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Deformations in a Symmetric Member in Bending

Deformations in a Symmetric Member in Bending

When analyzing the deformation of a symmetric prismatic member subjected to bending by equal and opposite couples, it becomes clear that as the member bends, the originally straight lines on its wider faces curve into circular arcs, with a constant radius centered at a point known as Point C. This phenomenon helps to understand the stress and strain distribution within the member more clearly.
When the member is segmented into tiny cubic elements, it is observed that the primary stress...

Relation between Poisson's ratio, Modulus of Elasticity and Modulus of Rigidity

Relation between Poisson's ratio, Modulus of Elasticity and Modulus of Rigidity

Deformation occurs in axial and transverse directions when an axial load is applied to a slender bar. This deformation impacts the cubic element within the bar, transforming it into either a rectangular parallelepiped or a rhombus, contingent on its orientation. This transformation process induces shearing strain. Axial loading elicits both shearing and normal strains. Applying an axial load instigates equal normal and shearing stresses on elements oriented at a 45° angle to the load axis.

Bending of Curved Members - Strain Analysis

Bending of Curved Members - Strain Analysis

The mechanics of deformation in curved members, such as beams or arches, under bending moments, involve complex responses. When such a member, symmetric about the y-axis and shaped like a segment of a circle centered at point C, is subjected to equal and opposite forces, its curvature and surface lengths change significantly. This alteration results in the shift of the curvature's center from C to C', indicating a tighter curve.
The important part of bending analysis for such a member is the...

Deformation of Member under Multiple Loadings

Deformation of Member under Multiple Loadings

When a rod is made of different materials or has various cross-sections, it must be divided into parts that meet the necessary conditions for determining the deformation. These parts are each characterized by their internal force, cross-sectional area, length, and modulus of elasticity. These parameters are then used to compute the deformation of the entire rod.
In the case of a member with a variable cross-section, the strain is not constant but depends on the position. The deformation of an...

Generalized Hooke's Law

Generalized Hooke's Law

The generalized Hooke's Law is a broadened version of Hooke's Law, which extends to all types of stress and in every direction. Consider an isotropic material shaped into a cube subjected to multiaxial loading. In this scenario, normal stresses are exerted along the three coordinate axes. As a result of these stresses, the cubic shape deforms into a rectangular parallelepiped. Despite this deformation, the new shape maintains equal sides, and there is a normal strain in the direction of the...

Plastic Deformations

Plastic Deformations

It is essential to understand how structural members behave under plastic deformation when the bending stress exceeds the material's yield strength. This state of deformation permanently alters the shape of the member, in contrast to the linear elastic behavior observed before yielding. The strain at any point in the member is expressed in terms of maximum strain. Notably, the neutral axis, which coincides with the centroid during elastic bending, shifts away from the centroid under plastic...

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Related Experiment Video

Updated: May 15, 2026

Quantification of Strain in a Porcine Model of Skin Expansion Using Multi-View Stereo and Isogeometric Kinematics

Quantification of Strain in a Porcine Model of Skin Expansion Using Multi-View Stereo and Isogeometric Kinematics

Published on: April 16, 2017

Analytic regularization of uniform cubic B-spline deformation fields.

James A Shackleford¹, Qi Yang, Ana M Lourenço

¹Department of Radiation Oncology, Massachusetts General Hospital, Boston, MA 02114, USA.

Medical Image Computing and Computer-Assisted Intervention : MICCAI ... International Conference on Medical Image Computing and Computer-Assisted Intervention

|January 5, 2013

Summary

This study introduces a faster analytic method for calculating thin-plate bending energy in 3D B-spline image registration. This technique ensures physically plausible medical image deformations and significantly speeds up computation compared to traditional methods.

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Last Updated: May 15, 2026

Quantification of Strain in a Porcine Model of Skin Expansion Using Multi-View Stereo and Isogeometric Kinematics

Quantification of Strain in a Porcine Model of Skin Expansion Using Multi-View Stereo and Isogeometric Kinematics

Published on: April 16, 2017

Control of Cell Adhesion using Hydrogel Patterning Techniques for Applications in Traction Force Microscopy

Control of Cell Adhesion using Hydrogel Patterning Techniques for Applications in Traction Force Microscopy

Published on: January 29, 2022

Area of Science:

Medical Imaging
Computational Anatomy
Biomedical Engineering

Background:

Non-rigid image registration is crucial for medical applications but often ill-posed, lacking unique solutions.
Physically unsound deformations in medical image registration must be avoided to ensure clinical validity.
Thin-plate bending energy is a common regularization method for non-rigid image registration.

Purpose of the Study:

To develop an exact, analytic method for computing the bending energy of 3D B-spline deformation fields.
To accelerate the computation of regularization terms in medical image registration.
To ensure physically plausible deformations in non-rigid image registration.

Main Methods:

Developed an analytic method to compute 3D B-spline bending energy.
Expressed bending energy as a quadratic matrix operation on spline coefficients.
Validated the method on ten thoracic case studies.

Main Results:

The analytic solution for bending energy computation was successfully developed.
The analytic method demonstrated significant speed improvements.
Achieved computational speedups ranging from 61x to 1371x compared to numerical methods.

Conclusions:

The analytic method provides a computationally efficient way to calculate bending energy for 3D B-spline deformations.
This advancement can accelerate medical image registration while ensuring physically sound results.
The findings have implications for improving the speed and accuracy of non-rigid medical image analysis.