6: Bending Response of Plates and Optimum Design
- Page ID
- 21510
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 6.1: Beam Deflection Equation
- This page covers the equations for plate bending, detailing geometry, equilibrium, and elasticity. It derives a fourth-order linear inhomogeneous differential equation, \(D\nabla^4w = p\), for plate bending and discusses in-plane response equations, noting the uncoupling in traditional plate theory. It highlights the significance of finite rotation and presents an extended governing equation for moderately large deflections, emphasizing its importance in the analysis of plate behavior.
- 6.2: Deflections of Circular Plates
- This page analyzes a circular plate under axisymmetric loading using the Laplace operator in polar coordinates. It derives the bending equation and solution for the plate's deflection under uniform loading, considering boundary conditions for clamped edges. The differences in deflection between clamped and simply supported plates are discussed, along with their formulas and stiffness ratios.
- 6.3: Equivalence of Square and Circular Plates
- This page compares the bending responses of clamped square and circular plates under uniform loading, emphasizing stiffeners' role. It discusses total potential energy and conditions for maximum deflection, focusing on the center. The Ritz method is used to relate load to deflection, revealing that both plate types can exhibit comparable stiffness if dimensions are correctly matched. The findings indicate slight differences between exact solutions and approximations.
- 6.4: Design Concept for Plates
- This page covers design considerations for transverse-loaded plates, emphasizing stiffness, strength, and fracture. It defines stiffness and provides a calculation formula for clamped plates. Key factors like material choice, thickness, and support distance are crucial for optimizing stiffness relative to weight. Comparison of steel and aluminum indicates little advantage in using aluminum for stiffness.
- 6.5: Sandwich Plates
- This page discusses sandwich plates, composed of face sheets and a lightweight core that handles shear stresses while face plates manage tension or compression. It covers calculations for bending and axial stiffness using specific equations. Although sandwich plates enhance stiffness, they also present failure risks like yielding, buckling, and delamination. To improve performance and reduce failure chances, the optimization of core thickness \(H_{opt}\) is recommended.
- 6.6: Stiffened Plates
- This page explains stiffened plates as a lightweight construction method, comparing them to symmetric sandwich structures. It highlights their asymmetrical nature, with the neutral axis outside the plate profile, and includes interactions between beams and uniform thickness plates. The response to transverse loads is examined, along with stiffness calculations based on beam and plate theories.
- 6.7: Plates versus Grillages
- This page compares plates and grillages, emphasizing the behavior of a square plate with intersecting beams under point loads. It indicates that the beam system's stiffness overshadows that of the plate, leading to specific stiffness relationships. The analysis finds that while grillages are effective for point loads, they are inadequate for distributed pressure, necessitating the use of plates or stiffened plates in those scenarios.
- 6.8: The Concept of Equivalent Thickness
- This page covers plate bending mechanics with weak stiffeners, emphasizing the calculation of equivalent thickness \(h_{eq}\) based on equal moments of inertia. It explores how the neutral axis position relates to stiffener height and aspect ratio, noting that stiffness increases parabolically with height while weight increases linearly, optimizing the stiffness-to-weight ratio. Additionally, it introduces shear lag to guide efficient stiffener design.
- 6.9: Shear Lag
- This page examines the challenges of in-plane displacements at the interface of a beam and a plate, emphasizing discrepancies during separate bending. It proposes stretching the plate to align with the beam as a solution, which leads to high shear stresses in a localized area referred to as "effective breadth." The page also notes that existing literature often employs the shear lag approach for analyzing these interface stresses.