If expected lower bound of material strength Section 3107F.2.1.1 Equations (7-2a, 7-2b, 7-2c) are used in obtaining the nominal shear strength, a new nonlinear analysis utilizing the upper bound estimate of material strength Section 3107F.2.1.1 Equations (7-3a, 7-3b, 7-3c) shall be used to obtain the plastic hinge shear demand. An alternative conservative approach is to multiply the maximum shear demand, V_{max} from the original analysis by 1.4 (Section 8.16.4.4.2 of ATC-32 [7.8]):

(7-13)

If moment curvature analysis that takes into account strain-hardening, an uncertainty factor of 1.25 may be used:

(7-14)

Shear capacity shall be based on nominal material strengths, and reduction factors according to ACI 318 [7.7].

As an alternative, the method of Kowalski and Priestley [7.9] may be used. Their method is based on a three-parameter model with separate contributions to shear strength from concrete (V_{c}), transverse reinforcement (V_{s}), and axial load (V_{p}) to obtain nominal shear strength (V_{n}):

(7-15)

A shear strength reduction factor of 0.85 shall be applied to the nominal strength, V_{n}, to determine the design shear strength. Therefore:

(7-16)

The equations to determine V_{c}, V_{s} and V_{p} are:

(7-17)

**where:**

k = factor dependent on the curvature ductility , within the plastic hinge region, from Figure 31F-7-6. For regions greater than 2D_{p}(see Equation 7-18) from the plastic hinge location, the strength can be based on μ_{Φ} = 1.0 (see Ferritto et. al. [7.2]).

f'_{c} = concrete compressive strength

A_{e} = 0.8A_{g} is the effective shear area

Circular spirals or hoops [7.2]:

(7-18)

**where:**

A_{sp} = spiral or hoop cross section area

f_{yh} = yield strength of transverse or hoop reinforcement

D_{p} = pile diameter or gross depth (in case of a rectangular pile with spiral confinement)

c = depth from extreme compression fiber to neutral axis (N.A.) at flexural strength (see Figure 31F-7-7)

c_{0} = distance from concrete cover to center of hoop or spiral (see Figure 31F-7-7)

θ = angle of critical crack to the pile axis (see Figure 31F-7-7) taken as 30° for existing structures, and 35° for new design

s = spacing of hoops or spiral along the pile axis

Rectangular hoops or spirals [7.2]:

(7-19)

**where:**

A_{h} =total area of transverse reinforcement, parallel to direction of applied shear cut by an inclined shear crack

Shear strength from axial mechanism, V_{p} (see Figure 31F-7-8):

(7-20)

**where:**

N_{u} = external axial compression on pile including seismic load. Compression is taken as positive; tension as negative

F_{p} = prestress compressive force in pile

α = angle between line joining centers of flexural compression in the deck/pile and in-ground hinges, and the pile axis

Φ = 1.0 for existing structures, and 0.85 for new design