ABSTRACT
It shows that the compound key strata will not break if the condition above is satisfied.
The bending moment of rock strata in the depth of i is expressed by
M qE I
F F
E I x
q E I F
q E I F
F E I
l
( )x = ⎛⎝⎜ ⎞ ⎠⎟
− −
cos
cos ⎛ ⎝⎜
⎞ ⎠⎟
⎛ ⎝⎜
⎞ ⎠⎟
⎛ ⎝⎜
⎞ ⎠⎟
+
si sn in
sin
F E I
l F E I
x
q E I F
F E I
l ⎛ ⎝⎜
⎞ ⎠⎟
⎛ ⎝⎜
⎞ ⎠⎟sin
F E I
x (5)
The equation above could be simplified as
E I F I
M( )x = ∑ ∑ ( )x (6) where, Mi (x) is the bending moment of cross section of the rock strata in the depth of i; Ii is inertia moment of neutral rock strata in the depth of i. Ei is the elastic modulus of the rock strata in the depth of i. Thus
i i q E I
q E I
F E I
F E I
l( )x = ∑ − ⎛ ⎝⎜
⎞ ⎠⎟∑ +
⎛ ⎝⎜
⎞ ⎠⎟ −
⎡ ⎣ ⎢⎢
co cxs os 1 ⎤ ⎦ ⎥⎥
⎛ ⎝⎜
⎞ ⎠⎟
⎛ ⎝⎜
⎞ ⎠⎟si sn in
F E I
l F E I
x (7)
That is
E I E I
q( )x = ∑ ∑ ( )x (8) where ∑qi (x) = q1 (x) + q2 (x) + ... + qi (x) + ... + qn (x), qi (x) is the distribution load functioned on the rock strata in the depth of i.
