—Warping stresses and deformation

DESIGN OF SLABS-ON-GROUND 360R-55
In 1938, Bradbury (1938) extended Westergaard’s work
with a working stress formula referred to as the Wester-
gaard-Bradbury formula. This formula is still in use today
(Packard 1976). In 1939, Kelley (1939) used the Wester-
gaard-Bradbury formula to calculate the warping stresses
shown in 
Fig. 13.5 
for 6 and 9 in. (150 and 230 mm) slabs-
on-ground. Note that Kelley calculated a maximum stress of
approximately 390 psi (2.7 MPa) for a 9 in. (230 mm) slab
with a length of 24 ft (7.3 m).
In 1959, Leonards and Harr (1959) calculated the warping
stresses shown in 
Fig. 13.6
, presented here for general under-
standing. The upper center set of curves in 
Fig. 13.6
 shows a
maximum warping stress of approximately 560 psi (3.9 MPa)
for almost the same assumptions made by Kelley when he
computed a stress of 390 psi (2.7 MPa)
.
The only significant
difference is that Kelley used a 27 °F (15 °C)
change in
temperature across the slab while Leonards and Harr used a
30 °F (17 °C)
temperature difference across the slab thickness.
Adjusting for this gradient difference by multiplying by the
ratio, 30/27, Kelley’s stress would be 433 psi (3.0 MPa) instead
of 390 psi (2.7 MPa). Leonards and Harr’s 560 psi (3.9 MPa)
stress, however, is still 29% greater than the stress Kelley
calculated due to their better assumptions. Walker and Holland
(1999) had similar results to Leonards and Harr (1959).
Leonards and Harr (1959) calculated warping stress with a
form of computer modeling that permitted the slab to lift off
the subgrade if the uplift force was greater than the gravity
force. 
Figure 13.7 
shows their vertical deflection curves for
the same six cases of slabs whose warping stresses are shown
in 
Fig. 13.6
. The upward slab edge lift and downward slab
center deflection shown in 
Fig. 13.7
 is the usual case for
slabs inside buildings. True temperature gradient is very
small for slabs inside a building, but the moisture gradient
can be equivalent to about a 5 °F per in. (2.8 °C per 25 mm)
of slab thickness temperature gradient for such slabs under
roof. Leonards and Harr assumed a 30 °F (17 °C) gradient
across all the slabs, shown in 
Fig. 13.6 
and 
13.7
, no matter
what the thickness. They also assumed a cold top and a hot
slab bottom, which is not a usual temperature gradient, but it
is a usual equivalent moisture gradient for slabs inside build-
ings with a very moist bottom and a very dry top.
The conflict between the Westergaard assumption of a
fully supported slab-on-ground and the reality of either
unsupported slab edges or supported slab centers is documented
in Ytterberg’s 1987 paper. Because the three commonly used
slab thickness design methods (PCA, WRI, and COE) all are
based on Westergaard’s work and on the assumption that the
slabs are always fully supported by the subgrade, they give
erroneous results for slab thickness where the slab is not in
contact with the subgrade (referred to as the cantilever
effect). The thickness of the outer 3 to 5 ft (0.9 to 1.5 m) of
slab panels on ground might be based on a cantilever design
when warping is anticipated.
Another anomaly is that the three current slab thickness
design methods permit thinner slabs as the modulus of
subgrade reaction increases. The fact is, however, that a higher
subgrade reaction modulus will increase the length of
unsupported curled slab edges because the center of the slab
is less able to sink into the subgrade.
Thus, curling stress
increases as the subgrade becomes stiffer, and resultant load-
carrying capacity decreases for edge loadings; however, higher
k with loadings away from the edges allows thinner slabs.
Proper design should take all of these factors into account.
ACI Committee 325 (1956) recommends that highway slabs-
on-ground
be designed for a 3 °F (1.7 °C) per in. (25 mm)
daytime positive gradient (downward curl) and a 1 °F (0.6
°C) per in. (25 mm) nighttime negative gradient (upward curl).
Enclosed slabs-on-ground should be designed for a negative
gradient (upward curl) of 3 to 6 °F per in. (1.7 to 3.4 °C per
25 mm), according to Leonards and Harr (1959).
The Westergaard-Bradbury formula (Yoder and Witczak
1975) concluded that warping stress in slabs is proportional
to the modulus of elasticity of concrete, and partially
proportional to the modulus of elasticity of aggregates used
in a particular concrete. Therefore, to reduce slab warping,
low-modulus aggregates, such as limestone or sandstone, are
preferable to higher-modulus aggregates, such as granite and
especially traprock; however, if no hardener or topping is
used, many low-modulus aggregates will not be as durable
for some applications.

tải về 2.35 Mb.

Chia sẻ với bạn bè của bạn: