A Beam Theory for Large Global Rotation, Moderate Local Rotation, and Small Strain

1988 ◽  
Vol 55 (1) ◽  
pp. 179-184 ◽  
Author(s):  
D. A. Danielson ◽  
D. H. Hodges
Keyword(s):  
Cross Section ◽  
Variational Principles ◽  
Beam Theory ◽  
Small Strain ◽  
General Context ◽  
Equilibrium Equations ◽  
Cross Sectional ◽  
Local Rotation ◽  
Material Points

Kinematical relations are derived to account for the finite cross-sectional warping occurring in a beam undergoing large deflections and rotations due to deformation. The total rotation at any point in the beam is represented as a large global rotation of the reference triad (a frame which moves nominally with the reference cross section material points), a small rotation that is constant over the cross section and is due to shear, and a local rotation whose magnitude may be small to moderate and which varies over a given cross section. Appropriate variational principles, equilibrium equations, boundary conditions, and constitutive laws are obtained. Two versions are offered: an intrinsic theory without reference to displacements, and an explicit theory with global rotation characterized by a Rodrigues vector. Most of the formulas herein have been published, but we reproduce them here in a new concise notation and a more general context. As an example, the theory is shown to predict behavior that agrees with published theoretical and experimental results for extension and torsion of a pretwisted strip. The example also helps to clarify the role of local rotation in the kinematics.

A Geometrically Exact Rod Model Including In-Plane Cross-Sectional Deformation

2010 ◽  
Vol 78 (1) ◽  
Author(s):  
Ajeet Kumar ◽  
Subrata Mukherjee
Keyword(s):  
Energy Density ◽  
Cross Section ◽  
Strain Energy Density ◽  
Strain Energy ◽  
Three Dimensional ◽  
Rigid Motion ◽  
Transversely Isotropic ◽  
Equilibrium Equations ◽  
Cross Sectional ◽  
Deformation Problem

We present a novel approach for nonlinear, three dimensional deformation of a rod that allows in-plane cross-sectional deformation. The approach is based on the concept of multiplicative decomposition, i.e., the deformation of a rod’s cross section is performed in two steps: pure in-plane cross-sectional deformation followed by its rigid motion. This decomposition, in turn, allows straightforward extension of the special Cosserat theory of rods (having rigid cross section) to a new theory allowing in-plane cross-sectional deformation. We then derive a complete set of static equilibrium equations along with the boundary conditions necessary for analytical/numerical solution of the aforementioned deformation problem. A variational approach to solve the relevant boundary value problem is also presented. Later we use symmetry arguments to derive invariants of the objective strain measures for transversely isotropic rods, as well as for rods with inbuilt handedness (hemitropy) such as DNA and carbon nanotubes. The invariants derived put restrictions on the form of the strain energy density leading to a simplified form of quadratic strain energy density that exhibits some interesting physically relevant coupling between the different modes of deformation.

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.

Sandwich Beam Analysis

1972 ◽  
Vol 39 (3) ◽  
pp. 773-778 ◽  
Author(s):  
D. Krajcinovic
Keyword(s):  
Cross Section ◽  
Static Load ◽  
Sandwich Beam ◽  
Beam Theory ◽  
Simple Algebra ◽  
Sandwich Beams ◽  
Consistent Theory ◽  
Entire Cross Section ◽  
Cross Sectional ◽  
Beam Analysis

A consistent theory of sandwich beams subjected to static load is presented. The theory is developed under the assumption that the Bernoulli’s hypothesis is valid for each lamina independently but not for the entire cross section as a whole. It is shown that the generalized displacement may be chosen in such a way that the set of equations governing the motions for which the beam remains straight on one, and a set of equations describing bending and shear types of motions on the other hand are independent. Furthermore, after some simple algebra, separate equations for each generalized displacement are derived. The normal stress is given in the from which is familiar from strength of materials with two additional terms embodying the influence of the cross-sectional distortion (deviation from classical beam theory).

A Beam Theory for Anisotropic Materials

1985 ◽  
Vol 52 (2) ◽  
pp. 416-422 ◽  
Author(s):  
O. A. Bauchau
Keyword(s):  
Basic Assumption ◽  
Variational Principles ◽  
Axial Stress ◽  
Beam Theory ◽  
Short Review ◽  
Cross Sectional ◽  
Out Of Plane ◽  
In Series ◽  
Axial Stress Distribution ◽  
Energy Principles

Beam theory plays an important role in structural analysis. The basic assumption is that initially plane sections remain plane after deformation, neglecting out-of-plane warpings. Predictions based on these assumptions are accurate for slender, solid, cross-sectional beams made out of isotropic materials. The beam theory derived in this paper from variational principles is based on the sole kinematic assumption that each section is infinitely rigid in its own plane, but free to warp out of plane. After a short review of the Bernoulli and Saint-Venant approaches to beam theory, a set of orthonormal eigenwarpings is derived. Improved solutions can be obtained by expanding the axial displacements or axial stress distribution in series of eigenwarpings and using energy principles to derive the governing equations. The improved Saint-Venant approach leads to fast converging solutions and accurate results are obtained considering only a few eigenwarping terms.

On a One-Dimensional Theory of Finite Torsion and Flexure of Anisotropic Elastic Plates

1981 ◽  
Vol 48 (3) ◽  
pp. 601-605 ◽  
Author(s):  
E. Reissner
Keyword(s):  
Linear Theory ◽  
Beam Theory ◽  
Bending Moment ◽  
Elastic Plates ◽  
Equilibrium Equations ◽  
Dimensional Theory ◽  
Cross Sectional ◽  
One Dimensional ◽  
Slender Beams ◽  
Anisotropic Elastic

Equations for small finite displacements of shear-deformable plates are used to derive a one-dimensional theory of finite deformations of straight slender beams with one cross-sectional axis of symmetry. The equations of this beam theory are compared with the corresponding case of Kirchhoff’s equations, and with a generalization of Kirchhoff’s equations which accounts for the deformational effects of cross-sectional forces. Results of principal interest are: 1. The equilibrium equations are seven rather than six, in such a way as to account for cross-sectional warping. 2. In addition to the usual six force and moment components of beam theory, there are two further stress measures, (i) a differential plate bending moment, as in the corresponding linear theory, and (ii) a differential sheet bending moment which does not occur in linear theory. The general results are illustrated by the two specific problems of finite torsion of orthotropic beams, and of the buckling of an axially loaded cantilever, as a problem of bending-twisting instability caused by material anisotropy.

A Study of the Effect of Geometry Changes on the Structural Stiffness of a Composite D-Spar

2010 ◽  
Author(s):  
Swaroop B. Visweswaraiah ◽  
Damiano Pasini ◽  
Larry Lessard
Keyword(s):  
Cross Section ◽  
Beam Theory ◽  
Leading Edge ◽  
Structural Stiffness ◽  
Cross Sectional ◽  
Composite Blade ◽  
The Matrix ◽  
Coupling Parameters ◽  
Laminate Theory ◽  
The Impact

The paper examines the impact of varying two geometric cross-section parameters of an advance composite D-spar on its structural stiffness. For a given blade topology, the orientation of the D-spar web with respect to the beam axis and the distance of the D-spar web from the leading edge of the blade have been selected here as the variables of study, as they govern the elastic properties of the composite cross-section. A code has been developed to calculate the matrix terms of the Euler-Bernoulli cross-sectional stiffness utilizing the closed form expressions of the structural properties formulated by assuming both Thin-Walled composite Beam theory (TWB) and Classical Laminate Theory. The code has been validated through the Variational Asymptotic Beam Sectional analysis (VABS) for the cross-sectional stiffness matrix. Two cases have been studied for a quasi-isotropic laminate D-spar. The first is for a symmetric airfoil, whereas the second is for an unsymmetrical airfoil. The variation of the stiffness parameters for the quasi-isotropic D-spar including the coupling parameters has been visualized into parametric maps. The paper also examines the impact that these geometric variables have on the stiffness-to-mass ratio to show that along with the ply orientations they play a major role in the aeroelastic tailoring and structural optimization of a composite blade.

ON THE INFLUENCE OF MATERIAL COUPLINGS ON THE LINEAR AND BUCKLING BEHAVIOR OF I-SECTION COMPOSITE COLUMNS

2007 ◽  
Vol 07 (02) ◽  
pp. 243-272 ◽  
Author(s):  
N. FREITAS SILVA ◽  
N. SILVESTRE
Keyword(s):  
Shear Deformation ◽  
Cross Section ◽  
Plane Deformation ◽  
Beam Theory ◽  
Laminated Plates ◽  
Element Formulation ◽  
Equilibrium Equations ◽  
Composite Columns ◽  
Buckling Behavior ◽  
Section Analysis

This paper presents the incorporation of shear deformation effects into a Generalized Beam Theory (GBT) developed to analyze the structural behavior of composite thin-walled columns made of laminated plates and displaying arbitrary orthotropy. Unlike other existing beam theories, the present GBT formulation incorporates in a unified fashion (i) elastic coupling effects, (ii) warping effects, (iii) cross-section in-plane deformation and (iv) shear deformation. The main concepts and procedures involved in the available GBT are adapted/modified to account for the specific aspects related to the member shear deformation. In particular, the GBT fundamental equilibrium equations are presented and their terms are physically interpreted. An I-section is used to illustrate the performance of GBT cross-section analysis and the mechanical properties are explained in detail. With the purpose of solving the GBT system of differential equilibrium equations, a finite element formulation is briefly presented. Finally, in order to clarify the concepts involved in the formulated GBT and illustrate its application and capabilities, the linear (first-order) and stability behavior of three composite I-section members displaying non-aligned orthotropy are analyzed and the results obtained are thoroughly discussed and compared with estimates available in the literature.

scholarly journals Natural plant stems modelling in a three-point bending test

2019 ◽  
Vol 252 ◽  
pp. 07001
Author(s):  
Bartosz Kawecki ◽  
Jerzy Podgórski ◽  
Aleksandra Głowacka
Keyword(s):  
Numerical Modelling ◽  
Cross Section ◽  
Beam Theory ◽  
Bending Test ◽  
Basic Material ◽  
Cross Sectional ◽  
Natural Plant ◽  
Cosine Series ◽  
Three Point Bending ◽  
Miscanthus Giganteus

The paper presents an approach to natural plant stems numerical modelling in a three-point bending test. Introduced subject was connected with elaborating more efficient systems for harvesting energetic plants. There were modelled, and laboratory tested two types of stems – sida hermaphrodita and miscanthus giganteus. Course of proceedings for obtaining natural cross-sectional dimensions with graphical data processing was described in detail. Basing on dozens of stems slices from random parts of plants, three different cross-section approximations were proposed and computationally implemented – a circular pipe, an elliptical pipe (symmetrical cross-section) and a sine-cosine series pipe (asymmetrical cross-section). Analytical formulas for calculating a cross-sectional area and moments of inertia for each approximation were given. Basic material parameters as an elastic modulus and yielding stress was obtained from simply supported beam theory and laboratory force – the deflexion relation. FEM models were created in Simulia Abaqus software using C3D20R elements. Preliminary approach to modelling damage with perfect plasticity was done basing on several samples bended to failure in laboratory tests. Conclusions for future work with numerical modelling natural plant stems were drawn.

Sign in / Sign up

Export Citation Format

Share Document