Evaluation of the shear capacity of precast-prestressed hollow core slabs: numerical and...

ORIGINAL ARTICLE

Evaluation of the shear capacity of precast-prestressedhollow core slabs: numerical and experimental comparisons

E. Brunesi • D. Bolognini • R. Nascimbene

Received: 5 February 2013 / Accepted: 7 January 2014 / Published online: 28 January 2014

� RILEM 2014

Abstract Since eighties, 400 and 500 mm thick

precast-prestressed concrete hollow core slabs, char-

acterized by increasingly optimized cross-sections

with non-circular voids, became very common. How-

ever, deeper slabs with long spans, which have to resist

high line loads acting close to the supports, are

subjected to initial web shear cracking and may fail at

loads less than those predicted by traditional codes

prescriptions. The shear strength capacity of these

members without transverse reinforcement is evalu-

ated through a campaign of detailed nonlinear finite

element analyses, matching experimental test data

collected from past programs. Constitutive models,

based on nonlinear fracture mechanisms, are consid-

ered to numerically reproduce the experimental

response of single span, simply supported, isolated

hollow core units, highlighting web-shear failure

mechanism, due to short development length and lack

of transverse reinforcement. The adopted diffuse

smeared fixed cracking constitutive model allows a

reliable prediction of shear stress distributions and

crack patterns for these members in their inelastic

branch. The presence of a variable inclined strut is

clearly evident. Peak shear stress is localized at the

bottom side of the cross-section, rather than at the

level of the centroid. The experienced brittle web-

shear failure mechanism is governed by hollow core

shapes with circular or non-circular voids, as evi-

denced by the evolution of the principal tensile strain

distributions. Typically, less inclined, more rounded,

diagonal crack, controlled by the smooth web width

variation along depth, is opposed to a fairly constant

variation of the fracture angle inclination, governed by

the abrupt and irregular web width drop.

Keywords Hollow core slab � Shear strength �Prestressed concrete � Precast members � Non-

circular voids

Abbreviations

a Shear span

a1 Distance between two line loads

a1 Ratio of distance from end of member

to transfer length

Alow Area of lower strands

Aup Area of upper strands (if present)

bw Sum of web widths

D Cross-section depth, assumed as HS

Di/Di,max Normalized depth

E. Brunesi (&)

ROSE Programme, UME School, IUSS Pavia, Pavia, Italy

e-mail: [email protected]

URL: http://www.umeschool.it

D. Bolognini � R. Nascimbene

European Centre for Training and Research in Earthquake

Engineering (EUCENTRE), Pavia, Italy


URL: http://www.eucentre.it

R. Nascimbene


URL: http://www.eucentre.it

Materials and Structures (2015) 48:1503–1521

DOI 10.1617/s11527-014-0250-6

Dlow Diameter of lower strands

Dup Diameter of upper strands (if present)

Dz Mid-span vertical displacement

dp,low Average distance from lower strand to

the soffit

dC50 Standard deviation of fC50,mean

E1 Principal tensile strain

E3 Principal compressive strain

Fz Shear force at the support

fc Concrete compressive strength

fC50,mean Mean concrete compressive strength

by cylinder tests on 50 mm cores

drilled from the tested specimens

fctd Design tensile strength of concrete per

EC2

ft Concrete tensile strength

/ Diameter of strand

H Nominal slab depth

HH Average depth of hollow core

HS Average depth of slab cross-section

I Moment of inertia of the section about

the centroidal axis

LF Normalized load level

Ls Distance between slab end and centre

of support

Lt Transfer length of prestressing force in

strands, assumed as 55/ (slow release)

Nlow Number of lower strands

Nup Number of upper strands (if present)

Ri Ratio between experimentally observed

and predicted shear strength capacity

S First area moment of the section about

the centroidal axis

Slow Largest slippage of lower strands

Sup Largest slippage of upper strands (if

present)

rcp Fully effective concrete compressive

stress at the centroid due to prestressing

rp Prestress in the strands within the

transfer region

rp0 Fully effective prestress in the strands

outside the transfer region

rp0,low Initial fully effective prestress in lower

strands outside the transfer region

rp0,up Initial fully effective prestress in upper

strands (if present) outside the transfer

region

tb Length of bearing

tlow Average concrete thickness below the

hollow core

syz Shear stress

Vobs Experimentally observed shear strength

capacity

Vpre Predicted shear strength capacity

x Coordinate along the longitudinal axis

of the member

x/D Ratio between longitudinal distance

from the support and cross-section depth

1 Hollow core features and criticalities

The use of precast concrete in flooring system is a

widespread practice in many countries. Precast-pre-

stressed hollow core (PPHC) units are commonly

spread products of precast industry, finding wide

applications in building construction, particularly as

floor and roof components. Short construction times,

high level of quality and consistent reduction of self-

weight are achievable, allowing manufacturers to

cover spans up to 20 m. The upper surface of the

hollow core slabs is usually levelled with a cast-in situ

concrete topping. Normally, for long spans, advantage

is not taken of this topping layer as a composite

construction because of the uncertainty concerning the

bond strength between the precast and cast-in situ

concrete. Proper shear strength at the interface is

required for the composite action; the horizontal and

vertical shear strength of the joint between adjacent

hollow core units has to be high enough to transmit the

forces due to horizontal diaphragm action and load-

sharing of concentrated loads.

Through the years, 200 and 265 mm thick units,

characterized by cross-sections with circular voids, were

replaced by deeper units (320, 370, 400 and 500 mm),

presenting cross sections with non-circular voids. Stan-

dardized cross-section shapes, usually characterized by

relevant void ratios, mostly depending on the thickness

of the slab itself, were increasingly optimized. However,

experimental research works available in literature [9,

13] evidenced that PPHC slabs, with deeper cross

sections, could be subjected to initial web shear cracking

at the end regions, resulting into critical brittle failure

mechanisms. Comparisons with traditional codes pre-

scriptions, American Concrete Institute (ACI) [1] and

Eurocode (EC2) [5], highlighted that these members

1504 Materials and Structures (2015) 48:1503–1521

without transverse reinforcement, generally not feasi-

ble, can fail, due to web-shear cracking in the ‘‘dis-

turbed’’ end regions, at loads less than those predicted by

conventional design methods.

Although PPHC slabs have been widely adopted,

some aspects of the commonly-used design approaches

against shear are still a cause of concern; short bearings

and not fully developed prestressing force influence

their response. Furthermore, their common production

system (i.e. extrusion) may result in features not

properly in agreement with those of the standard

concrete strength monitoring procedures. PPHC have

undergone extensive experimental tests [2, 8, 12, 21] to

verify their compliance with the provisions of tradi-

tional Codes and Standards and classify the possible

failure modes. Walraven and Mercx [21] also noticed

that the observed shear resistance was lower than the

predicted one when the voids were non-circular.

Furthermore, Yang [23] observed that the maximum

tensile stress is not located at the level of the centroid,

as expected by EC2, which, additionally, ignores the

shear stresses due to the prestress force. Recently,

Hawkins and Ghosh [9] have also shown that PPHC

slabs deeper than 320 mm can fail in shear at loads less

than those predicted by ACI recommendations. Addi-

tional explanations for the observed mismatch have to

be researched in the different contributions to shear

transfer mechanism with respect to traditional normal-

strength concretes [7]. Shear failure surfaces in high-

strength concrete members are smoother than in

normal-strength concrete ones, showing cracks that

can be propagated through coarse aggregate

particles rather than around them, as shown by Cladera

and Marı [6].

0 100 200 300 400 500 600 700 8000

0.5

1

1.5

2

2.5

Vpre

[kN]

Ri =

Vob

s / V

pre [−

]

200265320370400500V

obs = V

pre

0 100 200 300 400 500 600 700 8000

0.5

1

1.5

2

2.5

Vpre

[kN]

Ri =

Vob

s / V

pre [−

]

200265320370400500V

obs = V

pre

0 100 200 300 400 500 600 700 8000

0.5

1

1.5

2

2.5

Vpre

[kN]

Ri =

Vob

s / V

pre [−

]

200265320370400500V

obs = V

pre

0 100 200 300 400 500 600 700 8000

0.5

1

1.5

2

2.5

Vpre

[kN]

Ri =

Vob

s / V

pre [−

]

200265320370400500V

obs = V

pre

a b

c d

Fig. 1 a–d Ri versus Vpre for each considered depth, according to EC2, Yang’s approach, ACI and CSA, respectively

Materials and Structures (2015) 48:1503–1521 1505

1.1 Experimental test database and accuracy

of current design methods

Past experimental data, collected by Pajari [13], have

been re-examined to obtain an experimental database

of single span, simply supported, isolated PPHC units,

loaded with transverse and uniformly distributed line

loads, highlighting web-shear failure mechanism, due

to short development length and lack of shear rein-

forcement. The 49 specimens were characterized by

six nominal slab depths, five hollow core shapes with

circular and non-circular voids, different void ratios,

several prestressing steel strands arrangements and

levels of initial prestress. The imposed vertical loads

acted at x/D [ 2.4. No cast-in situ filling concrete was

provided in the voids at the end regions. Idealized

pinned ends were mechanically materialized by rollers

and wedges, properly tightened to prevent torque.

To quantify their effectiveness, three commonly-

known Codes prescriptions, EC2, ACI and Canadian

Standards Association (CSA) [3], as well as a research

approach [23] against web-shear failure mechanism,

were adopted to predict Pajari experimental database.

In particular, Eq. 6.4 of EC2, reported below,

Eqs. 11–12 of ACI and Eqs. 11-6, 11-10, 11-11 and

11-13 of CSA were used.

Fig. 2 Conceptual experimental test set up: details of wedges, rollers and examples of tested hollow core units (adapted from [13])

Table 1 Cases of study: model and slab type, nominal depth (H), number, diameter, area, initial prestress and largest slippage of

upper (Nup, Dup, Aup, rp0,up, Sup) and lower strands

Model

type

Slab

type

H

(mm)

Nup

(–)

Dup

(mm)

Aup

(mm2)

rp0,up

(MPa)

Sup

(mm)

Nlow

(–)

Dlow

(mm)

Alow

(mm2)

rp0,low

(MPa)

Slow

(mm)

200 Circular 200 – – – – – 7 12.5 93 1100 0.4

265 Circular 265 – – – – – 10 12.5 93 1100 1.1

320a Non-circular 320 – – – – – 13 12.5 93 1100 2.5

320b Non-circular 320 – – – – – 11 12.5 93 1000 1

320c Circular 320 2 9.3 52 900 0.2 11 12.5 93 1000 0.6

370 Non-circular 370 – – – – – 12 12.5 93 1000 0.6

400a Non-circular 400 – – – – – 13 12.5 93 1100 1.3

400b Non-circular 400 – – – – – 13 12.5 93 1000 1.9

500 Non-circular 500 – – – – – 16 12.5 93 1000 0.8


Vpre;EC2 ¼Ibw

S

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

fctdð Þ2þ a1rcpfctd

q

ð1Þ

As evidenced in Fig. 1, analytical approaches

roughly lead to unconservative (EC2 and Yang), or

over-conservative predictions (ACI and CSA), accord-

ing to different nominal depths and geometry types,

not directly accounting for the influence of local

hollow core shape. Therefore, the inaccuracy of

current codes is reaffirmed, evidencing a particularly

critical scenario, if they are adopted outside of their

‘‘implicit field of applicability’’.

2 Nonlinear monotonic solid FE analyses

The critical scenario depicted by analytical predictions

suggests the need of a detailed campaign of numerical

analyses, as in past research works to assess the

performance of RC [10] and steel structures [11, 15,

22], addressed to investigate the effects of the hollow

core shape on the shear response of such members. To

properly reproduce the experimental test set up,

conceptually schematized in Fig. 2, refined nonlinear

3D FE models, based on nonlinear fracture mecha-

nisms, were developed and monotonically loaded till

the brittle web-shear failure mechanism is reached.

Nine specimens, whose main features are presented

in Tables 1 and 2, have been considered. In particular,

for 320 mm deep slabs, three specimens have been

selected: two characterized by non-circular and one by

circular voids, to quantify the hollow core shape

influence on the shear strength capacity for members

with similar depths. Similarly, two 400 mm deep units

have been analysed, since they evidence non-circular

hollow core shapes with various levels of irregularity.

2.1 Proposed numerical approach

To predict brittle web-shear failure mechanism, the

total strain crack model, developed along the lines of

the modified compression field theory (MCFT), orig-

inally proposed by Vecchio and Collins [19] and then

extended to the three-dimensional case by Selby and

Vecchio [16], was adopted. The classical Von Mises

yielding criterion with strain hardening was used for

Table 2 Cases of study. length, span, mass, bearing (tb), shear

span (a) and distance between two line loads (a1), average

depth of section (HS), sum of individual web widths (bw),

average depth of hollow core (HH), average concrete thickness

below hollow core (tlow), average distance from lower strand to

the soffit (dp,low), mean core strength (fC50,mean) and relative

standard deviation (dC50)

Model

type

Length

(mm)

Span

(mm)

Mass

(kg)

tb(mm)

a(/a1)

(mm)

HS

(mm)

bw

(mm)

HH

(mm)

tlow

(mm)

dp,low

(mm)

fC50,mean

(MPa)

dC50

(MPa)

200 6,643 6,603 2000 40 917 200 239 153 19 39 54.8 3.8

265 5,253 5,213 2240 40 724 260 228 175 42 36 71.3 4.2

320a 7,198 7,118 3810 80 1,160 315 309 222 38 43 61.8 2.4

320b 7,995 5,800 3880 80 760 321 289 216 36 38 64.3 2.6

320c 5,985 5,945 3100 40 826 310 263 213 40 51 46.8 2.0

370 7,002 6,942 4060 60 960/300 359 272 237 54 40 72.2 2.1

400a 7,626 7,546 4510 80 1,048 402 286 299 49 38 71.5 4.0

400b 8,390 7,150 4580 100 1,150 390 293 310 38 41 67.3 1.7

500 8,492 8,412 6000 80 1,475/300 503 324 398 40 47 76.4 2.6

−6 −5 −4 −3 −2 −1 0 1

x 10−3

−80

−70

−60

−50

−40

−30

−20

−10

0

10

Strain ε [−]

Str

ess

σ [M

Pa]

200265320a320b320c370400a400b500

Fig. 3 Calibration of the solid FE models: uniaxial tensile and

compressive constitutive laws


prestressed steel strands, represented as embedded

elements. Two types of prestress distribution, linear,

according to codes provisions (ACI and CSA) and

parabolic, according to Yang [23], were considered to

evaluate their influence on the shear strength. In

particular, Eq. 2, proven by full-scale tests [17],

expresses the parabolic curve selected.

rp¼0 0�xþ Ls� 5/

1� Ltþ5/�x�Ls

Lt

� �2� �

rp0 5/� xþ Ls�5/þ Lt

rp0 xþ Ls� 5/þ Lt

8

>

<

>

:

ð2ÞTherefore, no interface elements were introduced to

represent strands-concrete interaction, since implicitly

captured by an equivalent prestress distribution.

According to Pajari [13], two loss of prestress ratios,

5 and 15 %, were adopted, due to the little level of

knowledge about concrete mix and storing conditions,

also providing an upper and lower threshold and

showing that the shear capacity of PPHC units is not

sensitive to small variations in the losses. Similar

range is observed by direct computations, as con-

firmed by Palmer and Schultz [14] for analogous

Fig. 4 Specimen 320b:

example of detailed solid

FE models developed

Table 3 Total number of bricks and DoF for each specimen

Model type #Bricks #DoF

m1 m2 Strip m1 m2 Strip

200 124,960 88,770 13,640 257,187 181,566 28,927

265 73,920 52,800 9,600 153,800 110,120 20,871

320a 190,060 129,688 30,160 365,533 250,027 59,259

320b 155,100 109,604 25,016 303,727 215,155 50,337

320c 187,872 131,016 29,468 350,891 245,321 56,436

370 253,640 173,072 40,832 461,977 316,015 76,107

400a 247,680 165,120 37,920 467,279 312,299 72,903

400b 224,112 151,984 34,220 427,631 290,711 66,699

500 350,976 230,328 52,416 636,664 418,864 96,714

Table 4 Combinations of prestress distributions and losses of

prestress

Combination

ID

Prestress

distribution

Loss of

prestress (%)

m2p1 Parabolic 5

m2p2 Parabolic 15

m2p3 Linear 5

m2p4 Linear 15


PPHC slabs, even though more conservative levels

(20–25 %) may be assumed in the design process of

such members to safely represent the long-term

effects. The diffuse smeared fixed cracking constitu-

tive law assumed for concrete exploits an exponential

softening decay curve and Thorenfeldt et al. [18]

model for uniaxial tensile and compressive behavior,

respectively. Figure 3 presents the calibration of the

adopted uniaxial tensile and compressive constitutive

laws, for each selected specimen, according to CEB-

FIP Model Code [4].

Confinement effect was neglected, while full shear

retention and lateral crack effect [20] were potentially

accounted, as well as inward slippage of the strands.

Newton–Raphson iteration scheme was adopted with

an energy-normalized convergence criterion. Six-

node brick elements were used and mesh sensitivity

was investigated by doubling the extruding offset. For

a

b

Fig. 5 Comparison between experimentally observed and numerically predicted web-shear failure mechanism: a experimental failure

of specimen 320b (adapted from [13] )and b numerical crack pattern at ultimate conditions


each specimen, an equivalent strip of the PPHC unit

was modelled to evaluate the effectiveness of this

assumption. An example of the FE mesh is depicted in

Fig. 4, while Table 3 summarizes the total number of

brick elements and degrees of freedom employed for

each specimen. Finally, considering the four combi-

nations of prestress distributions and losses, detailed in

Table 4, a total of 54 models were created.

2.2 Results and discussion

In this section, FE results are synthesized to outline the

influence of the hollow core shape on the web-shear

failure mechanism and, hence, on the shear strength of

such members. First, the proposed numerical approach

is validated by focusing on a single PPHC unit, taken

as reference. Therefore, the accuracy of the FE

predictions obtained for the nine specimens analysed

is quantified in comparison with experimental data.

Finally, the observed trends are justified by consider-

ing the evolution of local quantities, such as principal

tensile strain and shear stress, in order to depict

behavioral changes as consequence of geometrical

variations in the cross-section shape of such slabs.

2.2.1 Model validation: specimen 320b

The adopted modeling approach evidences a consis-

tent match with experimental test results; Fig. 5

presents a comparison between the experimentally

observed and numerically predicted web-shear failure

mechanism, for a 320 mm deep PPHC unit, charac-

terized by non-circular voids (specimen 320b). Prin-

cipal tensile and compressive strains, crack pattern and

shear stress distribution at failure are in close agree-

ment with the collapse mechanism evidenced by the

experimental damage pattern at the end of the test. A

smooth, inclined crack, whose width reduces as the

imposed displacement monotonically increases,

emerges from both principal tensile strains and

numerical crack pattern. Simultaneously, an inclined

compressive diagonal strut arises according to the

principal tensile strain distribution, resulting in the

undergone failure mode, also confirmed by a cut-off in

the shear stress flow.

According to mesh sensitivity analysis, strip

modeling assumptions and the considered combina-

tions of prestress distributions and losses, six capacity

curves (i.e. shear force at the support, Fz, vs. mid-span

vertical displacement, Dz) are graphed in Fig. 6.

Independency with respect to adopted mesh refine-

ments is achieved through a proper calibration of the

crack bandwidth in each discretization zone, since the

observed response is essentially governed by tensile

behavior. The choice of parabolic or linear prestress

distribution in strands leads to negligibly small global

differences in terms of shear resistance (\0.1 %),

while larger influence is expected to occur if local

parameters, as stresses and strains distributions, are

considered. More appreciable difference is due to the

loss of prestress ratio. The adoption of 15 % loss

implies a reduced shear strength capacity, of about

5 %, if compared to 5 % loss; furthermore, the

capacity curves obtained for 15 % evidences a

smoother nonlinear response for lower levels of shear

force at the support, being also characterized by a

reduced ultimate mid-span vertical displacement

capacity of about 5 %. In this case, the strip modeling

assumption, referred as ‘‘HypS’’, cannot be accounted

either for design purposes, as the convergence of the

analysis failed when the capacity curve came into its

nonlinear branch, resulting in an over-conservative

estimation of the shear strength capacity. The intrinsic

lack of ductility reserves is confirmed by an evidently

low displacement ductility capacity, up to 1.3,

achieved for a maximum mid-span vertical drift of

*0.2 %.

The evolution of the shear stress distributions in the

end region of the specimen is presented in Fig. 7.

Specimen 320b, whose non-circular hollow core shape

−6 −4 −2 0 2 4 60

50

100

150

200

250

Vertical Displacement Dz [mm]

She

ar F

orce

Fz

[kN

]

320bm1p1320bm2p1320bm2p2320bm2p3320bm2p4HypS320bm2p1

Fig. 6 Specimen 320b: capacity curves, according to mesh

sensitivity, strip modeling assumption and the four combina-

tions of prestress distributions and losses


presents an evidently abrupt and irregular web width

drop along the depth (Fig. 4), highlights the presence

of a strongly-disturbed zone. High shear stress

concentrations are observed in the first LFs, since

the section shown in Fig. 7a is taken in correspon-

dence of a strand, longitudinally, and, transversally at

x/D = 0.4, where the prestressing force develops at a

high force variation per length. This concentration is

mostly due to the prestressing force and is mitigated as

x/D increases (see Fig. 7b). As LF increases, the shear

stress develops in the linear branch of the capacity

curve, evidencing a second discontinuity below the

mid-depth, where the web width drop is localized. For

further LF increments, the shear stress peaks at the

strand level slightly increase, while a much more

evident increment is detected in the upper discontinu-

ity. At LF roughly equal to 50 %, the peak shear stress

overall the depth rises up from the strand to the web

width drop level. This concentration remains pro-

nounced up to LF = 85 %, when the first cracking

appears. For further LF increments, the ratio between

the current and previous peaks increases at a lower

rate, in accordance with the reduced slope of the

capacity curve into its nonlinear branch and the related

shear stress redistribution. Finally, at LF = 100 %,

the peak is further uplifted, in accordance with the

shear stress cut-off experienced. This behavior reveals

the complex interaction between cross-section geo-

metric features, in terms of hollow core shape, and the

transfer mechanism of the prestressing force. The two

−7 −6 −5 −4 −3 −2 −1 0 1 2 30

50

100

150

200

250

300

350

Shear Stress τYZ

[MPa]

Dep

th [m

m]

LF=19%LF=37%LF=48%LF=59%LF=70%LF=78%LF=85%LF=93%LF=96%LF=100%

−7 −6 −5 −4 −3 −2 −1 0 1 2 30

50

100

150

200

250

300

350

Shear Stress τYZ

[MPa]

Dep

th [m

m]

LF=19%LF=37%LF=48%LF=59%LF=70%LF=78%LF=85%LF=93%LF=96%LF=100%

−7 −6 −5 −4 −3 −2 −1 0 1 2 30

50

100

150

200

250

300

350

Shear Stress τYZ

[MPa]

Dep

th [m

m]

320bm2p1320bm2p2320bm2p3320bm2p4

a b

c

Fig. 7 Evolution of the shear stress distributions at a x/D = 0.4 and b x/D = 0.7; c comparison between the four combinations of

prestress distributions and losses (x/D = 0.4)


discontinuities of the indented, fairly parabolic shear

stress distributions are particularly visible for sections

close to the support (x/D \ 0.56). Their evolution,

according to the experienced web-shear failure mech-

anism, results into an inclined crack. Their peak is

forced to rise up, increasing both longitudinal distance

from the support and imposed vertical displacement,

according to hollow core shape and inherent web

width drop. In fact, sections particularly close to the

support (x/D \ 0.56), are characterized by a localiza-

tion of the shear stress peak evidently below with

respect to the centroidal axis, about 25 %, while for

sections, placed at 0.56 \ x/D \ 1.68, this mismatch

tends to reduce, reaching a difference of *12 %.

Furthermore, at x/D = 2, the maximum shear stress is

localized at the level of the centroid of the PPHC unit.

If shear stresses at ultimate conditions are consid-

ered (see Fig. 7c), the comparison between the two

prestress distributions, for the same prestress loss,

leads to negligibly small differences in terms of peak

shear stress experienced overall the whole unit (2 %),

while the comparison between the two loss of prestress

ratios, for the same prestress distribution, evidences

slightly more appreciable mismatch (6 %).

2.2.2 Accuracy of FE predictions

Figure 8a presents the capacity curves determined for

the nine specimens, assuming 5 % loss of prestress;

shear force at support versus mid-span vertical drift

rather than displacement curves are shown to compare

units with different lengths. The effectiveness of the FE

models developed, accounting both for geometrical and

material nonlinearities, is reaffirmed. An accurate

agreement with experimentally observed shear strength

capacities is achieved, since 0.96\ Ri \ 1.05, with a

mean value of 0.998, characterized by a reduced

standard deviation of 2.8 %, as shown in Tables 5 and

6 and graphed in Fig. 8b. Herein, a comparison with

analytical predictions (EC2, Yang, ACI and CSA) is

presented as well. As a general tendency, slightly

unconservative predictions are observed for PPHC units

with circular voids and small nominal depths (200 and

265 mm), while, as the slab depth increases ([300 mm),

the accuracy of the FE results tends to increase. The level

of non-conservatism (EC2 and Yang) or over-conser-

vatism (ACI and CSA) disappears and an evidently

lower standard deviation demonstrates the robustness of

the proposed modeling approach.

Similar trends are obtained for 15 % loss of

prestress; in fact, the adoption of 15 % loss confirms

the previously mentioned reduction of shear resis-

tance, approximately ranging from 4 to 10 %, accord-

ing to specimen type. The capacity curves, obtained

for 15 % loss, are again characterized by a smoother

nonlinear branch for lower levels of shear force and, in

some cases, lower ultimate displacement capacity, of

about 10–13 %, is even noticed. However, a weak

sensitivity of the failure mode with respect to the

prestress losses emerges, as, for all the specimens, the

web-shear failure mechanism is experienced at a

distance from support, which varies between one and

two cross-section depths, where the prestressing force

is not yet fully developed.

For mid-span vertical drifts, ranging from 0.1 to

0.4 %, the considered capacity curves reach their

−0.5 0 0.5 1 1.50

50

100

150

200

250

300

350

400

450

500

Vertical Drift [%]

She

ar F

orce

Fz [k

N]

200m2p1265m2p1320am2p1320bm2p1320cm2p1370m2p1400am2p1400bm2p1500m2p1

0 100 200 300 400 500 600 700 8000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Vpre

[kN]

Ri =

Vob

s / V

pre [−

]

EC2YangACICSAFEMV

obs = V

pre

a

b

Fig. 8 Shear capacities of the nine specimens selected: a shear

force at the support versus mid-span vertical drift curves and

b experimentally observed versus numerically (FEM) and

analytically predicted shear strength


nonlinear branches, with different slope and post-

yielding stiffness. Evidently low displacement ductil-

ity levels, approximately ranging from 1.3 to 2.3,

according to different hollow core shapes and cross-

section features, are observed. Each specimen col-

lapses by brittle web-shear mode, thus confirming the

intrinsic lack of ductility reserve for this type of

member. According to FE results, PPHC units with

circular voids typically exhibits a relatively more

ductile response, reaching a displacement ductility of

about 2, while a less uniform trend is detected in

deeper cross-sections characterized by non-circular

voids. In fact, displacement ductilities up to *2 are

determined for slabs presenting a relevant web width

and concrete thickness above and below the hollow

core itself (specimens 500 and 370) or an almost

regular web width variation along the hollow core

depth (specimen 400a). By contrast, specimens char-

acterized by an abrupt and irregular web width drop

(specimen 400b) or reduced web width, if compared to

concrete thickness above and below the hollow core

(specimens 320a and 320b), ensure evidently lower

displacement ductility of about 1.3. Therefore, the

non-circularity of the voids, the inherent web width

variation along depth and concrete thickness above

and below the hollow core are observed to govern the

displacement ductility capacity of PPHC units, and,

hence, the fragility of the collapse mechanism

experienced.

2.2.3 Influence of hollow core shape: 400b

versus 400a and 320c versus 320b

This trend is corroborated by the evolution of principal

tensile and compressive strain distributions, presented

in Figs. 9 and 10, for specimen 400b. The principal

tensile strain directly governs the failure mode and the

presence of a variable inclined strut and tie, whose

width and inclination angle evolves during the mono-

tonic analysis, is clearly evident. As shown in Fig. 9,

tie width reduces as the imposed vertical displacement

increases, reproducing the actual shear stress flow,

characterized by peaks placed in correspondence of

the bottom side of the cross-section, rather than at the

Table 5 Comparison between experimental, analytical and FEM results

Model type Loss of prestress = 5 %

EC2 Yang ACI CSA FEM

Vobs Vpre Ri Vpre Ri Vpre Ri Vpre Ri Vpre Ri

(kN) (kN) (–) (kN) (–) (kN) (–) (kN) (–) (kN) (–)

200 90.2 145.7 0.619 128.0 0.704 98.5 0.916 114.8 0.786 94.3 0.956

265 219.4 223.8 0.980 223.7 0.981 155.8 1.408 158.1 1.388 223.2 0.983

320a 297.1 378.7 0.785 322.9 0.920 266.8 1.114 240.4 1.236 290.8 1.022

320b 232.5 359.7 0.646 317.2 0.733 253.1 0.919 236.5 0.983 221.4 1.050

320c 253.9 246.6 1.030 239.7 1.059 174.8 1.453 166.0 1.530 252.8 1.004

370 306.2 396.2 0.773 373.8 0.819 277.7 1.103 255.4 1.199 315.4 0.971

400a 291.3 475.4 0.613 444.0 0.656 363.3 0.802 309.0 0.943 293.9 0.991

400b 444.0 500.9 0.886 405.7 1.094 348.8 1.273 278.9 1.592 443.3 1.002

500 471.2 702.5 0.671 565.7 0.833 550.7 0.856 415.6 1.134 469.8 1.003

Vobs, Vpre and relative Ri for each specimen, according to EC2, ACI, CSA provisions, Yang’s approach and FEM

Table 6 Mean and standard deviation of Ri, according to EC2, Yang, ACI, CSA and FEM approaches

Model type Loss of prestress = 5 %

EC2 Yang ACI CSA FEM

Mean SD Mean SD Mean SD Mean SD Mean SD

All 0.778 15.7 % 0.867 15.7 % 1.094 24.1 % 1.199 27.1 % 0.998 2.8 %


Fig. 9 Evolution of the principal tensile strains E1 for monotonically increased vertical displacement steps imposed (specimen 400b)


level of the centroid. A brittle web-shear failure

mechanism results from an inclined diagonal crack,

which, for this specimen, with a sharp web width

variation along the depth, almost linearly propagates

from the support to zones upper than the centroidal

axis at a distance of about two cross-section depths.

Accordingly, a diagonal strut, whose width increases

as the imposed vertical displacement increases, con-

nects the transverse line load and the support, as

depicted in Fig. 10. This failure mode results into an

incipient change in the shear stress distribution,

localized at a distance roughly equal to the hollow

core depth, where the maximum tensile strain overall

the whole member is undergone.

In addition, the evolution of the principal tensile

strain distributions and inherent damage pattern

experienced by non-circular PPHC units with different

regularity of web width variation along depth is

compared, by taking specimens 400a and 400b as

reference. Specimen 400a, more regular than 400b,

presents very highly concentrated tensile strains in

correspondence of the minimum of the web width, as

shown in Fig. 11. For this type of cross-section

configuration, an inclined crack appears at the support

and, almost linearly, extends to approximately twice

the depth of the considered PPHC slab, since its

propagation along the longitudinal axis finds no

significantly abrupt web width variation. Conversely,

in specimen 400b, the principal tensile strain was

shown to be more evidently lumped at the level of the

sharp and irregular web width drop. As the distance

from the support increases, the experienced peak rises

up from the bottom of the cross-section, evidently

below with respect to the centroid, for sections close to

the support, to zones upper than the centroidal axis, for

sections at two depths. The evolution of the fracture

angle significantly differs from that shown by speci-

men 400a. Even though the inclination of both remains

fairly constant, at increased distances from the

support, specimen 400b highlights a more prominent

brittle web-shear failure mechanism, if compared to

400a, since the shear fracture more suddenly appears

and more rapidly evolves, as confirmed by their

capacity curves (Fig. 8a). As a result, it occurs a less

inclined and slightly more rounded fracture, whose

propagation along the longitudinal axis of the member

and evolution, according to the imposed monotonic

loading history, is controlled by a more abrupt and

irregular change in the hollow core shape.

To further validate this trend, a comparison

between the evolution of the principal tensile strain

distributions, according to the monotonically increased

vertical displacements imposed, is presented in

Fig. 12, for units with circular (specimen 320c) and

non-circular (specimen 320b) hollow core shapes.

Specimen 320b evidences a failure mode in consistent

agreement with that previously discussed for specimen

400b, being the two PPHC units characterized by

similar cross-section geometric features, particularly

in terms of non-circularity of the voids. High principal

tensile strains concentrations are again depicted in the

inner constant part of the web width, causing a fairly

linear crack, which expands from the lower part of the

hollow core, in the proximity of the support, to upper

ones, in sections at about two slab depths. The fracture

angle observed visibly differs from those experienced

by PPHC units with circular voids, e.g. specimen

320c. According to its capacity curve, a more rapid

and brittle failure mechanism takes place in specimen

320b, highlighting a more inclined, almost linear,

fracture angle, whose spread follows the sharp and

prominent web width drop. On the contrary, in

specimen 320c, principal tensile strain peaks are

forced to occur in accordance with the gradual and

regular web width variation along the depth. In fact,

their evolution, according to the inherent crack pattern

at ultimate conditions, reveals a more manifest

variation of the fracture angle at further distances

from the support, resulting in a clearly more rounded

and less inclined diagonal crack, which links the

support and the centroidal axis at a distance of about

two hollow core depths.

Therefore, in Fig. 13, the shear stress distribu-

tions at ultimate conditions, obtained by assuming

parabolic prestress distribution along strands, com-

bined with 5 % loss of prestress, are compared, for

each specimen, at commonly selected distance from

the support. The nine PPHC slabs analysed experi-

ence their peaks overall the whole unit at different

sections, anyway all included within 0 \ x/D \ 0.5.

The fairly parabolic shear stress distributions col-

lected reaffirm the behavior previously discussed, in

terms of fracture angle inclination. Cross-sectional

geometric features, interacting with the transfer

mechanism of the prestressing force, govern their

evolution and shape, confirming that the web-shear

failure mode of such PPHC units depends on the

irregularity of the voids. Slabs characterized by


Fig. 10 Subsequent evolution of the principal compressive strains E3 for monotonically increased vertical displacement steps

(specimen 400b)


Fig. 11 Different evolution of E1 in specimen 400a, still characterized by non-circular voids, but with increased regularity in terms of

web width variation along depth


Fig. 12 Comparison

between the evolution of E1

for different hollow core

shapes, characterized by

a circular (specimen 320c)

and b non-circular voids

(specimen 320b)


circular voids show a nearly parabolic slope, with

peaks closer to the level of the centroid, while more

prominent and irregular distributions are observed

for sections presenting non-circular voids, in accor-

dance with a different web width variation along

depth. Furthermore, discontinuities at the pre-

stressed strands level are particularly evident for

deeper cross-sections (specimens 400a, 400b and

500) which locally evidence strongly-disturbed and

indented slopes characterized by maxima localized

clearly below the centroidal axis of the member. For

further longitudinal distances from the support, the

local disturbance is visibly mitigated, resulting in

peaks close to the level of the centroid.

3 Conclusions

A modeling approach, based on nonlinear fracture

mechanisms, was proposed to predict the shear

response of PPHC slabs. Numerical observations were

discussed to investigate the failure mode of the units

analysed and to depict behavioral changes as conse-

quence of geometrical variations in their cross-section

shape. The following observations and conclusions

were drawn from the numerical study:

• In contrast of the mismatch shown by conventional

codes, a satisfying fit with experimental results

was achieved in terms of shear capacity, faithfully

reflecting the behavior of such high-strength

−8 −6 −4 −2 0 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Shear Stress τYZ

[MPa]

Nor

mal

ized

Dep

th D

i / D

i,max

[−]


−8 −6 −4 −2 0 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Shear Stress τYZ

[MPa]

Nor

mal

ized

Dep

th D

i / D

i,max

[−]


−8 −6 −4 −2 0 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Shear Stress τYZ

[MPa]

Nor

mal

ized

Dep

th D

i / D

i,max

[−]


−8 −6 −4 −2 0 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Shear Stress τYZ

[MPa]

Nor

mal

ized

Dep

th D

i / D

i,max

[−]


a b

c d

Fig. 13 Shear stress distributions at ultimate conditions versus normalized slab depth for increased distance from the support,

assuming parabolic prestress distribution and 5 % loss of prestress


concrete members (0.96 \ Ri \ 1.05, with mean

of 0.998 and standard deviation of 2.8 %).

• The complex contribution due to the transfer

mechanism of prestressing force, which interacts

with the hollow core features, was highlighted by

the detailed FE analyses performed in a nonlinear

fashion. Furthermore, the influence of prestress

distributions and losses was quantified.

• Intrinsic lack of ductility reserves was confirmed for

these units without transverse reinforcement and a

trend was established, according to the regularity of

the voids. Displacement ductility of about 2 was

observed for specimens with circular voids, while

levels slightly larger than unity were determined in

deep cross-sections with pronounced flat webs that

create discontinuities in the shear stress path.

• The brittle web-shear failure mechanism experi-

enced was proven to be controlled by hollow core

shapes and related non-circularity of the voids,

inherent web width variation along depth and

concrete chords above and below the void, as

confirmed by the evolution of the principal tensile

and compressive strains, developing a variable

inclined strut and tie mechanism.

• The presence of a less inclined, more rounded,

diagonal crack, controlled by the smooth web width

variation along depth, is generally opposed to a fairly

constant variation of the fracture angle inclination,

governed by the abrupt and irregular web width drop.

• Shear stress distributions confirmed the key role

played by cross-section features. Deeper specimens,

whose cross-sections are characterized by non-

circular hollow core shapes, reveal more prominent,

irregular and indented shear stress distributions,

whose peak is localized evidently below the centroid,

at the level of the sharp change in their web width.

• The numerical outcome of this research may be

directly applicable to develop alternative design

methods against web-shear failure of PPHC units,

saving the inaccuracy of previous proposals,

especially the various codes, by explicitly account-

ing for the influence of the hollow core shape.

References

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