Euler–Bernoulli beam

Node.js examples of eulerBernoulliBeam, in examples/eulerBernoulliBeam. Run the commands below from that folder (see running an example). All results are from FEAScript-core 0.4.0.

The beam elements are cubic Hermite elements, with the deflection $w$ and the rotation $\theta$ at each node. The deflection curves below are drawn with the Hermite shape functions between the nodes; the markers are the nodes.

Clamped and spring-supported beam

The "Bending of a Beam" example of J. N. Reddy, An Introduction to the Finite Element Method, 3rd ed., 2006: a 10 m beam, $EI = 2 \times 10^6$ N·m², clamped at $x = 0$, on a roller at midspan with an applied moment of 1250 N·m, and on a spring ($k = 200$ N/m) at the free end with a point load of −2500 N. A uniform load of −1000 N/m acts on the clamped span.

node clampedSpringSupportedBeam1D/clampedSpringSupportedBeam1D.js

Source: clampedSpringSupportedBeam1D.js

Geometry and mesh

The mesh has 2 elements, with nodes 1, 2 and 3 at the clamp, the roller and the spring.

Results

The example prints the deflection and rotation at each node:

Node |    x (m) | Deflection w (m) | Rotation theta (rad)
-----|----------|-------------------|----------------------
  1  |     0.00 |         0.0000e+0 |           4.5475e-18
  2  |     5.00 |        9.4739e-18 |           -5.6791e-3
  3  |    10.00 |        -8.0145e-2 |           -2.1204e-2

The deflection and rotation at the clamp, and the deflection at the roller, are zero to round-off.

Cantilever beam

A 4 m cantilever, $EI = 10^5$ N·m², clamped at $x = 0$, with a point load $P = -1000$ N at the free end. The exact tip deflection and rotation are $w(L) = P L^3 / (3 EI)$ and $\theta(L) = P L^2 / (2 EI)$, which the cubic Hermite elements reproduce exactly.

node cantileverBeam1D/cantileverBeam1D.js

Source: cantileverBeam1D.js

Geometry and mesh

The mesh has 4 elements, with nodes 1 to 5; the load is applied at node 5.

Results

The example prints the tip deflection and rotation next to the exact values:

Tip deflection w(L): -2.133333e-1 m (exact -2.133333e-1 m)
Tip rotation θ(L): -8.000000e-2 rad (exact -8.000000e-2 rad)