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Curved Bar Nodes

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42
SEPTEMBER 2008
/
Concrete international
BY GARY J. KLEIN
Curved-Bar Nodes
A detailing tool for strut-and-tie models
S
ince reinforcing bars were first used in concrete, they have been designed with bends for anchorage and transfer of forces in connection regions. Examples include beam-column and wall-slab junctions, where bent bars at the outside of a frame corner resist closing moments. The forces in these corners can be modeled using the strut-and-tie model (STM) shown in Fig. 1. In this example, the curved region of the bar is modeled as a series of segments with tension forces opposed by a series of fan-shaped compression struts. In this article, I’ll explain how this curved region can be modeled as a curved-bar node. In the general case, a curved-bar node is the bend region of a continuous reinforcing bar (or bars) where two tension ties are in equilibrium with a compression strut in an STM. Specific recommendations for design and detailing using STMs with curved-bar nodes in frame corners and dapped-end beams are offered.
STRUT-AND-TIE MODEL
The tie forces at a curved-bar node must be equilibrated by one or more struts. In most cases, the intersection of two ties and a strut at the curved-bar node form a compression-tension-tension (C-T-T) node. Several additional examples of curved-bar nodes in concrete connection regions, or so-called D-regions, are shown in Fig. 2. To simplify analyses of C-T-T nodes such as shown in Fig. 1, 2(a), or 2(b), the curved region can be modeled as a single node at the intersection of the centerlines of the straight ties (Point a in Fig. 1).As shown by the example C-T-T node in Fig. 3, nodal zones are generally too small to allow development of tie forces through bond alone. If conservative design guide-lines for the use of curved-bar nodes in D-regions are developed, curved-bar nodes can provide a cost-effective, simpler alternative to separate mechanical anchorages. Although Appendix A of ACI 318-08
1
does not yet recognize curved-bar nodes, its provisions can be used to develop design recommendations.
COMPRESSIVE STRESS AT CURVED-BAR NODES
In a typical case, a strut bisects the angle formed by the ties extending from the curved-bar node. For 90-degree corners with equal tie forces, the strut angle is 45 degrees, and, using a pressure vessel analogy, the compressive stress acting in the curved region of the bar depends only on the radius of the bend and the tensile force in the bar. It follows that no bond stresses are required within the curved region itself—only a uniform,
Fig. 1: Strut-and-tie model of forces due to closing moment at a frame corner
45
º
C
1
C
1
A
ts
f
y
A
ts
f
y 12
C 2C
=
a
Closing moment Closing moment
Worked example designs for strut-and-tie models using curved-bar nodes are available with the online version of this article at www.concreteinternational.com
Concrete international
/
SEPTEMBER 2008
43
(a) (b)
(c)(d)20
º
WWR throughoutPrestressing strand(not shown in elevation
)
radial compression stress is required to maintain equilibrium within the region. By placing conservative limits on the compressive stresses acting on the bar, an equilibrium (lower bound) model for the node can be established. Compressive stress
f
cu
at a curved-bar node is limited by the yield strength of the tie reinforce-ment. Thus, the maximum value is given by
r
b
b
y
(1)where
A
ts
is the area of nonprestressed tie reinforcement,
f
y
is the specified yield strength of the tie reinforcement,
r
b
is the inside radius of the reinforcing bar bend, and
b
is the width of the strut transverse to the plane of the STM.The minimum bend radius in terms of allowable nodal stress,
f
ce
,
can be derived by reorganizing Eq. (1) as follows
bf
ce
r
b
(2)
Fig. 2: Strut-and-tie models with curved-bar nodes: (a) column corbel; (b) dapped- end beam; (c) hammer-head bridge pier with rounded end; and (d) pocket in a spandrel beamFig. 3: Development of the tie reinforcement in nodal zones cannot rely on bond alone (from Reference 1)
NODE STRENGTH
ACI 318 requirements for C-T-T nodes
ACI 318 limits the compressive stress at nodes to 0.85
β
n
f ′
c
. The
β
n
values reflect the degree of disruption of the nodal zone due to the incompatibility of tension strains in the ties and compression strains in the struts. For C-T-T nodes, where there is tension strain in two directions,
β
n
is 0.6, which is less than the 0.8 value for compression-compression-tension (C-C-T) nodes. Thus, the allowable compressive stress for a typical curved-bar node (a C-T-T node) is 0.85 × 0.6 = 51% of the specified compressive strength,
f ′
c
.
Frame corners tests
To confirm that the ACI 318 limits are appropriate, the performance of frame-corner connections with curved-bar nodes was reviewed. Several researchers
2-6
have studied the flexural strength of reinforced concrete corners. In most cases, conventional corner reinforcement (Fig. 1) is sufficient to develop the full flexural strength of the adjacent members. One study
6
of the flexural strength of corners reinforced with two No. 6 (No. 19) bars spaced
44
SEPTEMBER 2008
/
Concrete international
3 in. (75 mm) apart in 6 in. (150 mm) square members, however, reported that all of the specimens (except for one specimen with a welded diagonal stiffener) failed at a moment below the nominal flexural strength. The report noted that, “It is likely that the failures were caused by bearing failure of the concrete in the diagonal compression zone between the bends in the tensile and compressive reinforcement.”
6
In these tests, the calculated compressive stress at ultimate load given by Eq. (1) varied from 1.08 to 1.60 times the measured concrete compressive strength. Thus, the limiting strength given in ACI 318 of 0.51
f
c
′
appears safe, if not overly conservative. Using the ACI 318 strength limit for C-T-T nodes (
β
n
= 0.6), the minimum radius becomes
bf
c
' r
b
2
≥
(3a)
DAPPED-END BEAM TESTS
Tests
7
have also been performed on a series of dapped-end tee beams to investigate several reinforcement schemes for the end region. A typical reinforcement scheme (Specimen 1B) is shown in Fig. 4(a). Specimen 1B failed at a load of 27 kips (120 kN) when a diagonal tension crack developed and extended through the lower corner of the full section. Although the end region was not designed using an STM, the cracking pattern clearly indicated a diagonal strut extending upward from the lower corner of the full section. The lower end of strut was equilibrated by the inclined and horizontal extensions of the continuous reinforcing bars [1 No. 4 and 1 No. 3 (1 No. 13 and 1 No. 10)] at the lower corner of the full section. Thus, the bend region is a curved-bar node. Shear reinforcement was not used inside the end region, so there were no ties at the upper end of the strut. The expected shear resistance of the full section, however, was developed, and the overall performance was considered satisfactory. Therefore, although the diagonal crack extended through the curved-bar node, specimens exhibiting this behavior should be considered to have failed in shear.An alternate scheme (Specimen 2B), in which a single inclined hanger bar was used, is shown in Fig. 4(b). This specimen did not exhibit the typical failure behavior exhibited by Specimen 1B. At a load of only 20 kips (89 kN), the web split at the lower corner of the full section. This failure at the curved-bar node is attributable to the difference in the reinforcement details, as the single No. 5 (No. 16) hanger bar had only 5/8 in. (16 mm) clear side cover. The eccentricity can be accounted for by considering an effective width for the curved-bar node equal to twice the distance from the center of the bar to the nearest concrete surface—about
9
18
726
L 5
x
2
x
x
3
14383878
x
4
x
1
58587
8
2 No. 3
2 No. 3
2 No. 3No. 4
x
4 in.No. 4
x
4 in.
Bars A
Specimen No.
Bars A
L
1A 2 No. 4 13 1B and 1C No. 4 + 1 No. 3 22 ¼
L
60º3d
b
Bars A
PL
CL CL
PLx
6
x
6
38
PLx 3 x 4
9
18
726
3878
1
1½1½1½¾½ in. strand(typ)
X
C
2 No. 3½ in. strand(typ)2 No. 3
Bar ABar A
Dimensions in in. Deformed Bars per ASTM A706Specimen No.
Bar A L C X
2A 1 No. 5 16 ¾ ¾ 1 1/8 2B and 2C 1 No. 5 28 ½ ¾ 1 1/8 2D 1 No. 5 28 ½ 1 1
L
60º
No. 5
x
17 in. (2A only)
3d
b
PLx 6 x 4
No. 5 (2A only)
a)
b)
Fig. 4: Reinforcement for double-tee specimens with dapped ends (based on Reference 7) (note: 1 in. = 25.4 mm)
2 in. (50 mm) in this case. Based on strain measurements, the stress in the No. 5 (No. 16) bar was about 30 ksi (200 MPa) just before failure. The corresponding compressive stress over the reduced effective web width was 2700 psi (1806 MPa), or about 48% of the measured concrete compressive strength. Thus, the limiting strength given in ACI 318 of 0.51
f
c
′
may be unconservative for bars with shallow cover.Two other tests
8
on dapped-end beams with curved-bar nodes contained a reinforcement scheme similar to that shown in Fig. 2(b). Before reaching ultimate load, a diagonal crack extended downward to the lower corner of the beam. The calculated ultimate compressive stress at the curved-bar node was 1.50 and 1.89 times the measured concrete compressive strength.
Concrete international
/
SEPTEMBER 2008
45
C-C-T curved bar nodes
A C-C-T node is formed by a curved-bar node with a 180-degree bend. The upper right nodes shown in Fig. 2(b) and (c) are examples of such C-C-T nodes. Unlike the general case, where ties exiting the curved bar node create disruptive tensile strains in two directions, parallel ties provide confinement. As such, Eq. (3a) is too conservative for curved-bar nodes with 180-degree bends. Using the ACI 318 strength limit for C-C-T nodes (
β
n
= 0.8) the minimum radius from Eq. (2) becomes
bf
c
' r
b
1.5
≥
(3b)where
A
ts
is the area of nonprestressed tie reinforcement at one
end of the 180-degree bend.
Multi-layer curved-bar nodes
Where more than one layer of reinforcement is used in the plane of the STM, nodal zone stresses are increased in proportion to the number of layers. Figure 5 illustrates the use of two layers of reinforcement at a frame corner. In these cases, Eq. (3a) or (3b) may be used provided
A
ts
is taken as the area of tie reinforcement in all layers, and
r
b
is taken as the bend radius at the inside layer.
BOND STRESS AT CURVED-BAR NODES
In some cases, for example a junction between a wall and a slab with different effective depths (Fig. 6), the tie forces are not equal. The compressive stress on the inside radius of the bar must therefore vary, and circumferential bond stress develops along the bar.
9
The maximum nodal compressive stress occurs at the point of tangency for the tie carrying the greater force. Assuming this tie yields, the compressive stress is given by Eq. (1). At 90-degree corners, the resultant force in the strut is
A
ts
f
y
/cos
θ
c
, where
θ
c
is the smaller of the two angles between the strut (or the resultant of two or more struts) and ties extending from a curved-bar node. The bond force that must be developed along bend length
l
b
is
A
ts
f
y
(1 – tan
θ
c
). In view of the high contact stress on the inside of the bend, it would seem that the development length
l
d
for straight bars can be conservatively applied to the bend region at a curved-bar node. Accordingly, at 90-degree corners, the ratio of curve length to development
r
b
A
ts
ℓ
b
Resultant
= C
3
bac
Circumferential bond force per unit length
= A
ts
f
y
(1-tan
θ
c
)
/
ℓ
b
Variable radial compressive stress. Maximum at point
c
= A
ts
f
y
/br
b
A
ts
f
y
tan
θ
c
C
2
= A
ts
f
y
tan
θ
c
A
ts
f
y
tan
θ
c
A
ts
f
y
A
ts
f
y
C
3
= A
ts
f
y
/cos
θ
c
C
1
= A
ts
f
y
θ
c
θ
c
r
b
Free Body Diagram of Curved Bar
Fig. 5: Frame corner with two layers of reinforcementFig. 6: Unequal tie forces in a frame corner result in bond stress along the circumference of the bend. The radius of the bend must produce a bend length
l
c
adequate to develop the required bond force
length
l
d
should be at least (1 – tan
θ
c
); that is
l
b
≥
l
d
(1 – tan
θ
c
)
(4a)In terms of
r
b
, Eq. (4a) becomes
d
b
r
b
2
l
d
≥
(1 – tan
θ
c
)2
–
(4b)
EFFECTIVE WIDTH OF CURVED-BAR NODES
The effective width of a node is usually taken as the width of the member transverse to the plane of the STM. This assumption is usually valid for curved-bar nodes, but there are three potential concerns: 1) excessive compressive stress under the bend region of the bar; 2) transverse eccentricity of the bars relative to the member; and 3) side splitting of bars with shallow side cover.The minimum bend radii provided in ACI 318 are sufficient to avoid crushing under the bend region of the bar. Assuming the minimum bend radius provisions are met, this condition need not be checked, even when large, widely-spaced bars are used.

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