Overview of solution procedure

The five-spring model of Fig. 3 is solved under increasing deflection Δ to obtain the complete shear force vs. deflec-tion response for the shear spans of deep beams. Since the deflection of the shear span is imposed, the only kinemat-ic unknown of the model is the elongation of the set of four parallel springs Δc (elongation of flexural spring Δt

equals Δ-Δc). Elongation Δcand shear force V are obtained by solving equilibrium equation Eq. (13). The solution of this equation for a given Δ is explained with the help of Fig. 5 prepared for a sample deep beam. The horizontal axis of the figure shows the full range of possible values of Δcfrom zero to Δ, normalized with respect to Δ. The verti-cal axis reproduces the shear forces in Eq. (13) obtained from Eqs. (14), (15), (17), (20) and (21). These forces are functions of the DOFs of the kinematic model Δc and εt,avg, where the latter DOF is obtained from the elonga-tion of the flexural spring Δtas shown in Fig. 3.

When Δcis zero, DOF εt,avghas a maximum and the deformed shape of the beam is described by the upper de-formation pattern in Fig. 2a only (pure flexural deforma-tions). Since εt,avgis a maximum, the tension in the bottom reinforcement T and the corresponding shear T(0.9d)/a al-so have a maximum, as is evident from the thick black line in Fig. 5. Contrary to this, the sum of the shear forces transferred across the critical crack ΣVi(thick red line) has a minimum as these forces depend mainly on DOF Δc. For example, the shear V_cicarried by aggregate interlock is ze-ro because the critical crack is open and slip at the crack is zero. The only non-zero shear component is V_ssince the stirrups are strained when Δc = 0. In the other limit case when Δc= Δ, DOF εt,avgis zero and the deformed shape of the beam is described only by the lower deformation pat-tern in Fig. 2a (pure shear deformations). In this case the thick black line is at zero whereas the shear resistance ΣVi

is either in the pre- or post-peak regime depending on the magnitude of Δ. Since the thick black and red lines repre-sent the two sides of equilibrium equation Eq. (13), the in-tersection of the lines corresponds to the solution of the

0 0.2 0.4 0.6 0.8 1

Fig. 5. Equilibrium of forces in five-spring model for an applied displace-ment Δ

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equation. The intersection is found iteratively by using the bisection method. The ordinate of the intersection point is the shear force V corresponding to the imposed deflection Δ. The bisection method is applied with increasing values of Δ to compute the complete V-Δ response of the mem-ber.

4 Comparisons of predicted and measured behaviour 4.1 Specimens S1M and S1C

The five-spring model is used to predict the behaviour of two deep beam specimens, S1M and S1C, tested to failure at the University of Toronto [19]. These beams were sim-ply supported and loaded with a point load in the middle of the span. The only difference between the two tests was that specimen S1M was loaded monotonically whereas S1C was subjected to reversed cyclic loading. The effective depth of the section was d= 1095 mm (h = 1200 mm) and the shear-span-to-depth ratio a/d was 1.55. The beams had symmetrical top and bottom longitudinal reinforcement with a ratio ρl = 0.70  % and stirrups with a ratio ρv = 0.10 %. The compressive strength of the concrete on the day of beam testing was f_c= 33.0 MPa. Table 1 summarizes the properties of the specimens as well as the measured and predicted failure loads. Both specimens failed in shear along critical diagonal cracks prior to yielding of the flex-ural reinforcement. As the beams were symmetrical, the five-spring model is used to model one-half of the beams (one shear span).

The upper plot in Fig. 6 shows the measured and pre-dicted responses of specimens S1M and S1C. The hori-zontal axis is the midspan deflection of the beams, the ver-tical axis is the shear force. It can be seen that the two green curves, which represent the envelopes of the mea-sured responses, almost overlap. This shows that the load reversals applied to specimen S1C did not cause signifi-cant strength or stiffness degradation. It can also be seen that the red prediction curve matches the experimental curves well in both the pre-peak and post-peak regimes.

The prediction curve consists of a non-linear part pro-duced by the five-spring model and a tri-linear curve for the initial response. The latter curve models the behaviour prior to the development of the deformation patterns as-sumed in the kinematic model. The first point along this curve corresponds to the flexural cracking at the section with maximum moment (shear V_cr,fl). This point is ob-tained based on flexural beam theory by using trans-formed sectional properties. The cracking is assumed to occur when the stress in the concrete at the bottom of the section reaches 0.63√⎯fc [6]. The flexural cracking is fol-lowed by the propagation of the first radial flexure-shear crack at a distance s_crfrom the midspan section along the bottom reinforcement, see Fig. 2a. Further load incre-ments result in the propagation of more radial cracks away from the flexural crack until the critical diagonal crack forms under a shear force V_cr,sh. This load level cor-responds to the second point in the initial tri-linear re-sponse. The radial cracks and the critical crack initiate at the bottom of the section in the zone influenced by the flexural reinforcement. In beams without web reinforce-ment, these cracks propagate almost instantaneously to the vicinity of the load. Therefore, it is assumed that the

shear at diagonal cracking V_cr,sh is proportional to the cracking force N_crof the zone influenced by the bottom reinforcement:

(22)

where

(23) The factor of 1.5 in Eq. (22) accounts approximately for the load increase between the occurrence of the first radi-al crack and the propagation of the criticradi-al diagonradi-al crack.

The deflection Δcr,sh under shear V_cr,sh is obtained from Eq. (12) of the kinematic model by taking Δc= 0 and as-suming a shorter cracked length along the bottom rein-forcement:

(24)

The last point from the initial tri-linear response corre-sponds to the breakdown of beam action and the transi-tion to the deformatransi-tion patterns assumed in the five-spring model. This point is defined by the sectional shear capacity V_sect obtained from the Canadian code or fib Model Code 2010 [4], [13]. If V_sectis larger than the peak resistance obtained from the five-spring model, the beam is considered slender and the five-spring model is not ap-plicable. It can be seen from the plot that the tri-linear curve matches the initial response of specimens S1M/S1C well.

The upper plot in Fig. 6 also shows the predicted shear resistance mechanisms and how they vary with in-creasing deflection. The main contribution to the shear re-sistance is provided by the critical loading zone (V_CLZ) whose failure is predicted to trigger the shear failure of the beam. This result is consistent with the main assumption underlying the 2PKT: the peak shear response of the mem-ber coincides with the peak response of the CLZ. Signifi-cant shear resistance is also provided by the stirrups (shear V_s), which are predicted to yield at a deflection of about 3.5 mm. The aggregate interlock mechanism V_ci reaches its maximum when the member is in the post-peak regime, and vanishes at a deflection of about 14.5 mm. At this deflection the critical crack is very wide (w≈ 10 mm) and thus the crack surfaces are not in contact any more, regardless of the large slip at the crack (s≈ 7 mm). Finally, the dowel action shear V_d is predicted to have an effect mainly on the post-peak behaviour of beams S1M and S1C.

The lower plot in Fig. 6 shows the evolution of the deflections in specimens S1M and S1C associated with the two deformation patterns of the kinematic model. Ra-tio Δt/Δ shows the portion of the deflection due to the flexural deformation pattern, whereas Δc/Δ is the portion due to the shear pattern. It can be seen from the predic-tion line that, initially, the deformapredic-tions are mostly flexur-al. The shear deformations begin to develop rapidly just prior to shear failure, and at the end of the test they ac-count for almost 90 % of the total deflection. It can also be

V N d

B. Mihaylov · Five-spring model for complete shear behaviour of deep beams

Table 1.Test specimens investigated Beama/dbdhalb1*V/PρlnbfyagfcρvfyvMmax/MnVexpVpredΔexpΔpred (mm)(mm)(mm)(mm)(mm)(%)(MPa)(mm)(MPa)(%)(MPa)(kN)(kN)(mm)(mm) Mihaylovet al. [19], [20] S1M1.554001095120017003000.50.7066522033.00.104900.80941.0909.37.77.5 S1C1.554001095120017003000.50.7066522033.00.104900.80943.0909.38.47.5 S0M1.554001095120017003000.50.7066522034.200.61721.0770.36.46.7 S0C1.554001095120017003000.50.7066522034.200.981162.0770.310.96.7 L1M2.284001095120025003000.50.7066522037.80.104900.82663.0652.414.212.6 L1C2.284001095120025003000.50.7066522037.80.104900.79642.0652.413.712.6 L0M2.284001095120025003000.50.7066522029.100.52416.0404.610.09.6 L0C2.284001095120025003000.50.7066522029.100.62492.0404.611.19.6 SB1.594001070120017003000.50.6016522030.500.74715.7724.7**9.58.9** Salamyet al. [21] B-20.5024040047520010012.0253762036.200.61775.0761.43.21.0 B-30.5024040047520010012.0253762036.20.43760.60768.0761.44.81.0 B-40.5024040047520010012.0253762031.30.83760.78975.5699.61.91.1 B-61.0024040047540010012.0253762031.300.84525.0452.22.82.5 B-71.0024040047540010012.0253762031.30.43760.94590.5464.82.82.6 B-81.0024040047540010012.0253762037.80.83761.17750.5–3.3– B-10-11.5024040047560010012.0253762029.200.75308.0293.63.84.1 B-10-21.5024040047560010012.0253762023.000.90351.5259.85.34.5 B-111.5024040047560010012.0253762029.20.43761.24512.5–14.7– B-121.5024040047560010012.0253762031.30.83761.39580.5–7.1– B-10.3-11.5036060067590015012.1193882037.800.95980.0743.46.66.0 B-10.3-21.5036060067590015012.1193722031.200.93893.5664.58.66.1 B-13-11.50480800905120020012.07103982031.600.841492.51191.611.98.0 B-13-21.50480800905120020012.07103982024.000.671128.51028.19.38.6 B-141.5060010001105150025012.04143982031.000.721984.51790.09.310.1 B-171.5060010001105150025012.04143982028.70.43980.962607.02295.411.912.4 B151.5072012001305180030011.99184022027.000.712695.02330.911.912.2 B-161.5084014001505210035012.05183942027.300.572987.53195.810.614.1 B-181.5084014001505210035012.05183982023.50.43980.834198.04232.415.817.6 * lb2=150 mm for tests by Mihaylovet al. and lb2=lb1for tests by Salamyet al. ** Obtained with α1=50° and lt=1850 mm based on experimental observations.

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seen that the model matches well the experimental points from test S1C, with the exception of the first point, which is overpredicted. The sudden increase in shear deforma-tions between the first and second points can be attributed to the breakdown of beam action at V= Vsect.

Fig. 7 shows the complete deformed shapes of speci-men S1C for four displacespeci-ment levels. The triangular meshes show the measured deformed shapes and the green dots show the predicted locations of the vertices of

the triangles. The first three diagrams correspond to the second, third and fifth experimental points in the lower plot of Fig. 6. The diagram at bottom right shows the pre-dicted deformations only since measurements of de-formed shapes were not perde-formed in the post-peak regime of the beam. It can be seen that the five-spring model provides excellent approximations for the mea-sured deformed shapes of specimen S1C from the stage of breakdown of beam action to the shear failure of the beam. The first deformed shape resembles the flexural de-formation pattern of the kinematic model since, initially, DOF Δcis relatively small. The shear deformation pattern emerges clearly in the diagram at bottom left, which corre-sponds to the shear failure of the beam. In the post-peak regime, the shear deformations continue to increase, whereas the bottom reinforcement unloads and the flexur-al deformations decrease.