Walkw ay (^1) Walkw ay 2
q 1 q 2 q^3 q 4 q 5 q 6
L 1 L 2 L 3 L 4 L 5 L 6
E 1 ,e 1 E 2 ,e 2 E 3 ,e 3 E 4 ,e 4 E 5 ,e 5 E 6 ,e 6
Ks1 Ks2 Ks3 Ks4 Ks5 Ks6
Ke1 Ke2 Ke3
Figure 4: Panel type 1.
Walkw ay 1
q 1 q 2 q^3 q 4 q 5
L 1 L 2 L 3 L 4 L 5
E 1 ,e 1 E 2 ,e 2 E^3 ,e^3 E 4 ,e 4 E 5 ,e 5
Ks1 Ks2 Ks3 Ks4 Ks5
Ke1 Ke2
Figure 5: Panel type 2.
The application of the boundary and compatibility con-
ditions gives rise to the system of ( 6 ); where푛will be 6 for
paneltype1and5thepaneltype2.
푦 1 (푥 0 )=0,
푦耠 1 (푥 0 )=0,
푦푖(푥푖)=푦푖+1(푥푖), 푖=1,2,...,푛−1,
푦푖耠(푥푖)=푦耠푖+1(푥푖), 푖=1,2,...,푛−1,
퐾푡푖⋅푦푖耠耠(푥푖)=퐾푡푖+1⋅푦푖+1耠耠 (푥푖), 푖=1,2,...,푛−1,
퐾푡푖⋅푦 1 耠耠耠(푥 1 )=퐾푡푖+1⋅푦 2 耠耠耠(푥 1 )
퐾푡푖⋅푦푖耠耠耠(푥푖)−퐾푡푖+1⋅푦耠耠耠1+1(푥푖)=퐾푒푖⋅푦푖(푥푖),
푖=2,...,푛−2,
퐾푡푛−1⋅푦푛−1耠耠耠 (푥푛−1)=퐾푡푛⋅푦耠耠耠푛(푥푛−1),
푦푛(푥푛)=0,
푦푛耠(푥푛)=0,
(6)
where
푥푖=
푖
∑
푗=1퐿푖, 푖=1,2,...,푛. (7)
Replacing the values of the deflection and its derivatives in
( 6 ) and expressing the system in matrix form, the following
is obtained:푀⋅푈=퐵, (8)where the transposed matrix of푈is given by푇푈=[푇푈
1푇푈
2 ⋅⋅⋅
푇푈
푛], (9)
where
푇푈
푖=[퐴푖 퐵푖 퐶푖 퐷푖], 푖=1,2,...,푛. (10)Therefore, the not null elements of the matrices푀(4푛×4푛) =
{푚푖푗}and퐵(4푛×1)={푏푖}are obtained (the Appendix) for the
paneltype1.Inpaneltype2,thematrixes푀and퐵are equal
to the first type in its first 14 rows, with the last ones also
being equal with the exception of the following changes of
indexes: indexes 5 and 6 of the problem type 1 transform into
the indexes 4 and 5, respectively, of the problem type 2.4. Calculating the Factor of Safety of the
Workshop
Once the efforts have been calculated, it is possible to
calculate the factor of safety (FS) of the panel dividing the
tensile strength of each section(휎푡(푥))for the normal stress
(휎(푥))( 14 ). If this coefficient is bigger than one, the roof
of the panel is capable of supporting the stresses to which
it is subjected, and, therefore, the work is realised in safe
conditions.
TocalculatethisFSineachsectionoftheroof,theshear
stress ( 11 ) and the bending stress ( 12 ) are calculated as follows:휏(푥)=
4⋅푉(푥)
3⋅푒⋅푏
, (11)
휎푓(푥)=
6⋅푀(푥)
푏⋅푒^2
. (12)
From ( 11 )and( 12 ), the normal stress is calculated as휎(푥)= 0.5 ⋅ [휎푓(푥)+√휎푓^2 (푥)+4⋅휏^2 (푥)]. (13)
Oncethestressesarecalculated,theFSisgivenbyFS=
휎푡(푥)
휎(푥)
. (14)
5. Practical Case
A panel in the take-off phase is analysed with two config-
urations commonly used in the mines of Castilla-Leon and ́
more specifically in the Feixolin mine [ 17 ], with two walkways
(panel type 1) and with one walkway (panel type 2). Figures
4 and 5 show that in the phase of take-off, the first and last
spans rest elastically on the coal, whereas the halfway sections
rest on siltstone with an apparent density of 2.75 t/m^3 .The
hydraulic props are of the type EA 25 manufactured by
Salzgitter and reach a maximum extend length of 2.5 m.