Distribution of Stress in Built-In Beams of Narrow Rectangular Cross Section

1942 ◽  
Vol 9 (3) ◽  
pp. A108-A116
Author(s):  
F. B. Hildebrand ◽  
Eric Reissner
Keyword(s):  
Cross Section ◽  
Elementary Theory ◽  
Rectangular Cross Section ◽  
Beam Theory ◽  
Degree Polynomial ◽  
Height Ratio ◽  
Load Curve ◽  
Normal Stress Distribution ◽  
Stress Curve ◽  
Distribution Of Stress

Abstract This paper deals with the problem of the distribution of stress in cantilever beams of narrow rectangular flanged cross section with one end of the beam rigidly built-in. Since an exact solution of this plane-stress problem appears difficult to obtain, an approximate solution is derived by applying the principle of least work. Instead of the linear normal stress distribution of the elementary beam theory, a third-degree polynomial is assumed, and the spanwise variation of this stress curve is determined by means of the calculus of variations. Numerical results are obtained with regard to the stresses at the built-in end of the beam, in their dependence upon (a) the span-height ratio of the beam, (b) the flange area-web area ratio of the beam, (c) Poisson’s ratio of the material, and (d) the distribution of load along the span. It is found that the deviations from the results of the elementary theory may be appreciable when the distance of the center of gravity of the load curve from the built-in end of the beam is less than twice the height of the cross section of the beam.

Bending Vibrations of Variable Section Beams

1956 ◽  
Vol 23 (1) ◽  
pp. 103-108
Author(s):  
E. T. Cranch ◽  
Alfred A. Adler
Keyword(s):  
Cross Section ◽  
Rectangular Cross Section ◽  
Circular Cross Section ◽  
Beam Theory ◽  
Cantilever Beams ◽  
Constant Depth ◽  
Bending Vibrations ◽  
Variable Section

Abstract Using simple beam theory, solutions are given for the vibration of beams having rectangular cross section with (a) linear depth and any power width variation, (b) quadratic depth and any power width variation, (c) cubic depth and any power width variation, and (d) constant depth and exponential width variation. Beams of elliptical and circular cross section are also investigated. Several cases of cantilever beams are given in detail. The vibration of compound beams is investigated. Several cases of free double wedges with various width variations are discussed.

A Refinement of the Theory of Buckling of Rings Under Uniform Pressure

1955 ◽  
Vol 22 (1) ◽  
pp. 95-102
Author(s):  
A. P. Boresi
Keyword(s):  
Cross Section ◽  
Classical Result ◽  
Rectangular Cross Section ◽  
Finite Thickness ◽  
Beam Theory ◽  
Elastic Stability ◽  
External Pressure ◽  
Uniform Pressure ◽  
Variational Theory ◽  
Uniform External Pressure

Abstract A general variational theory of elastic stability that was originated by E. Trefftz (1) is applied to the problem of buckling of rings of rectangular cross section subjected to uniform external pressure. The theory is believed to be more rigorous than previous treatments of the problem, since it avoids conventional assumptions of curved-beam theory, such as the assumptions that plane sections remain plane and that radial stresses vanish. The classical result of Levy (2) is confirmed for a ring of infinitesimal thickness. New results are obtained which show the effect of the finite thickness of a ring on the coefficients in the buckling formula.

A simple single variable shear deformation theory for a rectangular beam

2016 ◽  
Vol 231 (24) ◽  
pp. 4576-4591 ◽  
Author(s):  
Rameshchandra P Shimpi ◽  
Rajesh A Shetty ◽  
Anirban Guha
Keyword(s):  
Shear Deformation ◽  
Cross Section ◽  
Governing Equation ◽  
Deformation Theory ◽  
Rectangular Cross Section ◽  
Shear Deformation Theory ◽  
Beam Theory ◽  
Single Variable ◽  
Bernoulli Beam ◽  
Euler Bernoulli Beam

This paper proposes a simple single variable shear deformation theory for an isotropic beam of rectangular cross-section. The theory involves only one fourth-order governing differential equation. For beam bending problems, the governing equation and the expressions for the bending moment and shear force of the theory are strikingly similar to those of Euler–Bernoulli beam theory. For vibration and buckling problems, the Euler–Bernoulli beam theory governing equation comes out as a special case when terms pertaining to the effects of shear deformation are ignored from the governing equation of present theory. The chosen displacement functions of the theory give rise to a realistic parabolic distribution of transverse shear stress across the beam cross-section. The theory does not require a shear correction factor. Efficacy of the proposed theory is demonstrated through illustrative examples for bending, free vibrations and buckling of isotropic beams of rectangular cross-section. The numerical results obtained are compared with those of exact theory (two-dimensional theory of elasticity) and other first-order and higher-order shear deformation beam theory results. The results obtained are found to be accurate.

Timoshenko Beam Theory Is Not Always More Accurate Than Elementary Beam Theory

1977 ◽  
Vol 44 (2) ◽  
pp. 337-338 ◽  
Author(s):  
J. W. Nicholson ◽  
J. G. Simmonds
Keyword(s):  
Cross Section ◽  
Plane Stress ◽  
Timoshenko Beam ◽  
Virtual Work ◽  
Rectangular Cross Section ◽  
Weighted Average ◽  
Beam Theory ◽  
Vertical Displacement ◽  
Timoshenko Beam Theory ◽  
Displacement Fields

A counterexample involving a homogeneous, elastically isotropic beam of narrow rectangular cross section supports the assertion in the title. Specifically, a class of two-dimensional displacement fields is considered that represent exact plane stress solutions for a built-in cantilevered beam subject to “reasonable” loads. The one-dimensional vertical displacement V predicted by Timoshenko beam theory for these loads can be regarded as an approximation to either the exact vertical displacement v at the center line, or a weighted average of v over the cross section, or a quantity defined to make the virtual work of beam theory equal to that of plane stress theory. Regardless of the interpretation of V and despite the presence of an adjustable shear factor, Timoshenko beam theory for this class of problems is never more accurate than elementary beam theory.

A Photoelastic Study of Maximum Tensile Stresses in Simply Supported Short Beams Under Central Transverse Impact

1957 ◽  
Vol 24 (4) ◽  
pp. 509-514
Author(s):  
A. A. Betser ◽  
M. M. Frocht
Keyword(s):  
Satisfactory Agreement ◽  
Cross Section ◽  
Transverse Section ◽  
Elementary Theory ◽  
Rectangular Cross Section ◽  
Approximate Theory ◽  
Transverse Impact ◽  
Tensile Stresses ◽  
Simply Supported ◽  
Wide Range

Abstract Simply supported short Castolite beams of uniform rectangular cross section were subjected to central transverse impact by a heavy mass. Photoelastic streak photographs were taken of the transverse section of symmetry for a wide range of spans, widths, and impact velocities at exposures of less than 1 microsec. The maximum tensile stresses were determined. Comparison with the elementary theory for long beams shows that while this theory is satisfactory for long beams, it does not agree with the results from short beams. An approximate theory for short beams under central impact is developed which gives satisfactory agreement. The duration of impact also was determined and the appearance of isotropic points is discussed.

Contribution to beam theory based on 3-D energy principle

2017 ◽  
Vol 23 (5) ◽  
pp. 775-786 ◽  
Author(s):  
Erick Pruchnicki
Keyword(s):  
Potential Energy ◽  
Cross Section ◽  
Rectangular Cross Section ◽  
Three Dimensional ◽  
Elastic Beam ◽  
Beam Theory ◽  
Displacement Fields ◽  
Energy Principle ◽  
Lateral Boundary Conditions ◽  
The First Variation

This paper presents a general elastic beam theory, which is consistent with the principle of stationary three-dimensional potential energy. For the sake of simplicity we consider the case of a rectangular cross section. The series expansion of the displacement field up to fourth-order in h (dimension of the cross section) is defined by 45 unknowns. The first variation of the potential energy must be zero but we only impose that each term guarantees an [Formula: see text]error. By adding supplementary lateral boundary conditions and on two extremities end cross section of the beam, we finally arrive at a well posed system of unidimensional differential equations. A linear algebraic dependence with respect to 16 displacement fields allows us to reduce the unknown to 19 displacement fields. To our knowledge this work is the first contribution to this end when the beam problem is completely three-dimensional.

Sign in / Sign up

Export Citation Format

Share Document