Pallares and Hajjar

204 L. Pallarés, J.F. Hajjar / Journal of Constructional Steel Research 66 (2010) 198–212
Fig. 2. Assessment of anchor strength using the minimum of steel and concrete failure formulas in [2] Appendix D and [6].
α is equal to 0.55, given by Ravindra and Galambos [51];
β is the reliability index; and
VR =
�
V 2
F
+ V 2
P
+ V 2
M , where VF is the coefficient of varia-
tion on fabrication and is taken as VF = 0.05 as recommended
by Ravindra and Galambos [51], reflecting the strong control characteristics
of stud manufacturing; VP is the coefficient of variation
of Rm
Rn ; VM is the coefficient of the variation of the materials and is
taken as VM = 0.09 based on test data from [52–54].
Ravindra and Galambos [51] recommend a reliability index β
of 3 for members and 4.5 for connections. In this work, a reliability
index of 4 has been targeted to compute the resistance factors.
Resistance factors for steel strength prediction using only tests
that failed in the steel are computed for values of the Cv coefficient
equal to 1.00, 0.75 and 0.65 (Table 6). Values of the resistance factor
for a β value of both 3 and 4 are presented. Eq. (2) presents a sample
calculation for Cv = 1.00 and β = 4.
φv = Rm
e
Rn
(−0.55βVR) (−0.55·4.0·0.160)
= 0.933e = 0.65. (2)
With Cv = 0.65 the resistance factor computed by Eq. (2) is larger
than 1.0, so it should be taken as 1.0.
Table 6
Resistance factors computed for CvAsFy for Cv = 1, Cv = 0.75 and Cv = 0.65 for
steel strength based on steel failure in tests.
202 tests Cv µ σ C.O.V. φ
β = 3 β = 4
S1 1.00 0.933 a
S2 0.75 1.224 a
S3 0.65 1.436 a
0.150 0.161 0.68 0.61
0.200 0.161 0.89 0.80
0.231 0.161 – 0.94
a With measured values reported by authors or nominal values if the measured
steel strength was not reported.
6. Formulas for concrete failure
The concrete failure formulas of ACI 318-08 and PCI 6th Edition
are geared for general conditions for preventing failure of headed
steel anchors, especially cases where free edges may be close to
the stud. Such free edges rarely occur in composite construction.
Thus, several alternative formulas were developed in this work
to compute the concrete strength surrounding headed studs for
conditions commensurate with composite construction. These
formulas are compared with the AISC and EC-4 formulas in Table 7.
It can be seen that the mean value of the test-to-predicted ratios for
the AISC formula in particular is quite low, coupled with a relatively
large coefficient of variation. Both the optimized formula and