Beam Problem 4 first created 23/10/09 - last modified 23/10/09 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 in the vertical down direction. Find the vertical and horizontal deflections 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.

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Figure 1 - A curved beam problem




Choose your own material values and dimensions.
I've chosen to use 50x50x5 (mm) SHS steel beam with an end force of 1kN.
  • The second moment of area I: 3.075E5 mm^4
  • Young's Modulus for mild steel E: 2.05E5 N/mm^2

    The analytic theory for bending is derived in most textbooks and referring to Timoshenko's [1] work the vertical deflection, delta is:

    $delta = pi/4 (P*R^3)/(E*I) = pi/4 (1000*1000^3)/(3.075E5*2.05E5) = 12.5 [mm] $

    and the horizontal deflection, delta1 is:

    $delta_1 = (P*R^3)/(2*E*I) = (1000*1000^3)/(2*3.075E5*2.05E5) = 7.9 [mm] $
    Z88i1.txt

    2 10 9 30 1 0 1 0 Z88I1.TXT,typed in by Ty
    1 3 +1.00000E+003 +1.00000E+003 +0.00000E+000 node #1
    2 3 +8.26352E+002 +9.84808E+002 +0.00000E+000 node #2
    3 3 +6.5798E+002 +9.39693E+002 +0.00000E+000 node #3
    4 3 +5.00000E+002 +8.66025E+002 +0.00000E+000 node #4
    5 3 +3.57212E+002 +7.66044E+002 +0.00000E+000 node #5
    6 3 +2.33956E+002 +6.42788E+002 +0.00000E+000 node #6
    7 3 +1.33975E+002 +5.00000E+002 +0.00000E+000 node #7
    8 3 +6.0307E+001 +3.42020E+002 +0.00000E+000 node #8
    9 3 +1.51922E+001 +1.73648E+002 +0.00000E+000 node #9
    10 3 +0.00000E+000 +0.00000E+000 +0.00000E+000 node #10
    1 13 element #1
    1 2
    2 13 element #2
    2 3
    3 13 element #3
    3 4
    4 13 element #4
    4 5
    5 13 element #5
    5 6
    6 13 element #6
    6 7
    7 13 element #7
    7 8
    8 13 element #8
    8 9
    9 13 element #9
    9 10
    1 9 +2.05000E+005 +3.00000E-001 1 +9.000E+003 0 0 +3.075E+005 +2.50000E+001 0 0
    Z88i2.txt

    4 , typed in from ty 10 1 2 0.00000E+000 10 2 2 0.00000E+000 10 3 2 0.00000E+000 1 2 1 -1.00000E+003 4 , typed in from ty
    10 1 2 0.00000E+000
    10 2 2 0.00000E+000
    10 3 2 0.00000E+000
    1 2 1 -1.00000E+003


    text here
    Figure 2 - Z88P graphical plot of the deflection


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

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

    1 +7.9214925E+000 -1.2380823E+001 -1.5803207E-002
    2 +7.6826342E+000 -9.6505242E+000 -1.5563121E-002
    3 +6.9948599E+000 -7.0836130E+000 -1.4850158E-002
    4 +5.9411143E+000 -4.8237758E+000 -1.3685976E-002
    5 +4.6485175E+000 -2.9776575E+000 -1.2105953E-002
    6 +3.2729742E+000 -1.6020197E+000 -1.0158113E-002
    7 +1.9803732E+000 -6.9683842E-001 -7.9016074E-003
    8 +9.2663770E-001 -2.0537423E-001 -5.4050123E-003
    9 +2.3886172E-001 -2.0992127E-002 -2.7441954E-003
    10 +0.0000000E+000 +0.0000000E+000 +0.0000000E+000

    References

    (1) Timoshenko, Strength of Materials: Part 2 Advanced Theory and Problems, D. Van Nostrand Company, Inc., 2nd, Torento: (1941), 79 - 80.

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