L.-Y. Zhou/Z. Liu/Z.-Q. He ¡ Further investigation of transverse stresses and bursting forces in post-tensioned anchorage zones
the FEA results. Similarly, the transverse stress distribution using the modified equations of He et al. is also far from satisfactory when compared with the FEA results.
3 3.1
Updated equations for ILCs Equations for concentric loading
Fig. 5. Finite element model for the anchorage zone (30 Ă— 60 elements)
For the same problem as depicted in Fig. 4, in accordance with St. Venant’s principle, section CD is the interface between the zone of disturbed stresses and the uniformly distributed stress zone, which means that the transverse stresses at this section should have diminished. As a result, the rate of transverse stress change along the X-axis should approach zero as it approaches section CD, i.e.
Table 2. Bursting forces and their locations
wV y
Method
Bursting force Tb /P
Location db /h
Sahoo et al. He et al. FEA
0.14 0.13 0.15
0.79 0.67 0.5
Noting that the transverse stress is proportional to the curvature of the ILCs, this leads to d3 y dx3 x
modelled by four-node plane stress elements. The bearing plate ratio a/h is taken to be 0.4, Young’s modulus of concrete as 3.0 × 104 MPa and Poisson’s ratio as 0.2. The anchor force is applied as a uniform load within the footprint of the anchor plate, and the boundary conditions at the far-end are restrained as shown in Fig. 5. The bursting forces and their locations calculated by the equations of Sahoo et al and He et al are summarized in Table 2. It can be seen that the bursting forces estimated by either Eq. (11) or Eq. (16) are in good agreement with the FEA results. However, the locations of the bursting forces obtained by Eqs. (12) and (17) are significantly different from the FEA results. The relative error is 58 % and 34 % for the equations of Sahoo et al. and He et al. respectively. Fig. 6 shows the distributions of the transverse stresses obtained along the tendon axis. The diagram indicates that there is a substantial discrepancy between the transverse stresses given by the equations of Sahoo et al. and
(21)
0
wx
(22)
0 h
According to the theory of elasticity [1], the compatibility equation and equilibrium differential equations for the planar stress problems can be obtained as follows: w2W xy w2 w2 ( V – PV ) ( V – PV ) 2(1 P ) x y y x wx w y wx 2 wy2 wW xy
wV y
Y wy wV x X wx
wx wW xy wy where: Îź Ďƒx Ďƒy – X – Y
(23)
(24)
Poisson’s ratio normal stress in x direction normal stress in y direction sum of external forces in x direction sum of external forces in y direction
0.50
ĎƒT /Ďƒ 0
0.00 Sahoo et al. He et al. FEA
-0.50 -1.00
hP
a
x
-1.50
Ďƒ 0 = P / (ht)
l h
-2.00 0
0.1
0.2
0.3
0.4
0.5
Relative distance,
0.6
0.7
0.8
0.9
1
x/h
Fig. 6. Distribution of transverse stresses along the tendon axis
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