Second Cycle, 30 Credits Stockholm, Sweden 2019
The effect of pre-stressing location
on punching shear capacity of
concrete flat slabs
Author: Saeed Vosoughian
KTH ROYAL INSTITUTE OF TECHNOLOGY
The effect of pre-stressing
location on punching shear
capacity of concrete flat slabs
S
AEED VOSOUGHIAN
TRITA-ABE-MBT-19686. Master Thesis 2019. KTH School of ABE SE-100 44 Stockholm SWEDEN © S. Vosoughian 2019
Royal Institute of Technology (KTH)
Abstract
Implementing pre-stressing cables is a viable option aiming at controlling deformation and cracking of concrete flat slabs in serviceability limit state. The pre-stressing cables also contribute to punching shear capacity of the slab when they are located in vicinity of the column. The positive influence of pre-stressing cables on punching capacity of the concrete slabs is mainly due to the vertical component of inclined cables, compressive in-plane stresses and counter acting bending moments near the support region. The method presented in Eurocode 2 to determine the punching capacity of the pre-stressed concrete flat slabs considers the in-plane compressive stresses but totally neglects the effect of counter acting moments. The effect of vertical forces introduced by inclined cables is only considered when they are within the distance 2d from the face of the column. This area is called basic control area in the Eurocode 2.
In this master thesis nonlinear finite element analysis is carried out to study the effect of pre-stressing cables on punching shear capacity of concrete slabs respecting the distance of cables from the face of the column. To attain this objective, the concrete damage plasticity model is implemented to model the concrete. The results indicate that until the distance of 6d from the face of the column the contribution of pre-stressing cables in punching shear capacity of slabs is significant. Furthermore, comparing the numerical results with the punching shear capacity of slabs predicted by Eurocode 2 reveals that Eurocode tremendously underestimates the punching shear capacity when the cables are located outside the basic control area.
Sammanfattning
Inläggning av spännkablar är ett möjligt alternativ som syftar till att begränsa deformation och sprickbildning i plana, pelarunderstödda betongplattor i bruksgränstillståndet. Spännkablarna bidrar också till stanskapaciteten hos plattan när de är belägna i närheten av pelaren. Spännkablarnas positiva påverkan på stansningskapaciteten hos betongplattorna beror främst på den vertikala komponenten i lutande kablar, tryckspänningar i planet och motverkande böjmoment nära stödområdet. Den metod som finns i Eurokod 2 för att bestämma stansningskapaciteten hos förspända betongplattor beaktar tryckspänningarna i planet men negligerar helt effekten av motverkande moment. Effekten av vertikala krafter som ges av lutande spännkablar beaktas endast när kablarna är placerade inom avståndet 2d från pelarens yta. Detta område begränsas av det s.k. grundkontrollsnittet enligt Eurokod 2.
I detta examensarbete genomförs icke-linjär finitelementanalys för att studera effekten av spännkablar på betongplattors stansförmåga med avseende på kablarnas avstånd från pelarens yta. För att uppnå detta mål implementeras en icke-linjär plasticitetsmodell för betongens brottstadium (CDP, concrete damage plasticity model). Resultaten indikerar att fram till avståndet 6d från pelarens yta är bidraget från spännkablarna till betongplattans stanskapacitet betydande. En jämförelse av de numeriska resultaten med stansningskapaciteten hos plattor som beräknas med Eurokod 2 visar att Eurokod kraftigt underskattar stansningskapaciteten när kablarna är belägna utanför grundkontrollsnittet.
Preface
This master thesis is initiated by the department of Civil and Architectural Engineering at Royal Institute of Technology (KTH) and is result of collaboration with CBI Betonginstitutet AB.
In advance, I would like to express my deepest gratitude to my supervisors Professor Johan Silfwerbrand at KTH and Mr. Ghassem Hassanzadeh Lic. Tech., for all their support, guidance and valuable comments throughout my work. Their dedication and constant encouragement were inspirational while keeping me on the right track to attain determined objectives.
Secondly, I wish to express my sincere appreciation for Adj. Prof. Mikael Hallgren. He contributed with valuable advice and generously shared his experience and knowledge in the field.
Furthermore, I want to thank Dr. Richard Malm for providing worthwhile comments and help about the finite element method.
Last but not the least, I want to send tons of love and appreciation to my family for their support throughout my entire study at KTH.
Stockholm, October 2019
Abstract... i Sammanfattning ... iii Preface ...v 1 Introduction ...1 1.1 Problem definition ...1 1.2 Background ...2 1.3 Aim ...3
1.4 Scope and delimitations ...3
2 Punching shear models ...5
2.1 Kinnuen and Nylander model ...5
2.2 Andersson model ...8 2.3 Hallgren model ... 10 2.3.1 The model... 10 2.3.2 Failure criterion ... 15 2.4 Menétrey model ... 17 2.4.1 General assumptions ... 17
2.4.2 Concrete tensile force ... 18
2.4.3 Dowel effect ... 19
2.4.4 Shear reinforcement contribution ... 19
2.4.5 Pre-stressing tendon contribution ... 21
2.4.6 Relation between punching and flexural capacity ... 21
2.5 Georgopoulos model ... 21
2.6 Bond model for concentric punching shear ... 22
2.7 Muttoni model ... 23
2.8 Discussion about mechanical models ... 25
3 Effect of pre-stressing on punching capacity of flat slabs ... 27
3.1 Effect of pre-stressing... 27
3.2 Eurocode 2, approach ... 29
4.1 Introduction ... 31
4.2 Concrete material behaviour ... 33
4.2.1 Compressive behaviour ... 33
4.2.2 Tensile behaviour ... 34
4.2.3 Multiaxial behaviour ... 37
4.3 Concrete model ... 38
4.3.1 Yield surface ... 38
4.3.2 Defining damage in CDP model... 42
4.3.3 Longitudinal bars and pre-stressing tendons ... 43
4.4 Finite element model ... 46
4.5 Verification ... 48
5 Result and Discussion ... 51
5.1 Results ... 51
5.2 Discussion ... 52
5.3 Future investigations ... 53
1.1.PROBLEM DEFINITION
1
Introduction
1.1 Problem definition
Concrete flat slab supported by columns is a widely used structural element in buildings and bridges. It has the edge over the other forms of slab systems from both aesthetic and economic points of view. This is due to flexibility of planning and elimination of beams and girders which result in creating additional floor space for a certain height of the building. However, flat slabs are more prone for brittle punching shear failure either next to the column or below a concentrated force compared to beam-slab systems.
In design of flat slabs to avoid punching failure, usually huge amount of reinforcement is required in critical zones. It goes without saying that the high amount of flexural and shear reinforcement makes casting concrete cumbersome. Beside the need to have opening in vicinity of columns, which especially is required when reconstructing old buildings to install new apparatus, motivates to find an alternative method to normal reinforcement in order to increase punching shear capacity of the flat slabs.
Previous researches have shown that the idea of pre-stressing is a suitable technique to reduce the amount of reinforcement for flat slabs with large spans or slabs subjected to huge concentrated forces. This is because:
i. Pre-stressing exerts in-plane compressive stresses on concrete which leads to increase in punching shear capacity.
ii. Bending moments caused by eccentricity of the tendons counteract those of external forces leading to smaller crack openings in failure region and thus increase of capacity of the concrete to carry shear force.
iii. Inclined pre-stressed tendons introduce a vertical force in the punching failure surface opposing the forces caused by external loads.
Reflecting on the formulas presented by design codes such as ACI 318 [1] and Eurocode 2 [2] indicates that they attempt to acknowledge the effect of pre-stressing on punching shear capacity based on empirical investigations and only account for in-plane stresses and deviation force. Accordingly, the bending moment which is introduced due to pre-stressing eccentricity is not taken into account [3] furthermore, despite both numerical and empirical investigations conducted in the past, the contribution of active known pre-compression force is still not well understood.
Considering these facts necessitates more detailed and accurate research in this field to attain reliable design criteria.
The other interesting issue is that the influence of pre-stressing on punching failure of a flat slab is taken into account by conventional design codes only if the pre-stressed cables are stretched in a certain area around the perimeter of the column. Series of experimental investigations have been done in the past studying this phenomenon [4,5]. However numerical models are more flexible in dimensioning and alteration of the position of pre-stressed cables. Thereby, with a reliable numerical analysis, a more profound comprehension of influence of cable arrangement is achieved.
1.2 Background
It seems first Elstner and Hognestad [6] noticed the shear capacity of thin concrete slabs. They conducted an experiment and studied the influence of concrete strength, tension and compression reinforcement amount as well as position and amount of shear reinforcement on capacity of reinforced concrete flat plates. They concluded that increasing tension and compression reinforcement over the column does not affect the load bearing capacity of the slab. In contract with longitudinal reinforcement, they observed shear reinforcement contributes to load capacity of the slab and could enhance it up to 30%. Nevertheless, they realized that although achieving flexural failure rather than shear failure is desirable, it is impractical by increasing shear reinforcement. Accordingly, they proposed to have small amount of shear reinforcement by choosing appropriate values for concrete strength, slab thickness and column stiffness as well.
Kinnunen and Nylander [7] in 1960 developed a model for punching capacity of flat slabs. To attain this model, they performed series of experiments and studied the concrete segment between radial flexural cracks and conical shear crack. Based on this model, punching failure occurs when simultaneously, radial inclined compression stress and tangential compression strain tend to critical values.
In 1961, Johannes Moe [8] conducted a thorough experiment to investigate the effect of hole in vicinity of column, effect of eccentric loading, effect of shear and tensile reinforcement and some other variables on load bearing capacity of flat slabs. In this research he also shed more light on the concept of eccentricity of shear force. His design provisions for openings in vicinity of columns and for eccentric loading are still implemented by building codes.
Sundquist [9] in 1978 investigated the effect of dynamic loads and presented a theoretical model for capacity of flat plates when they are subjected to impulsive loads.
1.3.AIM
The size effect was studied by Tolf [13]. He concluded that the increase in specimen size leads to considerable decrease in punching shear stress.
Hallgren in 1996 [14] conducted a comprehensive study about punching shear behaviour of high strength concrete flat slabs. For this purpose, he did both empirical investigations and created numerical models as well. He improved a model presented by Kinnunen and Nylander [7] for punching shear capacity of flat slabs implementing linear fracture mechanics approach. The model presented by Kinnuen and Nylander was improved by Broms [15] taking into account the size effect and brittleness of the concrete. He showed that the critical value for concrete compression strain at which the punching failure occurs is significantly less than the generally accepted 0.0035 for one way structural elements under bending. He also prescribed a novel reinforcement concept through which punching shear failure is avoided and slab acquires ductile behaviour.
Muttoni [16] in 2008 implemented critical shear crack theory to describe punching shear phenomenon. He developed a criterion for punching shear on the basis of rotation of the slab and verified it with experimental testing. Furthermore, he came up with an interesting result asserting that despite to the conventional belief, punching shear capacity of a flat slab is more dependent on the length of the span rather that slab thickness.
In 1998, Hassanzadeh [4] conducted an experiment to investigate the influence of pre-stressing cables on punching shear capacity of the concrete flat plates. His research indicated a large discrepancy between theoretical and practical results. He asserted that theoretical methods presented by design codes for calculating punching shear capacity of pre-stressed slabs are strongly conservative.
1.3 Aim
The aim of this master thesis is to investigate the effect of pre-stressing on punching shear capacity of flat slabs. Furthermore, the influence of cable arrangement is studied. What is meant by the term cable arrangement is distance of cables from the column. In other words, the area around a column in which pre-stressing cables contribute to enhancement of punching shear capacity is determined.
1.4 Scope and delimitations
The outline of the thesis falls into two categories. The first section sheds more light on punching shear phenomenon and surveys the literature. It is worth mentioning that in this part the treatment prescribed by ACI 318 [1] and Eurocode 2 [2] to calculated punch shear capacity and the effect of pre-stressing is discussed.
numerical model is going to be satisfied by doing a verification with outcome of the empirical study conducted by Hassanzadeh at KTH university [4].
2.1.KINNUEN AND NYLANDER MODEL
2
Punching shear models
Over the years, researchers have attempted to explain punching shear phenomena in concrete slabs. They implemented theoretical concepts such as fracture mechanics, conducted experimental investigations and created numerical modes to demystify punching shear behaviour. Some of these investigators developed theoretical models through which a more profound comprehension of the issue is possible. In this chapter more light is going to be shed on some of these models.
2.1 Kinnuen and Nylander model
Conducting series of experiments on polar-symmetrical concrete slabs without shear reinforcement Kinnuen and Nylander [7] observed that the concrete segment inscribed by conical and radial cracks rotates like a rigid body. With regard to this fact, they satisfied the equilibrium of forces for the segmental part of the concrete bounded between two radial cracks with the angle of Δϕ and one shear crack, see Figure 2.1. Furthermore, the root of the shear cracks was assumed as the centre of the rotation for the considered segment. The forces affecting the mentioned concrete segment are illustrated in Figure 2.1(b). Where 𝑃∆∅/2𝜋 is the fraction of the external force acting on the segment, R2 and R4 are resultants from the forces in radial reinforcement and in tangential reinforcement respectively. The ultimate load Pu for a two-way reinforced slab without shear reinforcement is achieved considering the equation of vertical projection 2 2 ( ) ( ) 1 2 1 1 . 1 f t d B y B y d y B Pu (2.1)
Where y is the distance between root of the shear crack and bottom surface of the slab, α is inclination of the conical shell, B is diameter of the column, d is effective depth of the slab and
σ(t) is the stress in conical shell. Kinnuen and Nylander took into account the effect of dowel
action by introducing the factor 1.1 in Eq. (2.1). Function f (α) is defined as
The angle α can be achieved from the following equation:
d y d B d c B y Ky 5 . 0 5 . 0 ln 1 35 . 2 2 1 tan 1 tan 1 1 tan 2 (2.3)Ky is a dimensionless parameter which is expressed as
3 2 5 . 0 y d B c Ky (2.4)
In the Eq. 2.4, c stands for diameter of the area above the column in which the radial bending moment is negative.
2.1.KINNUEN AND NYLANDER MODEL
Considering the experimental results, Kinnuen and Nylander proposed an approximated value for the stress in conical segment, σ(t). It depends on tangential strain at the horizontal radius
r = B/2 + y and is expressed as
t
2
.
35
E
c
cT,r
(2.5) The failure criterion in this model is the tangential strain at the distance of B/2 + y from centre of the column reaching a critical value as calculated belowFor B/d ≤ 2 d B y B r cT, /2 0.0035 1 0.22 (2.6a) For B/d > 2
0019
.
0
2 / ,rB y
cT
(2.6b) It is worth mentioning that beside Eq. (2.1), the equation of moments with regard to the pointQ1 illustrated in Figure 2-1 also yields ultimate capacity of the slab For rs ≤ c0 B c y d c c r d f Pu sy s 1.1 4 . 1 ln 0.5 3 0 (2.7a) For rs> c0 B c y d r c r d f P s s sy u 1.1 4 . 1 ln 0.5 3 (2.7b)
Where ρ is mean ratio of reinforcement and fsy is yield stress of the reinforcement. Furthermore, rs manifests the radius of the area in which the reinforcement reaches yield stress and c0 stands for the distance between centre of the column to the concentric shear crack. rs and
c0 are approximated to
d
B
c
0
0
.
5
1
.
8
(2.8) d d y f E r sy s s 1 (2.9) The angle ψ indicates the rotation of the slab outside the shear crack when ultimate load is reached and can be calculated asAs mentioned before y is the distance between root of the shear crack and bottom surface of the slab which is calculated by the Eq. (2.4).
The model developed by Kinnuen and Nylander was a base for later important investigations. For instance, Andersson [17] modified this model to take into account the effect of shear reinforcement. The model developed by Andersson is presented in the following.
2.2 Andersson model
Andersson conducted a series of experimental tests to study the punching behaviour of slabs with shear reinforcement. He realized that adding shear reinforcement alters the behaviour of the slab at failure load in a way that its deflection varies linearly from the column to edge of the slab. Accordingly, he proposed a mechanical model to take into account the effect of shear reinforcement by modifying the Kinnuen Nylander model. His model is illustrated in Figure 2-2. Going into the depth, his model assumes a bi-linear and perfectly elasto-plastic behaviour for the concrete in tangential direction. The failure criterion in this model is that the failure occurs when the strain at a certain distance cpl from the centre of the column reaches the limit of plastification εpl. This distance is determined as
For B/d ≤ 2.5 d d B cpl 0.74 1.45 (2.11a) For B/d >2.5 d d B cpl 0.5 2.05 (2.11b)
There is an upper limit for cpl depending on the type of shear reinforcement. This value for slabs in which longitudinal bars are bent to function as shear reinforcement is from the centre of the column to outermost part of the shear bar. While the upper limit of cpl for slabs with stirrups is the distance between the centre of the column and 3d/4 outside the most far stirrups. In failure stage, when εpl tends to 0.002 at the point of cpl, the stress of concrete in tangential direction σcT and the stress in conical shell σt are approximated to
15 3 . 0 35 . 0 25 c,cub cT f
[MPa] (2.12) cT t
1
.
9
(2.13) Looking at the isolated segment of the slab depicted in Figure 2.2(b) the resultant force in tangential direction is expressed as2.2.ANDERSSON MODEL
Where, fsyv and Ash are yield stress and the cross-sectional area of shear reinforcement intersecting the tangential flexural crack. Δϕ stands for the angle illustrated in Figure 2.2. Furthermore, rs is determined by Eq. (2.9). However, it should be noticed that the angle of rotation ψ for a segment of the concrete which behaves like a rigid body while failure is occurring, in Andersson’s model is different from the value according to the Kinnuen model and is expressed as y cpl pl
(2.15)Figure 2-2 Mechanical model by Andersson [17].
In Eq. (2.15),
κ
1 is reduction factor taking into account the effect of reinforcement when the slab is not polar symmetrical.κ
1 is function of 2rs/c and is determined through the graph presented in Figure 2.3.The force R4 indicated in Figure 2.2 is the projection of the sum of the compressive stresses
The influence of the conical shell on the rigid segment as shown in Figure 2.2 is the compression force T which can be calculated as below
0
.
5
B
y
tcos
T
(2.17) Establishing the equations of the equilibrium for the isolated rigid body yields three equations from which three unknowns including the ultimate load Pu, distance y and angle α shown in Figure 2.2 are acquired 2 sin cos 72 . 0 2 2 cos 2 3 2 1 4 4 u u P T R R R R T B c y d R B c y d T P (2.18)
Figure 2-3 Reduction factor κ [17].
2.3 Hallgren model
2.3.1 T he model
2.3.HALLGREN MODEL
the Kinnuen and Nylander model which implements semi-empirical expressions, Hallgren used fracture mechanics considering size effect and concrete brittleness.
Hallgren model is illustrated in Figure 2.4 in which 1.5α is the angle of shear crack and c0 is the distance between the center of the column and the shear crack at the level of the flexural reinforcement. The amplification factor of 1.5 which is used for α is achieved based on test observations and finite element analysis results and c0 is determined as
1
.
5
tan
2
0x
d
x
B
c
(2.19)Figure 2-4 Hallgren model [14].
As shown in Figure 2.5 test results and numerical analysis unveiled the fact that strain in tangential direction for the concrete and steel, respectively on compression side and tension side of the slab are inversely proportional to the distance from the center of the slab. Furthermore, the tangential compression zone x is calculated based on the state of the material as below
If concrete and steel are elastic
In the case that steel is in the elastic state while concrete has yielded, x is determined as
d
E
f
f
E
x
cT s cc c cc c cT s
1
4
1
2
0 0 0 0
(2.22) 0 0 1 cT c c
(2.23)Where concrete strain in tangential direction at the point c0 and at ultimate load is expressed as 0 0 2 c y B cTu cT (2.24) Where
ε
cTu is the ultimate strain of concrete in tangential direction. Hallgren considered this parameter as failure criterion and achieved it by fracture mechanics.Figure 2-5 Concrete and steel strain in tangential direction [14].
For the condition in which at the point c0 concrete is in elastic state while steel has been yielded
x is given by c E cT sy E k f d x 0 2 (2.25)
2.3.HALLGREN MODEL cc c sy f f d x 0 (2.26)
Considering compressive stress distribution of the concrete in tangential direction which is shown in fig. 2.6 the projection of the resultant force of concrete compressive stresses in tangential direction RcT acting on the rigid wedge as shown in Figure 2.4 is calculated as
If concrete is in elastic state
y B c y B x E RcT c cTu ln 2 2 1 (2.27)
If concrete has yielded
2 2 4 1 2 2 ln 2 1 4 3 y B r y B r c r x f R c c c cc cT (2.29)
rc indicates the yield radius and is given as
B y r cy cTu c 2 (2.30)
Figure 2-6 Variation of stress in concrete according to Hallgren [14].
Distribution of stress in reinforcement at the state of punching failure is depicted in Figure 2.7. If the flexural bars are in elastic state, the force caused by flexural reinforcement acting on rigid segment RsT in tangential direction is calculated as
0 2 ln 1 c r c r f d R s s sy sT
(2.32) Where rs indicates a radius along which steel has plastic behaviour and is determined as B y x d r sy cTu s 2 1 (2.33)
Figure 2-7 Steel stress distribution of flexural reinforcement in tangential direction [14]. To determine the radial force caused by reinforcement which is denoted by RsR in Figure 2.4, the strain of steel in radial direction at the point intersecting the shear crack needs to be known. Hallgren assumes it to be equal to the strain of steel in tangential direction and proposes the following set of equation to determine RsR
If rs < c0 B y x d E d RsR s cTu 2 1 2
(2.34) If rs ≥ c0 sy sRd
c
f
R
2
.
0.
(2.35) In Hallgren’s model the dowel force effect D is taking into account by implementing semi-experimental expression presented by Hamadi and Regan [18] as2.3.HALLGREN MODEL
Where Φ is the diameter of the steel bars in [mm]. It worth mentioning that the dowel force significantly decreases when the steel is yielded. Thereby and for the sake of simplicity this force is neglected if reinforcement bars are yielded at r = c0.
Finally establishing the equilibrium equations for the rigid segment illustrated in Figure 2.4 leads to two equations presented below through which the punching capacity of the slab is determined. This process is iterative where the angle α is iteration variable. The failure load is achieved when the compressive strain of the concrete at distance y from the column as shown in Figure 2.4 reaches the critical value of
ε
cTu. This is the failure criterion in Hallgren’s model and is explained in the following.
R
R
R
D
P
u
sR
2
sT
cTtan
(2.37)
c B
x x R x R R R x B c D x d R R P cT cT sT sR sR sT u 5 . 0 3 2 2 cos 2 2 2 2 0 2 (2.38)2.3.2 Failure criterion
Numerical studies done by Hallgren [14], [19] revealed that at the column-slab root the concrete is under tri-axial compression while at r=B/2+y, and in the compression zone it is under biaxial compression, see Figure 2.8. By increasing the external load, the stress σ111
decreases and leads to an unstable state for tri-axial compressive stress. At this moment the shear crack propagates through the compression zone resulting in punching failure.
By neglecting the effect of shear deformation Hallgren realized that at the point r=B/2+y, the tangential strain εcT and the radial strain εcR are equal. Furthermore, considering the experimental
investigation conducted by Kupfer et al. [20], when a concrete specimen is under biaxial compression, at the level of ultimate stress where the macro cracks begin to develop, the compressive strains are equal to the transverse tensile strain. Based on these facts and with regard to the fictitious crack model proposed by Hillerborg et al. [21] ultimate tangential strain at the point r=B/2+y in Hallgren’s model is defined as
x wc cTu
Figure 2-8 State of stress [14].
Hallgren implemented bi-linear tension-softening curve proposed by Petersson [22] to calculate the critical crack width, see Figure 2.9, which gives
ct F c f G w 3.6 (2.40)
Figure 2-9 Bi-linear tension-softening curve [21].
Based on previous studies the fracture energy GFis size dependent which for an element with
very large size is considered as the area under the tension-softening curve illustrated in Figure 2.9. thereby, Hallgren used the multifractal scaling law to take into account the size effect, which gives 2 / 1 . 1 x d G G F a F F (2.41)
Where dα is the maximum size of the aggregate in concrete, αF is an empirical factor which with
2.4.MENÉTREY MODEL
Based on RILEM Draft Recommendation [24] setting α equal to 13 achieves fracture energy for infinite size as 2 / 1 13 1 R a R F F d d G G (2.42)
Where dR is the height of the beam and R
F
G
is the fracture energy according to RILEM. Accordingly, considering Eqs. (2.39), (2.40), (2.41) and (2.42) the failure criterion for Hallgren’s model is written as2 / 1 13 1 . 6 . 3 x d f x G a ct F cTu (2.43)
2.4 Menétrey model
2.4.1 General ass umptions
Figure 2-10 Representation of punching shear capacity of a general reinforced slab [23].
2.4.2 Concrete tensile force
While studying the influence of concrete on punching shear capacity Menétrey neglected the effect of friction which arises due to aggregate interlock along the shear crack. Furthermore, he implemented non-linear fracture mechanics to calculate tensile stress in the concrete and assumed at ultimate limit state as depicted in Figure 2.10 an inclined crack formed at a distance
r1 and r2which are given as
tan 10 1 1 d r r s (2.45)
tan 2 d r r s (2.46)Menétrey et al. studied tensile force created in concrete while cracking by non-linear fracture mechanics and realized that
F
ct
f
ct2/3[26]. What is more, in the Menétrey model the effect oflongitudinal reinforcement, slab thickness and the radius of punching crack initiation on tensile force of the concrete are taking into account by introducing parameters ξ, μ and η respectively. So the tensile force is expressed as
2.4.MENÉTREY MODEL 2 / 1
1
6
.
1
ad
d
(2.49) Where da is maximum aggregate size. 5 . 2 625 . 0 5 . 2 5 . 0 25 . 1 5 . 0 1 . 0 2 h r h r h r h r s s s s (2.50)
Where h is slab thickness and rs is radius of the column.
2.4.3 Dowel effect
The resisting force produced by longitudinal reinforcement is known as dowel effect. This force significantly enhances the punching shear capacity of the concrete flat slabs. As the study done by Regan and Braestrup [26] indicates, up to 34% of the punching capacity of reinforced slabs is due to dowel force contribution. In the Menétrey model this force is calculated as
)sin 1 ( 2 1 2 2 i syi c bars i i dow f f F
(2.51)Where Φi is diameter of the reinforcement crossing the shear crack and ζi is given as
sy s i
/
f
(2.52) Stress in the reinforcement which is denoted by σs is determined by dividing horizontal component of the force in compressive strut, illustrated in Figure 2.10, by the total area of the longitudinal reinforcement crossing the punching crack which gives
bar i si u s A P tan (2.53)It is worth mentioning that in Eq. (2.51) the reduction factor of 1/2 is an approximation taking into account the fact that the reinforcement bars are orthogonal and as a consequence don’t cross the punching crack at right angle.
2.4.4 Shear reinf orcement contribution
i. In the case that a punching crack occurs between the column face and the shear reinforcement the ultimate load is calculating by Eq. (2.57) where the interaction of punching and flexural stiffness is considered. What is more, the inclination of the crack in this failure mode is defined as
h r rswi s 1 tan (2.54) Where, as illustrated in Figure 2.11., rswi is the distance between the shear stud and the
center of gravity of the column and s is inclined length of the punching crack which is determined as
2
2 1 2 r 0 d.9 r s (2.55)ii. In the case that punching crack happens outside the shear reinforcement as depicted in Figure 2.11 b), the punching capacity of the slab is determined as for the case without shear reinforcement except the radius of column rs, is considered rsc, which is the radius of the punching crack initiation and calculated by Eq. (2.45). To elucidate on, in this case the punching capacity of the slab increases due to having a more stiff core around the column.
iii. When the punching crack crosses through the shear reinforcement the bond properties are decisive in the failure mechanism. So that, for the cases with poor bond strength shear reinforcement doesn’t contribute to the punching capacity of the slab. However, the added punching capacity due to shear reinforcement for the case with sufficient bond strength is determined as c sw sw sw
A
F
F
sin
(2.55)αc is the angle between the shear stud and the horizontal direction shown in Figure 2.11 c).
2.5.GEORGOPOULOS MODEL
2.4.5 Pre-stressing tendon contribution
Menétrey takes into account the effect of pre-stressing simply by considering the vertical component of force in tendons crossing the punching crack. It is expressed as
) sin(
p tendons i p p A F
(2.56)Where β is the inclination of the tendon at the place it intersects the punching crack.
Menétrey asserts that when tendons are bonded to the surrounding concrete even additional enhancement of the punching capacity. It is due to the fact that bonded tendons in addition to applying counter acting moment contributing to the flexural stiffness of the slab function in the same way as longitudinal reinforcement. Accordingly, in this case while calculating ζ by Eq. (2.48) the area of pre-stressing tendons intersecting punching cracks should be added to the area of longitudinal reinforcement.
2.4.6 Relation between punching and flexural capacity
It goes without saying, of paramount importance is the geometry of the failure surface in determining the failure load for a slab. This issue has been studied experimentally by Menétrey [26]. He concluded that the inclination of the crack α is a key parameter through which flexural and punching failure loads are linked to each other. He suggested the following expression
90 30 45 2 3 sin ) ( 2 / 1 u flex u fail P F P F (2.57)
Contemplating on Eq. (2.57) indicates that for α=30̊, Ffail=Pu, while for α=90̊, Ffail=Fflex. It is worth mentioning that, for design purpose, Pflex can be computed by yield-line theory.
2.5 Georgopoulos model
Georgopoulos [27] developed a method to predict the inclination of the punching crack and the punching capacity of flat slabs without shear reinforcement. In his model the mechanical reinforcement ratio and the tensile strength of the concrete are the main variables. In this model the inclination of the crack is determined as
3 . 0 056 . 0 tan (2.58) Where ω is mechanical reinforcement ration and is defined as
The model presented by Georgopoulos is illustrated in Figure 2.12 where the punching capacity of the slab is approximated to
0.20 0.35cot 2 cot 13 . 4 2 d d d Pu s (2.60)
Where ds is diameter of the column, and σ is maximum principle tensile stress of the concrete in the punching crack which is determined by the following expression
2/3 , 17 . 0 fckcub
(2.61)Figure 2-12 Load bearing model of flat slab developed by Georgopoulos [27].
2.6 Bond model for concentric punching shear
Alexander and Simonds [28] developed a mechanical model relying on radial arch action and critical shear stress for beams. They assumed that punching capacity of the slab is mainly yielded from the resistance of four strips branching out of the column and extended parallel to the reinforcement direction as shown in Figure 2.13. The length of the strips in this model, denoted by lw, is from the edge of the column to the point where shear force is zero. These stirrups are assumed as cantilever beams having the bending capacity of Ms. Assuming q, as the shear force that can be transferred by the quadrant of the slab adjacent to the cantilever strip,
Ms is determined by implementing equilibrium as
2
2
w2 sql
M
(2.62)On the other hand, by isolating a single strip and writing the equilibrium equations the axial force created in column due to the effect of one strip, P1 is determined as
w
ql
2.7.MUTTONI MODEL
q M
Pu 8 s (2.64) Looking at Eq. (2.64) indicates that by determining q, which is the maximum shear force that the strips can bear, the punching capacity of the slab is calculated. Alexander and Simonds claim that using one-way shear capacity which is known as inclined crack load to determine the
q leads to satisfactory results. Accordingly, they propose
c
f d
q0.1667 [kN/m] (2.65) Where fc is in MPa and d is in mm.
Figure 2-13 layout of radial strips in bond model [28].
2.7 Muttoni model
Muttoni applied critical shear crack theory to determine the punching failure capacity of concrete slabs [29]. As illustrated in Figure 2.14, the inclined crack propagates through the slab into the compression strut transferring the force to the column and disturbs its performance. As the crack becomes wider the capacity of inclined concrete strut decreases which eventually leads to punching failure. In other words, based on critical shear crack theory, for concrete elements without shear reinforcement the roughness and width of the inclined crack developing through the concrete compressive strut govern the shear strength.
Walraven et al. [30] indicated that roughness of the crack can be taken into consideration by dividing the width of the inclined crack by the quantity (dg+16). Where dg and dg0 are maximum aggregate size and the reference size of 16 mm, respectively. On the other hand, Muttoni and Schwarts [31] showed that the width of the critical shear crack is proportional to product of slab rotation ψ and effective height d as illustrated in Figure 2.14. Accordingly, Muttoni suggests the punching failure criterion for flat slabs without shear reinforcement as
0 0 1 15 4 / 3 g g c u d d d f d b P [MPa, mm] (2.66)
Where b0 is the perimeter of the control area which is at the distance d/2 from the face of the column. Furthermore, Muttoni by analytical approach indicated that the relationship between load and rotation in flat slabs is expressed as
2 / 3 2 5 . 1 flex s y V V E f d B (2.67) Where Vflex is the flexural capacity of the slab and V is the acting shear force. By implementing yield-line theory flexural capacity of uniformly reinforced slab loaded as Figure 2.15 is determined as c B c c B B c r m V q r flex 4 8 sin 8 cos 4 2 2 (2.68)
Where mr is the moment capacity. In Muttoni model to calculated the punching shear capacity, failure criterion expressed by Eq. (2.66) and load-rotation relationship presented by Eq. (2.67) should be solved simultaneously, see Figure 2.16.
2.8.DISCUSSION ABOUT MECHANICAL MODELS
Figure 2-16 Load-rotation curve and failure criterion [29].
2.8 Discussion about mechanical models
Mechanical models which have been developed in history by studying the punching behaviour of flat slabs are based on equilibrium and compatibility concepts. Each model has its own limitations and best fits for special cases. Some don’t take into account the effect of reinforcement and pre-stressing while some attempt to consider these factors. Furthermore, failure criteria implemented in different models are of significant importance which sometimes vary dramatically. In the author’s opinion it could be an interesting field of research to improve existing mechanical models based on modifying the failure criterion. Alongside with that, the role of concrete tensile strength is not acknowledged by all mechanical models. Generally speaking, how mechanical models are taking into account the role of concrete tensile strength in punching capacity of slabs fall into two categories as illustrated in Figure 2.17. Some models neglect this effect while some models attempt to take it into account. However, implementing nonlinear fracture mechanics and creating numerical models provide an opportunity through which more comprehension of this issue could be grasped.
3.1.EFFECT OF PRE-STRESSING
3
Effect of pre-stressing on punching
capacity of flat slabs
3.1 Effect of pre-stressing
Concrete is a material with relatively high compressive capacity and low tensile strength. Alongside with that, concrete has very low deformability and cracks in the case that shrinkage is prevented. Accordingly, devising methods to tackle such drawbacks are quite natural. Engineers came up with two solutions. They suggest either implementing pre-stressing technique or reinforcing the concrete.
Generally speaking, pre-stressing in flat slabs leads to reduction of cracking and deformation under serviceability loading. Furthermore, it gives an opportunity to reduce dead weight of the slab having slender section which results in an economic design. However, in such slabs usually punching failure is dominant in the ultimate limit state due to the limited thickness of the slab. On the other hand, although punching failure is a local damage, it can yield in progressive damage by overloading adjacent columns. These facts necessitate a deep and more detailed reflection on the issue.
As mentioned in chapter one, both experimental and numerical studies have been conducted over the years intending to unveil the influence of pre-stressing punching behaviour of slabs. They stress some potential beneficial effects as
i. In vicinity of the column, implementing inclined tendons results in vertical forces assisting punching resistance.
ii. Pre-stressing leads to increase of the compressive stress in the concrete which has been reported to enhance the punching capacity.
iii. The counter acting moments produced by eccentricity of the tendons also have been reported to boost punching capacity.
In 2012, Clément et al. [3] attempted to have a thorough look at the issue by applying the critical shear crack theory. They considered the failure criterion proposed by Muttoni, presented by Eq. (2.66) and tried to investigate the influence of pre-stressing. They assert that the vertical component of the pre-stressed tendon reduces the shear force transferred by the concrete in vicinity of the column so that
P E P
c R q dA V V V
V
(3.1) Where q is the external load exerted on the punching cone, R is the reaction force, VE is thw acting shear force and VP is the vertical component of the pre-stressing force at the place where tendons intersect the punching crack, see Figure 3.1. Considering Eq. (2.66) punching failure is reached when PU=Vc=VE-VP. It is all transparent that in this situation part of the load is carried by the pre-stressing tendons resulting in achieving higher punching capacity.Figure 3-1 Reduction of shear force due to tendon inclination [3].
Alongside with that, Clément et al. shed more light on the influence of pre-stressing on general behaviour of the slab which is expressed by rotation of the slab, ψ. They concluded that pre-stressing contributes to the amount of slab rotation through two distinct mechanisms. First one is due to the effect of the normal compressive stresses and the second effect is caused by tendons eccentricity. To elucidate on, as illustrated in Figure 3.2, normal compressive stresses increase the rotational stiffness of the slab which leads to less ultimate rotation and higher punching capacity.
3.2.EUROCODE 2, APPROACH
neglected. While they only take into account the contribution of vertical components of pre-stressing tendons. In the following the design approach proposed by Eurocode 1992 is presented.
Figure 3-3 Influence of bending moment caused by pre-stressing on punching capacity [3].
3.2 Eurocode 2, approach
While determining the punching capacity of concrete flat slabs, Eurocode 2 [2] acknowledges only the vertical components of tendons Vp,EC, stretching within the area at distance of 2d from edge of the column, see Figure 3.4. and neglects the effect of eccentricity and in-plane compressive stresses. The criterion presented by Eurocode 2, considering punching failure of pre-stressed flat slabs is expressed as
k f
b dV V
VE p,EC R,EC 0.18 100 c 1/3 0.1c EC (3.2) Where bEC as shown in Figure 3.4 is perimeter of the control area, k and ρ are size effect factor and reinforcement ration, respectively, which are expressed as
2 200 1 d k (3.3) 02 . 0 lx ly (3.4)
4.1.INTRODUCTION
4
Methodology
4.1 Introduction
In this chapter the process of creating numerical model of a pre-stressed concrete flat slab is presented and the theoretical concept behind each step is discussed. Furthermore, validity of the created numerical model is going to be controlled by experimental investigation conducted by Hassanzadeh [4].
From economical point of view pre-stressed flat slabs made of normal strength concrete are suitable for medium spans where the advantage of implementing pre-stressing tendons overweighs the self-weight of the slab. Accordingly, in this investigation diameter of the slab,
c, corresponding to the area around the column in which the radial bending moment is negative
is assumed to be 2.5 m which makes sense for medium length spans of about 10 m, see Figure 4-1. Going into the depth, in order to study the punching shear capacity of the slab resting on group pf columns, only a circular part of the slab a long circumference of which the bending moment is zero may be modelled. Alongside with this idealization, an appropriate boundary condition should be assigned for the isolated circular part of the slab in vicinity of the column. The slab can be considered to have simply supported boundary condition by accepting the simplifying assumption that the vertical displacement of the slab would be zero at the same place where negative radial bending moment vanishes. This assumption has been implemented in this investigation.
Figure 4-1 Diameter of the modeled slab in relation to bending moment diagram [14].
As presented in Table 4-1 properties of the concrete and reinforcement steel in the model is assumed to be identical to concrete class C30/37and steel B500B based on Eurocode 2 [2]. Furthermore, pre-stressing cables consist of 7 wire strands of type Y1670S7 based on prEN 10138 [32]. Figure 4-2 illustrates the geometry of the slab which is going to be studied in this investigation.
Table 4-1 Material properties
Density (kg/m3) E (GPa) fcm (MPa) fctm (Mpa) fy (MPa) fu (MPa) εcu (%) εc1 (‰) concrete 2400 33 38 2.9 - - 0.35 2.2 reinforcement 7850 200 - - 500 540 5 - cable 7850 195 - - 1640 1860 2.5 -
Figure 4-2 Reinforcement and dimensions of the slab.
4.2.CONCRETE MATERIAL BEHAVIOUR
4.2 Concrete material behaviour
In this part, first the mechanical properties of the concrete is presented and then the mathematical model implemented to define concrete in numerical analysis is discussed.
4.2.1 Compressive behaviour
Concrete has completely distinct response when it is subjected to compressive or tensile force. Accordingly, two different stress-strain curves should be introduced to represent compressive and tensile behaviour of the concrete.
Concrete shows highly non-linear behaviour while applying uniaxial compressive load so that only up to around 40% of the ultimate compressive strength, fcm, the stress-strain curve remains linear. At this point micro cracks appear and propagate due to increasing the load. This phenomenon which results in gradual decrease of modulus of elasticity continues approximately up to 70% of ultimate compressive strength. After this stage, bond cracks form between cement paste and aggregates due to increasing the external load which leads to dramatic reduction of the modulus of elasticity. Then, gradually the number of these bond cracks increases and matrix cracking begins. This process is continued until the maximum compressive strength is achieved. Finally, the softening stage initiates where stress-strain curve descends and failure occurs at ultimate strain. Eurocode 1992 [2] presents an empirical formula to determine nonlinear stress-strain curve of concrete when it is subjected to uniaxial compressive force. Figure 4.3 illustrates the idealized uniaxial compressive behaviour of the concrete introduced by Eurocode [2].
Figure 4-3 Uniaxial compressive of concrete [2].
1 c c
(4.2) cm c cm f E k 1.05
1 (4.3)It is worth mentioning that Abaqus software assumes the stress at last point of the stress-strain curve is maintained for strains higher than εcu1, which means neglecting the crushing of concrete. Therefore, to have more realistic behaviour one point is added to stress-strain curve where stress is equal to zero1. Figure 4-4 depicts the stress-strain curve for concrete C30/37 which is assumed while creating numerical model.
Figure 4-4 Stress-strain curve for concrete C30/37.
4.2.2 Tensile behaviour
Concrete has brittle behaviour while subjected to tensile force. As illustrated in Figure 4-5 the response of the concrete remains linear up to just below the tensile strength which is denoted by point b. At this stage increasing the external load leads to formation of micro cracks resulting in reduction of modulus of elasticity and as a consequence nonlinear behaviour. This continues until point c which represents the tensile strength of the concrete. Before reaching the tensile strength crack propagation is ceased if the external load remains constant. However, after this point cracks becomes unstable meaning that the released strain energy is adequate to cause crack propagation without increasing the external load. Accordingly, after point c, in order to have stable crack growth the external load has to be decreased. Descending part of the
1 While modeling nonlinear material abaqus software does not allow to introduce a point with zero stress after
yield point. Thus the stress at last point is considered around 1% of compressive strength.
0 5 10 15 20 25 30 35 40 0 0.001 0.002 0.003 0.004 Str es s Mpa strain uniaxial compressive response of concrete C30/37
4.2.CONCRETE MATERIAL BEHAVIOUR
strain curve confirms this fact. During this stage, micro cracks which are accumulated in limited areas named fracture process zone are merged together forming macro cracks. This results in softening behaviour of the concrete in tension [33].
Figure 4-5 Behaviour of concrete under uniaxial tensile loading [33].
In numerical analysis, to have a more realistic tensile behaviour of concrete especially in descending part of uniaxial tensile response, introducing stress-strain is not an appropriate format. This is due to the fact that in this stage total displacement of the concrete consists of sum of elastic strain in un-cracked part of the concrete and displacement caused by increasing of crack’s width. Accordingly, as illustrated in Figure 4-6, it is proper to divide the tensile curve into two parts; un-cracked part and cracked part. the response of the un-cracked concrete is elastic and defined by stress-strain curve while crack opening stage in which the width of the crack starts to increase is more suitable to be defined in terms of stress-displacement (σ-w) [34]. This is because displacement resulted by crack opening is independent of specimen size and rather is related to fracture energy which is a material property.
Figure 4-6 Crack propagation in concrete at uniaxial tensile loading [34].
It is worth mentioning that fracture energy which is equal to the area under stress-displacement curve represents the amount of energy required to establish a stress free crack. As given in Eq. (4.4) International concrete design standard model code 2010 [35] presents a formula in which fracture energy of the concrete is only function of compressive strength.
There are several crack opening curves defining the uniaxial nonlinear tensile behaviour of the concrete while crack propagation among which linear, bi-linear and exponential curves are common, see Figure 4-7.
Figure 4-7 three common crack opening curves [36].
Deciding about which curve to be used in numerical analysis depends on the required accuracy. As depicted in Figure 4.7 implementing linear curve overestimates the stiffness and tensile capacity of the concrete. In other words, exponential and bi-linear curves result in quicker depletion of strain energy just after crack initiation. At the same time Figure 4-7 indicates that the higher tensile strength results in more ductile behaviour. This can be realized in Figure 4.7 by observing steeper response in unloading stage for concrete samples with higher tensile strength.
Figure 4-8 Crack opening law and tensile strength effect on tensile response of concrete [37].
In this study the exponential curve is chosen to define crack opening behaviour of the concrete. Cornellissen et al. [38] suggested the formula given below to define the exponential curve
4.2.CONCRETE MATERIAL BEHAVIOUR Where
c c w w c w w c w f 2 3 1 exp 1 (4.6)wc is amount of displacement which is required to occur in a crack to be considered as stress-free and is approximated to
t f c f G w 5.14 (4.7)
Where Gf is fracture energy and is calculated by Eq. (4.4).
In Eq. (4.6) c1 and c2 are material constants and are considered equal to 3.0 and 6.93, respectively.
Figure 4.9 depicts the crack opening curve which is introduced in Abaqus software for concrete material in this investigation.
Figure 4-9 Crack opening curve for concrete material.
4.2.3 M ultiaxial behaviour
While creating a numerical the method multiaxial behaviour of the concrete should not be neglected. Concrete indicates more ductile behaviour when it is subjected to multiaxial compression compered to uniaxial loading. In other words, the ability of the material for energy consumption is increased. Furthermore, in the condition of multiaxial compression the compressive strength of the concrete is increased. This is due to the phenomenon called confinement. Figure 4-10 indicates the biaxial behaviour of the concrete.
The other feature of biaxial response of concrete which can be inferred by reflecting on Figure 4-10 is that applying tensile and compressive stress simultaneously, decreases the capacity of the sample. Based on what alluded to above, about the multiaxial behaviour of concrete it is of
in numerical analysis. Conventionally, Von Mises theorem is used to define the behaviour of elasto-plastic materials when they indicate analogous behaviour under tensile and compressive stress which is not the case for concrete material the behaviour of which dramatically is distinct under compression and tension. Accordingly, in this study concrete damage plasticity (CDP) model is implemented to define behaviour of the concrete. In CDP model modified Drucker-Prager strength hypothesis is taken into account to define yield surface of the material.
Figure 4-10 Failure envelope under biaxial loading [36].
4.3 Concrete model
4.3.1 Yield s urface
As mentioned above CDP is used to define concrete behaviour. This model is a continuum plasticity based damage mode for concrete. the main assumption in this model is that the concrete fails either due to compressive crushing or tensile cracking. In this model the yield function which was defined by Lubliner et al. [39] and later modified by Lee and Fenves [40] is determined as
pl c c pl p q F 3 ~ ~ 1 1 max max (4.8)Where maxis maximum eigenvalue of stress tensor and
pl c~ is compressive plastic strain.
/
1 2 1 / 0 0 0 0 c b c b
(4.9)4.3.CONCRETE MODEL
the amount of σb0/σc0 is suggested to be considered equal to 1.16 [41]. is Macauley bracket, defined as
x x
x 2 1 (4.10)
1
(1 ) ~ ~ ~ pl t t pl c c pl (4.11)In Eq. (4.8)p indicates hydrostatic pressure and q stands for Von Mises equivalent stress
which are determined as σ ( 3 1 tr p ) (4.12) 2 3J q (4.13) Where σ is stress tensor and J2 second invariant of deviatoric stress which is calculated as
S S j : 2 1 2 (4.14)
The function (:) stands for doubly contracted product and S indicates deviatoric stress which is defined as
I p S
I is identity tensor.
γ is an empirical parameter which is determined by conducting triaxial test and is equal to
1 2 ) 1 ( 3 c c K K
(4.15)Kc is interpreted as the ratio of distance between hydrostatic axis and respectively compression meridian and tension meridian in the deviatoric cross section. However, in lack of experimental data parameter Kc is suggested to be assumed as 2/3 in CDP model [41]. By calculating γ which
