Lessons learned and extension to the development of an analytical model…

affected by many factors. In general, the seismic behavior of a structural member depends very much on the interaction between flexural and shear strengths. While the flexural strength can be easily calculated using flexural analysis by assuming plane remains plane behavior, the shear strength of a coupling beam is rather complicated.

Therefore, before deriving analytical model to predict the shear strength of a coupling beam, major findings related to the important factors related to the structural behavior are summarized in the following:

1. Force transfer mechanism

Observations of crack patterns provided useful information to understand the force transfer mechanism in a structural member. Crack patterns of specimens with small clear span-to-depth ratio ( A_n h≤2.0 ) are different to those of specimens with intermediate/slender clear span-to-depth ratio (A_n h>2.0). For example, one may notice a clear single direct strut mechanism for specimens withA_n h=1.0 CB10-2 (Fig.

3.4a) and CB10-1 (Fig. 3.4b). But, for specimens CB30-2 and CB30-7 with A_n h=3.0, compression fans were observed at both beam ends (Figs. 3.4f and 3.4g).

2. Crushing of concrete indicated distinct drop of strength

Test observation strongly indicated that crushing of concrete is the major reason for the final failure of the majority of test specimens. This observation holds regardless of the failure mode experienced by each specimen, i.e.: shear, flexure-shear, or flexure failure.

For example: specimens failing from shear failure, such as CB10-2 (Fig. 3.4a) crushing

Similarly, for specimens CB30-14 and CB30-19 which failed from flexure-shear, shear behavior became dominant at DR of 4.4% and 4.8% and caused distinct drop in the hysteretic loop of CB30-14 (Fig. 3.30d) and CB30-19 (Fig. 3.30e), respectively. At those particular DRs, crack patterns also indicated crushing of concrete at beam ends as shown in Fig. 3.31d and Fig. 3.31e for CB30-14 and CB30-19, respectively.

3. Concrete participation in flexure and shear strength

Test results showed that concrete contributed to both flexural and shear strength of a specimen. For example: consider the test parameters of CB30-14 and CB30-16 which were governed by flexural strength as listed in Table 3.3. The ratio of total reinforcement for these two specimens ( ρ =2.5%) and transverse reinforcement _t (ρ =1.3%) were similar. However, only concrete strength was different, 52.4 MPa and _v 86.6 MPa for CB30-14 and CB30-16, respectively. The theoretical calculation of the nominal flexural strength (Mn) for these two specimens indicated that specimens with higher concrete strength had higher flexural strength than that with lower concrete strength. Test results (Table 3.5) also showed similar result: The Vmax of CB30-16 was 907.4 kN, higher than that of CB30-14, 815.4 kN.

The participation of concrete in the shear strength can also be clearly observed from specimen CB30-7 which was detailed using traditional layout. At large drift ratio, it was the concrete which contributed to the shear strength. The contribution of concrete to the shear strength can also be identified from crack patterns. Although cracks became severe at large drift ratio, it still contributed to the shear strength as long as sufficient confinement and internal support are provided.

4. Longitudinal reinforcement as flexural reinforcement

Strain gage measurement of longitudinal reinforcement located at top and bottom of the beam for majority of the specimens showed that they yielded and showed in the

hysteretic loops. Of course, it should be kept in mind that for all specimens tested in NTU, the longitudinal reinforcement was well anchored to the foundation block, different from the ACI 318-14 provision that does not allow development of yielding strength.

5. Diagonal reinforcement as both flexural and shear reinforcement

Strain gage measurement of diagonal bars also indicated yielding of diagonal reinforcement. For example, take specimens CB30-16 and CB30-17 as a comparison.

These two specimens had similar parameters, except for the yielding strength of diagonal bars (Table 3.3). The yield strengths of diagonal bar for CB30-16 and CB30-17 were 461 MPa and 723 MPa. Consequently, the calculated Mn and Vmax for CB30-17 were higher than those of CB30-16. The Mn of CB30-17 was 666 kN-m, while that of CB30-16 was 606 kN-m (Table 3.3). The Vtest of CB30-17 was 942.6 kN, while that of CB30-16 was 907.4 kN (Table 3.5).

The presence of diagonal bars also contributed to the total shear strength of a beam. It is generally accepted that the shear strength contributed by concrete deteriorates as the cracks become abundant. The loss of shear strength can be compensated by the presence of diagonal reinforcement. In addition, it was also observed during the test that the more amounts of diagonal reinforcement being proportioned, the better the structural behavior;

but the construction became more difficult. Therefore, providing a model to quantify a proper amount of diagonal reinforcement is of major interest.

6. Role of stirrups

During the experimental study, the role of stirrups was not of major concern since the requirement for confinement of the entire beam region is the controlling equation. The role of stirrups is discussed more in details when deriving a shear strength model as discussed in Section 4.3 and 4.4. In Section 4.3 and 4.4, three major roles of stirrups are:

(1) hold the integrity of concrete, especially at high displacement level; (2) as tension ties so that concrete sub-struts can be formed; and (3) to provide internal support for development of concrete strut, especially for beams with intermediate/slender span-to-depth ratio.

Concluding remarks

Based on the aforementioned description, an analytical model which can estimate the shear strength of a coupling beam is if interest. The shear strength model should be able to consider both the concrete and diagonal bar contribution. In addition, the model should also be able to reflect the concrete shear strength degradation phenomenon.

Hence, Chapter 4 focuses on the development of the analytical model for a coupling beam.

CHAPTER 4

ANALYTICAL FORMULATION FOR SHEAR STRENGTH

The observations from the experimental study suggested that in order to achieve good seismic behavior, it is very important to make sure that the shear strength of a coupling beam is adequate along its deformation history (at low and high displacement level).

Typically, the shear strength of a member degrades along with the deformation increment as the crack propagates and widens. Meanwhile, the strain hardening of tensile main reinforcement causes the increment of flexural response. The interaction between these two curves may result in three possible failure modes, which are: flexural failure, flexural-shear failure, and shear failure.

Figure 4.1a shows the desirable flexure failure response of a typical reinforced concrete element, including a coupling beam. In Fig. 4.1a, the flexural capacity developed fully because the shear capacity of the member is always kept larger than the flexural demand along the deformation history. However, when the shear capacity at the high displacement level is lower than the flexural requirement possibly developed due to strain hardening, a flexure-shear failure would occur as shown in Fig. 4.1b. Finally, shear failure would occur if the member can not even achieve the flexural strength at low displacement level due to the inadequacy of the shear strength (Fig. 4.1c).

The test observations also indicated that the crushing strength of diagonal concrete strut played significant role in contributing to the total shear strength of a coupling beam.

Hence, a semi-rational model based on strut-and-tie model is proposed in this chapter to estimate the shear strength at both low displacement level and high displacement level.

In order to develop a model for a coupling beam subjected to reversed cyclic loading, this study begins with a simpler case, which is beam subjected to static loading. A series

determining a shear strength model. In order to verify this analysis, only beams failed in shear failure, i.e.: deep beams are considered. Using the findings from the shear strength modeling of deep beam specimens, a semi-rational model is then developed for a coupling beam subjected to reversed cyclic loading.

Part I: Static Behavior

As previously mentioned, a strut-and-tie model is considered as an appropriate tool to derive a shear strength model. The strut-and-tie model (STM) of ACI 318 Code (ACI 318-14) which is based on the lower bound theory of plasticity, has served as a design and analytical tool for structural members, especially those with complicated flows of forces or deep members. This STM has been widely used by both engineers and researchers because the method provides a clear load path and is simple in terms of both the solution algorithm and the equations involved.

By contrast, several analytical models, which satisfy Navier’s three principles have also been proposed. These analytical models allow engineers to produce one unique solution that satisfies not only force equilibrium but also strain compatibility and a stress-strain relationship for cracked reinforced concrete. Two of the available models include a compatibility-based strut-and-tie that uses the secant stiffness formulation (Park and Kuchma 2007) and the softened strut-and-tie model (Hwang and Lee 1999 and 2000, Hwang et al. 2000). However, these analytical models require rigorous computational effort and may not be practical for engineering practice.

It is commonly assumed that sophisticated solution algorithms produce better accuracy when predicting the shear capacity of deep beam specimens compared to the simple ACI 318 strut-and-tie equations. This assumption may be a misconception because not all of the available algorithms use the same macro model. In this case, the macro model corresponds to the idealized visualization of the load path and the geometry of the struts and ties that reflect the major parameters influencing the physical behavior of a deep beam.

This study begins with the strut-and-tie model (STM) provision described in ACI 318 (2014), which gives a shear strength prediction that is too conservative when gauged

against a group of deep beam databases. A further investigation of this over conservative prediction suggests that it does not come from the simplicity of the solution algorithm, but rather from the improper modeling of the physical parameters inherent within the macro model. Therefore, the original macro model of ACI 318 is modified to include a structural behavior that represents the testing conditions and behavior of the deep beam specimens tested in the laboratory as well as with the proper justification of the most probable failure modes.

In this study, two groups of deep beam specimens with different loading conditions are used. The first group, which consists of a larger number of test specimens (118 specimens) is the deep beam specimens which testing conditions involved actuator acting upon a steel bearing plate as illustrated in Fig. 4.2a. In this group, the discontinuity is determined by the force discontinuity.

However, for the case of a coupling beam, its boundary condition is determined from the geometry discontinuity (Fig. 4.2b). Due to the major difference on the location of critical section, in order to simulate the similar boundary condition, the second group of deep beam specimens which consists of 36 specimens tested with actuator acting upon a column stub as shown in Fig. 4.2c is examined.

4.1 Deep beam with bearing plate

In order to verify the accuracy of the STM of ACI 318 as presented in Fig. 2.28 (Analysis 1), a database of deep beam specimens with bearing plate that failed in shear was collected from literatures. The specimens used in this study were collected from the available literature, ensuring that the complete information of the test setup is provided.

The databank covers a wide range of concrete compressive strengths f ′c with different layouts of reinforcements. The first eight columns of Table 4.1 present general information of each deep beam. Meanwhile, Table A.1 of Appendix A shows a more

detailed information of each one of them. Similarly, columns (9) to (17) of Table 4.1 simply show the strength ratios obtained from each analysis. Detailed calculations are shown in Tables A.2 to A.9 of Appendix A.

Using the macro model illustrated in Fig. 2.28 and repeated in Fig. 4.3a, the calculation results using original ACI 318-11 STM (Analysis 1) is presented in column (9) of Table 4.1. The shear strength ratios (Vtest Vcalc) are plotted in Fig. 4.3, with each specimen represented by a number to indicate the predicted failure mode. The STM defined using the ACI 318 parameters predicts that the majority of specimens would fail in the upper part of the strut (failure mode 1). The ACI 318 STM also provides a very conservative and scattered strength prediction as indicated by its average value of 1.54 and a coefficient of variation (COV) of 0.41. Detailed calculations can be found in Table A.2.

The authors argue that the main reason for this over conservative prediction is the fact that the current ACI 318 macro model does not reflect the physical behavior of deep beam specimens tested in the laboratory.

In the following section, a series of parametric analyses with each analysis representing a macro model closer to the physical behavior of deep beam specimens is performed.

These analyses include the effect of force 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 steel loading plate on the dimension of the nodal zone (Analysis 4), and the selection of the probable failure mode (Analysis 5 and 6). In addition, using the same macro model as that used in the latest analysis (Analysis 6), this research also verifies the applicability of a sophisticated strain-compatible model, i.e. a softened strut-and-tie model as shown in Analysis 7 to 9.

Consideration of the force discontinuity (Analysis 2)

Because perfect bonding does not exist between the steel plate and the concrete surface, ACI 318-14 indicates that the D-region of the deep beam specimens tested using two-point loading is bounded by the discontinuity of force. In the original ACI 318 model (Analysis 1), the geometry of the nodal zone at the upper part of the diagonal strut is determined using the entire width of the loading plate ap (Eq. 2.18). However, taking the entire width of the loading plate may not be appropriate.

Because the inclined force diagonal transfers the force, the shear strength should be determined by the nodal/strut properties associated with the inclined force diagonal.

Considering that the actuator is applied at the middle of the steel plate at the upper part (Fig. 4.4a), half of the plate’s width (a_p 2) to the left of the actuator belongs to the constant moment (zero shear) region. In a constant moment region, only horizontal concrete stresses can be found and no shear stress exists. Therefore, the left half of the bearing plate should have no influence on the nodal/strut dimension. Only the other half of plate to the right of the actuator, which is located within the shear transfer region, is considered to be effective as the strut area. Consequently, the area of the upper strut is adjusted to:

Meanwhile, at the lower node, where the moment is zero, the stress is consequently more uniformly distributed. Therefore, the width of strut area at the support region remains unchanged (Eq. 2.19):

(

w

)

b

A

Acs,bot = cn,bot = tcosθ +Absinθ (2.19) Analysis 2 predicts that all specimens failed due to the crushing of the upper strut and

indicates that its accuracy is reduced compared to Analysis 1, as indicated in Fig. 4.4b

and column (10) of Table 4.1. For the majority of specimens tested by Kong et al.

(1970), the strength predictions are scattered and too conservative. The average shear strength ratio of the collected 118 specimens increases up to 2.12, with a COV of 0.56.

Detailed calculation of Analysis 2 is presented in Table A.3. This result suggests that the geometrical modeling of the strut area at the upper part still neglects some important considerations. In the two subsequent parametric studies (Analysis 3 and 4), the authors suggest two important factors should be considered when modeling the width of the strut and the dimensions of the nodal zone in the upper part. These factors are the consideration of the elastic behavior of a deep beam and the effect of a steel loading plate.

Consideration of the elastic compression zone (Analysis 3)

As mentioned before, Analysis 2 provides poor accuracy and scattered predictions of the shear strength. One of the possible explanations of this performance is the inappropriate use of plastic compression zone depth as the width of the horizontal strut ws. In all of the collected specimens, the beams were reinforced with a large amount of flexural reinforcement to ensure that the shear capacity is reached prior to flexural failure, as reported by the test results. In addition, a simple conventional flexural analysis indicates that the majority of collected specimens failed before reaching their nominal flexural strength Mn which is defined as the bending moment calculated when the strain in the extreme concrete compression fiber reaches 0.003. As a consequence, the use of plastic compression zone may underestimate the width of horizontal strut since the concrete in flexure is still primarily in the elastic range.

In Analysis 3, the width of the horizontal strut is taken as the elastic compression zone kd, as given by Eq. (4.2):

kd

w = (4.2)

where k is derived from a singly reinforced beam section: k =

( )

nρ ²+2nρ −nρ, n is the elastic modulus ratio of steel to concrete, and ρ is the tensile steel ratio.

A similar method for choosing the elastic compression zone kd has been used by Park and Kuchma (2007). Following this modification, the macro model of the STM is re-defined as shown in Fig. 4.4c and the strut inclination angle changes to:

⎟⎠

The strut area at the upper part is modified, as given by Eq. (4.4):

a b

The average shear strength ratio becomes 1.63, with a COV of 0.33, as shown in Fig.

4.4d and column (11) of Table 4.1. Detailed calculation can be found in Table A.4.

Although the average shear strength ratio is still too conservative, the consideration of the elastic compression zone to model the geometry of the strut area at the upper part slightly improves the accuracy of the strength prediction. The main reason for this over-conservatism can be argued as being the result of neglecting the beneficial effect of the loading plate. The presence of a steel loading plate should smooth out the spread of force and enlarge the dimensions of the nodal zone. This effect is considered in the following analysis (Analysis 4).

Consideration of the effect of a steel loading plate (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 how the load spreads throughout the steel plate into the beam specimen. As the load is applied from the actuator, a very high stress concentration occurs at the nodal zone beneath it. This stress concentration would spread throughout the steel plate to the beam specimen and reach a more uniform stress

at the middle of the diagonal strut. Hence, a proper redefinition of the nodal zone at the upper and lower parts of the beam with consideration of the spreading of the force may play an important role because failure may occur at these points.

In Analysis 4, the authors assume that the force spreads out with a ratio of 1:2 (vertical : horizontal). A similar ratio can also be obtained from the principle stress trajectory using finite element analysis for a deep beam performed by Cook and Mitchell (1988).

The strut area is defined at the location where the spreading of the vertical load reaches the resultant horizontal compressive force C, as illustrated in Fig. 4.4e. Correspondingly, a larger area of concrete below the loading plate participates in resisting the concrete crushing. The strut area is then defined accordingly:

θ

Similarly, the strut area at the lower part is:

( )

The accuracy of the shear strength prediction using Analysis 4 is found to be greatly

The accuracy of the shear strength prediction using Analysis 4 is found to be greatly

在文檔中 鋼筋混凝土剪力連接梁之剪力行為與設計 (頁 97-0)