2: The Concept of Stress, Generalized Stresses and Equilibrium
- Page ID
- 21482
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\dsum}{\displaystyle\sum\limits} \)
\( \newcommand{\dint}{\displaystyle\int\limits} \)
\( \newcommand{\dlim}{\displaystyle\lim\limits} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\(\newcommand{\longvect}{\overrightarrow}\)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 2.1: Stress Tensor
- This page covers the concepts of stress in one- and two-dimensional contexts, introducing the stress tensor via a prismatic bar under tensile forces. It explains surface traction, complex stress states, and the Cauchy formula, alongside the significance of shear loading and failure orientation. The page details the symmetry of the stress tensor and conditions for plane stress, as well as equilibrium equations.
- 2.2: Advanced Topic - Local Equilibrium from the Principle of Virtual Work
- This page covers the derivation of the local equilibrium equation using the principle of virtual work in continuum mechanics, highlighting the importance of the divergence theorem and calculus of variations. It explains the connection between internal strains and external forces, leading to boundary conditions. Additionally, it discusses bending moments in beams, introducing tensile and compressive stresses, and illustrating the relationship between bending moment signs and beam deformation.
- 2.3: Generalized Forces and Bending Moments in Plates
- This page explains the in-plane stress components of the stress tensor in plates, focusing on \(\sigma_{xx}\), \(\sigma_{yy}\), and \(\sigma_{xy}\). It defines generalized forces and couples linked to these stresses, with equations for moments, normal forces, and shear forces. The integration of these quantities over the plate's thickness yields dimensions expressed "per unit length."
- 2.4: Advanced Topic - Principle of Virtual Work for Beams
- This page explores the principle of virtual work in beam theory within a one-dimensional stress context. It derives a crucial expression linking internal and external work in beams through displacements and rotations, encompassing both internal forces and external loads. This fundamental result aids in developing solutions and equilibrium equations for beam problems, serving as a basis for more advanced analysis in structural mechanics.
- 2.5: Derivation of Equation of Equilibrium for Beams from the Principle of Virtual Work
- This page covers integration by parts in beam theory, emphasizing the principle of virtual work. It presents key equations on beam curvature and axial strain, leading to equilibrium conditions, including local forms. The page also details boundary conditions, differentiating between static and kinematic conditions, and their implications at beam ends. Finally, it addresses symmetry considerations under specific loading conditions.
- 2.6: Advanced Topic - Mathematical Theory of Beams
- This page details the derivation of equilibrium equations for a rectangular beam under planar bending, employing 3-D equilibrium principles. It emphasizes crucial stress components and boundary conditions, particularly addressing the impact of pressure loads. The derivation is presented in three steps, resulting in fundamental force and moment equilibrium relationships. The final result aligns with established structural mechanics equations, confirming the validity of the approach taken.
- 2.7: Equilibrium in the Theory of Moderately Large Deflections of Beams
- This page explains how finite beam rotation introduces non-linear axial strain and modifies the effective shear force \(V^*\), which combines cross-sectional shear and axial force contributions. Although this addition alters the force equilibrium equation, the overall equations of equilibrium remain unchanged. A new governing equation emerges for beams experiencing moderately large deflections, emphasizing the interrelationship between static and kinematic variables in beam behavior analysis.
- 2.8: Equilibrium of Rectangular Plates
- This page derives equilibrium equations and boundary conditions for rectangular plates using tensor notation. It introduces the equation \(M_{\alpha \beta,\alpha \beta} + p = 0\) and clarifies bending moment dimensions. The page relates wide beams as a subset of rectangular plates and aligns their boundary conditions, detailing specific requirements for moments, shear, and normal forces at edges, thus highlighting the interconnections between plates and beams.
- 2.9: Circular Plates
- This page covers the derivation of equilibrium equations for a circular plate through virtual work, focusing on bending and in-plane responses. It outlines the equilibrium and boundary conditions related to radial and circumferential curvatures, leading to governing equations. Additionally, it examines in-plane loading, highlighting that the plate remains flat and simplifying in-plane equilibrium derivations.