CHAPTER 5 NEW ANALYTICAL MODELS FOR FULLY GROUTED JOINTS
5.2 A NALYTICAL FAILURE MODEL FOR A FULLY GROUTED X- JOINT SUBJECTED TO AXIAL
5.2.4 New equations for the ultimate strength of fully grouted X-joints
As discussed in Section 5.2.3, the development of the new equation for the ultimate strength of a fully grouted X-joint under brace axial tension depends on the calculation of kT and kV. According to Equation (5.13), to calculated kT and kV, the value of Ψ at any value of ρ under the maximum brace load needs to be determined. However, as mentioned earlier, the dihedral angle Ψ along the brace-to-chord intersection changes significantly after the grouted joint is loaded. Thus it is difficult to determine the actual dihedral angel along the brace-to-chord intersection at failure for a fully grouted joint.
Alternatively, the un-deformed joint geometry is adopted to determine the distribution of the values of the dihedral angle Ψ along the brace-to-chord intersection for the following calculations. Since for a fully grouted joint, the value of the dihedral angle at any position along the brace-to-chord intersection before test is smaller than that at failure, the tensile
force portion in the resistant force will be under-estimated under such calculations.
Consequently, a conservative prediction will be made.
90 100 110 120 130 140 150 160 170 180
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 Valuse of ρ
Values of Dihedral Angle Ψ
Figure 5-9 Distribution of Dihedral Angle Ψ (θ=90o)
The dihedral angle distributions along the brace-to-chord intersection of a tubular joint with any θ are provided in AWS D1.1 (AWS, 1998). Figure 5-9 shows the actual distribution of the values of Ψ with respect to ρ for a joint with θ=90o. Ψ cannot be expressed as a single function of ρ explicitly. Thus, it is difficult to conduct integrations directly for above equations using the actual Ψ distribution curves.
However, for a tubular X-joint with a given geometry, the values of the dihedral angles at crown and saddle points are known (Ψcrown=π/2, Ψsaddle =π-cos-1β). Thus, there are two ways to simplify the integration procedure, which give the two different equations as follows:
Equation I
The first way to simplify the procedure is to use the dihedral angle at the saddle point to represent the dihedral angle anywhere else (e.g., Ψ =Ψsaddle).Thus,
Values of β
1.0 0.9
0.8 0.70.6
0.5 0.40.3
0.2
⎪⎩
⎪⎨
⎧
β
−
⋅
= β
− π
= π ρ
= ψ π ρ
= ψ
β
⋅
= β
− π
−
= π ρ
− ψ
= π ρ
− ψ
=
∫
∫
∫
∫
π −
π
π −
π
2 0
2 1
saddle 2
P 0
2 0
1 saddle
2 P 0
1 P ) cos sin(
2 P sin P 2
sin V P
P ) cos cos(
2 P cos P 2
cos T P
d d
d d
(5.14)
Since the full brace-to-chord intersection is considered to be effective, the maximum shear stress and the corresponding normal stress can be calculated as,
⎪⎪
⎩
⎪⎪⎨
⎧
⋅
⋅ π
⋅ β
−
= ⋅
= τ
⋅
⋅ π
⋅ β
= ⋅
= σ
= τ
0 a
2
effective P max
0 a
effective P
T d K
1 P A
V
T d K
P A
T
(5.15)
Combining Equation (5.7) and (5.15) and let σ_ =Fy leads to,
3 2 K 2 T F
P
2 a 2
0
y −
β γ
= π (5.16)
In Equation (5.14), as β approaching 1.0, TP→P and VP→0, while as β approaching 0, TP→0 and VP→P. This implies that for the joints with a large β ratio, the shear action tends to be cancelled out and the brace load is resisted only by the tensile force. This obviously over-estimates the tensile force portion in the resistance force, since for the joint with β =1.0, the shear resistance along the brace-to-chord intersection is still quite significant. Thus Equation (5.16) may provide over-estimated predictions for the capacity of a fully grouted joint and probably results in an upper bound for the joint capacities.
Equation II
The second way to simplify the integration procedure is to use a linear interpolation to represent the distribution of the dihedral angle between a saddle point and a crown point.
Based on the linear distribution assumption, it can be deduced that,
) 2 2 1
( ) (
f saddle π
+ ρ π −
= ψ ρ
=
ψ , 0 ≤ρ≤
2
π (5.17)
Due to symmetry of the joint, Equation (5.12) can be re-written as,
⎪⎪
⎩
⎪⎪⎨
⎧
π ρ
= ψ π ρ
= ψ
π ρ
− ψ
= π ρ
− ψ
=
∫
∫
∫
∫
π π π π
d d
d d
2 0 2
P 0
2 0 2
P 0
2 sin 4 P
2 sin V P
2 cos 4 P
2 cos T P
(5.18)
Combining Equation (5.17) and (5.18) gives,
⎪⎪
⎩
⎪⎪⎨
⎧
β⋅
− π
= β ψ ⋅
− π
= ψ π ρ
= ρ π ρ
= ψ
β ⋅
− π
β
−
= − π ⋅
− ψ
ψ
= − π ρ
−
= ρ π ρ
−
= ψ
− π
π
− π
π
∫
∫
∫
∫
cos P 2 P 2
2 cos 2 2
)) ( f sin(
4 P 2
sin 4 P
V
cos P 2
) 1 1 ( P 2 2
) sin 1 ( 2 2
)) ( f cos(
4 P 2
cos 4 P
T
1 saddle
saddle 2
0 2
P 0
1 2
saddle saddle 2
0 2
P 0
d d
d d
(5.19)
Thus,
⎪⎪
⎩
⎪⎪⎨
⎧
β
− π
⋅ β
⋅
⋅ π
= ⋅
= τ
β
− π
β
−
⋅ −
⋅
⋅ π
= ⋅
= σ
= − τ −
1 0
a effective
P max
1 2
0 a
effective P
cos 2
2 T
d K
P A
V
cos 2
) 1 1 ( 2 T d K
P A
T
(5.20)
Combining Equation (5.7) and (5.20) and let σ_ =Fy lead to,
) 1
1 ( 2
) cos 2 ( K T
F P
2 2
1 a
2 0
y − −β +β
βγ β
− π
= π − (5.21)
In Equation (5.19), as β approaching 0, TP→0 and VP→P, while as β approaching 1.0, TP→2P/π and VP→2P/π. This suggests that when the brace diameter approaching zero, the brace load is transferred purely through punching shear, which is close to the real scenario. When the brace diameter is close to the chord diameter, the brace load is resisted equally through the punching shear and the membrane action. Thus the punching shear portion is not neglected as in Equation (5.14). However, this formula is based on the un-deformed joint shape, while the dihedral angle along the brace-to-chord
intersection of the grouted joint increases significantly at the failure. As a result, by using this formula, the tensile force portion in the resistance will be under-estimated since larger dihedral angle results in higher tensile force portion in the resistance force.
Consequently, equation (5.21) may provide under-estimated prediction for the capacity of a fully grouted joint and thus probably results in a lower bound for the joint capacities.
Equation III
If the actual dihedral angle values are adopted, kT and kV will need to be determined through numerical integration based on the dihedral angle distribution curves shown in Figure 5-9. Once TP and VP are determined, through the same procedure as in the two previous cases, it can be deduced that the capacity of a fully grouted joint under brace axial tensions can be written as
2 V 2 T
a 2
0
y k 3k
K T
F P
+ βγ
= π (5.22)
Numerical integrations have been carried out for β=0.2, 0.3, 0.4, 0.5, 0.6 0.7, 0.8, 0.9 and 1.0 based on Figure 5-9 and Equation (5.13). Figure 5-10 shows the distribution of the calculated values of kT and kV with respect to β. The capacity of a fully grouted joint thus can be calculated using Equation (5.22) for any β value through the interpolation between the calculated points. Note that the calculations here are still based on the un-deformed joint shape and consequently the prediction by the corresponding equation may also under-estimate the real joint capacity.
Figure 5-10 also includes the calculated values of kT and kV based on the linear Ψ distribution assumption. The differences between the values of kT based on the actual dihedral angles and those on the linear distribution assumption are marginal and so does the value of kV. The differences only become significant when the value of β is larger
than 0.9 (for kV) or 0.8 (for kT) and yet the differences are still within very limited range.
This means that the predictions of the grouted joint capacities from equation (5.21) and (5.22) are close to each other. Thus the linear Ψ distribution assumption is a reasonable approximation to the actual distribution. Later, the comparisons between the predictions from the proposed equations and the test results also confirm this.
0.0 0.2 0.4 0.6 0.8 1.0 1.2
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Value of β Value of kT and kV
Figure 5-10 Distributions of kT and kV against β