Problem Statement
Figure 1. Schematic of a simply supported beam under load. Coupling dP and kinematics generates the Euler-Bernoulli beam theory statement
dP:
where y is vertical displacement, E is Young’s modulus, I is cross-section area moment of inertia, p(x) is a distributed load and P is a point load. Admissible boundary conditions at x = 0 and x = L include deflection: y(x)
slope:
moment:
shear:
The FE implementation of GWSh for (1) produces
GWSh =
The Euler-Bernoulli beam theory possession of a fourth derivative requires the FE basis in (3) to support two derivatives. Thus a minimum of a quadratic polynomial is required. One can modify this problem statement such that a linear basis can be used. Noting the definition of moment M can be written as a displacement differential equation, separating dP and kinematics leads to a pair of conservation statements
dP:
kinematics:
For two degrees of freedom per node, i.e, {Q}e = {M, Y}eT the associated GWSh now produces the familiar syntactical form
{WS}e = {DIFF}e{Q}e
- {b}e
(6)
The eligible boundary conditions remain as stated in (2), and the linear FE basis is adequate for this
formulation. Recalling the template syntax statement {WS}e = (const)(avg.data)e{dist.data}e(det)[FE matrix]{Q or data}e (7) the template
pseudo-code essence for (6) is {WS(M)}e = ( ) ( ) { } (-1) [A211] {M} – ( ) ( ) { } (1) [A200] {P} -Pd+ + BC {WS(Y)}e = (EI) ( ) { } (-1) [A211] {Y} + ( ) ( ) { } (1) [A200] {M} + BC (8) Recall now the Timoshenko beam theory statement
DP:
kinematics:
where r is transverse plane rotation, w is vertical displacement, E is Young’s modulus, I(x) is area moment of inertia of the beam cross-section for A(x), p(x) is a distributed load, P is a point load, G is shear modulus and k is a theory constant. For two degrees of freedom per node, the GWSh produces the familiar expression {WS}e = {DIFF}e{Q}e - {b}e (11) The eligible boundary conditions are constraints on r and w, or their normal derivatives and the linear FE basis is again adequate. The template pseudo-code for uniform I for (9)-(10) is {WS(R)}e = (EI) ( ) ( ) {-1} [A211] {R} - (kG) (A) { } (0) [A201] {W} + (kG)
(A) { } (1) [A200] {R} + BC {WS(W)}e = (kG) (A) { } (-1) [A211] {W} + (kG) (A) { } ( ) [A201] {R} + (12) + ( ) ( )
{ } (1) [A200] {P} +P{d}
+ BC
The GWSh for the Timoshenko beam theory produces an
algorithm that is “too stiff.” The use of under integration introduces
an artificial diffusion mechanism, which produces a more accurate solution
on a coarse mesh. From the lecture, under integration of the offending term
in (10) produces the additional diffusion term {WS(R)}e = {WS(R)}e + (kG/12) (A) { } (1) [A211] {R} (13) The following data
specification completes the problem statement assuming AISI 1018 cold-rolled
steel.
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