Tests of Hardened
Concrete
Stress

Balance for equilibrium
 loads
= external forces
 internal forces = stress
Axial tension
 eng
P

A0
 true
P
   '   eng (1   eng )
A
for VOL  0
'
Strain

deformation (elastic or permanent)
 load
 change
in temperature
 change in moisture

unit deformation = strain
( l  l0 ) 
 eng 

l0
l0
Axial
l 
 true    ln    ln(1   eng )
 l0 
for VOL  0
'
Strain
Strength
Envelope
For Concrete
Effect of Confinement
Affect of Water Cement Ratio
Compressive Testing
brittle
 stronger in compression
 cross-sectional area
 cylindrical, cube
 ends must be plane & parallel
 end restraint
apparently higher strength

Loaded Compressive Specimen
Elastic Properties
E = modulus of elasticity = Young’s modulus = slope
Strain energy per unit volume = area

Linear Elastic
Stress
E
()
 E
1
Strain ()

Nonlinear Elastic
Elastic Properties
E  33w
3
2
E  0.043w

f c' ( psi)  57000 f c'
3
2
f c' (GPa)
Poisson’s ratio =- (radial strain/axial strain)
Poisson’s Ratio (u)

ratio of lateral strain to axial strain
u

 y
x

z
x
0.15 to 0.50
axial
 steel
0.28
 wood 0.16
 granite 0.28
 concrete 0.1 to 0.18
 rubber 0.50
deformed
Flexure (Bending)
Compression
Neutral
Axis
Tension
How would the cross-section deform?
Flexure (Bending)
Compression
Neutral
Axis
Tension
Laboratory Measuring Devices

Dial gage:

Measure relative deformation between two points.
 Two different pointers: one division of small pointer
corresponds to one full rotation of the large pointer.
Laboratory Measuring Device

Linear Variable Differential Transformer (LVDT)
 Electronic device for measuring small deformations.
 Input voltage through the primary coil
 Output voltage is measured in the secondary coil
 Linear relationship between output voltage and
displacement.
zero voltage
Secondary
coil
Core attached
to point B
Primary
coil
Secondary
coil
Shell attached
to point A
LVDT Schematic
zero voltage
Secondary
coil
Negative voltage
Primary
coil
Secondary
coil
Positive voltage
Longitudinal Displacement
Gage length
LVDT
Radial Displacement
LVDT
Electrical Strain Gage





Measure small deformation within a certain gage length.
A thin foil or wire bonded to a thin paper or plastic.
The strain gage is bonded to the surface for which
deformation needs to be measured.
The resistance of the foil or wire changes as the surface
and the strain gage are strained.
The deformation is calculated as a function of resistance
change.
wire
Surface
Load Cell




Electronic force measuring device.
Strain gages are attached to a member within the load
cell.
An electric voltage is input and output voltage is obtained.
The force is determined from the output voltage.
Strain gages
Strain gages
Data Acquisition Setup
8 Channel LVDT
Input Module
8 Channel Universal
Strain/Bridge Module
2 Voltage Inputs from
the controller (Stroke
LVDT, and Load Cell)
6 strain Gauges
Strength
Tensile Testing

Indirect: brittle
 cylindrical

Direct: ductile
 cylindrical,
prismatic
 reduced section @
center

 splitting
tension /
diametral
compression
Test Parameters
 surface
imperfections
 rate of loading
 temperature (ductile)
 specimen size
c
t
Flexure (Bending)
Compression
Neutral
Axis
Tension
elastic range
My
 
I
for y  c ,   modulus of rupture  flexural strength
Flexural Testing
Three-point
(center point)






Four-point

constant moment, no shear
in center

localized loading stresses
(3 vs. 4 pt)
load symmetrically
smaller specimens
higher flexural strength (size
effect)
shear may be a factor
General


shear effects ignored as long
as l/d > 5
apply load uniformly across
width

Correlation of Concrete Strengths
Torsion


torque
pure shear
strain (g)
cylindrical (radius r)  G=shear modulus

  Gg
Tr

J
s
g 
l
G


T = torque, twisting moment
J = polar moment of inertia
g = angle of rotation
l
g

s

E
2(1  u)

for isotropic materials
Standards & Standard Tests




allow comparison
ensure design =
construction
standard specifications
for materials
properties specified in
design, measured with
standard tests

Standards
Organizations
 ASTM
 AASHTO
 ACI
 State Agencies
 Federal Agencies
 Other
Scanning Electron Microscope
Impact Hammers
Ultrasonics
Pulse Velocity Testing
ASTM C 597
 Velocity of sound wave from transducer to
receiver through concrete relates to
concrete strength
 Develop correlation curve in lab
 Precision to baseline cylinders: ±10%

Pulse Velocity
1,500
10
8
1,000
6
4
500
2
0
0
2
0
4
6 8 10 12 14
(1000 ft/s)
1
2
3
4
Pulse Velocity (1000 m/s)
Compressive Strength (psi)
Semi-direct mode
Compressive Strength (MPa)
12
Concrete Strength Models
Compressive Strength
f  a log( t )  b
'
c
COV a and COV b
Modulus of Elasticity
E  33
3
2
'
c
f ( psi )
Tensile Strength
ft  6.7 f
'
c
COV  f c'  and COV   
 
COV f
'
c
Hitting Target Strengths
Variability of Strength
VARIABILITY


n
 xi
measured properties not exact
x  i 1
n
always variability
 material
n
 sampling
 testing

probability of failure
s
2
x

x
( i )
i 1
n 1
x
COV 
s
mean, standard deviation (s), coefficient
of variation (COV)

DESIGN / SAFETY FACTORS

design strength = f(material, construction variables)
f c'design  N * f c'  f c'  Z R * sd ;COV  f  
 sd
f c'
f c'design  f c' (1  Z R *COV  ) ; N  (1  Z R *COV  f )

working stress = f(y)
w 

y
N
N = 1.2 to 4 = f(economics, experience, variability
in inputs, consequences of failure)
Variability-Specification
Using the normally distributed tensile test data for
concrete, determine the mean and standard
deviation for both MoR & ft. In order to maintain a 1
in 15 chance that ft ≤ 320 psi, what average ft must
be achieved?
Specimen
MoR (psi)
ft(psi)
1
580
319
2
578
322
3
588
331
4
588
352
Crack Growth
Crack path
around
aggregates
(a)
Crack path
through
aggregates
(b)
Stress Intensity Factor
y
a
KI
 N  1/ 2
2r
x
Crack Tip
Stress Distribution
Fracture Mechanics

KI = stress intensity factor = F(C)1/2
F



is a geometry factor for specimens of finite size
KI = KIC OR GI=GIC
unstable fracture
KIC= Critical Stress Intensity Factor
= Fracture Toughness
GI=strain energy release rate (GIC=critical)
K IC 2
G IC 
plane stress
E
(
)
K IC 2 1  u 2
G IC 
plane strain
E
Fracture Mechanics

Three modes of
crack opening

Focus on Mode I
for brittle materials
2d
c
KI
c
2a
Alpha = a/d
F
Alpha
KI = N (a)1/2 F(
)
Failure Criterion
log(N )
Strength theory
Along the solid curve:
geometrically similar specimens
LEFM
Size-effect of
concrete structures
2
Along the vertical line:
specimens of the same size
with variable notches
log(d)
1
Linear Fracture Mechanics
K If 1
N 
C g
Non-Linear Fracture Mechanics
N 
cn K If
g '()c f  g ()d
a
cf
Crack
Process
Zone
KI
d
Alpha = a/d
Fracture specimens
P=2pt
R
R
R
2t
2t
2a
P=2pt
2a
P=2pt
2t
r
Specimen Apparatus
Specimen
Preparation
Test Specimens
Failure Criterion
log(N )
Strength theory
Along the solid curve:
geometrically similar specimens
LEFM
c n K 1f
 N  Size-effect
of
g ' ( ) c f  g ( ) d
2
concrete
structures
2
 g ' ( ) c f   N g ( ) d  c n2K 1f Along the vertical line:
specimens of the same size
2
g ( )
c
 N2 c f   N2
d  n K 12f with variable notches
g ' ( )
g ' ( )
2
N
B  AX  Y
g ( )
c n2
Y 
K 12f ; X 
d
g ' ( )
g ' ( )
A  slope   N2 ; B  Ac f
log(d)
1
Fracture Spread Sheet
c n K 1f
g ' ( ) c f  g ( ) d
 N2 g ' ( ) c f   N2 g ( ) d  c n2K 1f
g ( )
c n2
 N2 c f   N2
d 
K 12f
g ' ( )
g ' ( )
cf
1 g ( )
c n2

d 
K 12f
K 12f g ' ( )
g ' ( )  N2
B  AX  Y
g ( )
c n2
Y 
;
X

d
g ' ( )  N2
g ' ( )
A  slope 
1
K 12f
; B  Ac f
Split Tension Test
1.000E-14
y = 2E-11x + 4E-15
R2 = 0.996
8.000E-15
6.000E-15
Series1
4.000E-15
Regression
Y
N 
2.000E-15
0.000E+00
0.00000
0.00005
0.00010
0.00015
X
0.00020
0.00025
Fracture Spread Sheet
Spec
#
1
2
3
b
(in)
3
3
3
d
(in)
6
6
6
2a0
(in)
0
1
4
F() g() g'() X (in)
(g/g') d
0.964 0.000 2.92 0.0000
0.999 0.523 3.60 0.8711
1.645 5.699 10.02 3.4125
P
(lb)
13000
10000
3500
Y (psi.in1/2)
1/(g'2)
1.620E-06
2.219E-06
6.512E-06
c nP
- split tensile
lr
cP
 N = n - beam
bd
N =

0
0.167
0.667
Fracture Spread
2
2
Sheet
g ( )  c n F ( )
Split Tensile
Plain
#
#
Slotted
Hole and
Slot##
Cn

F ( )
g ( )
g ' ( )
2
0.0
0.964
0.0
2.9195
0.1667
0.9994
0.5230
3.6023
0.6667
1.6497
5.6997
10.0214

2

2

F ( )  0.964  0.026  1.472 2  0.256 3
##
F ( )  2.849  10.451  22.938 2  14.940 3
#
Beam###
Cn
1.5
###
s
d

a
d
F ( )  1.122  1.40  7.33 2  13.08 3  14.0 4
Applications of Fracture Parameters
Strength Determination - Beam
4 .0 0
N 
 N (M P a )
3 .0 0
c n K 1f
g ' ( ) c f  g ( ) d
 N2 g ' ( ) c f   N2 g ( ) d  c n2K 1f
2 .0 0
1 .0 0
0 .0 0
0 .0 0
2
g

(
)
c
n
 N2 c f   N2
d 
K 12f
g ' ( )
g ' ( )
cf
1 g ( )
c n2
 2
d 
2
K 1f K 1f g ' ( )
g ' ( )  N2
0 .2B
0  AX
0 .4 0 Y
0 .6 0
0 .8 0
1 .0 0
g ( )
c n2 = a /d
Y 
; X 
d
2
g ' ( )  N
g ' ( )
Applications of Fracture Parameters
Strength Determination
Size effect on strength
( 0 = 0.2; Bfu = 3.9 MPa = 566 psi; da = 25.4 mm = 1 in)
log (d/da)
Specimen or structure size
d (mm or inch)
log (N / Bfu)
N
(MPa or psi)
0.70
127 or 5
- 0.18
2.57 or 373
1.00
305 or 12
- 0.26
2.15 or 312
1.30
507 or 20
- 0.35
1.75 or 254
Durability