The Use of Advanced Optical Measurement Methods for the Mechanical Analysis of Shear Deficient Prestressed Concrete Members
International Journal of Concrete Structures and Materials
The Use of Advanced Optical Measurement Methods for the Mechanical Analysis of Shear Deficient Prestressed Concrete Members
K. De Wilder
G. De Roeck
This paper investigates on the use of advanced optical measurement methods, i.e. 3D coordinate measurement machines (3D CMM) and stereo-vision digital image correlation (3D DIC), for the mechanical analysis of shear deficient prestressed concrete members. Firstly, the experimental program is elaborated. Secondly, the working principle, experimental setup and corresponding accuracy and precision of the considered optical measurement techniques are reported. A novel way to apply synthesised strain sensor patterns for DIC is introduced. Thirdly, the experimental results are reported and an analysis is made of the structural behaviour based on the gathered experimental data. Both techniques yielded useful and complete data in comparison to traditional mechanical measurement techniques and allowed for the assessment of the mechanical behaviour of the reported test specimens. The identified structural behaviour presented in this paper can be used to optimize design procedure for shear-critical structural concrete members.
experimental mechanics; prestressed concrete; coordinate measurement machine; stereo-vision digital image correlation
Despite the long-established and worldwide research
effort, a widely accepted theory for determining the shear
capacity of a structural concrete member remains open for
(Bala´zs 2010; Collins 2010)
. This can be mainly
attributed to the complex nature of shear in structural
concrete. Indeed, after the occurrence of inclined cracking,
various shear transfer mechanisms are activated (Jeong and
Kim 2014). The aforementioned mechanisms are interrelated
and highly susceptible to various parameters such as the
geometry, the amount of prestressing, material properties,
the amount and type of shear and longitudinal reinforcement
and the loading conditions. As a consequence, numerous
analytical modelling approaches can be found in literature.
An overview of recent approaches to shear in structural
concrete elements can be found in Fe´de´ration Internationale
du Be´ton (fib) (2010).
The on-going debate on how to deal with shear in structural
concrete members is also reflected by current codes of
(European Committee for Standardization 2004;
Canadian Standards Association 2004; American Concrete Institute
which propose different design provisions resulting in
varying design shear capacities and take factors affecting the
shear capacity into account in a different way. Due to our
incomplete knowledge and the brittle failure modes typically
associated with shear, current codes of practice generally
propose highly conservative shear provisions, specifically in
the case of prestressed concrete ele
ments (Nakamura et al.
). For the design of structural concrete members, these
design equations lead to excessive material usage and
corresponding construction costs. Reversely, using current codes of
practice to determine the shear capacity, existing concrete
structures are often found to be unable to withstand the
applied service loads whereas no structural problems are
reported in reality
(Valerio et al. 2011; Lantsoght et al. 2013)
Improving analytical modelling approaches for shear in
structural concrete members thus remains of utmost
importance to optimize the design and analysis of structural
concrete elements. A prerequisite for developing suitable models
is a clear understanding of the actual mechanical behaviour
observed during experimental tests. Traditional measurement
techniques, i.e. linear variable differential transformers
(LVDTs) or demountable mechanical strain gauges
(DEMEC), usually provide limited test data in one or two
directions. If the actual mechanical behaviour is to be
understood, more elaborate experimental data is required.
This paper therefore investigates on the use of advanced
optical measurement methods, i.e. 3D coordinate
measurement machines (CMMs) and Stereo-vision digital image
correlation (3D DIC), for the mechanical analysis of
sheardeficient structural concrete elements. The main focus in this
paper will be on prestressed concrete beams. Firstly, the
experimental program is elaborated. Secondly, the adopted
measurement methods are presented and their corresponding
setup is described. Thirdly, the experimental results are
presented including the results of the precision assessment of the
measurements and the full-scale experimental results. Finally,
the mechanical behaviour of the reported test beams is
investigated based on the acquired measurement data and
compared to current modelling approaches found in codes of
2. Experimental Research
2.1 Geometry and Materials
Nine test specimens were constructed and labelled with
the descriptive letter B followed by a number ranging from
101 to 109. All specimens were characterized by an
Ishaped cross section, were 7000 mm long, 630 mm high.
The flange width was equal to 240 mm whereas the web
width was 70 mm wide. All specimens were prestressed
using 8 or 4 seven-wire strands (nominal diameter 12.5 mm)
at the bottom and two seven-wire strands at the top (nominal
diameter 9.3 mm). The initial strain given to each
prestressing strand was 7:5 10 3 mm/mm (rp0 ¼ 1488 MPa)
for specimens B101–B103 and B107–B109 whereas the
stress level rp0 in the strands of beams B104–B106 was
equal to 750 MPa (initial strain equal to 3:8 10 3 mm/
mm). Shear reinforcement was provided in six specimens
consisting of open single-legged stirrups with a nominal
diameter of 6 mm (center-to-center distance equal to
150 mm). To avoid splitting failure at the end of each
specimen due to the gradual development of the prestressing
force over the member’s height, splitting reinforcement was
provided in each specimen (nominal diameter 8 mm, center
to center distance equal to 50 mm). An overview of the
geometry of the reported test specimens is given in Figs. 1a
A self-compacting concrete mixture, designed to have a
characteristic cylindrical compressive strength fck equal to
50 MPa, was used to cast each specimen. The concrete
mixture consisted of cement CEM I 52.5 R (16.0 wt%),
limestone gravel 2/12 (43.8 wt%), sand 0/2 (26.5 wt%), water
(6.7 wt%), limestone filler (6.7 wt%) and a high-range water
reducer (0.3 wt%). The expected density of this mixture was
equal to 2378 kg/m3. Standardized tests were performed to
determine the mean cylindrical compressive strength fcm as
well as the mean cube compressive strength fcm;cube.
Moreover, the mean flexural tensile strength fctm;fl and the secant
modulus of elasticity Ecm were determined for each specimen.
The testing procedures can be found in
Standardisation NBN (2009
a, b, 2014) The results are presented in
Table 1. The age of each specimen at the day of testing is also
indicated in the aforementioned Table 1.
Standard tensile tests were performed on the reinforcement
bars to determine the mean yield stress fym, the modulus of
elasticity Es, the mean ultimate tensile strength ftm and the
corresponding strain at failure su. The same characteristics
were adopted from the technical information provided by the
manufacturer of the prestressing strands. The results are
shown in Table 2.
2.2 Experimental Setup
All test specimens are subjected to a four-point bending test,
schematically shown in Fig. 2. A force-controlled loading
scheme was followed using a hydraulic press (Instron,
maximum capacity 2.5 MN). The load exerted by the hydraulic
press was transferred to two point loads using a steel transfer
beam. The total load was applied at a rate equal to 0.250 kN/s
(shear force rate V_ ¼ 0:125 kN/s). The distance between both
support points was equal to 5000 mm. The distance outside the
support points was therefore equal to 1000 mm. This allowed
the authors to investigate on the mechanical behaviour of shear
in the presented test beams outside the length needed for the
prestressing force to gradually develop over the member’s
height. The shear span a, i.e. the distance between the support
point and the load point, was equal to 1600 mm (specimens
B101, B104 and B107) or 2000 mm (B102–B103, B105–
B106, B108–B109). Given the geometry and reinforcement
layout, this resulted in shear span-to-effective depth ratios
(daratio) between 2.91 and 3.91. An overview of the investigated
parameters per specimen is given in Table 3.
2.3 Adopted Measurement Methods
2.3.1 CCD-LED Coordinate Measurement Machine (CMM)
As an optical motion tracking system, coordinate
measurement machines [CMMs, also referred to as dynamic
measurement machines (DMMs)] have proven to be an ideal
tool for a wide range of applications, including biomechanical
(Burg et al. 2013)
(Deconinck et al. 2004)
civil engineering applications
(De Roeck et al. 2004; Sun
. The basic principle behind this technique is to
determine the three-dimensional location of a discrete amount of
points by means of triangulation. Here, two Krypton K600
CMM (Nikon Metrology, formerly known as Metris, Leuven,
Belgium) systems were used for the experiment on specimen
B101. The points of which the three-dimensional coordinates
were to be determined were infra-red (IR) light emitting
diodes (LEDs). In total 112 LEDs were used and each LED
was placed onto an orthogonal grid covering both zones
where a shear force occurred, refer to Fig. 3a, and was glued
onto the concrete surface by using a thermoplastic adhesive.
Since the LEDs were attached to the material underneath, the
displacement of each LED, from which deformations can be
derived, is the same as the material under investigation. The
IR light emitted by each LED was seen by a camera unit
consisting of three 2048 px charge-coupled device (CCD)
line-element cameras, as shown in Fig. 3b. All LEDs were
connected in series to a camera control unit which also acted
as an interface between the cameras and the DAQ laptops.
(b) B103 , B106, B109
d6 @ 150
* Effective depth.
** Longitudinal reinforcement ratio bAwsdl with Asl area of longitudinal reinforcement and bw web width.
*** Shear reinforcement ratio qw ¼ bAwsws with Assw the area of shear reinforcement per unit length.
To assess the accuracy (closeness of the measurement to
the true value, related to systematic errors) and precision (the
degree to which repeated measurements under unchanged
conditions show the same results, related to random errors)
of the CCD-LED system, a reference measurement was
performed. During the aforementioned reference
measurement, each coordinate of each LED was measured during
50 s at a measurement frequency of 20 Hz. Typical results of
the variation of each coordinate around its initial value ui
(with i ¼ x; y; z) for one LED are shown in Figs. 4a to 4c.
The noise on the measurement data can assumed to be
approximately normally distributed. Therefore, Figs. 4d to
4f show the fitted normal probability density functions
(PDF) of ui (i ¼ x; y; z). The mean value and corresponding
standard deviation are also indicated in the aforementioned
Based on the coordinate measurements obtained during the
aforementioned reference measurement, in-plane strain
values (both horizontal x and vertical y) can be derived between
two consecutive LEDs. The standard deviation of the in-plane
horizontal and vertical strain value, s x and s y can then be
calculated as a function of the surface coordinates. The results
are presented in Figs. 5a and 5b. From the aforementioned
Fig. 5a, it can be seen that the value of the standard deviation
of the horizontal strain s x is well below the typical strain
values occurring for concrete in compression (maximum
compressive strain cu ’ 35 10 4 mm/mm). Figure 5b
indicates that less precise measurements of the vertical strain
are obtained near the top flange of the beam but good overall
results are acquired in the web of the specimens. Indeed, the
values of the vertical strain will primarily be used to assess the
strain and thus stress levels in the shear reinforcement
elements. Yielding of the shear reinforcement occurs at nearly
29 10 4 mm/mm, refer to Table 2 whereas rupture occurs
at a strain value equal to 273 10 4. Therefore, the expected
vertical strains are well outside the obtained noise levels. The
variation of the derived strain values as a function of the
surface coordinate is primarily due to the different relative
positions of each IR LED with respect to the optical centre of
the camera control units.
2.3.2 Stereo-Vision Digital Image Correlation (3D-DIC)
As an optical-numerical full-field measurement method,
digital image correlation (DIC) has proven to be an ideal tool
for a wide range of applications, including the identification
of the mechanical material behavior through inverse
(Cooreman et al. 2007, 2008)
, structural health
monitoring (Sas et al. 2012) and the study of the deformation
characteristics of a wide range of materials
(Ivanov et al.
2009; Van Paepegem et al. 2009; Srikar et al. 2016)
the testing of specimens B103–B109, the stereo-vision
digital image correlation technique (3D-DIC) has been adopted
Fig. 3 Experimental setup for the adopted coordinate
measurement machine (CMM): a IR LEDs (indicated with
red rectangle); b Camera units mounted on tripod.
to assess the displacements and deformations during the
loading procedure. Stereo-vision implies the use of two
cameras to allow for the reconstruction of the
three-dimensional geometry and measurement of the three-dimensional
displacements opposed to single-camera measurements
which yield only two-dimensional data. The basic principle
behind this technique is to calculate the displacements on the
surface of an object by taking images of a random speckle
pattern in undeformed and deformed state. As the speckle
pattern is attached to the material underneath, the
displacement and deformation of the speckle pattern is the same as
the surface material of the object under investigation. To
follow the displacement and deformation evolution during
the full-scale experiments, a series of images of the object is
made and the speckle pattern displacement and deformation
in the series of images can be followed. The grey-value
images of the speckle pattern are captured with a CCD
camera. Similar to the CMM measurements, both zones
where a shear force occurs were investigated using two DIC
systems, each consisting of two cameras. This setup allowed
for stereo-vision measurements. Both cameras of each DIC
system take simultaneously a picture of the investigated
object, in the following denoted as a frame. Each
stereovision DIC system consisted of two 8-bit CCD cameras
(AVT F201 B; 1628 px 1236 px resolution) with
wideangle lenses (focal length equal to 12 mm) mounted on a
tripod. Each zone under investigation measured
approximately 1500 mm 630 mm. The cameras were located at a
perpendicular distance of approximately 2700 mm from the
web of the specimen. To ensure good lighting conditions and
allow for small exposure times, two 500 W quartz iodine
lamps were provided per investigated zone. The image
acquisition rate of each camera was equal to 2 Hz. All
images were transferred to a desktop computer and
synchronized with the analogue data of the hydraulic press (i.e.
applied load and corresponding displacement). The
experimental setup is depicted in Figs. 6a and 6b.
Fig. 4 Accuracy and precision assessment of the Krypton K600 CMM for the upper left LED (refer to Fig. 3a): a horizontal
inplane displacement ux; vertical in-plane displacement uy; out-of-plane displacement uz; fitted normal PDF for d ux; e uy; f uz.
Fig. 5 Precision assessment of the adopted Krypton K600 CMM for the derived strain measurements: a standard deviation of
the horizontal strain s x as a function of the location on the specimen; b standard deviation of the vertical strain s y as a
function of the location on the specimen.
The analysis of the frames taken during the loading was
done with specialized software. In this work, the in-house
code MatchID (KU Leuven, Campus Ghent)
(Lava et al.
2009, 2010, 2011; Wang et al. 2011)
was used. To reveal the
displacements and deformations of an object during an
experiment, typically a square subset of (2M þ 1) pixels (px)
from the undeformed image is taken and its location in the
deformed image is traced. The principle of the stereo-vision
DIC algorithm is clearly explained by
Lava et al. (2011)
Matching of two (2M?1) px subsets in the undeformed
image Fðxi; yjÞ and deformed image Gtðxi; yjÞ at a certain time
t (i.e. load step) is performed by adopting an optimization
routine for a degree of similarity expressed by a correlation
criterion. Here, the Zero Normalized Sum of Squared
Differences (ZNSSD) correlation criterion was adopted. This
correlation criterion is independent of scale and offset in
(Sutton et al. 2009)
and is therefore the most
suitable correlation criterion to yield accurate results, especially in
the zones where the lighting conditions are difficult to control,
i.e. transition zones between web and flange of the I-shaped
specimens. The mathematical formalism is clearly explained
in the reference work by
Sutton et al. (2009)
states that it has already been shown that
the subset size is a critical parameter in the correlation
(Knauss et al. 2003)
. On the one hand it should be
chosen small enough to allow for a reasonable linear
approximation of the displacement field, within the region of
the subset. On the other hand the subset size should not be
chosen too small, to avoid correlation problems due to the
non-uniqueness of the subset information content. This
indicates the importance of an adequate speckle pattern for
digital image correlation. From a solely black or white
pattern, no valuable displacement information can be
gathered so that the subset size should be larger than the speckle
states that to isolate the effect of the
speckle pattern, there should be a way of applying the
speckle pattern in a controlled way, e.g. controlling the
speckle size, the speckle size distribution, the grey value
distribution and the actual colour of the black and white
paint. Therefore, to be able to generate suitable DIC speckle
patterns, a numerical technique recently proposed by
was adopted. In his work,
defines two concepts which are used to assess the suitability
of a speckle pattern for DIC measurements. Firstly, the
autocorrelation peak sharpness radius of a pre-processed
image of the considered pattern is proposed to quantitatively
evaluate how a particular strain sensor pattern influences the
sensitivity of a DIC measurement. Secondly, the
autocorrelation margin is proposed to evaluate how that pattern
influences the robustness of the DIC measurement. The
former is related to the measurement precision whereas the
latter is correlated to the measurement accuracy. Ideal
patterns for DIC would combine a sharp autocorrelation peak
with a well-defined autocorrelation margin. For simple
patterns, these characteristics vary in direct proportion to each
proposes a method based on
morphological image processing and Fourier transform to
synthesize a DIC pattern with wide autocorrelation margins
even though the autocorrelation peaks are sharp. Such
patterns are exceptionally well-suited for DIC measurements. A
detail of the numerically generated speckle pattern is shown in
Fig. 7a. The generated pattern is then applied onto each
specimen where the DIC technique was used by adopting a
heat-sensitive stencil printing technique which consists of
three layers: a vinyl base layer, the inverse of the speckle
pattern and a top protective heat-sensitive polypropylene
layer. The printed speckles have a precalculated oversampling
of at least five pixels in order to avoid aliasing effects in the
obtained results due to the expected small magnitude of the
displacement and deformation field. Given the camera sensor
properties and the dimensions of the field of view, speckles
with a diameter of nearly 5 mm were required. Obtaining
Maximum mean value of the standard deviation of the in-plane horizontal and vertical displacement; mean value of the standard deviation of
the out-of-plane displacement.
a Physical dimension of subset.
Fig. 7 a Detail of numerically generated speckle pattern; b Detail of the speckle pattern shown in (a) applied onto specimen B105;
c indication of area of interest (AOI) and subset size in red for the left measurement field of specimen B103.
large speckles is nearly impossible with traditional speckle
techniques (i.e. spray painting). However, the adopted
numerical technique allows for the generation of a speckle
pattern tailored to the needs of the experiment. Figure 7b
shows the same detail as presented in Fig. 7a applied onto
beam B105. Figure 7c shows the speckle pattern, area of
interest and adopted subset size. Since full-field displacement
data is readily available, Green-Lagrange strains can be easily
derived from the aforementioned displacement data.
Therefore, the displacement data is smoothed over a certain zone to
damp out the effect of noise and local uncertainties. A bilinear
plane can be fitted through the displacement values in the
points around the center of the strain window.
To assess the precision of the 3D-DIC setup for the
presented experiments, a number (N ¼ 50) of reference images
were taken prior to each test under a zero-loading condition.
The subset size was taken equal to 27 px whereas the stepsize
was chosen equal to 3 px. Given the experimental setup, the
physical dimension of 1 pixel approximated 1 mm. The
subset was allowed to undergo an affine transformation thus
taking into account translation, rotation, shear and normal
straining. As the displacements from one frame to another
may be smaller than one pixel, the subset in the image of the
deformed state is not likely to fit on the pixel grid and an
interpolation method between the pixels is needed. Therefore,
a bicubic interpolation scheme has been adopted in the
MatchID software during the analyses. Additionally,
Gaussian prefiltering (5 5 px kernel size) of the subset
information was adopted as a low pass-filter to attenuate
highfrequency signals and allow for proper interpolation. Based
on the measured displacements ui (with i ¼ x; y; z),
GreenLagrange strains can be determined by means of
bilinearquadrilateral smoothing of the displacement field. This
method was adopted for the analyses of the frames during the
loading procedure. For the purpose of assessing the precision,
strains were derived directly from the displacement data with
similar initial base lengths L0 as the CMM and without
smoothing of the displacement field to allow for a reasonable
comparison between both adopted systems. The base length
to determine the horizontal strains L0 was chosen
approximately equal to 200 mm whereas the base length to determine
vertical strains Lj0 was taken roughly equal to 75 mm. An
overview of the adopted DIC settings is presented in Table 4.
Similar to the CMM system, Figs. 8a and 8b present the
standard deviation of the horizontal and vertical strain s i
(with i ¼ x; y) based on the N pairs of reference images
obtained from both DIC systems for specimen B103.
Comparable results were found for the remaining specimens where
the DIC technique has been adopted.
From the aforementioned Figs. 8a and 8b, it can be firstly
seen that similar values for the standard deviation of both the
horizontal and vertical strain are found if the data of the 3D DIC
system is compared to the data obtained from the Krypton
K600 system, refer to Figs. 5a and 5b. Secondly, it can be seen
that less precise measurements of the horizontal strain x occur
near the edge of the area of interest, refer to Fig. 8a. This can be
attributed to optical aberration effects near the edge of the field
of view. However, the value of the expected strains during the
loading procedure well exceed the presented values of the
standard deviation of the horizontal strain. Thirdly, due to the
shaded area near the transition between the top flange and the
web, refer to Fig. 7c, less precise measurements of the vertical
displacement are obtained resulting in less accurate vertical
strain measurements near the top flange. However, vertical
strains will primarily be investigated in the web of the
specimens to assess the strain and stress levels in the shear
3. Results and Discussion
Figure 9a–c depict the experimentally observed
load-displacement response curves of the presented test specimens.
The onset of bending ( ) and diagonal (M) cracking is also
Fig. 9 Experimentally determined load-displacement response
curves measured at 1200 mm from the support point:
a specimens B101, B104 and B107; b specimens B102,
B105 and B108; c specimens B103, B106 and B109
(note onset of bending and diagonal cracking indicated
with (circle) respectively (triangle).
indicated in the aforementioned Figs. 9a to 9c. All I-shaped
test specimens but beams B107 and B108 failed due to shear
in a very brittle manner. The aforementioned specimens
exhibited severe diagonal web cracking leading to excessive
yielding and sudden rupture of the shear reinforcement bars.
Crushing of the diagonal compressive struts was not
observed. Beams B107 and B108 exhibited a ductile
bending failure mode leading to rupture of the prestressing
strands. Typical experimentally observed failure modes of
the presented test beams are shown in Figs. 10a to 10d. A
number of observations can be made based on Figs. 9a to 9c
and 10a to 10d:
1. Prior to the onset of cracking, the stiffness in the elastic
regime is comparable for all specimens with the same
shear span and overall height. Indeed, prior to the
occurrence of cracking, the response of the test
specimens to the applied load is governed by the bending
stiffness EI. Due to the comparable secant modulus of
elasticity, refer to Table 1, and the negligible influence
of the area of longitudinal reinforcement on the second
moment of inertia, it can be concluded that the bending
stiffness is similar for the reported test specimens.
2. The occurrence of cracks determines the transition
between linear and nonlinear behaviour. All specimens
exhibited both bending and web cracks. The load at
which web cracks occurs, is function of the amount of
prestressing and the concrete tensile strength respectively.
3. Specimens where shear reinforcement was provided and
which failed in shear (B101–B102, B104–B105)
exhibited a significant post-cracking stiffness and
postcracking bearing capacity resulting in a brittle shear
failure mode due to diagonal tension. Specimens B107
and B108 which failed in bending, show a highly ductile
behaviour with a limited post-cracking bearing capacity.
Finally, beams without shear reinforcement (B103,
B106 and B109) failed immediately after the occurrence
of the first inclined web crack.
The failure load and failure mode of each specimen is
summarized and presented in Table 5. The experimentally
determined failure load and failure mode can be compared to
analytical predictions of the bearing capacity according to the
shear design procedures outlined in Eurocode 2 (EC 2)
(European Committee for Standardization 2004; Bureau for
Standardisation NBN 2010)
. For the calculation of the shear
capacity of a structural concrete member, a distinction is to be
made between members with and without shear reinforcement.
Structural concrete members with shear reinforcement are to be
designed according to EC 2 using the variable angle truss
model (VATM). This approach assumes that the behaviour of a
structural concrete member near failure can be idealized by
means of a parallel chord truss. The bottom and top flanges
resist the applied bending moment whereas the combination of
a compressive stress field with constant inclination and vertical
tension bars resist the applied shear force. The design shear
capacity VRd of a member with shear reinforcement is the
minimum of the shear force required to obtain yielding of the
shear reinforcement VRd;s and the shear force which causes
crushing of the compressive struts VRd;max, refer to Eq. (1).
VRd ¼ min VRd;s; VRd;max
The expressions for VRd;s and VRd;max can be derived from
equilibrium conditions of the adopted truss model where the
Shear failure mode due to diagonal tension.
Obtained using Eq. (12).
a Coefficient of Variation.
angle between the horizontal and the inclined compressive
stresses is denoted by h, refer to Eqs. (2)–(3).
VRd;s ¼ Assw zfywd cot h
VRd;max ¼ cot h þ tan h
strength of the shear reinforcement bars. Factors acw and m1
take into account the stress distribution of the compressive
chord respectively the effect of lateral tensile straining on the
ultimate compressive strength. The width of the web is
denoted by bw. The inclination angle h can be chosen freely
between certain limits as presented in Eq. (4).
In Eqs. (2)–(3), Asw=s denotes the area of shear
reinforcement per unit length, z is the internal lever arm
equal to 0.9d whereas fywd is the design value of the yield
The maximum value of cot h, thus the minimum allowable
value for the angle h can be determined using Eq. (5)
(Bureau for Standardisation NBN 2010)
Equation (5) clearly indicates that for highly prestressed
members (average normal stress due to prestressing rcp), a
lower angle of inclination is allowed with a minimul value of
h ¼ 18:4 .
In the case of structural concrete members without shear
reinforcement, a distinction has to be made between
members cracked respectively uncracked due to the acting
bending moment, refer to Eqs. (6)–(7).
VRd ¼ VRd;c1
100 bAwsdl fck
In the previous equations, cc is a partial safety factor (equal
to 1.5), Asl is the area of longitudinal tensile reinforcement, I
is the second moment of area whereas S is the first moment
of area. Finally, fctd is the design value of the characteristic
uni-axial concrete tensile strength. If Eqs. (1)–(7) are used to
estimate the actual failure load, partial safety factors should
be omitted and average material properties are to be used
rather than characteristic or design values. Eqs. (2)–(3) and
Eqs. (5)–(7) are then transformed to Eqs. (8)–(12).
VR;s ¼ Assw zfym cot hmax
þ 0:15rcp bwd
In Eq. (12), the mean value of the uni-axial tensile strength is
written as a function of the experimentally determined mean
flexural tensile strength. The adopted relation is given in
M u¨ller et al. (2013
). The results of the aforementioned
calculation using Eqs. (8)–(12) are also indicated in Table 5.
The theoretical failure load required to obtain a bending
failure mode, derived from a general plane section analysis
Vu;bend; is also indicated in the aforementioned Table 5.
Based on the results presented in Table 5, the following
observations and preliminary conclusions can be made:
1. In general, a poor correlation is found between the
experimental results and analytical calculations
according to EC 2 for all specimens apart for beams B103,
B106 and B109 without shear reinforcement. Even if all
partial safety factors are omitted and average material
strength properties are used rather than characteristic or
design values, an average experimental-to-predicted
failure load ratio of 1.61 is found with a coefficient of
variation (COV) equal to 55.4 %.
Fig. 12 Experimentally determined direction and magnitude of the principal compressive strain field 2 of specimen B104 as a
function of the surface coordinates and the applied load level (Note line where the angle jh 2 j was determined in Fig. 11b
indicated with a dotted line).
2. The failure mode is correctly predicted for all specimens
apart for beams B107–B109. Specimens B107 and B108
failed due to bending despite having a lower shear
capacity in comparison to the corresponding bending
strength. Indeed, the experimental failure load correlates
well with the theoretical load required to obtain the
bending capacity for the aforementioned specimens.
Moreover, the wrongly predicted failure mode for
specimen B109 can be attributed to the small difference
in the analytically calculated shear and bending capacity.
3. As expected, increasing the shear reinforcement ratio,
increases the shear capacity (B102–B103, B105–B106,
4. Increasing the shear span-to-effective depth ratio while
keeping all other investigated parameters constant,
consistently decreases the experimentally observed
shear failure load (B101–B102, B104–B105).
5. Increasing the prestressing force while keeping the
longitudinal reinforcement ratio approximately constant,
increases the shear capacity of specimens with (B101–
B104, B102–B105) and without shear reinforcement
6. Decreasing the longitudinal reinforcement ratio while
keeping the prestressing force constant does not
significantly influence the failure load (B104–B07, B105–
B108 and B106–B109). However, the failure mode
shifts from a brittle shear induced failure mode towards
a more ductile bending induced failure mode for
specimens with shear reinforcement.
The found discrepancy between the experimentally
observed and analytically calculated failure load in the case
of prestressed concrete members with shear reinforcement is
certainly to be further investigated. Therefore, the
experimentally observed mechanical behaviour is assessed based
on the acquired data of the optical(-numerical) measurement
methods. Figure 11a, b present the experimentally
determined angle jh 2 j between the principal compressive strain 2
with respect to the horizontal at mid-depth determined from
the Krypton K600 CMMs respectively 3D-DIC as a
function of the location along the x-axis and the applied
load. The measured value for jh 2 j is compared to the
adopted value for h used for the presented strength
calculations, refer to Table 5. Similar results were obtained for the
remaining test specimens.
Figures 11a and 11b indicate that the value of the
experimentally determined angle jh 2 j in the middle of the shear span
(x ¼ 1800 and 5200 mm) correlates well with the adopted
value of the angle h in the presented shear strength calculations,
refer to Table 5. However, the value of jh 2 j is not constant
along the longitudinal x-axis of the specimen contrary to the
assumption made by the VATM approach. Instead, a parabolic
course of the angle jh 2 j is observed which does not tend to
change significantly if the loading is furthermore increased.
This observation if clarified by Figs. 12a and 12b which
presents the full-field magnitude and direction of the principal
compressive strain field 2 for specimen B104 as a function of
the surface coordinates and the applied load. Similar results
were observed for the remaining specimens where the DIC
technique has been adopted.
Based on the direction and magnitude of the principal
compressive strain field 2 presented in Figs. 12a and 12b, it
can be observed that a direct compression strut is developed
between the support point and the load application point
which carries a significant amount of the applied load. The
remaining part of the applied load fans out towards the
bottom of the specimen. A similar observation can be made
for the support point load. The force in both aforementioned
fan regions is equilibrated by the vertical force in the shear
reinforcement elements. A schematic representation of the
identified structural behaviour as presented in Figs. 12a and
12b is depicted in Fig. 13. The identified structural
behaviour as described above corresponds with the observed
parabolic course of the direction of the principal
compressive strain jh 2 j. In view of the mechanical behaviour of the
presented test specimens presented in Figs. 12a, 12b and 13,
the experimentally observed failure mode should be
interpreted as a splitting failure mode due to the forces in the fan
Direct strut action is generally considered to be an
important bearing mechanism for beams with a shear
spanto-effective depth ratio less than or equal to 2.5
et al. 1998)
. However, the presented test beams were
characterized by a shear span-to-effective depth ratio varying
between 2.91 and 3.91, refer to Table 3. The identified
structural behaviour, as presented in Figs. 12a, 12b and 13,
provides a plausible explanation for the following
experimentally observed phenomena.
1. The possibility of carrying the applied shear force by
means of direct strut action significantly increases the
shear carrying capacity in comparison to the shear
capacity obtained using the variable angle truss model
as proposed by Eurocode 2. This provides a possible
explanation why the current sectional shear design
provisions found in EC2 performed poorly in predicting
the shear capacity of the presented prestressed concrete
2. Figure 14 shows the typically observed profile of the
horizontal strain x at the top flange of the presented test
beams. Figure 14 clearly shows that the horizontal strain
at the top of the presented beams rapidly decreases to
relatively low strains, and thus relatively low stresses,
away from the loading point. Due to the inclined strut
action, it is indeed expected that low strain values occur
at the top of the specimen near the support point.
This paper aims to investigate on the use of advanced
optical(-numerical) measurement methods for the
mechanical analysis of shear-critical prestressed concrete beams.
Therefore, an experimental program consisting of nine
fullscale prestressed I-shaped beams was drafted. The main
investigated parameters were the amount of prestressing, the
amount of longitudinal reinforcement and shear
reinforcement and the shear span-to-effective depth ratio respectively.
All specimens were subjected to a load-controlled four-point
bending test until failure. During the experimental research,
the use of two advanced optical(-numerical) measurement
methods, i.e. 3D coordinate measurement machines (CMM)
and stereo-vision digital image correlation (3D-DIC), was
explored. Firstly, the experimental setup was elaborated in
detail. Specifically in the case of the DIC technique, a novel
technique to apply numerically synthesised strain sensor
patterns, i.e. speckle patterns, in a controlled way was
presented. The presented technique allows for the application of
tailor-made strain sensor patterns to virtually any given
object’s surface. A reference measurement was performed in
unloaded state to asses the measurement precision. Both
techniques were found to be comparable in terms of
displacement and strain resolution. The maximum standard
deviation of the in-plane displacements was equal to
approximately 20 10 3 mm for the CMMs whereas a
value of approximately 30 10 3 mm was found for the
DIC technique. Moreover, the expected value of the strains
occurring during the experiments well exceeded the
observed noise levels on the in-plane horizontal and vertical
strains. It can thus be concluded that both techniques are
well suited for assessing the structural behaviour of the
reported test specimens. However, due to the brittle and
highly energy releasing failure modes observed during the
tests on specimens failing in shear, the DIC technique is
preferred over the CMM technique since the latter technique
requires relatively expensive IR LED sensors to be glued
onto the concrete side surface which can sustain damage at
the moment of failure.
All specimens were designed to fail in shear. However,
seven specimens failed in a brittle manner due to shear
(diagonal tension failure mode) whereas two specimens
failed due to bending in a ductile manner. The experimental
results were compared to analytical calculations according to
the current design procedures found in Eurocode 2 (EC 2).
Based on the work presented in this paper, it can be
concluded that EC 2, adopting the variable angle truss model, in
general significantly underestimates the experimentally
determined failure load. Omitting all partial safety factors
and using average material strength properties rather than
characteristic values resulted in an average
experimental-topredicted failure load ratio equal to 1.61 (coefficient of
variation equal to 55.4 %). Based on the extensive amount
of experimental displacement and deformation data, it was
found that the applied load was primarily carried by means
of a direct compression strut in combination with fan regions
contrary to the model adopted by EC 2 which assumes that a
compression field with constant inclination along the
member’s axis is developed. The identified structural behaviour
for the reported test specimens can be used to optimize
current shear design provisions as proposed by codes of
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