Chapter 4. Experimental Results & Analytical Modelling: Large-Scale Slab Specimens
5.3 N ear Surface M ounted CFRP Strips
5.3.2.4 Verification o f the Analytical M odel
The analytical model is verified by comparing the calculated shear stress from Equation 5.37 to that obtained from the finite element analysis at the cutoff point. The finite element modelling described in this section was conducted using the ANACAP program (Version 2.1). It is well established by many researchers that mesh size is a key parameter that influences the shear stress distribution in externally bonded FRP reinforcement especially at cutoff points [Malek et al., 1998]. Consequently, the first step towards developing a reliable finite element model is to determine the optimum size o f elements that should be used in the analysis.
5.3.2.4.1 Modelling of Test Specimens
Two test specimens, B3 and B4, were selected from the experimental program to be modelled using finite element analysis. Failure o f both beams was due to debonding o f the near surface mounted CFRP strips at the cutoff points. Taking advantage o f the symmetry of the specimens, only one quarter o f the beams was modelled. The concrete, epoxy and the CFRP strip were modelled using 20-node isoparametric brick elements with a 2x2x2 reduced Gauss integration scheme. Each node has three translational degrees o f freedom. The load was applied as a uniform pressure acting on an area o f
100x150 mm.
Various numerical simulations were conducted by varying the size o f the brick elements around the strip cutoff point in the longitudinal as well as in the vertical directions. One
______________ 5. Experimental Results & Analytical Modelling: Bond Specimens
element was used within the thickness o f the CFRP strip and the adhesive as shown in Fig. 5.28.
1350
Load | i| C.L
Strip cu toff point V ertical direction
Height o f the CFRP strip All dimensions are in mm
40
LoadC.L
.72.5
Strip
A dhesive --- ► Longitudinal direction
Fig. 5.28 General mesh dimensions o f the test specimens
The influence o f the mesh size on the predicted shear stress at the cutoff points was noticeable. In general, increasing the size o f the elements results in a proportional increase in the distances among the integration points within the element. Therefore, the induced shear stresses at the strip cutoff points are averaged over a large distance and are considerably less than the true values. Decreasing the size o f the elements results in a substantial increase in the maximum shear stress up to a certain limit beyond which no further increase in the shear stresses is observed. The size o f the elements at this transition stage is termed the “optimum size”.
The optimum size o f the elements in the longitudinal as well as in the vertical directions was determined as shown in Fig. 5.29 and 5.30, respectively. Further refinement o f the
______________ 5. Experimental Results & Analytical Modelling: Bond Specimens
mesh around the cutoff points increased the predicted shear stress by less than 0.5 percent.
0.3
P = 10 kN L=500 mm 0.27
0.24 -- 0.21 - -
I 0.18 -- J-j
0.15 -- s-a
■§ 0.12 4
£§ 0.09 -- C utoff point 0.06 --
^ 0.03 -- Optimum size
9 10
2 3 4 5 6 7 8
1
Size o f brick element in the longitudinal direction (mm)
Fig. 5.29 Influence o f element size in the longitudinal direction on maximum shear stress at cutoff point
0.24
P = 1 0 kN L=500 mm 0.22 - -
cd
0.2 - - 0.18 J j 0.16 -
CA
B
3 0.14 -
C utoff point
Optimum size
10 12.5 15 17.5 20 22.5 25
0 2.5 5 7.5
Size o f brick element in the vertical direction (mm)
Fig. 5.30 Influence o f element size in the vertical direction on maximum shear stress at cutoff point
300. , 300
5 .Experimental Results & Analytical Modelling: Bond Specim ens
The final mesh dimensions used for test specimens B3 and B4 with embedment lengths o f 500 mm and 750 mm, respectively are shown in Fig. 5.31.
B3 (L=500 mm)
1350
Load C.L
1. 160 I, 91 |.52| j, , 3 1 6 3 1, 109 I 100 I, 150 I,
200 280
1350
Load C.L
(N
16 4 0 i 2 8 k U 5 1
I, 9 5 1,4 Ol. 41, 57 , 80 , 114 „ 159 i. 100 l, 150 |,
200 230
B4 (L=750 mm)
1 2 x 3 3 ? All dimension are in mm
75 Loadi
mC.L
75 Loadi
O
1.9
0.6 \
Fig. 5.31 Mesh dimensions used in finite element analysis
_______________ 5 . Experimental Results & Analytical Modelling: Bond Specim ens
5.3.2.4.2 Comparison with Finite Element Analysis
The interfacial shear stress distributions for specimens B3 and B4 are calculated at two different load levels and compared to those predicted using finite element analysis. The first selected load level was less than the cracking load o f the test specimens to validate the analytical model at the elastic stage. The second selected load level matched the cracking load at cutoff points for specimens B3 and B4. The shear modulus o f the epoxy adhesive was set to 1230 M Pa as reported by the manufacturer. Based on a groove width o f 5 mm. the thickness o f the epoxy adhesive used in the analysis was set to 1.9 mm. The results o f the finite element analysis together with the proposed analytical model are shown in Fig. 5.32 at the elastic stage. The results, at the onset o f cracking are shown in Fig. 5.33.
0.25
Proposed m odel (B 3) Finite elem ent method (B 3)
0.2
P=10 kN 0.15
t*c/f CD 3 GO
Proposed m odel (B 4)
Finite elem ent method (B 4) 0.05
" v
90 100 60 70 80
30 40 50
10 20
0
Distance from cutoff point (mm)
Fig. 5.32 Comparison o f the proposed model to finite element method before cracking o f the concrete at cutoff point
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The interfacial shear stress distribution, predicted using the analytical model, is in a good agreement with the results o f the finite element analysis before cracking o f the concrete.
Finite elem ent method (B 3) Proposed
m odel v (B3)
'e?
Finite element method (B 4) Proposed
- m odel (B 4)
1 1
25* "~3tF--- -0.5
Distance from cutoff point (mm)
Fig. 5.33 Comparison o f the proposed model to finite element method at the onset o f cracking o f the concrete at cutoff point
At the onset o f cracking o f the concrete at the cutoff points, the calculated shear stress distribution deviates from the finite element results. This behaviour is attributed to the possible redistribution o f shear stresses around the cracks. Such a phenomenon is accounted for in the finite element analysis and not considered in the proposed model.
However, the maximum shear stress calculated using the analytical model is in a good agreement with the finite element results. Using the proposed model for the effective transformed moment o f inertia showed good agreement in predicting the maximum interfacial shear stresses at the cutoff points.
_______________ 5 . Experim ental R esu lts & Analytical M odelling: B ond S p e c im e n s
5.3.2.4.3 Comparison with Experimental Results
Debonding loads are predicted using the proposed analytical model (Equations 5.37 and 5.43) for different embedment lengths o f the CFRP strips as shown in Fig. 5.34.
90
80 Rupture o f CFRP strips 70
L=850 mm 60
L=750 mm 50
40
L=500 mm 30
L=250 mm 20
(Equation 5.43) 10
0
2 2.5 3 3.5 4
0.5 1 1.5
0
Shear stress at cutoff point (MPa)
Fig. 5.34 Maximum shear stress at cutoff point vs. applied load
The measured debonding loads for different embedment lengths are also shown in Fig.
5.34. In general, the predicted loads at the maximum shear stress for each length are in a good agreement with the measured debonding loads. The predicted debonding loads for specimens B2, B3 and B4 underestimated the measured values by less than 7 percent.
The mid-span section o f the test specimen was analyzed using a strain compatibility approach to predict the flexural behaviour up to failure. The predicted failure load due to rupture o f the CFRP strip is 76 kN. Failure o f specimen B5, with an embedment length o f 850 mm was due to rupture o f the CFRP strip at a load level o f 79 kN, which is 4 percent higher than the predicted value. Fig. 5.34 shows also that the minimum embedment
_______________ 5 . E xperim ental R esu lts & Analytical M odelling: B ond S p e c im e n s
length needed to rapture the CFRP strips used in this program is greater than 750 mm and less than 850 mm, which coincides with the experimental results.
The development length is highly dependent on the dimensions o f the strips, concrete properties, adhesive properties, internal steel reinforcement ratio, reinforcement configuration, type o f loading, and groove width. The proposed model in Equations 5.37 and 5.43, can be used to estimate the development length o f near surface mounted strips o f any configuration as follows:
1. Use the proposed Equations 5.37 and 5.43 to determine the debonding load o f the strip for different embedment lengths as shown in Fig. 5.35. The resulting curve represents a failure envelope due to debonding o f the strip at cutoff point.
Near surface mounted strip
13_Q
Rupture o f CFRP strips
4=i
Development length
Embedment length, L
Fig. 5.35 General procedures to calculate the development length for near surface mounted strips
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2. Use a cracked section analysis at sections o f maximum induced normal stresses and determine the ultimate load required to rupture the strip as shown in Fig. 5.35.
3. Determine the development length at the intersection o f the line corresponding to flexural failure o f the strip with the curve representing debonding failure at the cutoff point.
The calculated development length will preclude brittle failure due to debonding o f the strips and will ensure full composite action between the strip and concrete up to failure.