64
M.-L. Bes , liu-Gherghescu et al.
Table 1 Values of parameters for different values of the exponent n
n
a n
H [N]
h [m]
θ f
0
2
0.3881913
161.4413419
0.0349372
13.11066004
3
1.1266249
164.9456759
0.0304189
16.91928297
4
3.3977716
171.0226902
0.0275220
20.15105633
5
10.4834847
177.4731071
0.0254749
23.00479786
6
32.8259942
183.7708766
0.0239301
25.57591923
7
103.8427031
189.7718622
0.0227104
27.91953344
8
330.9715613
195.4460183
0.0217150
30.07375196
9
1060.9348536
200.8014071
0.0208824
32.06596502
10
3416.1783028
205.8586803
0.0201722
33.91713753
20
465,758,204.67
244.5944225
0.0162400
47.27300159
in the area of the cross section for each beam A = 9 × 10
−6 m
2 , and the common
moment of inertia I = 6.75 × 10
−12 m
4 .
In Table 1, we determined the mean parameters a n , H , h and θ f in function of
the exponent n. One may easily observe that increasing the exponent, the coefficient
a n , the axial force H and the rotation angle at the end of each cantilever beam
θ f increase too, while the displacement of the free ends of each cantilever beam
h decreases. These are expected results due to the shape of the deformed beams,
y = a n x
n . Moreover, the results for n = 2 are in excellent agreement with those
reported in [11], where the integrals are directly calculated. In Fig. 2, we have drawn
the variations of main parameters for different values of the exponent n. It is easy
to observe that these variations are similar to the variation of the force P. When the
exponent n increases, the variations of the main parameters increase too.
5 Conclusions
The paper presents a nonlinear model of calculation for the case of two cantilever
beams jointed by a cylindrical joint. The shapes of the deformed beams are considered
to be given by monomial forms for the simplicity of calculation. More general forms
may use Airy type functions (in [7] is presented an example for a shell). It is proved
that the results obtained by classical theory of the strength of materials are only
approximate ones, so this method has to be extended to other practical cases.
M.-L. Bes , liu-Gherghescu et al.
Table 1 Values of parameters for different values of the exponent n
n
a n
H [N]
h [m]
θ f
0
2
0.3881913
161.4413419
0.0349372
13.11066004
3
1.1266249
164.9456759
0.0304189
16.91928297
4
3.3977716
171.0226902
0.0275220
20.15105633
5
10.4834847
177.4731071
0.0254749
23.00479786
6
32.8259942
183.7708766
0.0239301
25.57591923
7
103.8427031
189.7718622
0.0227104
27.91953344
8
330.9715613
195.4460183
0.0217150
30.07375196
9
1060.9348536
200.8014071
0.0208824
32.06596502
10
3416.1783028
205.8586803
0.0201722
33.91713753
20
465,758,204.67
244.5944225
0.0162400
47.27300159
in the area of the cross section for each beam A = 9 × 10
−6 m
2 , and the common
moment of inertia I = 6.75 × 10
−12 m
4 .
In Table 1, we determined the mean parameters a n , H , h and θ f in function of
the exponent n. One may easily observe that increasing the exponent, the coefficient
a n , the axial force H and the rotation angle at the end of each cantilever beam
θ f increase too, while the displacement of the free ends of each cantilever beam
h decreases. These are expected results due to the shape of the deformed beams,
y = a n x
n . Moreover, the results for n = 2 are in excellent agreement with those
reported in [11], where the integrals are directly calculated. In Fig. 2, we have drawn
the variations of main parameters for different values of the exponent n. It is easy
to observe that these variations are similar to the variation of the force P. When the
exponent n increases, the variations of the main parameters increase too.
5 Conclusions
The paper presents a nonlinear model of calculation for the case of two cantilever
beams jointed by a cylindrical joint. The shapes of the deformed beams are considered
to be given by monomial forms for the simplicity of calculation. More general forms
may use Airy type functions (in [7] is presented an example for a shell). It is proved
that the results obtained by classical theory of the strength of materials are only
approximate ones, so this method has to be extended to other practical cases.