Part I: Static Behavior
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 (ap 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)
bA
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 improved over the previous versions of the models, with an average value of 1.35 and a COV of 0.21, as indicated in Fig. 4.4f and Column (12) of Table 4.1. Detailed calculation can be found in Table A.5.
The failure mode prediction of Analysis 4 indicates that approximately one-third of the collected 118 specimens failed due to the crushing of concrete at the lower strut (failure mode 2). Examination of the details of these specimens indicates that the longitudinal flexural reinforcement bars were properly anchored to the steel plate (Kong et al. 1970 and Yang et al. 2003). With the presence of the anchorage plate, the stress can be more uniformly distributed towards the location of the anchorage plate. In reality, this
for example, for the last term of Eq. (4.6), Ab 2 could be further increased by considering a spreading of the reaction towards the anchorage plate and the corresponding widening of the strut area. However, for simplicity, the authors assume that the crushing of the diagonal strut at the lower part (V2 and V4) is less likely to govern the overall behavior of a deep beam specimen.
Selection of the most probable failure mode (Analysis 5)
Analysis 5 adopts the same macro model as the one described in Analysis 4. However, these models differ by the selection of the probable controlling failure mode for the majority of deep beam specimens. As mentioned before, the presence of a plate to anchor the longitudinal flexural reinforcement bars can help widen the strut area and spread out the stress at the lower node. In addition, the moment diagram also suggests a zero moment at the lower part (support reaction). This observation implies that the stress concentration at the lower part (CCT node) is relatively small compared to that at the upper part (CCC node) and is therefore less critical. Hence, in Analysis 5, failure modes 2 and 4 are excluded. The shear strength of a deep beam is then given by Eq.
(4.7):
{
1, 3, 5}
min
Vn = V V V (4.7)
The shear strength prediction of Analysis 5 shows an improvement in the accuracy. The average strength ratio is 1.29 with a COV of 0.19, as shown in Fig. 4.5 and column (13) of Table 4.1.
It should be noted that the definition of shear strength governed by the yielding of flexural reinforcement (V5) has changed from the original ACI 318’s definition. In the original ACI 318 model (Analysis 1), V5 is the shear of a deep beam corresponding to its nominal flexural strength. However, as the macro model (Fig. 4.4e) has been modified to suit the elastic concrete behavior, V5 reflects the shear corresponding to the
yielding flexural strength of a deep beam. Since it is more appropriate to represent the flexural capacity of a beam using its nominal flexural strength (Mn) rather than its yielding strength (V5), in the following analysis (Analysis 6), the authors consider the nominal flexural strength of a deep beam as one of the governing strengths.
Consideration of nominal flexural strength (Analysis 6)
In the original macro model (Fig. 4.3a), ACI STM uses the plastic compression zone as the depth of the compression zone. This macro model suggests that the shear corresponding to the yielding of flexural reinforcement (V5) is similar to the nominal flexural strength Mn of the beam. However, the use of an elastic compression zone in Analysis 4 suggests that the yielding strength of flexural reinforcement actually corresponds to the yielding moment My of a deep beam. Since the nominal flexural strength Mn, is more representative as one of the governing strengths for a reinforced concrete member, in Analysis 6, the definition of yielding of flexural reinforcement adopts the ACI 318 original intention; namely, the beam nominal flexural strength. The Vf is calculated using:
Vf =M an (4.8)
where Mn is the nominal flexural strength calculated when the outermost concrete compressive fiber reaches 0.003, and a is the shear span.
Therefore, the strength of a simply supported deep beam in Analysis 5 is determined by either the crushing of the strut and nodal zone at the upper part or the shear corresponding to the nominal flexural strength Vf as given by Eq. (4.9) :
{
1 3}
Vn =min V V V, , f (4.9)
The shear strength prediction of 118 specimens using Analysis 6 shows similar accuracy to the results from Analysis 5. The average strength ratio is 1.28 with a COV of 0.19
It is observed in Fig. 4.6 that there was quite a scattered strength prediction result for 29 specimens tested by Yang et al. (2003). The average strength ratio for these 29 specimens is 1.05 with a COV of 0.30. The main reason for this result is argued to be due to the absence of vertical and horizontal shear reinforcements within the deep beam.
The absence of these reinforcements could possibly cause premature failure of the specimen prior to the mature development of the strut-and-tie mechanism. Thus, with the exclusion of these 29 specimens, the average strength ratio given by Analysis 6 for the remaining 89 specimens is 1.32 with a COV of 0.14.
Consideration of a strain-compatible rational model (Analysis 7)
Using the macro model described in Fig. 4.4e, the softened strut-and-tie model (SST) as presented in section 2.4 is adopted in Analysis 7 to calculate the shear strength of the collected deep beam specimens. Since the SST solution algorithm only calculates the crushing strength of a strut at the most critical location, namely at the upper strut, the governing strength is determined by:
{ }
min ;
n SST f
V = V V (4.10)
The result, as shown in Fig. 4.7 and column (15) of Table 4.1, shows that the average shear strength ratio is 1.27 with a COV of 0.16. Table A.7 gives detailed calculations for Analysis 7.
Results of Analysis 6 and 7 suggest that the ACI 318 STM provisions to calculate the strength of a diagonal concrete strut provides similar accuracy compared to a rigorous SST procedure, as long as the appropriate physical parameters are considered into the macro model. A simple comparison indicated in Fig. 4.8 between empirically determined strut efficiency factors using ACI 318 with the softening coefficient suggests that the average ratio of these two factors is 1.12 with a COV as small as 0.10.
This finding is consistent with the report by Quintero-Febres et al. (2006), which
concludes that the strut efficiency factor recommended by ACI 318 is reasonable for normal strength concrete. In the ACI 318 STM, the role of vertical and horizontal reinforcements to restrain cracking is included in the strut efficiency factor β . s Meanwhile, vertical and horizontal reinforcements in the SST model not only provide crack restraining, but also provide additional load paths by including more concrete sub-struts through vertical and horizontal mechanisms.
The macro model proposed in this research is also able to appropriately model the presence of a wide loading plate used in specimens MT, MR, CT, and CT tested by Alcocer and Uribe (2008). Their calculation results using the original macro model of the SST provided an average shear strength ratio as low as 0.75 (Alcocer and Uribe 2008). However, by using the macro model proposed in this research (Fig. 4.4e), the contribution of the loading plate can be properly attributed. The average shear strength ratio for these four specimens becomes 0.92.
The solution procedure using SST general approach (Fig. 2.37) in Analysis 7 requires a rigorous computational effort. So, in the following analysis, the SST simplified approach (presented in section 2.4.2) is adopted.
Consideration of a simplified SST procedure (Analysis 8)
The accuracy and reasoning for the use of the macro model (Fig. 4.4e) developed in this thesis was justified already through the acceptable accuracy of the strength ratio regardless of the use of the ACI 318 STM (Analysis 6) or a sophisticated SST model (Analysis 7). When the robust SST solution procedure is simplified as in Analysis 8, the accuracy of the strength prediction remains similar, with an average strength ratio of 1.30 and a COV of 0.17 (Fig. 4.9 and column (16) of Table 4.1). Detailed calculations are presented in Table A.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 eight analysis (Analysis 1 – Analysis 8), the definition of the strut area was such that it was perpendicular to the inclination angle. For a practical reason, it may not be so convenient for engineering use. In the original paper of the SST general (Hwang and Lee 1999, 2000) and simplified approach (Hwang et al. 2002), the strut area may not need to be perpendicular to the inclination angle. So, in Analysis 9 the definition of strut area is redefined as in Eq. 4.11 and illustrated in Fig. 4.10a:
2
With a simpler definition of strut area, the accuracy of Analysis 9 is slightly more conservative than Analysis 8. The average strength ratio is 1.35 with a COV of 0.18, as shown in Fig. 4.10b and column (17) of Table 4.1. Detailed calculations are presented in Table A.9.
Summary on major findings for deep beam specimens with bearing plate
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 118 deep beam specimens, this study indicates that:
1. The macro model should reflect the structural behavior of a deep beam.
2. The most important structural parameters used to model a deep beam specimen tested in the laboratory are the definition of a shear element that is consistent with the force 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. The size of strut area is affected by the depth of the compression zone and size of the bearing plate.
4. Failure at lower part is less likely to occur because the stress can be easily widespread through the steel bar detailing.
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.
6. The use of a simple definition of strut area as shown in Eq. (4.11) also resulted in reasonable strength accuracy.