Plate buckling

Simply supported edges —> No deflection and moment at the edge

(Assumption: Moment only with D11 & D22 because we assume that under uniaxial load, no twisting arises)

In the condensed plate equilibrium, load p is dropped, but additional loads due to Normal force flows in 2nd order are introduced.

Substituting Moments with deflections via the constitutive law, the buckling differential equation is created

A similar approach to the Navier solution in plate analysis is used, however not as a series expansion —> Case studies must be conducted:

The Ansatz

is inserted into the buckling differential equation, yielding:

—> To avoid the trivial solution, the Bracket term is set to 0 yielding:

n: appears only in the nominator —> lowest n (1) technically relevant

m: appears in the nominator and denominator —> different approaches can be utilized to finde lowest Nxx:

• Extremum problem: Differentiate w.r.t. m, set to zero

• Case study: Investigate until lowest load found

• Graphical representation: Draw the Gierlandenkurve, utilize lower boundary (or just use first low minimum vor all plate sizes —> on the conservative side)

psi (-) tensile psi (+) compressive