Shear resistance of beams based on the effective shear depth

Ir. A.F. Pruijssers

T H Delft

Section of Concrete Structures Stevin Laboratory Delft University of Technology

Shear resistance of beams based on the effective shear depth

by

Ir. A.F. Pruijssers

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Mailing address: S T E V I I M

Delft University of Technology ^ ^ '^Kal Concrete Structures Group

Stevin Laboratory II Stevinweg 4

2628 CN Delft LABORATORY

The Netherlands CO«C».TE STPUCTU-ES

No part of t h i s report nay be published without written permission of the author.

Technische Hogeschool ^ Bibliotheek Afdeling: Civiele Techniek

^TL S'-Sé"/. Stevintveg I

' postbus 5048 2600 CA Delft

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Summary

1. Introduction 1

2. Existing theoretical models and empirical formulas 3

3. Mechanism of shear resistance based on basic material properties

3.1 Introduction 11 3.2 Mechanism of shear failure for a/d ) 2.5 12

3.3 Mechanism of shear failure for a/d < 2.5 23

4. Comparison of the model with existing formulas

4.1 Introduction 27 4.2 Comparison of Rafla's formula and the model derived 29

4.3 Comparison of Zsutty's formula and the model derived 31 4.4 Comparison of the Bazant/Kim formula and the model derived 33

5. Beam behaviour according to the model with effective shear depth

5.1 Introduction 35 5.2 Characteristic lower bound of the shear resistance 35

5.3 Time-dependent behaviour of beam with web reinforcement 37

5.4 Shear resistance of T-beams 38

5.5 Distributed loads 40 5.6 Lightweight concrete 44

6. Synopsis 47

8. References 51

Appendix I 55 Appendix II 57 Appendix III 63 Appendix IV 67

expressions. A reliable empirical formula is derived by Rafla [10]. This formula is based on 442 experimental results.

In this report no experiments are described. It is endeavoured to make a new contribution to the theoretical study of shear strength. The model presented in this report is based on the idea that apart from the effective depth of the compressive zone there is an effective shear depth. The model developed is compared with empirical expressions defined by Rafla and other investigators. The results are promising.

With the model the time-dependent behaviour of the shear resistance, the strength criterion and the durability are discussed. For beams with web reinforcement it is explained which mechanism causes the almost constant contribution of the 'compressive zone' to the shear resistance in spite of the increasing deformations.

According to the model derived, a formula is presented for lightweight concrete by substitution of the proper material properties into the general expressions for the shear resistance. This formula is in good agreement with the experinents reported by other investigators.

Due to the complex internal equilibrium in the case of (partially) prestressed concrete the influence of a normal force on the shear strength is not discussed in detail. The general principle is presented and an example is given. The observed experimental behaviour can be explained with the model derived.

1. INTRODUCTION.

Extensive experimental and theoretical studies into the strength of beams subjected to bending have been carried out. As a result of these investigations there is a clear understanding of the mechanisms affecting the flexural strength of reinforced concrete beams. Contrary to this, the mechanisms affecting the shear failure of beams are not yet clearly defined. The brittle nature of shear failure makes this type of failure very dangerous and requires reliable formulas in the building codes. Because the theoretical understanding of this important failure mechanism is not complete, all the formulas in the current building codes are empirical. The influence of all the important parameters, such as concrete grade, reinforcement ratio of the longitudinal bars and depth of the beam, have been investigated by many research workers in the past decades. There is uo lack of experimental results, but of theoretical understanding. Therefore no new experimental results are reported in this study. The main goal of this report is to make a fresh attempt to formulate the mechanisms which cause shear failure. The model described in this report will be compared with existing experimental data. Because of the impossibility of collecting all the experimental results reported by former research workers, the model will be compared with the empirical formulas which show close agreement with the experimental results. The contribution of web reinforcement to the shear resistance is fairly well described by the truss analogy. Therefore most attention is paid to the mechanisms which contribute to the shear resistemce in beams without stirrups. For a simple understanding of the model presented here, beams subjected to point loads are studied. Some coaments will be made on beans siibjected to distributed loads.

The expressions derived in this report will be used to describe the shear strength of T-beams and lightweight concrete beams. For these beams the shear resistance according to the model shows a close agreement with experimental results obtained in tests ceu-ried out by Leonheurdt [11].

In this report no extension is made to (partially) prestressed concrete. In the case of (partially) prestressed concrete the normal force acting in the cross-section influences the depth of the ccmpressive zone. According to the

model derived in this report the shear resistance depends upon the depth of the compressive zone. Although the model provides a qualitatively good description of the observed behaviour of the beam, the model is too complex to be presented briefly in this report.

2. EXISTING THEORETICAL MODELS AND EMPIRICAL FORMULAS.

The theoretical modelling of the mechanism of shear failure in beams without shear reinforcement was based on the assumption that the whole cross-section of the beam contributes to the shear resistance Vcu. This assumption is true for uncracked beams and is presented in the form:

tu = Vcu/bd (2.1)

In spite of the increasing knowledge of the actual behaviour of a beam subjected to a shear load, expression (2.1) is still used in the current building codes. A model which gave a fairly good description of the experimental observations was Kani's comb-analogy [1,2]. See Fig. 2.1.

Fig. 2.1 Kani's comb-analogy [1,2].

In this model cracks develop due to the bending mcnent, which exceeds the cracking moment. The concrete 'teeth' between these cracks are considered as cantilevers, fixed to the uncracked arch. Such a cantilever is loaded due to

the bond between the concrete and the longitudinal reinforcenent (Fig. 2.2).

arch

AF = F, • F2 = ^i-'^2 = M

When the tensile stress at the root of the tooth exceeds the tensile strength, the flexural crack will be transformed into a shear crack which separates the cantilever from the arch. In general the remaining arch is not capable of resisting the shear force. Although there is good agreement with the observed behaviour of the beam, the model underestimates the measured shear capacity.

For this reason Kani's model was extended by Fenwick, Paulay [3] and Taylor [4]. The cracks, which cause shear failure are curved (see Fig. 2.3). Due to the deformation of the beam the opposing crack faces slide over each other.

Fig. 2.3 Beam with shear cracks.

The rough surface of the crack will resist this displacement. This mechanism is called aggregate interlock and is described by the model of Walraven [5]. There is also a contribution by dowel action of the longitudinal reinforcing bars. The magnitude of the dowel force can be calculated with the empirical formula given by Baumann [6]:

Fdowel = 1.64 * bn * • * V f c c » (2.2) with bn = b - 2(*)

The experimental work of Taylor [4] showed that approximately 25-45* of the total shear force is transferred by the uncracked compressive zone (arch), 35-505K by the mechanism of aggregate interlock and 15-25* by dowel action.

indicated that the ratio between the crack width and the shear slip of the crack at the bottom of the beam is approximately 3. The crack opening path is shown in Fig. 2.4.

.separation [mm

0 0.01 0.02 0.03 ODL 0 0.05 0.10 slip [mm] slip [mm]

Fig. 2.4 Crack opening path in beams subjected to shear forces [4,18]

According to Walraven's model [5] the shear stress transferred by the mechanism of aggregate interlock is about 0.5 [N/mm^], which is a rather low

value. Furthermore, the upper half of the crack tends to open normal to the crack plane because of the position of the point of rotation. Hence the shear transfer across the crack makes a small contribution to the shear resistance of the beam.

Bazant and Kim [7] made an analysis of the shear failure taking into account the structural size effect. This effect, reflecting the non-linear elastic behaviour of the beam, is expressed by:

(1 +d/(Dn.ax*Xo))°-5 _(2.3)

in which d = depth of the beam

Daax = maximum particle size Ao = empirical parameter

The horizontal projection of the inclined crack was taken as equal to the effective depth d of the beam. A straightforward derivation of the internal equilibrium of the forces yields:

rcu = ki*pP*(fc'»+k2*yp/(a/d)'")*(l+cl/(Dmax*Xo))-o-^ (2.4)

in which ki, ka, Xo, p, q, r are empirical constants.

Substitution of the empirical constants lead to:

rcu = 10*3yp'*(2/f^+3000*Vp/(a/d)s)*(l+d/(Dmax*25))-o-5.[psi] (2.5)

= 0.16*3yp^*(2yfr77+27.9*^po/(a/d)^)*(l+d/(Dn,ax*25))-o-5 [N/mm^]

This formula is similar to the equation used in the ACI code and shows close agreement with the results of 296 experiments. See Fig. 2.5.

calculoted shear s t r e n g t h [ p s i ]

Fig. 2.5 Comparison of the formula of Bazant/Kim with experimental data [7].

The work of Bazant and Kim [7] is both theoretical and empirical. Now some empirical formulas are presented for comparison with the model developed in this report.

Zsutty [8,9] derived the following empirical expressions for the inclined cracking strength tcr on the basis of 377 experiments: Note that the shear stress calculated with eq. (2.6)-(2.7) is the cracking strength Tcr. For that stress level the flexural crack is transformed into a shear crack. The cracking strength is 0-15% lower than the ultimate shear stress.

Tcr,a/d)2.5 = 60*3/fc*p*(d/a) [psi] (2.6) 0.44*3/fcci«*Po*(d/a) [N/mm2]

rcr,a/d<?..b '- 150*3yfc*p*(d/a)4 [psi] = 0.96*3/fc*po*(d/a)'* [N/mm2]

(2.7)

Zsutty draw a distinction between the shear resistance of beams with a low shear slenderness ratio a/d < 2.5 and beams with a high shear slenderness ratio (> 2.5). The same distinction is drawn by Kani [1,2] in his theoretical model. For point loads close to the support Kani defined another mechanism which caused shear failure. After the separation of the concrete teeth, the remaining arch is still capable of resisting the shear force (See Fig. 2.6). This mechemism will be discussed in detail in Chapter 3.

Fig. 2.6 The remaining arch according to Kani [1,2].

The marked increase in shear resistance for low shear slenderness ratios is clearly shown by the empirical formula of Rafla, which is in close agreement with 442 experimental results [10]. This expression is derived for the actual ultimate shear stress:

2.0 10 iO SM SO lOa/h SO

tu = ttu * d-o-2& * yj^ * 2/fr^ [N/rama] (2.8)

The parameter ttu takes into account the influence of the shear slenderness ratio a/d and is presented in Fig. 2.7.

For experiments on beams with shear reinforcement the observed behaviour deviates from the behaviour of a beam without stirrups. For beams with stirrups the cracks in the shear zone of the beam start vertically at the bottom of the beam. Directly above the longitudinal reinforcing bars the

cracks undergo an abrupt change of direction. See Fig. 2.8.

^

1

1 i>M1i^l^/

F

\\^^#kK^^

All the cracks propagate parallel to one another. This observation yielded the formulation of the truss analogy by Hitter and Mbrsch. According to this model the shear force is transferred by concrete compressive struts and steel tensile bars. See Fig. 2.9.

However, experimental results [llj indicated that the lattice overestimates the steel stress in the stirrups. See Fig. 2.10.

t h e o r y 3500 P, 3000 \7500 •5 POOD &;5a7 ^7000 •*» 500 0 -500 n j t t u iLULi^p.-p/ '"'83-36 'i'^SZ i ' • \ ^ '

fr

/ '

^<t^r /1/ i /I

Fig. 2.10. Experimental results obtained by Leonhardt [11].

The difference in shear resistance A^ between the lattice theory and the experimental results appeared to be equal to the shear resistance of a beam without stirrups.

Discussion

The difference in shear resistance between beams with and without stirrups can be described by the lattice theory. For this reason attention is focussed on the mechanisms affecting the shear resistance in beams without stirrups. According to Fenwick, Paulay [3] and Taylor [4], the shear resistance of beams without shear reinforcement is largely based upon the mechanisms of aggregate interlock and dowel action. The difference between the measured shear force and shear force calculated according to the lattice theory, the force ^V, should depend upon the mechanisms described by Fenwick,Paulay and Taylor. Because /)V equals the shear resistance ( or the cracking strength) the contributions of aggregate interlock and dowel action to the shear resistance must be the same for beams with and without stirrups. The same holds true for the crack-opening path for these types of beam. However,

experimental observations showed that cracks in beams with shear reinforcement open perpendicularly to the crack direction. So no aggregate interlock or dowel action is to be expected.

The same observations showed that the magnitude of A\ remained almost constant with increasing deformations of the beam. Increasing crack deformations should result in a greater contribution of aggregate interlock and dowel action to the shear resistance.

It is the author's believe that the mechanism of aggregate interlock in the crack and the dowel action of the longitudinal reinforcement contribute to the shear resistance by bringing about a redistribution of shear stress. This redistribution of stress occurs during failure of the compressive zone of the beam. Therefore this contribution remains very small.

Kani's comb-analogy [5] provides an explanation for the initiation of shear failure by separation of the concrete teeth. However, there remains the question as to which mechanism is responsible for the transfer of the shear

force to the support. This mechanism has to be in agreement with the above statement that increasing deformations hardly influence the shear resistance. Therefore, the relation between the shear force, bending moment and the strain of the compressive zone will be estimated in Chapter 3.

3. MECHANISM OF SHEAR RESISTANCE BASED ON BASIC MATERIAL PROPERTIES

3.1 Introduction

In this Chapter an attempt is made to extend the Kani's comb-analogy [1,2] in order to derive an expression for the shear resistance of a beam with longitudinal reinforcement. As Kani already pointed out, there exist several mechanisms of shear failure. From observations during experiments it appeared that the parameter influencing the type of failure is the shear slenderness ratio a/d.

For large ratios of a/d () 2.5) a beam fails due to the separation of the concrete cantilevers defined by Kani. The depth of the compressive zone is diminished by the propagation of the shear crack and is no longer capable of

resisting the shear force. See Fig. 3.1.

320 270

190 Fig. 3.1 Shear failure for a/d ^ 2.5 [11]

For small ratios of a/d (< 2.5) the remaining arch can resist the shear force after separation of the concrete teeth. Now failure occurs by crushing of the compressive zone. See Fig. 3.2. The boundary between both failure mechanisms depends upon material properties. Therefore, the shear resistance according

Fig 3.2. Shear failure for a/d < 2.5 [11].

For very small ratios of a/d (< 1.5) splitting failure may occur. Because this mechanism is not a typical shear failure, no attention will be paid here to this subject. In the next section the mechanism for large shear slenderness ratios will be defined.

3.2 Mechanisn of shear failure for a/d ^ 2.5.

Kani's comb-analogy [1,2] provides a good description of the behaviour of beams subjected to shear. However, the predicted shear resistance underestimates the measured strength. Because of the easy-to-understand description of the shear mechanism with this model the comb-analogy is used here. The model is adjusted to the effective shear depth defined by the author.

In Fig 3.3 the comb-analogy according to Kani [1,2] is represented. It is' obvious that shear failure is caused by the separation of the concrete teeth from the arch. The remaining arch is not capable of transmitting the shear force to the supports.

Fig 3.3 Kani's comb-analogy [1,2].

However, the cantilevers transmitted no shear force to the support. The uncracked arch resists the shear force. Therefore a detailed representation of the arch is given in Fig. 3.4.

At the onset of shear failure the crack propagates into the compressive zone, decreasing the effective depth of the uncracked area. In front of the crack

Fig. 3.4 Detailed representation of the arch

tip there is a zone, in which the tensile strain exceeded the fracturing strain. This zone is called the fracture zone (or tension-softening zone) and consists of concrete intersected by small micro-cracks. See Fig.3.5.

fictitious crack or tension-softening zone T visible crack

I

I

-7^-Fig. 3.5 The fracture zone

Due to the development and opening of the micro-cracks the strain in this zone increases with a decreasing tensile stress. This phenomenon is presented in Fig. 3.4 as well as in Fig. 3.6.

^0 [N/mm^]

0,

w

A

/ V

f ^

0 Q(

35 0.1

0 0'

15 0.2

A(mnn

Fig. 3.6 Stress-deformation relation in tension-softening zone

For calculations this relation is approximated by the dotted line in Fig. 3.6. The ultimate strain in the tension-softening zone (just in front of the crack tip) is estimated to be eleven times the cracking strain, which can be approximated as lO"*:

Ectu = 1.1 * 10-3 (3.1)

For this value of ectu the fracture energy is in good agreement (125 [N/m] with the experimental results of Körmeling [20]. In Fig. 3.7 a detailed

representation of the concrete teeth is given. It is obvious that this tension-softening zone hardly influences the normal stress distribution in the cross-section of the beam.

Fig. 3.7 Concrete teeth

However, the concrete teeth deform due to the bond between the concrete and the longitudinal reinforcing bars. At the root of the teeth the tensile strain is reached and another tension-softening zone will develop. Due to this softening the teeth will slide over each other close to the root, where the crack width is very small. The micro-cracks in zone I (See Fig. 3.7) undergo a shear deformation. In [12] it is shown that the model of aggregate interlock can be used to describe the shear stiffness of the tension-softening zone. For large shear deformations the contribution of the tension-softening zone to the shear transfer is considerable. This is shown in Fig. 3.8. In this diagram three situations are presented; (Tnn = 3.0, 2.0 and 1.0 [N/ram^]. For these nohnal stresses the normal strain of the element Enn is kept constant during the increase in the shear deformation Y- For increasing ratios of y/tnn the shear- and normal stress are shown. According to the Walraven's model [5] the magnitude of the shear force transferred by the tension-softening zone is influenced by the concrete grade, gradation of the particles, maximum particle size and the crack-opening path. In [12] a regression analysis made on the theoretical model of Walraven [5] yielded the following expressions for the stresses transferred by aggregate interlock:

Cv. (N/mm'l

3 - 0„„=3 see tig. 5.2 / : ^ " ^ < ^ o„„

! i 1

/ '"""-"--^i 0„n = ' swtiq. SA]

/ i . 1 . 1 . .. ^ 0 1 2 3 4 5 6 7 8 9 10 n 12 formula Rots [6] — eq. chapter 5.1 . - . eq. chapter 5.2 t.o

L

Fig. 3.8. Shear transfer in the tension-softening zone [12]

ragar = Dinax°-l''*;^^^.„#(0.76*Y/Cnn-0.16*(l-exp(-6*Y/£nn))) (3.2)

(Tagcr = DmaxOl'»*2/fecm*(0.29*Y/Enn-0.13*(l-exp(-4*Y/enn))) (3.3)

The derivation of the eq. (3.2) and (3.3) is given in [12]. Bazant and Kim [7] mentioned the influence of the particle size upon the shear resistance of a beam. This influence is reflected by a factor defined in eq. (2.3) (Chapter 2). However, according to the eq. (3.2) and (3.3) the influence of the variation of the maximum particle size is limited to about 5-lOX. Hence no attention is paid to this minor influence of the maximum particle size. The factor defined by Bazant and Kim may reflect the scale effect mentioned by

Heinhardi [15,16]. This will be discussed in this Section relating to the flexural tensile strength.

Due to the fact that the tension-softening zone transfers shear stress, the depth of the arch is not the depth hx of the uncracked compressive zone, but equals the so-called effective shear depth h^ • See Fig. 3.9.

ea,. = 1.1x10 strain normal stress real shear stress

a.x,

Fig. 3.9 Representation of the effective shear depth h^

The shear r e s i s t a n c e of a beam can now be defined by the simple r e l a t i o n :

(2/3*a) * to • b * h,

in which To = maximum shear stress

a = parameter representing the effectiveness of the tension-softening zone in transferring shear stress.

h^ = (lEcl +|£ctu| )/|Ec| * hx (3.4a)

It is obvious that the weakend tensile zone has a low shear stiffness compared with the compressive zone. The tension-softening zone also reflects the shear-softening zone. This softening is shown in Fig. 3.10. The parameter a is used to translate the real shear stress distribution into a parabolic distribution with the same total shear force.

0 1

axis

Fig. 3.10 The shear stress - normal strain relation in the tension-softening zone.

This effectiveness of the tension-softening zone can be related to the stiffness of the compressive zone by:

a = ( IECI + l Y * S c t u l ) / ( l £ c l + i E c t u I ) (3.5)

in which Y ~ parameter taking into account the stiffness of the

tensile zone related to the stiffness of the compressive zone.

In this equation the effectiveness parameter Y is implicit. For the use in this report the effectiveness parameter is written explicitly. By means of a regression analysis this relation is approximated by:

a = Z/y-* ( l E c l + | E c t u | ) / | £ c l + | € c t u | )

a = 2/y- _(3.6)

The magnitude of tt is shown in Fig. 3.11 for the equations (3.5) and (3.6). The effectiveness parameter Y is varied between 0.3 and 0.7. The magnitude of Y depends upon the deformation of the tension-softening zone. This deformation is not known beforehand.

Q8 0.6 0.A 02 0.(

?

1 i

b ^

strains important for

3.5 3.6 the me «ctianisn

1

Fig. 3.11. T h e parameter a in eq. (3.4)

Therefore Y cannot b e calculated on a purely theoretical basis. However, in Appendix IV an attempt is m a d e to calculate Y according to the theory presented in [ 1 2 ] . Within the limits o f the assumptions m a d e for this calculation it appeared that the magnitude of Y is approximately 0.44. In Chapter 4 it will b e shown that this value is in good agreement w i t h the experimental results. It must b e noted that the contribution of the mechanism of aggregate interlock according to Walraven [5] to the shear transfer in the tension-softening zone is less than 10S5. The area of the tension-softening zone, which tranfers t h e tensile stress, transfers about 9 0 % o f the shear stress. (See Appendix I V ) .

The maximum shear stress is calculated according to the failure criterion of Mohr-Coulomb. See Appendix I:

tu - (fcm*fctm)/(fcm+fctm) (3.7) 25 \ OS

n

It is a well known fact that the tensile strength for a beam subjected to a flexural moment is influenced by the depth of the beam. See Fig. 3.12. This is due to the gradient of the strain in the tensile zone and the non-linear stress-strain relation. To take into account the influence of the depth the following empirical relation is used:

CTcr.fj = (0.6 + 0.4 * h - o s ) * (Tcr (3.8)

with h [m]

This so-called flexural tensile strength is used in linear-elastic calculations. The tensile strength is size-dependent. The parameter defined by Bazant and Kim may reflect the influence of the beam depth upon the tensile strength. The actual nonlinear distribution of the tensile stress will influence the real distribution of the shear stress. The distribution of

the shear stress in the tensile zone will be nonlinear due to the nonlinearity of the tensile stress distribution. In a linear-elastic

situation, relation (3.7) is valid. Teiking into account that the compressive strength is about 80 percent of the mean cube compressive strength and the

tensile is about 10 percent of the mean compressive strength, this yields:

tu = (0.8*fccm*0.1*fccm)/(0,8*fccm+0.1*fccm) = 0.09* fccm

= approx. 0.9*fctm (3.9)

For the nonlinear stress-situation the fictitious flexural tensile strength 0"cr,fi is used instead of fctm. The most convenient way to take into account the nonlinear distribution of the shear stress is the definition of a

fictitious flexural shear strength in analogy to (3.8):

tu.fi = (0.6 + 0.4 * h-o-6) * Tu (3.10)

In linear elastic analysis the flexural shear strength tu,fi is used instead of the pure shear strength tu. Therefore:

to = tu,ƒ 1 (3.11) (See Fig. 3.9)

With eq. (3.9) and (3.10) this yields:

to - approx. 0.9*(0.6+0.4*h-°-6)*fctiD (3.12)

Now eq. (3.4) c£ui be written as:

Vcu = 2/3*^*0.9*(rcr, f i*h.j.*b (3.13)

Eq. (3.13) is the basic formula for the calculation of the shear resistance of beams. This expression is in close agreement with the linear elastic shear stress distribution in an uncracked beam:

Vcu = 2/3*tmax*h*b (3.14)

Now expression (3.13) can be made valid for various materials by substitution of the proper material properties.

In the linear-elastic stress-strain theory, there is a straightforward expression for the depth of the compressive zone hx:

hx = d*(-np+2/(np)2+2np ) (3.15)

This expression is derived neglecting the tensile strength. Because of the minor influence of this tensile strength the relation hardly changes by taking into account the tensile strength. For use in this report all the relations are written as power functions. Therefore, in Appendix II a simple expression is derived for hx:

hx =^ 0.521*fccm-°-i=^i*po'3-377*d (3.16)

In the same Appendix an expression is defined for the flexural tensile strength: (two-point loads)

(7cr.fi = 5.62*p*cltE-°-2i>*0.277*fccm°-s67 (3.17)

with P = time-dependent parameter =^ 1.0 for t = 0

= 0.6 for t = oo

dts = 2 * depth of the tension-softening zone

For the depth of the tension-softening zone dts an expression is derived in Appendix II: dts = 1 . 1 4 6 * (d/lEcl )*po°-3''''*fccii.-°i5i (3.18) Finally we obtain: V c v i . m h j hx Cc r , f 1 d t s V c u . . 3 . L 4 3 . 4 a 3 . 1 6 3 . 1 7 3 . 1 8 3 . 1 9 0.6 / 7 ( U l + U c t u D / J E l 0..521 1.56 p d t s - o ^i-1 . ^i-1 4 6 - 0 ^ 5 |e|o 2 5 Cc r , f I ti -r b hx f c c . - ° ' 5 ' f c c » ° « ^ o ' ' f c c . u o ^ a p „ 0 . 3 7 7 PJJ - Ü , Ü 9 4 d ( 1 - Ü . 2 5 0 . 4 8 ( | £ | + lEctuI ) / | E p ^ 5 P / Y f e e . " •!>=>•> p o O ^ a a d - o - 2 * bd Vcu.n. ^ 0.48*a*d-'^-25#p„0.283*f^^„0.5b4*b*d (3.19)

with a = p*i'Y*(lec*103H-l.l)/|Ec*103 |o.75 (3.20)

It is obvious that the strain of the compressive zone Ec is a function of material properties, reinforcement ratio, geometry and the shear slenderness ratio. For linear-elastic material behaviour the following relation is valid

(modular ratio method):

In Appendix II it is shown that eq. (3.21) can be used to write eq. (3.20) in the form:

tt = 2.44*Y-°-^i*p-°-22*(a/d)-o-22*doo5*fc^„-o.o6*po0.oi (3.22)

Now the mean shear resistance Vcu.m can be written as:

Vcu,« a V c u , . 3 . 1 9 3.-J2 3 . 2 3 ^ ^ 0 . 4 8 ( 1 6 | + | E c t . l ) / 1 6 p ^ 5 P / T 2 . 4 4 Y"°- " p - ' ' - ^ ^ ( a / d ) - o - 2 i 5 f c c . ° " 4 f c c . ' ° ° ^ p „ o . 2 a 3 p „ o . o i d-0 25 (jO . 0 b 4 bd 1.17 po •'8 Y ° ^ * ( a / d ) ° 2 1 * f c c . ° " « • pe» 2«'' d-o 196 bd Vcu,n.= au*d-0-2*p„0.3*f^.^„0.5)(tb*d (3.23) with ttu = 1.17*po-8*Y° '»*(a/d)-o-2

p = 1.0 for t= 0 p = 0.6 for t- oo

3.3 Mechanism of shear failure for a/d < 2.5

Kani [1,2] already indicated that the separation of the concrete teeth does not initiate shear failure for small shear slenderness ratios. In that case the remaining arch is still capable of carrying the shear force. Kani stated that due to the separation of the teeth there is a transformation of the reinforced beam into a tied arch. See Fig. 3.13. During this transformation the cracks develop according to the internal stress trajectories (lines in the direction of the principal stresses), thus influencing the shape of these trajectories. The development of the cracks is shown in an idealized form in Fig. 3.14. The crack nearest to the point load will arrive at the neutral axis at 45 degrees, while the internal stress conditions are hardly changed by the first crack. However, the development of this crack disturbs the internal stress conditions. Therefore, the next crack will have a different shape. The depth of the remaining arch decreases due to the development of

the shear cracks.

Fig. 3.13 The remaining arch

^(-^-^

Fig. 3.14 The development of the cracks.

In the idealized form as shown in Fig. 3.14 all the cracks have the same intersection point. It must be noted that usually only the last crack develops fully. However, the decrease of the depth of the compressive zone remains the same. The ratio between the depth h'^ of the remaining arch and

the depth hx of the compressive zone can be expressed by: (see Fig. 3.14)

hV / hx = d/(a-s+hx) _(3.24)

Note that the depth h^ differs from the effective shear depth used in Section 3.2.

According to Kani the distance of the first crack from the support s is of the same magnitude as the effective depth of the compressive zone. With s almost equal to hx this yields:

h!j / hx - d/a (3.25)

Kani assvuned that the distribution of the stress in the remaining arch is almost the same as it is in the original compressive zone. Contrary to this, we asssume that due to the separation of the teeth the remaining arch is no longer forced to rotate under the flexural moment. Instead of this beam-like model there is a system of a centrically loaded compressive and tensile zone. See Fig. 3.15.

Fig. 3.15. System of centrically loaded compressive and tensile zone.

In that case the distribution of the normal stress is constant. The ultimate compressive force can be expressed by:

Nc u , m - Cc u * h'- * b (3.26)

U s i n g e q . ( 3 . 2 5 ) and (Tcu = 0.8*fccin t h i s y i e l d s :

Ncu.m = 0 . 8 * ( d / a ) * f c c m * h x * b (3.27)

compressive force. With eq. (3.IB) and eq. (3.27) the following expression is found for the sheiir resistance:

Vcu.m = 0.42*p'*(d/a)2*po'^-^''''*fccm'^«5*b*d (3.28)

in which P' = time-dependent parameter = 1.0 for t = 0

4. COMFAHISON OF THK MODEL WITH EXISTING FOHMULAS.

4.1 Introduction

The model developed in Chapter 3 is based on two mechanisms. The shear resistance can therefore be calculated with two expressions;

For a/d > 2.5:

Vcu.m = au*d-o-2*po0.3j(cf^c„o.b#5*d (3.23)

in which ttu = 1 . 1 7 * Y O " * P ° 8*(a/d)-o-22 p = time-dependent parameter

= 1.00 for t = 0 = 0.60 for t = oo

For a/d < 2.5:

Vcu.n. = 0.42*p'*(d/a)2*po0.3a*f^^„o.85*b*d (3.28) in which P' - time-dependent parameter

:^ 1.00 for t ^ 0 =0.85 for t = oo

According to the theory presented in Chapter 3 the mechanism described by eq. (3.28) becomes active after failure according to eq. (3.23). The same subdivision into two expressions is found in all the empirical formulas mentioned in Chapter 2, except for the expression defined by Bazant and Kim

[7].

The most simple way of comparing the different expressions is by substituting concrete grade, reinforcement ratio and geometry into the formulas. In order to keep the comparison as general as possible, all the expressions are written in the same form. In this report the formula defined by Rafla [9] is chosen because it is based on over four hundred experimental results. An even stronger reason is the clear presentation of the two mechanism as shown in Fig. 4.1

Momenten - SchuO - Vernottnrs

Fig. 4.1 The parameter «u [10]. (Fig. 2.5)

In Fig. 4.I the influence of the shear span to depth ratio upon the shear resistance is shown using parameter ttu. In the expression defined by Rafla three regions of a/d are used [10]:

tu = ttu * d-°-25 * 3jJ^ * ^/fcir [N/mm2] (2.8)

with

for a/d < 2 ffu = 6 - 2.2*a/d

for 2 < a/d < 3.5 ttu = 0.795 + 0.293*(3.5-a/d)2 s for a/d > 3.5 ttu = 0.9 - 0.03*a/d

The region 2 < a/d < 3.5 can be considered as an interconnection between the two main regions.

In the following Sections the expressions defined by Bazant and Kim [7], by Zsutty [8,9] and according to the model developed in this report, are written in the form of eq. (2.8). The ttu.com, which is a result of the translation of the formulas, will be compared with the parameter ffu represented in Fig. 4.1.

4.2 C<Nq>arison of Uafla's formula and the model derived

The eq. (3.23) is written in the form of eq. (2.8). This yields for the parameter oCu, c o m: V c v , . . V R a f La ttu,COB 3 . 2 3 2 . 8 4 . 1 1 . 1 7 P" 7 0 Y O . 3 9 ( a / d ) - 0 21S ttu , c 0 m f c c . 0 ' > 9 4 f e e . ° - 50 P„0 2B4 p „ 0 . 3 3 3 d - 0 . 1 9 6 ( 1 - 0 . 2 5 bd bd 1 . 1 7 PO-TB Y O . 3 9 ( a / d ) 0 . 2 1 £ . f ^ ^ , - O . O 0 6 p „ - O . 0 4 B d D . 0 ! . 4 b d

ttu.CO» = 1.17*Y0-392*p0.785*d0054#(a/d)-0-215 (4.I)

The eq. (3.28) is written in the form of eq. (2.8). This yields for the parameter Cu, c om:

V c u , . V R a f 1 a ttu . C O . 3.28 2 . 8 4 . 2 0 . 4 2 p ' ( a / d ) - 2 OCu , c o B f e e . ' ^ a s f c c m " - 5 0 p < , 0 . 3 7 7 p „ 0 . 3 3 3 d - 0 2 5 bd bd 0 . 4 2 f ( a / d ) - 2 f e e . " a s po" o** # " 2 5 bd ttu.com = 0.42*p'*dO-25*(c[/a)2*fccin°-35 (4.2)

The eq. (4.1) and (4.2) will be compared with the ttu according to Rafla. For a proper comparison an estimate is made of the time-dependent losses in the experiments used by Rafla. In general, an experiment leisted at least one or two hours. In Fig. 4.2 the time-dependent behaviour of the tensile strength is represented. During the first houres the decrease of the tensile strength is considerable. Therefore, the time-dependent parameter P is taken equal to 0.75. This is 63% of the total losses. Taking into account this 63* for the time-dependent parameter p', this parameter is 0.90 (= 1-0.63*0.15).

The parameter ttu is still dependent upon the concrete grade and the effective depth and the shear stiffness parameter Y- In Fig- 4.3 mean values

10 100 1000 10 000

RUPTURE TIME-h

Fig. 4.2 Time dependent behaviour of the tensile strength [22]

8

10

a/d

Fig. 4.3 Comparison of Rafla's equation and the model

for the conrete grade and the beam depth have been used:

fccm = 39 N/ram2 d = 500 mm

For Y = 0.49 there is close agreement between the model and Rafla's equation. This value of Y is almost equal to the theoretical value (0.44) derived in Appendix IV. The difference of approximately 10% may be caused by the contribution of dowel action and aggregate interlock. Note that the

intersection point between the lines described by the eq. (4.1) and (4.2) is a/d - 2.84 , which nearly equals the familiar value of 2.5.

In Appendix III an analysis is made of the sensitivity of the model for different concrete grades and beam depths. It is shown that the model is insensitive for changes in the concrete grade and the beam depth for higher shear slenderness ratios.

4.3 Comparison of Zsutty's formula and the model derived

The eq. (2.6) is written in the form of eq. (2.8). In this case the parameter ttu.com is expressed by:

f c r T R a f 1 a • « , C O . 2 . 6 2 . 8 4 . 3 0 . 4 4 ( a / d ) o - 3 3 3 (tu , C O B f c c n , 0 - 3 3 3 f e c . o SO p < , 0 . 3 3 3 p < , 0 . 3 3 3 d - O . 2 5 0 . 4 4 ( a / d ) 0 3 3 3 f e e . - ° 1 6 7 J O . 2 5 ttu.com(a/d ) 2.5) = 0.435*do-25*(a/d)-o.33*f^^„-o.i67 (4.3)

The cracking shear stress according to eq. (2.7) yields a parameter ttu.com:

r e r TRaf l a O u . C O . 2 . 7 2 . 8 4 . 4 1 0 . 9 6 ( a / d ) i -333 Au . c 0 D f e e . 0 3 ^ 3 f e c . ° s ° p „ 0 . 3 3 3 poO 3 3 3 ( J - 0 . 2 b 0 . 9 6 ( a / d ) l - 3 3 3 f e e . - 0 . 1 6 7 d O . 2 5 ttu.com(a/d < 2.5) = 0.96*d0-25*(a/d)-i.33*f^^„-o.i67 (4.4)

In Fig. 4.4 a comparison is made between the formulas (4.3)-(4.4) and the model with the effective shear depth derived in this report. The concrete

grade is 39 N/mm2 and the depth is 500 mm.

0 2 A 6 8 10

a/d

Fig. 4.4 Comparison of equation of Zsutty and the model.

In accordance with the difference between the cracking strength and the ultimate stress, the model with the effective shear depth (eq. 4.1) provides shear stresses 10% higher than the stresses obtained with formula (4.3) of Zsutty. The difference between the model and Zsutty for low shear slenderness

ratios (eq. 4.2 and 4.4) is larger. It is obvious that although the parameter ttu according to the eq. (4.3)-(4.4) is only slightly influenced by the concrete grade, there is a strong influence of the beam depth. On the other hand eq. (4.2) is strongly influenced by the concrete grade. The majority of the experiments used by Zsutty [8,9] had a beam depth in the range of 200-400 mm. For this depth the shear resistance is another 10 percent lower than for

4.4 Coni>arison of the Bazant/Kim formula and the model derived

The eq. (2.5) is written in the form of eq. (2.8). The parameter ttu.com is expressed by: 1 Tcr T R a f l a Ku , c o . 2 . 5 2 . 8 4 . 5 Q.16/y(l+d/25U»ax) l u . e o . f e e . 0 - 5 0 p<,0.333 PoO 333 C/fec. + 2 7 . 8 5 ; C P o / ( a / d ) 5 ) ) 1 d - O . 2 5 0.16/y(l+d/25D.ax) do 2s ^ ^ ^ , + 2 7 . 8 5 / ( P o / ( a / d ) s ) ) | au.com==0.16*d" 2S/2^1+d/(25*Dmax)*(l+27.9*2ypo/(fccm*(a/d)!') (4.5)

In Fig. 4.5 a comparison is made between the formula (4.5) and the model with the effective shear depth derived in this report. The concrete grade is 39 N/mm2 and the depth is 500 mm. The maximum particle size is 16 mm and the reinforcement ratio is 0.01.

Fig. 4.5 Comparison of equation of Bazant/Kim and the model derived.

Note that the shear strength according to the equation of Bazant and Kim is even less than the stresses obtained with Zsutty's formulas. This is caused by the strong influence of the beam depth upon the parameter ttu. The majority of the experiments used by Zsutty [8,9] had a beam depth in the

range of 200-400 mm. For this depth the shear resistance according to the expression of Bazant/Kim is iO percent higher than shear resistance according

to Zsutty's formulas. For this range of beam depths and a low concrete grade the comparison with the model is presented in Fig 4.6. There still is a difference between the model and the formulas of Bazant/Kim and of Zsutty. However, the difference is far less than the variation of the shear strength due to the variation of the tensile strength.

a

u

1

, . . , 1

s.

; 1

a/d

Fig. 4.6 Comparison of eq. (4.5) and the model for d = 200 mm and fccm = 19 N/mm2

5. BEAM BKUAVIOUH ACCOBDING TO THK MODEL WITH EFFECTIVE SHEAR DEPTH

5.1 Introduction

The model derived in Chapter 3 is based on material properties. Changc;s in material properties due to time-dependent losses will cause a reduction of the shear resistance. In this Chapter the following topics will be discussed:

1. Characteristic lower bound of the shear resistance.

2. Time-dependent behaviour of beams with web reinforcement. 3. Shear resistance of T-beams.

4. Distributed loads. 5. Lightweight concrete

5.2 Characteristic lower bound of the shear resistance

The shear resistance expressed by the eq. (3.23) and (3.28) is the strength corresponding to the average material properties. The strong influence of the

tensile strength upon the shear resistance yields a lower bound for the mean shear resistance according to the probabilistic distribution of the tensile strength. According to Rusch [17], Carino and Lew [13] and Heilmann [27] the scatter of the tensile strength is considerable. Therefore, the mean value of

the tensile strength is 1.20 times the characteristic lower bound of the strength. Although the reduction of the tensile strength may be too large for

laboraty tests, the lower boundary of the shear resistance is 1/1.20 = 0.83 of the mean strength. The eq. (3.23) becomes:

Vcu(a/d>2.5) = 0.83 * Vcu , m(a/d)2.5)

= au*d-o 2 3,:pj,o. 3*f^^„0.5jt:b*d (5.1) with ttu = 0.98*Y°^°*P°-öo*(a/d)-o 22 (5.2)

The shear strength according to the eq. (3.28) is influenced by the concrete

Vcu(a/d<2.5) = 0.85 * Vcu,m(a/d<2.5)

= 0.36*P'*(d/a)2*poO^'''*fccm"-ö^*b*d (5.3)

The eq. (5.1) and (5.3) can be used in strength criteria for short-term loads. For this purpose the expressions are written in a more convenient form:

Vcu(a/d)2.5) = ttu *d-o • 2 * 3 ^ i^zjf^^„)|cb*d with ttu = 0.74*po-8*(a/d)-"-22

(5.1a) (5.2a)

Vcu(a/d<2.5) = 0.36*P'*(d/a)2*3/^»cf<-cn.oöi.*b*d (5.3a)

The time-dependent parameters P and P' reflect the total losses of the tensile strength resp. compressive strength. Therefore, p equals 0.60 and p' is equal to 0.85. In Fig. 5.1 the time-dependent behaviour of the tensile strength is shown. This behaviour is taken into account by means of the parameter P. 1 0 0 9 0 8 0 7 0 6 0 5 0 4

- T.me jnder lead

p< 5i \ 1 77 m,n - , \ 6 70 rnin N O 40 rron ^ S s ^ h 4 n -r>in ^~~~ 1 ~ - ^ ^ ^ ^ - ^ _ _ _ 1 — . no ïign of fitlure after 9 (nonths 100 !000 10 000 R U P T U R E T I M E — h

5,3 Time-dependent behaviour of beams with web reinforcement

In beams with shear reinforcement the ultimate shear resistance of the effective shear zone is reached and the stirrups are strained. Due to time-dependent losses or repeated loading the contribution of the compressive zone decreases and the steel stress in the stirrups increases. When the beam is subjected to an increasing shear load the section which is influencend by the high tensile stress is changing. Now a part of the tensile zone, which is not affected by time-dependent influences, is strained. Therefore, the contribution of the shear zone will be 100 percent of the short-term shear capacity as soon as the beam is subjected to a higher shear load and increasing deformations. This is in good agreement with experimental results. See Fig. 5.2. [11] iou.Q 1.75zut.Q Fig. 5 . 2 . B e a m s s u b j e c t e d t o r e p e a t e d l o a d s . [ 1 1 ] H o w e v e r , b e c a u s e o f t h e dureibility o f t h e s t r u c t u r e t h e s h e a r r e s i s t a n c e u s e d in a s t r e n g t h c r i t e r i o n m u s t b e c a l c u l a t e d w i t h e q . ( 5 . 1 ) a n d ( 5 . 3 ) . T h e r e d u c e d s h e a r r e s i s t a n c e leads t o a n i n c r e a s e in s t e e l s t r e s s . T h e s t e e l s t r e s s i n f l u e n c e s t h e c r a c k w i d t h a n d t h e d u r a b i l i t y . F o r p r a c t i c a l u s e e q . (5.1) a n d ( 5 . 3 ) a r e v a l i d f o r b o t h t h e s t r e n g t h a n d t h e d u r a b i l i t y c r i t e r i o n .

Now the steel stress in the stirrups is overestimated.

For beams without stirrups the shear resistance is influenced by the strain in the compressive zone according to eq. (3.20). However, in beams with web reinforcement the deformation increases with an almost constant contribution of the compression zone. Therefore, the influence of the strain in the compression zone upon the shear strength must remain very small. In Fig. 5.3 it is shown that the parameter (IEC*103| +1.1*Y)/lec*1031 o•^6 ig rather insensitive to changes of Ec.

3.0

2.0

liD

0.0

(|e.1O^U1.1^0/lG.1O^I°^^

—important for shear failure

• •

05

ID

1.5 2D

e(10'^)

Fig. 5.3 The influence of the strain Ecupon the shear resistance.

5.4 Shear resistance of T-beams

The model has been derived for rectangular beams without stirrups. For T-beams the shear resistance found in experiments is lower than the resistance of rectangular beams with the same width as the flange of the T-beams. There is a strong dependence of the ratio between the width of the slab and the width of the beam be/bo. See Fig. 5.4.

In general the compressive zone of a T-beam is situated in the flange of the T~beam. The beam itself is ruptured. Now the same situation is apparent as in a cracked rectangular beam. In the eq. (3.23) and (3.27) the reinforcement

ratio depends upon the cross-section of a rectangular beam with width be

Nebenwirliungen

•~ b,

-,5-4/*.

Fig. 5.4 Influence of ratio be/bo upon the shear resistance [II]

We obtain:

For a/d ^ 2 . 5 :

V C U . m. T - b e a m - {\>o /\>e)^ • ^'^^''^^c\i , m , rect . ( 5 . 4 )

For a / d < 2 . 5 :

V c u . m . T - b e a n = ( b o / b e ) ° • ^ ' ' ' ' * V c u , m , r e c t ( 5 . 5 )

The difference in shear resistance of various T-beams is shown in Fig. 5.5, where the shear resistance of the rectangular beam with width be is taken as

reference. The beams have no shear reinforcement.

1.0

05

Vj/Vo

DO

r

/(5.5)1

1 2 3 ^ 5 6 7 8

be/bo

5.6 Distributed loads

The model derived is valid for point loads. For practical use an extension to distributed loads is useful. It is obvious that in the case of distributed loads a pari of the load is close to the support and this part causes no shear problems. See Fig. 5.6.

'C

^A

L

Fig. 5.6. Distributed loads

The shear resistance of beams subjected to distributed loads can be compared with beams subjected to point loads if the relation between the a/d-ratio and the L/d-ratio is known. Reckoned from the middle of the beam towards the support the shear resistance increases far less than the shear force. Therefore the weakest section of the beam is located in relation to the support at the shortest distance at which there is no influence of the support. The weakest section of a beam is shown in Fig. 5.7. The shear load

V=(V2-L-a)q 1 t t

m

'zzznm

—

at this section can be calculated with eq. (3.23). The shear load consists of the distributed load between the two point loads. In that case the total reaction in the support is:

V c u . d i s t r . = L/2 / ( L / 2 - a ) * V c u . p o i n t .

^ L/d /(L/d-2*a/d)*(a/d)-o-22*f(c|,fccm,p) (5.6)

For the distributed loads the ratio a/d is not known, but with eq. (5.6) the lower bound can be determined. The result is shown in Fig. 5.8.

I - 2a

10 20 30 ^0 mechonism 1 ( a / d ^ 2 . 5 )

*" l/d

Fig. 5.8. The parameter L/d /(L/d-2*a/d)*(a/d)-o 22 for distributed loads.

It appeared that the lower bound can be approximated by the line described by the expression:

Vcu.distr. = 0.95*L/d/(L/d-2.5) * f(d,fccm,p) (5.7)

The expression (5.7) is almost the same as the formula used by Walraven [14] for the shear resistance parameter ttu according to Rafla [10]. Now the shear

resistance parameter becomes:

ttu.distr. = 1 . 1 7 * 0 . 9 5 * p o « * Y " '»*L/d/(L/d-2.5) * f(d,fccm,p) (5.8)

For beams with a low shear slenderness ratio (L/d < 10) shear failure depends upon the strength of the remaining arch. See Fig. 3.14. In case of distributed loads the remaining arch is no longer centrically loaded; see Fig. 5.9. Now there is a bending moment in the arch. This gives a reduction of the total ultimate normal force in the compressive zone:

Ncu.distr. = approx. 2/3 * Ncu.point. (5.9)

Fig. 5.9. Remaining arch with distributed loading.

Therefore:

Vcu.distr. = approx. 2/3 * Vcu.point. (5.10)

According to this mechanism the location of the weakest section of the beam is influenced by the following expression:

V c u . d i s t r . = L / 2 / ( L / 2 - a ) * V c u . p o i n t .

= L/d /(L/d-2*a/d)*(a/d)-2*f(d,fccm,p) (5.11)

(4-)

a/d=1.25 D - D Z . 0 ( l / d ) " ^ ' ^

0 10 20 30 ^0 mechanism 2 { a/d < 2,5 )

l/d

Fig. 5.10. The parameter L/d /(L/d-2*a/d)*(a/d)-2

The lower bound can be expressed by:

L/d /(L/d-2*a/d)*(a/d)-2 ::: 40 * (L/d)-2i5 (5.12)

The shear resistance parameter «u according to the described mechanisms is shown in Fig. 5.11. As for beams subjected to point-loads the mechanism

according to eq. (5.7)-(5.8) must fail before the mechanism according to eq. (5.12) can develop. Because of the fact that even for small ratios of L/d the line describing the ttu according to eq. (5.8) provides a good approximation of the actual shear resistance, just one simple formula is used in describing

the shear resistance of beam subjected to distributed loads:

Vc u . d i s t r . - ttu *§^ccm * %fpo /^/d

(5.13)

with

fn a strength criterion the shear resistance must be reduced by a factor 0.83.

A ctu

0 10 20 30 1,0

l/d

Fig. 5.11. The parameter ttu in the case of distributed loads. fccm = 39 N/mm2;d = 500 mm;Y = 0.49;p = 0.75;P'= 0.90

5.7. Lightweight concrete

The model derived in Chapter 3 can be used for lightweight concrete by substituting the proper material properties in the basic model. In lightweight concrete the strength of the aggregate particles is of the magnitude of the strength of the matrix material. Hence, lightweight concrete is a far more homogeneous material than ordinary concrete. According to the linear elastic fracture mechanics [15,16] the size-effect of a brittle material is proportional to 1/^/5^ Applying this factor to the flexural

tensile strength of lightweight concrete this yields:

C c r . f l = l / ^ * f c t m

(5.15)

fctm - 0 . 2 7 7 * f c c m " ' ^ ' ' ' ( 5 . 1 6 )

The relation (5.16) is in good agreement with the experimental results of Walraven [18]. Note that the tensile strength can be approximately 30 percent

lower due to drying shrinkage. The modulus of elasticity can be calculated with the formula of Pauw [19]:

Eic = 0.04 * 2/pg3*f^_<.„ (5.17)

with pg - specific gravity = approx. 1800 kg/m^

This yields:

Eic = 3055 * 2^ fccm (5.18) The substitution of these material properties into the model yields the

folllowing expressions for the shear resistance of lightweight concrete (Y -0.4, See App. IV):

Vcu.m, ic =au.lc * d-°-'»17*fccmO-^3*po°-22*b*d (5.19)

with ttu.ic = 4.65*Y°'»i*P°-2*(a/d)-o 165 (5.20)

In Table 5.1 the experimental results of Walraven [18] are compared with eq. (5.19). In general the calculated values are 30 percent too high. This can be explained by the influence of the drying shrinkage, which can cause a

reduction up to 30 percent in the tensile strength. The form of the expression is in good agreement with the empirical formulas derived by Walraven [18] and by Koch and Rostasy [24].

Walraven:

Tu = 3 * fspi * Po°-3 * d-O-* (5.21)

Kock/Rostasy:

with tu and fccm in kg/cm2 d in cm.

ttu according to Rafla.

Table 5.1. Experimental results of Walraven [18] and the calculated strength according to eq. (5.19).

Specimen Bl B2 B3 CI C2 C3 T c u , W a l r a v e n 1.19 N/ram2 0.72 0.55 1.40 1.07 0.76 Tc u , m o d e 1 1.40 0.82 0.64 1.63 1.00 0.77

I

6. SYNOPSIS

The theoretical model derived in this report describes the shear resistance of beams with longitudinal reinforcement by means of two mechanisms. These mechanisms are based upon material properties. Therefore, this model can be used as a theoretical basis for strength and durability criteria.

The main mechanism depends on the shear stiffness of the compressive zone and the tension-softening zone in front of the shear crack. The depth of the zone is called the effective shear depth. The tension-softening zone contributes to the shear resistance by means of the (low) shear stiffness of the uncracked cross-sectional area and the mechanism of aggregate interlock of the cracked cross-sectional area. Within the limits of the assumptions made in the calculation procedure it can be stated that the contribution of aggregate interlock to the shear stiffness of the tension-softening zone is about 10 percent. The mean shear stiffness of the tension-softening zone is approximately 40 percent of the shear stiffness of the compressive zone. The shear force transferred by the mechanism of aggregate interlock in the shear crack and by dowel action of the longitudinal reinforcement is approximately 10 percent of the total shear force.

For small shear slenderness ratios, there is a considerable increase in the shear strength. Now a mechanism emerges which is based upon centrically loaded compressive and tensile zones. Failure occurs due to crushing of the concrete in the compressive zone.

The basic model provides a quite simple expression for the shear resistance of a cracked beam. According to the model the shear resistance for high shear slenderness ratios can be described by a parabolic shear stress distribution

in a 'hidden' beam with depth h-^ . This depth is the effective shear depth. According to the linear-elastic theory the shear strength is expressed by:

Vcu.m = 2/3 * (a*ro) * h.^ * b (3.4)

Although the model has been derived for point loads it has been extended to comprise distributed loads. For ordinary concrete the shear resistsunce can be

c a l c u J a t e d w i t h :

Vcu.m -- ttu * ^ * 2 / f r ^ / ^ * b*d ( 3 . 2 3 )

with ttu,point = 0.71 * ( a / d ) - ^ 22 (3.23a) with ttu.distr = 0.68 * (L/d)/(L/d-2.5) (5.14)

For low a/d-ratios (< 2.5 ) the following expression becomes appropriate for point loads:

Vcu.m,point = 0.42 * (d/a)2 * 3jp^ * fccm^J-ss * b*d (3.28)

Note that these expressions provide mean values for the shear resistance,

The theoretical model is used for the determination of expressions for the time-dependent behaviour of the shear resistance, the effect of lightweight concrete and the influence of the shape of the cross-sectional area upon the shear strength. The resulting formulas showed close agreement with the experimental results.

7. NOTATION.

a = shear span; for point loads M/V [ram] b - width of the beam [mm]

bn - effective width of the beam [mm] d = effective beam depth [mm]

dts = effective depth of the tension-softening zone [mm] fcm = mean cylindrical compressive strength [N/mm2]

fccm = mean cubic compressive strength of concrete [N/mm2] fctm.o = mean tensile strength (Time = 0 ) [N/mm2]

fct.o = characteristic tensile strength (Time = 0) [N/mm2] fctm.00= mean tensile strength (Time = oo) [N/mm2]

fct.oo = characteristic tensile strength (Time = oo) [N/mm2] h = total depth of the beam [mm]

hx = depth of the compressive zone [mm] ki = empirical constant [-]

k2 = empirical constant [-] h = effective shear depth [mm] p = empirical constant [-] q - empirical constant [-] q = distributed load [N/m]

r - empirical constant [-]

s = distance between support and first crack [mm] t = time [h]

z = internal lever arm [mm]

Dmax = maximum size of aggregate particle [mm] Ec = modulus of elasticity of concrete [N/mm2]

G ~ shear modulus [N/mm2]

L - length of the beam [mm] P = point load [N]

V = shear force [N] Vcu.m = mean shear force [N]

Vcu = characteristic shear force [N] Vcr.m = mean cracking force [N]

Ncu.m - mean normal force IN] M = bending moment INnira] a - reduction factor (-]

ttu = shear resistance parameter according to Rafla [-] 6 -' strain (of compressive zone) [-]

p = time-dependent parameter [-] P' - time-dependent parameter [—] Ectu = ultimate strain o f tensile zone Enn = normal strain in the crack [-] Y - shear deformation [-]

Y - reduction factor for shfiar stiffness of the tensile zone [-] r = shear stress [N/inra2]

to = maximum shear stress at the neutral axis [N/mm2J tcu = shear stress [N/mni2 ]

tcu.fi = flexural shear stress [N/nmi2] tcr = shear cracking stress [N/mm2]

Taggr = shear stress due to aggregate interlock [N/mm2] (7aggr = normal stress due to aggregate interlock [N/mm2] CT - normal stress [N/mm2]

(7cr - tensile strength [N/mm2]

Ccr.fi = flexural tensile strength [N/mm2] /Vo - empirical constant [-]

<t> = b a r diameter [mm] p = A3/(bd) [-] po = 100S5 * As/(bd) [%]

As = cross-sectional area of steel [mm2]

8. REFERENCES.

[1] Kani, G.N.J. , The riddle of shear failure and its solution ACI-journal. Proceedings Vol. 61,(1964), No. 4 pp. 441-465.

|2] Kani, G.N.J., Basic facts concerning shear failure,

ACI-journal, Proceedings Vol. 63,(1966), No. 6 pp. 675-692.

[3] Fenwick, R.C., Paulay, T., Mechanisms of shear resistance of concrete beams , Journal of the structural division, ASCE, 94 (1969), pp. 2325-2350.

[4] Taylor, H.P.J., The fundamental behaviour of reinforced concrete beams in bending and shear, ACI-SP-42, Vol. 1,(1974), pp. 43-78.

[5] Walraven, J.C., Aggregate interlock: a theoretical and experimental analysis. Dissertation Delft university of Technology (1980), pp. 197.

[6] Baumann, T., Versuche zum Studium der Verdübelungswirkung der Biegezugbewehrung eines Stahlbetonbalkes, Deutscher Ausschuss für Stahlbeton, Heft 210, Berlin (1970), pp. 46-83.

[7] Bazant, Z.P., Kim, J.K., Size effect in shear failure of longitudinally reinforced beams, ACI-journal Vol. 81,(1984), No. 5 pp. 456-468.

[8] Zsutty, T.C., Beam shear strength prediction by analysis of existing data, ACI-journal Vol 65,(1968), No. 11 pp. 943-951.

[9] Zsutty, T.C., Shear strength prediction for separate catergories of simple beam tests, ACI-journal, Proceedings Vol. 68,

1 101 Hafla, K. , Empirische Formt;ln zur Berechnung der Schubföhigkeit von Stahlbetonbalken, Teil I; Einfeldrige Rechteckbalken ohne Schubbewehrung bei direkter Einleitung von Einzellasten, Strasse, Brucke, Tunnel 23,(1971), No. 12, pp. 311-320.

[11] Leonhardt, F., Schub bei Stahlbeton und Spannbeton - Grundlagen der neueren Schubbemessung, Beton- und Stahlbetonbau Vol. 72, (1977), No. 11 pp. 270-277,

Beton- und Stahlbetonbau Vol. 72,(1977), No. 12 pp. 295-302.

[12] Pruijssers, A.F., Description of the stiffness relation for mixed-mode fracture problems in concrete using the rough-crack model of Walraven, Stevin Report 59, Delft University of Techn.,

(1985), pp. 36.

[13] Carino, N.J. ,Lew, H.S.,Heeaximination of the relation

between splitting tensile and compressive strength of normal weight concrete, ACI-Journal Vol. 79,(1982), No. 3 pp. 214-219.

[14] Bruggeling, A.S.G., de Bruijn, W.A., Theorie en praktijk van het gewapend beton. Lecture notes Stichting Professor Bakkerfonds, VNC

's-Hertogenbosch,(1985).

[15] Reinhardt, H.W., Massstabseinfluss bei Schubversuchen im Licht der Bruchmechanik, Beton- und Stahlbetonbau Vol.(1981), No. 1 pp. 19-21.

[16] Reinhardt, H.W., Similitude of brittle fracture of structural concrete, Advanced Mechanics of reinforced concrete,

lABSE-colloqium, Delft,(1981), pp. 175-184.

[17] Rusch, H., Die Ableitung der charakteristischen Werte der Betonzugfestigkeit, Beton 2/75,pp. 55-58.

[18] Walraven, J.C., The influence of depth on the shear strength of lightweight concrete beams without shear reinforcement,

Stevin Report 5-78-4, Delft University of Techn.,(1978), pp. 36.

[19] Lightweight Aggregate concrete, CEB/FIP Manual of Design and Teclinology, the construction Press, London (1977), p. 169

[20] Körmeling, H.A., Impact tensile strength of steel fibre concrete, Experimental results of plain and steel fibre reinforced concrete under uniaxial impast tensile loading, Stevin Report 5-84-8, Delft University of Techn.,(1984), pp. 149,

[21] Regulations for concrete 1974/1984, NEN3880 (in Dutch), NNI, Delft,(1984), pp. 523.

|22] Al-Kubaisy, M.A., Young, A.G., Failure of concrete under sustained tension. Magazine of Concrete Research: Vol. 27, (1975), No. 92 pp. 171-178.

[23] Eibl, E., Ivanyi, G., Studie zum Trag- und Verformungsverhalten von Stahlbeton, Deutscher Ausschuss fur Stahlbeton, Heft 260,(1976), Berlin, pp. 335.

[24] Koch, R., Rostasy, F.S., Schubtragfahigkeit von Platten aus Stahlleichtbeton ohne schubbewehrung,

Beton- und Stahlbetonbau 2,(1978), pp. 42-46.

[25] Reinhardt, H.W., Beton als constructiemateriaal, Eigenschappen en duurzaamheid (in Dutch),

Delftse Universitaire Pers,(1985), pp. 315,

[26] Walther, R., Uber die Berechnung der Schubtragfahigkeit von Stahl-und Spannbetonbalken, - Schubbruchtheorie,

[27] Heilmann, H.G., Beziehungen zwischen Zug- und Druckfestigkeit des Betons, Beton (1969), Heft 2, pp. 68-70.

[28] Raphael, J.M., Tensile strength of Concrete, ACI journal Vol. 81,(1984), No. 2 pp. 158-165.

[29] CEB recomraandations I,

APPENDIX I. FAILURE CRITERION IN A BIAXIAL STRESS CONDITION

in a t)iaxia.l stress condition fciiluie occurs when the principal stresses exceed the yield criterion presented in Fig. I.la. The yield criterion in the compression-tension area can be described by several expressions. For the use in this report the yield criterion in the compression-tension area is approximated by a straight line. See Fig. 1.1b.

Fig. I.l Two-dimensional strength criterion [13,25]

In this approximation the following relation between the principal stresses is valid:

(T?. = fctm - f c t m / f c c m * (Tl ((Tz > 0) ( I . l )

According to the theory of Mohr the principal stresses can be written as a function of the actual stresses:

<7i ^ (Jc/2 - 0.5j(jc^ + 4 rc2 (1.2)

0-2 = crc/2 + 0.5/(rc2 + 4 Tc2 (1.3)

Substitution of the eq. (I.2)-(I.3) in (I.l) yields:

yCrc2+4Tc2 (l(-fccm/fctm)+Crc(l-fccm/fctm)-2fccm - 0 ( 1 . 4 )

This yields: Tc = y i / 4 A fccm2 + B fccm "» C w i t h A = ( f c c m - f c tin ) / ( f c c r a t-fc tm ) B = ( f c c m - f c t m ) * f c c m * f c t m / ( f c c m + f c t m ) 2 C = ( f c c r a * f c t m ) 2 / ( f c c r a + f c t m ) 2 (1.5)

This relation is represented in Fig. 1.2.

i ccmctm f +f 1 ', ^00171*^ m i , "c , ytxnt^ctm x.max" 2

i ^

Fig. 1.2. The relation between the normal stress and the shear stress.

For (Tc = 0 (neutral axis of the beam) the shear stress is expressed by:

tc = ( fc tm*fccm ) / ( f c t m+fccm ) (1.6)

Walther [26] described the envelope according to the theory of Mohr by a parabolic function. The yield criterion is shown in Fig. 1.3.

li:i'i4.

Fig. 1.3 Yield criterion according to Mohr [26]

Now the shear stress at the neutral axis can be described by [25,26]

APPENDIX II. MATERIAL PROPERTIES AND DEPTH OF THE COMPRESSIVE ZONE

11.1 Introduct i on

In this Appendix all the important material properties are presented in their original form. For the use in the model developed in this report all the formulas are written in the form:

y - a*x'> (II.1)

The material properties are presented in this form in order to obtain a relatively simple expression for the shear resistance. The material properties, which are important for the shear resistance, are:

- The Young's Modulus Ec. - The tensile strength fctm.

The cracking behaviour of beams is influenced by the depth of the beam. The influence is taken into accoimt by means of the flexural tensile strength

(Tc r , f 1 .

Apart from the material properties the geometry of the beam will be expressed in the form of (II.I).

11.2 The Young's Modulus Ec

The Yound's modulus of the concrete grades in the range of 19-59 N/mm2 are presented in Fig. II.I. The relation according to the Code NEN3880 [21] is presented by line 1 in the figure.

10000 30000 20000 10000 0 (211 in,21 y ^ - t r * * ^ — -10 20 » 40 50 60