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FIG. 7. Dimensionless Flexural Strength versus AS /b2
of Singly RC Rectangular Section Beam Subjected to Bending: Diagrams for
Various Theories for Constant Values of d/b (from 1 to 4)
J. Perform. Constr. Facil. 1999.13:67-75.
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FIG. 8. Dimensionless Flexural Strength versus d/b of Singly RC Rectangular Section Beam Subjected to Bending: Diagrams for
Various Theories for Constant Values of AS /b2
(from 0.01 to 0.04)
J. Perform. Constr. Facil. 1999.13:67-75.
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FIG. 9. 3D Diagrams (Maximum Stresses versus Depth and Reinforcement Area) for Koenen’s Theory of Bending (1886) Applied to
Rectangular Section Beam (b =40cm; M = 150 kN?m)
FIG. 10. 3D Diagrams (Maximum Stresses versus Depth and Reinforcement Area) for Demay’s Theory of Bending (1887) Applied to
Rectangular Section Beam (b =40cm; M = 150 kN?m)
FIG. 11. 3D Diagrams (MaximumStresses versus Depth andModular Ratio, Assuming AS =20cm2
;MaximumStresses versusDepth
and Reinforcement Area, with n = 20) for Melan’s Theory of Bending (1890) Applied to Rectangular Section Beam (b =40cm; M = 150
kN?m)
ing, following an allowable stresses criterion, has been pro-
posed in the diagrams shown in Figs. 7 and 8. The values of
modular ratios proposed by each author has been assumed,
while the ratio d/h has been assumed constant to 0.95.
In each diagram the dimensionless parameter d/b—values
from 1 to 4 in Figs. 7(a–d)—or the dimensionless reinforce-
ment parameter AS/b2
—values from 0.01 to 0.04 in Figs.
8(a–d)—is given. This allows one, for example, to control
easily the trend of allowable flexural strength when the depth
or the reinforcement amount is kept constant, for a beam of
constant width.
A more synthetic 3D representation of the verification ap-
proaches, already shown in Figs. 1–6, is also given in Figs. 9–
12 for the theories of Koenen, Demay, Melan and Coignet–
de Tedesco. Those have been applied to the same beam cross
section (width b = 40 cm; cover c = 3 cm), subjected to a
constant bending moment of 150 kN?m. Maximum concrete
stress and tensile reinforcement stress are represented versus
J. Perform. Constr. Facil. 1999.13:67-75.
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FIG. 12. 3D Diagrams (MaximumStresses versus Depth andModular Ratio, Assuming AS =20cm2
;MaximumStresses versusDepth
and Reinforcement Area, with n = 20) for Coignet and de Tedesco’s Theory of Bending (1894) Applied to Rectangular Section Beam(b
=40cm; M = 150 kN?m)
depth (h from 60 to 140 cm) and reinforcement area (AS from
10 to 30 cm2
), assuming, for the Melan (Fig. 11) and Coignet–
de Tedesco (Fig. 12) approaches, the modular ratios suggested
by them (n = 20), while the Koenen (Fig. 9) and Demay (Fig.
10) methods are independent by n. Figs. 11 and 12 also give
3D diagrams showing the maximum concrete stress and tensile
reinforcement stress versus depth (from 60 to 140 cm) and the
modular ratio (from 5 to 25), with AS =20cm2.
CONCLUSIONS
Today a large number of buildings and other structures built
at the turn of 19th century using the RC technique need to be
restored and rehabilitated. Thus, it is very important to know