Moment-rotation
05 O . .SA**85**5 BF-N0.50 ••••BF-N0.50-F1(5)-B1 = «BF-N0.50-F1(5)-B2 20 40 60 80 Deflection at mid-span (mm) 100 120
Figure 4.21 Load-deflection relative performance of beams: BF-NO. 50, BF-NO. 50-Fl (5)- BlandBF-N0.50-Fl(5)-B2 « O BF-H0.33 •••?•BF-H0.33-F1(5)-B1 BF-H0.33-F1(5)-B2 BF-N0.50 — — -BF-N0.50-F1(5)-B1 BF-N0.50-F1(5)-B2 40 60 80 Deflection at mid-span (mm)
Figure 4.22 Load-deflection relative performance of beams: BF-HO.33, BF-HO.33-Fl (5)- Bl, BF-H0.33-F1(5)-B2, BF-N0.50, BF-N0.50-F1(5)-B1 and BF-N0.50-F1(5)-B2
Chapter 5
Numerical Model
5.1 Introduction
The main objective of this chapter is to set up a numerical method that is capable of predicting the elastic and the post-yield behaviour of unstrengthened and strengthened deteriorated steel girders. This proposed method can be used by designers to calculate the reduction in the moment capacity of the deteriorated steel girders within a reasonable level of accuracy. The analysis approach is based on a moment-curvature analysis which satisfies equilibrium and compatibility. The following assumptions were considered in
this analysis: i) plane sections remain plane after deformation; ii) deformations are small;
iii) a simplified material characteristic was used for the steel wide flange beam; iv) linear elastic behaviour of CFRP; v) bilinear relation between the bond length and the ultimate
tensile strength in the bonded CFRP; vi) linear relation between the total number of CFRP layers and the minimum bonded development length; and vii) the stress in all CFRP layers is the same for the same tensile strain in the steel flanges (i.e. perfect bond until peel off of FRP). The results of the tested beams which presented in Chapter 4 were
used to validate this numerical method.
It is noted that the discussion in this chapter is limited to the flexural analysis of
unstrengthened and strengthened corroded steel girders using bonded CFRP. A bilinear
relation between the bond length and the ultimate tensile strength for Tyfo SCH-IlUP
and Tyfo UC was assumed in this numerical model. Based on the experimental results and modes of failure, only three modes of failure were considered in this analysis such as:
in-plane plastic buckling for deteriorated steel beams; unbonded and rupture of CFRP for
the rehabilitated/strengthened steel beams.
5.2 Section analysis in flexure
The mechanical properties for the steel material of the W-shape beams were
determined by testing three dog-bone coupon specimens which were cut from the web and the flanges as described in Chapter 3. The measured stress-strain curves for the coupons were used to generate a best fit stress-strain curve which was used as input for the analytical model as shown in Figure 5.1. A bilinear relation between the bond length per layer and the ultimate tensile strength in this layer is shown in Figure 5.2. The bond
length per layer was defined as:
Un
b/layer= -^
n(5.1)
where Lb is the total bond length and ? the total number of layers
Based on this bilinear relation, unbonded failure mode can be expected when
bonding Tyfo SCH-IlUP sheets to steel using Tyfo S and the bond length per layer is
less than 205 mm. When MB-3 is used instead of Tyfo S, the minimum bonded length
per layer which is required to have rupture in the CFRP sheets as a failure mode is only 90 mm. while, for Tyfo UC bonded by MB-3 the minimum bond length per layer
required to have rapture failure mode.
In this chapter, efforts are made to develop a numerical model that is capable of predicting the remaining flexural capacity of deteriorated steel girders as well as the magnitude of the yielding moment. The full moment-curvature relationship of any steel
cross-section is established by increase incrementally the strain at the extreme 109
compression fiber (see Figure 5.3). In the first step it is assumed that for a given strain, S c, the neutral axis depth is Y\ The corresponding curvature of this section can be
calculated as:
f = —^-
Qi-Y)(5.2)
Based on the assumed linear strain through the depth of the section as shown in
Figure 5.3, the strain at any level in the section, Gx, can be defined as:
Sx = 0 y
(5.3)
The strain is calculated at the inner and outer surfaces of the flanges of the steel
beam, at which the geometric properties have changed. Once the strain at the different
layers is known the stress distribution, fs(x), can be determined from the appropriate
stress-strain relationship of the steel material as:
€x E €x < 0.00155
fM=i310
0.00155 < €x < 0.004
Z54-)
1310+(6,-0.004XE1)
Q.Û04 < €x < Q.G14
.400 + (Cx- 0.014XE2) 0.014 < Cx < 0.045
where E is the elastic modulus of the steel, Ej and E2 are the second and third slope stiffness, respectively.
Once the force contribution of the steel cross-section is calculated, the total force on the
cross-section can be calculated as:
F = F1 + F2+ F3 + F4 (5.5)