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- https://eng.libretexts.org/Bookshelves/Mechanical_Engineering/Structural_Mechanics_(Wierzbicki)/03%3A_Development_of_Constitutive_Equations_of_Continuum%2C_Beams_and_Plates/3.04%3A_Hook%E2%80%99s_Law_in_Generalized_Quantities_for_Beamsand the one-dimensional Hook law, Equation (3.1.1), and the definition of the bending moment and axial force in the beam, Equations (2.2.16-2.2.18). Note that the strain of the middle axis \(\epsilon^...and the one-dimensional Hook law, Equation (3.1.1), and the definition of the bending moment and axial force in the beam, Equations (2.2.16-2.2.18). Note that the strain of the middle axis \(\epsilon^{\circ}\) and the curvature of the beam axis are independent of the \(z\)-coordinate and could be brought in front of the respective integrals.
- https://eng.libretexts.org/Bookshelves/Mechanical_Engineering/Structural_Mechanics_(Wierzbicki)/02%3A_The_Concept_of_Stress%2C_Generalized_Stresses_and_Equilibrium
- https://eng.libretexts.org/Bookshelves/Mechanical_Engineering/Structural_Mechanics_(Wierzbicki)/06%3A_Bending_Response_of_Plates_and_Optimum_Design/6.09%3A_Shear_LagThe question is how to remove the incompatibility of in-plane displacements between the beam and plate, shown in Figure (6.6.5). One way of making the incompatible edge displacement to vanish, \(\Delt...The question is how to remove the incompatibility of in-plane displacements between the beam and plate, shown in Figure (6.6.5). One way of making the incompatible edge displacement to vanish, \(\Delta u = 0\), would be to stretch the plate to match the tensile side of the beam. The finite region of the plate subjected to large in-plane shear is called the “effective breath”. Most of literature dealing with bending of stiffened plates took the approach called the shear lag.
- https://eng.libretexts.org/Bookshelves/Mechanical_Engineering/Structural_Mechanics_(Wierzbicki)/02%3A_The_Concept_of_Stress%2C_Generalized_Stresses_and_Equilibrium/2.08%3A_Equilibrium_of_Rectangular_PlatesA step-by-step derivation of the equation of equilibrium and boundary conditions for rectangular plates is presented in the lecture notes of the course 2.081 Plates and Shells. \[\frac{\partial^2 M_{x...A step-by-step derivation of the equation of equilibrium and boundary conditions for rectangular plates is presented in the lecture notes of the course 2.081 Plates and Shells. \[\frac{\partial^2 M_{xx}}{\partial x^2} + 2\frac{\partial^2 M_{xy}}{\partial x \partial y} + \frac{\partial^2 M_{yy}}{\partial y^2} + p = 0 \label{3.82}\] The boundary conditions for plates are similar to those for beams in the local coordinate system at the edges, \((n, t)\), Figure (\(\PageIndex{1}\)).
- https://eng.libretexts.org/Bookshelves/Mechanical_Engineering/Structural_Mechanics_(Wierzbicki)/01%3A_The_Concept_of_Strain/1.07%3A_Advanced_Topic-_Derivation_of_the_Strain-Displacement_Relation_for_Thin_Plates\[\epsilon_{\alpha \beta} = \frac{1}{2} [ u_{\alpha}^{\circ} - zw_{, \alpha}]_{, \beta} + \frac{1}{2} [ u_{\beta}^{\circ} − zw_{,\beta}]_{, \alpha} = \frac{1}{2} (u_{\alpha , \beta}^{\circ} + u_{\beta...\[\epsilon_{\alpha \beta} = \frac{1}{2} [ u_{\alpha}^{\circ} - zw_{, \alpha}]_{, \beta} + \frac{1}{2} [ u_{\beta}^{\circ} − zw_{,\beta}]_{, \alpha} = \frac{1}{2} (u_{\alpha , \beta}^{\circ} + u_{\beta , \alpha}^{\circ}) - \frac{1}{2} z [ w_{,\alpha \beta} + w_{,\beta \alpha}] \label{1.7.7}\]
- https://eng.libretexts.org/Bookshelves/Mechanical_Engineering/Structural_Mechanics_(Wierzbicki)/08%3A_Stability_of_Elastic_Structures/8.01%3A_Prelude_to_Stability_of_Elastic_StructuresAt the origin of the coordinate system \(u = 0\), so the first variation of \(\prod\) is zero no matter what the sign of the coefficient \(C\) is. The incremental change of the potential energy \(\Del...At the origin of the coordinate system \(u = 0\), so the first variation of \(\prod\) is zero no matter what the sign of the coefficient \(C\) is. The incremental change of the potential energy \(\Delta \prod = \prod (u) − \prod (u_o)\) upon small variation of the argument \(\delta u = u − u_o\) is Therefore, to the second term expansion, the sign of the increment of \(\prod\) depends on the sign of the second variation of the potential energy.
- https://eng.libretexts.org/Bookshelves/Mechanical_Engineering/Structural_Mechanics_(Wierzbicki)/zz%3A_Back_Matter
- https://eng.libretexts.org/Bookshelves/Mechanical_Engineering/Structural_Mechanics_(Wierzbicki)/00%3A_Front_Matter
- https://eng.libretexts.org/Bookshelves/Mechanical_Engineering/Structural_Mechanics_(Wierzbicki)/06%3A_Bending_Response_of_Plates_and_Optimum_Design/6.03%3A_Equivalence_of_Square_and_Circular_PlatesIn the section of Chapter 6 on stiffened plates, the analogy between the response of circular and square plates was exploit to demonstrate the effectiveness of stiffeners. It is interesting that the a...In the section of Chapter 6 on stiffened plates, the analogy between the response of circular and square plates was exploit to demonstrate the effectiveness of stiffeners. It is interesting that the approximate solution obtained by the Ritz method is very close to the exact series solution where the coefficient 47 in Equation \ref{7.42} should be replaced by 49.5.
- https://eng.libretexts.org/Bookshelves/Mechanical_Engineering/Structural_Mechanics_(Wierzbicki)/05%3A_Moderately_Large_Deflection_Theory_of_Beams/5.02%3A_Solution_for_a_Beam_on_Roller_SupportAt the same time, the nonlinear term in the vertical equilibrium vanishes and the beam response is governed by the linear differential equation In order to get a physical sense of the above result, th...At the same time, the nonlinear term in the vertical equilibrium vanishes and the beam response is governed by the linear differential equation In order to get a physical sense of the above result, the vertical and horizontal displacements are normalized by the thickness \(h\) of the beam To summarize the results, the roller supported beam can be treated as a classical beam even though the displacements and rotations are large (moderate).
- https://eng.libretexts.org/Bookshelves/Mechanical_Engineering/Structural_Mechanics_(Wierzbicki)/02%3A_The_Concept_of_Stress%2C_Generalized_Stresses_and_Equilibrium/2.02%3A_Advanced_Topic_-_Local_Equilibrium_from_the_Principle_of_Virtual_WorkThe principle of virtual work states that the incremental work of strains on the stresses over the volume of the body must be equal to the work of surface tractions in the incremental displacements ov...The principle of virtual work states that the incremental work of strains on the stresses over the volume of the body must be equal to the work of surface tractions in the incremental displacements over the surface of the body. The beam (or its portion) where the bending moment is negative is called the “smiling beam”. Therefore looking at the deformed shape of the beam one can determine immediately the sign of the bending moment.
