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# LEM Method

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# LEM Method

The limit equilibrium method consists in looking for solutions that are compatible only with the static aspect of the problem. Basically, we reason in terms of equilibrium of a rigid body, without minding the kinematic congruence of the displacements. The main calculation diagrams we will refer to are the following:

Calculation of the limit embedment depth

For unanchored bulkhead, the stability is guaranteed by the passive strength of the ground situated downhill from it; from the equilibrium of the moments as to the center of rotation, we obtain:

Where the symbols have the following meaning:

Sm = horizontal component of the active thrust

Bm = arm of Sm as to O center of rotation

Rv = horizontal component of the passive strength

Bv = arm of Rv  as to O center of rotation

Each term is a function of t, where t is the depth of the rotation center with respect to the downhill reference level (downhill natural surface level). The necessary length to guarantee the equilibrium of the horizontal translation is obtained by increasing t as follows:

where a=0.2 (Blum method)

Reference diagram for calculating the equilibrium of the bulkhead

Safety factor on the passive strength

The length of the embedment d, as determined above, refers to the limit condition of an incipient collapse through a coefficient F. It is possible to introduce a safety margin on the passive strengths; such reduction is made as follows:

Calculation of the limit embedment depth

The stability of the work is also guaranteed by the anchors fastened upon the bulkhead. In order to use the calculation scheme with free end, the bulkhead must be sufficiently short and rigid. The embedment length will be determined by imposing the equilibrium to the rotation on the origin of the anchor indicated as B1

Where the symbols have following meaning:

Sm = horizontal component of the active thrust

H = height of the ground to be supported

t = calculated embedment depth

Bm = arm of Sm as to the base of the bulkhead

Pm = ordinate of the point of application of the upstream anchor

Rv = horizontal component of the passive strength

Bv = arm of Rv

When t is known, Sm and Rv as well as the stress of the anchor will be determined.

Safety factor F on the passive strengths

The embedment length will be further increased so as to have a safety margin in working conditions by means of the safety factor F:

Calculation of the limit embedment depth

If the deepest section of the bulkhead does not translate or rotate, it can be assimilated to a fixed joint; in this case, the bulkhead is defined with fixed end. A procedure developed by BLUM allows obtaining the embedment depth (t+t'), imposing the kinematic conditions of null displacements on the base of the work and on the origin of the anchor (B1) and the static conditions of null moment and shear on the base of the bulkhead. We therefore reach a 5th degree equation in (t+t') which can be easily solved.

Safety factor F on the strengths

In order to increase the safety factor, values of the reduced passive strengths have been introduced in the numerical developments.