- Email: [email protected]
Stability analysis of tunnels using the program ADINA
Computers & Strucrures Printed in the U.S.A.
Vol.
21. No.
I/Z. pp. 341-351.
STABILITY
0045-7949185 $3.00 + .OO B 1985 Pergamon Press Ltd.
1985
ANALYSIS OF TUNNELS PROGRAM ADINA
USING THE
FRANZ TH. LANGER and KLAUS STOCKMANN Dorsch Consult, Miinchen, Federal Republic of Germany Abstract-A straightforward generalized computational scheme has been established to simulate rock tunnel support (RTS) behaviour under three-dimensional and time-dependent stress conditions. The scheme may be applied to all construction stages from initial conditions to final concrete lining. Calculated displacements are compared with observed values for a double-tracked railway tunnel under construction in West Germany, showing reasonably good agreement. The method represents a generalization of a scheme developed for specific application to a double tunnel construction[3], which also resulted in good agreement between computed and observed values.
‘F = nodal point force vector equivalent
to the element stresses at time I, U = vector of nodal point displacement increments from time t to time t + A r,
1. INTRODUCTION The structural
analysis of tunnel drivage requires a nonlinear three-dimensional calculation method to model the supports as well as the nonlinearities in the rock material. Unfortunately, the required software and hardware facilities for civil engineers today are still too expensive to carry out economically a full nonlinear three-dimensional calculation. The use of appropriate two-dimensional stress and strain approximations as a replacement for the three-dimensional stress state calculation is therefore much desired. During tunnel excavations a stress redistribution takes place, which creates an arch-like support in the longitudinal and transverse direction of the RTS. The size of the spatial support effect of the RTS is not known a priori. However, there is a way to simulate the three-dimensional support effect of the RTS by varying (1) the material constants of the rock and concrete lining and the “birth” and “death” of elements for successive construction states. 2. THE GOVERNING EQUATIONS USED
According to the incremental finite element equilibrium equations[l, 21 we use the following fundamental equation: (‘KF + ‘KA + ‘KS) U = ‘+“R
- ‘F.
(1)
where ‘KA = aJ-A’KA
+ (1 + (1 -
‘KS = b.‘-A’KS ‘+AQ
=
u=
Cr-Ar/2~
+
(1
_
n).‘+A’KA,
(2)
b).‘+A’KS,
(3)
+r+ArlZ~,
r+ArU _ ‘U,
(4) (5)
a = factor for variation of ‘KA, b = factor for variation of ‘KS, c = factor for variation of ‘+“R. The variation of a, b and c adjusted in an appropriate manner is used to model the three-dimensional support effect as a function of [l - (a, 6, c)] in the calculation method. The “true” values for a, 6, and c can only be obtained iteratively, by measurements taken in the RTS. However, the variation of “a” and “b” is easily achieved by the birth and death option available in ADINA[ 11. The variation of “c” is obtained by the method of incremental loads. The method of calculation utilizing the “timevarying” factors a and b will now be described (factor c is constrained to unity).
In eqn (1) the following notation is used: 3. PROJECT DESCRIPTION
‘KF = tangent stiffness matrix at time t for the surrounding rock material (excluding the tunnel opening), ‘KA = tangent stiffness matrix at time t for rock material within the tunnel opening (to be excavated), ‘KS = tangent stiffness matrix at time t for the supporting construction material (concrete lining, etc.), ‘+A’R = external load vector applied at time t + At,
A 4180 m long tunnel on the line between Hannover and Wiirzburg under the jurisdiction of the German Federal Railways is being driven. The double-tracked tunnel has an excavation cross section of 142 m2. The surrounding soil consists of redistributed material (detrital limestone, clay, and sand deposits). The soil overburden above the crown is about 25 m. The large cross section of the tunnel and the poor stability of the surrounding soil does not permit excavation in a single stage. This means 341
341
FKANZTH.LANGER~~~ KLALJS~TOCKMANN
that the whole section has to be driven in three parts (crown, bench, and invert segments). The tunnel is being driven by the so-called New Austrian Construction procedure. Depending on the geological situation which is instantly exposed to view during the crown drivage, the portal zone of the tunnel is stabilized by reinforced shotcrete and steel anchorage. In sections with extremely poor soil support properties, the width of the crown footing will be increased to stabilize the crown segment. The excavation of the crown will be advanced 100 to 150 m ahead of the bench excavation. This distance is necessary for the required excavation tools and facilities. The final lining in the invert segment will be up to 60 m behind the invert excavation. The supporting structure of the tunnel consists of a 25 cm reinforced shotcrete lining supported by steel arches and rock anchorage. The internal tunnel structure consists of a 30 cm reinforced concrete shell (not shown in this report).
5. FINITE ELEMENTS AND ALGORITHMS
USED
For the calculation we basically have to model four structural components (1) the surrounding rock material. (2) the excavation material, (3) the reinforced shotcrete lining (outer shell). and (4) the reinforced concrete lining (inner shell). The construction of a tunnel is a time-dependent process. Essentially. realistic calculations can be obtained if one is able to (I) simulate numerically the true time-varying procedure of the excavation and the construction of the supporting structure: (2) accommodate the proper interaction of the various structural components at all time steps, i.e. model complete interelement compatibility within adjacent element groups[2][4]; (3) include the nonlinearily of the material (plastification. etc.): (4) update iteratively the estimated parameters CL, h. and c according to appropriate measurements during construction, then perform new calculations, and revise the final design if required. 4.SYSTEM MODELING The algorithms and elements available in In Fig. 2 all construction states and correspondADINA can accommodate for all four requireing calculation steps are shown: ments The notation used is to (1) the possibilities of the birth and death option is a “clean” method for the simulation of exEGR = element group number, cavation and construction. (i.e. the loading GO = surrounding rock material, consisting of of new structural parts with existing nodal EGR (13), forces released during “older” construction Cl = rock material for the crown excavation, stages): consisting of EGR (7 + 8). to (2) rxchiw application of parabolic isoparaG2 = rock material for the bench excavation metric elements guarantees a priori the reconsisting of EGR (9 + lo), quired interelement compatibility[41; G3 = rock material for the invert segment ex- to (3) the “Drucker-Prager” material law is able cavation consisting of EGR (I 1 + 12). to handle the required stress conditions (including tension cutoff); whereby to (4) due to the fact that (i) all time step solutions are obtained in GX = (1 - a)GX + aGX, X stands for 1, one calculation. 2. and 3. (ii) start and end time for the calculation is (1 - a)GX = three-dimensional support effect of easily prescribed in the input data, the RTS at time t (GX) within the (iii) LI and h values are multiples related to excavation zone X. the material. and a GX = the supporting structure SX (shot(iv) the preparation and revision of the input Crete lining) at time t (a GX) within data is not a time-consuming task. the zone of X. will be loaded with The following finite elements were employed in the internal nodal point forces rethe calculation: leased by (a GX), (1) For the rock Sl = material for the supporting structure Eight-node isoparametric plane elements in the crown range, consisting of (plane strain), EGR (1 + 2), s2 = material for the supporting structure Material 1 in the bench range, consisting of EX elastic moduli, EGR (3 + 4), EV material for the supporting structure s3 = in the invert range, consisting of EX? EGR (5 + 6). Poisson ratios. E.YZ (1 - b).SX + b SX, X stands for 1. sX= 2, and 3.
1
Stability analysis of tunnels using the program ADINA Elastic modulus far the rock Elastic modulus for the shotcrete Specific wei# of the rock Angle of friction cotM?sion Coefficient for actual earth pressure Poisson’s ratio Hardening factor far shotcrete Factor for 3CLsupparting effect
Y t
: +10.359
343 Ev = 200 MN/m2 Eb = 10000 MN/m2 8 = 22 kN/m3 ‘0 = 17. i.=’ = 06735 MN/m2
t:
= OIL = 0,s
a
= 0.15, 0.3. 0.5
Fig. 1. System and input data for the calculation. G a C
Shear modulus, factor for 3D effect, etc., Cohesion, angle of friction,
Y Drucker-Prager -yield surface.
(2) For the shotcrete lining (outer shell) construction state Three-node parabolic isoparametric beam elements, Material b = hardening factor, E = elastic modulus, G = shear modulus, d = shell thickness.
(3) For the shotcrete lining (outer shell) collapse state Three-node parabolic isoparametric rod elements, Material E = elastic modulus, d = shell thickness, only axial forces are transmitted. (4) For the inner shell Three-node parabolic isoparametric ments, Material E = elastic modulus, G = shear modulus, d = shell thickness.
beam ele-
--l-l_-_l_-_.
-.
_... -._----_r_
11”11111----
for crownsettlement
.---_
Displacemeclts.
Comparing AW&ealcuiations with maaruromW?ta according ta conrtructkertates and catcufotlon-gtepr
-.--
P 1
--.-~_--_ll_--
--.,
-l.l...“lll.lll. ~..--“I I--_ _..- ----~ - -
I
_--
_
-----
__..__~.
.__
--^..-
__.~
345
Stability analysis of tunnels using the program ADINA
element
mesh: initial
state
-.
_
Fig. 3. Element mesh: initial state.
-
Y
FRANZ TH. LANCXR
346
k’
element
mesh:
and
crown
KL.AUS STOCKMANN
excavation
I I 50.00
I T
Y -_)
Fig. 4. Element
mesh:
crown excavation
Stability analysis of tunnels using the program ADINA
element
mesh: bench excavation
so,00
Fig. 5. Element mesh: bench excavation.
347
FRANZTH. LANCERand KLAUSSTOCKMANN
348
Z
element
mesh:
invert
excavation
50.00
--
Fig. 6. Element mesh: invert excavation.
Y
1
i
& 1 (I-blS2
51bS3
i
50 -
LO
30
20.
1 1
8oGl 9 Il-olt2
E 5
1
1.
ll-bIS3
6 7
Il-olG1
0
1
31bS2
111-blS1
1 [bSl
2
I I
1
I
I
Comparing ANNA-calculations with different a-Factors according to construction-states and calculation-steps for the moment at Point P 2
\
\ \ \
\
\ \
\ \
\
‘k
\
J-a
a . 0.3
___________
= OS ._._._. _._,_.._._._._._
‘L.J-zL!L?____
‘L
._
._.__.
__________
_.-.-.-.-.-.
Comporing ADINA-colculotions with different o-Factors according to construction-states and calculation-steps far the axial force nt Paint P 7
Stability analysis of tunnels using the program ADINA 6. RESULTS
For the sake of brevity, only a small extract of results is presented. Fig. 1 shows the geometry and tunnel description. Fig. 2 gives the construction states, birth and death timing, element groups, comparison of calculations of crown displacements with on site measured data. Fig. 3 shows the element mesh in its initial state, Fig. 4 gives the element mesh after crown excavation, Fig. 5 shows the element mesh after the bench excavation, Fig. 6 shows the element mesh after the invert excavation, Fig. 7 depicts the bending moments for the shotcrete lining, and finally Fig. 8 shows the axial forces for the shotcrete lining for different a factors.
The above results correspond to a reasonable calculation time. Comparisons of the calculated results with onsite measurements verify that the actual three-dimensional and time-dependent stress conditions can be modeled using the two-dimensional idealization. However, the only means of verifying the assumed parameters a and b, is the comparison of the calculated displacements with existing values on site during the period of construction. Acknowledgment-The authors gratefully acknowledge the kind assistance rendered by Mr. DipI.-Ing. Fr. Schrewe (Head of the Department of Tunnel Construction and project leader for HW-NORD, Construction Headquarters, German Federal Railways) for making on-site measurements available for this study. REFERENCES
7. CONCLUDINGREMARKS
The necessary input data for ADINA has been obtained using a preprocessor @‘REPRO) written in FORTRAN 77. The PREPRO reduces the necessary manual work to a minimum. The calculation was performed on a DEC 2060 Computer system, the CPU times read as follows:
1. ADINA-a
2.
3.
4.
generation computation
of the input data with PREPRO approx. 3 min., of eight time steps with ADINA 81 approx. 85 min.
351
5.
finite element program for automatic dynamic incremental nonlinear analysis. Report AE 811. ADINA Enaineerinn. Seotember. 1981. i. J. Bathe, Fynite Ele&ent’Proced&es in Engineering Analysis. Prentice-Hall, Englewood Cliffs, New Jersey (1982). K. Schikora, Calculation model and measuring results for a double tunnel with low overburden in quatemary soil. Tunnel 3/82, West Germany, p. 153ff. 0. C. Zienkiewicz. The Finite Element Method in Engineering Science. McGraw-Hill, London (1971). K. Wanka and F. Th. Langer, Vorfluterbelastung, iiber zwei neuartige Verfahren zur instationlren Berechnung der FlieSzusttinde in Kanalisationsnetzen. KfK-CAD 148, Kemforschungszentrum Karlsruhe, West Germany, March, 1980.
Vol.
21. No.
I/Z. pp. 341-351.
STABILITY
0045-7949185 $3.00 + .OO B 1985 Pergamon Press Ltd.
1985
ANALYSIS OF TUNNELS PROGRAM ADINA
USING THE
FRANZ TH. LANGER and KLAUS STOCKMANN Dorsch Consult, Miinchen, Federal Republic of Germany Abstract-A straightforward generalized computational scheme has been established to simulate rock tunnel support (RTS) behaviour under three-dimensional and time-dependent stress conditions. The scheme may be applied to all construction stages from initial conditions to final concrete lining. Calculated displacements are compared with observed values for a double-tracked railway tunnel under construction in West Germany, showing reasonably good agreement. The method represents a generalization of a scheme developed for specific application to a double tunnel construction[3], which also resulted in good agreement between computed and observed values.
‘F = nodal point force vector equivalent
to the element stresses at time I, U = vector of nodal point displacement increments from time t to time t + A r,
1. INTRODUCTION The structural
analysis of tunnel drivage requires a nonlinear three-dimensional calculation method to model the supports as well as the nonlinearities in the rock material. Unfortunately, the required software and hardware facilities for civil engineers today are still too expensive to carry out economically a full nonlinear three-dimensional calculation. The use of appropriate two-dimensional stress and strain approximations as a replacement for the three-dimensional stress state calculation is therefore much desired. During tunnel excavations a stress redistribution takes place, which creates an arch-like support in the longitudinal and transverse direction of the RTS. The size of the spatial support effect of the RTS is not known a priori. However, there is a way to simulate the three-dimensional support effect of the RTS by varying (1) the material constants of the rock and concrete lining and the “birth” and “death” of elements for successive construction states. 2. THE GOVERNING EQUATIONS USED
According to the incremental finite element equilibrium equations[l, 21 we use the following fundamental equation: (‘KF + ‘KA + ‘KS) U = ‘+“R
- ‘F.
(1)
where ‘KA = aJ-A’KA
+ (1 + (1 -
‘KS = b.‘-A’KS ‘+AQ
=
u=
Cr-Ar/2~
+
(1
_
n).‘+A’KA,
(2)
b).‘+A’KS,
(3)
+r+ArlZ~,
r+ArU _ ‘U,
(4) (5)
a = factor for variation of ‘KA, b = factor for variation of ‘KS, c = factor for variation of ‘+“R. The variation of a, b and c adjusted in an appropriate manner is used to model the three-dimensional support effect as a function of [l - (a, 6, c)] in the calculation method. The “true” values for a, 6, and c can only be obtained iteratively, by measurements taken in the RTS. However, the variation of “a” and “b” is easily achieved by the birth and death option available in ADINA[ 11. The variation of “c” is obtained by the method of incremental loads. The method of calculation utilizing the “timevarying” factors a and b will now be described (factor c is constrained to unity).
In eqn (1) the following notation is used: 3. PROJECT DESCRIPTION
‘KF = tangent stiffness matrix at time t for the surrounding rock material (excluding the tunnel opening), ‘KA = tangent stiffness matrix at time t for rock material within the tunnel opening (to be excavated), ‘KS = tangent stiffness matrix at time t for the supporting construction material (concrete lining, etc.), ‘+A’R = external load vector applied at time t + At,
A 4180 m long tunnel on the line between Hannover and Wiirzburg under the jurisdiction of the German Federal Railways is being driven. The double-tracked tunnel has an excavation cross section of 142 m2. The surrounding soil consists of redistributed material (detrital limestone, clay, and sand deposits). The soil overburden above the crown is about 25 m. The large cross section of the tunnel and the poor stability of the surrounding soil does not permit excavation in a single stage. This means 341
341
FKANZTH.LANGER~~~ KLALJS~TOCKMANN
that the whole section has to be driven in three parts (crown, bench, and invert segments). The tunnel is being driven by the so-called New Austrian Construction procedure. Depending on the geological situation which is instantly exposed to view during the crown drivage, the portal zone of the tunnel is stabilized by reinforced shotcrete and steel anchorage. In sections with extremely poor soil support properties, the width of the crown footing will be increased to stabilize the crown segment. The excavation of the crown will be advanced 100 to 150 m ahead of the bench excavation. This distance is necessary for the required excavation tools and facilities. The final lining in the invert segment will be up to 60 m behind the invert excavation. The supporting structure of the tunnel consists of a 25 cm reinforced shotcrete lining supported by steel arches and rock anchorage. The internal tunnel structure consists of a 30 cm reinforced concrete shell (not shown in this report).
5. FINITE ELEMENTS AND ALGORITHMS
USED
For the calculation we basically have to model four structural components (1) the surrounding rock material. (2) the excavation material, (3) the reinforced shotcrete lining (outer shell). and (4) the reinforced concrete lining (inner shell). The construction of a tunnel is a time-dependent process. Essentially. realistic calculations can be obtained if one is able to (I) simulate numerically the true time-varying procedure of the excavation and the construction of the supporting structure: (2) accommodate the proper interaction of the various structural components at all time steps, i.e. model complete interelement compatibility within adjacent element groups[2][4]; (3) include the nonlinearily of the material (plastification. etc.): (4) update iteratively the estimated parameters CL, h. and c according to appropriate measurements during construction, then perform new calculations, and revise the final design if required. 4.SYSTEM MODELING The algorithms and elements available in In Fig. 2 all construction states and correspondADINA can accommodate for all four requireing calculation steps are shown: ments The notation used is to (1) the possibilities of the birth and death option is a “clean” method for the simulation of exEGR = element group number, cavation and construction. (i.e. the loading GO = surrounding rock material, consisting of of new structural parts with existing nodal EGR (13), forces released during “older” construction Cl = rock material for the crown excavation, stages): consisting of EGR (7 + 8). to (2) rxchiw application of parabolic isoparaG2 = rock material for the bench excavation metric elements guarantees a priori the reconsisting of EGR (9 + lo), quired interelement compatibility[41; G3 = rock material for the invert segment ex- to (3) the “Drucker-Prager” material law is able cavation consisting of EGR (I 1 + 12). to handle the required stress conditions (including tension cutoff); whereby to (4) due to the fact that (i) all time step solutions are obtained in GX = (1 - a)GX + aGX, X stands for 1, one calculation. 2. and 3. (ii) start and end time for the calculation is (1 - a)GX = three-dimensional support effect of easily prescribed in the input data, the RTS at time t (GX) within the (iii) LI and h values are multiples related to excavation zone X. the material. and a GX = the supporting structure SX (shot(iv) the preparation and revision of the input Crete lining) at time t (a GX) within data is not a time-consuming task. the zone of X. will be loaded with The following finite elements were employed in the internal nodal point forces rethe calculation: leased by (a GX), (1) For the rock Sl = material for the supporting structure Eight-node isoparametric plane elements in the crown range, consisting of (plane strain), EGR (1 + 2), s2 = material for the supporting structure Material 1 in the bench range, consisting of EX elastic moduli, EGR (3 + 4), EV material for the supporting structure s3 = in the invert range, consisting of EX? EGR (5 + 6). Poisson ratios. E.YZ (1 - b).SX + b SX, X stands for 1. sX= 2, and 3.
1
Stability analysis of tunnels using the program ADINA Elastic modulus far the rock Elastic modulus for the shotcrete Specific wei# of the rock Angle of friction cotM?sion Coefficient for actual earth pressure Poisson’s ratio Hardening factor far shotcrete Factor for 3CLsupparting effect
Y t
: +10.359
343 Ev = 200 MN/m2 Eb = 10000 MN/m2 8 = 22 kN/m3 ‘0 = 17. i.=’ = 06735 MN/m2
t:
= OIL = 0,s
a
= 0.15, 0.3. 0.5
Fig. 1. System and input data for the calculation. G a C
Shear modulus, factor for 3D effect, etc., Cohesion, angle of friction,
Y Drucker-Prager -yield surface.
(2) For the shotcrete lining (outer shell) construction state Three-node parabolic isoparametric beam elements, Material b = hardening factor, E = elastic modulus, G = shear modulus, d = shell thickness.
(3) For the shotcrete lining (outer shell) collapse state Three-node parabolic isoparametric rod elements, Material E = elastic modulus, d = shell thickness, only axial forces are transmitted. (4) For the inner shell Three-node parabolic isoparametric ments, Material E = elastic modulus, G = shear modulus, d = shell thickness.
beam ele-
--l-l_-_l_-_.
-.
_... -._----_r_
11”11111----
for crownsettlement
.---_
Displacemeclts.
Comparing AW&ealcuiations with maaruromW?ta according ta conrtructkertates and catcufotlon-gtepr
-.--
P 1
--.-~_--_ll_--
--.,
-l.l...“lll.lll. ~..--“I I--_ _..- ----~ - -
I
_--
_
-----
__..__~.
.__
--^..-
__.~
345
Stability analysis of tunnels using the program ADINA
element
mesh: initial
state
-.
_
Fig. 3. Element mesh: initial state.
-
Y
FRANZ TH. LANCXR
346
k’
element
mesh:
and
crown
KL.AUS STOCKMANN
excavation
I I 50.00
I T
Y -_)
Fig. 4. Element
mesh:
crown excavation
Stability analysis of tunnels using the program ADINA
element
mesh: bench excavation
so,00
Fig. 5. Element mesh: bench excavation.
347
FRANZTH. LANCERand KLAUSSTOCKMANN
348
Z
element
mesh:
invert
excavation
50.00
--
Fig. 6. Element mesh: invert excavation.
Y
1
i
& 1 (I-blS2
51bS3
i
50 -
LO
30
20.
1 1
8oGl 9 Il-olt2
E 5
1
1.
ll-bIS3
6 7
Il-olG1
0
1
31bS2
111-blS1
1 [bSl
2
I I
1
I
I
Comparing ANNA-calculations with different a-Factors according to construction-states and calculation-steps for the moment at Point P 2
\
\ \ \
\
\ \
\ \
\
‘k
\
J-a
a . 0.3
___________
= OS ._._._. _._,_.._._._._._
‘L.J-zL!L?____
‘L
._
._.__.
__________
_.-.-.-.-.-.
Comporing ADINA-colculotions with different o-Factors according to construction-states and calculation-steps far the axial force nt Paint P 7
Stability analysis of tunnels using the program ADINA 6. RESULTS
For the sake of brevity, only a small extract of results is presented. Fig. 1 shows the geometry and tunnel description. Fig. 2 gives the construction states, birth and death timing, element groups, comparison of calculations of crown displacements with on site measured data. Fig. 3 shows the element mesh in its initial state, Fig. 4 gives the element mesh after crown excavation, Fig. 5 shows the element mesh after the bench excavation, Fig. 6 shows the element mesh after the invert excavation, Fig. 7 depicts the bending moments for the shotcrete lining, and finally Fig. 8 shows the axial forces for the shotcrete lining for different a factors.
The above results correspond to a reasonable calculation time. Comparisons of the calculated results with onsite measurements verify that the actual three-dimensional and time-dependent stress conditions can be modeled using the two-dimensional idealization. However, the only means of verifying the assumed parameters a and b, is the comparison of the calculated displacements with existing values on site during the period of construction. Acknowledgment-The authors gratefully acknowledge the kind assistance rendered by Mr. DipI.-Ing. Fr. Schrewe (Head of the Department of Tunnel Construction and project leader for HW-NORD, Construction Headquarters, German Federal Railways) for making on-site measurements available for this study. REFERENCES
7. CONCLUDINGREMARKS
The necessary input data for ADINA has been obtained using a preprocessor @‘REPRO) written in FORTRAN 77. The PREPRO reduces the necessary manual work to a minimum. The calculation was performed on a DEC 2060 Computer system, the CPU times read as follows:
1. ADINA-a
2.
3.
4.
generation computation
of the input data with PREPRO approx. 3 min., of eight time steps with ADINA 81 approx. 85 min.
351
5.
finite element program for automatic dynamic incremental nonlinear analysis. Report AE 811. ADINA Enaineerinn. Seotember. 1981. i. J. Bathe, Fynite Ele&ent’Proced&es in Engineering Analysis. Prentice-Hall, Englewood Cliffs, New Jersey (1982). K. Schikora, Calculation model and measuring results for a double tunnel with low overburden in quatemary soil. Tunnel 3/82, West Germany, p. 153ff. 0. C. Zienkiewicz. The Finite Element Method in Engineering Science. McGraw-Hill, London (1971). K. Wanka and F. Th. Langer, Vorfluterbelastung, iiber zwei neuartige Verfahren zur instationlren Berechnung der FlieSzusttinde in Kanalisationsnetzen. KfK-CAD 148, Kemforschungszentrum Karlsruhe, West Germany, March, 1980.