Class #27.2
Civil Engineering Materials – CIVE 2110
Concrete Material
Stress vs. Strain Curves
Steel Reinforcement
Fall 2010
Dr. Gupta
Dr. Pickett
1
Stress-Strain Curve for Compression

Slightly ductile shape of Stress-Strain curve


f c' is reached
A descending branch exists after
Due to redistribution of load to un-cracked regions with less stress,
(MacGregor, 5th ed., Fig. 3-26)
2
Stress-Strain Curve for Compression

Strength of Reinforced Concrete structures controlled by,
Size of members,
 Shape of members,
 Stress-Strain curves of; - concrete,
- reinforcement.
Five properties of Stress-Strain curves;
(1) - Initial slope, Ec
'
f
(2) - Ascending parabola c
(3) - Strain at max stress,
(4) - Descending parabola
(5) - Strain at failure

(Fig. 3-18, MacGregor, 5th ed.)
3
Stress-Strain Curve for Compression

(1) - Initial Slope, Ec ;

ACI 318, Sect. 8.5, 8.6


sensitive to Eaggregate , Ecement .
For normal weight concrete;

wc  145 Lb Ft 3

Ec  psi  57,000 f c'

For other weight concrete;



90 Lb Ft3  wc  160 Lb Ft3

Ec  psi  wc1.5 33 f c'

Defined as the slope
of a line drawn from
  0 to   0.45 f c'
As water increases, Ec decreases,
because cement paste becomes
more porous, there is less aggregate.
(MacGregor, 5th ed., Fig. 3.17)
4
Stress-Strain Curve for Compression

Lightweight Concrete ;

ACI 318, Sect. 8.5, 8.6



sensitive to Eaggregate .



90 Lb Ft3  wc  120 Lb Ft3

Ec  psi  wc1.5 33 f c'
f c'
For all parameters involving
Each parameter shall be multiplied by a modification factor

  0.85 for sand-lightweight conc. For normal weight concrete the average
f ct  6.7 f c'
splitting
tensile
strength
is;
  0.75 for all-lightweight concrete







'
If splitting tensile strength, fct , is specified, then   f ct / 6.7 f c  1.0
This accounts for the reduced
capacity of lightweight concrete
due to aggregate failure;
Such as:



Shear strength
Splitting resistance
Concrete-rebar bond
(MacGregor, 5th ed., Fig. 3.26)
5
Stress-Strain Curve for Compression

(2) – Ascending Parabola;


'
(3) – Strain (  0 ) at f c ;



Strain at max stress increases
as f c' increases.
(4) – Slope of descending branch;
'
 Less steep than ascending branch, f c  6ksi
'
 Slope increases as f c increases.
(5) – Strain (  cu ) at failure;


Curve becomes steeper
as f c' increases.
Decreases with increases in f c'
(4 and 5) – depend on;

Specimen size; Load, type, rate
6
(Fig. 3.18, MacGregor, 5th ed.,)
Stress-Strain Curve for Tension

Tensile strength of concrete:

Determined by one of 2 tests:


P
H
(1) Flexure (Modulus of Rupture) test,
(2) Split Cylinder test, fct
P
B


8”
(1) Flexure (Modulus of Rupture) test;

Load until failure due to cracking on tension side,

ASTM C78 or ASTM C293,
 H = 6”, B = 6” L = 30”
3”
 
3”
P
V
My
f r   Flexure 
I
M H
2
fr 
BH 3
12
6M
fr 
BH 2
8”
8”
0
-P
PL
M
3
0
7
Stress-Strain Curve for Tension

(2) Split Cylinder test, fct ;

Load in compression along long side,

ASTM C496,
 a standard 6”x12” cylinder is placed on side,
 Outside surface area, Area  2rl  dl

Load is resisted by only half of surface area,
Re sistingArea  ld
2
f ct   
P
Re sistingArea
P
f ct 
ld


2P
f ct 
ld
2
8
(MacGregor, 5th ed., Fig. 3.9)
Stress-Strain Curve for Tension
Concrete always cracks
on plane of  MaxTension
Split Cylinder Test
Bi-Axial Stress
 maxTension   1
 2   ApC
2x90˚
Tension


Compression
 max 
 ApC
2
9
Stress-Strain Curve for Tension

Tensile strength of concrete:

Determined by one of 2 tests:



(1) Flexure (Modulus of Rupture) test,
(2) Split Cylinder test, f ct
Tensile strength from
Split Cylinder test
is less than that from
Flexure (modulus of Rupture) test
because;


H
fr
P
P
B
f r  1.5 f ct
In Flexure test, only bottom of beam reaches  Tension Max
In Split Cylinder test, majority of cylinder reaches  Tension Max
10
Stress-Strain Curve for Tension

Results from various Split Cylinder
tests vs. f c' are plotted in Fig. 3.10

The mean Split Cylinder strength is:
f ct  6.4 f c'

ACI 318, Sect. R8.6.1 states;
f ct  6.7 f c'

The mean Modulus of Rupture
strength is:
'
f r  8 .3 f c

(MacGregor, 5th ed., Fig. 3.10)
ACI 318, Sects. 8.6.1 & 9.5.2.3 state,
for deflection calculations:
  1.0 for norm al weight concrete
f r  7.5 f c'

f ct
6.7 f
'
c
 1.0
  0.85 for sand  lightweight concrete
  0.75 for all  lightweight concrete
11
Stress-Strain Curve for Tension

From: 00..5 fftt' ' 
ft 'ft '
Tensile strength of concrete:
ft '  0.08  0.15 f c'


Concrete tensile failure is BRITTLE.

Same factors affect ft ' as f c' ;

Water/Cement ratio,

Type of Cement,
Type of Aggregate,
Curing Moisture conditions,
Curing Temperature,
Age,
Maturity,
Loading rate.
'







'
t
1.8 f t '

Ec
where
f t '  f ct  6.4
0.5 f t '
f c'
(MacGregor,
5th ed., Fig. 3-21)
From: 0  0.5 f t '
'
 t'  f t E
c
t MAX
 0.0001 for
'
t MAX
 0.00014 0.0002 for
pure tension
12
flexure
Steel Reinforcement in Concrete

In any beam (concrete, steel, masonry, wood):
 Applied loads produce
Internal resisting Couple,
 Tension
and Compression
forces form couple.
In a concrete beam:
- Cracks occur in areas of Tension,
- Beam will have sudden Brittle failure
unless Steel reinforcement
is present to take Tension.
MacGregor, 5th ed.
Fig. 1-4
13
Mohr’s Circle Method – Failure Modes
Brittle concrete fails on plane of max normal (tension) Stress.
Failure stress located at: 2x90˚=180˚on Mohr Circle
Concrete
Brittle
 tension
 tension
90˚
 min   ApC
Neutral Axis
2x90˚
 max  slightTension Plane of

2x45˚
 max 
Stress
Shear
ApC
2

Normal Stress
max
Tension
Principal
Stress
14
Steel Reinforcement in Concrete

Steel Reinforcement:
 Hot-Rolled deformed bars (rebars)
ASTM A 615:
 Welded wire fabric
Reinforcement Bars (Rebars):
ASTM specs specify;
- diameter, cross-sectional area
- made from steel billets
- most commonly used
ASTM A 706:
- made from steel billets
- for seismic applications
- better - ductility
- bendability
- weldability
- sizes in terms of 1/8 inch
- #4 rebar, diameter = 4/8 in.
- metallurgical properties
- mechanical properties
- Grade  min. Tensile Yield Strength
- Grade 60, Yield Strength = fy = 60 ksi
15
Steel Reinforcement in Concrete
Reinforcement Bars (Rebars):
(MacGregor, 5th ed., Table 3-4)
Upper Limit on
UltimateTensileStrength  1.25 f y   actualYieldStrength 
16
Steel Reinforcement in Concrete
(MacGregor, 5th ed., Fig. 3-30)
Rebars in metric units:
- just numerical conversions
of US customary sizes.
11"
35.8m m
d

 1.409"
- # 11  d 
 1.375"  1.41"
- #36 
25
.
4
m
m
1
"
17
8
- Grade 420,  f y  420MPa
Rebars in US customary units:
- Grade 60,  f y  60ksi
Steel Reinforcement in Concrete
Reinforcement Bars (Rebars):
(MacGregor, 5th ed., Table A-1)
18
Steel Reinforcement in Concrete
Reinforcement Bars (Rebars):
(MacGregor, 5th ed., Table A-1M)
19
Steel Reinforcement in Concrete
Reinforcement Bars (Rebars):
- modulus of Elasticity,
ES = 29,000,000 psi
ACI 318, Sect. 8.5.2
- for rebars with
fy > 60,000 psi
must use
fy = ES x ( S  0.0035)
ACI 318, Sect. 3.5.3.2
(MacGregor, 5th ed., Fig. 3-31)
20
Steel Reinforcement in Concrete
Reinforcement Bars (Rebars):
- at temperatures > 850˚F
(MacGregor, 5th ed., Fig. 3-34)
fy and fultimate
drop significantly
- concrete cover
over the rebars
helps to delay loss
loss during fires
21
Steel Reinforcement in Concrete
Fatigue Strength of rebars:
- Bridge decks subjected to large number of load cycles
- Stress Range, Sr =  MaxTensileStress in a cycle   MinStress in same cycle
- Fatigue failure may
occur if at least one
stress is tensile
and Sr > 20 ksi
- Fatigue failure will not
occur if;
(MacGregor, 5th ed., Fig. 3-33)
 Max  20ksi inf inite cycles
 Max  any 20,000 cycles
- Fatigue strength
reduced at: Bends, Welds
22
Steel Reinforcement in Concrete
Fatigue Strength of rebars:
- Stress Range, Sr =  MaxTensileStress in
a cycle
  MinStress in
same
Example: Fatigue Failure possible;
S r  5ksi in
a
cycle
  16ksi in
same
cycle
S r  21ksi
Example: Fatigue Failure not possible;
S r   5ksi in
a
cycle
  26ksi in
same
cycle
S r  21ksi
23
cycle
Steel Reinforcement in Concrete
Welded-Wire Reinforcement:
- used in: Walls, Slabs, Pavements.
- due to cold-working process used in drawing the wire
strain-hardening occurs, so wire is BRITTLE.
- Plain wire;
ASTM A82; A185;
ACI 318, Sect. R3.5.3.6  fy = 60,000 psi
- mechanical anchorage in concrete provided by
- cross-wires
- Deformed wire; ASTM A496; A497;
ACI 318, Sect. R3.5.3.7
 fy = 60,000 psi
- mechanical anchorage
in concrete provided by
- cross-wires
- deformations
24
Steel Reinforcement in Concrete
Welded-Wire Reinforcement:
- Wire diameter = 0.125”  0.625”
- Wire area  increments of 0.01 in2 .
(MacGregor, 5th ed., Table A-2a)
- Plain wire;
W
- Deformed wire: D
- ACI 318, Sect. 3.5.3.5
D-4  wire size  D-31
area = 0.04 in2
area = 0.031 in2 .
-
25
Steel Reinforcement in Concrete
Welded-Wire Reinforcement:
- Wire area  increments of 0.01 in2 .
- Wire center-center spacing  a x b , inches
- Plain wire;
(MacGregor, 5th ed., Table A-2b)
W
-
26