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Numerical simulation of dolos drop tests
Com~~ers & Struciures Vol. 40, No. 2, pp. 243-255, Printed in Great Britain.
NUMERICAL
US-7~9191 $3.00 + 0.00 Pergamon Press pk
1991
SIMULATION
OF DOLOS DROP TESTS
J. W. TEDE~CO,?B. T. Ro~x~N~ and W. G. MCDOUGALL tCivi1 Engineering Department, Auburn University, Auburn, AL 36849, U.S.A. $Ocean Engineering Program, Civil Engineering Department, Oregon State University, Corvallis, OR 97331. U.S.A. Abstract-A three-dimensional finite element method (FEM) analysis is conducted to simulate doles drop tests. The FEM analysis employs a nonlinear concrete material model and utilizes a contact surface at the base of the vertical fluke. The results of the analysis predict the dynamic states of stress in the dolos and the pattern of cracking in the unit.
To assess the impact strength of the dolos, physical testing of the units has been ~dertaken by Burcharth [IO] and Hall et al. [ 1I]. A schematic representation of the physical doles drop test is shown in Fig. 2. This test is representative of the rigid body rocking mode illustrated in Figure la. In an effort to emulate the drop tests numerically, a finite element method (FEM) analysis 1121, which in~~orated a nonlinear concrete material model and contact surface was conducted. The finite element model is shown in Fig. 3.
BACKGROUND Doles concrete armor units are employed worldwide in the construction of rubble mound structures [I]. Armor layer design for rubble mound structures is presently based on hydrodynamic stability design parameters with little or no consideration given to the structural performance of individual armor units. Reports of recent failures of dolos armored breakwaters attributed in part to dolos breakage [2,3] has prompted a need for the assessment of the structural behavior of dolosse. Dolosse are essentially subjected to three types of loading 141:(1) static, (2) dynamic pulsating, and (3) dynamic impact. Static loads are due to unit selfweight and unit-unit wedging. Dynamic pulsating loads result from wave action on the structure. Dynamic impact loads occur when units undergo rigid body rolling or rocking resulting in unit to unit impacts. Researchers in the field of breakwater design and construction, including Burcharth [S], Zwamborn et al. (61, Tedesco and McDougal[‘7J, and task forces at the Coastal Engineering Research Center (a division of the U.S. Army Corps of Engineers Waterways Experiment Station) [8] have indicated the need for information relating to the st~~ural response of the doles to both static and wave induced loads. Impact loads result from collision with pieces of other units that may be tossed about in severe wave action and from wave induced rigid body motion. This occurs in most breakwater designs when a nonzero displacement criteria is used. Typical breakwater designs allow up to 5% of the units to displace [9J. Two of the rigid body rocking modes that a doles may undergo are illustrated in Fig. 1. Figure la illustrates the dolos pivoting about an axis along the horizontal fluke with the vertical fluke directed seaward. Figure 1b illustrates the dolos pivoting about an axis ~~ndicular to the vertical with the horizontal fluke directed seaward. Often, for large units, minimal rigid body motion can result in unit failure since the mass moment of inertia of the unit is large.
RIGID BODY ROTATION
Lagrange’s equation can be used to determine the equation of motion for a dolos undergoing a rigid body rotation about a given point. The position variable 0 is used to define the angular displacement of the unit with respect to a reference axis as shown in Fig. 4. The governing differential equation of motion I131 is & + k cos 0 = 0,
(1)
where
and m is the mass of the armor unit, g is the acceleration due to the gravity, H is the length of the unit and 1, is the mass moment of inertia of the dolos about the pivot point. The expression for the angular velocity at impact, 0, follows from the equation of motion R = (O,o,O) = [O,(2Wh/Z,)“2,0],
(3)
where h is the vertical drop of the center of gravity, W is the weight of the unit, and w is the only unconstrained angular velocity for this one degree of freefall motion. A dolos undergoing a rigid body rotation and impacting an adjacent unit was simulated by specifying initial nodal point velocities in the FEM model.
243
244
J. w.
-hDESCCl
et al.
(4
Seaward side
/ Armor layers
Fig. 3. Finite element model of doles.
W Seaward side
Fig. 1. Dolos subject to rigid body rotation. (a) vertical fluke
seaward, (b) horizontal fluke seaward. Fig. 4. Definition sketch for dolos rotation.
These initial nodal point velocities are equivalent to the angular velocity at impact, calculated for a drop from a given height. Initial velocities at a given nodal point are determined from the kinematic relationships of circular motion, v=Rxr,
(4)
where v is the angular velocity vector at impact and I is the position vector of the nodal point relative to the pivot point. For a given impact angular velocity,
the resulting nodal point velocities were determined for each node in the FEM model. The direction of motion and boundary conditions in the impact analysis with the vertical fluke seaward is illustrated in Fig. 5 (also refer to Fig. la). In the FEM analyses, the pivot point was modeled as a fixed point restraint. The impacting dolos, or object body, was modeled with a three dimensional finite element mesh consisting of 1934 20-node isoparametric elements. The
OBJECT
BODY
ton doles
- .. .
. . . ;:’ *.
.
.
*
.:.
*.
“‘..*...., . .
.
.’
._ *
,:
eoncrsto
bum
. .
Fig. 2. Schematic of doles drop test.
Fig. 5. Computer model of dolos with vertical fluke forward.
245
Numerical simulation of dolos drop tests
Table 1. Linear concrete material properties 28-day compressive strength Elastic modulus Density Unit weight Poisson’s ratio
uu
Fig. 6. Dolos specifications.
impact surface was modeled with a rigid contact surface. An angular velocity at impact, w, was
applied to the armor unit, and the resulting structural response was determined for a 42-ton dolos. The dolos specifications are presented in Fig. 6, in which A= 3.06ft, B=4.89ft, C= 15.29ft, and D = 0.87 ft. To determine the locations of the peak stress regions in the dolos at impact, a linear constitutive relationship was used to describe the concrete behavior. Once the critical regions of the unit were identified, these regions were modeled with a nonlinear constitutive concrete model. The nonlinear material model allowed for tensile cracking, including postcracking behavior, and compression strain softening. The properties for the linear material model are presented in Table 1, and the properties for the nonlinear material model are presented in Table 2. In all of the impact simulations, the impacted surface was modeled as rigid. This idealization of the impact surface requires that all of the energy generated from the impact be directed into the impacting unit. In reality, some of the energy would be dissipated in the deformation of the impacted surface. therefore, the predicted critical impact velocities yield conservative estimates of the impact resistance of the unit. The rigid impact surface was modeled with a contact surface. The rigid contact model supports the impacting dolos in compression only. That is, no tensile forces are allowed to develop on the contact surface. Numerically this involves a computationally intensive algorithm that insures
5000psi 4,286,826psi 0.00022465lb &/in4 150.0lb/ft3 0.21
that the deflected shape of the impacting surface (dolos) is compatible with the contact surface (which is rigid in this case). This condition also requires that equilibrium iterations be performed at each time step to satisfy the contact surface compatibility requirements. The implementation of a contact surface requires a nonlinear analysis, even if the constitutive material properties are elastic, In addition, the concrete material properties were based on static strength values, therefore the apparent strengthening of the material under dynamic loads was not considered. The nonlinear finite element equations were solved using the Newton-Raphson iteration method. Equilibrium at a time step t + At is obtained through a process of successive displacement approximations. Displacement increment approximations are obtained in each iteration until convergence is reached. The dynamic response was calculated using the Newark method of direct time integration. A lumped mass formulation of the mass matrix was employed in each dynamics analysis. In a dynamic analysis, the selection of the integration time step is important for accuracy and stability in the solution. The critical structural response time was determined to be the time increment required for a stress wave to pass from the point of impact of the fluke to the free end of the fluke and back again to the impact point (i.e. the impact duration time). A stress wave is generated when the dolos impacts an adjacent unit or other body. The resulting stress wave travels through the dolos until it reaches a free boundary from which it reflects and/or refracts. This process is continued until the energy is absorbed by the body and/or transmitted to other bodies. From elementary wave propagation theory, the speed at which a longitudinal stress wave passes
Table 2. Nonlinear concrete material model Parameter
Specified value
Initial tangent modulus (psi) Uniaxial cut-off tensile strength (psi) Uniaxial maximum compressive stress (psi) Uniaxial ultimate compressive stress (psi) Compressive strain at uc Uniaxial ultimate compressive strain Principal stresses in directions 1, 2, 3, respectively (psi) Uniaxial cut-off tensile stress under multiaxial conditions (psi) Uniaxial compressive failure stress under multiaxial conditions (psi)
4,286,826 psi 530 psi 5000 psi 4500 psi 0.002 in/in 0.003 in/in (Fig. 16) (Fig. 16) (Fig. 16)
J. W. TEDFXOet al.
246
through an isotropic, Kolsky [14] as c=
elastic medium
(1-UN -2u)
pE
1fU
[
RESULTS OF FEM ANALYSES
is given by
1’ 1’2
The critical impact velocity for a unit was defined to occur when a drop resulted in the formation of cracks through approximately 30% of the dolos cross-section. Cracking to this extent should render the unit relatively ineffective in resting additional impact loads. In order to ascertain the critical velocity for a particular unit, comprehensive FEM analyses of drop tests on a wide range of armor unit sizes were conducted [13]. The resulting critical velocity data was fit with a quadratic poynomial resulting in the following expression
(5)
where E is Young’s modulus, u is the Poisson’s ratio, and p is the material mass density. In the preliminary impact tests the concrete was modeled as a linear elastic material with the properties given in Table 1. Based on these values, the time required for a stress wave to travel the length along the vertical fluke and back to the point of impact can be calculated from T = 2H/c,
w = 0.6-3.899
x 10-3H + 1.0 x 10-5H2
in which H is the length of the fluke in inches. For the particular case of the 42-ton dolos unit, the critical impact velocity as calculated from eqn (7) was 0.223 rad/sec. This corresponds to a drop height, h, of 0.62 inches (refer to Fig. 4). Before proceeding with the full nonlinear analyses (contact surface and material nonlinearity), a series of anlayses employing a linear material models, with contact surface, were conducted in order to establish the locations of critical (tensile) stress in the unit. Time histories for normal stress at eight locations throughout the unit are illustrated in Figs 7-14. These eight locations are identified in Fig. 3 as points A-H. Close scrutiny of Figs 7-14 indicates that a critical state of (tension) stress occurs in the vertical fluke at
(6)
where H is the fluke height and c is the velocity of a longitudinal stress wave. The structural response of the dolos was determined for the time interval required for the stress wave to travel the length along the vertical fluke and back. The time step selected was based upon the time required for the wave to travel between the integration points of the element. For these analyses, a time step corresponding to approximately one-third of the time required for the wave to travel between the integration points was used [15], At = 6.0 psec.
0.0
-250.0
+ Tension
-750.0
-
1000.0
0.000
-
(7)
Compression
I
I
0.001
0.002
Time (Seconds)
Fig. 7. Stress history response at point A of linear material FEM model.
Numerical simulation of ddos drop tests
-
Compression
300.0 c ::
z
iiie
200.0
5 E ‘0 z 100.0
0.0
-100.0
/ 0.000
0.001
0.002
Time (Seconds)
Fig. 8. Stress history response at point B of linear material FEM model.
0.0
- 100.0
p‘ ::
-200.0
% z iii ;d E t;
z
-3oo.c
I
-4oo.c
) -
-500s
f
Tension
-
Compression
-
)-
0.000
0.001 Time (Seconds)
Fig. 9. Stress history response at point C of bear
material FEM model.
J. W. TEDEXOet al.
248
2000.0
I
1
I
+ Tension -
Comoression
1500.0
CI :: D $
1000.0
a E tj z
500.0
0.0
0.000
0.001
0.002
0.003
0.004
Time (Seconds)
Fig. 10. Stress history response at point D of linear material FEM model.
2000.0
1
I
I 0.001
0.002
I
1500.0
h g
1000.0
E
s VI
a
E $ z
500.0
0.0
’ 0.000
-500.0
1
I 0.003
Time (Seconds)
Fig. Il. Stress history response at point E of linear material FEM model.
0.004
Numerical simulation of dolos drop tests
249
0.0
0.001
0.002
Time (Seconds)
Fig. 12. Stress history response at point F of linear material FEM model.
0.0
-500.0
c 8.
- 1000.0
-2000.0
-2500.0 0.000
0.001
0.002
Time (Seconds)
Fig. 13. Stress history response at point G of linear material FEM model.
J. W.
250
'~DFSCO et al.
0.c
+ Tension -
Compression
- 1500.0
0.000
0.001
0.002
Time (Seconds)
Fig. 14. Stress history response at point H of linear material FEM model
Tension
Compression
-/ %
t-
-_j_L_---___-__ _
Fig. 15. Uniaxial stress-strain
relation used in concrete model.
-u
‘; t
251
Numerical simulation of dolos drop tests
Y,=0
.
T=O
co,:;,z; )
c0,o.z; 1
Fig. 16. Three-dimensional tensile failure envelope of concrete model. alized to take biaxial and triaxial conditions into account. The model employs three basic features to describe the material behavior: (i) a nonlinear stress-strain relation including strain softening to allow for weakening of the material under increasing compressive stresses; (ii) a failure envelope that defines cracking in tension and crushing in compression; and (iii) a strategy to model post-cracking and crushing behaviour of the material. In order to establish the uniaxial stress-strain law accounting for multiaxial stress conditions, an
the fluke-shank juncture (point E) and also at the top of the shank at the fluke-shank juncture (point D). The maximum normal tensile stress at both of these locations is in excess of 1500 psi as illustrated in pigs 10 and 11, for points D and E, respectively. It is at these locations that cracking in the units will be initiated and the material nonlinear analysis was focused. The concrete material model employed in the nonlinear analysis is a hypoelastic model based on a uniaxial stress-strain relation (Fig. 15) that is gener0.0
-250.0
+ Tension
-750.0
-
1000.c
-
I -
0.000
Compression
1
0.001
I
0.002
Time (Seconds)
Fig. 17. Stress history response at point A of nonlinear material FEM model. CAS 40,2-E
J. W.
252 1000.0
~DEWOet
al.
I
I
+ Tension -
Compression
250.0
-250.0
0.000
0.001
0.002
Time (Seconds)
Fig. 18. Stress history response at point D of nonlinear material FEM model.
750.0
500.0
+ Tension -
Compression
250.0
-250.0
-500.0
0.000
0.001
0.002
Time (Seconds)
Fig. 19. Stress history response at point E of nonlinear material FEM model.
Numerical simulation of doles drop
tests
253
0.0
-250.0
A
& :: e $5
-5OCLO
b E
k
z
- 750.0
I
- 1000.0
/
0.002
0.001
0.000
Time
(Seconds)
Fig. 20. Stress history response at point F of nonlinear material FEM model,
-
---.+-----
0.0
20.0
(Point Fig.
0)
Shank
Time Xme
= 0,001380 = 0.002OCO
sbc set
40,o CWSS
Section
(inches)
2 1+ ProlFrlcsfor norrrml stress along the shank cross-section4
60.0
(Paint
A)
J. W. mDEsC0
254
el al.
600.0
.-_-.---
I
-600.0 0.0 (Point
Fluke
= 0.001788 = 0.0021 84
set set
I
40.0
20.0 F)
Time Time
Cross
Section
60.0
(inches)
(Point
E)
Fig. 22. Profiles for normal stress along the fluke cross-section. approximate failure envelope must be employed. Since failure of the unit is tension dominated, the failure envelope depicted in Fig. 16 was used in the concrete model. To identify whether the material has failed, the principal stresses are used to locate the current stress in the failure envelope. These tensile strength of the material in a principal direction does not change with the introduction of tensile stresses in other principal directions. However, the compressive stresses in the other principal directions alter the tensile strength. The pertinent material parameters for the failure envelope and the uniaxial stress-strain relation are summarized in Table 2.
Time histories of the normal stress at points A, D, E, and F are illustrated in Figs 17, 18, 19 and 20, respectively. Profiles for the normal stress across the vertical fluke at the fluke-shank juncture (section E-F) are presented in Fig. 21 for time t = 1.788 and t = 2.184 msec. Similar profiles for the normal stress across the shank at the shank-fluke juncture (section D-A) are illustrated in Fig. 22 for time t = 1.38 and 1 = 2.0 msec. The crack pattern at failure (previously defined as cracking through 30% of the dolos cross-section) is illustrated in Fig. 23. This condition corresponds to a contact velocity of 0.223 rad/sec. The cracks initiate at approximately the same time at locations D and E. However, the cracks in the vertical fluke (section E-F) propagate at a faster rate than those in the shank. Consequently, the failure condition is attained by extensive cracking in the vertical fluke. The pattern of cracking predicted by the numerical analysis is consistent with that observed in the experimental studies of Burcharth [lo]. CONCLUSIONS
Fig. 23. Fracture crack pattern for doles with vertical fluke forward.
The results of a dynamic finite element analysis of dolos drop tests are presented. A 42-ton dolos was modeled using linear material properties, and a threedimensional contact surface; stress histories were obtained at eight locations throughout the unit to determine the locations of critical (tensile) stress. The maximum normal tensile stresses occurred in the
Numerical simulation of dolos drop tests vertical fluke and shank at the shank-fluke interface; these locations indicated where cracking initiates. In
the regions of high tensile stress, nonlinear concrete elements were employed to determine stress intensities and crack patterns. The cracks in the vertical fluke propagate at a faster rate, and are more extensive, than those in the shank. Also, the cracking pattern predicted by the numerical study is consistent with observed experimental studies.
mour unit stresses including specific results related to static and dynamic stresses in Dolosse. Proceedings from seminar on Stresses in Concrete Armor Units, Waterways Experiment Station, Vicksburg, MS (1989). 5. H. F. Burcharth, The lessons from recent breakwater failures. Developments in breakwater design. Proc. 2Oth Internaiional Conference on Coastal Engineering, Taipei, Taiwan (1986). J. A. Zwamborn, D. E. Bosman and J. Moes, Dolosse past, present, future? Coastal Structures 1948-1976 (1980).
work described in this paper was conducted as part of the Crescent City Prototype Dolosse Study of the Coastal Engineering Research Center, Waterways Experimental Station, U.S. Army Corps of Engineers. Permission to publish this paper was granted by the Chief of Engineers. Computational resources for the numerical analyses were provided by the Alabama Supercomputer Network. Acknowledgements-The
10. REFERENCES
11.
I.
H. F. Burcharth, The way ahead. Proceedings, Breakwaters, Design and Construction Conference, Institution of Civil Engineers, published by Thomas Telford, London (1983). 2. B. L. Edge and 0. T. Magoon, A review of recent damages to coastal structures. Coastal Sfructures 333-349 (1979). 3. D. G. Markle and D. D. Davidson, Breakage of concrete armor units; survey of existing corps structures. Miscellaneous paper CERC-84-2 (1984). 4. H. F. Burcharth and L. Zhou, A general discussion of problems related to the determination of concrete ar-
255
12.
13. 14. 15.
J. W. Tedesco and W. G. McDougal, Nonlinear dynamic analysis of concrete armor units. Compur. Slruct. 21, 189-201 (1985). Comments from the Crescent City Prototype Dolosse Study Test Working Group Meeting, Coastal Engineering Research Center, Vicksburg, MS (1988). Shore Protection Manual, Coastal Engineering Research Center, Department of the Army, Waterways Experiment Station, Corps of Engineers, Vicksburg, MS (1984). H. F. Burcharth, Full scale dynamic testing of dolosse to destruction. Coastal Engineering 4, No. 3 (1981). R. L. Hall et al., Drop tests of dolos armor units. Proceedings of 1987 ASCE Structures Congress, Dynamics of Structures (1987). ADINA, A finite element program for Automatic Dynamic Incremental Nonlinear Analysis, Report ARD 87-1, ADINA R & D, Inc., Watertown, MA (1987). P. B. McGill, Analysis of dolos concrete armor units. Ph.D dissertation, Auburn University (1990). H. Kolsky, Stress Waves in Solids. Dover, New York (1963). A. B. Chaudhary and K. J. Bathe, A solution method for static and dynamic analysis of three dimensional contact problems with friction. Comput. Structures 24, 855-873
(1986).
NUMERICAL
US-7~9191 $3.00 + 0.00 Pergamon Press pk
1991
SIMULATION
OF DOLOS DROP TESTS
J. W. TEDE~CO,?B. T. Ro~x~N~ and W. G. MCDOUGALL tCivi1 Engineering Department, Auburn University, Auburn, AL 36849, U.S.A. $Ocean Engineering Program, Civil Engineering Department, Oregon State University, Corvallis, OR 97331. U.S.A. Abstract-A three-dimensional finite element method (FEM) analysis is conducted to simulate doles drop tests. The FEM analysis employs a nonlinear concrete material model and utilizes a contact surface at the base of the vertical fluke. The results of the analysis predict the dynamic states of stress in the dolos and the pattern of cracking in the unit.
To assess the impact strength of the dolos, physical testing of the units has been ~dertaken by Burcharth [IO] and Hall et al. [ 1I]. A schematic representation of the physical doles drop test is shown in Fig. 2. This test is representative of the rigid body rocking mode illustrated in Figure la. In an effort to emulate the drop tests numerically, a finite element method (FEM) analysis 1121, which in~~orated a nonlinear concrete material model and contact surface was conducted. The finite element model is shown in Fig. 3.
BACKGROUND Doles concrete armor units are employed worldwide in the construction of rubble mound structures [I]. Armor layer design for rubble mound structures is presently based on hydrodynamic stability design parameters with little or no consideration given to the structural performance of individual armor units. Reports of recent failures of dolos armored breakwaters attributed in part to dolos breakage [2,3] has prompted a need for the assessment of the structural behavior of dolosse. Dolosse are essentially subjected to three types of loading 141:(1) static, (2) dynamic pulsating, and (3) dynamic impact. Static loads are due to unit selfweight and unit-unit wedging. Dynamic pulsating loads result from wave action on the structure. Dynamic impact loads occur when units undergo rigid body rolling or rocking resulting in unit to unit impacts. Researchers in the field of breakwater design and construction, including Burcharth [S], Zwamborn et al. (61, Tedesco and McDougal[‘7J, and task forces at the Coastal Engineering Research Center (a division of the U.S. Army Corps of Engineers Waterways Experiment Station) [8] have indicated the need for information relating to the st~~ural response of the doles to both static and wave induced loads. Impact loads result from collision with pieces of other units that may be tossed about in severe wave action and from wave induced rigid body motion. This occurs in most breakwater designs when a nonzero displacement criteria is used. Typical breakwater designs allow up to 5% of the units to displace [9J. Two of the rigid body rocking modes that a doles may undergo are illustrated in Fig. 1. Figure la illustrates the dolos pivoting about an axis along the horizontal fluke with the vertical fluke directed seaward. Figure 1b illustrates the dolos pivoting about an axis ~~ndicular to the vertical with the horizontal fluke directed seaward. Often, for large units, minimal rigid body motion can result in unit failure since the mass moment of inertia of the unit is large.
RIGID BODY ROTATION
Lagrange’s equation can be used to determine the equation of motion for a dolos undergoing a rigid body rotation about a given point. The position variable 0 is used to define the angular displacement of the unit with respect to a reference axis as shown in Fig. 4. The governing differential equation of motion I131 is & + k cos 0 = 0,
(1)
where
and m is the mass of the armor unit, g is the acceleration due to the gravity, H is the length of the unit and 1, is the mass moment of inertia of the dolos about the pivot point. The expression for the angular velocity at impact, 0, follows from the equation of motion R = (O,o,O) = [O,(2Wh/Z,)“2,0],
(3)
where h is the vertical drop of the center of gravity, W is the weight of the unit, and w is the only unconstrained angular velocity for this one degree of freefall motion. A dolos undergoing a rigid body rotation and impacting an adjacent unit was simulated by specifying initial nodal point velocities in the FEM model.
243
244
J. w.
-hDESCCl
et al.
(4
Seaward side
/ Armor layers
Fig. 3. Finite element model of doles.
W Seaward side
Fig. 1. Dolos subject to rigid body rotation. (a) vertical fluke
seaward, (b) horizontal fluke seaward. Fig. 4. Definition sketch for dolos rotation.
These initial nodal point velocities are equivalent to the angular velocity at impact, calculated for a drop from a given height. Initial velocities at a given nodal point are determined from the kinematic relationships of circular motion, v=Rxr,
(4)
where v is the angular velocity vector at impact and I is the position vector of the nodal point relative to the pivot point. For a given impact angular velocity,
the resulting nodal point velocities were determined for each node in the FEM model. The direction of motion and boundary conditions in the impact analysis with the vertical fluke seaward is illustrated in Fig. 5 (also refer to Fig. la). In the FEM analyses, the pivot point was modeled as a fixed point restraint. The impacting dolos, or object body, was modeled with a three dimensional finite element mesh consisting of 1934 20-node isoparametric elements. The
OBJECT
BODY
ton doles
- .. .
. . . ;:’ *.
.
.
*
.:.
*.
“‘..*...., . .
.
.’
._ *
,:
eoncrsto
bum
. .
Fig. 2. Schematic of doles drop test.
Fig. 5. Computer model of dolos with vertical fluke forward.
245
Numerical simulation of dolos drop tests
Table 1. Linear concrete material properties 28-day compressive strength Elastic modulus Density Unit weight Poisson’s ratio
uu
Fig. 6. Dolos specifications.
impact surface was modeled with a rigid contact surface. An angular velocity at impact, w, was
applied to the armor unit, and the resulting structural response was determined for a 42-ton dolos. The dolos specifications are presented in Fig. 6, in which A= 3.06ft, B=4.89ft, C= 15.29ft, and D = 0.87 ft. To determine the locations of the peak stress regions in the dolos at impact, a linear constitutive relationship was used to describe the concrete behavior. Once the critical regions of the unit were identified, these regions were modeled with a nonlinear constitutive concrete model. The nonlinear material model allowed for tensile cracking, including postcracking behavior, and compression strain softening. The properties for the linear material model are presented in Table 1, and the properties for the nonlinear material model are presented in Table 2. In all of the impact simulations, the impacted surface was modeled as rigid. This idealization of the impact surface requires that all of the energy generated from the impact be directed into the impacting unit. In reality, some of the energy would be dissipated in the deformation of the impacted surface. therefore, the predicted critical impact velocities yield conservative estimates of the impact resistance of the unit. The rigid impact surface was modeled with a contact surface. The rigid contact model supports the impacting dolos in compression only. That is, no tensile forces are allowed to develop on the contact surface. Numerically this involves a computationally intensive algorithm that insures
5000psi 4,286,826psi 0.00022465lb &/in4 150.0lb/ft3 0.21
that the deflected shape of the impacting surface (dolos) is compatible with the contact surface (which is rigid in this case). This condition also requires that equilibrium iterations be performed at each time step to satisfy the contact surface compatibility requirements. The implementation of a contact surface requires a nonlinear analysis, even if the constitutive material properties are elastic, In addition, the concrete material properties were based on static strength values, therefore the apparent strengthening of the material under dynamic loads was not considered. The nonlinear finite element equations were solved using the Newton-Raphson iteration method. Equilibrium at a time step t + At is obtained through a process of successive displacement approximations. Displacement increment approximations are obtained in each iteration until convergence is reached. The dynamic response was calculated using the Newark method of direct time integration. A lumped mass formulation of the mass matrix was employed in each dynamics analysis. In a dynamic analysis, the selection of the integration time step is important for accuracy and stability in the solution. The critical structural response time was determined to be the time increment required for a stress wave to pass from the point of impact of the fluke to the free end of the fluke and back again to the impact point (i.e. the impact duration time). A stress wave is generated when the dolos impacts an adjacent unit or other body. The resulting stress wave travels through the dolos until it reaches a free boundary from which it reflects and/or refracts. This process is continued until the energy is absorbed by the body and/or transmitted to other bodies. From elementary wave propagation theory, the speed at which a longitudinal stress wave passes
Table 2. Nonlinear concrete material model Parameter
Specified value
Initial tangent modulus (psi) Uniaxial cut-off tensile strength (psi) Uniaxial maximum compressive stress (psi) Uniaxial ultimate compressive stress (psi) Compressive strain at uc Uniaxial ultimate compressive strain Principal stresses in directions 1, 2, 3, respectively (psi) Uniaxial cut-off tensile stress under multiaxial conditions (psi) Uniaxial compressive failure stress under multiaxial conditions (psi)
4,286,826 psi 530 psi 5000 psi 4500 psi 0.002 in/in 0.003 in/in (Fig. 16) (Fig. 16) (Fig. 16)
J. W. TEDFXOet al.
246
through an isotropic, Kolsky [14] as c=
elastic medium
(1-UN -2u)
pE
1fU
[
RESULTS OF FEM ANALYSES
is given by
1’ 1’2
The critical impact velocity for a unit was defined to occur when a drop resulted in the formation of cracks through approximately 30% of the dolos cross-section. Cracking to this extent should render the unit relatively ineffective in resting additional impact loads. In order to ascertain the critical velocity for a particular unit, comprehensive FEM analyses of drop tests on a wide range of armor unit sizes were conducted [13]. The resulting critical velocity data was fit with a quadratic poynomial resulting in the following expression
(5)
where E is Young’s modulus, u is the Poisson’s ratio, and p is the material mass density. In the preliminary impact tests the concrete was modeled as a linear elastic material with the properties given in Table 1. Based on these values, the time required for a stress wave to travel the length along the vertical fluke and back to the point of impact can be calculated from T = 2H/c,
w = 0.6-3.899
x 10-3H + 1.0 x 10-5H2
in which H is the length of the fluke in inches. For the particular case of the 42-ton dolos unit, the critical impact velocity as calculated from eqn (7) was 0.223 rad/sec. This corresponds to a drop height, h, of 0.62 inches (refer to Fig. 4). Before proceeding with the full nonlinear analyses (contact surface and material nonlinearity), a series of anlayses employing a linear material models, with contact surface, were conducted in order to establish the locations of critical (tensile) stress in the unit. Time histories for normal stress at eight locations throughout the unit are illustrated in Figs 7-14. These eight locations are identified in Fig. 3 as points A-H. Close scrutiny of Figs 7-14 indicates that a critical state of (tension) stress occurs in the vertical fluke at
(6)
where H is the fluke height and c is the velocity of a longitudinal stress wave. The structural response of the dolos was determined for the time interval required for the stress wave to travel the length along the vertical fluke and back. The time step selected was based upon the time required for the wave to travel between the integration points of the element. For these analyses, a time step corresponding to approximately one-third of the time required for the wave to travel between the integration points was used [15], At = 6.0 psec.
0.0
-250.0
+ Tension
-750.0
-
1000.0
0.000
-
(7)
Compression
I
I
0.001
0.002
Time (Seconds)
Fig. 7. Stress history response at point A of linear material FEM model.
Numerical simulation of ddos drop tests
-
Compression
300.0 c ::
z
iiie
200.0
5 E ‘0 z 100.0
0.0
-100.0
/ 0.000
0.001
0.002
Time (Seconds)
Fig. 8. Stress history response at point B of linear material FEM model.
0.0
- 100.0
p‘ ::
-200.0
% z iii ;d E t;
z
-3oo.c
I
-4oo.c
) -
-500s
f
Tension
-
Compression
-
)-
0.000
0.001 Time (Seconds)
Fig. 9. Stress history response at point C of bear
material FEM model.
J. W. TEDEXOet al.
248
2000.0
I
1
I
+ Tension -
Comoression
1500.0
CI :: D $
1000.0
a E tj z
500.0
0.0
0.000
0.001
0.002
0.003
0.004
Time (Seconds)
Fig. 10. Stress history response at point D of linear material FEM model.
2000.0
1
I
I 0.001
0.002
I
1500.0
h g
1000.0
E
s VI
a
E $ z
500.0
0.0
’ 0.000
-500.0
1
I 0.003
Time (Seconds)
Fig. Il. Stress history response at point E of linear material FEM model.
0.004
Numerical simulation of dolos drop tests
249
0.0
0.001
0.002
Time (Seconds)
Fig. 12. Stress history response at point F of linear material FEM model.
0.0
-500.0
c 8.
- 1000.0
-2000.0
-2500.0 0.000
0.001
0.002
Time (Seconds)
Fig. 13. Stress history response at point G of linear material FEM model.
J. W.
250
'~DFSCO et al.
0.c
+ Tension -
Compression
- 1500.0
0.000
0.001
0.002
Time (Seconds)
Fig. 14. Stress history response at point H of linear material FEM model
Tension
Compression
-/ %
t-
-_j_L_---___-__ _
Fig. 15. Uniaxial stress-strain
relation used in concrete model.
-u
‘; t
251
Numerical simulation of dolos drop tests
Y,=0
.
T=O
co,:;,z; )
c0,o.z; 1
Fig. 16. Three-dimensional tensile failure envelope of concrete model. alized to take biaxial and triaxial conditions into account. The model employs three basic features to describe the material behavior: (i) a nonlinear stress-strain relation including strain softening to allow for weakening of the material under increasing compressive stresses; (ii) a failure envelope that defines cracking in tension and crushing in compression; and (iii) a strategy to model post-cracking and crushing behaviour of the material. In order to establish the uniaxial stress-strain law accounting for multiaxial stress conditions, an
the fluke-shank juncture (point E) and also at the top of the shank at the fluke-shank juncture (point D). The maximum normal tensile stress at both of these locations is in excess of 1500 psi as illustrated in pigs 10 and 11, for points D and E, respectively. It is at these locations that cracking in the units will be initiated and the material nonlinear analysis was focused. The concrete material model employed in the nonlinear analysis is a hypoelastic model based on a uniaxial stress-strain relation (Fig. 15) that is gener0.0
-250.0
+ Tension
-750.0
-
1000.c
-
I -
0.000
Compression
1
0.001
I
0.002
Time (Seconds)
Fig. 17. Stress history response at point A of nonlinear material FEM model. CAS 40,2-E
J. W.
252 1000.0
~DEWOet
al.
I
I
+ Tension -
Compression
250.0
-250.0
0.000
0.001
0.002
Time (Seconds)
Fig. 18. Stress history response at point D of nonlinear material FEM model.
750.0
500.0
+ Tension -
Compression
250.0
-250.0
-500.0
0.000
0.001
0.002
Time (Seconds)
Fig. 19. Stress history response at point E of nonlinear material FEM model.
Numerical simulation of doles drop
tests
253
0.0
-250.0
A
& :: e $5
-5OCLO
b E
k
z
- 750.0
I
- 1000.0
/
0.002
0.001
0.000
Time
(Seconds)
Fig. 20. Stress history response at point F of nonlinear material FEM model,
-
---.+-----
0.0
20.0
(Point Fig.
0)
Shank
Time Xme
= 0,001380 = 0.002OCO
sbc set
40,o CWSS
Section
(inches)
2 1+ ProlFrlcsfor norrrml stress along the shank cross-section4
60.0
(Paint
A)
J. W. mDEsC0
254
el al.
600.0
.-_-.---
I
-600.0 0.0 (Point
Fluke
= 0.001788 = 0.0021 84
set set
I
40.0
20.0 F)
Time Time
Cross
Section
60.0
(inches)
(Point
E)
Fig. 22. Profiles for normal stress along the fluke cross-section. approximate failure envelope must be employed. Since failure of the unit is tension dominated, the failure envelope depicted in Fig. 16 was used in the concrete model. To identify whether the material has failed, the principal stresses are used to locate the current stress in the failure envelope. These tensile strength of the material in a principal direction does not change with the introduction of tensile stresses in other principal directions. However, the compressive stresses in the other principal directions alter the tensile strength. The pertinent material parameters for the failure envelope and the uniaxial stress-strain relation are summarized in Table 2.
Time histories of the normal stress at points A, D, E, and F are illustrated in Figs 17, 18, 19 and 20, respectively. Profiles for the normal stress across the vertical fluke at the fluke-shank juncture (section E-F) are presented in Fig. 21 for time t = 1.788 and t = 2.184 msec. Similar profiles for the normal stress across the shank at the shank-fluke juncture (section D-A) are illustrated in Fig. 22 for time t = 1.38 and 1 = 2.0 msec. The crack pattern at failure (previously defined as cracking through 30% of the dolos cross-section) is illustrated in Fig. 23. This condition corresponds to a contact velocity of 0.223 rad/sec. The cracks initiate at approximately the same time at locations D and E. However, the cracks in the vertical fluke (section E-F) propagate at a faster rate than those in the shank. Consequently, the failure condition is attained by extensive cracking in the vertical fluke. The pattern of cracking predicted by the numerical analysis is consistent with that observed in the experimental studies of Burcharth [lo]. CONCLUSIONS
Fig. 23. Fracture crack pattern for doles with vertical fluke forward.
The results of a dynamic finite element analysis of dolos drop tests are presented. A 42-ton dolos was modeled using linear material properties, and a threedimensional contact surface; stress histories were obtained at eight locations throughout the unit to determine the locations of critical (tensile) stress. The maximum normal tensile stresses occurred in the
Numerical simulation of dolos drop tests vertical fluke and shank at the shank-fluke interface; these locations indicated where cracking initiates. In
the regions of high tensile stress, nonlinear concrete elements were employed to determine stress intensities and crack patterns. The cracks in the vertical fluke propagate at a faster rate, and are more extensive, than those in the shank. Also, the cracking pattern predicted by the numerical study is consistent with observed experimental studies.
mour unit stresses including specific results related to static and dynamic stresses in Dolosse. Proceedings from seminar on Stresses in Concrete Armor Units, Waterways Experiment Station, Vicksburg, MS (1989). 5. H. F. Burcharth, The lessons from recent breakwater failures. Developments in breakwater design. Proc. 2Oth Internaiional Conference on Coastal Engineering, Taipei, Taiwan (1986). J. A. Zwamborn, D. E. Bosman and J. Moes, Dolosse past, present, future? Coastal Structures 1948-1976 (1980).
work described in this paper was conducted as part of the Crescent City Prototype Dolosse Study of the Coastal Engineering Research Center, Waterways Experimental Station, U.S. Army Corps of Engineers. Permission to publish this paper was granted by the Chief of Engineers. Computational resources for the numerical analyses were provided by the Alabama Supercomputer Network. Acknowledgements-The
10. REFERENCES
11.
I.
H. F. Burcharth, The way ahead. Proceedings, Breakwaters, Design and Construction Conference, Institution of Civil Engineers, published by Thomas Telford, London (1983). 2. B. L. Edge and 0. T. Magoon, A review of recent damages to coastal structures. Coastal Sfructures 333-349 (1979). 3. D. G. Markle and D. D. Davidson, Breakage of concrete armor units; survey of existing corps structures. Miscellaneous paper CERC-84-2 (1984). 4. H. F. Burcharth and L. Zhou, A general discussion of problems related to the determination of concrete ar-
255
12.
13. 14. 15.
J. W. Tedesco and W. G. McDougal, Nonlinear dynamic analysis of concrete armor units. Compur. Slruct. 21, 189-201 (1985). Comments from the Crescent City Prototype Dolosse Study Test Working Group Meeting, Coastal Engineering Research Center, Vicksburg, MS (1988). Shore Protection Manual, Coastal Engineering Research Center, Department of the Army, Waterways Experiment Station, Corps of Engineers, Vicksburg, MS (1984). H. F. Burcharth, Full scale dynamic testing of dolosse to destruction. Coastal Engineering 4, No. 3 (1981). R. L. Hall et al., Drop tests of dolos armor units. Proceedings of 1987 ASCE Structures Congress, Dynamics of Structures (1987). ADINA, A finite element program for Automatic Dynamic Incremental Nonlinear Analysis, Report ARD 87-1, ADINA R & D, Inc., Watertown, MA (1987). P. B. McGill, Analysis of dolos concrete armor units. Ph.D dissertation, Auburn University (1990). H. Kolsky, Stress Waves in Solids. Dover, New York (1963). A. B. Chaudhary and K. J. Bathe, A solution method for static and dynamic analysis of three dimensional contact problems with friction. Comput. Structures 24, 855-873
(1986).