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Constituents of Concrete
Cement Types
•
•
•
Sand
Aggregate
Water
•
Cement
• Limestone
• Cement rock
• Clay
• Iron ore
• + (after firing and grinding)
• gypsum
• Type 1
normal portland cement. Type 1 is a
general use cement.
• Type 2
is used for structures in water or soil
containing moderate amounts of sulfate,
or when heat build-up is a concern.
• Type 3
high early strength. Used when high
strength are desired at very early periods.
• Type 4
low heat portland cement. Used where the
amount and rate of heat generation must
be kept to a minimum.
• Type 5
Sulfate resistant portland cement. Used
where water or soil is high in alkali.
• Types IA, IIA and IIIA are cements used to
make air-entrained concrete.
University of Michigan, TCAUP
Structures II
Slide 2/23
Effect of r
The behavior of the beam at failure (mode of failure)
is determined by the relative amount of steel present
– measured by r.
As
r
bd
r=0
No steel used. Brittle (sudden) failure.
r min
Just enough steel to prevent brittle failure
200
r min 
fy
r < r balance
Steel fails first – ductile failure (desirable)
0.18 f c'
r 
fy
r balance = r max
r max  r balanced
Steel and concrete both stressed to allowable limit
r > r balance
Concrete fails first – brittle failure (not desirable)
University of Michigan, TCAUP
Structures II
Slide 3/23
Calculate r balance
Procedure:
1. Draw stress diagram using allowable
stresses fc and fs/n
2. Use similar triangles to find x and
bar_xs
3. Find bar_xc = x/2
4. Use moments of areas on transformed
section to solve for As.
5. Calculate r bal = As/bd
University of Michigan, TCAUP
Structures II
Slide 4/23
As
r
bd
“Internal Couple” Method
• Uses the internal force couple T & C to
determine the moment
• Defines factors k and j that can be
used to find depth of stress block and
moment arm of couple
• Provides equations for analysis or
design.
University of Michigan, TCAUP
Structures II
Es
n
Ec
Slide 5/23
Analysis by “Internal Couple”
Example :
1.
2.
3.
4.
5.
Find r =As/bd
Find k
Calculate j
Calculate either force T or C
Calculate M using either T or C
University of Michigan, TCAUP
Structures II
Slide 6/23
Flexure Design
Procedure:
1. Determine load conditions.
• choose material grade, f’c
• calculate n = Es/Ec
• estimate size, choose b and
estimate d
1 b 2
 
2 d 3
• determine loads (+ member DL)
• calculate moment
2. Choose a target steel ratio, ρ.
3. Sketch the stress diagram with
force couple.
4. Calculate d based on the required
moment.
5. Calculate As.
6. Choose bar sizes and spacing.
7. Choose beam size and revise
(back to step 1 with new b, d and ρ)
. University of Michigan, TCAUP
Structures II
Slide 7/23
Example – Flexure Design
Source: University of Michigan, Department of Architecture
1. As a simplification the
moment is given = 200 ft-k.
d will be determined based
on the moment.
f'c is given as 4000 psi
n is found = 8.
University of Michigan, TCAUP
Structures II
Slide 8/23
Example – Flexure Design
cont.
2. Steel ratio, As/bd is taken
as balanced for this
problem.
3. Using similar triangles,
determine depth of
reinforcement, D in
relationship to depth of
compression zone, x.
Calculate the compression
zone resultant, Rc in terms
of x
Rc = fcBx/2
4. Use the internal moment
couple
M = Rc(D-x/3)
to solve for x and D.
University of Michigan, TCAUP
Source: University of Michigan, Department of Architecture
Structures II
Slide 9/23
Example - Flexure Design
cont.
5. Calculate As using
Rc = Rt and
Rt = Asfs
6. Choose bar sizes and
spacing.
•
•
•
•
Area >= As
c.g. = D
must be symetric
minimum spacing
7. Choose cover,
recalculate dead load,
iterate with new
moment.
Source: ACI-318-05
University of Michigan, TCAUP
Structures II
Slide 10/23