The elasticity of fluid phase lipid membranes can be characterized using the Helfrich Hamiltonian, which depends only on the square of the total curvature and the Gaussian curvature and the corresponding curvature moduli: the mean bending modulus and the Gaussian curvature modulus. Even at large curvatures approaching the inverse bilayer thickness, the effect of higher order terms of the elastic energy are minimal. Recently, a method has been developed to derive the bending modulus for fluid membranes from the stress-strain relationship of a buckled membrane [1,2]. The method also predicts the shape of the membrane, an Euler Elastica, and this serves as a check that the membrane indeed follows quadratic curvature elasticity. Using a coarse-grained lipid model, we analyze the shape of a buckled membrane in a gel phase and show that it does not behave like an Euler Elastica, even at low curvatures. The deviation from the theory suggests that higher order terms of the total curvature reduce the energy penalty for high curvatures. We present an extended version of the Helfrich Hamiltonian that captures this effect and show that it describes the shapes that we observe in simulations. We then calculate the bending modulus, as well as the modulus describing higher order corrections.[1] Noguchi H., “Anisotropic surface tension of buckled fluid membranes”, Phys. Rev. E 83, 061919 (2011).[2] Hu M., Diggins P., and Deserno M., “Determining the bending modulus of a lipid membrane by simulating buckling”, J. Chem. Phys. 138, 214110 (2013).