Lecture

Mechanics of Slender Structure: Loaded Beam Approximations

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In course

ME-411: Mechanics of slender structures

Analysis of the mechanical response and deformation of slender structural elements. ...

Description

This lecture covers the matched approximations of a loaded beam, focusing on strain approximation and constitutive models. The instructor discusses the conditions for small rotations and linear extensibility, as well as the exact non-linear strain. The lecture emphasizes the importance of establishing equilibrium using the principle of virtual works and Lagrange multipliers. The instructor also delves into the inextensibility condition and the variations of strain, leading to the derivation of equilibrium equations. The lecture concludes with the consideration of different regimes based on the imposed strain and the constitutive law, highlighting the significance of geometric approximations and the assumption of inextensibility.

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Instructor

Pedro Miguel Nunes Pereira de Almeida Reis

Pedro M. Reis is a Professor of Mechanical Engineering at the École Polytechnique Fédérale de Lausanne (EPFL) in Switzerland, where he is the Director of the Institute of Mechanical Engineering. Prof. Reis received a B.Sc. in Physics from the University of Manchester, UK (1999), a Certificate of Advanced Studies in Mathematics (Part III Maths) from St. John’s College and DAMTP, University of Cambridge (2000), and a Ph.D. in physics from the University of Manchester (2004). He was a postdoc at the City College of New York (2004-2005) and at the CNRS/ESPCI in Paris (2005-2007). He joined MIT in 2007 as an Instructor in Applied Mathematics. In 2010 he moved to MIT’s School of Engineering, with dual appointments in Mechanical Engineering and Civil & Environmental Engineering, first as the Esther and Harold E. Edgerton Assistant Professor and, since the summer of 2014 as Gilbert W. Winslow Associate Professor. In October 2013, the Popular Science magazine named Prof. Reis to its 2013 “Brilliant 10” list of young stars in Science and Technology. He has received the 2014 CAREER Award (NSF), the 2016 Thomas J.R. Hughes Young Investigator Award (Applied Mechanics Division of the ASME), the 2016 GSOFT Early Career Award for Soft Matter Research (APS), he is a Fellow of the APS, and he is the 2021 President of the Society of Engineering Science (SES).

Official source

Ontological neighbourhood

Chemical engineering

Materials science: Deformation of materials

Related concepts (32)

Infinitesimal strain theory

In continuum mechanics, the infinitesimal strain theory is a mathematical approach to the description of the deformation of a solid body in which the displacements of the material particles are assumed to be much smaller (indeed, infinitesimally smaller) than any relevant dimension of the body; so that its geometry and the constitutive properties of the material (such as density and stiffness) at each point of space can be assumed to be unchanged by the deformation.

Strain-rate tensor

In continuum mechanics, the strain-rate tensor or rate-of-strain tensor is a physical quantity that describes the rate of change of the deformation of a material in the neighborhood of a certain point, at a certain moment of time. It can be defined as the derivative of the strain tensor with respect to time, or as the symmetric component of the Jacobian matrix (derivative with respect to position) of the flow velocity. In fluid mechanics it also can be described as the velocity gradient, a measure of how the velocity of a fluid changes between different points within the fluid.

Deformation (engineering)

In engineering, deformation refers to the change in size or shape of an object. Displacements are the absolute change in position of a point on the object. Deflection is the relative change in external displacements on an object. Strain is the relative internal change in shape of an infinitesimally small cube of material and can be expressed as a non-dimensional change in length or angle of distortion of the cube. Strains are related to the forces acting on the cube, which are known as stress, by a stress-strain curve.

Stress–strain analysis

Stress–strain analysis (or stress analysis) is an engineering discipline that uses many methods to determine the stresses and strains in materials and structures subjected to forces. In continuum mechanics, stress is a physical quantity that expresses the internal forces that neighboring particles of a continuous material exert on each other, while strain is the measure of the deformation of the material. In simple terms we can define stress as the force of resistance per unit area, offered by a body against deformation.

Stress–strain curve

In engineering and materials science, a stress–strain curve for a material gives the relationship between stress and strain. It is obtained by gradually applying load to a test coupon and measuring the deformation, from which the stress and strain can be determined (see tensile testing). These curves reveal many of the properties of a material, such as the Young's modulus, the yield strength and the ultimate tensile strength. Generally speaking, curves representing the relationship between stress and strain in any form of deformation can be regarded as stress–strain curves.

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