CHAPTER I INTRODUCTION
1.3 Organization of thesis
This thesis is divided into six chapters. Chapter 1 outlines the research background, motivation, and objectives for this research. A more detailed study of available experimental study of reinforced concrete coupling beams is described in Chapter 2. In addition, Chapter 2 also illustrates the available analytical model for strength prediction.
Chapter 3 discusses the major finding of the five year experimental study of coupling beam conducted in NTU. Complete test matrix along with the test parameters and test results is given, but in-depth discussions are provided to key specimens only. These key specimens are grouped based on their test objectives. At the end of each group, based on the test parameters, comparisons between the test specimens and benchmark specimens are given. The last section of Chapter 3 summarizes the major findings during this five year research project and their extensions in developing analytical models.
Deep beam and coupling beam elements are parts of structural elements subjected to very high shear. The most prominent difference between these two is the loading condition, where the first is subjected to monotonic loading while the latter is subjected to cyclic loading. Hence, before developing the shear strength model for a coupling beam, the first part of Chapter 4 deals with a parametric study on the shear strength model for a deep beam database to identify the most important parameters affecting the accuracy of a strength model. Later on, based on the test observation and the parametric study, an analytical model for shear strength prediction of a coupling beam is developed
in the second part of Chapter 4. The accuracy of the proposed model is then gauged with available database of coupling beam specimens.
Chapter 5 extends the analytical model into the design recommendation for engineers.
This design flowchart is used to determine the appropriate amount of diagonal bars when designing a coupling beam. A series of parametric study was also carried out to evaluate the proposed design procedures. Finally, Chapter 6 provides the major conclusions of these series of study.
CHAPTER 2
LITERATURE REVIEW
The literature study is separated into three sections. The first section reviews the experimental work of coupling beams and its related issues. The second and third sections focus on the available analytical tools for strength prediction of a beam. The analytical tools consists not only the ones adopted in ACI 318-14 (2014) code provision, but also a softened strut-and-tie (SST) model. The softened strut-and-tie model is considered as a rational model satisfying force equilibrium, strain compatibility, and constitutive equations.
2.1 Experimental research of coupling beam
The first research of coupling beam, to the author’s knowledge, was conducted in the University of Canterbury (Paulay 1969). As many as 12 specimens was detailed using conventional beam layout with major test parameters being clear span-to-depth ratio (An h) of 1.02, 1.29, and 2.0, amount of vertical shear reinforcement (0.41% - 2.52%), and type of loading (monotonic or cyclic loading) as shown in Fig. 2.1. Test results indicated that most of these beams were subjected to axial elongation, causing a reduced moment arm and ineffectiveness of compression reinforcement. Also, most of these specimens were unable to reach their theoretical ultimate moment capacity due to the inadequacy of their shear capacity.
To solve this issue, Paulay and Binney (1974) published test results comparing coupling beams with conventional layout (An h=1.02)and diagonal layout (An h=1.29)as shown previously in Fig. 1.5. From the hysteretic loops, it was clear that the seismic behavior of coupling beam with diagonal reinforcement was much better compared to
that with conventional layout. Similar results were also concluded by Barney et al.
(1980), Tassios et al. (1996), and Galano and Vignoli (2000) as discussed below.
In 1980, researchers in Portland Cement Association (PCA) conducted eight coupling beam specimens (Barney et al. 1980) as shown in Fig. 2.2. The first six specimens (C1 to C6) had An h=2.50 and the last two had An h=5.0. In relation with the behavior of beam with short clear span-to-depth ratio, they compared the hysteretic loops of specimen C5 and C6 (Figs. 2.3 and 2.4). The comparisons showed that the seismic behavior of coupling beam with diagonal layout (C6) was much better than that with conventional layout (C5). Similar conclusions were also obtained by comparing specimens CB-1A and CB-2A with An h=1.0 (Tassios et al. 1996) as shown in Fig.
2.5 and specimens P02 and P07, both with An h=1.50 (Galano and Vignoli 2000) as show in Fig. 2.6. The hysteretic loops of CB-1A and CB-2A (Fig. 2.7) and P02 and P07 (Fig. 2.8) indicated that specimens with diagonal layout possessed better deformation capacity and energy absorption.
Following the advantages of having diagonal reinforcement within a coupling beam with low span-to-depth ratio, several detailing alternatives were also proposed. Within the eight specimens tested by Barney et al. (1980), three specimens (C1, C3, and C4) were detailed by having diagonal bars only in the hinge region (Fig. 2.9). Their test results showed that their original intention of putting diagonal reinforcement within the hinge region to eliminate the sliding shear was ineffective as shown by the hysteretic loops shown in Fig. 2.10. The main reason for this unsatisfactory improvement was because of the detailing failure at the location of bar bending (Fig. 2.11). This kind of detailing was also criticized to have improved the complexity in the steel assemblage.
Tegos and Penelis (1988) introduced the rhombic layout (Fig. 2.12) and followed by Galano and Vignoli (2000) in Fig. 2.13. They reported a satisfactory seismic behavior of
coupling beam with rhombic layout as shown in the hysteretic loops (Figs 2.14 and 2.15).
Naish et al. (2013a) tested specimens FB33 and CB33F with An h=3.33 as shown in Fig. 2.16. Their test results showed diagonally reinforced specimen (CB33F) performed much better compared to traditionally reinforced specimen (FB33), but FB33 still met the standard as shown in Fig. 2.17. However, the presence of diagonal bar affected significantly the overall energy dissipation. It is also noteworthy that the acting shear stress for FB33 was quite low (<0.33 f MPac′
( )
).The difference in the deformation capacity for a specimen with diagonal layout and a specimen with traditional layout becomes less significant as the clear span-to-depth ratio (An h) increases. Specimens C7 and C8 (Fig. 2.18) with An h=5.0 by Barney et al.
(1980) showed little improvement on the deformation capacity of a coupling beam detailed with diagonal layout as compared to beam detailed with conventional layout (Fig. 2.19).
Although the use of diagonal reinforcement layout has been proven to bring beneficial effect on the overall seismic behavior of a coupling beam, but its constructability has been a major concern. Harries et al. (2005) claimed that although diagonally reinforced coupling beam provides good seismic behavior, but it was simply impossible to design a practically constructible one when the acting shear stress approaches the maximum ACI 318 stress limit, i.e. 0.83 f MPac′( ). Meanwhile, Canbolat et al. (2005) and Naish et al.
(2013a) were concern about the constructability of confinement reinforcement on the groups of diagonal bars.
As the advancement of material science, new material was developed to enhance the tensile capacity of concrete. Canbolat et al. (2005) introduced the use of
high-reinforcement detailing of a coupling beam with diagonal bars. This HPFRCC improved not only the tensile capacity of concrete which leads to improved deformability, but also provided confinement effect. They also suggested bending the diagonal bars once they are within the mass concrete block. Figure 2.20 shows specimen 1 (with ordinary concrete) and specimen 4 (with HPFRCC). The hysteretic loops of these two specimens showed comparable seismic behavior (Fig. 2.21).
The use of HPFRCC was further investigated by Lequesne et al. (2010). They claimed that due to the improvement of tensile capacity of concrete, hence the required amount of diagonal bars can be reduced to 30% - 40%. Although no quantification method was proposed, this concept leads to an alternative detailing for a coupling beam with a partial amount of diagonal reinforcement.
Since 2008, the ACI 318 committee introduced two alternatives for the confinement detailing. The first alternative was the one in which confinement was provided for the diagonal bar groups only to prevent bar buckling, and the second alternative is to provide confinement on the entire beam section. In 2009, researchers in University of California at Los Angeles (UCLA) conducted the first test to evaluate these two alternatives (Naish et al. 2013a). Specimen CB33D (Fig. 2.22) only confined the diagonal bar groups, while specimen CB33F (Fig. 2.16b) confined the entire beam section. They found that beams with confinement on the entire section (CB33F) had slightly better seismic behavior compared to those with confinement on the diagonal bar groups (CB33D) as illustrated in Fig. 2.17b.
2.2 ACI 318-14 (2014) strengths formulation
In the design process, engineers must provide an adequate strength to a coupling beam when resisting the acting force. In general, two types of strength are of concern when designing a coupling beam, i.e.: flexural strength and shear strength. This section mainly introduces the ACI 318-14 (2014) approaches in calculating the flexural strength and shear strength of a flexural member. The flexural strength theory in ACI 318-14 which adopted the Bernoulli’s strain compatibility and the shear strength theory which adopted incomplete truss analogy are usually applicable for beams other than deep beams. Especially for coupling beams with diagonal reinforcement, the ACI 318-14 code also provides a specific equation to estimate its strength.
For deep beams, the ACI 318-14 recommends the strut-and-tie model (STM) provided in Chapter 23. However, the ACI 318-14 STM derived following the lower bound theory only requires that the force equilibrium is satisfied and neglects the strain compatibility and the material’s stress-strain relationship.
2.2.1 Flexural strength formulation based on ACI 318-14
The flexural strength of a beam can be calculated based on the well-developed flexural analysis. This flexural analysis was developed based on the Navier’s three principles in which force equilibrium, strain compatibility, and constitutive laws of material are satisfied. In many cases, except for deep member, the strain compatibility can be assumed to follow Bernoulli’s plane section remains plane before and after bending.
Given a beam section (Fig. 2.23) and by using the flexural analysis, one can develop a moment-curvature (M −φ) diagram based on the solution procedure illustrated in Fig.
2.24.
A beam section with beam width b, beam depth h, effective beam depth d calculated A
distance d ′from extreme compression fiber to centroid of compression reinforcement A′s are given for a flexural analysis. For each calculation point in M −φdiagram, one can firstly assume the extreme compression fiber of concrete εcand an arbitrary value for depth of compression zone c. Then, by assuming Bernoulli’s strain compatibility, the strain in compression reinforcement εs′and tensile reinforcement ε can be found s using similar triangle principle. Next, by using a given constitutive law for concrete and mild steel, the previously calculated strains can be transformed into stresses ( fc, f ′s , and fs). At this point, by iterating the depth of the compression zone c, equilibrium in the horizontal forces must be sought as given in Eq. (2.1):
=0
where Cc is the compressive force of concrete, C′s is the force of compression reinforcement, and T is the force of tension reinforcement, each obtained by multiplying the respective stress with the area. Finally, the moment and curvature can be calculated using Eqs. (2.2) and (2.3), respectively:
(
d y)
C(
d d)
where y is the centroid of concrete compressive force measured from the extreme compressive fiber of concrete.
However, when evaluating the flexural strength, one simply needs to calculate the nominal flexural strength Mn as illustrated in Fig. 2.25. The ACI 318-14 specifies that the Mn is calculated when the extreme compression fiber ε equals 0.003 and the c concrete compressive stress can be simplified using Whitney’s stress block. Using the Whitney’s stress block, the average concrete stress is taken as 0.85f ′c with depth of
block a equalsβ , where 1c 0.65≤β1 =0.85−
(
fc′−28)
7×0.05, where f ′ is in MPa. In c this way, the nominal moment capacity Mn can be calculated using Eq. (2.4):(
d d)
2.2.2 Shear strength formulation based on ACI 318-14
In general, a structural member can be divided into two regions, i.e.: B-region and D-region. A B-region is a region where the stress distribution is more uniform and the assumption of Bernoulli’s principal can be applied. On the other hand, D-region is the region where some stress concentration may exist. The D-region may be caused due to a stress concentration (such as a point load) or abrupt change in geometry. In ACI 318-14, the D-region can be determined as one beam depth h from the source of discontinuity.
Figure 2.26 illustrates the determination of B- and D-regions for a simply supported beam subjected to a point load at the midspan, while Fig. 2.27 illustrated the B- and D-regions for a beam with fixed ends subjected to double curvature bending.
Since the stress distribution between these two regions is different, their shear strength mechanisms are also different. In ACI 318-14, the shear strength of B-region can be estimated using the incomplete truss analogy, while, that of D-region can be estimated using Strut-and-Tie model (STM). Then, the shear strength of a beam can be theoretically defined as the minimum of these two strengths.
Shear strength of B-region based on ACI 318-14
The ACI 318 shear strength provision for beam in B-region adopts the incomplete truss analogy. The incomplete truss analogy acknowledges the shear contributed by concrete Vc and shear contributed by vertical shear reinforcement Vs as given in Eq. (2.5):
s c
n V V
V = + (2.5)
The shear strength contributed by concrete was determined empirically and can be calculated using Eq. (2.6):
1 ( )
c 6 c
V = f MPa bd′ (2.6)
Meanwhile, the shear strength contributed by vertical shear reinforcement is estimated using truss analogy by assuming a 45o crack as given in Eq. (2.7):
s d f
Vs = Avt y (2.7)
where Avt is the area of vertical shear reinforcement within spacing s, and s is the center-to-center spacing of vertical shear reinforcement along beam axis.
Shear strength of D-region based on ACI 318-14 Strut-and-Tie Model
The solution procedure of ACI 318 STM can be presented in a simple form, as shown in Fig. 2.28. Once all of the beam dimensions, reinforcement detailing, material properties, and testing parameters are known, one may develop any macro model of a strut-and-tie that satisfies force equilibrium. Then, based on the built macro model, the shear strength of a deep beam (Vn) is determined as being the smallest of the following: the strengths of the diagonal strut at the top (V1) and at the bottom (V2), strengths of the nodal zone at
The strengths of the struts at the top and the bottom are determined from Eq. (2.9) and (2.10), respectively:
(
0.85β)
sinθV1 = sf ′c Acs,top (2.9)
(
0.85β)
sinθV2 = sf ′c Acs,bot (2.10)
Where βs is the strut efficiency factor, Acs,top and Acs,botare strut area at top and bottom, and θ is the inclination angle of strut.
The strut efficiency factor βsis calculated based on the amount of provided vertical and horizontal shear reinforcements as given in Eq. (2.11) and shown in Fig. 2.29:
003
If the provided vertical and shear reinforcement satisfies Eq. (2.11) and the concrete compressive strength fc′<42MPa, the βscan be taken as 0.75 and if not satisfying the criteria, the βsis taken as 0.6. For other cases, the βs is only allowed to be 0.4.
Meanwhile, the strut areas at top and at bottom as well as the inclination angle of strut are determined following the developed macro model.
The strengths of the nodal zone at the upper part (V3) and the lower part (V4) are determined using Eqs. (2.12) and (2.13):
(
0.85β)
sinθV3= nf ′c Acs,top (2.12)
(
0.85β)
sinθV4 = nf ′c Acs,bot (2.13)
where the efficiency factor of the nodal zone βn for the CCC node (top node) is 1.0 and CCT node (bottom node) is 0.8, respectively.
Finally, the strength of the tension tie (V5) is taken as the yielding strength of flexural reinforcement as shown in Eq. (2.14):
θ tan
V5 =Astfy (2.14)
where Ast is the total area of tension reinforcement.
Macro Model for ACI 318 STM
One of the challenges of using strut-and-tie model is how to choose an appropriate force
transferred from the loading point (actuator) to the support. One may use a direct STM, in which the load is transferred directly from the loading plate to the reaction plate or other truss models to consider additional load paths due to the presence of vertical stirrups. The simplest STM, also adopted in this paper, uses a direct transfer mechanism in which the force from the loading actuator is directly transferred to the support reaction (Fig. 2.28). Brown and Bayrak (2008) also concluded that the direct transfer mechanism might be considered as an appropriate mechanism, especially for deep beams with a shear span to depth ratio that is less than 2.
After determining the load path, one needs to define the macro model of the strut-and-tie, which includes the determination of the width of the horizontal strut ws, the width of the horizontal tie wt, and the inclination angle of the diagonal strut θ . The aforementioned width of the horizontal strut represents the depth of the compression zone at the constant bending moment region. According to Tjhin and Kuchma (2002), the depth of the compression zone is taken as the plastic compression zone shown in Eq.
(2.15):
where Astfy is the yield strength of the main flexural reinforcement. Additionally, the width of the horizontal tie is calculated by following the recommendation in ACI 318 Appendix A as shown in Eq. (2.16):
where β in this case is the efficiency factor of the nodal zone taken at the lower part n
(CCT node).
The inclination angle relative to the horizontal axis θ of this concrete strut is given in Eq.
where jd is the force lever arm, a is the shear span, A is the width of the bearing plate, b
ap is the width of the loading plate, and a′ is the clear shear span.
Finally, after these three parameters are determined, the geometry of the nodal zone and the strut area at the top and bottom parts of the diagonal strut can be determined as illustrated in Fig. 2.30. The nodal zone areas (Acn,top;Acn,bot), which are the same as the strut areas (Acs,top;Acs,bot), are defined such that they are perpendicular to the inclination angle: 2.2.3 ACI 318 strength formulation for diagonally reinforced coupling beam
The shear strength for diagonally reinforced coupling beam for coupling beam mainly follows the recommendation by Paulay and Binney (1974). The nominal shear strength of a diagonally reinforced coupling beam is:
( )
2 sin 0.83
n vd y c
V = A f α ≤ f MPa bd′ (2.20)
where Avd is the area of one group of diagonal bars and α is the inclination angle between diagonal bar with respect to beam longitudinal axis as illustrated in Fig. 2.31.
The ACI 318 design concept for coupling beam relies solely on the resistance from diagonal bars as described in Section 2.3.
2.3 ACI 318-14 design procedure of a coupling beam
The ACI 318-14 design procedure of a coupling beam depends very much on the span-to-depth ratio (An h) and shear demand. Moehle et al. (2011) and Moehle (2015) summarized the ACI 318-14 design concept using a figure as shown in Fig. 2.32. Figure 2.32 categorized possible design alternatives for a coupling beam into six regions as follows:
a. Coupling beams with An h≥4.0
The use of diagonal reinforcement becomes inefficient due to the large span-to-depth ratio, which will consequently cause a smaller inclination angle of diagonal bars α . The ACI 318 suggests that traditional beam layout with main longitudinal reinforcement placed at top and bottom is a good design option. The design of this traditional beam layout follows mainly the requirement for beam in special moment resisting frame (SMRF). Barney et al. (1980) concluded that for beam with large span-to-depth ratio, traditionally reinforced coupling beam would give satisfactory seismic performance.
Figure 2.33 illustrates the design flowchart for this traditionally reinforced coupling beam. The amount of tension reinforcement Ast is determined satisfying the flexural design:
fMn Mu
φ ≥ (2.21)
where Mn is the nominal flexural strength calculated using the sectional analysis as described in Fig. 2.25, Mu is calculated from the structural analysis at the Design Based Earthquake (DBE) level, and φf =0.9 for flexure. During the Maximum Considered Earthquake (MCE), the beam is expected to undergo a plastic deformation level where the steel reinforcement may undergo strain-hardening stage. Hence, the plastic shear demand Vp is determined from the probable moment developed Mpr using:
In order to ensure that the flexural strength can be fully developed, a capacity design
In order to ensure that the flexural strength can be fully developed, a capacity design