽  
Rectangular Cross Section ◽  
Circular Cross Section ◽  
Bending Moment ◽  
Internal Variables ◽  
Memory Recovery ◽  
The Cross ◽  
The Mathematical Model

The effect of the shape of the cross-section of a bent prismatic bar on its shape memory recovery moment is investigated. The analysis is based on the mathematical model of the first author (Kafka, 1994a, 1994b, 2001). The area of the cross-section of the bar is assumed to be constant, the shape of the cross-section is varied. The investigated shapes are rectangles with various relations of their sides, and a circular cross-section. It is assumed that the rod is bent above elastic limit and unloaded at room temperature, which results in macroscopic residual stresses giving zero bending moment, and in residual internal variables descriptive of the change of the state of the material. Then, the resulting form is held fixed and temperature of the rod is raised. Due to the increase of temperature there arise recovery stresses resulting in recovery moments. These moments—depending on the shape of the cross-section—are calculated, and in this way the effectiveness of the shape of the cross-section is evaluated. In the case of a rectangular cross-section the effect of the relation of the sides is strongly non-linear, the effect of the circular cross-section is lower by 20% than that of a square cross-section.

On the Determination of Stresses and Deflections for Anisotropic Homogeneous Cantilever Beams

1976 ◽  
Vol 43 (1) ◽  
pp. 75-80 ◽  
Author(s):  
S. Nair ◽  
E. Reissner
Keyword(s):  
Cross Section ◽  
Lower Bounds ◽  
Basic Assumption ◽  
Rectangular Cross Section ◽  
Upper And Lower Bounds ◽  
Cantilever Beams ◽  
Normal Stresses ◽  
Complementary Energy ◽  
Minimum Potential

We analyze the effect of anisotropy on beam flexibility by the derivation of upper and lower bounds, through use of the principles of minimum potential and complementary energy, for the load-deflection ratios of narrow rectangular cross-section cantilever beams. The basic assumption is a class of stress-strain relations of such nature that normal strains are caused not only by normal stresses but also by shearing stresses, and shearing strains are caused not only by shearing stresses but also by normal stresses.

On the Experimental Analysis of Temperature Influence on Stiffness of Reinforced Concrete Beams

2015 ◽  
Vol 6 (1) ◽  
pp. 49-58 ◽  
Author(s):  
Robert Kowalski ◽  
Michal Glowacki ◽  
Marian Abramowicz
Keyword(s):  
Reinforced Concrete ◽  
Cross Section ◽  
Cross Sections ◽  
Concrete Beams ◽  
Rectangular Cross Section ◽  
Thermal Strain ◽  
Bending Moment ◽  
Reinforced Concrete Beams ◽  
Stiffness Reduction ◽  
Tensile Zone

The paper presents results of experimental research whose main topic was determination of stiffness reduction in bent reinforced concrete beams in two cases: when only tensioned or only compressed zone was exposed to high temperature. Twenty four reinforced concrete beams with rectangular cross-section were prepared for the experiment. Eight groups of beams were prepared in total: 2 with reinforcement ratio - 0.44 and 1.13% x 2 levels of load - 50 or 70% of destructive force ensuring the constant value of bending moment in the centre part of heated beams x 2 static schemes. Three beams were used in each group. Significant cross-section stiffness reduction was observed in beams where the tensile zone was heated. This was due to considerable elongation of the bars where the steel load elongation summed up with the free thermal strain. In beams where the compressed zone was heated the stiffness reduction was observed only after the time where the tensile zone heated cross-sections were already destroyed.

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