Stress wave propagation in reinforced concrete piles during driving

DEPARTMENT OF CIVIL ENGINEERING

s a È S M S ^ K i ü ü i l ü •-:»s«;wmsxiWKj8^5gKi»' xiBssEaaai

Report 5-83-21

Stress wave propagation

In reinforced concrete

plies during driving

Dr.-Ing. Nils F. Zorn

STEVIN LABORATORY

CONCRETE STRUCTURES

IX^'

Delft University of Technology Department of

Civil Engineering

Report No. 5-83-21 December 1983

Stress wave propagation in reinforced concrete piles during driving

ct

Dr.-Ing. Nils F. Zorn

Mail ing address:

Technische Hogeschool Delft Vakgroep Betonconstructies Stevinlaboratorium II Stevinweg 4 2628 CN Delft The Netherlands Technische Hogeschool ylp*- Bibliotheek

Afdeling: Civiele Techniek

bóOO GA Delft

-^J • N » ^ « ^ *

ACKNOWLEDGEMENT

This report was prepared during a NATO research fellowship of the author to Delft University of Technology. The financial aid provided by the DAAD, Bonn FRG, and the encouragement of the head of the concrete section of

Stevin Laboratory, Prof.Dr.-Ing. H.W. Reinhardt, is gratefully acknowledged. The author would also like to thank Ir. J.A. den Uijl , Stevinlaboratory

II, for the valuable comments he made during discussions with him.

No part of this report may be published without written permission of the author.

CONTENTS

Summary

1. INTRODUCTION

2. WAVE PROPAGATION IN PILES

2.1 Introduction to one dimensional wave 2.2 Example pile and stress wave duration

3. STRESS WAVE GENERATION 3.1 General remarks 3.2 Analytical models

3.2.1 Direct impact of a rigid mass

3.2.2 Impact on a linear elastic pile cap 3.2.3 Impact of rods with equal impedance 3.3 Analysed stress waves

4. STRESS WAVE BEHAVIOUR AT PILE END

4.1 General remarks

4.2 Analytical models 4.2.1 Free end condition 4.2.2 Fixed end condition

4.2.3 Rigid-plastic soil behaviour 4.2.4 Linear-elastic soil behaviour 4.2.5 Viscous soil behaviour

4.3 Concluding remarks to wave reflection

5. REFLECTED TENSILE WAVES IN PILES

5.1 Concrete tensile failure, spall ing 5.2 Reinforced and prestressed piles 5.3 Postcracking behaviour

5.4 Prestressed concrete piles

6. EFFECT OF EXISTING CRACKS

SUMMARY

The aim of this report is to give an insight into what happens in a rein-forced concrete pile during driving, and to contribute to design require-ments for the loading case pile driving. Since the behaviour of stress waves in piles is influenced by a various number of parameters, the fewest of which result in linear relations, models are presented to calculate the initial stress wave, the reflection at the pile tip and the behaviour at a crack in the concrete. These models lead to differential equations that can be solved analytically for assumed linear material behaviour. This allows to follow the stress wave in the pile to every point and time of special interest. Case studies for different wave shapes are presented and the influence of nonlinearities discussed. Nonlinear material behaviour may be introduced, however this makes numerical solutions necessary and requires quantitative knowledge of the different parameters.

Special interest is given to reflected tensile waves that may lead to cracks in the concrete. Different crack criteria are discussed and the steel stress in the reinforcement crossing the crack is calculated. The necessary amount of reinforcement then can be evaluated.

INTRODUCTION

Foundation piles generally once in final position only experience compres-sive stresses, however before they reach their final position they have to

be handled and driven into the soil. During driving rather high tensile stresses can occur and produce cracks in the concrete that may lead to corrosion of the reinforcing steel, and significantly reduce the load bearing capacity. A high number of different integrity tests and models to derive the load bearing capacity have been developed but few interest was given to the circumstances that may cause concrete cracking during driving and how high stresses are induced into the reinforcing steel crossing the crack. An approach to analyse the^e phenomena is presented in this study beginning with a short introduction to one dimensional wave theory.

WAVE PROPAGATION IN PILES

Introduction to one dimensional wave theory

Stress waves are transmitted through piles when different surfaces are not in equilibrium, i.e. when one end is struck by a drop weight and the other end is stress free and at rest. The information that a force is applied to one end is transferred through the pile with a speed that is characteristic for the elasticity and density of the pile material, the wave speed. The acting force also induces movements of material particles, the particle speed, which is a function of the stress level and therefore differs from the wave speed. The one dimensional wave equation can be quickly derived from equilibrium conditions for an element dx cut out of a rod or pile with cross-section A (Fig. 1) that is subjected to an initial stress a at one end.

A.o

A(o.i^.dx)

5x

dx

+

Fig. 1 Rod element dx, one dimensional forces,

The inertia force of the element resisting to the applied force is

f . dx . A = A

dx

u being the displacement in x direction, p the density of the material Thus follows

6x

6^u

(2)

E • t "'

and differentiating it follows

^ = E ^ (4)

óx ^ • 5x2 ^'+^

which can be used in (2)

r

Better known in following form as wave equation

Ó^U 2 52u

where c = J— is the longitudinal wave speed.

The complete solution to (6) for the displacement u is

u = f(x-ct) + F(x+ct) (7)

in which f and F are independent functions representing waves travelling

in opposite directions. The displacement therefore generally results from

the superposition of two waves, f and F, one of which may be zero.

In linear problems the superposition of results can take place in every

stage of the calculations, so it is common to analyse the displacement

velocity or accelleration of two waves seperately, and superpose the

results.

The force acting in a cross-section due to a propagating wave can be

evaluated as follows

c A C

i\ Ü ^^

A.E 5u A.E

F = A.E e = A . E ^ = — . ^ = — . V (8)

where v is the particle velocity and c the wave speed. The factor AE/c

is referred to as impendance of the bar denoted ? in the following

It can be interpreted as a measure for the mass of the pile being crossed in a unit time by the propagating wave.

In general pile wave analysis is performed using only one dimensional wave theory, ignoring the dispersion of the wave due to lateral inertia forces. A comparison with approximate equations taking into account the transverse radial motion by Love and Rayleigh [2] and very elaborate exact solutions by Pochhamrer Chree [1] shows that for the common wave lengths and pile geometries this results in an error of below 5% and therefore can be accepted.

2.2 Example pile and stress wave duration

The analysis in this study is performed for an example pile with the following properties:

A = 0.4 . 0.4 m = 0.16 m^ concrete cross section

c _3

A = 20 izi 8 = 1.005 . 10 m^ steel cross section 9

E = 45 . 10 N/m^ dynamic modulus of e l a s t i c i t y concrete Eg = 0.21 10^^ N/m2 modulus of e l a s t i c i t y steel a = 5 5 N/mm^ Oj = 5 N/mm^ dynamic value (F -j. = 800 kN) p = 2400 kg/m^ p = 7800 kg/m^ c^ = 4330 m/s c = 5188 m/s 1 C7 in6 Ns Z = 1.57 10 -— c = 0.04 lO" — s m n = - i = 0.024 ^c

The value of n represents the relation between the dynamic force trans-mitted in the reinforcing steel and in the concrete. Since only 2.4% of

the force is carried by the reinforcement in an uncracked pile the steel contribution to the pile properties may be neglected in this case.

It is further assumed that there is no skin friction acting, this may later be included in the model.

The stress wave used in the analysis is assumed to have the following properties:

c = 4330 m/s T = 10 ms

A = c.T = 43.3 m

The wave length and the assumption that no superposition of initial stress wave and reflected stress wave takes place at the pile top - this can

result in amplification of the stress at the top by a factor of 2.135 [21] - requires the pile to have a length L

L i I = 21.65 m

r

The maximum amplitude of the striking force F is assume to be

FQ = 2500 KN a = 15.6N/mm2

which will be used for the different wave shapes analysed. For triangular wave shapes it is assumed that the value of F is reached by the wave during the given duration.

When compressive waves are reflected as tensile waves it is of interest to know the relation between the maximum amplitude of the reflected tensile force and the ultimate tensile force of the concrete cross-section.

"^0 2500 , .^r

^ = - r = -8üïï= ^-^2^

tu

So the r e f l e c t e d maximum t e n s i l e amplitude exceeds the t e n s i l e strength of the concrete cross-section by a f a c t o r of 3.125.

3. STRESS WAVE GENERATION

3.1 General remarks

Generally the driving stress wave can be calculated as the result of the impact of a rigid mass on the pile top. The duration and shape of the ini-tial stress wave can be modified in various manners to obtain optimal per-formance of the piling apparatus. This however is not within the scope of this study and therefore only three cases are considereded: the impact of a mass directly to the pile top, the impact to an elastic cap on the pile

top and the impact of two rods with equal impedance.

For other more special cases the generated wave shape is generally known and can be directly used in the following models for wave reflection etc.

3.2 Analytical models

3.2.1 Direct impact of a rigid mass

Fig. 2 shows the impact of a rigid mass m to an elastic pile, impedance ?, with an initial velocity v . The equilibrium equation is

dt

Vo

^

Fig. 2 Rigid mass impacting pile

Resulting in the following integral equation that can be integrated noting that V = V at t = 0

0

m

0 0

for the velocity v at the top of the pile. This results in an initial force

F^ = V . ? = VQ . ? .e

^ t m

which is valid for the case of no interaction between striking and reflected wave force, since no reference to the boundary conditions at the pile tip

is made.

3.2.2 I"rract on a linear elastic vile cap

Equilibrium for the system shown in fig. 3 leads to the following equations

(13) m.y + k(y-x) = 0

^A + k(y-x) = 0

resulting in the following differential equation for the displacement of the pile top

X - ii X + - X = 0

? m (14)

Fig. 3 Rigid mass impacting pile with cap.

Solving this for the initial conditions ^(o) = v = X{o)-^ x(o) and )<(o)=0 this results in

, V .k

-at 0 • J.

X = e . s i n ÜJ t

^ .0)

with a=-^ , 6 2 = - ^ and u ^ J F ^ , given 6^^ a^.

The induced stress wave force then has the following form

V . k _ . _ .

F = r.-k= . e '^ . s i n t ü t = F .e '^ . s i n cot

^ 0 ) 0

(15)

(16)

3.2.3 Impact of rods with equal impedance

A rectangular stress wave force can be induced if striking body and pile have the same impedance.

F. = F = const = v^.i;

1 0 0

The duration of the stress wave force being t = 2L/c,

In addition to these three wave shapes introduced other geometrical shape, saw tooth and triangle will be used evaluating possible tensile waves in the following study.

3.3 Analysed stress waves

Since very different stress wave shapes can be induced depending on the driving apparatus and the nonlinear behaviour of the pile cap more than the three analytically derived shapes will be used in the following

analy-sis. The shapes of the initial stress waves, respectively their resulting forces are shown as a function of time in fig. 4.

a)

-h

b)

T

c)

Fig. 4 Initial wave shapes F..

They include a) F.=FQ=const, b) F . ^ F ^ d - q t ) , c) F.=qt, d) F.=F .e'

4 STRESS WAVE BEHAVIOUR AT PILE END

4.1 General remarks

The transmission and reflection of an initial stress wave at the end of the pile is of special interest for the following analysis. Since the exact

boundary conditions are seldom known several approaches to model the soil behaviour are presented and it is advised to apply the model that fits best to the special conditions. This may in case of stiff soil and slow motion be the dashpot model, and for soft soil and large movements of the pile be the spring model. In each case the conditions vary between a free and a. fixed end of the pile which are both rather unrealistic but used for demon-stration.

4.2 Analytical models

4.2.1 Free end condition

A free end boundary condition for the initial stress wave, i.e. its Force F., reaching the pile end would mean reaching an airvoid. However the case of passing through the bottom of a rather stiff soil layer above a very • weak layer may get close to the free end condition. The physical boundary

condition for a free end is no resulting stress at this surface. This leads to the reflection of a compressive wave F- as a tensile wave of same amplitude and shape.

F, = - F. (18)

Since the particle velocity in a tensile wave is in opposite direction to the wave propagation the particle speed reaches twice the initial value at the free surface.

4.2.2 Fixed end condition

This can only occur reaching bedrock of infinite stiffness. It however approximately often happens in Scandinavia where piles are always driven to base rock even if this means pile lengths of 100 m [3]. The physical condition for a fixed boundary is zero velocity at the pile end. This leads to the reflection of a wave of same size and shape.

F = F.

A result of which is that the stress amplitude is doubled at the fixed sur-face due to superposition of the initial and reflected waves.

4.2.3

Rigid-plastic soil behaviour

A mixture of the two first boundary conditions is assuming that the soil behaviour can be modelled as rigid-perfectly plastic, a model commonly used in pile driving. The idea is that the pile will only penetrate into the soil when the force exceeds the soil resistance. This means that only forces above this level really drive the pile deeper into the soil.

So as long as the surface stress of the pile end doesnot exceed a - this means a-iO.Sa (see 4.2.2) the soil creates a fixed end condition. The stress component exceeding this value experiences a free-end condition and the reflected stress wave results from the superposition. Given an initial compressive wave the resulting stress in the reflected wave will be tensile

only if

a;^>a^-A/^- Fo

- A ^

"U

U

F i g . 5 shows the r e f l e c t i o n of a t r i a n g u l a r stress wave f o r these assump-t i o n s

Linear-elastic soil behaviour

Another possibility in representing the soil reaction at the pile tip is to assume linear elastic behaviour. This is a rather gross assumption for stress wave analysis since it is not capable of modelling the high frequen-cy behaviour but is a possibility to model the soil reaction. Fig. 6 shows the system in which the stress waves are represented by their result-ing Forces F, the index i and r stand initial and reflected respectively. tively.

-A-E q u i l i b i r u m at the p i l e t i p leads to

F^(t) + F . ( t ) = Fg(t) (20) r

Inserting the following expressions for the forces F r F. 1 c(x-y) ex K3.y (21) F i g . 6 P i l e - s o i l model f o r 1inear e l a s t i c s o i l .

r e s u l t s in the f o l l o w i n g d i f f e r e n t i a l equation f o r the displacement of the p i l e t i p

K_

y + — y = 2x

which has initial conditions y(o) = 0 and ^(o) = x(0)

(22)

This differential equation can be solved analytically for the different initial wave shapes choosen for this study.

Case A rectangular wave F. = F

^ 1 0 2 F. _{"^s t 2 F^} — . t 0 e ? + -Ü— K F, = F ^ ( 1 - 2 e " T ^ ) (23) (24)

Fig. 7 shows the shape of the reflected wave including the extreme values for K . The time for F =0 is evaluated from

t(F^=0)= ^ . In2 (25)

-F.

Ks = co

Fig. 7 Reflection of initial wave F.=F =const. , 1 0

Case B F.=q.t triangular wave

A .t

K^. (26)

K

F, = 2q X (e- f

'

Fig. 8 shows the reflected wave.

o

-Fo

;=10 N/mirock)

t

The properties of the tensile reflection can be evaluated as follows t(F ) . ? . In 2 ^ r max & K s F ^ = q • T^ (1 - ""n 2) r max ® K K

--it

\

and t(F =0) iteratively from e z, - l + ^ r r - t ^ O

T^

i t

F = F 4 (1 + - ^ ^

r 0 V 6-a

..) e-«^ - J l _ e - ^ ^

The bounding values for K=0 and K=cxDare -F. and F. respectively. Fig. 9 shows the reflected wave for a striking mass of m=3000kg. Note that the duration of the tensile spike depends on the values of a and 6.

Fo

Ks=10^N/m{rock)

— t

- a t

Case D F. = F . e " "^ ^ s i n u j t s e t t i n g e = — and N = (2a6 - a^

2F„.u _ 2F^(a-B) . 2F u

0 -St 0 -at . . 0 -at y = - ^TTi— e "^ + —e-ïi e sinwt + ^r-^i- e ^ sinwt

?.N

?.N

C.N

- 6^ - u)2)

F =F .e r o •at . . r. 2a(a-B)^2Lü^l ^ ^^^ sinwt [ 1 + — f. J + coscj 2F^Sw o N •Bt

t[

_{— n}

_^}

Fo^

Ks = 10 N/m (rock)

t

at

Fig. 10 Reflection of F^=F .e sinut

4.2.5 Viscons soil behaviour

The shortcoming of the spring soil representation, being valid only for ^ low frequencies and in addition even knowing a modulus of elasticity

-which effective length is to be used to calculate the spring constant, can be overcome using a dashpot to model the soil behaviour. The dashpot can directly model the impedance of the soil. This however means that the interface between pile and soil is interpreted as an impedance change only. The model used is shown in fig. 11.

Equilibrium conditions lead to

2F.

which r e s u l t s i n F = F. {1 • ^ -r 1 c+b,

T.

Fi

T.

y/////////y^

A

Fig. 11 Pile-soil model for viscous soil behaviour.

(35)

The reflected wave represents the initial wave however with a rescaled amplitude. In case of impedance matching (b =^) no wave is reflected. T free and fixed surface boundary conditions can also be modelled setting b =0 and CC respectively. This model cannot represent low frequency eff i.e. the slow movement of the pile deeper into the soil.

4.3 Concluding remarks to wave reflections

As the examples presented show tensile waves may result from wave reflec-tion at the pile end in many ways. The amplitude and durareflec-tion of the ten-sile waves depend very much on the model adopted and the properties used. The spring model always beginns with a tensile reflection since the

resistance has to be built up by displacement. The dashpot model assumes there is only an impedance change between pile and soil, no relative dis-placement is possible. Both models have shortcomings in modelling the reality, and cause considerable problems when evaluating the stiffness and

impedance properties respectively. If the conditions are well known it

however is no problem to introduce stress dependent nonlinearities in the time domain. It however may generally be sufficient to assume rigid plastic soil behaviour to evaluate the amplitude and shape of a reflected tensile wave. In order not to make the following analysis to complicated the free end condition is adopted for the following analysis.

5 REFLECTED TENSILE WAVES IN PILES

5.1 Concrete tensile failure, spall ing

Generally the crack or failure criteria for concrete will be the exceedence of the tensile material strength. But in contrast to static loading cases when analysing stress waves it is important to bear in mind that cracking takes time. The sole exceedence of a stress level may therefore not be a sufficient criteria. The failure of a concrete cross section can be described in four steps [4]:

i) generation of a large number of microcracks at different locations i.e. the randomly distributed regions of low strength

ii) growth of these microcracks

iii) connection or conjunction of adjacent microcracks

iv) formation of a continuous fracture plane through the cross section - spal1 ing

Each of these steps requires a finite amount of time, so that a single tensile spike exceeding the material strength will not necessarily cause complete failure if the duration is short enough. So for not monotonie or generally randomly shaped stress waves an additional criteria must be introduced taking the energy or the duration into account.

This would also apply for the reflected tensile spike shown in Fig. 7. For the case of rather monotonie stress waves the strength exceedence criteria however can be used since once cracking has begun a lower stress level can complete it to fracture. This case can also be veryfied by observations and experiments quoted in [2].

One example is the cracking and displacement of a ryolite rock above a nuclear blast in 275m depth. The spall that resulted from exceedence of the low tensile strength of 2.07 N/mm^ by the reflected tensile wave rose 22.8cm. Approximating the wave shape as a saw tooth, scaling the maximum amplitude according to the accelerometer measurements and analysing the stresses results in a 64m thick spall that would rise 25.4cm against gravity and fall back.

A second experiment [5] refers directly to concrete subjected to stress waves. A concrete bar is subjected to a blast load at one end that

produces an approximately vertical wave front. Upon reaching the free end the compressive wave is reflected as a tensile wave and when the resulting stress exceeded the tensile strength a crack was formed. This produces a new free surface for the still propagating wave in the bar and the spall as a result of the momentum trapped in it displaces with a velocity of 5.5 m/s in horizontal direction from the rod. This process is repeated until the failure criteria cannot be reached by the reflected tensile wave. The last spall had a velocity of 0.61 m/s. Knowing the spall

velocity spall lengths and the failure criteria the nonlinear wave front of the initial wave could be constructed (fig. 12)

2C42L32L2 2 L i

Fig. 12 Wave shape, spall lengths.

This case of multiple spall ing will be analysed for a saw tooth wave

reflected at a free surface. Fig. 13. It is assumed that the wave has a ^; maximum amplitude of n-F^^j, F^^ being the tensile strength of the material (nsl), the velocity c and the length L. At time t, = ^J=— after

I 2n.c

reaching the free surface the failure criteria - resulting force equal to tensile strength is reached and the first crack produced.

The spall has a length of L.=t..c=p- and a momentum resulting to

1 1 2n.c •1 "1

^4P--^\f^

Fo="Ft

Ft

Fig. 13 Saw tooth wave reflection and pile failure,

Neglecting the time necessary to develop failure the rest of the wave will at t=t^ find a new free surface and be reflected there. So again after At=t^ or counting from t=0, after t 2 = 2 t ^ = ^ the failure criteria is reached again and a second spall produced. Its momentum results to

^2 =

(2n-3)L

2n.c (37)

This procedure can be applied to every wave shape reflected from the pile tip, it however for nonlinear wave shapes becomes more complicated. In every case it is necessary to evaluate the location in the pile where the failure criteria is reached, then the transferred momentum into the spall can be calculated using

Li

I = m.v - ^ r F (x) dx (38)

O

for the instant the spall is produced.

A very simple and also extreme case is the rectangular wave reflected from a free end. The failure criteria is reached after t^ = p— when the total momentum is transferred into the spall. It results to

I = ^ . F^ (39)

which is considerably larger thag I. (36) for the saw tooth shaped wave. Reviewing the other initial waves used it can be qualitatively stated that the triangular wave will exceed the failure criteria only once if ever, the exponential wave will show similar results as the saw tooth wave but due to the nonlinear amplitude decay have reduced spall thicknesses and the exponential harmonic wave will be similar to the triangular wave in the beginning and after the peak is reflected act similar to the saw tooth.

Like in the presented examples the shape and behaviour of the trapped wave in the spall is not further considered. It is important to state clearly: after the fracture occurs the system is changed, the relation between pile diameter and wave length is changed. This means that one dimensional wave theory is not applicable any more. The effect of the lateral inertia, i.e. dispersion of the wave would have to be considered. This requires a frequency analysis of the trapped wave and results in different velocities for different wave lengths. The transferred momentum cannot disperse, it can only be altered by forces acting for finite dura-tions and therefore is chosen as characteristic and physically important parameter of the spall.

5.2 Reinforced and prestressed piles

Now when analysing reinforced concrete piles the failure criteria for the concrete cross-section may be exceeded and a crack produced but the displacement due to the trapped momentum is restricted by the reinforce-ment. So the combination of trapped momentum and restricted movement will produce forces that act in the reinforcement and alter the momentum, e.g. reduce it to zero.

In case of prestressed concrete piles the failure criteria must take the prestressing into account, i.e. include it. The failure criteria then is that the resulting tensile force exceeds the sum of concrete strength and prestressing force. Reviewing the cracking of the pile it becomes necessary to consider the bond between the prestressing steel and the concrete. If it is not assumed to be perfect the cracking produces two free surfaces and releases prestressing force as an unloading pulse. The situation is shown in fig. 14. The shaded concrete stress is released and travels as an unloading unit pulse. The time and amplitude of this pulse cannot be easily determined since the stress travels with the same speed as the information 'a crack occurred'. The only description that can be given is the integral, so the pulse can be referred to as a Heavyside step function.

crQckplone

Or

0

Tr-c-s

Fig. 14 Concrete and bond stress at crack of prestressed pile,

Due to the short duration the second failure criteria, referring to an energy content of duration must be applied to this pulse. Since the stress amplitude will be higher than the failure criteria on the one hand and continuous cracking of prestressed piles has not be noticed on the other hand, this shows how important the second failure criteria is for randomly shaped waves.

5.3 Postcracking behaviour

As soon as the resulting force of a stress wave exceeds the failure

criteria two coupled systems are the result for prestressed or reinforced concrete piles: the spall and the rest of the pile. They are generally connected by steel reinforcement, fig. 15.

Fig. 15 Coupled system - pile rest and spall.

These systems will be analysed for reinforced concrete piles, the effects of prestressing will be considered later, as done for crack generation.

For simplicity a triangular initial wave is assumed and a free surface at the pile tip, the example pile and an example stress wave T=5ms are used. The failure criteria for this wave is reached only once, for

L4

t. = —K-— = 3.3 ms (40)

this corresponds to a distance from the pile tip.

*-$ " '^1^ " 14.289 m

The stress distribution is shown in Fig. 16

KP^

The momentum of the spall is equal to the total wave momentum trapped at the instant the failure criteria is exceeded.

I = /mvdx = 10890 Ns

The mean velocity of the spall can now be easily calculated knowing the length

- ^ 10890 Ns

'o 5486.9Kg 1.98 m/s

This is the mean velocity of the pile being driven into the ground for

r

the assumed conditions. The system of interest in this case however is the spall connected with reinforcement to the pile, or vice versa (Fig. 15).

Depending on assumed boundary conditions it is a spall moving away from the pile that is being held back by the soil or - in case of a crack above the soil during driving^a pile - rest being pulled down by the spall.

Not knowing how long the pile is used for this example it is assumed that the pile rest can be modelled as a fixed boundary condition. This first approach allows to use the system shown in fig. 17 for analysis.

Vo

/.

MA

m

Using this simple system and assuming linear elastic behaviour of the reinforcement results in the following expression for the steel strain e , denoting the displacement with x

max

max

m.l

ETA

max

^o\|on-= 0-0^

m being the mass of the spall, EA the properties of the reinforcement and 1. the free length of the steel.

This value initially is almost z^ro and then grows according to the force - bond slip relation of the reinforcement. Assuming a constant free length of 0.2m results in a stress of 4740 N/mm^ for the example pile reinforce-ment. It must be noted, that this model ignores the fact, that the wave that crossed the section cracked later also induces a particle velocity. However this is of short duration compared with the time necessary to reach the maximum force in the steel. Assuming l.=0.2m the maximum force is reached after 3.6ms - a period in which the wave travels 15.5m.

Neglecting the influence of skin friction for the moving part of the pile in addition to the assumption of a free surface boundary condition is considered a too simple and unrealistic assumption. This shortcoming however can be overcome by introducing viscous damping. This results in a frequency reduction and an exponential amplitude decay. Assuming a critical damping the maximum steel stress for l.=0.2m then becomes 1744 N,

The assumption of critical damping means that there will be no harmonic oscillation, the mass moves away from the original position due to initia velocity and back to the old position (Fig. 18). A behaviour of the

spall of this sort can be expected in reality.

The triangular wave shape is rather artifical for pile driving problems, and in addition results in a very long spall which means high momentum due to the involved mass. It however has the advantage that only one craci will occur.

-1.0

-ÖMAt.B)

Fig. 18 Displacement time history undamped and critical damping.

A more realistic initial wave shape is the exponential wave F.=F .e " that will be analysed next. This wave similar in its shape to the saw tooth wave will produce more than one spall upon being reflected from a free surface, however due to the nonlinear shape the spalls will have different thicknesses. The failure criteria is reached for the first time for

'^1^0 -0

n (42)

At this instant the wave front of the reflected tensile wave exceeds the initial wave by the value of the failure criteria

1

t = h - . In n = 1.02ms

a

The length of the spall is L. = c.t. = 4.45 m And the momentum of the spall results from

I = -^^^^ F e'^^^dt = 1948 Ns

C 0

The mean spall velocity v

v- = i

I

Using the single degree of freedom (SDOF) system again with a free length 1. results in

e - 3.24 10"-^ r max 1 .

The steel stress for a free length l.=0.2m then for the undamped case is

Og^ - 1523 N/mm2

Assuming critical damping leads to a reduction of the steel stress to

a^^ = 560 N/mm^

The corresponding values for 1 .=0.4m are a^=1077 and a =396 N/mm^ respectively. The latter value is within the elastic range of reinforcing steel, the resulting force is equal to 49% of the assumed failure force for the concrete.

It however must be rememberred that this is valid only for the assumption of critical damping - for an undamped system the resulting force is 1.35 times the assumed failure force.

5.4 Prestressed concrete piles

Generally, prestressed concrete behaves like reinforced concrete subjected to an additional load. In the case of piles this leads to an increase of the tensile failure criteria of the size of the prestressing force. How-ever, after the system is changed by cracking the equilibrium of

pre-stressing force in the steel and the concrete is not given at the crack surface. This means that the simple model of free vibration of a SDOF system is not applicable for prestressed piles. The then constantly acting prestressing force has to be considered as external load. This will reduce the maximum displacement of the spall and the hereby induced steel stress but on the other hand the steel is not unstressed when the failure

occurs. For better comparison the same values and criteria are used as for the reinforced concrete piles, however it is assumed that the failure criteria for tensile loading is produced by a concrete tensile strength of 2 N/mm^ and 3 N/mm^ prestressing.

-at

For F.=F .e the inital conditions are the same: 1 0

V = 1.14 m/s m = 1709 kg

The active prestressing force in the crack plane can be modelled by a constant force

F =480 kN P

The spall now no longer resembles a SDOF in free motion, since the constant force F is applied. The solution to the differential equation of motion is:

V F F

X = — s i n w t + - 2 - coscüt ^ (43)

0) c c

in which c is the spring constant of the reinforcement andw=J—. The

^ im

maximum displacement and resulting stress and strain are evaluated for a value of 1 . = 0.2m

S a x = 5.32 10-3

S a x = 1 1 1 9 N/mm^

Assuming critical damping for this case leads to a reduction of the resulting maximum steel stress to

S a x - 457 N/mm^

Considering the initial stress in the steel the maximum total steel stress in the last case is

a^Q^ = 935 N/mm^

which is a value below the yield stress of prestressing steel. The pre-stressing force however may be choosen higher which leads to an increase of the trapped momentum and mass, to a reduction of w and therefore to an

A further refinement of the model is to evaluate the bond-slip relation-ship between steel and concrete realistically. This makes it necessary to solve the equation of motion numerically since the free length of the steel 1. will then be a function of the force and bond. A qualitative result of this effect is that the force-time behaviour will show a steeper rise in the beginning since the effective spring constant is much higher. Over all it may lead to a reduction of the maximum steel stress compared to the assumption of a fixed free length. Exact numerical results may be obtained in a later report.

EFFECT OF EXISTING CRACKS

After analysing how a crack is produced and how high the steel stress is during the propagation of the crack initiating wave, it is of interest what this crack means for following stress waves.

It is assumed that the reinforcing or prestressing steel crossing the crack was only elasticly stressed, the two crack surfaces are in contact,

i.e. the permanent crack width is zero.

This means a crack will not influence a compressive wave in which particle speed and wave propagation have the same direction. In case of a tensile wave an existing crack defines the weakest link and will open for a resulting tensile force and then change the system producing a free

surface condition. So in general the same thing happens as in the case of crack initiation, but the crack defines the geometrical properties, not only the shape of the initial wave. So if the resulting force is tensile in a cracked cross-section only the mean velocity of the spall has to be calculated, the mass and the length of the spall is already defined. In case of prestressed piles the criteria to activate the crack is that the resulting force is higher than the prestressing force.

For the case that one dimensional wave theory is applicable also to the spall i.e. if the boundary conditions do not influence the behaviour of the separated parts, and the wave propagation is stationary, the lineary elas-tic splice model [3] may be used (Fig. 1 9 ) .

X y

Ft

Fr

Fig. 19 Linear elastic splice.

Equilibrium requires

F + F. = F,

r 1 T

which leads to the differential equation

y + f ^ = f . F. (44)

Solving this differential equation for the boundary conditions y(0)= ^(0)=0 leads to

2K

F^ = F.(1-e"T ^) (45)

Note again the force F-j. has to be built up by displacement, it initially is zero and the equilibrium condition does not consider any inertia force as essential to model the displacement of a spall. For a stationary

F.=F the transmitted force in the spring and on the other side of the crack will reach the initial value F rather quickly.

LITERATURE

1. Goldsmith, W.

"Impact, the theory and physical behaviour of colliding solids", Edward Arnold,London

2. Johnson, W.

"Impact strength of materials" Edward Arnold, London 1972.

3. Bredenberg, H., Broms, B.B.,

"Joints used in Sweden for precast concrete piles", pp 11-22 in Recent developments in the design and construction of piles,ICE, London,

1979.

4. Zukas, J.A. et al ,

"Impact dynamics", A Wiley-Interscience publication, John Wiley & Sons, New York, 1982.

5." Landon, J.W., Quinney, H.,

"Experiments with the Hopkinson Pressure Bar", Proc. R.Soc. A 103, pp 622 (1923)

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