Concrete structures under impact loading rate effects

Dr. ir. A . J . Zielinski

Report 5-84-14

Concrete structures

under impact loading

Rate effects

-14

^^r^L^k I " \ f^ I T ^ Department of Civil Engineering

I I • L / v 5 l I L Concrete structures

Rapp

^ ^ Technische Hogeschool Delft

B e t o n

8 4 - 0 2

,'N1

/^'

Delft University of Technology Department of Civil Engineering

Report 5-84-14

Research No. 2.3.84.07 December 1984

Technische Universiteit Delft

Faculteit CiTG

Bibliotheek Civiele Techniek

Stevinweg 1

2628 CN Delft

CONCRETE STRUCTURES UNDER IMPACT LOADING RATE EFFECTS

dr,ir. A.J. Zielinski

Mailing address:

Delft University of Technology Stevin Laboratory Stevinweg 4 2628 CN DELFT The Netherlands

Kcpp

Cf

Technische Hogeschool ^ Bibliotheek

AfdeKng: Civiele Tcc-iaiek

C, ~/_ ^_ é^^. /L^ Stevinweg 1

postbus Sr^*} 2600 GA Delft

i'

This research program has been carried out with financial support recieved from the Foundation for the Technical Sciences (STW Project DCT 11.0166), which is gratefully acknowledged and with the stimulus of valueable

CONTENTS page

SUMMARY

INTRODUCTION

MATERIAL PROPERTIES IN STATIC LOADING

2.1 Concrete 8

2.1.1 Uniaxial loading °

2.1.2 Multiaxial loading 13

2.1.3 Shear 18

2.2 Steel 20 2.3 Interaction between steel and concrete 21

RATE EFFECTS 25 3.1 Concrete 25 3.1.1 Uniaxial loading 25 3.1.2 Multiaxial loading 32 3.1.3 Shear 34 3.2 Steel 36 3.3 Bond between steel and concrete 39

BEHAVIOUR OF STRUCTURES AT HIGH RATES OF LOADING 44

4.1 Load categories 44 4.2 Local response 46 4.3 Overall response 47 4.4 Assessment methods 47 4.4.1 Experimental investigations 47 4.4.2 Analytical studies 47 4.4.3 Numerical methods 48 4.4.4 General strategy 49 4.4.5 Material modelling ^'

5 CONCLUSIONS 63 6 REFERENCES 64

SUMMARY

Various structures may be subjected to accidental, impact loading. The be-haviour of structures under that kind of extreme loading conditions is of vital importance for people's safety and environmental damage prevention. There is a need for reliable assessment methods for the local and the over-all response of concrete structures to impact. The numerical methods - in particular the finite element method - enable extensive parameter studies to be carried out and should be treated as complementary to experimental methods.

Both the structures and the materials have to be idealized for a numerical analysis. A dynamic analysis of simple mechanical models with non-linear elastic material characteristics may provide a rough approximation of the global response of the structure to impact.

For a more accurate dynamic analysis advanced finite element codes should be used together with realistic models for reinforced concrete. The material models should account for the rate sensitiveness of materials discussed in this report. Through sophisticated dynamic FE analysis involving various parameters better insight into the global and the local response of concrete structures to impact can be obtained and may provide a basis for the devel-opment of simplified approaches and design rules.

INTRODUCTION

Offshore structures, nuclear reactor containtments and highway structures, among others, must meet high requirements with respect to their safety not only under normal loading conditions during their service life but also in cases of accidental overloading. For instance, the above-mentioned struc-tures may be subjected to severe impact loading due to earthquake, explo-sions and ship, aircraft or vehicle colliexplo-sions. In many cases, however, there is insufficient experience concerning the behaviour of structures un-der complex, dynamic loading conditions. The appropriate methods are re-quired for studying the response of structures to impact. These methods are necessary for reliable assessment of the safety of structures which allows rational, economical design. Behaviour of structures or structural members may be determined by means of:

- experimental investigations, - theoretical analysis.

Experiments provide good insight into global performance of structural mem-bers, their bearing capacity and force-displacement relations.

They are, however, less effective in determining local stress-strain condi-tions and internal damage. Testing structural members of their scale models is very laborious and requires special laboratory facilities. Only a limited number of loading conditions and other variables can be investigated.

Developments in the finite element method and other computer-aided numerical methods have extended possibilities for refined analytical studies on

struc-tural behaviour involving influences of various parameters.

It should be emphasized that these methods require, beside codes and compu-ters, appropriate material models for particular problems. With respect to the behaviour of concrete structures under dynamic loading numerical ana-lysis becomes very complex due to time-dependent structural response and rate effects upon material properties.

In this report the rate effects are discussed in view of their importance to studies on the behaviour of concrete structures under impact loading.

Knowledge of the mechanical properties of concrete and steel, as well as the principles of the interaction between them, is necessary for studying the behaviour of reinforced concrete or prestressed concrete structures.

The reason for reinforcing or prestressing concrete members is that the ten-sile strength of concrete is very low, approximately one order of magnitude less than its compressive strength,

When tensile stresses occur due to external loading or imposed deformation in concrete members, the material may crack and fail. In reinforced concrete these tensile stresses are resisted by reinforcing bars crossing crack

planes and providing integrity of the members.

In prestressed concrete compressive stresses are introduced in order to compensate for tension.

In the following treatment of the subject those features of concrete and steel will be briefly reviewed which are essential for analysing behaviour of concrete structures.

Concrete

Concrete is a complex composite material consisting of aggregate particles which are dispersed and embedded in hardened cement paste. It contains a large number of air voids, bond microcracks at interfaces of particles and microcracks within the cement matrix. Many of these microcracks are due to manufacturing procedures and physico-chemical precesses taking place during the hardening of cement-based composites (differential thermal movement and shrinkage during hydration and drying). The propagation of microcracks is a key to understanding the non-linear behaviour of concrete under various loading conditions.

Uniaxial loading

Typical stress-strain relationships for concrete subjected to uniaxial loading are shown in Fig. 2.1. It emerges that both the maximum stress and the corresponding strain have higher values in compression than in tension. The G-e relationships for concrete depend on various factors, including the

structure of the material, the properties of its constituents and environ-mental conditions.

o(N/mrn )

2.45 1.75 1.05 -0.35 - , 1 -— j//y^ ïï ^ ^ m ^^ ' 1 1 1 2 \

n\\

\ \

\ V

1 Curve No. 1 2 3 4 5 V ^>« 1 Aggregate Type Granite Gravel Gravel Gravel Gravel Si2e 3 / 8 - 3 / 1 6 " 3 / 8 - 3 / 1 6 " 3/16-B.S.7 3 / 8 - 3 / 1 6 " 3 / 8 - 3 / 1 6 " '~~~~~-• 1 1 Age (months) 2 2 3 2 1 -— ^ -0.010 0.020 0.030

£(%)

otN/mm^)

/ ; - 67 N/mm' F i g . 2.1 S t r e s s - s t r a i n r e l a t i o n s h i p for concrete a t u n i a x i a l l o a d i n g . a. tension [l] b. compression [2]

Compression

Experimental investigations on concrete behaviour in compression [3, 4, 5, 6, 7] among others have shown that it is closely associated witli the develop ment and propagation of cracks. Linear-elastic behaviour of concrete can be observed up to the stress level of '^'30% of the compressive strength (see Fig. 2.1).

The first cracks start from interfaces of aggregate particles and extend parallel to the direction of loading (see Figs. 2.2 and 2.3). Load increase causes higher stress intensity at stiff aggregate particles, and cracks grow in tension and shear. At about 70-90% of the ultimate load the cracks tend to propagate through the mortar matrix and the stress-strain curves continue to bend towards the horizontal. The volumetric strain, which grad-ually decreases up to this stress level, then rapidly increases. Up to that stage of loading Poisson's ratio v has a fairly constant value, normally ranging from 0.18 to 0,22,

At about the ultimate load seperate concrete columns are formed.

Further loading leads to shear-compression failure of the concrete associat-ed with development of cone-like pieces and buckling of separatassociat-ed columns. The decreasing load-carrying capacity of concrete results in the descending branch of the stress-strain curve.

r— — T

r_

Fig. 2.2 Relevant stress components in the vicinity of a spherical relatively stiff inclusion in a homogeneous matrix [6].

a)

c)

b)

d)

Fig. 2.3 Uniaxial compression-crack pattern obtained from X-ray examination [5], a. prior to loading b. 65% of ultimate load c. 85% of ultimate load d. failure load Tension

The shape of tensile stress-strain curves of concrete has also been explain-ed with the aid of cracking mechanisms [1, 8, 9 ] .

Concrete behaves quite elastically for stresses lower than '^^60% of the ten-sile strength. Thereafter, extension of microcracks takes place at weak interfaces between aggregate particles and the cement matrix. Crack growth is caused by high intensity of stresses at tips of bond cracks (see Fig. 2.4).

Or/O

Fig, 2,4 Radial stresses along interfaces of a stiff bonded inclusion and a partially debonded inclusion [13],

a) b)

Fig, 2.5 Extension of bond cracks resulting in tensile fracture (schematic) a. prior to loading

At higher stress levels cracks penetrate through the matrix. Usually tensile failure rapidly occurs, due to cracks linking one with another (see Fig. 2.5) and separation of material. The descending branch of a-e curves can be

determined in deformation-controlled tests.

The cracking process in concrete is not limited to the major cracks only, but also includes extensive microcracking in the highly stressed zones ahead of the tips of macrocracks.

The investigations [10, 11, 12] focusing upon the phenomenon of microcracking of concrete have revealed great complexity of crack growth in this heteroge-neous material.

2.1,2 hhiltiaxial loading

The mechanisms associated with the failure of concrete at uniaxial loading are governed by the local state of stress and strain.

An additional loading applied perpendicularly to the main direction of loading may therefore significantly influence the behaviour of material. In-troduction of compressive stresses will result in crack arrest, whereas ten-sile stresses may accelerate propagation of cracks [5] .

The above means that the strength, deformation and stiffness of concrete depend upon the multiaxial state of loading. Fig, 2.6 shows several failure modes of concrete which were observed in biaxial tests on concrete [14]. Nelissen [14] gives an extensive review of earlier multiaxial tests on con-crete. The results obtained by Kupfer and Hilsdorf [15] are shown in Fig. 2.7. The plots give information on characteristics points of the stress-strain behaviour of concrete.

In Fig. 2,8 the course of strains is displayed for biaxial loading conditions, With respect to the strength of concrete the following can be concluded: - under biaxial compression the compressive strength is higher than the

uni-axial compressive strength; the maximum strength increase ('\'25%) is obser-ved at a stress ratio a : a = 1 : 0.5;

- under biaxial tension-compression loading regime there is an almost linear relationship between the strength in one direction of loading and the stress applied in the other direction;

- under biaxial tension the strength does not significantly differ from the uniaxial tensile strength.

Fig. 2.6 Failure of biaxially loaded concrete [14]

Ci/ffc

Fig, 2.7 Stresses at the elastic limit (1), inflection of volumetric strain (2), minimum volume (3) and failure (4) of concrete subjected to biaxial stress states [15].

compression - compression O = O|/02 0.000/-1 = 0.000 0.052/-1 =-0.052 0.103/-1 =-0.103 0.204/-1 = -0.204 compression - tension o = oJa2 =0.00/1.0 = 0.00 =0.55/1.0 = 0.55 = 1.00/1.0= 1.00 tension - tension -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10

Concerning concrete ductility the following is observed:

- under biaxial compression larger strains are achieved than under uniaxial compression;

- under biaxial tension-compression the principal compressive strain and the principal tensile strain decrease at increasing tensile stress;

- under biaxial tension the maximum principal strain is about equal to the strain at the ultimate stress under uniaxial tension.

Nelissen did not find any pronounced effect of the loading path upon the behaviour of concrete under biaxial loading,

Oj • 03 - - 1 7 0 MPa

0 - 0 . 0 1 - 0 . 0 2 - 0 . 0 3 - 0 . 0 4 - 0 . 0 5 Axial strain

Fig, 2.9 Triaxial stress-strain relationship for concrete [16]

The behaviour of concrete under triaxial loading has usually been investi-gated by means of axial tests on confined specimens or under hydrostatic pressure.

The plots of stress-strain curves shown in Fig. 2.9 indicate that increasing confinement increases both the compressive strength of concrete and the cor-responding strain. This is due to opposing bond cracking and therefore more extensive cracking of cement paste instead of cleavage failures. Figs. 2.10 and 2.11 are schematic representations of failure envelopes for concrete under triaxial loading. Isotropy of concrete is assumed for presenting the limit of elasticity under triaxial loading.

0=6Ö*

crack

initiation

v^o/fl

limit of

/ elasticity ^—

4 v/3 Co/f,

4-«-Fig. 2.10 Triaxial compression (©=60°) and extension (0=0°) test of con crete, Rendulic stress plane [17],

Elastic

limit

surface

Fig. 2.11 Schematic representation of the elastic limit and failure sur faces of concrete in the three-dimensional principal stress space,

From the experimental results it appears that concrete has a fairly consis-tent failure surface. The cross-sections perpendicular to the axis of prin-cipal stresses (deviatoric planes) of the failure surface are convex and non-circular for small hydrostatic pressures. For increasing hydrostatic pressures the deviatoric sections of the failure surface are almost circular,

i.e. fairly independent of the third stress invariant. In general, however, three stress invariants are needed for describing the failure surface.

2.1.3 Shear

Transfer of shear stresses across cracks in concrete is due to the roughness of the crack faces (see Fig. 2,12). According to the mechanisms of aggregate

interlock [18] the shear resistance of cracked concrete depends upon the

structure of the concrete (aggregate distribution and content) and the mechanical properties of its constituents (crushing strength of aggregate and the mortar matrix), The shear stress x is influenced by the normal stress a , the crack width w and the crack displacement A - see Fig, 2.13.

Fig. 2.12 Aggregate interlock.

Increasing shear force causes growing contact area between the crack faces for increasing A and W. If the crack opening is opposed, for instance by reinforcing bars, the normal stress also will increase. After reaching the contact, for a certain combination of a and W, the maximum shear resistance of concrete is achieved. The magnitude of the normal stress a and the crack opening W depend upon axial restraint stiffness of the reinforcement

1

o o. « 1.0 15 2.0 ?S W ( m m ) separation 0 5 10 15 2.0

U

^

w

ftj =37 6 N / m m ' D ^ j j 5 l 6 m m

m

!V

_II

1

₁

/ :

-f

- ^ ^ ^ ^

:::^—1

f „ =37 6 N / m m ' 0 _ . . = ' 6 m m m a l l / n / « « 1 / 0 / 1 G ij^iiiS 1 • ^ ^ • 1 / ! / » 1 / i M

r~-

IH-0.5 1.0 - * & (mm) stip 01 02 03 a4 05 06 07 08 OS Ifl 1.1 12 > W(mm) separation " 1.5 2.0 25

Fig. 2.13 Results of shear tests [18]. a. slip-crack relation

b. shear stress-slip relation

c. normal stress-crack width relation.

Furthermore, the behaviour of cracked reinforced concrete members subjected to shear is affected by another mechanism — dowel action. The latter will be discussed in the section on concrete-steel interaction.

Walraven [18] proposed the following relations for x and a as functions of the crack width W, displacement A and the compressive strength of concrete:

, % + {1.8 W - ° - ^ ^ (0.234 W-°-^°^ - 0.20) f }A

JU cc f'

o = ^ + { 1.35 W"°-^^ + (0.191 W °-^^^ - 0.15) f • } A

20 cc These relationships are in good agreement with his experimental results,

2.2 Steel

The behaviour of steel in tension is most important, considering its func-tion in the reinforcing or prestressing of concrete members.

Fig. 2.14 shows typical stress-strain diagrams for steels normally used. A pronounced yielding is exhibited by hot rolled self-hardening reinforcing steel. The yield stress a is approximately 220 and 420 N/mm^ for ordinary and for high tensile steel respectively.

In the o-£ curve of cold-worked reinforcing steel in tension the 0.2% proof stress o^ „ ranges from 200 to 500 N/mm^ .

^ „ „ C(N/mm2) l U U U 17nr ifinn i«;nn iz.(in 1300 1200 11 nn i f i n n onn Q n n 7 n n 600 500 400 9 0 0 lUU 0 / y f

iy

/ \ E -7^ sts / . - X . ^ ,*--e 00/S50RI'

Kj

i - ^ "^ BS t 4201500 RU / S^ \ ,^ B St 420/500 RK r-)St2 1 ^ 1 1 20/3tflGU/RU ~~" _{" ^ a} \ pO.2 --0 2 4 6 8 1--0 12 14 16 18 2--0 22 24 26 28 05 ÏD

Fig. 2.14 Stress-strain diagrams of steels

a,b - self-hardening reinforcing steel; c,d - cold-worked reinforcing steel; e - prestressing steel

The shape of the a-e curves of prestressing steel is similar to that of cold-worked reinforcing steel, but both the elastic limit and the tensile strength are much higher and often exceed 1400 N/mm^, The 0.1% proof stress

o_ is of major importance for the design of prestressed concrete members. Plain and deformed bars are usually used as reinforcement, whereas for wires, strands and cables as well as bars can be used as prestressing tendons.

2.3 Interaction between steel and concrete

Bond between steel and concrete is essential to the behaviour of reinforced and prestressed concrete members. The following fundamental bond mechanisms are to be distinguished:

1) a d he s io n ; 2) f r i c t i o n ;

3) anchoring due to ribs;

4) anchoring due to lack of fit and restrained twist,

The first two mechanisms govern the bond of plain reinforcing bars and prestressing tendons. These mechanisms, together with the fourth, ensure the bond of prestressing wires. For deformed reinforcing bars and pre-stressing tendons the bond resistance due to ribs is most important for transmitting forces to the concrete.

It should be pointed out that in the case of unbonded prestressing tendons end anchorage arrangements are necessary.

Fig. 2.15 shows the main difference in stress-slip curves between plain and deformed bars and different crack patterns in concrete.

bond stress

deformed bar

plain bar

displacement ( a )

smooth bar d e f o r m e d bar

(b)

Fig. 2.15 a) Characteristic difference in bond stress-slip curves between plain and deformed bars.

The distribution of the steel stress a , the concrete stress o , the bond

s c' stress X and the slip A are shown in Fig. 2.16 for the vicinity of the crack

in a tensile member.

Higher bond resistance of bars subjected to higher radial pressures p o around them, see Fig. 2.17, is not surprising in the view of the bond mech-anisms , steet stress concrete stress bond stress slip transmission zones

Fig. 2.16 Realistic stress distribution around a crack in a tension member (schematic). 15 10 [ N /mml 1 f, =r 36 N/mm 1 d s 16 mm 1 vertical po«it fp«0.06S /

r

ion ^ 0 ^ / : ^ — ^ H= 5 B=1S NAwn^l — a = 10

oos

0.10 015 [mm] 0 20

Fig. 2.17 Influence of external radial pressure p on the average bond stress-slip relationship [19]. °

Noakowski [20] proposed the following exponential relationship between the bond stress x and the relative displacement ó between steel and concrete;

X = C . 6 ^

where the C and N are coefficients which depend upon the compressive strength of concrete f' , bar diameter é, relative rib area f , position of the bar

cc r in the reinforced concrete member. He found good agreement between the

results of tests performed on concrete members reinforced with deformed bars (f =0.065) when the values of C=0.28 f' and N=0.16 were used,

r cc The reinforcing bars which cross cracks oppose parallel displacement of the crack faces (A) - see Fig, 2,18, This is called dowel action,

The dowel force depends upon the deformation of both the free and the em-bedded parts of bars.

Fig, 2.18 Dowel action,

The mechanism of dowel action and its contribution to the shear resistance of concrete members are reviewed by Walraven [18], According to his formula, the dowel force F, is mainly influenced by the concrete quality ( f ) , bar

d cc diameter («i) , crack width (W) , shear displacement (A) :

F, = 10 (W + 0,2)"^ . A ° - ^ ^ é^-''^ f 0.38

a cc From the experiments it emerges that dowel action is of minor importance for the total shear resistance of cracked concrete members (see Fig. 2.19).

x ( N / m m ) 12

Ql 0.2 0.3 0.4 0.5 A (mm)

F i g . 2.19 C o n t r i b u t i o n of dowel a c t i o n , to the t o t a l shear s t r e s s in crack [ 1 8 ] .

3. RATE EFFECTS

The influence of high stress rates upon the mechanical properties of mate-rials is considered in this chapter. The increasing interest in investiga-tions on rate effects is associated with safety requirements for special structures which may be subjected to rapid, accidental loads. Insufficient insight into the behaviour of materials at higher rates of loading may lead to overestimating safety of structures and to uneconomical design.

Table 3.1 shows load categories which can be distinguished with respect to the strain rate (ê).

Table 3.1 Load categories è(l/s) load category <10 quasi-static 10 -10 intermediate 10^-10^ high 4 >10 shock waves 3.1 Concrete

Experimental investigations show that concrete is a rate-sensitive material, Most research programs have focused on the behaviour of concrete specimens subjected to compressive, flexural and tensile loading at high stress rates. Information on concrete behaviour under multiaxial impact loading is scarce. A brief review of some phenomena observed is given in the following.

3.1.1 Uniaxial loading

Tension

In Fig. 3.1 the results obtained by various researchers are plotted in the same diagram with, on the horizontal axis, the stress or strain rate and, on the vertical axis, the impact tensile strength standardized to the static strength. The relationship between tensile strength and stress rate, based on results of an extensive experimental program carried out on 26 concrete mixes in the Stevin Laboratory [21], can be linked with the formula for

s t r e s s r a t e e f f e c t s proposed by Mihashi and Izurai [ 2 2 ] :

1

( f / f ) = {bib ) ^ * ^

o o ( 3 . 1 )

in which the exponent -;—- is 0.042.

This relationship is shown in Fig. 3.2 in which both the tensile strength f and the stress rate a are standardized to the static case .

(f =3.1N/mm2; a =10~'^N/mm2ms) o ' o f/f„ Komlos HcilrTwm Takeda Kviritudze Soeikin Hatano Birkimer Kormclrng el at

J

\<y 10" lO"' 10" 10" 10' F5- •pr 10' ó(N/mm'msl to' É(l/s)

n

10 10 10' 10" 10'

Fig. 3.1 Influence of the stress rate on the uniaxial tensile strength of concrete [21],

A general conclusion may be drawn that higher loading rates result in in-crease of tensile strength of concrete.

This increase may be influenced by some factors such as the composition and properties of constituents of concrete.

u Z5 2 1.5 1 0.75 f/fo " " __^ « ^• — , - - ' — — - ^ •^ ' • ' • ' ^X'''— -" " 5 V . — — -ons% _ , -^—'• . — — ' ' ^ _ , -— n . 32) 10 10' lo-" 10' 10' 10" Ó/Óo

Fig. 3.2 Increase in tensile strength of concrete with increasing stress rate: In f = 1.51 + 0.042 In a cic CI x0,042 f/fo = Kolo )

-4°

a = 1 0 N/ram^ms o f = 3 , 1 N/mm2

Fig. 3.3 shows stress-strain relationships determined in uniaxial impact and static tensile tests respectively. It emerges that not only the tensile strength but also the corresponding strains are greater under impact than under static loading conditions.

0 I N / m m ' l 0 ( N / m m 2 ) ^ /

k

iJr

Iff / s l a t i MOli ^ im -•^ TAR tact 0(N/(nm>) 01 02 03 0.4 as 06 0 0.1 0.2 0.3 0.4 05 0.6 0 Ql Q2 03 04 05 06 elVoo)

In recent tests Kormeling [23] has obtained descending branches of

a-e-diagrams (see Fig. 3.4). It is to be noted that the energy absorbed under impact loading is much greater than under static loading.

c(N/mm2)

5 0.20

A(mm)

Fig. 3.4 Complete stress-displacement curves for concrete in uniaxial ten-sion [23].

A model for tensile fracture of concrete at high rates of loading [13] ex-plains the above-mentioned phenomena by considering the amount of simulta-neous cracking of concrete and the paths of single cracks as being stress rate dependent (see Fig. 3.5), At high stress rates more extensive cracking takes place and cracks are forced to propagate through tough aggregate par-ticles instead of along weak interfaces between the parpar-ticles and the matrix. The model predicts higher tensile strength and fracture energy for concretes made with tough, well-bonded aggregate particles, which is in fair agreement with experimental results showing higher tensile strength of concretes made with low water-cement ratios and relatively small aggregate particles. The properties of interfaces are essential to the tensile behaviour of concrete at low loading rates, whereas the properties of aggregate are essential to the behaviour at high rates of loading.

STATIC _microcrocks mocrocrocks IMPACT oggregot* B particles cement matnx'

Fig. 3.5 Fundamental differences between static and impact tensile fracture [13].

Compression

Fig. 3.6 shows results of compression tests conducted by several researchers on various concretes. Despite the considerable scatter exhibited by the experimental results it can be concluded that the compressive strength of concrete increases with the rate of loading.

This increase is less than in the case of tensile strength. At high loading rates (ê > 0.2/s) the compressive strength increases very rapidly. A similar effect can be observed for tensile strength (see Fig, 3,1).

Dargel [24] suggested relationships:

f'/f' = 1.10 + 9.06.10 In ê for è < 0.191 1/s o

(3.2) f'/f' = 1.30 + 13 In i

o for ê > 0.191 1/s

It fits the experimental results rather well. Note that the relationship between a and ê involves the E-modulus which is rate-dependent. Here again

the increase in strength depends on the composition and properties of con-stituents of concrete. It was observed [25, 26] that under impact loading conditions more aggregate particles were fractured than under static com-pressive loading. Concretes with smaller maximum size of aggregate particles and a better interfacial bond with the cement matrix exhibited higher

that impact compressive loading forced cracks to undergo rapid extension through tougher material zones.

2.0

t.8

f/fn

1.6

1.4

12

1.0

0.8

7

-•«

_v^

»^v

4-Jj

10-8 10-7 10-6 10-5 10-^ 10-3 10-2 10

102

é(l/s)

Fig. 3.6 Influence of the loading rate upon uniaxial compressive strength of concrete [24]. 30 20 10 O (N/mm') 2. ^ ^ — ^ 5 • ^ ^ 4 3 2 3 EC/..) E (1/s) 130 6-10' 7 10

sio'-Fig, 3.7 Stress-strain curves for concrete in compression at different strain rates [27].

The stress-strain relationships for concrete under static and impact rates of loading manifest differences in strains corresponding to the maximum stresses, see Fig. 3.7.

They show the stiffness of concrete to increase with the rate of loading. The increase in the E-modulus can be described by expressions to those for the compressive strength [24]. Fig. 3.8 shows the proposed relationships together with the experimental results. Here again a considerable scatter in results is to be noted,

1.6

U

1.2

1.0

0.8

E/Eo

T • • ^

r^ öi /

^ » ?_ t - ^

r* jt * ^ » (>

10'

Co

v 4 r2

10'" 10 10 10

<-1

1 10

ê(1/S)

Fig. 3,8 Influence of the loading rate upon the E-modulus [24],

The influence of the stress rate upon Poisson's ratio is shown in Fig, 3.9. The value of Poisson's ratio increases with increasing rate of tensile lead ing and decreases with increasing rate of compressive loading.

a3 02 Ql X tension # compression

°^

10-= 10' K' 10' é/s

Fig. 3.9 The effect of strain rate upon Poisson's ratio for a/a = 40%

^ r •% max

3.1.2 Multiaxial loading

To the author's knowledge there has been only one reported experimental in-vestigation on concrete subjected to multiaxial states of stress and high rates of loading [31].

In these tests cylindrical specimens were subjected first to a confining pressure and were then axially loaded in either tension or compression at the static rate of straining (S:é=10 /s) and two high rates of straining (III:é=10~^/s and I:ê=1/s).

Fig. 3.10 shows stress-strain relationships for different combinations of confinement and loading rates. In Fig. 3.11 the results are shown in the three-dimensional stress-space,

It can be concluded in general that the rate effect on tensile strength and compressive strength of concrete was consistent for all levels of confine-ment,

The strains corresponding to the yield stresses appear to increase with both the rate of loading and the confining pressure,

Axial stress (N/mm^)

A 8.0| 1 8.0

Fig. 3.10,a Influence of axial strain rate and magnitude of confining pres-sure on the axial stress-strain curves of concrete [29],

100

C(N/mm2)

e(10 ^)

G(N/mm2)

_-9.3

e(10"1

'em-f

Fig, 3,10,b Influence of axial strain rate and magnitude of confining pres-sure on the axial stress-strain curves of concrete [29],

Compressive test. 100 Ci(N/mm2) C2aG3(N/mm2) 30 ^ocUCo 025 1.0 0.75 0.5 0.25 \ -0.1 -0.2 — -0.3 y

y

0 0 \ .

y

/ .5 1.( ^oct/Co 3 1.5 s — l i i X Fig. 3.11a

Variation of failure criterion of concrete in the triaxial tests conducted in the three stages of axial straining rate.

Fig. 3.11b

Relation between octahedral shearing stress (x ) and

oc t

octahedral normal stress (o ) at various strain-rates [29].

3.1.3 Shear

Stress rate effects upon the resistance of concrete to shear loading cannot be discussed in terms of fundamental relationships between x and x-results of shear tests depend on boundary conditions. The type of specimen and the manner of loading must rank first among all other factors affecting shear behaviour.

Results of experiments performed by Chung [30] on concrete joints indicate significant increase in shear strength with the rate of loading. He found

that at a stress rate of about 12.10 N/mm^s the shear strength was increas-ed by a factor of about 1.8 with respect to the static loading conditions. The shear joint investigated by Chung is shown in Fig. 3.12.

• ,1QQ> i~F o <£> (O 200

<P

\ : V

Si

j3a.

r

^

X

J

yy.

5mm dJQ.stirmps (senesB onLy) 5 mm dig bars 10mm d i g . b a r s unhatched " p r e c g s t ' p g r t hgtched " i n s i t u " p a r t F i g , 3,12 D e t a i l s of t e s t specimens [ 3 0 ] ,

Shear t e s t s a t high r a t e s of loading have been c a r r i e d out a l s o on d i f f e r e n t specimens. F i g . 3.13a shows specimen used by Takeda a t a l [ 2 8 ] . F i g s . 3.13b and 3.14 show the t e s t r e s u l t s . The impact shear s t r e n g t h of m a t e r i a l s t e s t e d i n c r e a s e d s i g n i f i c a n t l y with the r a t e of l o a d i n g . Another i n t e r e s t i n g f e a t u r e appearing from t h i s i n v e s t i g a t i o n s i s t h a t the displacement a t which the maximum shear s t r e s s i s obtained d e c r e a s e s with the i n c r e a s i n g r a t e of l o a d i n g . F i g . 3.14 shows i t q u i t e c l e a r l y .

It can presumably be explained in terms of fracturing of aggregate particles at high rates of shear loading instead of crushing of the cement matrix by the particles at low rates of loading,

LOADING I HEAD

'"i

2 : ^

LOAD CELL POTENT IO^tTER,

T-\j

T

m

SPECIMEN 15xl5x50cm)

F i g , 3,13a Shear t e s t of c o n c r e t e [ 3 1 ] , Set up,

l(N/mm2)

20

10

0

1 2

A(mm)

T(N/mm2)

20

10

^

w

E

A(mm)

TlN/mm^)

20

10

J

0 1

Jjf \ , , : 1

1

2.

A(mm)

( a )

T(N/mm2)

KN/mm^)

20

10

_{7' >J} / ' \-.'^ it \

H

0 1 2

A(mmj

i(N/mm2)

0 1 2

A(mm)

( b )

Fig, 3,13b Shear stress-displacement curves.

a. mortar

A max/Ao. max

0.75

0.50

0.251

10-^ 10-3 10-2 10-1

10J 102

A(cm/5)

MORTAR CONCFtlE REINF. (OM.Y) WITHOUTIWITH REIh#^. REINF. 1 O n • • A

Fig. 3.14 Changes of max.displacement (displacement at max. shear stress) of mortar and concrete specimens with or without reinforcement, and reinforcement only, tested at various loading rates [31].

3.2 Steel

The behaviour of reinforcing steel at high rates of loading is of consider-able importance, since it influences deformations of concrete structures and their ability to absorb energy at impact loading.

Despite extensive research activities in steel testing, little has been done with respect to the behaviour of reinforcing bars under impact loading.

Valuable investigations have been carried out in Germany and are reviewed by Dargel [24] - see Fig. 3.15.

It emerges that the stress corresponding to the strain of 0.2% increases with the rate of loading.

Berner's [32] results are of particular value, since he carried out his

tests on ^22 bars which had not been subjected to any treatment before

being tested.

Fig. 3.16 shows stress-strain relationships for a ribbed cold worked reinfor-cing steel (BST 420/500 RK), a ribbed self-hardening reinforreinfor-cing steel (BST 420/500 RU) and a ribbed high-tensile steel (BST 1080/1320).

It is to be noted that the effects of high rates of loading are not the same for the reinforcing bars tested.

1.5

1.4

1.3

12

1.1

1.0

0.9

Po 2/420

_{BST 420/500 RK} • o H j o r t h . 05 Bemer «22 1 steiner 1 D^IO oóQ <> B A —• ""^ .^^ • • - - • O t 1 " • • • - 6 - * k «

é(1/s)

10"^ 10"^ 10'^ 10'^ 10"^ 1 10^ 10^

3Q2/420

1.4

1.3

1.4

1.2

1.0

0.9

BST 420/500 • Hjorth * 5 1 Bemer [ <^^22 ^Henseleit 0 U 1 1 1 1 ^ ' ••" • • . - r — RU < 0 » » ^ ^ • • • ~ 1

è(1/s)

'lO-^ 10-^ 10-3 10-2 lO-"" 1 10M02

Fig. 3.15 Influence of the rate of strain upon the steel stress corres-ponding to 0.2% plastic strain [24],

In the case of BST 420/500 RK steel the tensile strength and elongation characteristics increased almost linearly with the logarithm of the rate of strain,

The tensile strength of BST 420/500 RU steel increased linearly with the rate of stressing, whereas the ductility was little affected by the rate of loading.

The behaviour of BST 1080/1320 steel was hardly affected by the rate of strain,

800 600 400 200 O

C{N/mm2)

f

• -1 i -1 -1 • est 400/500RK \

t

\

k

K\

2 =•3 f O 2 4 6 8 10 12 14 16 18

£(%)

1000 800 600 400 200

01

4

_

N/r

71 nr BSt4 1 / r

L-^ l2) 20 /50 ORU ^ ^ ' ^ «

L—

i -O 2 4 6 5 10 12 14 16 18

£(%)

1400 12 00 1000 800 600 400 200

°(

^{

/

f

' : Mymm2)

e

> ^

i

S t i 4 e \ > ^ \^3

^z

080/1320 ) ( i 1 0 1 2 1 4 1 6 18

Fig. 3.16 Stress-strain curves for steel at several strain rates [32] é =5.10-^/s é =0.2/s ê =2.0/s ê =8.5/s

The influence of the rate of strain upon the tensile strength a and the uniform elongation e (at onset of necking) can be expressed by the formulae:

o =A + B In c

(3.3) e = C + D In è

The approximate values of the coefficients A, B, C and D are summarized in the Table 3.1.

Table 3.1 Coefficients for rate effects on steel properties coefficients A B C D

steel

BST 420/500 RK 580 6,03 11.5 0.78 BST 420/500 RU 820 6.27 12.2 0.19 BST 1080/1320 - - 5.5 0.19

3.3 Bond between steel and concrete

Hjorth [27] performed pull-out tests on plain and deformed reinforcing bars of 16mm diameter.

The embedment length 1 varied between 16 and 160mm. The results of his

V

tests are shown in Fig, 3,17. It emerges that the bond resistance of deform-ed bars (BST 420/500 RK steel) was increasdeform-ed by the higher rates of loading, whereas the resistance of plain bars (BST 220/340 GU steel) was hardly

affected by the rate of loading.

Bond stress-displacement curves were determined for high loading rates in tests performed at the Stevin Laboratory [33]. The tests were carried out on plain and deformed bars of lOmra diameter and on 9.5mm (3/8 ) prestressing strands. The embedment length of 30mm was kept constant.

Fig. 3.18 shows results obtained on deformed bars. The effect of loading rate is clear. It is to be noted that the increase of bond resistance due to high loading rate is more pronounced for low-strength concrete.

,^max/fc

10^ 10-^ 10-3 10-210-1 10 101

T(N/mm2ms)

Fig. 3.17 Bond strength at various rates of loading [27]

0.3» « • '

002 0O4 Q06 008 010 0.12 014 0.16 018 020 5(mm)

Fig. 3.18a Bond stress-slip curves of deformed bars at various loading rates [33].

T(N/mm')

O.JilO '

° 002 004 006 008 QK) Q12 014 016 018 020

6(mm)

Fig. 3.18b Bond stress-slip curves of deformed bars at various loading rates [33],

- High-strength concrete

Fig, 3,19 confirms Hjorth's conclusions concerning insensitiveness of bond of plain bars to the rate of loading. The x-a curves determined for strands indicate that their bond with concrete is not significantly affected by the impact loading (see Fig, 3,20).

These phenomena are explained by considering the bond mechanisms.

In the case of plain bars and strands the bond is governed by the adhesion. When the adhesion is destroyed, plain bars and strands can be easily pulled out by forces exceeding the resistance due to friction,

.T(N/mm') ^. ^::^^ /y^^ / ^ ~ -t^aiSN/mm' • 10. plain t — (ON/mnt'mt — 03 W ' N / m m ' n i ^^ s 0.05 0.10 0.15 0.20 6(mm)

Fig. 3.19 Bond stress-slip curves of plain bars for a low and a high loading rate [33].

X (N/tnm'l -f g . 5SN/mm' -t — 40N/mm ms - 0.3lO'N/mm»m» 0.05 0.10 0.15 0.20 6(nim)

Fig. 3.20 Bond stress-slip curves of strands for a low and a high loading rate [33],

For deformed bars, however, the ribs oppose the bar displacements in con-crete. Due to very high intensity of stresses under the ribs crushing and cracking of the concrete takes place. Since both the compressive strength and the tensile strength of concrete are stress rate dependent, the bond of deformed bars is also influenced by the rate of loading.

The following formula is suggested for the bond strength x and the rate of loading x:

- ^ = ( ^ ) ' ^ (3,4)

X X

o o

-3

in which x corresponds to the static rate of loading of 0,3,10 N/mm^ms, The exponent n is a function of the compressive strength of the concrete f' and the displacement 6 between the bar and the concrete:

c

0.7 (1-2.5) n =

f'0.8 c

This relationship is assumed to hold for deformed bars with a relative rib surface area f ranging from 0.065 to 0.1 and displacements 6 < 0.2 mm. It should be realized that the stress rate effects described above have on-ly an indicative character and that the conditions at testing and the struc-ture of the materials considered may differ from those in actual strucstruc-tures.

More detailed information on stress rate effects and governing mechanisms is to be found in the references,

It is of essential importance to recognize that the stress rate effects on material behaviour may either increase or decrease the safety of concrete

structures under rapid loading. For instance, crushing and cracking of con-crete under impact loading will occur at higher stresses than under static loading. On the other hand the higher bond strength of deformed bars may lead to fracture of reinforcing bars under impact loading at deformation of structural members which can be statically imposed without exceeding the strain capacity of the steel bars in the cracked areas,

BEHAVIOUR OF STRUCTURES AT HIGH RATES OF LOADING 4,1 Load categories

The loading conditions can be categorized with respect to the ratio of load duration (t ) to characteristic response time (T) as proposed in

o Table 4,1.

Table 4,1 Load categories t /T

o load category type of loading > 4 4 - i i - 10 -6 < 10 quasi-static quasi-impact impuls ive/impac t shock loads conventional transient blast, impact high-energy exposives

At high rates of loading the response of structures is affected by propaga-tion of stress waves and mopropaga-tion-generated inertia forces. The dynamic re-sponse of a structure to impact loading can be analysed in terms of a local (primary) response in the direct vicinity of the applied load and of an overall (secondary) response of a structural member or the entire structure (see Fig. 4.1).

i

^Mf-^-Perforation and spalling Perforation, spalling and scabbing

0'M./>-^

* : V. Perforation Punchinrt sliear

O

£

^y displacement

Development of bending moments in the course of impact

moment

V ^

. ^

£0

V ^

shear

static mode

first mode

third mode

Fig. 4.2b Overall response to impact loading.

The behaviour of a system consisting of the structure and an impacting body depends on their masses and rigidities, sizes and shapes of contact surfaces, initial velocities, stiffness of structural members and supporting conditions, among other factors.

Local response

The local response of concrete structures involves crushing, punching shear, spalling and scabbing of concrete. In the case of hard missile impact signi-ficant penetration and even perforation of concrete members may occur. These phenomena are governed by stress wave propagation and behaviour of sound and cracked concrete under multiaxial, high rate loading conditions.

Due to the complex nature of the local response of concrete structures reliable and economical analysis, so that empirical formulae often have to be used in design procedures. Several formulae and results of hard missile impact tests are reviewed by Hughes [34],

The following formulae for the penetration depth x^ and for the barrier thicknesses h and h preventing scabbing and perforation respectively:

s p x^ = 0,19 K I/S , d h = 1,74 x^ + 2,3 d s f h = 1,58 x^ + 1.4 d P f

are based on the model developed by Hughes. The impact parameters are: K - nose shape coefficient (= 1; 1.12; 1.26 and 1.39 for

flat, blunt, spherical and very sharp noses respectively); I - impact parameter (= ——jj-) ;

r * m - missile mass;

V - impact velocity;

f - flexural strength of concrete; d - missile diameter;

S - strain rate factor (= 1 -F 12.3 In (1 + 0,03 I)).

The above formulae are valid for I < 3500 and normalweight concrete barriers with 0-1.5% two-way front face reinforcement and 0.3-1.7% two-way back face

bending reinforcement,

Recent investigation on penetration of concrete by rigid and deformable projectiles is reported in [31.2 and 31.3].

4.3 Overall response

In considering the overall response of concrete structures interest is focused on the time-dependent state of deformation of the structure and energy absorption. Vibrations due to concentrated impact loads are trans-mitted into neighbouring members and result in a complex state of motion of

the entire structure. The overall response is governed by non-linear behav-iour of materials affecting stiffness of structural members. Cracking of con-crete and yielding of steel influence natural frequences of concon-crete struc-tures and redistribution of internal forces,

4.4 Assessment methods

Actually, three methods for assessing the behaviouj: of concrete structures under dynamic loading can be distinguished:

- experimental investigations; - analytical studies;

- numerical methods,

In the following it will be discussed which methods can be applied to stu-dying particular aspects of the dynamic behaviour of concrete structures.

4.4.1

Experimental investigations

This method requires special laboratory facilities and is very laborious. Therefore, even in tests on scale models of structures, only a small number of variables can be involved. The models must satisfy scaling laws [35] which refer not only to geometry but also to mechanical properties of

mate-rials, Appropriate scaling of concrete structures or parts of them is still difficult, so that experiments have often to be carried out on rather large members,

Such experiments provide sufficient insight into global force-displacement-time relations and the bearing capacity of the structural member or the en-tire structure considered.

The internal state of stress or strain and the internal damage cannot be determined with high accuracy. This method is of particular importance for studying the local response of concrete structures doe to impact, which cannot as yet be properly investigated by other methods,

With respect to the overall response of concrete structures the results of experiments are normally applied to the verification of analytical and numerical solutions,

2 Analyviaal studies

This method was of great importance in the period before the widespread applications of computers. In some cases it is still convenient for making preliminary predictions of the global response of structures under dynamic loading and for the verification of results obtained with the aid of ad-vanced numerical studies, A great disadvantage of this method is that only a very limited number of relatively simple structures and loading conditions can be studied.

Several analytical approaches for beams, plates, frames and shells subjected to ideal impulsive and impact loading are discussed in [36].

Fig. 4.2 shows beam deformation due to impulsive loading, calculated accord-ing to different analytical methods.

Wave propagation problems are discussed in [37].

48 MoWf

Mb^o2

method of separation of variables

upper limit solution exact solution

method of series expansion ace. to eigenfunctions lower limit solution

Fig,

02 0.4 0.6 08 1.0

b/l

4.2 Residual deformation of beam under impulsive loading I Comparison of analytical approaches [36] .

Simple mechanical models consisting of masses and springs may be used for roughly estimating the time-dependent displacement of idealized structures. The equations for systems consisting of more than two masses and/or n o n -linear springs (see Fig. 4.3) cannot be solved in closed form and therefore require numerical integration. The use of computers makes possible the dy-namic analysis of more complex mass-spring models which better represent actual structures and non-linear behaviour of reinforced concrete,

No = triy.g

Fig. 4.3 Mechanical model for a column subjected to a horizontal impact [3.4]

N = longitudinal force in column 3 Numerical methods

Many cases of idealized structures subjected to dynamic loading could have been described by writing the governing equations of continuum dynamics into differential form and solving them with the aid of computers.

Besides the finite difference method (FDM) the finite element method (FEM) should be mentioned as most widely applied in the advanced analysis of con-crete structures. In the FEM an actual structure is idealized by a system of discrete elements of defined load-deformation characteristics. The ele-ments are interconnected at nodal points and the behaviour of the assembly of these finite elements approximates the behaviour of the actual structure. The fundamentals of the FEM can be found in [ 3 8 ] , and the recent achievements in the field of concrete structures are reported in [39] and [ 4 0 ] .

The governing equations of continuum dynamics can be expressed in the FEM as follows:

[M] [A] + [K] [A] = [F(t)] - [F^(t)] where;

[M] = mass matrix;

[A] = displacement matrix; [A] = velocity matrix;

[E] = acceleration matrix; [K] = stiffness matrix; [F(t)] = external load matrix; [F (t)] = damping matrix (=a[M] [A]).

Various techniques may be used for mesh description (Langrangian, Eulerian or hybrid) and for time integration (implicit, explicit or mixed), depending upon the particular subject of the dynamic analysis and upon capabilities of available computer codes.

In Langrangian codes the computational grid is fixed in the material and

follows its motions and distortion whereas in Eulerian codes the computational grid is fixed in space so that the material passes through it.

The Langrangian codes are widely applied in the dynamic analysis of concrete structures. In large deformation problems, for instant perforation of con-crete members, rezoning becomes necessary.

Explicit methods of time integration enable displacements to be determined at any particular time t + At even if the accelerations at that time step are not known in implicit methods, the displacements at any particular time t + At cannot be calculated without a knowledge of the accelerations at occurring at that time. Implicit methods such as the Newmark g-method or the Wilson 0-method are unconditionally stable and deserve particular at-tention for dynamic loading problems, especially with many load reversals (earthquakes), Explicit methods are more appropriate for wave propagation problems (hard impact),

The above-mentioned techniques for dynamic FEM analysis have several

advantages and disadvantages - see for instance [41]. It must therefore be carefully considered which approach is most appropriate for analysing particular problems of local and overall response of structures to impact loading. For this purpose the knowledge and experience of analysts appear to be of a primary importance for obtaining the required information at low cost of computation. It should be emphasized that intensive investigations

on applications of the FEM to the dynamic analysis of concrete structures are being carried out in several research institutes and universities. A number of programs developed is described in [40].

4 General strategy

It is essential to choose a correct strategy for analysing the behaviour of concrete structures under rapid loading.

First of all it should be investigated whether a dynamic analysis is neces-sary. A comparison of load duration (t ) with characteristic response time (T) may lead to the conclusion that a quasi-static analysis will be sufficient. This comparison may be carried out with the aid of analytical methods or

numerical methods applied to idealized structures. For this purpose the simple mechanical models are of great importance, since the cost of dynamic analysis increases rapidly with the complexity of structural models and non-linearity of material behaviour. Figure 4,4 shows a simple model which

is very convenient for impact loading.

The impacting object (1) has a certain mass (m) and velocity (v ) and the contact zone (2) may deform permits transfer of compressive forces. The elements (3) and (4) should represent the structure treated locally

(structural member) and globally (entire structure) ,

Tlie elements (3) and (4) have characteristic response times (T , < member T ^ ^ ) associated with material behaviour represented by

spring-structure '^ J f b

stiffness. The spring-stiffness may represent linear-elastic, elastic-perfectly plastic of non-linear elasto-plastic behaviour of materials. The computations must take the conservation laws for energy and impulse into account. The initial kinetic energy associated with the impact loading is E = i m.v^ and the impulse I = m.v .

o '^ o

In some cases of dynamic loading a force-time relation can be given instead of elements (1) and (2). A rough estimation of the dynamic response of the structure is limited to the overall behaviour. If the preliminary analysis indicates that the dynamic effects cannot be neglected, the structure should be analysed by means of refined approaches resulting in the determination of stresses and strains in it.

Refined approaches are usually based upon the FEM or the FDM. It is diffi-cult to give general rules for the schematization of structures and material modelling. They depend very much upon available codes, type of structure,

loading conditions, required information and accuracy of calculations. Examples of advanced dynamic analysis of concrete structures can be found in [31]. Behaviour of beams is dealt with in [31.5, 6, 7 ] , whereas shells, slabs and plates are analysed in [31.8, 9, 10, 11, 12],

As already mentioned, determining local response of structures to impact loading may require experimental investigations. Examples of impact test on concrete members can be also found in [31], For beams see [31.13, 14], for slabs and plates [31,15, 16, 17, 18], for shells [31.19],

^Ch-a

77/7/ /77T7

_m

TT

/7777? ////// ///////

Fig. 4.4 A simple model of impact 1 - impacting object 2 - contact zone 3 - structure locally 4 - structure globally

Material modelling

It would be ideal to have universally applicable material models for various types and rates of loading. For the present, the effects of the multiaxial state of stresses, previous loading conditions, unloading and reloading re-quire further experimental investigations. These should take into account the structure of concrete and the environmental conditions. The results should be incorporated in general failure criteria for concrete and models including the post-failure behaviour.

The behaviour of concrete is complex (see Chapter 2 and 3 ) , and it is difficult to describe it by means of a single model. It is more convenient and effective to choose appropriate simple material models for analysis of the behaviour of a given structure under certain loading conditions,

Modelling of reinforced concrete for finite element analysis depends on the object of such an analysis and the schematization of a structure or struc-tural member. Several models can be considered - a comprehensive review is given in [42], The fracture models of concrete can be classified as follows: - linear-elastic; - non-linear-elastic; - elastic-perfectly plastic; - elastic-hardenin-plastic,

"4^

;

\r\:\

e

Fig, 4.5 Uniaxial stress-strain diagrams for concrete - linear-elastic

- non-linear-elastic

- elastic-perfectly plastic - elastic-hardening plastic

Figure 4.5 shows the corresponding stress-strain diagrams for uniaxial loading. Beside brittle fracture models, more realistic models for concrete have been developed which take account of stress-softening in both

compres-sion and tencompres-sion. The models should be combined with failure criteria applic-able not only under uniaxial stress conditions but also under multiaxial stress conditions.

The Mohr-Coulomb criterion with tension cut-off (see Fig. 4,6) is one of the most widely applied in the finite element analysis of concrete structures and often provides satisfactory agreement with experimentally determined behaviour of various structural members under multi-axial loading conditions. Even better agreement can be obtained with more sophisticated models such as five-parameter model of William and Warnke.

G2

No-tension

collapse regime

^ft=85ft Ci

/salension cutoff

ft = 44kgf/cm2

Mohr-Coulomb

c = 78.5kgf/cm2

0=41.8°

Fig. 4.6 Mohr-Coulomb with tension cut-off failure criterion

In general, three approaches to crack modelling in concrete are to be distinguished (see Fig. 4.7):

- discrete crack models; - smeared crack models; - fracture mechanics models.

= 1 1 '

) b 6 6—

o 6 o

b—

I

^t

P o p

> b (1

-+4—}

o p p -o -O 9 ? ^'

t $ ^ ^"-r=4 1 i i

' p n o—•— T

6 6 6- 6 -6 6 6 o

Cc

" W V , , , , w , , , , , , ' f

Stress free ^ ^Inelastic stress distribution Elastic stress distributiqp

True crock o ^ i ^ Fictitious crack Aa ^

Process zone

The smeared crack models incorporate cracking into stiffness properties of elements and are often used for global structural analysis. The discrete crack models are more appropriate for obtaining crack patterns and for stu-dying local behaviour of structural members, especially when aggregate inter-lock, dowel action, bond and yielding of reinforcement are essential to their behaviour.

Crack models based upon concepts of fracture mechanics can be used for spe-cial problems in which crack propagation in concrete members is of particu-lar interest.

Models for reinforcing steel (see Fig. 4.8) can be classified as: - elastic-perfectly;

- elastic-hardening plastic,

C 0

Fig, 4,8 Models for reinforcing steel

Figures 4.9 and 4.10 show simple models for bond and shear resistance

respectively. In many cases perfect bond is assumed and the shear resistance after cracking is approximated by a reduced shear modulus of concrete.

/

Fig. 4.10 Shear model

It would be outside the scope of this report to discuss in detail the model-ling of reinforced concrete and the computing techniques for finite element analysis. More information can be found in [39], [40] and [42],

It should be pointed out that sophisticated models provide better approxi-mations of the behaviour of reinforced concrete members of complex geometry under multiaxial loading conditions. This, however, is associated with a time-consuming and expensive FE analysis. Dynamic analysis requires much more computing time than static analysis. For efficiency of dynamic analysis of concrete structures it is therefore essential to start with simple models of structures and materials.

In the case of a dynamic analysis of simple mass-spring systems, the force-displacement relation for springs can be approximated with the aid of the stiffness of structural members - see Fig. 4,11.

.iS-^^^

w

mr^^^^^^

w

The impact zone can be characterized by a contact law. For instance a Hartezian contact law relates the impact force F to the deformation a of

the impact zone as follows:

k.a 3/2

where K is the stiffness of the impact zone.

It must be pointed out that assuming linear-elastic behaviour of concrete leads to overestimating forces and stresses generated by impact - see Fig,

4 . 1 2 .

4000

30002 0 0 0

-FlkN/m)

1500

1000

load cell

" -_K6.-6^.^

^ ^ ^ Ï ^ ^ K I

150 200

f (Hz)

Fig. 4.12 Comparison between physically non-linear respectively linear-elastic analysis and experiment on reinforced concrete slabs

[31.11].

It may be advisable to use in dynamic analysis first the material proper-ties and models which are usually applied in static analysis. Then, depend-ing upon the results of the preliminary analysis, a refined quasi-static or dynamic analysis may be carried out for the whole structure of critical regions. The FEM provides means for realistic idealization of structures.

Tlie rate sensitiveness of materials can be partially accounted for in special cases where the global response of structures to impact must be studied

with high accuracy and the local response is essential in view of safety requirements.

In such cases, the modelling of structures and materials must be as realistic as possible. The constitutive laws should take account of all major features of material behaviour,

On the other hand the constitutive equations have to be as simple as possi-ble for the sake of efficiency of calculations,

Nilsson [43] has developed a constitutive model for concrete which describes elastic-viscoplastic-plastic/brittle behaviour of this material in the multi-axial state of stress. This model accounts for rate effects by means of a single rate hardening parameter:

H = F {C + C„ ln(ê®^) + C.[ln(ê^^)]M r CU 1 2 3

where the parameters C , C en C are obtained by fitting H to experimental .ef

data, and e is the effective stram rate.

The values of the parameters C were approximated with the aid of results of compressive tests:

C^ = 1.6 C^ = 0.104 C = 0.0045

The analysis carried out by Nilsson for steel sphere - concrete rod impact (see Fig. 4.12) clearly show the great importance of realistic material modelling for achieving satisfactory insight into the local response of structures to impact.

König and Dargel [31.20] have presented a constitutive model for concrete taking into account rate effects described by the equations (3.1), (3.2) and (3.3).

The results of uniaxial tests are transmitted in the multiaxial stress space. The octahedral strain rate Y is used for the multiaxial state of

o stresses:

Y = I 2(1 - u).£

The shear stress-shear strain relation (x -y ) and the stress-octahedral o o

strain relation (a -e ) are formulated with the aid of results of multi-o multi-o

axial tests. The shear modulus G and the bulk modulus K are rate-dependent, It is suggested that this complex model can be used for parameter studies improving the understanding of structural behaviour under impact loading. The results of such sophisticated studies should contribute to the develop-ment of approximate methods for design procedures.

A2m/s/ a) SteeL sptiere _ , 012.7 mm ^"*"

21

533 mm Concrete rod ^19 mm

4-

-*^ b) I f I • , , ^ I L. — — •' I N I I I

3

-c)

£ ( % )

10 20 30 40 50 60 70

t(MS)

Fig. 4.13 Sphere-rod impact [43]

a. experiment by Goldsmith et al. b. FE idealization

c. course of axial strain in the concrete rod 89mm from the contact face

For studies focused upon the crack propagation fracture mechanics models should be applied. Besides the fictitious crack model of Hillerborg et al.

[44] the smeared crack models accounting for tension softening [45] are most appropriate for concrete. The latter approach is incorporated in the DIANA package developed by TNO-IBBC [46]. It seems that these models can be adopted for analysing crack growth under impact loading. The results of in-vestigations [13, 23] can be used for determining impact stress-strain diagrams - see Fig. 4.14.

static Static

Impact

Fig. 4.14 Idealized stress-strain diagrams for static and impact loading. Both the tensile strength f and the fracture energy G^ under impact can be approximated from the static values (for f see eq. (3.1), G /G^^«(f/f^)M