Deep beam with column stub

Previous analysis in Section 4.1 strongly showed that a good construction of a macro model is very important to obtain reasonable shear strength prediction. However, as mentioned previously in the beginning of chapter 4, the boundary condition which defined the discontinuity between a deep beam loaded through bearing plate and a coupling beam is totally different. The discontinuity criterion of a deep beam loaded through bearing plate is a force discontinuity, while that of a coupling beam is a geometry discontinuity. So, before going further to establish a shear strength model for a coupling beam, we need to re-adjust the macro model and re-verify the findings obtained from previous analysis (deep beam with bearing plate). One way to do it is through an analysis of deep beam specimens loaded through column stub (Fig. 4.2c).

The boundary condition of a deep beam with column stub resembles that of a coupling beam. The D-region of both of these members is determined from their geometries.

Foster and Gilbert (1998) tested deep beam specimens loaded through a column stub and through bearing plates. They found out that the failure location for the majority of

deep beam specimens loaded through a bearing plate occurred right beneath the bearing plate. Meanwhile, the failure location of deep beam specimens loaded through a column stub occurred right near the corner between a column stub and a beam as illustrated in Fig. 4.11. Similar finding was also obtained by O’Malley (2011).

O’Malley (2011) conducted experimental and analytical studies of reinforced concrete pier caps as illustrated in Fig. 4.12. Before conducting the experimental tests, he performed a finite element analysis to evaluate the influence of boundary condition to the structural behavior of a pier cap. In model 1 (Fig. 4.13), the force from bridge column was represented as a uniform loading acting on a steel plate. The finite element analysis of this model showed that stress concentration would occur at the location beneath the steel plate. In model 2 (Fig. 4.14), the bridge column was represented by a column stub, upon which the force was acted. The finite element analysis gave a totally different conclusion, which is stress concentration occurred at the location where the geometry discontinuity existed, instead of directly beneath the column. The location of failure obtained from the finite element analysis was verified in the experimental analysis (Fig. 4.15).

The findings by Foster and Gilbert (1998) and O’Malley (2011) gave us hints that:

1. Previous finding indicating that the size of bearing plate affected the size of strut area is reasonable because the critical failure location occurred beneath the bearing plate.

2. The size of shear element and also the size of strut area for a deep beam with column stub should not be affected by the size of the column stub because the critical failure location occurred outside of the column stub.

Parametric analyses

In order to do a similar parametric analysis as that in section 4.1, as many as 36 specimens are collected as shown in Table 4.2. However, due to the limited database,

not all of these specimens failed in shear or crushing of the compression strut. So, unless indicated otherwise in column (2) of Table 4.2, the specimen was reported to fail in shear. Similar to the analysis of deep beam with bearing plate, these 36 deep beam specimens with column stub also go through nine analyses as presented in the following.

Detailed specimens’ parameters and calculations can be found in the Appendix B.

Due to the presence of column stub, the macro model of the original ACI 318 STM (Analysis 1) is slightly different. The typical macro model of strut-and-tie model for deep beam with column stub is shown in Fig. 4.16. So, modifications on the strut area at top Acs,top and strut inclination angle θ are adjusted as in Eqs. (4.12) and (4.13), respectively following the recommendations by Wight and Parra-Montesinos (2003) as follows:

where ap in deep beam with column stub represents the column stub dimension, upon which the actuator acted, in the direction of beam longitudinal axis (Figs. 4.16 and 4.17a). Meanwhile, the strut area at lower part is calculated using Eq. (2.19):

(

)

A

Acs,bot = cn,bot = tcosθ +Absinθ (2.19) The calculation results using original ACI 318-14 STM (Analysis 1) is presented in

column (9) of Table 4.2 and Table C.1. The shear strength ratios (Vtest Vcalc) are plotted in Fig. 4.17b with average value of 1.47 and a coefficient of variation (COV) of 0.20.

Similar to the treatment in deep beam with bearing plate, a series of parametric analyses with each analysis representing a macro model closer to the physical behavior of deep beam specimens is proposed. These analyses include the effect of geometry

discontinuity on the appropriate determination of the shear element (Analysis 2), the selection of the compression zone to represent the width of the strut (Analysis 3), the influence of the force spreading ratio on the dimension of the nodal zone (Analysis 4), and the selection of the probable failure mode (Analysis 5 and 6). In addition, this research also uses a sophisticated strain-compatible model, i.e. a softened strut-and-tie model to provide verifications that the same macro model used in the simple ACI 318 STM is also applicable to a rational model, as shown in Analysis 7 to 9.

Consideration of geometry discontinuity (Analysis 2)

In the beginning of Section 4.2, it has been shown that the location of failure occurred outside of the column stub. It implies that the size of column stub should not affect the shear strength of the deep beam. Consequently, the size of the shear element represented by θ is redefined as shown in Eq. 4.14. Similarly, the strut area at the upper part is modified according to Eq. (4.14).

Meanwhile, although the discontinuity at the lower part is also defined by the geometry discontinuity, however, the stress concentration is less severe. So, the strut area at the lower part remains unchanged (Eq. 2.19):

(

)

A

Acs,bot = cn,bot = tcosθ +Absinθ (2.19) The illustration of the macro model is given in Fig. 4.18a. The strength ratios of each

specimen are given in column (10) of Table 4.2 and Table B.3. The average shear strength ratio of the collected 36 specimens increases up to 3.14 with a COV of 0.36 (Fig. 4.18b). This result is too conservative and is argued due to the inappropriate choosing of width of horizontal compressive strut.

Consideration of the elastic compression zone (Analysis 3)

In Analysis 3, the width of the horizontal strut is taken as the elastic compression zone kd as given in Eq. (4.2). Following this modification, the macro model of the STM is

re-defined as shown in Fig. 4.18c and the strut inclination angle changes to:

1 3

tan _b 2

d kd

θ = ^{− ⎛}⎜⎝a′ +⁻A ^⎞⎟⎠ (4.16)

While the equation of the strut area at the lower part remains unchanged (Eq. 2.19), the strut area at the upper part is modified, as given in Eq. (4.17):

(

)

A_cs,top = cosθ (4.17)

The average shear strength ratio improved significantly to 1.41, with a COV of 0.34 as shown in Fig. 4.18d and column (11) of Table 4.2 and Table B.4. This result shows that although the term kd implies to a stress condition of the occurrence of the first yielding of flexural reinforcement, however it is a good approximation as the depth of a compression zone for a deep beam specimens.

Consideration of the effect of a 1:2 force spreading (Analysis 4)

The main underlying concept for the redefinition of the macro model of the strut-and-tie in Analysis 4 is the consideration of the load spreading ratio of 1:2 as illustrated in Fig.

4.18e. In the analysis for deep beam with bearing plate, this load spreading ratio is considered to be a good approximation. Correspondingly, a larger area of concrete below the loading plate participates in resisting the concrete crushing. The strut area is then defined accordingly:

Similarly, the strut area at the lower part is:

( ) ( )

A =b w⎡ − −h d ⎤ θ+b^⎡ h d− +A ^⎤ θ

The strength ratios are given in column (12) of Table 4.2 and Table B.5, with an average value of 1.19 and a COV of 0.16 as indicated in Fig. 4.18f. We can notice that majority of the predicted failure modes shifts to failure at the bottom part (failure mode 2).

Selection of the most probable failure mode (Analysis 5)

By neglecting the failure at bottom part, the shear strength prediction of Analysis 5 shows an improvement in the accuracy. The average strength ratio is 1.11 with a COV of 0.15, as shown in Fig. 4.19 and column (13) of Table 4.2 and Table B.6.

Consideration of nominal flexural strength (Analysis 6)

By exchanging the yielding strength of tie with the flexural strength, the average shear strength ratio of 36 specimens using Analysis 6 shows increases to 1.18 with a COV of 0.14 (Fig. 4.20 and column (14) of Table 4.2 and Table B.6).

Consideration of a strain-compatible rational model (Analysis 7)

The average shear strength ratio obtained using SST general approach shows similar accuracy with that using ACI 318 STM. As shown in Fig. 4.21 and column (15) of Table 4.2 and Table B.7, the average shear strength ratio of Analysis 7 is 1.22 with a COV of 0.12.

Consideration of a simplified SST procedure (Analysis 8)

Using the simplified SST solution procedure, the results of Analysis 8 show similar accuracy with that of Analysis 7 which uses general approach. The average strength ratio is 1.20 and a COV of 0.12 (Fig. 4.22 and column (16) of Table 4.2 and Table B.8).

This comparison again supports the finding that proper parameter modeling plays a more important role than the use of a robust solution procedure.

Consideration of a simpler definition for strut area (Analysis 9)

In the previous analyses, the strut area was determined following the ACI 318’s concept that it should be perpendicular to the strut inclination angle. In Analysis 9, a simpler

b kd

A_cs,top = × (4.20)

With a simpler definition of strut area, the average strength ratio becomes 1.16 with a COV of 0.12, as shown in Fig. 4.23 and column (17) of Table 4.2 and Table B.9.

Summary on major findings for deep beam specimens with column stub

A series of parametric studies were evaluated to determine the most suitable macro model to reflect the shear strength behavior of a deep beam. Using data collected from 36 deep beam specimens, this study indicates that:

1. The size of the shear element and the macro model of deep beam specimens with column stub should consider the effect of the geometry discontinuity. This analysis also showed that the size of the column stub does not affect the important structural parameters.

2. The most important structural parameters identified are the definition of a shear element that is consistent with the geometry discontinuity, the consideration of the elastic behavior in defining the width of a strut, the effect of a steel loading plate on the spreading of the force, and the proper selection of the probable failure modes.

3. Failure at lower part is less likely to occur because the stress can be easily widespread through the steel bar detailing.

4. The use of a simple definition of strut area as shown in Eq. (4.20) resulted in reasonable strength accuracy.

5. Using the same macro model, the simple and straightforward solution procedure of ACI 318-14 STM yields agreeable accuracy compared with the sophisticated and rigorous procedure of a strain-compatible model, such as the softened strut-and-tie model.

4.3 Lessons learned and extension to the development of analytical model for