Home › GATE ME › Mechanical Engineering › platetheory › For a SIMPLY-SUPPORTED rectangular plate (a × b)…
For a SIMPLY-SUPPORTED rectangular plate (a × b) under uniform transverse load q₀, Navier's solution expresses the deflection w(x,y) as:
A{'text': '$w(x,y) = \\sum_{m,n} W_{mn} \\sin(m\\pi x/a) \\sin(n\\pi y/b)$ — a double sine series', 'label': 'B'}
B{'text': '$w = q_0 \\cdot x$', 'label': 'C'}
C{'text': '$w = $ constant', 'label': 'D'}
D{'text': '$w = q_0 a^2/D$', 'label': 'A'}
Answer & Solution
Correct answer: A. {'text': '$w(x,y) = \\sum_{m,n} W_{mn} \\sin(m\\pi x/a) \\sin(n\\pi y/b)$ — a double sine series', 'label': 'B'}
Navier's double Fourier solution: $w(x,y) = \sum_{m,n} W_{mn} \sin(m\pi x/a)\sin(n\pi y/b)$. Substituting into $D\nabla^4 w = q$ gives $W_{mn} = q_{mn} / [D\pi^4 (m^2/a^2 + n^2/b^2)^2]$.
Related questions
The TRANSITION from beam theory to plate theory is required when:For a thin plate, increasing thickness t by 2× changes flexural rigidity D by what factor?In the laminate ABD constitutive equation, the COUPLING matrix B is computed as:The STRESS RESULTANTS used in plate theory are integrals through the thickness. The bendinFor an ANISOTROPIC plate (e.g., laminate), the governing equation involves which sub-matriThe biharmonic equation $\nabla^4 w = D^{-1} q$ in plate theory has the form:The PRESENCE of "concentrated corner forces" at the corners of a simply-supported rectanguAt a FREE edge of a thin plate (x = a), Kirchhoff's theory imposes how many independent bo