ACII STRUCTURAL AC STRUCTURAL JOURNAL JOURNAL
TECHNICAL PAPER
Title no. 109-S02
Experimental Study of Dapped-End Beams Subjected to Inclined Loa Lo ad by Krystyna Nagrodzka-Godycka and Paweł Piotrkowski This paper presents the results of an experimental investigation of the reinforced concrete (RC) dapped-end beams loaded with inclined forces, compared to identical ones loaded with vertical forces only. only. Such a load may occur occur in, for example, Gerber’s Gerber’s joints or in dapped-end beams supported on corbels, where the vertical gravitation force is additionally completed with horizontal forces caused by temperature t emperature differences, differences, shrinking, or creeping. The problem of the state of stresses, cracking, and load capacity of dapped ends subjected to inclined forces, has not been experimentally sufficiently recognized. In the authors’ opinion, the results of the presented investigation establish conclusions that are useful in the theoreti theoretical cal verification of that type of structure. st ructure. Keywords: crack patterns; dapped-end beam; failure mode.
INTRODUCTION The principal stress trajectories, presented in Fig. 1(a), show the scale of the stress concentration appearing in the elastic state in the reentrant corner of a dapped end. The essential difference between a corbel and a dapped end is that the diagonal compressive force C c4 in a corbel (Fig. 1(b)) finds a stable support in the component force C c5 acting in the compressed zone near the edge of the column. In the case of a dapped-end beam, the vertical tension force T s2 is taken over by flexible steel bars, transmitting it into the bottom corner of the beam, where the state of equilibrium of forces occurs together with the additional component forces C c3 and T s1. The first experimental investigations of the dapped-end beams loaded with vertical forces, and with a combination of vertical and horizontal forces, were carried out by Mattock and Chan 1 and Mattock and Theryo.2 The results of these investigations have formed a base to create principles of the design of dapped-end beams—both reinforced and prestressed— with a shear span-depth ratio av /d k k ≤ 1.0. The recent design recommendations recommend ations for dapped-end beams in the PCI Design Handbook,3 applying the shear-friction theory, are based on Mattock and Chan’s 1 and Mattock and Theryo’s 2 works. The following experimental investigations, in most cases, concerned dapped ends subjected to vertical forces. 4-7 The results of these tests are described in detail by Nanni. 8 RESEARCH SIGNIFICANCE Considering the present state of knowledge on dapped-end beams, the authors decided to carry out experime experimental ntal research on these elements from a viewpoint of the influence of an additional horizontal force on the morphology of cracks, the stress state, and on the load-bearing capacity. The majority of standards recommend the assumption 9 of the horizontal component as H ≥ 0.2F V V . ACI 318M-08 (like its earlier version from 2005) does not exclude the maximum horizontal force H = F V V . In the investigation, ACI Structural Structural Journal/January-February Journal/January-February 2012
Fig. 1—Dapped-end beam supported on corbel: (a) trajectory of principal stresses in elastic state; and (b) scheme of inte rnal forces after cracking. cracking. the authors adopted the inclined force resulting from the relation H /F Vu Vu = 0.5, which was similar to Mattock and 1 Chan’s studies. To compare the specimens loaded with inclined force, specimens subjected only to the vertical force F V investigated. ted. V were also investiga
BACKGROUND The majority of European design requirements and standards for reinforced concrete (RC) structures 10,11 do not give the details related to action of inclined forces. In PN-B-
ACI Structural Journal, V. V. 109, No. 1, January-February 2012. MS No. S-2008-344.R5 received July 5, 2011, and reviewed under Institute publication policies. Copyright © 2012, American Concrete Institute. All rights reserved, including the making of copies unless permission is obtained from the copyright proprietors. Pertinent discussion including author’s closure, if any, will be published in the November-D November-December ecember 2012 ACI Structural Journal if the discussion is received by July 1, 2012.
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Krystyna Nagrodzka-Godycka is an Associate Professor in the Faculty of Civil and Environmental Engineering at the Gdansk University of Technology, Gdansk, Poland. Her research interests include shear strength and the strengthening of reinforced concrete structures. Paweł Piotrkowski is an Assistant Professor in the Faculty of Civil and Environmental Engineering at the Gdansk University of Technology. His research interests include reinforced concrete structures, especially reinforced concrete dapped-end beams.
Fig. 3—Geometry, dimensions, and scheme of investigated dapped-end beam loads ( hk / h = 0.5). (Note: Dimensions in mm; 1 mm = 0.0394 in.)
Fig. 4—Cracking of dapped-end beam (WB-3-L) loaded with inclined force (authors’ research). (Note: 1 kN = 0.2248 kips.)
1
.
s
Fig. 2—STM of dapped-end beams according to PN-B03264:200212: (a) suspending vertical reinforcement; and (b) diagonal reinforcement. 03264:200212 and EC2,11 the problem was considered by making use of the framework of strut-and-tie models. The earlier experimental investigations of the authors 13 contributed to the recommendations for design of the dapped-end beams. These recommendations12 were based on two general truss schemes presented in Fig. 2. In the scheme shown in Fig. 2(a), vertical stirrups took over the bearing reaction, whereas in Fig. 2(b), the bent-up bars did so, together with horizontal reinforcement in both cases. In addition, the formulas for dimensioning reinforcement of dapped-ends are given. The design principles presented in the Polish standard 12 apply to dapped end meeting the following conditions: lk ≤ hk and 0.3h ≤ hk ≤ 0.7h, while the width of the nib and its depth are to satisfy the following conditions
F V,Sd < F V,Rd max = 0.28 f cd · b · d k
≈
Sd d
12
a
1
)
(2b)
where aV is the distance from force axis to the edge of the undercut (refer to Fig. 2 and 3); zk is the lever arm of internal forces of the nib, which may be assumed as zk = 0.8d k; and a′ is the distance from the vertical undercut edge to the center of gravity of the suspending reinforcement (refer to Fig. 2(a) and (b)). The cross section of the suspending reinforcement in the form of vertical stirrups should satisfy the following condition
Asw V
1.3
.
Sd
(3)
f wd
The calculated suspending reinforcement is to be distributed along the member over a distance not longer than 0.2 h, counting from the undercut edge, where h is the overall depth of the beam. While applying the diagonal reinforcement inclined to the beam axis at the angle α, the cross section of bent-up bars should satisfy the following condition
(1)
The calculated value of concrete cylinder compressive strength12 f cd , with coefficient αcc = 0.85 ( f cd 0.85 f c′), is to be applied in Eq. (1). The horizontal dapped-end beam reinforcement should satisfy the following conditions
As
Sd
d
(2a)
Ast
S
f
⋅s n
(4)
In the results from the authors’ investigations, in dapped-end beams subjected to the inclined forces, very early cracks occur and their propagation is, to a great extent, affected by the type of reinforcement. As a result, a complicated crack pattern occurs, as shown in Fig. 4. An improper arrangement of the reinforcement causes an earlier local destruction of concrete and steel yielding as well.
k
ACI Structural Journal/January-February 2012
Table 1—Test details Shear span to effective depth ratio
Series Beam number Dapped end Type of load aV /d k 1 I 2
3 II 4
WB-1- L
F V + H
WB-1- P
F V + H
WB-2 - L
F V
WB-2 - P
F V
WB-3 - L
F V + H
WB-3 - P
F V + H
WB-4 - L
F V
WB-4 - P
F V
WB-5 - L
F V + H
5
a′, mm (in.)
0.9
F V + H
WB-6 - L
F V
2φ12mm 226 (0.34) 0.9
60 (2.36)
0.8
100 (3.94) 4φ12mm 452 (0.69)
0.9
WB-6 - P
387.3 (56.16)
383.8 (56.65)
4φ10mm 314 (0.48)
8φ10mm 628 (0.95)
402.7 (58.39)
381.7 (55.35)
f y, st , MPa (ksi)
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
1φ16mm 201 (0.305)
60 (2.36) 2φ8mm 101 (0.15)
0.9
Main inclined reinforcement
f y, MPa f y, sV , MPa Ast , mm2 (in.2) (ksi) AsV , mm2 (in.2) (ksi)
100 (3.94)
III 6
AsH , mm2 (in.2)
Vertical suspending stirrups
60 (2.36)
0.8 WB-5 - P
Main horizontal reinforcement
60 (2.36)
577.1 (83.68)
4φ8mm 201 (0.31)
577.1 (83.68)
2 φ16mm 402 (0.61) 1 φ16mm 201 (0.305)
418 (60.61)
2 φ16mm 402 (0.61)
F V
2 2 Notes: Denotation of reinforcement of Fig. 5; denotationF V + H indicates a nib loaded with inclined force; φ is bar diameter; 1 mm = 2.54 in.; 1 mm = 0.00152 in. ; 1 MPa = 0.145 ksi.
This situation makes the application of the scheme’s strut-and-tie model (STM); the stresses in the compressed concrete diagonal strut crossed by the cracks are adopted, considering that the concrete efficiency factor ν is significantly lower than unity. 14-18 Various sources give differing values of the efficiency factor of the concrete strut.19 As the research20 indicates, the lower-limit value of ν, determining the effective compressive strength of concrete strut, may attain the minimal value ν = 0.3.
EXPERIMENTAL PROGRAM Details of specimens The experimental program included 12 dapped ends as both ends of six beams. The geometries of all the beams— denoted with WB-1 and WB-2, WB-3 and WB-4, and WB-5 and WB-6—were identical (Fig. 3). The dapped-end beams WB-1, WB-3, and WB-5 were loaded with inclined reaction, whereas the dapped ends WB-2, WB-4, and WB-6 (as references), were loaded with vertical forces. It was assumed the constant relation H /F V = 0.5 was one-half the maximal value ( H /F V , max = 1.0), according to the provisions in Section 11.8 of ACI 318M-08. 9 On the bottom of the nibs, a 20 mm (0.79 in.) thick steel bearing plate was strongly fastened by steel rods in concrete through which vertical and horizontal loads were transferred to the dapped end. To resolve the behavior, dapped-end beams with uniform geometries and with high and low amounts of reinforcement in the “a priori” research assumed the ratio of bending reinforcement ρSH 1 = 0.76% and ρSH 2 = 1.52%. When specified in this way, the horizontal force As f yd using STMs (Fig. 2) then calculated the adequate vertical force F V , Sd , which in turn has been dimensioned web reinforcement (vertical stirrups and bent-up bars). ACI Structural Journal/January-February 2012
Fig. 5—Types of reinforcement for dapped ends: (a) with vertical suspending reinforcement; and (b) with bent-up bars. The data concerning the description and denotations of reinforcement of the tested dapped ends are presented in Fig. 5, and the detailed data are given in Table 1. Reinforcement of Beams WB-1 to WB-4 with suspending vertical reinforcement —The main horizontal reinforcement •
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Fig. 6—Reinforcement of tested beams: (a) WB-1 and WB-2 ( ρ sH = 0.76%); (b) WB-3 and WB-4 ( ρ sH = 1.52%). Dashed lines denote potential inclined cracks. (Note: Dimensions in mm; 1 mm = 0.0394 in.).
Fig. 7—Dapped-end beams WB-5-P and WB-6-P reinforced with bent-up bars: Ast = Ø16 mm (5/8 in.) for left side and 2Ø16 mm (5/8 in.) for right side of beam. (Note: Dimensions in mm; 1 mm = 0.0394 in.). AsH of the dapped ends of Beams WB-1 and WB-2 consisted of one loop (2Ø12) having a diameter Ø = 12 mm (1/2 in.), whereas the suspending vertical reinforcement AsV of two stirrups (2 × 2Ø10) had a diameter Ø = 10 mm (3/8 in.) (Fig. 6(a)). The dapped-end reinforcement AsH of the next pair of beams, WB-3 and WB-4 (Fig. 6(b)), were twice as much as the companion pair, and were composed of two loops (2 × 2Ø12) having a 12 mm (1/2 in.) diameter, whereas the suspending vertical reinforcement AsV consisted of four stirrups 4 × 2Ø10 (Ø = 10 mm [3/8 in.]) (refer to Fig. 6(b)). The remaining reinforcement in both pairs of beams was identical. The bars of the flexural longitudinal reinforcement of the beam (2Ø20 + 2Ø16, where Ø = 20 mm [3/4 in.] and Ø = 16 mm [5/8 in.]) were anchored by means of steel plates 20 x 85 x 170 mm (0.79 x 3.35 x 6.7 in.). The shorter length of the anchorage of the nib flexural reinforcement AsH of Beams WB-1, -2, -3, and -4 on the left side of the beam (L) assumed the possibility of a crack running from the reentrant corner of the dapped beam. Longer reinforcement in the right side of the beam assumed the possibility of an inclined crack running from the bottom corner of the beam (diagonal 45-degree dashed line on the right side of the beam [Fig. 6]). Reinforcement of Beams WB-5 and WB-6 with bent-up bars —In Beams WB-5 and WB-6, the suspending diagonal reinforcement of the dapped ends was a variable parameter. On the left side of the beams (dapped ends WB-5-L and WB-6-L), the reinforcement consists of a single bent-up bar Ø16 mm in diameter (5/8 in.). On the right side (Beams WB5-P and WB-6-P), there were two bars Ø16 mm (5/8 in.) in diameter (refer to Fig. 7). The structural reinforcement in
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Fig. 8—Dapped-end beam reinforcement: (a) with vertical stirrups, WB-3-L; and (b) with bent-up bars, WB-6-L. bent-up bars consisted of two double-leg stirrups Ø8 mm (5/16 in.). Figure 8 shows examples of two modes of the dapped-end beam reinforcement.
Test setu p Dapped ends loaded with inclined force —To cause an additional horizontal tension, acting simultaneously with the vertical component, a test stand presented in Fig. 9 was designed and executed. The load of the beam was transmitted from two hydraulic jacks by means of a traverse causing the compressive inclined forces in two bent-up members in a shape of the letter “A.” These members passed on inclined forces F /sinα on the bottom steel plates of the nibs. Two single diagonal telescopic pipes during the test played only the role of the zero-force stabilizers. In the entire investigation program, the relation of the horizontal forces to the vertical ones was adopted as ~0.5, corresponding to α = 63 degrees. Materials The concrete compressive strength determined from cylinders 150/300 mm (6/15 in.) for Beams 1 to 6 varied between 35.7 to 38.8 MPa (5.18 to 5.63 ksi); hence, the mean value was f cm = 36.4 MPa (5.28 ksi). The experimentally deter mined yield strength of reinforcement was approximately 400 and 577 MPa (58.39 and 83.66 ksi), respectively. Testing pro cedure The tested beams were loaded with two concentrated forces (Fig. 9), whose value was monotonically increased from F = 0 until failure, with steps of every 5 kN (1.124 kips). For each load increment to failure, the reinforcement strains and concrete strains were measured in each dapped-end ACI Structural Journal/January-February 2012
Table 2—Experimental and calculated data of failure forces Experimental vertical and horizontal failure force
F vu,exp, Dapped-end symbol kN (kips)
H exp, kN (kips)
Cracking force F v,crack , kN (kips)
F Vu, cal,12 (Eq. (1) governs), kN (kips)
F Vu, cal12 F Vu, cal12 F Vu, cal3 = V n * (Eq. (1a) with (Eq. (2a), (2b) PCI ν = govern) (Eq. (4.6.3.1) 0.6(1 – f c′/250)), †(Eq. (2) + Eq. (4) governs), kN (kips) governs), kN (kips) kN (kips)
F Vu, cal‡ (Eq. (5) governs), kN (kips)
η = F Vu,exp/ F Vu,calc
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
WB-1- L
65 (14.61)
32.5 (7.31)
10 (2.25)
253.2 (56.9)
130.2 (29.3)
22.66 (5.1)
44.5 (10.0)
58.6 (14.3)
1.11
WB-1- P
70 (15.74)
35 (7.87)
10 (2.25)
253.2 (56.9)
130.2 (29.3)
21.63 (4.9)
44.5 (10.0)
58.6 (14.3)
1.19
WB-2 - L
90 (20.23)
0
15 (3.37)
255.3 (57.4)
131.1 (29.5)
34.78 (7.8)
65.2 (14.7)
76.8 (18.8)
1.17
WB-2 - P
90 (20.23)
0
20 (4.5)
255.3 (57.4)
131.1 (29.5)
34.78 (7.8)
65.2 (14.7)
76.8 (18.8)
1.17
WB-3 - L
103 (23.15)
50.2 (11.28)
10 (2.25)
258.2 (58)
132.3 (29.8)
45.56 (10.2)
82.1 (18.4)
93.2 (22.8)
1.11
WB-3 - P
103 (23.15)
50.2 (11.28)
15 (3.37)
258.2 (58)
132.3 (29.8)
45.56 (10.2)
82.1 (18.4)
93.2 (22.8)
1.11
WB-4 - L
135 (30.35)
0
15 (3.37)
243.3 (54.7)
125.9 (28.3)
57.44 (12.9)
107.7 (24.2)
115.8 (28.3)
1.17
WB-4 - P
140 (31.47)
0
15 (3.37)
243.3 (54.7)
125.9 (28.3)
57.44 (12.9)
107.7 (24.2)
115.8 (28.3)
1.21
WB-5 - L
86 (19.33)
42 (9.44)
7 (1.57)
275.2 (61.9)
139.5 (31.4)
66.78† (15.0)
63.8 (14.3)
93.3 (22.8)
0.92§
WB-5 - P
100 (22.48)
46.2 (10.39)
15 (3.37)
275.2 (61.9)
139.5 (31.4)
124.26† (27.9)
96.0 (21.6)
137.5 (33.7)
0.73§
WB-6 - L
119 (26.75)
0
15 (3.37)
249.7 (56.1)
128.7 (28.9)
82.55† (18.6)
87.7 (19.7)
116.0 (28.4)
1.03
WB-6 - P
160 (35.97)
0
15 (3.37)
249.7 (56.1)
128.7 (28.9)
141.93† (31.9)
131.9 (29.6)
171.0 (41.9)
0.94
*
Eq. (1a) → F V,Sd < F v,Rd max = 0.28 f cd · ν · b · d k. † As = Ash + cosα Ast . ‡ Failure force calculated from Eq. (5) and Fig. 23. § Anchorage failure.
beam. Measurements of the width of cracks, together with recording the crack propagation, were also made. Crack widths were measured using the microscope with a magnification of 40×.
RESULTS OF EXPERIMENTS Crack pattern The first crack in both dapped ends WB-1-P and WB-2-P occurred in the reentrant corner. On the right side of the beam (WB-1-P) loaded with inclined force (Fig. 10(a)), the first crack was caused by a force of 10 kN (2.22 kips)—that is, 14% of the failure force. In WB-2-P (Fig. 10(b)) under vertical load, that crack force was doubled and reached 0.22F Vu. The crack pattern caused by inclined force was definitely different from the pattern caused by vertical force. The inclined force brought about more vertical cracks concentrated in the reentrant corner zone, whereas from the vertical force, inclined cracks covered nearly entire area of the nib and their inclination was smaller (approximately 30 degrees), only the early cracks occurring in the reentrant corner had the inclination angle from the beam axis of approximately 50 degrees. In the case of dapped ends with high amounts of reinforcement (Beams WB-3 and WB-4, ρsH =1.52%), differences in cracks under inclined and vertical forces were similar to those with a low amount of reinforcement of dapped-end beams WB-1 and WB-2 with ρsH = 0.76%. The crack pattern of Beam WB-3-L, loaded with inclined force, is presented in Fig. 4, whereas that of its reference, Beam WB-4-L, was loaded with vertical force (refer to Fig. 11). The dapped-end beam, WB-3-L, with an inclined loaded, cracked under 10 kN (2.25 kips). The dapped ends of Beam WB-4, identically reinforced and loaded with vertical force, cracked under 15 kN (3.37 kips) on both sides. The load-bearing capacity of dapped-end beam WB-4, loaded with vertical force only, exceeded the capacity of ACI Structural Journal/January-February 2012
Fig. 9—Testing stand for dapped-end beam subjected to inclined force. (Note: Dimensions in mm; 1 mm = 0.0394 in.) dapped-end beam WB-3, which was inclined-loaded by more than 30%. The recorded force at the appearance of the first crack and the load capacity of dapped ends are given in Table 2. The dapped ends of Beam WB-5, reinforced with bent-up bars having a 16 mm (5/8 in.) diameter, (WB-5-L: 1Ø16, and WB-5-P: 2Ø16), loaded with inclined force, demonstrated cracking load F V,cr = 7 and 15 kN (1.57 and 3.37 kips), respectively, and failure load F Vu,exp, for WB-5-L (1Ø16) = 86 kN (19.33 kips), and with inclined 2Ø16 = 119.0 kN (22.48 kips) (Fig. 12). In the dapped ends of Beams WB-5-L and WB-6-L, the bent-up bars did not have the significant influence on the pattern of cracks. The clear difference occurred mainly in the upper corner where, in the case of the nib WB-5-L, inclined force resulted in a lack of cracks in that region. Next, the activity of a major vertical force F V, without participation of horizontal force H , resulted in a
15
Fig. 10—Dapped ends with suspending vertical stirrups and extended length of horizontal reinforcement anchorage ( ρ sH = 0.76%): (a) Beam WB-1-P, loaded with inclined force; and (b) Beam WB-2-P, loaded with vertical force. (Note: 1 kN = 0.2248 kips.)
Fig. 11—Dapped-end Beam WB-4-L with suspending vertical ( ρ sH = 1.52% and ratio aV / dk = 0.9): (a) crack pattern; and (b) photo after failure. (Note: 1 kN = 0.2248 kips.)
Fig. 12—Dapped-end beams with bent-up bars after failure: (a) WB-5-L loaded with inclined force (1Ø16); and (b) WB-6-L loaded with vertical force (1Ø16). (Note: 1 kN = 0.2248 kips). more intensive cracking of the upper corner, where inclined cracks of a smaller inclination to the axis of the member occurred. That crack pattern corresponded very closely to the trajectory of the principal compressive stress of the elastic state. In Fig. 13, the crack width development in the reentrant corner versus the F V /F Vu ratio is shown. In all tested dapped-ends, cracks with the small width in the reentrant corner appeared very early, which was expected due to the maximum concentration of the tension stresses in that region. The first cracks appeared in the concrete layer covering the reinforcement. The width of cracks in the concrete above the reinforcement was considerably smaller. Under the service load, which was assumed as 55% of the design load, the crack width in the reentrant corner was 1.1 mm (0.043 in.) for orthogonal reinforcement, whereas
16
at the bent-up reinforcing bars, it was 0.9 mm (0.035 in.). It was also noted that the amount of reinforcement did not significantly influence the development of crack width in the reentrant corner (Fig. 13).
Strains of reinforcing bars The strains of reinforcing bars were measured on every loading stage using electric resistance strain gauges with bases 10 mm (0.4 in.) in length. The readings were recorded on every loading stage. The location of the measuring gauges on the steel reinforcement are presented in Fig. 14 to 16. On stirrups and reinforcement loops, a gauge was placed on each bar in the same cross section (they were denoted with two numbers). The main horizontal reinforcement situated closest to the tensioned edge of nibs WB-1 and WB-2 were yielding at the ACI Structural Journal/January-February 2012
Fig. 13—Width of crack development in reentrant corner of WB beams versus load ratio FV / FVu.
Fig. 14—Location of strain gauges on reinforcement of dapped ends of Beams WB-1 and WB-2. (Note: Dimensions in mm; 1 mm = 0.0394 in.) half of the bearing load capacity (Gauges 1 and 2, Fig. 17). In turn, the increase of stress of vertical stirrups 4 × 2Ø10 mm (Ø10 mm = 3/8 in.), was not so significant (Fig. 18). The failure in Beam WB-3, at the significantly higher failure force, was proceeded by yielding of horizontal reinforcement and the stirrups (Fig. 19 and 20). In Fig. 16, the location of measuring bases of the left dapped ends of Beams WB-5 and WB-6 was presented. The main suspending reinforcement was, in this case, made of bent-up bars. Contrary to the bent-up bars and horizontal reinforcement, the stresses in stirrups, even at failure ( F V /F Vu = 1.0), did not reach yielding (Fig. 21).
Load-bearing capacity The failure forces for the dapped-end beams are presented in Table 2. In Fig. 22, the values of failure forces are given in six pairs of beams from WB-1 to WB-6, which vary according to the load acting on dapped ends. For the dapped ends of Beams WB-1, WB-3, and WB-5, loaded with inclined force, the relative percentage of the load capacity in reference to the dapped ends in Beams WB-2, WB-4, and WB-6, loaded with vertical force only, is given. Symbols on the horizontal axis indicate two dapped-ends of beam in a given pair, and the side “P” refers to the right and “L” to the left. For example, symbol WB-3/4-P indicates a pair of right-side dapped ends of Beam WB-3 loaded with inclined force, and WB-4 loaded with vertical force. The first (light) columns indicate odd-number beams (1, 3, and 5) loaded with ACI Structural Journal/January-February 2012
Fig. 15—Location of strain gauges on reinforcement of dapped ends of Beams WB-3 and WB-4. (Note: Dimensions in mm; 1 mm = 0.0394 in.)
Fig. 16—Location of strain gauges on reinforcement of dapped ends of Beams WB-5 and WB-6. (Note: Dimensions in mm; 1 mm = 0.0394 in.)
Fig. 17—Strains of horizontal reinforcement versus FV / FVu ratio for right dapped end of Beam WB-1. (Note: 1 kN = 0.2248 kips.) inclined force. The second (dark) column of each group of results shows failure forces of the dapped ends loaded with vertical force only. In the first pair of beams, the WB-1 and WB-2 dapped ends were weakly reinforced with vertical stirrups and horizontal loops (ρsH = 0.76%). The failure forces for both dapped ends WB-2-L and WB-2-P loaded with vertical reaction only were equal to 90 kN (20.23 kips). The imposition of an additional
17
Fig. 18—Strains of stirrups versus FV / FVu ratio for right dapped end of Beam WB-1. (Note: 1 kN = 0.2248 kips.)
Fig. 19—Horizontal reinforcement strains versus FV / FVu ratio for left dapped end of Beam WB-3. (Note: 1 kN = 0.2248 kips.)
Fig. 20—Strains of stirrups versus FV / FVu ratio for left dapped end of Beam WB-3. (Note: 1 kN = 0.2248 kips.) horizontal force H ≈ 0.5F V , which took place in Beam WB-1, caused the load capacity decrease in the left-side dapped end WB-1-L to F Vu = 65 kN (14.61 kips). On the right side of the beam (dapped end WB-1-P), at the horizontal force H u
18
Fig. 21—Strains of reinforcement versus FV / FVu ratio for dapped-end Beams WB-6-L: (a) horizontal and inclined (Gauge 4 defected); and (b) vertical stirrups. (Note: 1 kN = 0.2248 kips.)
Fig. 22—Failure forces of dapped ends in separate pairs of beams. (Note: 1 kN = 0.2248 kips.) = 35 kN (7.87 kips), the vertical component F Vu decreased to 70 kN (15.74 kips). That meant the load capacity caused by horizontal load decreased by an average of 25%. In the following pair of beams (WB-3 and WB-4), with double the reinforcement ratio ( ρsH = 1.52%), vertical
ACI Structural Journal/January-February 2012
•
•
•
•
•
Fig. 23—Internal forces system. (Note: Dimensions in mm; 1 mm = 0.0394 in.) •
failure forces for dapped ends increased, respectively, up to 135 kN (30.35 kips) for WB-4-L and 140 kN (31.47 kips) for WB-4-P. It was an increase by 50 to 55% in relation to the dapped ends with the reinforcement ratio ρsH = 0.76%. In turn, the load capacity for dapped ends with the reinforcement ratio ρsH = 1.52% (Beam WB-3) due to inclined force decreased by approximately 25%. In the case of beams reinforced with bent-up bars (Beams WB-5 and WB-6), the failure forces for dapped ends with a low amount of reinforcement 1Ø16 mm (5/6 in.) for Beams WB-5-L and WB-6-L were the following: for dapped ends subjected to the vertical force only, F Vu = 119 kN (26.75 kips), for dapped ends subjected to inclined force, F Vu = 86 kN (19.33 kips). Thus, for weaker reinforcement, the action of an additional horizontal force caused the load capacity to decrease by 28%. A double increase of the bent-up bars 2Ø16 mm (5/6 in.) for Beams WB-5-P and WB-6-P caused the load-bearing capacity for the inclined load to increase by 16%, whereas for vertical load, the increase reached 35%. The design procedure, according to the Polish standard given in the paper (Eq. (1) through (4)), makes it possible to separately dimension the different kinds of reinforcement. In reality, all kinds of reinforcement transfer the load simultaneously. The bearing load calculated from those formulas gives the results much lower than the values obtained from the experimental study. In the authors’ opinion, the scheme based on the crack’s morphology to calculate the load-bearing capacity could be used (Fig. 23). It provides an equation to calculate the load capacity
∑ T j F u cal =
j
j
H v
h
(5)
The experimental and predicted results of the failure load according to the Polish standard, 12 PCI provisions,3 and Eq. (5) are given in Table 2.
ACI Structural Journal/January-February 2012
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CONCLUSIONS In the dapped-end beam loaded with inclined force, the crack patterns were different from those observed in dapped ends loaded with vertical forces; The crack’s width wcr > 0.3 mm (0.012 in.) have appeared in dapped-end beams under the service load; Imposing an additional horizontal force equal to one-half of the component vertical force caused a 25% decrease of the load capacity; Doubling the horizontal reinforcement, as well as suspending vertical stirrups, regardless of the type of load, increased the load capacity only by approximately 50%; Experimentally, in the case of main reinforcement in the form of bent-up bars, an increase of number of bent-up bars from one Ø16 mm (5/8 in.) in diameter to two Ø16 mm (5/8 in.) increased the load capacity only by approximately 16% under inclined force, and by approximately 35% under vertical force; The failure of dapped ends with high amounts of reinforcement ( ρsH = 1.52%) most often occurred after the yielding of both the horizontal reinforcement and vertical stirrups; In the case of low amounts of reinforcement ( ρsH = 0.76%), the situation is different. The suspending reinforcement (bent-up bar and vertical stirrups) did not reach yielding; and In the authors’ opinion, the bearing load capacity F Vu,cal can be calculated using the internal forces system determined on the basis of the crack morphology. REFERENCES
1. Mattock, A. H., and Chan, T. C., “Design and Behavior of Dapped End Beams,” PCI Journal, V. 24, No. 6, Nov.-Dec. 1979, pp. 28-45. 2. Mattock, A. H., and Theryo, T. S., “Strength of Precast Prestressed Concrete Members with Dapped Ends,” PCI Journal, V. 31, No. 5, Sept.Oct. 1986, pp. 58-75. 3. PCI Design Handbook, Precast and Prestressed Concrete , sixth edition, Precast/Prestressed Concrete Institute, Chicago, IL, 2004. 4. Hamoudi, A. A.; Phang, M. K. S.; and Bierweiler, R. A., “Diagonal Shear in Prestressed Concrete Dapped-Beams,” ACI J OURNAL, Proceedings V. 72, No. 7, July 1975, pp. 347-350. 5. Steinle, A., “Zum Tragverhalten ausgeklinkter Trägerenden,” Vorträge Betontag, Deutscher Beton-Verein E.V., 1975, pp. 364-377. (in German) 6. Mader, D. J., “Detailing Dapped Ends of Pretensioned Concrete Beams,” MS thesis, University of Texas at Austin, Austin, TX, May 1990, 100 pp. 7. Lu, W.-Y.; Lin, I.-J.; Hwang, S.-J.; and Lin, Y.-H., “Shear Strength of High-Strength Concrete Dapped-End Beams,” Journal of the Chinese Institute of Engineers, V. 26, No. 5, 2003, pp. 671-680. 8. Nanni, A., and Huang, P. C., “Validation of an Alternative Reinforcing Detail for the Dapped Ends of Prestressed Double Tees,” PCI Journal, V. 47, No. 1, Jan.-Feb. 2002, pp. 38-49. 9. ACI Committee 318, “Building Code Requirements for Structural Concrete (ACI 318M-08) and Commentary,” American Concrete Institute, Farmington Hills, MI, 2008, 473 pp. 10. DIN 1045-1, “Tragwerke aus Beton, Stahlbeton und Spannbeton— Teil 1: Bemessung und Konstruktion,” Deutsches Institut für Normung, July 2001, 147 pp. (in German) 11. EN 1992-1-1, “Eurocode 2: Design of Concrete Structures—Part 1: General Rules and Rules for Buildings,” European Committee for Standardization, Brussels, Belgium, Dec. 2004, 225 pp. 12. PN-B-03264, “Design of Concrete Structures,” PKN, Polish Committee for Standardization, Warsaw, Poland, 2002, 142 pp. (in Poli sh) 13. Nagrodzka-Godycka, K., “Reinforced Concrete Corbels and Dapped-End Beams—Experimental Research, Theory and Desi gn,” Monograph No. 21, Wydawnictwo Politechniki Gdanskiej, Gdansk, Poland, 2001, 263 pp. (in Polish) 14. Vecchio, F. J., and Collins, M. P., “The Modified CompressionField Theory for Reinforced Concrete Elements Subjected to Shear,” ACI JOURNAL, Proceedings V. 83, No. 2, Mar.-Apr. 1986, pp. 219-231.
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15. Vecchio, F. J., and Collins, M. P., “Compression Response of Cracked Reinforced Concrete,” Journal of Structural Engineering, ASCE, V. 119, No. 12, Dec. 1993, pp. 3590-3610. 16. Cook, W. D., and Mitchell, D., “Studies of Disturbed Regions near Discontinuities in Reinforced Concrete Members,” ACI Structural Journal, V. 85, No. 2, Mar.-Apr. 1988, pp. 206-216. 17. Foster, S. J., and Gilbert, R. I., “The Design of Nonflexural Members with Normal and High-Strength Concretes,” ACI Structural Journal, V. 93, No. 1, Jan.-Feb. 1996, pp. 3-10.
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18. Park, J., and Kuchma, D., “Strut-and-Tie Model for Strength Prediction of Deep Beams,” ACI Structural Journal, V. 104, No. 6, Nov.-Dec. 2007, pp. 657-666. 19. Schlaich, J., and Schäfer, K., “Design and Detailing of Structural Concrete Using Strut-and-Tie Models,” The Structural Engineer , V. 69, No. 6, Mar. 1991, 13 pp. 20. Matamoros, A. B., and Wang, K. H., “Design of Simply Supported Deep Beams Using Strut and Tie Models,” ACI Structural Journal, V. 100, No. 6, Nov.-Dec. 2003, pp. 704-712.
ACI Structural Journal/January-February 2012
Disc. 109-S02/From the January-February 2012
DISCUSSION ACI Structural Journal ,
p. 11
Experimental Study of Dapped-End Beams Subjected to Inclined Load. Paper by Krystyna Nagrodzka-Godycka and Paweł Piotrkowski Discussion by Andor Windisch ACI member, PhD, Karlsfeld, Germany
The authors should be complimented for their interesting and promising paper. The description given in the introduction, which reflects the current conception of the way dapped ends supported on a corbel work, needs some corrections: The stress trajectories shown in Fig. 1(a) reveal that the bottom corner of the beam is quite stressless. The vertical reinforcing bars carrying tensile force T s2 do not transmit any relevant tension there; thus, the inclined compression force C c3 does not exist at all. The main longitudinal reinforcement of the beam (T s1) could be bent up. Moreover, T s1 ≠ T s2; hence, C c3 might not be inclined under 45 degrees, even if it did exist. (The identical failure forces of dapped ends L versus P related to Beams WB-1 to WB4 and the relevant crack patterns shown in the paper prove this statement.) If the column is deep enough to anchor the reinforcing bars resisting T s4, then no deviation occurs and, hence, the existence of C c7 is questionable as well. The compression force C c6 does not exist either. Which forces equilibrate C c6 at its other end? At the end of the background section, the authors refer to the confusion regarding the concrete efficiency factor ν, which reveals the shortages of the strut-and-tie model (STM). As long as the support plate is well-dimensioned, the strength of the compression strut will never govern the failure of any dapped end. When comparing the failure loads of the dapped ends with and without horizontal load, it should be considered that the horizontal force diminishes the horizontal reinforcement, which equilibrates the bending moment caused by the vertical load. Therefore, the references to the decreasing capacities found at the inclined-loaded dapped ends are irrelevant. The crack widths measured under the assumed serviceload level were quite high. Where were these crack widths measured? It should be noted that the first layer of the main reinforcement was placed quite high over the steel plates. Comparing the crack patterns with the locations of the strain gauges on the reinforcing bars, the discusser can conclude that most of the measured strains shown in Fig. 17 through 21 are not relevant. Only the strains measured with those gauges in or near the cracks can display the real participation of the reinforcing bars. Therefore, all conclusions concerning the stresses in the stirrups should be revised. The authors are correct that the crack pattern depends on the loading pattern. Their opinion that “the bearing load capacity can be calculated using the internal forces system determined on the basis of the crack morphology” is correct. Nevertheless, the form of the free body—that is, the angle α1, as shown in Fig. 23—also depends on the loading •
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ACI Structural Journal/November-December 2012
pattern, as proven previously by the discusser.21 The authors are encouraged to continue their promising research.
REFERENCES 21. Windisch, A., “Das Modell der charakteristischen Bruchquerschnitte— ein Beitrag zur Bemessung der Sonderbereiche von Stahlbetontragwerken (The Model of Characteristic Fracture Cross-Sections—A Contribution to the Design of the D-Regions of Reinforced Concrete Structures),” Betonund Stahlbetonbau, V. 83, No. 9 and 10, Sept. and Oct. 1988, pp. 251-255 and 271-274. (in German)
AUTHORS’ CLOSURE The authors would like to thank the discusser for his interest in the paper and provide clarifications to the comments and questions raised. Figure 1(a) shows the stress trajectories in the elastic state. In Fig. 1(b), the authors introduced the internal forces scheme after cracking of the dapped end reinforced with vertical suspended stirrups working together with the main horizontal longitudinal reinforcement. The bottom corner of the beam is stressless only in the elastic state on the assumption that its body is isotropic. The tension force T s2 (in suspending vertical stirrups) and the compression force C c3 exist in this dapped end. The C c3 inclination obviously does not have to be equal to 45 degrees. The authors’ test measurements of the strains and crack patterns confirm this. This type of scheme (Fig. 1(b)) was also used in German works. 22,23 According to Schlaich,23 the force T s2 can increase considerably (even to 1.5F v) in the case of action of a vertical force with participation of the horizontal force H . The authors agree with the discusser’s statement that diagonal compression force C c3 and tensile force T s2 do not exist if the main longitudinal reinforcement T s1 is bent up.22 Bent-up reinforcement controls the width of the inclined cracks more effectively. The compression force C c7 appears in the case of a one-sided corbel. Additionally, it is caused by the load of the column. The crack patterns of corbels tested by Niedenhoff 25 confirmed this kind of scheme. This is also assumed by Schlaich.23 Compression forces C c5 and C c6 in the bottom compression node of the corbel are components of the inclined force C c4 existing in the compression strut of the corbel.11,24 Support plate dimensions are one of a few reasons causing concrete compression failure in the support zone and concrete strut. The width of the main cracks was measured by visual observations of the concrete surface using a microscope with a magnification of 40×. In this study of dapped-end beams with a low reinforcement ratio, the effective depth d k •
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is assumed constant. The wide cracks in the cover are attributed to its large thickness. The major strain gauges were located near the cracks; some of the gauges were in the cracks showing maximum strains or strains close to maximum. The authors agree with the discusser’s opinion that the inclination of the angle α1 between the diagonal compressive strut and the flexural reinforcement of the beam longitudinal axis also depends on the pattern of the load.
902
REFERENCES 22. Bachmann, H.; Steinle, A.; and Hahn, V., Bauen mit Betonfertigteilen im Hochbau, Ernst & Sohn, Be rlin, Germany, 2010. (in German) 23. Schlaich, J., “Zum einhei tlichen Bemessen von Sthalbetontragwerken,” Beton- und Sthalbetonbau, No. 4, 1984, pp. 89-96. (in German) 24. Structural Concrete: Textbook on Behaviour, Design and Performance, Fédération Internationale du Béton ( fib), V. 3, 1999, 338 pp. 25. Niedenhoff, H., “Untersuchungen über das Tragverhalten von Konsolen und kurzen Kragarmen,” dissertation, the Universität Karlsruhe (TH), Karlsruhe, Germany, 1961.
ACI Structural Journal/November-December 2012
