Page 1

Strut-and-Tie Models for Dapped-End Beams Proposed model is consistent with observations of test beams by Alan H. Mattock

T

he use of the strut-and-tie method (STM) to model the behavior of dapped-end beams can be very useful in design; however, the choice of an appropriate STM truss model is very important. Logically, the STM truss model chosen should be consistent with the observed behavior of dapped ends, modeling the flow of forces in the dapped end. The use of STM models in design is based on the assumption that a selected truss model behaves plastically at loads approaching its “nominal” strength. An appropriate STM model will minimize the redistribution of internal forces and the inelastic deformation necessary for a member to develop its design nominal strength. As a result, STM models that closely approximate the flow of forces naturally occurring in a member will lead to the selection of more efficient reinforcement, and the resulting member will have narrower service-load cracks. The desirability of choosing an STM model in which the flow of forces approximates the flow of forces naturally occurring in the member was noted by Schlaich et al.1 in their landmark 1987 paper. Referring to the necessary inelastic deformations approaching nominal strength, they state, “In highly stressed regions this ductility requirement is fulfilled by adapting the struts and ties of the model to the direction and size of the internal forces as they would appear from the theory of elasticity.” They further emphasize that, while deviations from elastic stress trajectories are acceptable, the development of basic STM models should be consistent with observed force paths. In structural concrete, the development of cracks in the concrete is a good clue as to the orientation of tension and compression forces in the member. Compression forces are generally in alignment with the cracks and tension forces are oriented approximately normal to the cracks. A typical cracking pattern for a suitably reinforced dapped end approaching failure is shown in Fig. 1.

Fig. 1: Typical cracking approaching failure of a suitably reinforced dapped end

In this article, two widely published STM models for the dapped end are examined and compared with behavior observed in tests of 16 dapped ends subjected to a variety of combined vertical and outward horizontal reactions. It is shown that these STM models lead to overestimates of the amount of reinforcement required for a given combination of vertical and horizontal reactions. A simplified STM model for the dapped end is proposed. This model more nearly corresponds to the flow of forces observed in dapped ends and requires a smaller amount of reinforcement than the two other models.

Review of STM Models for Dapped Ends

A model of concern Originally proposed by Cook and Mitchell,2 the STM truss model shown in Fig. 2 has since appeared in ACI SP-2083 and ACI SP-273.4 Comparing with Fig. 1, it can Concrete international February 2012

35


1.5 kip 2.2 kip (6.7) (9.8) B

7 in. (178) 2 in. 10 kip (51) (44.5) 10 in. (250)

A

θ1

43.6 kip (194)

θ2

C

2.9 kip (12.9)

1.90 in. (48)

E

D θ 4

13.48 in. (342)

θ3

F

3 in. 4 in. 4 in. 7 in. 5 in. (76)(102)(102) (178) (127)

3.63 in. (92) (mm or kN)

Fig. 2: Since originally proposed by Cook and Mitchell,2 this STM truss model has been included in examples in ACI SP-2083 and ACI SP-273.4 Compression struts and tension ties are respectively represented by dashed and solid lines (after Fig. 2-4 in Reference 3)

Interface between nib Potential diagonal tension cracks and full-depth beam

Nib hn d n

Ah As

A Nn

Vn

h

D

lv a

Av

Hanger reinforcement Ash (closed stirrups)

45°

Beam flexural reinforcement

Beam shear reinforcement

Fig. 3: Schematic of dapped-end test specimens (after Reference 5). Specimens were 5 in. (127 mm) wide and 24 in. (610 mm) deep reinforced concrete beams. The distance from the vertical reaction to the center of the hanger reinforcement a and the effective depth of the nib flexural reinforcement dn were varied

7 in. (178) 2 in. 10 kip (51) (44.5) 10 in. (250)

A

43.6 kip (194)

1.5 kip 1.7 kip 2.4 kip (6.7) (7.6) (10.7) B crack E θ1

D

C

θ2

F

3 in. 4 in. 4 in. 7 in. (76)(102)(102) (178) Fig. 4: STM model truss after Reference 3

36

February 2012 Concrete international

1.90 in. (48) 13.48 in. (342)

3.63 in. (92) (mm or kN)

be seen that the assumed compression Strut BD must cross the diagonal tension cracks in this region, almost at right angles. The assumption of such a strut is therefore inconsistent with the behavior observed in many tests of beams with dapped ends. Use of the truss model in the example shown in Fig. 2 yields a value of 76.9 kips (342 kN) for the force in the hanger reinforcement (Tie BC). This is more than 1.8 times the net support reaction of 42.1 kips (187 kN) acting on the nib. Part of the calculated extra tension in the hanger reinforcement results from the assumed horizontal reaction at Node A, but this does not account for the full difference. Further, the model does not agree with test observations. Conflicting results Never, in any of the many tests of dapped ends that I have witnessed, has the measured force in the hanger reinforcement been so much in excess of the end support reaction, as indicated in Fig. 2. In almost all cases, the force in the hanger reinforcement has been very close to the magnitude of the vertical support reaction. This has been true both for the case of vertical support reaction only and for the case of combined vertical and outward horizontal reactions. Reference 5 provides supporting data. Specimens in this study were 5 in. (127 mm) wide and 24 in. (610 mm) deep reinforced concrete beams. A schematic of the dapped end of a test specimen from the study is shown in Fig. 3, and test parameters and results are listed in Table 1. In the test program, the distance from the vertical reaction to the center of the hanger reinforcement a and the effective depth of the nib flexural reinforcement dn were varied. The force in the hanger reinforcement (Tie AD in Fig. 2) at yield of the nib flexural reinforcement Fy(test) was deduced from the strain measured in the hanger reinforcement. It can be seen that in all the 16 dapped ends tested, Fy(test) was very close to the vertical reaction at yield of the nib flexural reinforcement Vy(test), with an average ratio of 0.97 and a standard deviation of 0.071. The applied outward horizontal force Nn in these tests varied from zero to 50 to 60% of the vertical reaction Vn. Alternative model Figure 2-6 of Reference 3, reproduced herein as Fig. 4, shows an alternative STM truss model previously proposed by the FIP.6 This model takes into account the diagonal tension cracking shown in Fig. 1 and, for the same loading condition as that shown in Fig. 2, predicts a force in the hanger reinforcement of 40.4 kips (180 kN), which is 96% of the net support reaction acting on the nib. Section 3.4.6 of Reference 3 discusses the difference in the calculated tension in Tie BC in the STM models shown in Fig. 2 and 4. It also observes that the STM model in Fig. 4 corresponds to the observed cracking, concluding: “This is an acceptable strut-and-tie model solution.”


B

Flexural compression Development length

A Shear stirrups

Inclined compression forces

D

P2 P1

45°

C Fig. 6: Simplified STM truss model. The external tensile Restraint P1 is provided by the development length extension of Tie AD

Beam flexural reinforcement Fig. 5: Transfer of force over the development length of the extension of Tie AD (refer to Fig. 3 and 4)

In Table 1, the nominal shear strength Vn(calc) was calculated considering the static equilibrium of those pieces of the dapped end that would be cut off by two cracks running from the reentrant corner of the dap—one vertically upward and one at 45 degrees to the horizontal and a third crack running upward at 45 degrees from the bottom corner of the full depth beam, as shown in Fig. 3. This approach was proposed in Reference 7 and leads to the conclusion that the hanger reinforcement force is equal to the vertical reaction acting on the dapped end. This approach is equivalent to assuming an STM model truss similar to that shown in Fig. 4, but without Tie DF and Node D. This is because both approaches are based on the satisfaction of static equilibrium for the various parts of the dapped end. The use of a tie such as Tie DF implies that the total tension in Tie AD must be resisted at Node D by Strut CD acting with Tie DF as a truss. This would only be true if Tie AD terminated at Node D with some form of positive anchorage, so that the total force in Tie AD was resisted by Strut CD acting with Tie DF as a truss. In fact, Tie AD is continued past Node D by a length sufficient to develop the yield strength of Tie AD, taking into account the depth of concrete below Tie AD. The buildup of force in Tie AD takes place gradually over this development length, rather than suddenly at Node D and, hence, there is no need for a tie such as Tie DF. (In the example from Reference 3, the force in Tie DF is 38.0 kips [169 kN]—approximately the same as the force in the hanger reinforcement.) No such tie was provided in the tests5 reported in Table 1, yet no tensile distress occurred due to the omission of such a tie. The tensile stresses in the concrete due to the transfer of force from the extension of Tie AD are evidently resisted by the surrounding concrete and the normally designed shear web reinforcement, because no extra cracking was observed in this part of the beam in the tests of References 5 and 7. In these tests, the

concrete and stirrup reinforcement stresses due to the development of Tie AD were higher than what would occur in most practical cases because the horizontal force acting on the dapped end was 50 to 60% of the vertical reaction, compared to 20% of the vertical reaction required by ACI 318-118 for corbel design and commonly used in the design of dapped ends.

Concrete international February 2012

37


Table 1:

Data from dapped-end beam tests5

Specimen No.

Nib depth hn, in. (mm)

Shear span a, hn/h in. (mm) a/dn

Vy (test), Fy (test), Vn (test), Nn, kips Vn (calc), kips Vn (test)/ kips Vy (test)/ kips Fy (test)/ Vn (calc) Vn (calc) Vy (test) (kN) kips (kN) (kN) (kN) (kN)

1A

12 (305)

0.50

9 (229)

0.82

0

39.24 (175)

46.44 (207)

1.18

42.15 (187)

1.07

42.7 (190)

1.01

1B

12 (305)

0.50

9 (229)

0.82

20 (89)

36.78 (164)

42.19 (188)

1.15

39.06 (174)

1.06

38.4 (171)

0.98

2A

12 (305)

0.50

11 (279)

1.01

0

46.17 (205)

46.77 (208)

1.01

44.10 (196)

0.96

43.0 (191)

0.98

2B

12 (305)

0.50

11 (279)

1.02

25 (111)

38.68 (172)

42.58 (189)

1.10

39.54 (176)

1.02

37.8 (168)

0.96

3A

12 (305)

0.50

13.63 (346)

1.25

0

38.19 (170)

44.38 (197)

1.16

40.68 (181)

1.07

37.8 (168)

0.93

3B

12 (305)

0.50

13.63 (346)

1.26

20 (89)

35.68 (159)

42.53 (189)

1.19

34.76 (155)

0.97

33.9 (151)

0.98

4A

12 (305)

0.50

16.25 (413)

1.49

0

31.31 (139)

39.45 (175)

1.26

31.87 (142)

1.02

31.9 (142)

1.00

5B1

12 (305)

0.50

16.25 (413)

1.50

22 (98)

28.26 (126)

36.83 (164)

1.30

28.33 (126)

1.00

27.6 (123)

0.97

5B2

12 (305)

0.50

16.25 (413)

1.50

23 (102)

27.76 (123)

32.00 (142)

1.15

24.75 (110)

0.89

27.8 (124)

1.12

6A

16 (406)

0.67

9 (229)

0.60

0

39.10 (174)

39.49 (176)

1.01

35.50 (158)

0.91

34.3 (153)

0.97

6B

16 (406)

0.67

9 (229)

0.60

28 (125)

38.64 (172)

45.85 (204)

1.19

36.30 (161)

0.94

28.7 (128)

0.79

7A

16 (406)

0.67

18.50 (470)

1.25

0

39.48 (176)

45.11 (201)

1.14

40.50 (180)

1.03

37.0 (165)

0.91

7B

16 (406)

0.67

18.59 (472)

1.25

20 (89)

35.47 (158)

40.21 (179)

1.13

34.80 (155)

0.98

36.0 (160)

1.03

8A

19 (483)

0.79

10.75 (273)

0.60

0

39.08 (174)

45.52 (202)

1.16

40.90 (182)

1.05

41.4 (184)

1.01

8B

19 (483)

0.79

10.75 (273)

0.60

28 (125)

40.04 (178)

45.62 (203)

1.14

40.50 (180)

1.01

36.3 (161)

0.90

9B

19 (483)

0.79

22.25 (565)

1.25

20 (89)

36.87 (164)

39.32 (175)

1.07

35.08 (156)

0.95

33.5 (149)

0.95

Mean

1.15

1.00

0.97

Standard deviation

0.076

0.056

0.071

a = distance from reaction V to the center of the hanger reinforcement dn = effective depth of nib reinforcement Fy(test) = measured force in hanger reinforcement Ash (cross-section area of hanger reinforcement), at shear Vy h = total depth of beam (24 in. [610 mm]) hn = total depth of the nib Nn = outward horizontal force acting on the dapped end at nominal strength Vn = nominal shear strength of dapped end Vy = shear acting on dapped end at yield of the nib flexural reinforcement

38

February 2012 Concrete international


Figure 5 shows how anchorage of Tie AD probably occurs by strutting action from the extension of the reinforcement beyond Node D. The bar extension would be anchored by struts between both the flexural reinforcement at the bottom of the beam and to the flexural compression zone at the top of the beam. The vertical components of the diagonal strut forces are taken up by the shear stirrups, which were continued to the end of the full depth part of the beams. Splitting of the beam along the axis of the development length would also be resisted by the tensile strength of the concrete窶馬o such splitting was observed in any of the tests. Point D in Fig. 3 corresponds approximately to Node D in Fig. 4. The stress in Tie AD was monitored at Point D by strain gauges and was found to reach the yield strength of the bar at nominal strength of the dapped end. It can be seen in Table 1 that the nominal strength of the dapped ends was closely (but conservatively) predicted using the assumption that the hanger reinforcement force at nominal strength is equal to the vertical reaction acting on the dapped end.

to develop its yield strength, taking into account the depth of concrete below the reinforcing bar. It must also be positively anchored at Node A. The hanger reinforcement should preferably be in the form of closed

stirrups, grouped close together and located as close as possible to the end face of the full-depth beam. This minimizes the distance a from the vertical reaction to the centerline of the hanger reinforcement (refer to

A Simplified STM Model for the Dapped End

I propose that an appropriate STM truss model would be similar to that shown in Fig. 4, with the omission of Tie DF. It is assumed that at Node D, an external tensile restraint acts on Tie AD and a compressive reaction acts on Strut CD. Node D is the point at which Tie AD is crossed by a line from the bottom corner of the full depth beam inclined at 45 degrees to the horizontal (Fig. 6). The external tensile Restraint P1 is provided by the development length extension of Tie AD. It is also assumed that the inclined compressive Force P2 is part of the overall truss action in the full depth beam. It should be noted that the bar along Tie AD must be extended beyond Point D by a length sufficient Concrete international February 2012

39


Fig. 3). In turn, this minimizes the force in the nib flexural reinforcement in Tie AD (Fig. 6) due to the steeper inclination of Strut AB. The reduction in force in AD and consequent reduction in bar size means that the stresses in the concrete beyond D, due to the development of force in AD, will be reduced. The hanger reinforcement must be looped around the reinforcement comprising Tie CF (Fig. 4), and the bars in Tie CF must also be positively anchored at Node C. In the case of a reinforced concrete beam, Tie CF will be an extension of the main flexural reinforcement of the beam. In the case of a prestressed concrete beam, Tie CF must extend a sufficient distance into the beam to enable it to transfer to the prestressing strand a force equal to its yield strength.9 This is to ensure the integrity of the truss action of the web reinforcement resisting shear in the full depth portion of the beam. It is assumed that the required regular web reinforcement will be carried to the end of the full depth part of the beam. In addition to the primary reinforcement corresponding to the tie members of the STM truss model, it is necessary to provide horizontal reinforcement in the nib to stabilize Strut AB. In the test specimens reported in References 5 and 7, this reinforcement was proportioned in the same way that similar reinforcement in a corbel is designed, according to Section 11.8 of ACI 318-11.8 This reinforcement is extended into the full-depth part of the beam a distance sufficient to develop its yield strength. In those test specimens in which a/dn exceeded 1.0 (refer to Table 1), the dapped-end nib was designed for shear according to the deep beam provisions of Reference 10.

Conclusions

Although used in design examples in ACI SP-2083 and ACI SP-273,4 the STM truss model shown in Fig. 2 is not consistent with the observed behavior of beams with dapped ends. Because it overestimates the force to be carried by the hanger reinforcement, its use will require the provision of more hanger reinforcement than is really necessary. The STM truss model originally proposed by FIP6 correctly estimates the required amount of hanger reinforcement, but requires additional reinforcement to carry the force in Tie DF (Fig. 4)—the amount being about the same as that required for the hanger reinforcement. This is actually correct only if some form of positive anchorage is provided at Node D for the nib reinforcement in Tie AD. If the Tie AD reinforcement is anchored by extending it past Node D—a length sufficient to develop its yield strength—then Tie DF is not necessary. In conclusion, the simplified STM truss model shown in Fig. 6 is consistent with observed behavior of dapped ends and will lead to the minimum required amount of reinforcement.

40

February 2012 Concrete international

Acknowledgment

The author wishes to thank J. Breen for his helpful comments on an earlier draft of this paper.

References 1. Schlaich, J.; Schäfer, K.; and Jennewein, M., “Toward a Consistent Design of Structural Concrete,” PCI Journal, V. 32, No. 3, May-June 1987, pp. 74-150. 2. 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. 3. Sanders, D.H., “Example 2: Dapped-End T-beam Supported by an Inverted T-beam,” Examples for the Design of Structural Concrete with Strut-and-Tie Models, SP-208, K.-H. Reineck, ed., American Concrete Institute, Farmington Hills, MI, 2002, pp. 91-103. 4. Mitchell, D.; Cook, W.D.; and Peng, T., “Example 14: Importance of Reinforcement Detailing,” Further Examples for the Design of Structural Concrete with Strut-and-Tie Models, SP-273, K.-H. Reineck and L.C. Novak, eds., American Concrete Institute, Farmington Hills, MI, 2010, pp. 237-252. 5. Mattock, A.H., “Behavior and Design of Dapped End Members,” Proceedings, Seminar on Precast Concrete Construction in Seismic Zones, V. 1, Tokyo, Japan, Oct. 29-31, 1986, pp. 81-100. 6. FIP Recommendations, Practical Design of Structural Concrete, FIP Commission 3 “Practical Design,” Sept. 1996, SETO, London, England, Sept. 1999 (distributed by fib, Lausanne, Switzerland). 7. 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. 8. ACI Committee 318, “Building Code Requirements for Structural Concrete (ACI 318-11) and Commentary,” American Concrete Institute, Farmington Hills, MI, 2011, 503 pp. 9. Mattock, A.H., and Abdie, J.L., “Transfer of Force between Reinforcing Bars and Pretensioned Strand,” PCI Journal, V. 33, No. 3, May-June 1988, pp. 90-106. 10. Joint ACI-ASCE Committee 426, “Suggested Revisions to Shear Provisions for Building Codes (ACI 426.1R-77),” American Concrete Institute, Farmington Hills, MI, 1979, 82 pp. Received and reviewed under Institute publication policies.

ACI Honorary Member Alan H. Mattock is Professor Emeritus of Civil Engineering at the University of Washington, Seattle, WA. He received his BS, MS, and PhD from the University of London, England. He was a member of ACI Committee 318, Structural Concrete Building Code, and ACI 318 Subcommittees on Shear and Torsion (318-E) and Precast and Prestressed Concrete (318-G) for 20 years. He received the Wason Medal for Most Meritorious Paper in 1967 and the Alfred E. Lindau Award in 1970.


Millions discover their favorite reads on issuu every month.

Give your content the digital home it deserves. Get it to any device in seconds.