How reliable and important is the prediction of crack width in ferrocement in direct tension?

Cement &Concrete Composites 13 (1991) 3-12

How Reliable and Important is the Prediction of Crack Width in Ferrocement in Direct Tension? G. Singh & G. J. Xiong Civil Engineering Department, University of Leeds, Leeds LS2 9JT, UK (Received 6 November 1989; accepted 9 July 1990)

Abstract A critical review of some of the well known models for crack-width prediction shows that these models are not reliable. It is demonstrated that for most of the structural applications, including water retaining structures the crack width is not the dominant criterion of design. The steel stress which is the dominant criterion can fortunately be modelled with an acceptable level of reliability. Introspection is recommended to all those involved in the research and the design issues. Keywords: Ferrocement, design criteria, crack growth, tension tests, steel stress, structural engineering, composite materials, crack width, mathematical modelling, accuracy, concrete durability, stresses, strains. INTRODUCTION Whilst designing ferrocement structures it is normal practice to cater for a number of criteria, two of which are addressed in this paper. The first is that of stress in the reinforcement which is not allowed to exceed an acceptable level, and the second is that of the crack width which is also not allowed to exceed an acceptable value. The acceptable limits are functions of a variety of factors, the primary one being that of the nature of the application. For example, applications of ferrocement in corrosive environments call for tighter restrictions. Demand for crack control calls for means or models for crack prediction. Several researchers have responded to this call and have proposed various mechanistic or theoretical models for describing the tensile and flexural behaviour of

this composite material. The objective of the paper is to address only the tensile behaviour. At best, the models should not only provide reliable predictions but they should also provide clear understanding of the interactions between the various phases of the composite. As a matter of fact, reliable modelling can result only from clear understanding of the components and their interactions. Concurrence between a mathematical model and the real-life observations can only be fortuitous or cooked (through back calculations) if the model ignores the physical realities of the situation and/or if it is based on gross simplifications. Such a model cannot and should not be used as a general purpose formula. The view taken by the authors is that for most structural applications the dominant criterion of design is steel stress and not that the crack width is irrelevant; because it can be relevant. It is relevant, for example, in applications where close range appearance is important (particularly where the surface can be partly wet). And then there are issues of long term durability that need to be addressed and which are dependent, partly on crack widths. In other words, a reliable crack prediction model is desirable but not available at present. This state needs to be demonstrated through review of the existing models. What follows is a critical review of some of the well known predictive models and a discussion of their numerical reliability as well as their elegance as far as the qualitative portrayal of the interaction between the phases is concerned. Attempt is made to answer the questions raised in the title. A new qualitative model which is thought to reflect the behaviour of the composite more realistically is also presented.

Cement & Concrete Composites 0958-9465/91/$3.50 © 1991 Elsevier Science Publishers Ltd, England. Printed in Great Britain

4

G. Singh, G. J. Xiong

GENERAL DESCRIPTION OF THE BEHAVIOUR IN TENSION

This behaviour is often presented in the form of a plot between average composite stress and strain and this is seen to be made-up of three stages (1-3) as shown in Fig. 1. The complex physical realities of this composite are represented by a simplified physical model as shown in Fig. 2. Stage 1: Reinforcement can be assumed to be fully bonded with mortar and the response is mainly elastic. The distribution of stresses in the three phases (i.e. steel, mortar and interface) can be assumed to be even as shown in Fig. 2(A). Stage 2: The beginning of the second stage is roughly indicated by the appearance of first mortar cracks which may be audible and can be seen with a magnifying lens. During this stage the number of cracks keeps increasing with load but the width increase is only marginal. The stress in the mortar at the cracked section is obviously zero but halfway between the cracks it rises gradually to a maximum through the mortar-steel interaction. The specimen can be seen to be a member with local regions of softening causing a decrease in the overall stiffness of the member (termed as equivalent element) as shown in Fig. 2(B5). A significant number of visible and invisible cracks occur near the beginning of this stage due to a small increment of load. This is why the average composite stress remains more or less constant whilst the average strain increases significantly. Subsequently, although some new cracks are formed, one by one, yet the rate of increase in their number is slower. This is reflected in the stress-strain curve but, of course, the slope is much lower than that in Stage 1. Towards the end of this Stage 2 there is no increase in the number of cracks. Stage 3: During this stage the number of cracks remains almost constant. The crack width, however, increases gradually. The slope of the average stress-strain curve decreases more sharply and eventually the plot approaches that of the equivalent steel tested on its own in the air. During this stage whilst the stress in the wire at cracked sections keeps increasing, that in the mortar has to remain zero at these sections. This causes shear stress concentration between wire and mortar to be more severe, resulting in bond failure to progress from the crack sections towards the middle of the two adjoining cracks. In turn, this causes the

softening regions to grow longer and longer as shown in Fig. 2(C). This explanation is well supported by Somayaji & Shah's I tests results (Fig. 1). Because virtually no new cracks are formed in this stage it can be inferred that the decrease in the average stiffness of the specimen is caused by debonding which results in transfer of stress from mortar to steel, with the former experiencing stress relief. Therefore, it should be noted that one of the main factors which influences the crack width is the deformation recovery of mortar. This is also one of the main factors which causes the smaller crack spacings to be accompanied by narrower crack widths and causes decrease in crack widths with increase in specific surface of the reinforcement.

DISCUSSION OF THE REI J&BILITY OF SOME OF THE MECHANISTIC AND EMPIRICAL MODELS

Paul & Pama 2 presented the first mathematical model in 1978 (Fig. 3) in which they took crack width (W) to be the difference between deformation of wire and mortar W - A A w - AAm= 2

£(/2 (O'w~ I dx ~ Omx Em]

(1)

where A2w is the elongation of wire between two adjacent cracks, A)].m is the elongation of mortar over the same distance, l, Ew and E m are the elastic moduli of wire and mortar. Owx and Omx= stress in the wire and mortar respectively at distance x from the cracked surface. They assumed the bond stress rx to be sinusoidaUy distributed (Fig. 3(b)) rx= ru sin

2ytx l

(2)

where ru = ultimate bond stress. After a series of steps and simplifications they arrived at W = - -l [a* l Ew

Cb'ru'de(R+m)12 ] 2Atom

~: Omu SL Ew.G,.~

, Omo ] Ow-T(R+m)

(3)

The prediction of crack width in ferrocement

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Typical composite stress-strain diagrams, reinforced with meshes.l

FIRST ST"AGE

SECONDSTAGE

THIRD STAGE

AI

BI

CI

A2

B2

C2

in wire

Tensitesfress in morlvr

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Bond stress

AL,

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A5

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i

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85

C5

Modelling of wire -- mortar interaction.

where Cb is d/de in which d is diameter of wire and de is equivalent diameter dependent on mesh type, R is the ratio of volume of matrix to volume of wire, m is the modular ratio Ew/Em, S L is nearly equal to ~ r d e / A ~ in which A ~ is mortar area and trmu is the ultimate tensile strength of mortar. For a given member, the only variable in eqn (3) is the stress (a*) in the wire at the cracked section.

This equation implies that bond stress is not related to steel stress. Paul & Pama 2 imply that the bond stress between wire and mortar remains unchanged with increasing load in Stages 2 and 3. Therefore, debonding and mortar stress relief in Stage 3 are not taken into account. Somaya & Shah ~ put forward in 1981 their analytical model (Fig. 4) in which they took the sum of increment of deformation in steel and

6

C,. Singh, G. J. Xiong

OWX

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Stress distributions in the cracked portion in a composite model with aligned fibre.:

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Moh'ix sh'e~

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(b)

only if the relation (difficult to obtain) between local bond stress and slip was known, they proposed an exponential function to describe this bond stress distribution (Fig. 4(e)) d 2wx

d x 2 = A e x + Be-x + C

X

(c)

Strain

(4)

where wx is the slip at section x and A, B and C are constants. Therefore Cx 2

Wx = Ae x + Be-X +

b

)

(d)

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jr

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Load, strain, slip and bond stress distribution.

decrement of deformation in mortar to account for the crack width. They proposed a differential equation to describe the distribution of local bond stress. Realising that the equation could be solved

+Dx+E

2

(5)

The constants were obtained from boundary conditions. It is obvious that different values of steel stress at cracks, crack spacing (S) and bond transfer length (Lt) , shown in Fig. 4(c), will yield different values for these constants. They conducted pull out tests to obtain the relationship between applied load and L t and took it to be linear. Figure 5 shows their own typical comparison of measured crack widths with those predicted by them. In the authors' view the results of prediction are far from satisfactory. As can be seen, the pre, diction of crack width is very sensitive to the assumed value of S. It has to be stated that their 'theoretical lower limit' of S L t has no rational base or meaning. As distinct from the last model they introduced the bond stress tramfer length into their model =

The prediction of crack width in ferrocement but the reality of debonding was not incorporated. The convergence of the composite curve with that of the steel (Fig. 1) was explained with the statement that 'the contribution of matrix can be considered to be negligible'. Their model assumes the existence of zero bond stress in the middle region (Fig. 4). If this were true then the mortar stress within the region would be calculated to be greater than its strength at higher loads in Stage 3 where no new cracks are actually observed. Akhtaruzzaman & Pama 3 proposed, in 1988 a relationship between bond stress, bond slip and tensile strain difference (Fig. 6) between the two phases as follows:

Sx

Ux/k =e= d x - emx dx

=

100

200

I

I

7

where Sx is the local slip at section x, Ux is the bond stress at section x, k is the slip modulus (slope of idealised bond-slip curve), e= and emx are strains in steel and matrix at section x. After a series of steps they give (for 'elastic slip') bond stress

Ux =( - kfcr/C1E,) [(ec'(u2 +x)_ eC'(l/2- x))/(eC,, + 1 )]

(7) and crack width

W=(2fcr/C1Es)[(ec ' ' - 1)/(eC"+ 1)]

where fcr is the average steel stress at cracked section, E s is modulus of steel, C1 is a constant and l is the crack spacing.

(6) Steer stress, 300

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Steel stress, ksi Typical comparison of predicted and experimental average crack width. ~

xl

Ux

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EF-3

_J Fibre stress distribution

Fig. 6.

(8)

Matrix stress distribution

Cracking of member under axial tension?

Bond stress distribution

8

G. Singh, G. J. Xiong

Unlike the previous model this one (eqn 7) takes the bond stress, Ux, at cracked section to increase with increasing steel stress, fcr. This is reasonable as far as Stage 2 is concerned. The model, however, does not explain the convergence of the curves of the composite stress and the steel stress because it does not allow for debonding and mortar stress relief in Stage 3. This model also can lead to mortar stress being calculated to be more than its strength. ACI state-of-the-art report, 4 gives two equations for predicting maximum crack width in tension. These equations are based on some theoretical insight, calibrated by experimental results which are specific to the materials, fabrication processes, curving conditions and testing methods used. For steel stress --345 x specific surface (SRL)in the loading direction

Wmax=3500/Es

Equation (9), which is independent of the stress level, relates mainly to the higher end of the Stage 1 and Stage 2. Equation (10), which does show dependence on steel stress relates to Stage 3 wherein the crack widths increase with load but crack numbers (or spacing) remain practically unaltered. It is interesting to review Naaman's 5 comparison of his observed data with these equations (Fig. 7). The unreliability of prediction is obvious. Chen & Zhao 6 suggested in 1988 the use of the following models for composites with mesh reinforcement, based on statistical analysis of experimental data:

l= 3.0C +0.26 d/V r (11) Wmax~---(3"0 C + 0"26 d/Vs)/(E s Vf) x e °°2:<0+3.92fret) (12) where l is the crack spacing, C is the cover thickness, Vs is the percentage of reinforcement in the loading direction, Wmax is the maximum crack width, d is the wire diameter, E s is the modulus of the steel, fmt is the tensile strength of mortar and o is composite stress. One significant feature of this formula is the inclusion of the mortar cover thickness. Pama & Paul 7 also observed that the crack width increased

(9)

where Wmaxis the maximum crack width. (Please note that in the ACI report 4 the numerator is misprinted as 35 000) For steel stress > 345 x Sin. Wm~,= 20 [175 + 3"69 (steel stress - 345 SRL)]/Es

(10)

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350

/+00

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Typical variations of crack widths in tension versus steel stress2

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The prediction of crack width in ferrocement

9

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Typical tensile behaviour of ferrocement. 8

linearly with cover thickness. Although Chen & Zhao 6 have included this effect, the difference between the observed and the calculated values of crack widths can be unacceptably large as shown in Fig. 8. The discussion of the various models has

shown that they are not reliable in the quantitative and in the qualitative sense. They do not describe the behaviour of the composite satisfactorily. The next question that has to be addressed is how critical is the need for prediction of the crack width.

10

G. Singh, G. J. Xiong

Specific surface of reinforcement, loading direction, S L, 0 0.393 0.787 1.181 1.57/+ I

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O b s e r v e d average crack spacing at failure versus specific surface of r e i n f o r c e m e n t ?

D I S C U S S I O N O F T H E I M P O R T A N C E OF CRACK W I D T H AS A D E S I G N CRITERION

The following is based on the assumption that the recommendations contained in the ACI Report 4 are valid. The two primary criteria for structural design are (a) The stress in the reinforcement should not be allowed to exceed a certain safe value, e.g. 207 N / m m 2 is the value recommended by the ACI Committee for liquid retaining and sanitary structures and 0 . 6 x y i e l d strength is recommended for common structures. (b) The surface crack width should not be allowed to exceed a safe value, e.g. 0.05 m m is thought to be the maximum allow-

able width for water retaining structures. The limit for common structures is 0.1 mm. Fortunately the stress can be reasonably reliably modelled (although it can not be measured) using mechanistic approaches. The second criterion, the subject of this paper, addresses the important object of designing for durability. Unlike the first criterion, crack width can be measured but cannot be modelled reliably as already discussed. The design decisions, obviously, have to be based upon the dominant criterion. Fortunately the dominant criterion is that of steel stress, 8,9 as can be seen in Fig. 9 which is based on typical experimental results. In the following the authors present an analytical proof of their view in which they have deliberately chosen a method which overestimates the crack widths. The line of reason, ing is that if the crack widths so determined can

The prediction of crack width in ferrocement

11

Table 1. Crack widths calculated by different methods

Allowable stress (MPa) Effective modulus (103 MPa)

Woven square mesh

Welded square mesh

270 (207) a

270 (207) ~

Hexagonal mesh

Expanded metal mesh

Longitudinal bars

186

186

248.4 (207)"

138

200

104

138

200

0"042 0 0"080 2

0"029 0 0"055 4

0"050 1 0"073 3

0"037 7 0"055 3

0"029 0 0'050 9

0"017 6 0"045 9

0"012 1 0"031 6

0"021 0 0"042 0

0"015 8 0"031 6

0"012 1 0"029 2

0"017 3 0"049 2

0"011 9 0"033 9

0"022 7 0"064 3

0"017 1 0"048 4

0"011 9 0"033 8

0"025 4 0"096 0

0"017 5 0"066 2

0"033 6 0"067 7

0"025 4 0"051 0

0"017 5 0"058 2

Maximum crack width

(mm)~

Using eqns (11) and (13) SRL= 0"08 SRL----0"04 Using eqns (13) and (14) SRL=0"08 SRL= 0"04 Using eqn (12) SRL= 0"08 SRL= 0"04 Using eqns (9) and (10) SRL= 0"08 SRL= 0"04

a207 means the ACI limits the tensile stress to 207 MPa for water retaining structures. bSRL= 0"08 for water retaining structures, SRL----0"04 for common structures.

still be shown to be d o m i n a t e d by the stress criterion t h e n their view shall be verified. It will be a s s u m e d that at the allowable steel stress, fa, the two phases will be totally d e b o n d e d and that the m o r t a r will experience total stress relief and strain recovery. T h e r e f o r e , the elongation of the r e i n f o r c e m e n t b e t w e e n two adjacent cracks will be equal to the crack width

W=l×(L/Es)

(13)

Paul & P a m a 2 have p r o p o s e d the following relationship for /, based on test results by various w o r k e r s (Fig. 10) t=

1.5 1.6

x

1 SRL

(14)

NOW, consider the most unfavourable conditions. Given that SRL = 0"08 m m for water retaining and 0-04 m m for c o m m o n structures, Vy= 1.8% for water retaining a n d 0.9% for c o m m o n structures, cover thickness = 5 ram, wire diameter = 0-9 m m and m o r t a r strength = 4 MPa. T h e c r a c k spacing, l, can be calculated by using first e q n (11) and t h e n eqn (14) as follows: F r o m eqn ( 11 ) lw, for water retaining = 3 x 5 + 0-26 x 0.9/(1.8%) = 28 m m l c, for c o m m o n structures -- 3 x 5 + 0.26 x 0"9/(0"9%) -- 41 m m

F r o m eqn (14) lw, for water retaining = (1.5/1-6) ×(1/0.08) = 11.72 m m /c, for c o m m o n structures = ( 1.5 / 1-6) x (1/0.04) = 23.44 m m Overestimates of crack width, IV, can be n o w obtained f r o m eqn (13). T h e results are presented in Table 1 along with those obtained by using eqn (12) (by C h e n and Z h a o ) and eqns (9) and (10) (by ACI). As can be seen, all these overestimates of crack width are smaller (sometimes greatly) than the allowable values, for all applications as specified by ACI.

CONCLUSIONS T h e available m e t h o d s of predicting crack width are far f r o m reliable. Fortunately the dominating design criterion for most applications is not the crack width but the steel stress which can be reliably modelled. This presentation raises an obvious question: Should research efforts and publications reflect the a n s w e r to the question raised in the title of this p a p e r ? T h e a n s w e r has to be yes.

12

G. Singh, G. J. Xiong

REFERENCES 1. Somayaji, S. & Shah, S. P., Prediction of tensile response of ferrocement. International Symposium on Ferrocement, RILEM/ISMES, Bergamo, Italy, July 1981, pp. 1/73-

1/83. 2. Paul, B. K. & Pama, R. P., Ferrocement, IFIC Publication No. 5/78, IFIC Bangkok, Thailand, 1978, pp. 36-78. 3. Akhtaruzzaman, A. K. M. & Pama, R. P., Cracking behaviour of ferrocement in tension. In Proceedings of 3rd International Conference on Ferrocement, Roorkee, India, 1988, pp. 3-11. 4. ACI Committee 549, State-of-the-art Report on Ferrocemerit, Report No. ACI 549R-82, Concrete International, Detroit, USA, 4 (8)(1982) 13-38. 5. Naaman, A. E., Design predictions of crack width in ferrocement, Ferrocement -- Materials and Applications, SP61, American Concrete Institute, Detroit, 1979, pp. 25-42.

6. Chen, X. B. & Zhao, G. F., The calculation of crack width in ferrocement under axial tension. In Proceedings of 3rd Conference on Ferrocement, Roorkee, India, 1988, pp. 12-20. 7. Pama, R. P. & Paul, B. K., Study of tensile cracks and bond-slip in ferrocement, Ferrocernent -- Materials and Applications, Special Publication SP-61, American Concrete Institute, Detroit, 1979, pp. 43-79. 8. Singh, G., Venn, A. B., Lilian, I. P. & Xiong, G. J., Alternative material and design for renovating man-entry sewers, No-DIG89, In International Conference of International Society for Trenchless Technology, London, UK, 1989. pp. 2.3.1-2.3.7. 9. Venn, A. B. & Singh, G., The development of ferrocement for use in sewers and tunnels, No-DIG88, In International Conference of International Society for Trenchless Technology, Washington DC, USA, 1988, pp. 1-7.