Basic criteria for development of a shear strength model

Based on the test results discussed in Chapter 3 and findings from the parametric analyses in Sections 4.1 and 4.2, there are four major observations which need to be considered in developing a macro model for a coupling beam. They are:

1. the force transfer mechanism,

2. the degradation of shear strength along with the deformation increment, 3. shear strength is contributed by both concrete and steel bars, and

4. the integrity of concrete.

The force transfer mechanism

Crack patterns observations in Chapter 3 have clearly shown that the force transfer mechanism is strongly affected by the clear span-to-depth ratio. In this study, two

distinct shear strength models are developed. They are shear strength model for coupling beam with A_n h≤2.0 (deep coupling beam) and A_n h>2.0 (intermediate/slender coupling beam).

For deep coupling beams (A_n h≤2.0), the crack pattern suggests formation of a diagonal strut extending from one end of the beam to the other end of beam (Figs. 4.24a and 4.24b). This strut formation is especially clear for CB10-1 (Fig. 4.24a). As an idealization, direct force transfer mechanism as illustrated in Fig. 4.24c was adopted.

Moreover, the applicability of the direct force mechanism is justified during the parametric analyses of deep beams as given in Section 4.3.

Meanwhile, for coupling beams with intermediate/slender clear span-to-depth ratio (A_n h>2.0), the force transfer mechanism is illustrated in Fig. 4.25. Crack pattern of CB30-7 (Fig. 4.25a) indicated that at the end region, compression fan is more dominant;

while in the middle span, truss mechanism is more dominant. So, the force idealization is illustrated as shown in Fig. 4.25b. In this study, this force transfer mechanism is called DBD mechanism. One important concept for this DBD mechanism is the role of hoops to provide internal support for the development of concrete strut. More comment of this force transfer mechanism is discussed in Section 4.6.

Shear strength degradation

In the seismic design philosophy, a full development of flexural behavior is a desired failure mechanism for a ductile structural element. To achieve this, one needs to make sure that shear strength is always larger than the flexural strength at any displacement level (capacity design). However, as illustrated in Fig. 4.26, the shear strength degrades along with the opening of cracks. Hence, a shear strength model which is capable of predicting the shear strength degradation is needed.

To predict this shear strength degradation curve is a difficult task and may require a robust computational effort. So, in this study, the shear strength degradation curve is simplified into two points, i.e. at low displacement and high displacement level as illustrated in Fig. 4.26. In the seismic design terminology, the low displacement level refers to the DBE (design based earthquake) level; while the high displacement level refers to the MCE (maximum considered earthquake) level. The low displacement level is defined as the point where the beam is allowed to develop its nominal flexural capacity Mn obtained using sectional analysis suggested by ACI 318-14. The shear corresponding to the nominal flexural capacity Vmn is calculated using double curvature bending:

n mn n

V M A

=2 (4.21)

Meanwhile, the high displacement level is defined as the point where the tension bars of the beam reach strain hardening. It is always important to remember that concrete, longitudinal, and diagonal bars participate in the flexural behavior.

If the shear strength of a beam is maintained larger than the flexural strength at both low and high displacement levels (Fig. 4.1a), the beam is expected to possess good seismic behavior (flexure failure). At the other extreme, if the shear strength at low displacement level is smaller than its flexural strength (Fig. 4.1c), the beam is expected to fail in shear. In between, the beam might develop its flexural behavior but its deformation capacity might be limited. The latter failure mechanism is recognized as flexure shear failure (Fig. 4.1b).

Components of shear strength

Similar to the calculation of the flexural strength, the shear strength of a coupling beam is contributed by both concrete and diagonal steel bars (Fig. 4.27). When calculating the

incorporated using sectional analysis. But, in order to develop a purely analytical shear strength model which considers both concrete and diagonal reinforcement bars may be too complicated and not practical for engineering practice. So, this study proposes a semi-analytical shear strength model. The contribution of concrete is calculated using a softened strut-and-tie (SST) model. Meanwhile, contribution of diagonal bars to the shear strength is justified from the observed test behavior and the strain gage readings.

The macro model of the SST is mostly based on the findings in the strength analysis of deep beams with column stub. However, some modifications are made to suit the physical behavior of a coupling beam as will be discussed in details in the following sections.

Integrity of concrete

Questions related to the applicability of strut-and-tie model, especially at the high displacement level were often received. Some people argued that at a high displacement level, cracks would have been so large that the integrity of concrete can not be maintained; hence the applicability of a strut-and-tie approach is invalid. However, based on the crack observations and strain gage readings of forty coupling beam specimens, majority of the core concrete of the beams were still remained in integrity (Chapter 3) and the average reading of strain gages indicated that the stirrups were in elastic range. The complete plotting of stirrups’ strain gage readings of the forty test specimens can be referred to their respective authors (Cheng 2010, Wang 2011, Chang 2012, Tsai 2013, and Lin 2014). But, for convenience, Fig. 4.28 gives plots of strain gage measurement for four representative specimens of deep coupling beams and Fig.

4.29 presents strain gage plotting for four representative intermediate/slender coupling beams.

Strain gage readings of deep coupling beam specimens in Fig. 4.28 indicated that majority of the stirrups were still in elastic, even at high displacement level. Similar observations could also be found for coupling beams with intermediate span-to-depth ratio, especially CB30-7 (conventional layout), CB30-19 (hybrid layout with normal strength steel for stirrups), and CB30-18 (hybrid layout with high strength steel for stirrups) as shown in Figs. 4.29c to 4.29e. This reading implied qualitatively that crack width formation was still within control and integrity of concrete is preserved. So, using SST model which was developed based on the strut-and-tie concept is a reasonable approach.

Among five specimens presented in Fig. 4.29, the ratios of stirrups ρ for CB30-1 and _v CB30-2 were the smallest, which were: 0.4% and 0.7%, respectively (Table 3.3).

Meanwhile, the ratios of stirrups ρ for CB30-7, CB30-19, and CB30-18 were all _v similar, which was 1.3% (Table 3.3). As a result, yielding of stirrups for CB30-1 and CB30-2 was more severe compared to the others as can be seen in Figs. 4.29a and 4.29b.

Nevertheless, core concrete was still maintained in CB30-1 and CB30-2.