Experimental Study on Stress-Strain Curve of Concrete Considering

Experimental Study on Stress-Strain Curve of Concrete Considering Localized Failure in Compression Ken Watanabe1, Junichiro Niwa2, Hiroshi Yokota3 and Mitsuyasu Iwanami4

Received 1 December 2003, accepted 2 June 2004 Abstract One of the important factors for compressive stress-strain curves of concrete is the localization of failure. The stress-strain curve of concrete strongly depends on the aspect ratio of the concrete specimen; therefore, a unique stress-strain curve is not adequate to express the softening behavior of concrete. To overcome the problem related to the localization of failure, a series of uniaxial compressive tests of concrete specimens was conducted. From the measured energy distribution, the failed specimen was assumed to be composed of 2 or 3 zones. Then, an equation for an envelope curve involving a characteristic of compressive strength of concrete was formulated so as to match the experimental curve of each zone. Combining 2 or 3 proposed equations considering the extent of each zone could express the experimental stress-strain curve of the specimen regardless of the aspect ratio.

1. Introduction kulrat et al. (2001) quantified the localized compressive failure zone length (Lp) based on the consumed energy A compressive stress-strain curve is an important mate- distribution along the height of a concrete specimen. rial characteristic of concrete. Many studies (e.g., Kar- They concluded that localized failure in compression san and Jirsa 1969; Popovics 1973) experimentally occurs in concrete specimens having H/D (the ratio of clarified the influence of compressive strength and height to maximum width of cross-section) of 2 or kinds of coarse aggregate on the stress-strain curve of more. concrete. Because of various influencing factors and The objective of this study is to formulate a different conditions in experimental approaches, a gen- stress-strain curve in compression through matching of eral equation expressing the stress-strain curve has not the experimental results. During the formulation, the been proposed yet. localization and the compressive strength of concrete The localization of failure of concrete in compression were taken into account. By referring to the study of is one of the influential factors on the stress-strain curve. Lertsrisakulrat et al. (2001), a series of uniaxial Compressive failure is typically observed in reinforced one-directional repeated load tests was conducted for concrete (RC) deep beams having a shear span length to obtaining stress-strain curves at local portions of the effective depth ratio of less than 1, which show the specimen by the acrylic-rod method. To deal with prob- shear-compression failure mode. Lertsrisakulrat et al. lems related to the localization of failure, a specimen is (2002) confirmed that the localized failure was observed assumed to be composed of 3 different zones, a failure in RC deep beams. The localized failure governs the zone, a transition zone, and an unloading zone. Bazant load-deflection relationship in the post-peak region, and (1989) reported a series coupling model based on a also gives the size effect on the shear strength of deep concept similar to that of this study. The lengths of each beams, as reported by Walraven (1994). zone were quantified according to the consumed energy Localized failure in tension was examined according distribution proposed by Lertsrisakulrat et al. (2001). to the fracture mechanics and useful results were ob- Next, equations to express the stress-strain relationship tained (e.g., Hillerborg et al. 1976). Many researchers for each zone of the concrete specimen were formulated (Markeset and Hillerborg 1995; Bazant 1989; Nakamura so that the experimental results matched well the calcu- and Higai 1999) applied fracture mechanics to the study lated ones. Finally, it was confirmed that combining 3 of localization in compression. In particular, Lertsrisa- proposed equations considering the length of each zone satisfactorily describes the experimental stress-strain relationship of concrete, which is strongly affected by 1 Doctoral student, Dep. of Civil Engineering, Tokyo the aspect ratio and the compressive strength of the Institute of Technology, Tokyo, Japan. concrete specimen. E-mail: [email protected] 2 Professor, Dep. of Civil Engineering, Tokyo Institute of 2. Outline of experiment Technology, Tokyo, Japan. 3 Head, Structural Mechanics Division, Port and Airport (1) Specimen Research Institute, Yokosuka, Japan. The characteristics of the test specimens (cylinders 100 4 Center researcher, Structural Mechanics Division, Port mm in diameter (D)) are listed in Table 1. Two speci- and Airport Research Institute, Yokosuka, Japan. 396 K. Watanabe, J. Niwa, H. Yokota and M. Iwanami / Journal of Advanced Concrete Technology Vol. 2, No. 3, 395-407, 2004

Table 1 Test specimens. Water-to- Compressive Diameter of the Height-to- Maximum Height cement strength of cylindrical speci- diameter G stress (H) max ratio Designation*1 concrete*2 men (D) ratio (H/D) ' (σmax) (W/C) (fc ) (mm) (mm) (mm) (MPa) (MPa) 0.4 T20-0.4-2 47.3 54.1 200 2 20 0.6 A20-0.6-2 31.1 29.4 0.4 A20-0.4-3 46.1 48.1 300 3 20 0.6 A20-0.6-3 30.0 28.4 0.4 A13-0.4-4 47.3 48.4 0.5 A13-0.5-4 42.0 39.3 13 0.6 A13-0.6-4 32.2 29.4 0.7 A13-0.7-4 26.2 21.9 400 4 100 0.4 A20-0.4-4 48.4 47.5 0.5 A20-0.5-4 39.0 28.2 20 0.6 A20-0.6-4 36.7 30.3 0.7 A20-0.7-4 28.4 22.5 0.4 T20-0.4-6 46.6 48.4 600 6 20 0.6 T20-0.6-6 31.2 29.3 0.4 T20-0.4-8 46.6 44.7 800 8 20 A20-0.6-8 28.5 16.6 0.6 T20-0.6-8 31.2 29.9 *1 A: With AC-rod, T: Without AC-rod *2 Average cylindrical compressive strength at the age of 7 days mens were used for each test case. Table 2 Mixture proportion.

Mixture proportions of concrete are presented in Ta- 3 ble 2 and material properties used for the concrete are Weight per unit volume (kg/m ) G s/a*1 presented in Table 3 (a). Before casting of concrete, an max W/C Gravel (mm) (%) acrylic rod, on which strain gauges (3 mm length) were Water Cement Sand 5-13 13-20 attached at intervals of 40 mm, was installed vertically (mm) (mm) in the mold of a specimen named “Type A”. In speci- 0.4 43 182 455 736 493 493 0.5 45 185 370 799 494 494 mens named “Type T”, no acrylic rod was embedded. 20 The properties of the acrylic rod are listed in Table 3 0.6 47 188 313 853 487 487 (b). 0.7 49 191 273 903 475 475 To investigate the effect of the strength of concrete on 0.4 47 187 468 787 897 − 0.5 49 190 380 853 897 − the stress-strain curve, the water-to-cement ratio (W/C) 13 of concrete was changed to 0.4, 0.5, 0.6 and 0.7. Coarse 0.6 51 193 322 909 883 − 0.7 53 193 280 959 860 − aggregate with the maximum size (Gmax) of 13 mm or *1 20 mm was used. The height of the specimen was 400 : Volume ratio of sand to aggregate mm, which indicated that a very clear localized failure Table 3 Material properties of concrete and acrylic rod. would occur (Lertsrisakulrat et al. 2001). The compres- ’ (a) Concrete sive strength of concrete (fc ) ranged from 26.2 to 48.4 Water Fineness MPa at the time of the loading test, which was averaged Density using three standard cylindrical specimens of 200 mm in Designation absorption modulus height and 100 mm in diameter. Specimens with H/D=2, (kg/m3) (%) 3, 6 and 8 were additionally prepared to discuss the ef- Fine aggregate 2.59 1.94 2.51 fect of H/D on the stress-strain curve. These specimens (from Obitsu, Chiba) were made of concrete having W/C=0.4 and 0.6 and Coarse aggregate 2.64 0.93 7.00 Gmax=20 mm. (from Oume, Tokyo) All the specimens were cast concrete vertically and Blaine High-early strength cement 3.16 remolded at one day after casting. Immediately after 3550 (cm2/g) remolding, they were cured in water for 6 or 7 days until (b) Acrylic rod the loading tests. The top end of the specimen was pol- Specific gravity 1.19 ished to ensure a smooth horizontal surface. Tensile strength (MPa) 76 (2) Loading test and instrumentation Elastic modulus (MPa) 3200 Figure 1 shows the test setup. To reduce friction, fric- Compressive strength (MPa) 120 tion reducing pads, i.e., two Teflon sheets (0.05 mm Thermal expansion coefficient (1/°C) 7×10-6 K. Watanabe, J. Niwa, H. Yokota and M. Iwanami / Journal of Advanced Concrete Technology Vol. 2, No. 3, 395-407, 2004 397

Loading platen Peak point (σmax, εpeak) Stress (MPa) 40 Friction-reducing Envelope curve pad

Specimen 30 Displacement gauge Unloading curve

20 Reloading curve

Loading platen Friction-reducing 0 pad 0 2000 4000 6000 Average strain (×10-6) (a) Test setup (b) Traced and envelope curves (A13-0.5-4)

Fig. 1 Outline of experiment. thick) sandwiching silicon grease, were inserted be- leased until 0 kN (one-directional repeated loading). tween the specimen and loading platens. (Fig.1 (a)) After that, load application was started with a controlled During the loading test, the load (P) was measured by monotonous displacement rate of 0.002 mm/s until the load cell. The stress (σ) was obtained with P divided by load dropped to 10% of the maximum load. Figure 1 the cross-section area of specimen (Ac). Displacements (b) shows the experimental results of traced stress-aver- (d) of the specimen were externally measured by 4 dis- age strain curves (σ−εave) and their envelope curve. Kar- placement gauges. In specimen Type A, internal strains san and Jirsa (1969) stated that an envelope curve coin- were measured by strain gauges attached on the acrylic cides with the stress and strain curves of specimens rod (AC-rod method). These strains were local strains subjected to monotonously increasing strains. In the (ε), which were assumed to represent the uniform strain following discussion, empirical formulas to represent in the measuring point of 40 mm in length. The average stress-strain curves in different states of stress and strain strain (εave) for the whole length of the specimen was are formulated so as to match the envelope curve. obtained by averaging all the ε Values. However, in specimen Type T without the acrylic rod, the value εave 3. Experimental results was determined by dividing measured displacement d by the initial height of specimen (H). (1) Maximum stress of a concrete specimen (σmax) To capture the softening behavior of concrete, the When a concrete specimen is subjected to compression load was increased up to its maximum, and then re- without friction-reducing pads, existing friction at the interface between the ends of the specimen and loading /f ' σmax c Stress σ (MPa) σ max =46.4MPa

1.1 σ max =36.4MPa 50 σ max =31.3MPa 1.0 40 σ max =21.9MPa 0.9 30

0.8 20 Average: 0.95 10 0.7 c.v.: 5.0 % 0 0.6 0 1000 2000 3000 4000 5000 123456789 Αverage strain εave (μ) H /D Fig. 3 Stress-average strain curves with different σmax ’ Fig. 2 Ratios of σmax to fc vs. H/D. (H/D = 4). 398 K. Watanabe, J. Niwa, H. Yokota and M. Iwanami / Journal of Advanced Concrete Technology Vol. 2, No. 3, 395-407, 2004

H/D=2 Designation Length Strain σ/σmax H/D=3 Failure zone Lp εF H/D=4 Transition 1.0 LT εT H/D=6 zone Unloading L ε H/D=8 zone U U

0.5 LU

Lp 0.0 01234 ε/εpeak Fig. 4 Stress-average strain curves with varied H/D ’ (W/C = 0.6; fc = 30MPa). L Lp T

LT LT LU LT Lp Lp D = 100 mm (a) Cross-section (b) H/D = 2 (c) H/D = 3 (d) H/D = 4 (e) H/D = 6 (f) H/D = 8 (A20-0.6-2) (A20-0.6-3) (A20-0.6-4) (T20-0.6-6) (T20-0.6-8)

Fig. 5 Failure patterns of specimens with varied H/D (black lines indicate cracks observed clearly).

platens affects the maximum stress. When fc’ is defined (2) Stress-average strain curve of concrete speci- as the compressive strength of concrete measured using men (σ−εave) cylindrical specimen having a diameter (D) of 100 mm The effect of σmax and H/D on the stress-average strain and a height (H) of 200 mm, Kosaka and Tanigawa curve (σ−εave) is discussed. Figure 3 shows the ’ (1981) reported that, compared with fc , the maximum stress-strain curves of a specimen having H/D=4 and stress of the specimen gradually decreases as H/D in- various σmax values. With increases in σmax, the strain at creases and becomes almost constant, approximately the peak point (εpeak) grew larger and after the peak ’ 96% of fc , when H/D>4. This was because the specimen point, the descending branch grew steeper. Figure 4 ’ failed at the middle portion of its height, which was shows stress-strain curves of specimens with fc =30 MPa, little affected by the friction. This implies that if a con- in which stress and strain were normalized by their re- crete specimen is subjected to purely uniaxial compres- spective values at the peak point (σmax, εpeak). The ap- sion, the maximum stress will reduce to approximately pearance of specimens after the test is shown in Fig. 5. ’ 96% of fc . The descending branch dropped rapidly as H/D in- Figure 2 shows the relationship between the ratio of creased. In particular, the specimen with H/D=8 clearly ’ σmax to fc and H/D obtained by the loading tests. This exhibited snapback behavior and localized failure. ’ figure shows that σmax is approximately 95% of fc (co- ’ efficient of variation: 5%). The ratio σmax/fc was inde- (3) Failure, transition and unloading zones pendent of H/D and was nearly 95% in the region where The failure mode and the stress-local strain curve (σ−ε) H/D was greater than 4 as mentioned above. This sug- of specimen A13-0.7-4 are shown in Figs. 6 (a) and (b), gests that the friction between the specimen and loading respectively. In Fig. 6 (b), it is easy to see that the fail- platens can be effectively reduced by inserting fric- ure gradually localized in the pre-peak region and then tion-reducing pads. From this result, σmax can be esti- secondary cracks propagated down to the failure-free ’ mated from fc as follows: zone after the peak point. The distribution of the locally

’ consumed energy (AINTi) calculated by the area under the σmax = 0.95 fc (1) stress-local strain curve (σ−ε) is illustrated in Fig. 6 (c). K. Watanabe, J. Niwa, H. Yokota and M. Iwanami / Journal of Advanced Concrete Technology Vol. 2, No. 3, 395-407, 2004 399

P (kN) A + A + A Distance from lower surface 200 INT1 INT2 INT3 of specimen (mm)

Ai: area under the stress-local strain curve. 100 400 A σ σ σ A F (3) INT1 (1) (2) 0 350 A L INT2 p 0.0 0.5 1.0 A2 A A1 3 d (=εF × Lp; mm) ε ε ε 300 AINT3 Failure zone

σ σ σ A Transition zone (4) (5) (6) 250 INT4

AINT5 P (kN) A A 4 5 A6 200 ε ε ε AINT6 200 σ σ σ (7) (8) (9) 150 AINT7 LT 100 A7 A A8 9 100 ε ε ε 0 σ (MPa) 50 0.15×ΣAINTi 0.0 1.0 (1): 380 (6): 180 d (=εT × LT; mm) 20 (10) (2): 340 (7): 140 (3): 300 (8): 100 0 AINT4 + AINT5 + AINT6 + 10 A + A + A + A (4): 260 (9): 60 01530 INT7 INT8 INT9 10 A (5): 220 (10): 20 Local consumed energy AINTi (kNmm) INT10 0 ε (µ) 0 5000 10000 Strain gauge No. and Failure zone: AINTi ≥ 0.15 × ΣAINTi located height (mm). where, AINTi = Ai × Ac × 40: consumed energy in local portion, Σ AINTi: consumed energy in entirely of specimen.

(a) A13-0.7-4 (b) Stress-local strain curve (c) Local consumed energy (d) Load-displacement distribution (AINTi) curve in failure zone and transition zone Fig.6 Distinction between the failure zone and the transition zone (A13-0.7-4).

Table 4 Three crack displacement types. Opening mode (or mode I) Sliding mode (or mode II) Tearing mode (or mode III)

The displacement of the crack surfaces The displacement of the crack surfaces The displacement is in the plane of is perpendicular to the plane of the crack. is in the plane of the crack and perpen- the crack and parallel to the leading dicular to the leading edge of the crack. edge of the crack.

The locally consumed energy (AINTi) is the sum of ener- (H/D=2-4) into 2 parts; that is, a failure zone with aver- gies in 3 cracking modes as illustrated in Table 4. age strain εF and length LF and a transition zone with Lertsrisakulrat et al. (2001) stated that the AC-rod average strain εT and length LT. In the following discus- method is able to determine the failure zone quantita- sion, the values εF and εT were obtained by averaging tively: More than 15 percent of the total energy con- the local strain (ε) measured in the respective zones. sumed in the whole specimen is consumed there. This The appearance of specimens with H/=6 and 8 are energy criterion can be used to divide the specimen shown in Figs. 5 (e) and (f), in which no-crack zones 400 K. Watanabe, J. Niwa, H. Yokota and M. Iwanami / Journal of Advanced Concrete Technology Vol. 2, No. 3, 395-407, 2004

Table 5 Experimental values in failure and transition zones. Failure zone Transition zone

Designation σmax Lp εF0 AF GFc LT εT0 σΤ1/σmax εΤ1/εT0 c e (MPa) (mm) (µ) (kNmm) (N/mm2) (mm) (µ) A13-0.4-4* 48.7 120 2299 140.9 0.150 280 1577 0.33 0.60 − − A20-0.4-4* 47.5 120 2233 177.6 0.189 280 1853 0.34 0.55 − − A13-0.5-4 39.3 120 2601 150.7 0.160 280 1636 0.30 0.82 0.169 -2.58 A20-0.5-4 28.2 140 1816 131.6 0.117 260 848 0.42 0.64 0.281 -1.88 A13-0.6-4 29.4 140 2076 99.0 0.090 260 1442 0.43 0.92 0.336 -1.87 A20-0.6-4 30.3 120 1522 76.6 0.082 280 1388 0.30 0.72 0.205 -1.84 A13-0.7-4 21.9 140 1328 79.3 0.074 260 1165 0.45 0.79 0.311 -1.27 A20-0.7-4 22.5 160 1335 99.3 0.079 240 1184 0.46 0.82 0.371 -1.56

σ (MPa) σ (MPa) σ (MPa) σ (MPa) 30 30 18 30

20 20 12 20 L p LT LU 10 G × ( ) +10 × ( ) + 6 × ( )10= Fc H H H 0 0 0 0 εF ε ε ε 0 2500 5000 0 3500T 7000 0 2500U 5000 035007000 (a) σ−εF (the failure zone) (b) σ−εT (the transition zone) (c) σ−εU (the unloading zone) (d) σ−εave

GFc: energy consumed per unit volume of the failure zone

Fig. 7 Concept for stress-strain curve (Lp+LT+LU=H).

were observed in the lower or upper portion of the the conclusions, LT and LU were also determined by Ac. specimen. A stress-local strain curve measured in the Further discussion of the applicability of Eqs. (2a) to no-crack zone exhibited unloading behavior until the (2d) to specimens with D greater than 100 mm is end of the loading test; hence, the zone having length LU needed. and strain εU is called the unloading zone. Lp and LT Figure 7 shows the proposed stress-strain curve. The were almost constant regardless of H/D, when H/D was failure zone is coupled in series on to the transition and 4 or larger. This suggests that in a specimen having H/D the unloading zones, such that the stresses carried in greater than 4, LU increases as H increases, but Lp and these 3 zones are identical and strains there can be su- LT are constant with the same length measured when perimposed considering the length of each zone. H/D=4. Consequently, the cylindrical specimens (D=100 mm) 4. Stress-strain curve in failure zone consist of 2 or 3 zones depending of the mode of failure. The lengths of each zone are as follows: Table 5 lists the experimental results to define the shape of a stress-strain curve: 1) εF0: strain at the peak (a) Failure zone: length (Lp) can be determined by the point in the failure zone, 2) AF: consumed energy in the AC-rod method (Lertsrisakulrat et al. 2001). failure zone, 3) GFc: energy consumed per unit volume (b) Transition zone: length (LT) of the failure zone, 4) εT0: strain at the peak point of the = H−Lp; H/D ≤4 (2a) transition zone, 5) σT1/σmax: ratio between stress at the = 4D−Lp; H/D >4 (2b) changing point and stress at the peak point, 6) εT1/εT0: (c) Unloading zone: length (LU) ratio between strain at the changing point and strain at = 0; H/D ≤4 (2c) the peak point, 7) c and e: coefficients defined in Eq. =H−4D; H/D >4 (2d) (12). All of the values are discussed in the following sessions. Lp obtained through this experiment is listed in Table 5. Lp had a scatter of approximately 20 mm, which was (1) Popovics equation determined based on the strain gauge interval. Lertsri- Lots of research projects have proposed equations for sakulrat et al. (2001) concluded that value Lp is de- compressive stress-strain curves of concrete. In particu- pendent solely on the cross-sectional width of the lar, Popovics (1973) proposed an equation to express specimen, while the specimen height, H/D and the shape experimental curves including the post-peak region of the cross-section are less influential in Lp. fc’ and Gmax completely as follows: showed a very slight effect on Lp as well. By referring to K. Watanabe, J. Niwa, H. Yokota and M. Iwanami / Journal of Advanced Concrete Technology Vol. 2, No. 3, 395-407, 2004 401

ε F 2 nF ×( ) G Fc (N/mm ) σ ε F0 = (3) 0.20 σ ε n max F F c.v.: 0.82 nF -1+ ( ) ε F0 0.15 where, εF0: Peak strain of the failure zone 0.10 nF: Experimental coefficient

Figure 8 (a) shows the calculated results of Eq. (3) 0.05 • : GFc with various nF values, where the stress and strain are : Eq. (5) divided by their respective values at the peak point (σmax, εF0). The softening behavior was well represented by 0.00 changing nF appropriately. Next, a0 is defined as the 20 30 40 50 area under the curve as shown in Fig. 8 (a) until σ/σmax σ (MPa) reaches 0.1. Figure 8 (b), indicating the relationship max between a0 and nF, confirming that nF decreases as a0 Fig. 9 Energy consumed per unit volume of failure zone increases. a0 is related to the consumed energy in failure (GFc) with σmax. zone AF, which is measured as the area under the load-displacement curve in the failure zone (P−dF (dF=LpεF)). AF includes the energy dissipated to form ε ，ε (µ) • : εF0 cracks and consumed by the friction at the surface of F0 T0 : Eq. (6) cracks (Carpinteri and Pugno 2002b). Moreover, divid- 3500 □ : εT0 ing the AF value by the failure zone volume (=Ac×Lp) : Eq. (9) provides the energy consumed per unit volume of the failure zone (GFc). Lertsrisakulrat et al. (2001) con- 2500 firmed that GFc would be independent of the shape of the specimen but depend on the compressive strength of the concrete. Taking this fact into consideration, the relationship between GFc and σmax is plotted in Fig. 9. 1500 The fact that GFc increases in proportion to σmax was also confirmed, leading to the formulation of Eq. (5). Summarizing the above discussion, n , a and G can F 0 Fc 500 be calculated based on the experimental results as follows: 20 25 30 35 40 45 50

−1.8 σmax (MPa) nF=6.21×a0 +0.185×a0+1.15 (4) where, Fig. 10 Peak strain of failure zone and transition zone a0: Area under a relative stress-relative strain curve with σmax and calculation by Eqs. (6) and (9). ⎛ G ⎞ ⎜ Fc ⎟ for the failure zone ⎜= ⎟ ⎝ σ max ×ε F0 ⎠ nF GFc: Energy consumption per unit volume of the failure zone (N/mm2) 5

⎛ A ⎞ ⎜= F = 3.6×10−3 ×σ − 3.6×10−3 ⎟ (5) ⎜ max ⎟ 4 ⎝ L p × Ac ⎠ A : Energy consumed at the failure zone (Nmm). F 3 : n G is defined as a material property within the ex- F Fc : Eq. (7) periment of this study. However, other studies (Carpin- teri and Pugno 2002a, 2002b; Nakamura and Higai 2 1999; Markeset and Hillerborg 1995) drew various con- 20 25 30 35 40 45 50 clusions about the consumed energy for compressive σ (MPa) failure of concrete, because these conclusions are de- max pendent on experimental conditions. More experiments Fig. 11 Relationship between nF and σmax, and Eq. (7). 402 K. Watanabe, J. Niwa, H. Yokota and M. Iwanami / Journal of Advanced Concrete Technology Vol. 2, No. 3, 395-407, 2004

: Failure zone : Eqs. (3), (6), (7) σ(MPa) σ(MPa) σ(MPa) : Transition zone : Eqs. (8), (9), (10), (12) 40 nF = 3.69 40 40

30 30 nF = 3.05 30

n = 2.78 20 20 20 F

10 10 10

0 0 -6 0 -6 -6 εF, εT (×10 ) εF, εT (×10 ) εF, εT (×10 ) 040000400004000 (a) A20-0.5-4 (b) A13-0.6-4 (c) A13-0.7-4

Fig. 12 Comparison of experimental data with proposed curve.

σ may be required to confirm the applicability of Eq.(5) to Peak (σ max, εT0) concrete under various conditions. σ max (2) Strain at the peak point in the failure zone Eq. (8) The strain at the peak point is usually estimated by an ’0.25 ’ Straight line empirical formula in proportion to fc or fc (Kosaka Changing point: Eq. (10) and Tanigawa 1981). However, since the size of the σ specimen and the loading condition strongly affect the T1 strain value, the existing equation should be revised to Convex curve: Eq. (13a) express the strain at the peak point in the failure zone (εF0). Figure 10 shows εF0 and the strain at the peak point of the transition zone (εT0) plotted against σmax. 0 ε εT The figure indicated that εF0 increases with σmax. With T1 εT0: Eq. (9) increase in σmax, Young’s modulus of mortar would be getting closer to that of coarse aggregate, so that con- Fig. 13 Stress-strain curve in transition zone. crete behaved rather uniformly in the region of larger σmax. This uniformity would make it difficult for micro cracks inside concrete to initiate and propagate. 5. Stress-strain curve in transition zone εF0 can be estimated as:

2 2/3 −6 εF0 = (1.72 ×10 × σmax ) ×10 (6) (1) Introduction White squares in Fig. 12 show the stress-strain relationship measured in the transition zone. Its ascending (3) Experimental value, nF branch has a small curvature near the peak point. In the As mentioned in sessions 4.(1) and (2), nF can be post-peak region, the relationship shows unloading be- given by substituting σmax into Eqs. (4) to (6). The rela- havior at first, then, it changes to softening behavior tionship between nF and σmax, which is plotted in Fig. 11, (Fig. 6 (b)). Thus, the ascending, the unloading and the is simplified as: softening curves of the stress-strain relationship in the n = 3.00 ×10−4 × σ 2 +3.47×10−2 ×σ + 1.86 (7) transition zone (σ−εT) were considered separately as F max max shown in Fig. 13. In particular, the unloading behavior A stress-strain curve of the failure zone (σ−εF) is pre- in the post-peak region was assumed to be a straight line dicted by substituting σmax into Eqs. (3), (6) and (7). The from the peak point to the changing point (σΤ1, εΤ1). proposed curves are compared with experimental data in Beyond the changing point, the softening behavior was the failure and the transition zones as shown in Fig. 12. expressed by a convex curve. The proposed curve agreed well with the experimental data measured in the failure zone with various σmax val- (2) Ascending curve ues. Figure 14 shows the experimental results of the stress-strain curve in the transition zone for σmax=23 K. Watanabe, J. Niwa, H. Yokota and M. Iwanami / Journal of Advanced Concrete Technology Vol. 2, No. 3, 395-407, 2004 403

MPa and 40 MPa. The stress and strain there were nor- • : σ /σ σ /σ , ε /ε T1 max malized by respective value at the peak point (σmax, εT0). T1 max T1 T0 : Eq. (10a) Identical behavior was observed in the ascending curves 1.0 : εT1/εT0 as follows: : Eq. (10b)

⎧ ⎡ 0.4 ⎤⎫ ⎪ ⎛ σ ⎞ ⎛ ⎛ σ ⎞⎞ ⎪ ε = ε 0.7⎜ ⎟ + 0.3⎢1− ⎜1− ⎜ ⎟⎟ ⎥ (8) T T 0 ⎨ ⎜ ⎟ ⎜ ⎜ ⎟⎟ ⎬ 0.5 ⎪ ⎝ σ max ⎠ ⎢ ⎝ σ max ⎠ ⎥⎪ ⎩ ⎣ ⎝ ⎠ ⎦⎭ where, 0.0 εT: Strain of the transition zone, εT0: Peak strain of the transition zone. 20 25 30 35 40 45 50 σ (MPa) (3) Strain at peak point of transition zone max Fig.15 Relationship between σmax and experimental val- As shown in Fig. 10, the relationship between εT0 and ues of σΤ1, εΤ1 and calculation by Eqs. (10a) and (10b). σmax was almost linear like in the case of εF0. The dif- ference between ε and ε was attributed to the fact T0 F0 σ/σmax c=0.1 that the localization of failure began even in the c=0.2 pre-peak region (Fig. 6 (b)). The relationship is ap- 1.0 c=0.3 c=0.4 proximated by the least-square method as follows: c=0.5 1 2 −6 εT0=(2.4×10 ×σmax+5.77×10 )×10 (9) 0.5

(4) Changing point (σΤ1, εΤ1) and straight line in the post-peak region 0.0 Figure 14 shows that σ-εT behavior from the peak εT/εT0 point to the changing point follows an identical path. In 0.0 0.5 1.0 1.5 2.0 addition, the stress at the changing point (σ ) clearly Τ1 Fig. 16 Effect of value c on Eq. (12), where e=-1.9. decreased with increase in σmax. The specimen with large σmax can accumulate a large amount of energy up c to the peak. This large energy accumulated inside the 0.6 • : c specimen induced a rapid propagation of cracks in the :Eq. (13b) failure zone and stress decreased sharply. In this case, 0.4 the stress at which the secondary cracks began to propagate in the transition zone would become small. 0.2 Figure 15 shows the relationship between σmax and the experimental ratios of σΤ1 to σmax and εΤ1 to εT0, and the results calculated with Eqs. (10a) and (10b). Since the 0.0 straight line from the peak point to the changing point 20 30 40 50 was identical, the tendency of εΤ1/εT0 correlated closely σmax (MPa) with that of σΤ1/σmax. Therefore, it is concluded that the Fig. 17 Relationship between c and σmax as well as Eq. changing point (σΤ1, εΤ1) can be estimated as follows: (13b). −0.7 σT1/σmax=3.2×σmax +0.1 (10a) σ εT1/εT0=(σΤ1/σmax)+0.35 (10b) σ max The straight line from the peak to the changing point can be expressed as follows: σ−εU: Eq.(14a)

σ ⎛ εT ⎞ Identical initial tangent = a⎜ ⎟ + b (11) σ max ⎝ εT 0 ⎠ where, σ−εT

⎛ σ T1 ⎞ 1− ⎜ ⎟ εU，εT ⎜ σ ⎟ εU0: Eq.(14b) εT0: Eq.(9) a = ⎝ max ⎠ ⎛ ε ⎞ ⎜ T1 ⎟ Fig. 18 Stress-strain curves for unloading and transi- 1− ⎜ ⎟ ⎝ εT 0 ⎠ tion zones. 404 K. Watanabe, J. Niwa, H. Yokota and M. Iwanami / Journal of Advanced Concrete Technology Vol. 2, No. 3, 395-407, 2004

b=1-a because it failed rapidly. A trial calculation showed that e changed slightly with the curvature. Since e was less (5) Softening curve after changing point influential on the softening curves, the averaged value By referring to the equation proposed by Popovics of e was set to −1.9 in this study (Eq. (13b)). (1973), a convex curve was applied for expressing the On the other hand, Fig. 16 shows the effect of c on softening behavior as follows: the shape of the convex curve. Taking into consideration the effect of σmax on the stress at the changing point σ/σ =c×(ε /ε )e+f (12) max T T0 (σT1) (Session 5.(4)), c would also be related to σmax. where, Figure 17 shows the relationship between c and σmax, c, e, f: coefficients. which is assumed by Eq. (13a). The value f was ar- ranged to match the experimental changing point (Eq. Values c and e, for which Eq. (12) agreed well with (13c)). The coefficients of Eq. (12) are summarized as experimental curves, are listed in Table 5. Values c and follows: e of the specimen with W/C=0.4 were not measured 1 −1.15 c = 1.2×10 ×σmax (13a)

Experimental result Proposed curve σ(MPa) σ(MPa) σ(MPa) σ(MPa) 50 60 50 50 A20-0.6-2 A20-0.4-2 A20-0.6-3 A13-0.4-3 40 40 40 40 30 30 30 20 20 20 20 10 10 10 0 0 0 0 020004000 0 2000 4000 6000 0 2000 4000 0 2000 4000 -6 -6 -6 εave (×10 ) -6 εave (×10 ) εave (×10 ) εave (×10 ) (a) H/D=2 (b) H/D=3

σ(MPa) σ(MPa) σ(MPa) σ(MPa) 50 50 50 50 A13-0.7-4 A20-0.6-4 A13-0.5-4 A13-0.4-4 40 40 40 40 30 30 30 30 20 20 20 20 10 10 10 10 0 0 0 0 020004000 0 2000 4000 0 2000 4000 020004000 -6 -6 -6 -6 εave (×10 ) εave (×10 ) εave (×10 ) εave (×10 ) (c) H/D=4

σ(MPa) σ(MPa) σ(MPa) σ(MPa) 50 50 50 50 T20-0.6-6 T20-0.4-6 T20-0.6-8 T20-0.4-8 40 40 40 40 30 30 30 30 20 20 20 20 10 10 10 10 0 0 0 0 0 2000 4000 0 2000 4000 020004000 020004000 -6 -6 ε (×10-6) -6 εave (×10 ) εave (×10 ) ave εave (×10 ) (d) H/D=6 (e) H/D=8

Fig. 19 Comparison of stress−average strain curve of specimen (σ−εave) with proposed curve. K. Watanabe, J. Niwa, H. Yokota and M. Iwanami / Journal of Advanced Concrete Technology Vol. 2, No. 3, 395-407, 2004 405

e = −1.9 (13b) εave: Average strain, e f = σT1/σmax −c×(εT1/εT0) (13c) εF: Strain of the failure zone, εT: Strain of the transition zone, (6) Comparison of proposed curve with the ex- εU: Strain of the unloading zone, perimental curve Lp: Length of the failure zone (mm) (=120 mm), In Fig. 12, the proposed curve is compared with the LT: Length of the transition zone (mm) experimental curve in the transition zone. The proposed (= H−Lp ; H/D ≤ 4), curve agreed well with the experimental curve including (= 4D−Lp ; H/D > 4), the peak strain, the changing point, and the softening LU: Length of the unloading zone (mm) behavior. (= 0 ; H/D ≤ 4), (= H−4D ; H/D > 4), 6. Stress-strain curve in unloading zone D: Diameter of cross-section (mm), H: Height of a specimen (mm). In this study, the specimen with H/D=8 was too slender to obtain sufficient numbers of experimental results for The strain (εpeak) at the peak point can be obtained by formulating the stress-strain curve of the unloading zone. replacing Eq. (15) by Eqs. (6), (9) and (14b). To handle this difficulty, the curve σ−εU was assumed to Figure 19 shows the comparison of the proposed be a straight line, the respective initial tangents of which curve with the experimental result of σ−εave, where the are the same as the curve of the transition zone (Fig. 18). solid lines are the proposed curves and the circles show Furthermore, the applicability of Eqs. (14a) and (14b) to experimental data with H/D=2, 3, 4, 6 and 8, and with the expression of experimental data will be examined in W/C=0.4 and 0.6. The proposed curves were the results session 7. The calculation for the secant modulus of Eq. of calculation by Eq. (15) with σmax. Since Lp had a (8) at the origin of the coordinate axes gives the follow- slight scatter, there were small differences between the ing equation for the unloading zone: proposed curves and the experimental values. However, the proposed curve represented well the experimental ⎛ σ ⎞ ε = ε ⎜ ⎟ (14a) result, particularly for εpeak and softening behaviors. For U U0 ⎜ ⎟ ⎝ σ max ⎠ specimens with H/D=6 and 8 having the unloading zone, Figs. 19 (d) and (e) also demonstrate that Eq. (14a) where, proposed in session 6 adequately expresses the experi- εU: Strain of the unloading zone, mental curve of σ−εU. εU0: Peak strain of the unloading zone The proposed equations were examined to match the =1/1.12×εT0 experimental results for the conditions summarized in 1 2 −6 Table 6. By referring to the previous studies, the =(2.14×10 ×σmax+5.15×10 )×10 (14b) stress-strain curve of a concrete specimen having the 7. Stress-strain curve considering localized following characteristics is different from the curve ob- failure tained in this study. Therefore, the proposed equations will be improved for use in the different kinds of con- Finally, based on the ratios of the length of each of the 3 crete. Further experimental or theoretical discussion on zones to specimen height (H), the proposed equations the localized failure of concrete in compression will be required, particularly regarding the length of each zone, for each strain, which are determined only by σmax, are the strain at the peak, and the shape of curve. combined to formulate an equation to express εave as follows: a) Concrete mixed with different kinds of aggregate: e.g. lightweight aggregate (Kosaka and Tanigawa

L p LT LU 1981). εave = εF × +εT × +εU × (15) H H H b) Confined concrete or concrete reinforced by stirrups or fibers (Hirano et al. 2002; Sfer et al. 2002). where, c) Concrete with compressive strength of higher than

Table 6 Experimental conditions for proposed equations. ’ Compressive strength (fc ) in MPa 20 to 50 Maximum size of coarse aggregate (G ) in mm 13, 20 Material properties of max Material of course aggregate crashed graywacke concrete Curing method in water Curing days 6 to 7 Shape of cross-section Circle Size and shape of Diameter in mm 100 specimen Height-to-diameter ratio (H/D) 2 to 8 Friction reducing pads use Loading condition Loading rate in mm/s 0.002 406 K. Watanabe, J. Niwa, H. Yokota and M. Iwanami / Journal of Advanced Concrete Technology Vol. 2, No. 3, 395-407, 2004

60 MPa. 773-782. d) Concrete in the un-matured stage before reaching its Hirano, T., Nakamura, H., Saito, S. and Higai, T. (2002). design strength (Yi et al. 2003). “Experimental evaluation on compressive fracture e) Concrete subjected to high speed loading of more behavior of confined concrete under uniaxial than 0.002 mm/s (Hujikake et al. 1999). loading.” Proceedings of Japan Concrete Institute, f) Concrete with H/D less than 2, if a loading test is 24(2), 145-150. conducted without friction reducing pads (Sangha Hujikake, K., Shinozaki, Y., Ohno, T., Mizuno, J. and and Dhir 1972). Suzuki, A. (1999). “Post-peak and strain-softening behaviors of concrete materials in compression under 8. Conclusions rapid loading.” Journal of Materials, Concrete Structures, Pavements, JSCE, 44(627), 37-54. To investigate the effect of the localized failure on the Karsan, I. D. and Jirsa, J. O. (1969). “Behavior of stress-strain curve of concrete in compression, a series concrete under compressive loading.” ASCE, Journal of uniaxial compression tests of concrete specimens was of the structural division, 95(ST12), 2543-2563. conducted. By analyzing the test results, the following Kosaka, Y. and Tanigawa, Y. (1981). “Mechanical conclusions were drawn in this study: property of concrete.” in: K. Okada and H. Muguruma Eds. Concrete engineering handbook. (1) Experimental equations of the stress-strain curve for Tokyo: Asakura-shoten, 376-377. (in Japanese) the failure and the transition zones were proposed Lertsrisakulrat, T., Watanabe, K., Matsuo, M. and Niwa, taking into consideration the effect of the maximum J. (2001). “Experimental study on parameters in stress of the concrete specimen (σmax). By using only localization of concrete subjected to compression.” σmax, the proposed equations were able to express Journal of Materials, Concrete Structures, Pavements, the experimental curve for the respective zones. The JSCE, 50(669), 309-321. stress-strain relationship for the unloading zone was Lertsrisakulrat, T., Niwa, J., Yanagawa, A. and Matsuo, approximated as a straight line. M. (2002). “Concepts of localized compressive (2) A combination of the 3 proposed curves based on the failure of concrete in RC deep beams.” Journal of length of each zone could predict the strain at the Materials, Concrete Structures, Pavements, JSCE, peak point and the softening behavior of the 54(697), 215-225. stress-strain curve for the whole length of the con- Markeset, G. and Hillerborg, A. (1995). “Softening of crete specimen regardless of H/D and σmax. The ap- concrete in compression localization and size plicability of the proposed equation was confirmed effects.” Cement and Concrete Research, 25(4), by comparing the experimental curve obtained from 702-708. the concrete specimen. Nakamura, H. and Higai, T. (1999). “Compressive fracture energy and fracture zone length of concrete.” Acknowledgements JCI-C51E Post-Peak Behavior of RC Structures The authors would like to extend their appreciation to subjected to Seismic Loads, 2, 259-272. the Research & Development Center of Taiheiyo Ce- Popovics, S. (1973). “A numerical approach to the ment Corporation and Kuraray Co., Ltd. for their kind complete stress-strain curve of concrete.” Cement and support for the loading tests and for supplying acrylic Concrete Research, 3(5), 583-599. rods. Sangha, C. M. and Dhir, R. K. (1972). “Strength and complete stress-strain relationships for concrete References tested in uniaxial compression under different test Bazant, Z. P. (1989). “Identification of strain-softening conditions.” Matériaux et constructions, RILEM, constitutive relation from uniaxial tests by series 5(30), 361-370. coupling model for localization.” Cement and Sfer, D., Carol, I., Gettu, R. and Etse, G. (2002). “Study Concrete Research, 19(6), 973-977. of the behavior of concrete under triaxial Carpinteri, A. and Pugno, N. (2002a). “One-, two- and compression.” Journal of Engineering Mechanics, three-dimensional universal laws for fragmentation 128(2), 156-163. due to impact and explosion.” Journal of Applied Walraven, J. C. (1994). “Size Effects: Their Nature and Mechanics, 69, 854-856. Their Recognition in Building Codes, Size Effect in Carpinteri, A. and Pugno, N. (2002b). “Fractal Concrete Structure.” E&FM SPON. fragmentation theory for shape effects of quasi-brittle Yi, S. T., Kim, J. K. and Oh, T. K. (2003). “Effect of materials in compression.” Magazine of Concrete strength and age on the stress-strain curves on Research, 54(6), 473-480. concrete specimens.” Cement and Concrete Research, Hillerborg, A., Modeer, M. and Petersson, P. E. (1976). 33(8), 1235-1244. “Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements.” Cement and Concrete Research, 6(6), K. Watanabe, J. Niwa, H. Yokota and M. Iwanami / Journal of Advanced Concrete Technology Vol. 2, No. 3, 395-407, 2004 407

Notation LT Length of the transition zone [mm] a, b, c, e, f Experimental value LU Length of the unloading zone [mm] a0 Area under a relative stress-relative nF Experimental value strain curve for the failure zone P Load [kN] 2 Ac Cross-sectional area [mm ] W/C Water-to-cement ratio AF Consumed energy in failure zone ε Local strain

[kNmm] εave Average strain Ai Area under the stress-local strain curve εF Strain of the failure zone 2 [N/mm ] εF0 Peak strain of the failure zone AINTi Consumed energy in local portion εpeak Peak strain of the specimen [kNmm] εT Strain of the transition zone D Diameter [mm] εΤ0 Peak strain of the transition zone * D Equivalent section width ( = AC ) [mm] εΤ1 Strain at the changing point [MPa] d Displacement [mm] εU Strain of the unloading zone ' fc Compressive strength of concrete [MPa] εU0 Peak strain of the unloading zone GFc Energy consumed per unit volume of the σ Average stress [MPa] failure zone [N/mm2] σmax Maximum stress of a specimen [MPa] Gmax Maximum size of aggregate [mm] σT1 Stress at the changing point [MPa] H Height of specimen [mm]

H/D Height-to-diameter ratio of a specimen

Lp Length of the failure zone [mm]