Journal of Mechanics of Materials and Structures vol. 5 (2010 ... - MSP

828 ELIA EFRAIM AND MOSHE EISENBERGER
h/R = 0.1 h/R = 0.2
Clamped-Free SS-Free Clamped-Free SS-Free
n 5 DOF Lit. Diff.% 7 DOF Lit. Diff.% 5 DOF Lit. Diff.% 7 DOF Lit. Diff.%
1.1353 1.139 –0.32 1.0722 1.072 0.02 1.2777 1.289 –0.88 1.0583 1.059 –0.06
0 2.0232 2.014 0.45 1.7280 1.726 0.12 2.0232 2.013 0.50 2.0223 2.014 0.41
2.1232 2.125 –0.08 2.0231 2.010 0.65 2.7504 2.747 0.12 2.4947 2.487 0.31
0.9833 0.984 –0.07 0.8890 0.887 0.22 1.1492 1.152 –0.24 0.8968 0.895 0.20
1 2.1026 2.106 –0.16 1.7584 1.756 0.14 2.3395 2.337 0.11 2.3347 2.332 0.12
2.4140 2.408 0.25 2.3790 2.369 0.42 2.8142 2.810 0.15 2.5781 2.565 0.51
0.7965 0.794 0.32 0.6507 0.648 0.41 1.0567 1.053 0.35 0.7540 0.751 0.40
2 2.2267 2.232 –0.24 1.8775 2.038 –7.88 2.8075 2.822 -0.51 2.6616 2.670 –0.31
3.0967 3.088 0.28 3.0846 2.764 11.6 3.1626 3.150 0.40 3.1131 3.087 0.85
0.7515 0.748 0.47 0.5844 0.581 0.58 1.1578 1.154 0.33 0.8790 0.879 0.00
3 2.4119 2.416 –0.17 2.0644 2.522 –18.1 3.0761 3.099 -0.74 2.9268 2.939 –0.42
3.3919 3.398 –0.18 3.3843 2.524 34.1 3.6591 3.634 0.69 3.5337 3.459 2.16
0.8764 0.873 0.39 0.7344 0.731 0.47 1.4400 1.441 –0.07 1.2250 1.230 –0.40
4 2.6411 2.643 –0.07 2.2964 2.307 –0.46 3.3779 3.407 –0.85 3.2616 3.278 –0.50
3.5287 3.532 –0.09 3.5250 3.477 1.38 3.8222 3.782 1.06 3.6590 3.586 2.03
1.1272 1.126 0.11 1.0160 1.014 0.20 1.8473 1.853 –0.31 1.6855 1.693 –0.44
5 2.8959 2.894 0.06 2.5589 2.546 0.51 3.7419 3.772 –0.80 3.6600 3.675 –0.41
3.7873 3.789 –0.04 3.7869 3.780 0.18 4.0686 4.025 1.08 3.8853 3.818 1.76
Table 2. Nondimensional frequency λ = ωR √ ρ/E for hemispherical shells with 60 ◦
cutout at the apex, with free boundary conditions at the cutout and different boundary
conditions at the base. The columns “5 DOF” and “7 DOF” give the result with the
present method (one exact element and the specified number of degrees of freedom).
“Lit.” refers to [Gautham and Ganesan 1992]. “Diff.%” is the ratio (λexact − λFE)/λFE.
60°
Figure 3. Left: hemispherical annular shell with linearly varying thickness; φ0 = 60
=
φ = ° φ = ° + = =
◦ ,
hb/he = 1/3 (results in Table 3, left). Right: spherical barrel shell with variable thickness;
φb = 45◦ , φe = 135◦ , (hb +he)/2R = 0.2, hb/he = 1/3, µ = 0.3 (results in Table 3,
right, and Figures 4–7).
45°
90 °