Static bending of a curved beam

In this benchmark problem, we examine the tip deflection of a tip-loaded curved beam, which is described in detail in [@Bathe-Bolourchi:1979]. The beam lies in the x-z plane with a tip force of 600 lbs acting in the positive y direction. The beam is curved on a radius of 100 inches in the positive x direction through a 45-degree arc.

Geometric properties

  • Cross section: 1 in x 1 in

Material properties

  • Elastic Modulus, E = 10**7 psi

  • Poisson Ratio, nu = 0

Diagonal stiffness matrix entries

  • kGA = 41.6667E+5

  • kGA = 41.6667E+5

  • EA = 100.E+5

  • EI_1 = 8.333E+5

  • EI_2 = 8.333E+5

  • GJ = 8.333E+5

Results

Tip displacement reported by [@Bathe-Bolourchi:1979]: (-13.4, 53.4, -23.5)

The following table show the tip displacement calculated with a single Legendre Spectral Finite Element (LSFE) with the number of nodes shown. N-Point Gauss-Legendre quadrature was employed, where N is the number of nodes. The results show the expected spectral convergence to a converged solution.

Number of Nodes (N)

u1

u2

u3

3

-1.390261363798350E+00

2.264633130098760E+01

3.168614893736530E+01

6

-2.362791403924360E+01

1.372486827629740E+01

5.356617784705440E+01

9

-2.364597006589220E+01

1.372080411971940E+01

5.357763167143680E+01

12

-2.364597159890480E+01

1.372080060483760E+01

5.357764506053460E+01

15

-2.364597160090470E+01

1.372080060400580E+01

5.357764506269870E+01

18

-2.364597160090480E+01

1.372080060400580E+01

5.357764506269890E+01

Note

This benchmark is included as a regression test in the Kynema code base. The test is implemented in:

Bathe, K.-J., and S. Bolourchi. 1979. “Large displacement analysis of three-dimensional beam structures.” International Journal for Numerical Methods in Engineering 14: 961-986.