3.7: Stress Formula for Plates
- Page ID
- 21735
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)\)In the section on beams, it was shown that the profile of axial stress can be determined from the known bending moment \(M\) and axial force \(N\), see Equation (3.4.8). A similar procedure can be developed for plates by comparing Equations (3.6.10-3.6.29) with Equation (3.6.1). The stress-strain curve for the plane stress can be expressed in terms of the middle surface strain tensor \(\epsilon_{\alpha \beta}^{\circ}\) and curvature tensor \(\kappa_{\alpha \beta}\) by combining Equations (3.6.1) and (3.6.5).
\[\sigma_{\alpha \beta} = \frac{E}{1 − \nu^2} [(1 − \nu)\epsilon_{\alpha \beta}^{\circ} + \nu\epsilon_{\gamma \gamma}^{\circ}\delta_{\alpha \beta}] \\ + \frac{E}{1 − \nu^2} [(1 − \nu)\kappa_{\alpha \beta} + \nu\kappa_{\gamma \gamma}\delta_{\alpha \beta}] z \]
From the moment-curvature relation, Equation (3.6.10):
\[(1 − \nu)\kappa_{\alpha \beta} + \nu\kappa_{\gamma \gamma}\delta_{\alpha \beta} = \frac{M_{\alpha \beta}}{D}\]
Similarly, from Equation (3.6.24)
\[(1 − \nu)\epsilon_{\alpha \beta}^{\circ} + \nu\epsilon_{\gamma \gamma}^{\circ}\delta_{\alpha \beta} = \frac{N_{\alpha \beta}}{C}\]
where \(D = \frac{Eh^3}{12(1 − \nu^2)}\) is the bending rigidity, and \(C = \frac{Eh}{1 − \nu^2}\) is the axial rigidity of the plate.
From the above system, one gets
\[\sigma_{\alpha \beta} = \frac{Ez}{1 − \nu^2} \frac{M_{\alpha \beta}}{D} + \frac{E}{1 − \nu^2} \frac{N_{\alpha \beta}}{C}\]
or using the definitions of \(D\) and \(C\)
\[\sigma_{\alpha \beta} = \frac{N_{\alpha \beta}}{h} + \frac{zM_{\alpha \beta}}{h^3/12}\]
The above equation is dimensionally correct, because both \(N_{\alpha \beta}\) and \(M_{\alpha \beta}\) are respective quantities per unit length. In particular stress in the case of cylindrical bending is
\[\sigma_{xx} = \frac{N_{xx}}{h} + \frac{zM_{xx}}{h^3/12}\]
Multiplying both the numerators and denominators of the two terms above by \(b\) yields
\[\sigma_{xx} = \frac{N_{xx}b}{hb} + \frac{zM_{xx}b}{bh^3/12}\]
Now, observing that \(N_{xx}b = N\) is the beam axial force, \(bM_{xx} = M\) is the beam bending moment, \(hb = A\) is the cross-section of the rectangular section beam, and \(\frac{bh^3}{12}\) is the moment of inertia, the familiar beam stress formula is obtained
\[\sigma = \frac{N}{A} + \frac{Mz}{I}\]