This chapter shall apply to the design of structural concrete members, or regions of members, where load or geometric discontinuities cause a nonlinear distribution of longitudinal strains within the cross section.
Any structural concrete member, or discontinuity region in a member, shall be permitted to be designed by modeling the member or region as an idealized truss in accordance with this chapter.
Strut-and-tie models shall consist of struts and ties connected at nodes to form an idealized truss.
Geometry of the idealized truss shall be consistent with the dimensions of the struts, ties, nodal zones, bearing areas, and supports.
Strut-and-tie models shall be capable of transferring all factored loads to supports or adjacent B-regions.
The internal forces in strut-and-tie models shall be in equilibrium with the applied loads and reactions.
Brackets and corbels with shear span-to-depth ratio av/d < 2.0 designed using strut-and-tie models shall satisfy 16.5.2, 16.5.6, and Eq. (23.2.9).
|Asc ≥ 0.04(fc'/fy)(bwd)||(23.2.9)|
For each applicable factored load combination, design strength of each strut, tie, and nodal zone in a strut-and-tie model shall satisfy ϕSn ≥ U, including (a) through (c):
(a) Struts: ϕFns ≥ Fus
(b) Ties: ϕFnt ≥ Fut
(c) Nodal zones: ϕFnn ≥ Fus
The nominal compressive strength of a strut, Fns, shall be calculated by (a) or (b):
(a) Strut without longitudinal reinforcement
|Fns = fceAcs||(23.4.1a)|
(b) Strut with longitudinal reinforcement
|Fns = fceAcs+ A'sf's||(23.4.1b)|
where Fns shall be evaluated at each end of the strut and taken as the lesser value; Acs is the cross-sectional area at the end of the strut under consideration; fce is given in 23.4.3; A's is the area of compression reinforcement along the length of the strut; and f's is the stress in the compression reinforcement at the nominal axial strength of the strut. It shall be permitted to take f's equal to fy for Grade 40 or 60 reinforcement.
Effective compressive strength of concrete in a strut, fce, shall be calculated by:
|fce = 0.85βs fc'||(23.4.3)|
where βs, in accordance with Table 23.4.3, accounts for the effect of cracking and crack-control reinforcement on the effective compressive strength of the concrete.
Table 23.4.3—Strut coefficient βs
|Strut geometry and location||Reinforcement crossing a strut||βs|
|Struts with uniform crosssectional area along length||NA||1.0||(a)|
|Struts located in a region of a member where the width of the compressed concrete at midlength of the strut can spread laterally (bottle-shaped struts)||Satisfying 23.5||0.75||(b)|
|Not Satisfying 23.5||0.60λ||(c)|
|Struts located in tension members or the tension zones of members||NA||0.40||(d)|
|All other cases||NA||0.60λ||(e)|
If confining reinforcement is provided along the length of a strut and its effect is documented by tests and analyses, it shall be permitted to use an increased value of fce when calculating Fns.
For bottle-shaped struts designed using βs = 0.75, reinforcement to resist transverse tension resulting from spreading of the compressive force in the strut shall cross the strut axis. It shall be permitted to determine the transverse tension by assuming that the compressive force in a bottle-shaped strut spreads at a slope of 2 parallel to 1 perpendicular to the axis of the strut.
Reinforcement required in 23.5.1 shall be developed beyond the extent of the strut in accordance with 25.4.
Distributed reinforcement calculated in accordance with Eq. (23.5.3) and crossing the strut axis shall be deemed to satisfy 23.5.1, if fc' ≤ 6000 psi.
where Asi is the total area of distributed reinforcement at spacing si in the i-th direction of reinforcement crossing a strut at an angle αi to the axis of a strut, and bs is the width of the strut.
Distributed reinforcement required in 23.5.3 shall be placed orthogonally at angles α1 and α2 to the axis of the strut, or in one direction at an angle α1 to the axis of the strut. Where the reinforcement is placed in only one direction, α1 shall be at least 40 degrees.
Compression reinforcement in struts shall be anchored to develop f's at the face of the nodal zone, where f's is calculated in accordance with 23.4.1.
Closed ties enclosing compression reinforcement in struts shall satisfy 25.7.2 and this section.
Spacing of closed ties, s, along the length of the strut shall not exceed the smallest of (a) through (c):
(a) Smallest dimension of cross section of strut
(b) 48db of bar or wire used for closed tie reinforcement
(c) 16db of compression reinforcement
The first closed tie shall be located not more than 0.5s from the face of the nodal zone at each end of a strut.
Closed ties shall be arranged such that every corner and alternate longitudinal bar shall have lateral support provided by crossties or the corner of a tie with an included angle of not more than 135 degrees and no longitudinal bar shall be farther than 6 in. clear on each side along the tie from such a laterally supported bar.
Spirals enclosing compression reinforcement in struts shall satisfy 25.7.3.
Tie reinforcement shall be nonprestressed or prestressed.
The nominal tensile strength of a tie, Fnt, shall be calculated by:
|Fnt = Ats fy+ Atp(fse+ Δfp)||(23.7.2)|
where (fse + Δfp) shall not exceed fpy, and Atp is zero for nonprestressed members.
In Eq. (23.7.2), it shall be permitted to take Δfp equal to 60,000 psi for bonded prestressed reinforcement and 10,000 psi for unbonded prestressed reinforcement. Higher values of Δfp shall be permitted if justified by analysis.
The centroidal axis of the tie reinforcement shall coincide with the axis of the tie assumed in the strut-and-tie model.
Tie reinforcement shall be anchored by mechanical devices, post-tensioning anchorage devices, standard hooks, or straight bar development in accordance with 23.8.3.
Tie reinforcement shall be developed in accordance with (a) or (b):
(a) The difference between the tie force on one side of a node and the tie force on the other side shall be developed within the nodal zone.
(b) At nodal zones anchoring one or more ties, the tie force in each direction shall be developed at the point where the centroid of the reinforcement in the tie leaves the extended nodal zone.
The nominal compressive strength of a nodal zone, Fnn, shall be calculated by:
|Fnn = fceAnz||(23.9.1)|
The effective compressive strength of concrete at a face of a nodal zone, fce, shall be calculated by:
|fce = 0.85βn fc'||(23.9.2)|
where βn shall be in accordance with Table 23.9.2.
Table 23.9.2—Nodal zone coefficient βn
|Configuration of nodal zone||βn|
|Nodal zone bounded by struts, bearing areas, or both||1.0||(a)|
|Nodal zone anchoring one tie||0.80||(b)|
|Nodal zone anchoring two or more ties||0.60||(c)|
If confining reinforcement is provided within the nodal zone and its effect is documented by tests and analyses, it shall be permitted to use an increased value of fce when calculating Fnn.
The area of each face of a nodal zone, Anz, shall be taken as the smaller of (a) and (b):
(a) Area of the face of the nodal zone perpendicular to the line of action of Fus
(b) Area of a section through the nodal zone perpendicular to the line of action of the resultant force on the section
In a three-dimensional strut-and-tie model, the area of each face of a nodal zone shall be at least that given in 23.9.4, and the shape of each face of the nodal zone shall be similar to the shape of the projection of the end of the strut onto the corresponding face of the nodal zone.