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Calculation of Bravo 25 Resistance
The original designers detailed Bravo 25 to be “underreinforced”
using the “working stress” method that was employed
in the first half of the twentieth century. The reinforcing steel
was detailed not to exceed an elastic limit (of 20 ksi (140mPa))
under design loads while the concrete stress remained below (an
assumed elastic limit) 0.45fc’ (1,350 psi (9.3mPa)). This
design procedure produces a highly “ductile” failure
that is preceded by large cracks and deformation. The ultimate capacity
is attained after the reinforcing steel yields followed by a concrete
“failure” (defined at strain levels of 0.3 percent).
The “ultimate strength” flexural resistance of the
existing sections is calculated from simple equilibrium analysis
while maintaining compatibility of strains. The ultimate resistance
of a conventionally steel reinforced section that is post reinforced
with a carbon rod or laminate is different from the conventional
section because the CFRP is linearelastic to failure and has no
plastic reserve. Given the underreinforced sections of Bravo 25,
the design objective was to obtain the maximum deck slab bending
resistance after the reinforcing steel yields. Reinforcing steel
yielding is followed by CFRP reinforcement failure and concrete
crushing (0.3 percent concrete strain). This assumes that there
is sufficient concrete strength to offset all tensile forces (under
reinforced), there is sufficient shear strength, and the carbon
composite and reinforcing steel retains firm bonding with the concrete
up to failure. Lower bending resistance results if the concrete
fails first (overreinforced), if shear failure occurs, or if anchorage
(bond) is compromised. The upgrade sections were checked to assure
that these design assumptions were valid.
The following constituent assumptions were applied to calculating
the bending resistance of a reinforced concrete cross section:
• 
Idealized stressstrain for concrete, steel and
CFRP. Steel strain hardening is ignored. 
• 
Concrete tensile force is ignored. 
• 
Strains are linearly distributed across the section in proportion
to distance from the neutral axis (section planes remain plane). 
• 
The position of the forces and neutral axis remain constant. 
Figure 46 shows the state of strain, stress, and
force for calculating the bending resistance of a post strengthened
reinforcedconcrete section with embedded carbon bars. Similar methodology
was applied to bending resistance of external carbon laminate where
the dimension, h, would be replaced by the full depth of the concrete
section.
Figure 46. Post strengthened cross section using embedded composite
bars. Assumed stress, strain, and internal forces for calculation
of bending resistance.
The strain relationships at maximum resistance are:
Where:
The value of a/2 is normally dependent on the concrete strength,
fc?. The average compressive stress in the concrete is 0.85 fc'
(the standard American Concrete Institute (ACI) Code allowance).
So
where “b” is the width of the section (or each unit
width of a slab). Changing the concrete strength of an under reinforced
section typical of those encountered in an existing, older Navy
pier has little effect on its flexural resistance, Mr, because the
value of a/2 is a small percentage of the total slab depth.
The above relationship should be valid as long as the steel yields
prior to a laminate failure and prior to the concrete strain reaching
0.003. This will be the case when the following relationship holds,
which is the case for Bravo 25:
In designing an upgrade to poststrengthen a section, the steel
stress should be limited to its yield value and the carbon laminate
stress should be limited to less than half of its measured strength.
This sets the value of the total tensile force of the internal couple,
which, in turn, sets the compression force. With the compression
force known, the Whitney compression stress block is defined and
the resisting moment can be determined with the equation above.
Setting the laminate stress also sets the laminate strain so a check
of neutral axis location and concrete strain can be made by compatibility
of strain requirements and since planes remain plane. The equations
above have been organized in an EXCEL® spreadsheet program to
design flexural members using CFRP (i.e., laminate, pultruded strips,
and embedded rods). The spreadsheet was used to detail the upgrade
reinforcement knowing the response from the FEA analysis of Bravo
25.
In order to control crack width, the strain in the laminate may
be restricted in the future to more than the limits listed above.
This is important when it is deemed necessary to protect existing
steel reinforcing from corrosion. For example, given a carbon laminate
with an ultimate strength of 300 ksi and a modulus of 20,000 ksi
(140 mPa), the laminate strain for a stress limit of 150 ksi (1,030
mPa) would be 0.0075 in./in. (m/m). The average crack width will
be almost 0.1 inch (0.25 cm) for average crack spacing of 12 inches
(30.5 cm) (larger for greater spacing). The ACI code (Section 9.4)
limits reinforcement design strength to 80,000 psi (550 mPa) to
control deformation and cracking. It would seem that similar restrictions
may be necessary for CFRP reinforcement in future designs. There
are no current guidelines limiting carbon laminate design stresses
for the purpose of limiting deformation and control cracks. However,
the ACI is formulating stress limits for carbon and other fiber
composite reinforcement. Until the ACI code provisions are approved,
NFESC recommends that the carbon fiber design stress never be allowed
to exceed one half of the ultimate strength.
