Geometric method (Euler)

The soil surrounding the micropile is represented in the program by the modulus of subsoil reaction E_p (Winkler constant k) defined by the user in the frame "Verification of cross-section". A model of a structure is displayed in the figure.

Model of structure

For a micropile in compression it is expected that a varying number of half waves occurs depending on the geometry and stiffness of the structure and surrounding soil, respectively. The solution of this case arises from the equation of bending of a straight beam.

After some manipulations the bending equation can be expressed as:

where:

Integration constants are found from four boundary conditions depending on the assumed end point supports.

Assuming hinges on both ends the following equation can be derived:

providing the number of half waves in the form:

where:	Ei	-	modulus of elasticity of ideal cross-section
	Ii	-	moment of inertia of ideal cross-section
	lp	-	micropile length
	Ep	-	modulus of subsoil reaction
	n	-	number of half waves

Assuming hinge on the one side and fixed end on the other side the following equation holds:

providing the number of half waves in the form:

where:	Ei	-	modulus of elasticity of ideal cross-section
	Ii	-	moment of inertia of ideal cross-section
	lp	-	micropile length
	Ep	-	modulus of subsoil reaction
	n	-	number of half waves

Force N_cr is determined from the following equation by iterations:

where:	Ei	-	modulus of elasticity of ideal cross-section
	Ii	-	moment of inertia of ideal cross-section
	Ncr	-	critical normal force
	lcr	-	buckling length of micropile cross-section in compression