Loads on tunnels (1)

• The most important potential loads acting on
underground structures are earth/rock pressures
and water pressure.

• Live loads due to vehicle traffic on the surface
can be safely neglected, unless the tunnel is a
cut and cover type with a very small depth of
overburden.

• It may be generally stated, that the dimensioning
of tunnel sections must be effected either
against the overburden weight (geostatic
pressure) or against loosening pressure (i.e. the
weight of the loosened zone, called also
protective or Trompeter's zone).

Design approach should include these elements:
• Experience, incorporating features of empiricism
based on an understanding of ground
characteristics and on successful practices in
familiar or similar ground.
• Reason, using analytical solutions, simple or
more complex as the situation may demand,
based on a comprehensive data on the ground
conditions
• Observation of the behaviour of the tunnel
during construction, developing into monitoring
with systematic pre-designed modification of
supports.

Types of rock pressure
• In nature, deep lying rocks, are affected by the weight of
the overlying strata and by their own weight. These
factors develop stresses in the rock mass. In general
every stress produces a strain and displaces individual
rock particles. But to be displaced, a rock particle needs
to have space available for movement. While the rock is
confined, thus preventing its motion, the stresses will be
accumulated or stored in the rock and may reach very
high values, far in excess of their yield point.
• As soon as a rock particle, acted upon by such a stored,
residual or latent stress, is permitted to move, a
displacement occurs which may take the form either of
'plastic flow' or 'rock bursts' (popping) depending upon
the deformation characteristics of the rock-material.
• Whenever artificial cavities are excavated in rock the
weight of the overlying rock layers will act as a uniformly
distributed load on the deeper strata and consequently
on the roof of the cavity.

• The resisting passive forces (shear strength) are scarcely
mobilized prior to the excavation of the cavity, since the
deformation of the loaded rock mass is largely prevented by
the adjacent rocks. By excavating the cavity, opportunity is
given for deformation towards its interior.
• In order to maintain the cavity the intrusion of the rock masses
must be prevented by supporting structures.
• The load acting on the supports is referred to rock pressure.
The determination of the magnitude or rock pressure is one of
the most complex problems in engineering science.
• This complexity is due not only to the inherent difficulty of
predicting the primary stress conditions prevailing in the
interior of the non-uniform rock mass, but also to the fact that,
in addition to the strength properties of the rock, the
magnitude of secondary pressures developing after
excavation around the cavity is governed by a variety of
factors, such as the size of the cavity, the method of its
excavation, rigidity of it support and the length of the
period during which the cavity is left unsupported.

• Rock pressures depend not only on the quality of rock
and on the magnitude of stresses and strains around the
cavity, but also on the amount of time elapsing after the
outbreak of the underground cavity.
• Within any particular rock the pressures to which it was
exposed during its history are best indicated by the
pattern of folds, joints and fissures, but it is difficult to
determine how far these pressures are still latent.

• According to Terzaghi secondary rock pressure, should
be understood as the weight of a rock mass of a certain
height above the tunnel, which, when left unsupported
would gradually drop out of the roof, and the only
consequence of installing no support props would be that
this rock mass would fall into the cavity. Successive
displacements would result in the gradual development
of an irregular natural arch above the cavity without
necessarily involving the complete collapse of the tunnel
itself.

• Earth pressure, on the other hand, would denote the
pressure exerted by cohesionless, or plastic masses on
the tunnel supports, without any pressure relief that
would, in the absence of supports, sooner or later
completely fill the cavity leading to its complete
disappearance.

• In general, the magnitude of earth pressure is independent of the
strength and installation time of the supporting structure and it is
only its distribution that is affected by the deformation of the latter.
The magnitude of rock pressures, on the other hand, is influenced
decisively by the strength and time of installation of props.

• This is because the deformation following the excavation of the
cavity in rock masses surrounding the tunnel is of a plastic nature
and extends over a period of time. This period required for the final
deformations and, thus for the pressures to develop, generally
increases with the plasticity of the rock and with the depth and
dimensions of its cross-section. The magnitude of deformations and
consequently that of stresses can, therefore, be limited by
sufficiently strong propping installed at the proper time.

• It should be remembered, however, that the intensity of plastic
pressures shows a tendency to decrease with increasing
deformations. Furthermore the loads are carried both by the tunnel
lining and the surrounding rock and every attempt should be made
to utilize this cooperation.

The reasons for the development of secondary rock
pressures can be classified according to Rabcewicz in
the following three main categories:
• Loosening of the rock mass
• The weight of the overlying rock masses and tectonic
forces
• Volume expansion of the rock mass, swelling due to
physical or chemical action.
These reasons lead in general to the development of the
following three types of rock pressure:
• Loosening pressure
• Genuine mountain pressure
• Swelling pressure
The conditions under which rock pressures develop, the
probability of their occurrence and their magnitude differ
greatly from one another and require the adoption of
different construction methods.

Rock pressure theories
There are various rock pressure theories; one
group of rock pressure theories deal, essentially,
with the determination of loosening pressure
since the existence of any relationship between
the overburden depth and mountain pressure is
neglected. The group of theories that does not
take the effect of depth into account are:
• Kommerell's theory
• Forchheimer's theory
• Ritter's theory
• Protodyakonov's theory
• Engesser's theory
• Szechy's theory

Another group of rock pressure theories takes
into account the height of the overburden above
the tunnel cavity. The group of theories that
takes the effect of depth into account are:
• Bierbäumer's theory
• Maillart's theory
• Eszto's theory
• Terzaghi's theory
• Suquet's theory
• Balla's theory
• Jaky's Concept of theoretical slope

Vertical Loading -Bierbäumer's Theory
• Bierbäumer's theory was developed during the construction of the
great Alpine tunnels. The theory states that a tunnel is acted upon by
the load of a rock mass bounded by a parabola of height, h = αH.

Two methods, yielding almost identical results, were developed for
the determination of the value of the reduction coefficient α.
One approach was to assume that upon excavation of the tunnel the
rock material tends to slide down along rupture planes inclined at
45° + ϕ/2

The weight of the sliding rock masses is counteracted by the
friction force
S = 2fE = 2 tan Φ tan² (45° -Φ/2) ·
developing along the vertical sliding planes and therefore a
rock mass of height αH only, instead of H, must be taken
into account during the calculations. Consequently, the
pressure on width b + 2m tan (45° -Φ/2) at the crown will be:
p = α1H γ

Taking into consideration the load diagram shown in Figure
3 the value of α1 is derived as follows:

P = Hγ [ b + 2m·tan(45° - Φ /2)] − H 2γ tan 2 ( 45° − Φ / 2) tan Φ
Since
P  tan Φ·tan 2 ( 45 − Φ / 2) H 
p= = Hγ 1 −
b + 2m·tan(45° - Φ /2)  b + 2m·tan(45° − Φ / 2) 
Thus
tan Φ·tan 2 ( 45° − Φ / 2) H
α1 = 1 −
b + 2m·tan(45° - Φ /2)

implying that geostatic pressure is diminished by the friction
produced by the horizontal earth pressure of the wedges EC
and DF acting on the vertical shear planes.
Values of the reduction coefficient for single and double-
track tunnels, as well as for various angles of internal friction
Φ and depths H are compiled in table 1.
The reduction coefficient α 1 has two limit values,
namely for very small overburden depths α = 1 and at
several hundred metres depth, whenever H > 5B, the
magnitude of α 1 is no longer affected by depth and
becomes
α1 = tan ( 45° − Φ / 2)
4

Terzaghi's vertical loading formula was
based on a series of tests:

γB
Pv =
2K tan Φ

B, Φ, and γ are the same as Bierbäumer.
K= 1 (based on his tests)

Determination of lateral pressures on tunnels
• The magnitude of lateral pressures has equal importance with the
vertical or roof pressures for structural dimensioning of tunnel
sections.
• The sidewalls of the cavity are often the first to fail owing to the less
favourable structural conditions developing in the rock there. In
some instances lateral pressure may play a more important role
than the roof load.
• Also its theoretical estimation is more involved than that of the roof
load, since its magnitude is even more affected by the extent of
deformations of the section, so that its value depends increasingly
on the strength of lateral support besides the properties of the rock
and dimensions of the cavity.
• Lateral pressures are also much more affected by latent residual
geological stresses introduced into the rock mass during its
geological history which are released upon excavation and whose
magnitude depends on the deformation suffered by, and the
elasticity of the rock, but it is unpredictable. Genuine mountain
pressure and swelling pressure that cannot be evaluated
numerically may act in full on the sidewalls.

Approximate determination of lateral pressures:
• Lateral pressures in soils are determined
approximately from earth pressure theory, as a
product of geostatic pressure, or roof load and the
earth pressure coefficient, respectively in terms of
the lateral strain.

• Lateral pressures are (Stini) in contrast to roof loads,
in a linear relationship with overburden depth. This
has been confirmed also by recent model tests and
in-situ stress measurements.

• At greater depths pressures on the springings are
usually higher than on the roof, but at the same time
frictional resistance is also higher.

Lateral pressures compiled according to practical experience are given in
the table below:

• According to Terzaghi, a rough estimate of lateral
pressure is given by the following formula:
Ph = 0.3γ (0.5m + h p )
• where hp is the height of the loosening core representing
the roof load; in granular soils and rock debris, on the
basis of Rankine's ratio
Ph = γH tan 2 ( 45° − Φ / 2)

• and finally, in solid rocks, relying on Poisson’s ratio
µ
Ph = pv
1− µ
• Lateral pressures should be assumed in a linear
distribution and should be based on the vertical pressure
estimated by one of the rock pressure theories instead of
on geostatic pressure.

The parabolic distribution shown in below figure should be assumed
for the vertical pressure having a peak ordinate corresponding to the
estimated roof load.

If the pressure ordinate of the
parabola an the vertical erected at
the side of the cavity is , then the
lateral pressure intensity at roof
level will be
e1 = p2 tan 2 ( 45° − Φ / 2) − 2c tan( 45° − Φ / 2)
and at invert level

e2 = ( p2 + mγ ) tan 2 (45° − Φ / 2) − 2c tan(45° − Φ / 2)
Owing to the favourable effect of
lateral pressure on bending
moments arising in the section,
cohesion, which tends to reduce
the magnitude of this pressure,
must not be neglected in the
interest of safety.
The lateral pressure coefficients
involved in the above formulae are
either
µ λ = tan 2 ( 45° − Φ / 2)
λ= or
1− µ

Bottom Pressures
• Bottom pressures should be essentially the counterparts of roof
loads, i.e. reactions acting on the tunnel section from below if the
tunnel section is a closed one having an invert arch.
• A certain part of this load is, however, carried by the surrounding
rock masses, so that this situation does not occur even in the case
of closed sections, and bottom pressures have usually been found
to be smaller than roof loads.
• Terzaghi quoted empirical evidence indicating that bottom pressures
are approximately one half and lateral thrusts one-third of the roof
load intensity.
• In the case of sections open at the bottom, i.e. having no invert arch,
pressures of different intensity develop under the side walls and
under the unsupported bottom surface. The pressures arising under
the solid side walls must be compared with the load-bearing
capacity, or ultimate strength of the soil, but do not otherwise affect
the design of the tunnel, The magnitude of rock pressure acting
upward towards the interior of the open tunnel section is, however,
unquestionably affected by these pressures .
• The development, distribution and magnitude of bottom pressures
are greatly influenced by the method of construction adopted, i.e. by
the sequence in which various structures and components of the
tunnel are completed.

Bottom pressure according to Tsimbaryevitch
He assumed that a soil wedge is displaced towards the cavity under
the action of active earth pressure originating from the vertical
pressure on the lateral parts. This displacement is resisted by the
passive earth pressure on the soil mass lying under the bottom of
the cavity.

• The active earth-pressure diagram at
the perpendicular of the corner point of
the excavated cavity is a trapeze. The
earth pressure at depth x will be
ea = ( p + xγ ) tan 2 (45° − Φ / 2) − 2c tan(45° − Φ / 2)

• At the same time the specific passive
earth pressure at depth x is
e p = xγ tan 2 (45° − Φ / 2) + 2c tan(45° − Φ / 2)

• Depth x: where ea = ep can be
computed by equating the above two
expressions. The layers above this
depth will be involved in bottom
pressure:
p tan 2 ( 45° − Φ / 2) − 2c[ tan( 45° + Φ / 2) + tan( 45° − Φ / 2)]
x=
[ ]
γ tan 2 ( 45° + Φ / 2) − tan 2 ( 45° − Φ / 2)
and
pλa − 2c( λp + λa )
x=
γ ( λp − λa )

The magnitude of the horizontal force acting towards the cavity above depth
x is given as the difference between the areas of the diagrams for ea and ep.
This force induces a set of sliding surfaces inclined at (45° - Φ/2) to develop
in the soil mass under the cavity.

The force of magnitude E = Ea - Ep may be resolved into components T
and S, parallel to the sliding surfaces and perpendicular to them,
respectively:

T = E cos (45º - Φ/2)
S = E sin (45º - Φ/2)

Force T tends to displace the soil and is resisted by the frictional component
of the normal force
T = S tan Φ
After trigonometric transformations and remembering that the soil is
displaced by forces acting from both corners, the magnitude of forces acting
on the bottom plane is obtained as:
sin 2 ( 45° − Φ / 2)
T0 = 2 E
cos Φ
The resultant To acts at the centre line and is vertical. This upward pressure
can be counteracted either by loading the bottom with the counterweight of
intensity qo, or by a suitably dimensioned invert arch.

The counter load qo must be applied over
a length y, which can be obtained from the
expression:
x
y=
tan( 45° − Φ / 2)

The pressure acting on the bottom of the cavity
in the practical case illustrated in the figure
below.

Assuming a granular soil, it can be
determined in the following way:

If the bottom re action under the side
Q
walls is p = 0 , the height of the q
soil
s
column at the side of the cavity=can be
H
y
obtained from the relation

Since tan 2 ( 45° − Φ / 2)
x=H
tan 2 ( 45° + Φ / 2) − tan 2 ( 45 − Φ / 2)

And
1 1
E = Ea − E p = γx ( x + 2 H ) tan 2 (45° − Φ / 2) − γx 2 tan 2 (45° + Φ / 2)
2 2

The pressure acting from below on the
cavity is sin(45° − Φ / 2)
T =E
cos Φ

T0 = 2T sin(45° − Φ / 2)

Closure of the section with a bottom slab and the
application of an internal ballast are the only possible
counter-measures to this pressure. In the interest of
safety a coefficient n = 1.3 to 1.5 has been specified.
b
−s
With 2 denoting the actual loading width, p s
and Pb the weight of the bottom slab and internal ballast,
respectively, the coefficient of safety can be determined
by comparing the resulting downwards stress and the
upward pressure N/y. In other words, it is required that
p s + Pb y
n= ≥ 1,3 − 1,5
T0 b
−2
2

Loads on tunnels (1)

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