Shear Strength Models for RC Columns

Based on the post-earthquake observations and experimental studies, it was

concluded that the shear failure of the RC columns may strongly limit the displacement

ductility of the existing RC structural system during an earthquake (Del Vecchio et al.,

2017). The conceptual model that illustrates the interaction between the shear strength

capacity and shear strength demand based on the displacement ductility demand was

69

Fig 3.9. In this equation, the dashed line represents the shear capacity of the RC column

while the sold line represents the shear demand. The brittle failure mode, Case #1,

happens when the shear demand is larger than the initial shear strength. In Case #2, a

flexural-shear failure happens where there is shear demand between the initial and

residual shear strength, causing a displacement ductility corresponding to the intersection

point between shear demand curve and shear strength curve. A ductile failure mode, Case

#3, is ensured when the maximum shear demand is less than the residual shear strength.

In order to determine the mode failure of the RC columns correctly, many researchers

(Priestley, M J Nigel, Verma, Ravindra, & Xiao, 1994; Kowalsky & Priestley, B. M.,

2000; Biskinis, Roupakias, & Fardis, 2004; Sezen & Moehle, 2004; Ghobarah &

Elmandoohgalal, 2004) have focused on proposing models for determining the

degradation of shear strength in RC columns with inelastic cyclic displacement.

70

Priestley, Michael J. N; Seible, F; & Calvi (1996) proposed a shear strength

model for RC columns that considers the effect of curvature ductility and axial load level.

The proposed equation is defined in following expression:

𝑉_𝑛 = 𝑉_𝑐+ 𝑉_𝑝+ 𝑉_𝑠

3.56 The values of 𝑉𝑐, 𝑉𝑠, and 𝑉𝑝 are calculated by using equations 3.40 through 3.42. In this model, the concrete shear strength capacity, 𝑉𝑐, was calculated based on the gradual reduction of the aggregate interlock along the flexural cracks.

In order to include the effect of the longitudinal reinforcement ratio and the

column aspect ratio, the concrete shear strength capacity, 𝑉𝑐 in the previous model was revised by Kowalsky & Priestley (2000):

𝑉𝑐 = 𝛼𝛽 𝑘√𝑓𝑐, 𝐴𝑒

3.57 Where:

𝛼 and 𝛽 are factors that account for the column aspect ratio and the longitudinal steel ratio, respectively, and are calculated by using the following equation:

1 ≤ 𝛼 = 3 − 𝐿𝑠

ℎ

≤ 1.5

3.58

𝛽 = 0.5 + 20 𝜌𝑙 ≤ 1

3.59 Based on a large database of experimental tests on RC columns and RC beams

with rectangular and circular sections, a new shear strength model was proposed by

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and transvers reinforcement was included. Based on the proposed model, the shear

strength is computed by using the following equation:

𝑉𝑛 = 𝑘(𝜇∆)(𝑉𝑐+ 𝑉𝑠) + 𝑉𝑝

3.60 Where;

𝑘(𝜇_∆) is the coefficient of the shear strength degradation with ductility demand, and is equal to:

0.75 ≤ 1.05 − 0.05𝜇_∆≤ 1

3.61 𝑉𝑐, 𝑉𝑠, and, 𝑉𝑝, are calculated by using the following equations:

𝑉𝑐 = [0.16 max(0.5,100𝜌_𝑡𝑜𝑡) (1 − 0.16 min (5,𝐿_ℎ𝑠)) √𝑓𝑐′𝐴𝑔+ 𝑉𝑠]

3.62

𝑉𝑠 =

𝐴_𝑠𝑠 0.9𝑑𝑓𝑦𝑤 3.63

𝑉𝑝 =

𝐻−𝑥 _2𝐿

𝑠 min (𝑃, 0.55𝐴𝑔𝑓𝑐) 3.64 For rectangular RC columns with light transverse reinforcement, a shear strength

model was proposed by Sezen & Moehle (2004). This model was developed based on the

large database of numerous column tests, and the model’s results showed improved

accuracy in predicting the shear strength compared with available models. The proposed

model included the contribution of the concrete and transvers reinforcement to the shear

strength as shown in following equation:

72 𝑉_𝑠 = 𝑘 𝐴𝑣𝑓𝑦𝑑 𝑠

3.66 𝑉_𝑐 = 𝑘

(

6√𝑓_𝐿 𝑐′ 𝑠 𝑑 ⁄

√1 +

𝑃 6√𝑓𝑐′𝐴𝑔

)

0.8𝐴𝑔

3.67 Where:

𝑘 is the factor that accounts for displacement ductility. The 𝑘 factor value is calculated by using the following expression:

0.7 ≤ 1.15 − 0.075𝜇∆≤ 1

3.68 Furthermore, the last model was adopted by ACI 369R-11 (2011); and

ASCE/SEI41-13 (2013) to determine the shear strength for RC columns based on the

required displacement ductility.

Ghobarah & Elmandoohgalal (2004) proposed a shear capacity model for RC

columns that includes the effect of the CFRP retrofit. The following equations illustrates

the shear capacity envelope based on displacement ductility limit, as shown in Fig. 3.10.

𝑉µ = [ 𝑉_𝑐+ 𝑉_𝑝+ 𝑉_𝑠+ 𝑉_𝑓 0 ≤ µ_𝛥 ≤ 2 . 1 3∗ (𝑉𝑐+ 𝑉𝑝) + 𝑉𝑠+ 𝑉𝑓 𝑎𝑡 µ𝛥 = 4 . 𝑉𝑠+ 𝑉𝑓 𝑎𝑡 µ𝛥 = 6

3.69 Where:

73

𝑉𝑐 is the shear strength of the confinement concrete, and the compressive strength of the confinement concrete, 𝑓_𝑐𝑐, , will be used instead of the unconfined compressive strength, 𝑓_𝑐,, as shown in following equation:

𝑉𝑐 = 0.3√𝑓𝑐𝑐, 𝐴𝑒

3.70 The equations that were used to calculate each of the 𝑉_𝑠, 𝑉_𝑓 and 𝑉_𝑝 are illustrated below:

𝑉_𝑠 = 𝐴𝑣𝑓𝑦𝑑 𝑠

3.71 𝑉_𝑓= 0.95(2𝑡_𝑓)(𝜀_𝑓𝑒 𝐸_𝑓)𝑑_𝑓

3.72 𝑉_𝑝 = 𝑘_𝑝 𝑃𝑡/2 𝐻

3.73 Where:

𝑘_𝑝 is the factor that equals 1 in a double curvature column, and 0.5 in a single curvature column. P, t, H are the axial load level, the total depth of the column, and the height of

the column, respectively. In this model, the contribution of the CFRP retrofit on the shear

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Figure 3.10: Shear Strength Envelope Proposed by (Ghobarah & Elmandoohgalal, 2004)

In document Behavior of Non-Ductile Slender Reinforced Concrete Columns Retrofit by CFRP Under Cyclic Loading (Page 95-101)