Euler–Bernoulli beam

Verification of eulerBernoulliBeam against closed-form solutions. All results are from FEAScript-core 0.4.0.

Cantilever beam

A cantilever of length $L = 4$, $EI = 10^5$, with a tip load $P = -1000$, on a single element.

The exact tip deflection and rotation are

$$w(L) = \frac{P L^3}{3 EI}, \qquad \theta(L) = \frac{P L^2}{2 EI},$$

which the cubic Hermite element reproduces.

Quantity	Exact	FEAScript	Absolute error	Relative error
Tip deflection w(L)	$-16/75$ (−0.213333…)	−0.2133333333333334	6.9e-17	3.2e-16
Tip rotation θ(L)	$-2/25$ (−0.08)	−0.08000000000000004	4.3e-17	5.4e-16
Clamped end w(0)	0	0	0	–
Clamped end θ(0)	0	0	0	–

The FEAScript values are printed in full, as the shortest decimals that identify each double precision number. The errors are measured against the exact fractions. The relative errors are 3 and 5 times the unit round-off of double precision, the number format of JavaScript, $u = 2^{-53} \approx 1.1 \times 10^{-16}$: the element is exact, and only the arithmetic rounds. The clamped values are exactly 0 because they are prescribed.

The test requires each value to match within 1e-9.

Element stiffness matrix

The test also compares the assembled stiffness matrix of a single element, entry by entry, with the textbook Hermite beam matrix, whose entries scale with $EI / L$, $EI / L^2$ and $EI / L^3$. It does this for two combinations of $EI$ and $L$, so that both scalings are checked.

$EI$	$L$	Largest entry ($4EI/L$)	Largest difference	Relative to largest entry
$2 \times 10^6$	5	$1.6 \times 10^6$	2.3e-10	1.5e-16
$7.5 \times 10^4$	2.5	$1.2 \times 10^5$	2.9e-11	2.4e-16

The test requires every entry to match within 1e-6: eulerBernoulliBeam1DCantilever.