A rectangular timber slab with dimensions Lx = 8.0 m and Ly = 6.0 m has stiffness values Dx = D11 = 3000 kNm, Dy = D22 = 750 kNm, Dt = 2D66
= 0 kNm (neglected). The distributed mass is: m = 200 kg/m2. Determine the fundamental frequency of the slab if
a) all four edges are simply supported,
b) one edge parallel to the y axis is free and the other three edges are simply supported (in the Figure EI = 0).
c) What should be the bending stiffness (EI =?) of the edge beam if we want the fundamental frequency to be the average of the frequencies
of case a) and case b)?
Solve Problem
Problem a) Natural frequency with hinged supports, fn [Hz]= Problem b) Natural frequency with one free edge, fn [Hz]= Problem c) Bending stiffness of the supporting beam, EI [kNm2]=Solve
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Steps
Problem a) Step 1. Give the natural frequency assuming hinged edges. f2=f2x+f2y+f2t=π2D114mL4x+π2D224mL4y+2π2D664mL2xL2y=π2D114mL4x+π2D224mL4y=π2×3000×1034×200×8.04+π2×750×1034×200×6.04f=4.02 Note that the eigenfrequencies due to the three stiffnesses are . The above expression is equivalent to the application of Southwell’s expression. Problem b) Step 1. Give the natural frequency assuming one free edge. Problem c) Step 1. Determine the necessary bending stiffness of the supporting beam to obtain the average of the above two frequencies. Step by step
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Results
Problem a) The natural frequency of the plate with four hinged edges is: Note that the eigenfrequencies due to the three stiffnesses are . The above expression is equivalent to the application of Southwell’s expression. Problem b) The natural frequency assuming one free edge: Problem c) The necessary bending stiffness of the supporting beam is calculated from its natural frequency, which results the average frequency of the plate of case a) and b). Worked out solution