MINING INFLUENCE ON THE SURFACE (a 2D case)

MINING INFLUENCE ON THE SURFACE (a 2D case)

Radius R causes:

redistribution of contact stress ∆r(x)

1. Design value

k wp p

d k k k

R R

⋅

= ⋅ ^{, R=}a given characteristic value kp = overloading factor (=1.3 usually)

kwp = working conditions factor (say ∼1.0 for beams) kk = directional factor (say ∼0.7 as an average 2. Subsoil deformation (mining subsidence) Mining-induced curvature of the ground surface causes an additional bending of the beam, i.e. a non-mining bending moment M(x) will be replaced by M(x)+∆M(x). By making use of the superposition law, the mining corrections ∆M(x) along the beam can be calculated as a separate mining action, i.e. ignoring the beam loading like P1,P2,P3,P4; therefore, the mining deflection y(x) of the ground causes changes (redistribution) of vertical reaction under the beam ∆r(x) and next ∆Q(x), ∆M(x). Assume:

=2 ∙

The origin x=0 coincides with the load center LC on the beam, so the point where the resultant of ΣPⁱ is applied, Ll +Lr = L= the total length of the beam; this is simply a geometrical center (Ll = Lr =L/2) if the situation is symmetrical or is shifted of eccentricity from the beam center otherwise.

Mining corrections ∆r(x) and next ∆Q(x), ∆M(x) can be estimated analytically (for the Winkler model) but computer-aided calculations are more universal. If the code ZEM_SIN is used¹, prepare first additional mining subsidences ∆i for center of each calculation segment i.

∆ = =

2 ∙

where xi is measured from LC to the center of each calculation segment i.

For example, x8 = 0 if the beam is symmetrical and is divided into 15 identical calculation segments.

Note that ∆Q and ∆M are zero at the beam ends but it is not so for ∆r.

Clearly, the change of the sign of the radius Rd will change the signs of all deltas ∆ⁱ, therefore automatically the signs of ∆Q and ∆M. In other words, only one run of the code is necessary to take into account mining actions.

Since the loads P1d,P2d,P3d,P4d denote the design values, and such is also Rd, so the solutions ∆r, ∆Q and ∆M are also of the “design type” – ready for use in the concrete design (STR).

3. Ultimate radius

For “small” Rd and long rigid beams it can happen that locally r+∆r < 0 which looks non physical on the contact between subsoil and foundation (tension is impossible!); it is a signal that the beam can loose contact with the subsoil and such a situation should be avoided by changing some design parameters.

Test yourself:

(1) Do you see that any reduction of the maximal mining subsidence wmax will also reduce∆M(x)? (abs.values) (2) Do you know the exact value of the integral of

∆

(3) Is the case of the depth H=400m less dangerous than that of H=800m in terms of

∆ Q

(4) Replace all

∆

∆

Guess the effects in terms of ∆M(x)^.

1 There are some arguments and evidences that mining deformations (tension first of all) can reduce stiffness of the subsoil, like the Young modulus which value should be decreased of (let us say) 20% – this is not analyzed here.

H w

= a⋅h

r

r

h

tgβ

R>0&ε>0 (convex) R<0&ε<0 (concave)

Ll Lr

0

y(x)

r(x)±∆r(x)

M(x)±∆M(x) for Rd>0

for Rd<0

±∆r(x)

±∆Q(x)

±∆M(x)

W.Brząkała, WUST, Wrocław/Poland