Page 1
Comparison of Design Methods for Composite Slabs Using Small-scale
Shear Tests
JOSEF HOLOMEK, MIROSLAV BAJER, JIŘÍ KALA
Institute of Metal and Timber Structures, Institute of Structural Mechanics
Brno University of Technology
Veveří 95, Brno 602 00
CZECH REPUBLIC
[email protected] , [email protected] , [email protected] , http://www.fce.vutbr.cz
Abstract: - Design methods for composite slabs still require full-scale bending tests which are expensive and
space demanding. Alternatively small-scale shear tests can be used to obtain shear characteristic of the
connection. Three of the methods which are using small-scale shear tests are presented and compared in this
paper: slip-bloct test, new simplified method and built-up bars method. Small-scale shear tests performed in
laboratory are used to calculate bending resistance using these methods. The resulting resistance is compared
with previously performed four-point bending tests.
Key-Words: Built-up bars, composite slab, experiment, longitudinal shear, new simplified method, slip-block
test, small-scale tests.
1 Introduction Composite slab is a suitable solution for horizontal
load-bearing structures. Steel sheeting serves as a
permanent formwork and as a tension bearing
member after hardening of concrete. Generally there
are three global failure modes: bending, vertical shear
and horizontal shear. Bending prevail for slim slabs
with large shear spans. Vertical shear occurs rarely
for short span thick slabs. Failure mode in horizontal
shear governs for the intermediate cases. In this case
the horizontal shear resistance is not sufficient to
preserve full composite action.
The horizontal shear resistance can be provided by
prepressed embossments, self-locking shape of the
sheeting or by end restraints.
Nowadays design codes use m-k method and partial
connection method. Both of them use full scale
bending tests to determine characteristics for design.
A less expensive alternative is to use small-scale
shear tests. This kind of tests cannot include all the
effects of bent slab such as clamping, load
distribution, strain in the sheeting, etc.
Three of the methods for designing composite slabs
are introduced and compared in this paper: slip-block
test, new simplified method and built-up bars method.
The results are calculated using experiments of slip-
block test arrangement and compared with bending
tests which were previously performed in laboratory.
2 Design methods using small-scale
shear tests 2.1 Slip-Block Test Slip-block test is a simple and transparent design
method. It uses test set-up consisting horizontal shear
force H and vertical clamping force V. The specimens
are fixed by the overlapped sheeting to the base plate
and concrete block is being pushed out of it. The
clamping force is transmitted through roller bearing
to the top surface of the concrete block. Magnitude of
V is changing during the test. The aim of the test is to
obtain corresponding shear resistance H for various
levels of clamping force V. The resulting dependence
forms a line when plotting H-V diagram which is
called a “friction line”. The slope of the line
determines friction μ of the specimen and intersection
with H axis determines shear resistance of
mechanical interlock Hrib. [1]
𝐻 = 𝑏1𝐻𝑟𝑖𝑏 + 𝜇𝑉 (1)
Fig. 1 Slip-block test set-up [2]
Advances in Mathematics and Statistical Sciences
ISBN: 978-1-61804-275-0 333
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The specimens used by the authors of the method had
dimensions bl = 300 mm, br = 300 mm. [2]
Two options of loading procedure were presented by
the authors: V was gradually changed in steps or the
V was continuously decreased by a constant
decreasing rate. The first option meant that at first V
was 27 kN to obtain fist slip which brakes chemical
bond. Then V was changed in steps 52 kN, 42 kN,
32 kN, 22 kN and 12 kN. The step was finished when
slip increased by 1 mm. It resulted in one friction line
per each specimen. [2]
Second option brakes the chemical bond at first as
well. Than the clamping force V was continuously
decreasing from 52 kN by a constant rate and
corresponding H was recorded. It enabled to perform
several cycles of loading per each specimens and thus
obtain more friction lines. Development of friction
and resistance of mechanical interlock in relation to
magnitude of slip could be therefore observed.
However, loading jack for H had to work in position
control mode. [2]
If the shear characteristics are determined one can
calculate equilibrium horizontal shear force in the
sheeting per unit width [3]:
𝑇 =𝐻𝑟𝑖𝑏
𝑏𝑟(𝑥 + 𝐿𝑐) + 𝜇𝑅𝑢 (2)
where 𝑥 + 𝐿𝑐 is the distance from the end of the slab
to the critical cross section and Ru is the reaction
above the support which is to be calculated
iteratively. The method enables easily to take into
account effect of end anchorage by adding the
resistance to the calculated shear force T. [3]
2.2 New Simplified Method New simplified method uses pull-out test
arrangement developed by Daniels (Fig. 2). In this
set-up two specimens were fixed together through
middle plate by constant clamping force 1.6 kN.
Concrete blocks were pulled out of the sheets by
precast rods. [4]
The method uses equivalent I section for sheeting and
rectangular section for concrete.
Fig. 2 Daniels test set-up [4]
Fig. 3 Phases of behavior according to New
Simplified Method [5]
The slab behavior is described in three phases in
moment-curvature diagram (Fig. 3). For each of the
phases is calculated ultimate moment and
corresponding curvature.
Phase I finishes when first cracks occurs in the
tension part of concrete.
𝑀𝑙𝑖𝑚,1 = 𝐸𝑎𝐼𝑎,𝑒𝑞,𝑦𝜙𝑙𝑖𝑚,1 (3)
The end point of Phase II is reached when first slip
occurs. This point is also ultimate bending resistance
for profiles with brittle behavior. The solution has to
be found by iteration when considering equality of
curvature calculated from maximum strain before
cracking and curvature calculated using value of
shear resistance corresponding to first slip from
small-scale shear tests.
Phase III is developed for ductile profiles only. Its
end point is the ultimate bending resistance of the
ductile profiles. The solution is to be found using two
iterations.
Despite its name the calculation procedure is
significantly more complicated than those of the
other two methods: elasto-plastic material behavior is
considered, cross-section characteristics for cracked
section are calculated and solution have to be found
using three iterations. A calculation procedure is
needed to be set in a spreadsheet program or similar.
The procedure is described in details in [6].
2.3 Built-up Bars Method derived from general built-up bars theory
determines the bearing capacity of composite section
by dividing into layers. [4]
𝑀𝑅𝑑 = ∑ 𝑀0,𝑖𝑛𝑖=1 + ∑ 𝑇𝑅𝑑,𝑖 ⋅ 𝑐𝑖
𝑛𝑖=1 , (4)
where 𝑀0,𝑖 is a bending stiffness of a separate layer,
𝑇𝑅𝑑,𝑖 is a shear stiffness of contact between the layers
and 𝑐𝑖 is a distance from centroid of the layer to the
contact layer. In case of composite slab the section is
divided into two parts (Fig. 5). Eq. 1 is therefore
written as follows [7]:
Advances in Mathematics and Statistical Sciences
ISBN: 978-1-61804-275-0 334
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Fig. 5 Scheme of forces used in the calculations [5]
𝑀𝑅𝑑 = 𝑀𝑝𝑎 + 𝑇(𝑥) ⋅ 𝑧𝑒𝑓𝑓 (5)
𝑀𝑝𝑎 is a bending resistance of the steel sheeting,
𝑇(𝑥) is a horizontal shear force and 𝑧𝑒𝑓𝑓 is lever arm
of internal forces. Value of the shear force of one of
the layers can be calculated using [7]:
𝑇(𝑥) =𝑀𝑅(𝑥)∙𝑧𝑒𝑓𝑓
𝛾(𝐸𝑝𝐼𝑝+𝐸𝑐,𝑒𝑓𝑓∙𝐼𝑐,𝑒𝑓𝑓)𝑘(𝑥). (6)
involves the influence of stiffness of each of the
layers [1]:
𝛾 =1
𝐸𝑝𝐴𝑝+
1
𝐸𝑐,𝑒𝑓𝑓𝐴𝑐,𝑒𝑓𝑓+
𝑧𝑒𝑓𝑓2
𝐸𝑝𝐼𝑝+𝐸𝑐,𝑒𝑓𝑓𝐼𝑐,𝑒𝑓𝑓 (7)
𝑀𝑅(𝑥) is a bending resistance of the section and 𝐸𝑝𝐼𝑝
is Young's modulus and moment of inertia of the
sheeting. 𝐸𝑐,𝑒𝑓𝑓 = 𝜔 ∙ 𝐸𝑐𝑚 is the Young's modulus of
concrete for calculation of bending resistance [7] and
𝐸𝑐,𝑒𝑓𝑓 = 𝜐 ∙ 𝐸𝑐𝑚 is the Young's modulus of concrete
for calculation of deflection [8]. 𝜔 is a coefficient for
the shape of the stress diagram in compression zone
of concrete flexural member (0.5 for triangular
distribution; 1.0 for rectangular distribution) and 𝜐 is
a concrete elasticity coefficient which involves
influence of cracks with increasing load [8].
𝐼𝑐,𝑒𝑓𝑓 =𝑏𝜔∙𝑥𝑚
3
3+
(𝑏−𝑏𝜔)∙ℎ𝑐3
12[1 + 12 (
𝑥𝑚
ℎ𝑐− 0,5)
2] (9)
𝑥𝑚 = 𝛽 ∙ 𝑑𝑝 (√𝜇2 +𝜇∙ℎ
𝑑𝑝− 𝜇) (10)
𝜇 = 𝜇𝑓 +𝐸𝑝∙𝐴𝑝
𝐸𝑐,𝑒𝑓𝑓∙𝑏𝜔∙𝑑𝑝 (11)
𝜇𝑓 =𝑏∙ℎ𝑐
𝑏𝜔∙𝑑𝑝 (12)
𝛽 = 1,9 − 2,6 (𝑀
𝑀𝑅𝐶)
2 (13)
𝛽 takes into account variation of equivalent
compression zone's depth (for bending resistance
calculation is equal to 1). 𝑘(𝑥) is a coefficient for
yielding of the layers and loading diagram of the slab.
For four-point bending can be expressed as [7]:
𝑘(𝑥) = 1 − [cosh((
𝑙
2−𝑎))
(∙𝑎) cosh(∙𝑙
2)
sinh( ∙ 𝑎)] (14)
is a coefficient defining the rigidity of the contact:
= √𝛼 ∙ 𝛾 (15)
𝛼 =𝑏∙𝐺𝜔
𝑧𝑒𝑓𝑓 (16)
𝐺𝜔 is the characteristic for the shear rigidity
determined from the shear tests using scheme in
Fig. 5 [9].
Fig. 5 Scheme to determine characteristics for the
shear rigidity [9]
If the characteristic for the shear rigidity is known it
is possible to calculate ultimate bending moment and
deflection for various level of loading:
𝑦 (𝑙
2) = 𝑀 (
𝑙2
8𝐸𝑒𝑓𝑓𝐼𝑒𝑓𝑓+
1
𝐷(
cosh(0,5∙𝑙)−1
2 cosh(0,5∙𝑙))) (17)
The method was derived using the results of
dovetailed sheeting tests.
The small-scale shear tests have the same
arrangement as for slip-block test. The specimens
consisted of 150 x 150 mm concrete block and
overlapped sheeting was fixed to the base plate. The
clamping force was transmitted through pneumatic
wheel. The magnitude of clamping force for the tests
taken into calculations was 5 kN. [7, 8, 9]
3 Experimental investigations Experiments were performed in laboratory of
Institute of Metal and Timber Structures in Brno
University of Technology (BUT) using the test set-
up similar to the slip-block test. Hand-operated
hydraulic cylinders were used to produce both
horizontal shear force H and vertical clamping force
V (Fig. 6). The clamping force V (was transmitted
through roller bearings.
Cofraplus 60 sheeting, 1 mm thickness, galvanized
was used in the experiments. The specimens have
width of two waves (414 mm) which enables to apply
load into one point in the middle wave. Length of
concrete block was 200 mm, length of sheeting
330 mm (Fig. 7). Thickness of the slab (height of the
specimen) was 110 mm. Sheeting was bolted to
a base plate and concrete block was pushed out of it.
Tests were performed in several options:
1) Clamping force 1.6 kN
2) Clamping force 5 kN
3) Slip-block test procedure
Advances in Mathematics and Statistical Sciences
ISBN: 978-1-61804-275-0 335
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Fig. 6 Test set-up before experiment (BUT)
Fig. 7 Specimen dimensions
3.1 Clamping force 1.6 kN Series with clamping force 1.6 kN was the first of the
tests with slip-block test set-up. The shear force was
applied into the half of the height of the slab which
caused an eccentricity (21 mm) of the load and
contributed to the uplift of the loaded side of concrete
for higher values of slip. The resulting diagram is in
Fig. 8. Shear resistance at the initiation of slip is taken
as 4 kN; maximum shear resistance 15 kN.
During the test of specimen 1 the clamping force
grew up to value higher than 4 kN. Resulting shear
resistance was calculated by subtracting the
difference in clamping force multiplied by coefficient
of friction 0.46 (as obtained from slip-block tests,
section 3.3). During the testing of specimens 2 and 3
the clamping force was held constant manually
(Fig. 9).
Fig. 8 Shear resistance – slip diagram for series 1
Fig. 9 Clamping force diagram for series 1
3.2 Clamping force 5 kN Resulting diagram for series with clamping force
5 kN is in Fig. 10. The shear resistance corresponding
to slip 0.5 mm was 6.6 kN and maximum shear
resistance 13.8 kN. There are two series, first with the
same eccentricity as in section 3.1 (specimens 1 – 3),
second without eccentricity (4 – 6).
Fig. 10 Shear resistance – slip diagram for series 2
0
5
10
15
20
0 2 4 6 8 10
Shea
r re
sist
ance
H[k
N]
Slip [mm]
Specimen 1
Specimen 2
Specimen 3
0,0
1,0
2,0
3,0
4,0
5,0
0 2 4 6 8 10
Cla
mp
ing
forc
e V
[kN
]
Slip [mm]
Specimen 1
Specimen 2
Specimen 3
0
2
4
6
8
10
12
14
16
18
0 2 4 6 8 10
Shea
r re
sist
ance
H[k
N]
Slip [mm]
Specimen 1
Specimen 2
Specimen 3
Specimen 4
Specimen 5
Specimen 6
Advances in Mathematics and Statistical Sciences
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Fig. 11 Clamping force diagram for series 2
Average value of maximum shear resistance with
eccentricity was 15.5 kN and without 13.8 kN. The
eccentricity has increased the shear resistance by app.
12%. The shear resistance was again calculated by
subtraction of clamping force V increment multiplied
by coefficient of friction.
3.3 Slip-block test procedure The values of clamping force were recalculated
according to ground plan of the specimens. The slip-
block test procedure was applied in several series. At
first the procedure of loading in steps was applied in
the same way as described by the authors. Next series
were performed in laboratory of Addis Ababa
Institute of Technology in Ethiopia where the
magnitude of slip for transition to next step was
decreased to 0.2 mm and the clamping force was
gradually increasing instead of decreasing. This
made possible to perform several series of loading
per specimen and obtain several friction lines for
each specimen. This comparison has shown large
dispersion of results for small slips. Therefore only
values corresponding to slip higher than 2 mm were
taken into calculations. Fig. 12 displays a friction line
coming from summation of the test results of all the
series with slip-block test procedure.
4 Comparison of results The test results are calculated considering a reference
slab loaded by four-point bending with width 1.08 m,
span 2 m and shear span 0.5 m. The real experiments
of such a slab were previously performed in
laboratory of BUT.
4.1 Slip-Block Test From the equation of friction line from Fig. 12
𝐻 = 12.95 + 0.46𝑉 (18)
Fig. 12 Friction line obtained by slip-block test
loading procedure
follows that μ = 0.46 and b1Hrib = 12.95 kN. The
length of the specimen bl = 200 mm, therefore
Hrib = 64.75 kN/m. The resultant tensile force in the
sheeting using the equation (2) is T = 181.1 kN and
considering magnitude of reaction at the support
Ru = 26.25 kN the ultimate bending moment results
in Mu = 13.12 kNm.
4.2 New Simplified Method From the results in section 3.2 follows that the input
value of shear stress corresponding to initial slip
τslip = 0.034 MPa and maximum shear stress
τmax = 0.127 MPa. Using the calculation procedure in
a spreadsheet the limiting bending moment for
phase 1 is Mlim,1 = 3.2 kNm and curvature
ϕlim,1 = 1.47E-06. The limiting bending moment for
phase 2 Mlim,2 = 3.5 kNm and curvature ϕlim,2 = 2.02E-
06. Finally the ultimate bending moment for phase 3
Mlim,3 = 17.76 kNm and curvature ϕlim,3 = 1.15E-04.
4.3 Built-up Bars Considering the results in section 3.3 and scheme in
Fig. 5 one obtains characteristic for the shear rigidity
𝐺𝜔1 = 32.0 𝑀𝑃𝑎 and 𝐺𝜔2 = 6.8 𝑀𝑃𝑎. Coefficient
for yielding of the layers k(1/4) = 0.29 and horizontal
shear force in the sheeting T(1/4) = 93.7 kN.
Considering the plastic bending moment for full
interaction calculated according to Eurocode 4
MRc = 32.2 kNm and plastic bending resistance of the
sheeting Mpa = 12.3 kNm we obtain ultimate bending
moment for the composite slab MRd = 18.1 kNm.
The built-up bars method enables easily to create load
deflection diagram. For this purpose concrete
elasticity coefficient 𝜐 and the lever arm of internal
forces zeff is calculated for each level of loading.
Resulting relationship in comparison to previously
performed four-point bending tests (static loading
and static loading after cycling) can be seen in
Fig. 15.
0
1
2
3
4
5
6
7
8
9
0 2 4 6 8 10
Cla
mp
ing
forc
e V
[kN
]
Slip [mm]
Specimen 1
Specimen 2
Specimen 3
Specimen 4
Specimen 5
Specimen 6
H = 12.95 + 0.46V
0
10
20
30
40
50
0 10 20 30 40 50 60
Shea
r re
sist
ance
H [
kN]
Clamping force V [kN]
Advances in Mathematics and Statistical Sciences
ISBN: 978-1-61804-275-0 337
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Fig. 15 Load-deflection curve calculated by built-up
bars in comparison with experiment data
Design method MRd [kNm]
Slip-Block Test 13.12
New Simplified Method 17.76
Built-up Bars 18.10
four-point bending test 19.51
Table 1 Ultimate bending moment comparison
4 Conclusions Three design methods for composite slabs which use
small-scale shear tests are presented and compared in
this paper. All of them are able to assess bending
resistance of composite slabs and can be very useful
for less-expensive design of composite slabs.
Slip-block test is simple and transparent but does not
reflect changes in curvature, effect of yielding of the
sheeting and cracking of concrete.
New simplified method includes bilinear material
model, effect of cracks, longitudinal shear stress
distribution, curvature, changes in cross section
characteristics due to slip, etc. The method gives
information about behavior before ultimate limit state
is reached. The solution has to be found iteratively.
On the other side the calculation procedure is very
complicated and has to be performed in a spreadsheet
or another calculation program.
Built-up bars method takes into account cracking of
concrete, yielding of sheeting, curvature, changes of
lever arm with increasing load, etc. The calculation
procedure is not as complicated as for new simplified
method and load-deflection diagram can be easily
obtained.
From the results comparison slip-block test is the
most conservative. New simplified method and built-
up bars method give results closer to the real
experiments.
Attention should be paid on test performance to
preserve clamping force constant or to subtract the
friction force increment; to avoid eccentricity in
loading; etc. The effect of loading speed is planned to
be investigated in the upcoming tests. Results from
slip-block test set-up does not show significant
differences from those of Daniels test set-up.
However, more precise comparison would help to
find a unified approach for design using small-scale
shear tests.
Acknowledgement
This paper has been worked out under the project No.
LO1408 "AdMaS UP - Advanced Materials,
Structures and Technologies", supported by Ministry
of Education, Youth and Sports under the „National
Sustainability Programme I" and under the project
GA14-25320S.
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0
5
10
15
20
25
30
35
40
45
0 10 20 30 40
Forc
e in
on
e lo
adin
g p
oin
t [k
N]
Deflection [mm]
Static 1
Static 2
After cycling 1
After cycling 2
After cycling 3
Built-up Bars
Advances in Mathematics and Statistical Sciences
ISBN: 978-1-61804-275-0 338