A curved beam with an out of plane end force first created 29/11/09 - last modified 29/11/09 Page Author: Ty Harness
A curved (quarter of a circle where R = 1000mm) beam shown in figure 1 is subjected to a end force,P out of plane at the free end. Find the vertical deflection and angle of twist at the free end using an analytic method and compare with any available finite element analysis program. Work together in groups of 2 or 3 and present your solutions at the next tutorial meet.

Use the following numerical values:

Force, P = 1000 N Young's Modulus,E = 2.05e5 N/mm^2 Rigidity Modulus,G = 7.88e4 N/mm^2 Use a solid circular cross section with a diameter of 50mm.

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Figure 1 - A curved cantilever with out of plane end force at the free end


Analytical Solution

Second moment of Area about the axis: $ I_(YY) = (pi*d^4)/64 = 3.068e5 [mm^4]$

Second moment of Area about the axis: $ I_(XX) = (pi*d^4)/64 = 3.068e5 [mm^4]$

Polar moment: $J = (pi*d^4)/32 = 6.1359e5 [mm^4]$

The analytical solution is derived in Gere & Timoshenko's Mechanics of Materials [1]:

$d_v = (pi*P*R^3) / (4*E*I) + ((3*pi - 8)*P*R^3)/(4*G*J)$

and

$phi = (pi*P*R^2)/(4*E*I) + ((pi - 4)*P*R^2)/(4*G*J)$

Inserting the numerical values for the above problem:

$d_v = 19.85[mm]$ and $phi = 8.052e-3$
FE solution

I've chosen to use Prof. Rieg's Z88 Finite Elements Program because it's free and unlimited and ideal for student use. The theory behind FEM is beyond the scope of this tutorial but you can learn a great deal from the Z88 manual. Using a Beam Element No. 2 we create a General Structure Data File, Z88i1.txt which basically consists of : 3D Analysis, 10 Nodes, 9 Elements, 6DOF per node 60 in total. Nodal co-ordinates, elements defined, DOF, and adjacent elements. All elements 1 to 9 have the same properties.
Z88i1.txt

3 10 9 60 1 0 1 0 Z88I1.TXT,typed in by Ty
1 6 +1.00000E+003 +1.00000E+003 +0.00000E+000 node #1
2 6 +8.26352E+002 +9.84808E+002 +0.00000E+000 node #2
3 6 +6.5798E+002 +9.39693E+002 +0.00000E+000 node #3
4 6 +5.00000E+002 +8.66025E+002 +0.00000E+000 node #4
5 6 +3.57212E+002 +7.66044E+002 +0.00000E+000 node #5
6 6 +2.33956E+002 +6.42788E+002 +0.00000E+000 node #6
7 6 +1.33975E+002 +5.00000E+002 +0.00000E+000 node #7
8 6 +6.0307E+001 +3.42020E+002 +0.00000E+000 node #8
9 6 +1.51922E+001 +1.73648E+002 +0.00000E+000 node #9
10 6 +0.00000E+000 +0.00000E+000 +0.00000E+000 node #10
1 2 element #1
1 2
2 2 element #2
2 3
3 2 element #3
3 4
4 2 element #4
4 5
5 2 element #5
5 6
6 2 element #6
6 7
7 2 element #7
7 8
8 2 element #8
8 9
9 2 element #9
9 10
1 9 +2.05000E+005 +3.00000E-001 1 +1.96349E+003 +3.06796E+005 +2.50000E+001 +3.06796E+005 +2.50000E+001 +6.13592E+005 +7.88462E+004
Z88i2.txt

7 , typed in from ty
10 1 2 0.00000E+000
10 2 2 0.00000E+000
10 3 2 0.00000E+000
10 4 2 0.00000E+000
10 5 2 0.00000E+000
10 6 2 0.00000E+000
1 3 1 -1.00000E+003
Z88P output

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Figure 2 - Z88P screen shot isometric view


Z88o2.txt

output file Z88O2.TXT : displacements, computed by Z88F V10
*************

Knoten U(1) U(2) U(3) U(4) U(5) U(6)

1 +0.0000000E+000 +0.0000000E+000 -1.9746853E+001 -8.0942122E-003 +1.8197204E-002 +0.0000000E+000
2 +0.0000000E+000 +0.0000000E+000 -1.6478013E+001 -8.0731595E-003 +1.7956568E-002 +0.0000000E+000
3 +0.0000000E+000 +0.0000000E+000 -1.3145911E+001 -7.9928450E-003 +1.7235454E-002 +0.0000000E+000
4 +0.0000000E+000 +0.0000000E+000 -9.9293059E+000 -7.7913499E-003 +1.6037015E-002 +0.0000000E+000
5 +0.0000000E+000 +0.0000000E+000 -6.9919596E+000 -7.4020624E-003 +1.4368933E-002 +0.0000000E+000
6 +0.0000000E+000 +0.0000000E+000 -4.4723400E+000 -6.7528061E-003 +1.2246660E-002 +0.0000000E+000
7 +0.0000000E+000 +0.0000000E+000 -2.4745412E+000 -5.7664522E-003 +9.6971320E-003 +0.0000000E+000
8 +0.0000000E+000 +0.0000000E+000 -1.0610971E+000 -4.3633003E-003 +6.7625844E-003 +0.0000000E+000
9 +0.0000000E+000 +0.0000000E+000 -2.4765429E-001 -2.4650337E-003 +3.5034493E-003 +0.0000000E+000
10 +0.0000000E+000 +0.0000000E+000 +0.0000000E+000 +0.0000000E+000 +0.0000000E+000 +0.0000000E+000
References

[1] Gere, J.,M., Mecahnics of Materials, PWS Engineering, 2nd, Boston: (1987), 671.

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