Nonlinear beam fixed at both ends. Level 2 SUGAR model derivation. w M(x,y) d L h Fy Fx M0 EIz x y z where (1) (2)

Solve for y(x) along beam Equation (1) is (3) Let constants A, B, & C be defined as (4) So (3) is of the form (5)

Solve y(x) (6) Complementary part (7) Particular part by undetermined coefficients Substitute into (8) Equating coefficients of x (9)

Boundary conditions on y(x) (9) To get a3 and a4, look at the slopes (10) To get M0 in terms of Fy AND Fx, look at the fixed end (11)

Simplify y(x) & get rid of M0 (12)

y(x) with Fx & Fy only (13) Sanity check against conventional non-coupled theory (Fx=0) at x=0 (14) (15)

Fx is unknown. Need another relationship. Change of length DL (16) (17) |dy/dx| << 1 binomial series (18) Since slope << 1, Fx is constant along beam.

Governing equations Putting it all together - integrating (18) gives Fy(Fx), (13) provides d(Fx), and (11) gives M0(Fx). Nondimensionalizing for generality gives the following Let  “Fx”, choose l then plug-in below  “Fy”  “d”  “M0”

Plots. “dnonlinear”,”dlinear” “Fy” and “M0” as functions of “Fx”