Within frame members with expected post-elastic deformations (having potential plas-tic zones), specific provisions have to be implemented in order to ensure their ductile behavior. In other words, potential brittle failure of these zones has to be prevented.
B e a m s
The frame beams are subjected to bending and shear force.
Normally, under bending moments a ductile behavior is naturally ensured, while shear force induces a brittle failure.
(a) Flexural ductility
Although the flexural behavior of beams is generally ductile, some rules have to be observed.
Potential plastic zones of beams have to be reinforced with ductile steel only.
The use of cold work steel in potential plastic zones has to be avoided.
Moderate longitudinal steel percentage has to be used (the recommended range is between 0.8 and 1.2% in beams).
A phenomenon which can lead to a premature lost of ductility is the post-elastic buckling of the compressed steel. Due to alternate loading, both longitudinal reinforcement layers (at the top and at the bottom of the beam) can be subjected to compression. Since post-elastic deformations are accepted within potential plastic zones of the beam, the post-elastic buckling of the compressed steel can occur, with the corresponding damage of the surrounding concrete. In order to prevent this buckling, stabilization stirrups spaced at less (4–6)φmin (φmin is minimum bar diameter of the section) have to be provided. This means that stirrups in potential plastic zones have two functions:
– Stabilization of compressed bars (to prevent buckling);
– Resist shearing forces (these stirrups will be calculated).
(b) Effect of the shear force
The ductile behavior of beams is effective when large flexural deformations (rota-tions) can be developed. So, the premature brittle failure through shear has to
Figure 6.11 Shear force associated to the yielding of beam-ends
be prevented in order to allow developing substantial plastic flexural deforma-tions. In other words, an appropriate over-resistance to shear (in comparison with bending) has to be observed.
This requirement is achieved by two complementary measures:
• Designing transverse reinforcement for maximum magnitude of shear force which can be developed within the plastic zone
• Using conservative procedures to design for shear.
The maximum magnitude of shear force, which can be developed in a beam, is related to the maximum bending capacity of the beam-ends. It is accepted that, at both beam-ends, plastic hinges are developed. Thus, the shear force associated to the bending moment capacity of plastic hinges, will be (Fig. 6.11):
Vas= 0|MRl| + |MrR|
l0 +(g+ 1.2p)l0
2 (6.5)
where: MlRand MrR are the flexural capacities of the left end, respectively right end sections of the member, determined with the real reinforcement area provided within the sections, taking into account the potential over strength of the reinforcement (fy∗= λ0fyd= 1.35fyd)
0– sectional over strength factor and l0– spacing between plastic hinges.
The material over strength (for reinforcing steel) is due to potential yielding limit increase through strain hardening and to statistical potential difference between design and average strength. The material over strength is propagated to sectional level generating a sectional capacity over strength.
The shear resistance of the beam is ensured by the stirrups and by the shear resistance of compressed concrete:
VR= Vstirr+ Vconcr (6.6)
The concrete within plastic zones can be damaged due to cyclic loading. Thus, a conservative shear capacity of the beam shall be determined accepting a diminished contribution of the concrete (in comparison with that corresponding to gravity loads).
The stirrups area and spacing will be determined from the inequality:
Vas ≤ VRs (6.7)
C o l u m n s
For columns, elastic behavior is expected if the capacity method is properly applied.
Accordingly, no special provisions should be observed for columns, excepting the potential plastic zones.
Within potential column plastic zones, ductile behavior shall be ensured.
Columns are subjected to bending moments and axial forces (M+ N) and to shear force.
The behavior for eccentric compression (M+ N) is governed by the magnitude of the dimensionless axial force (see Paragraph 2.3.3):
n= N Acfc
(6.8)
According to the magnitude of this parameter three behavior types until failure have been evidenced:
(a) For n= 0.0–0.2 – ductile behavior similar to that of elements subjected to pure bending
(b) For n= 0.3–0.5 – semi-ductile behavior; failure is caused by the compressed concrete crushing (ultimate concrete strain is exceeded)
(c) For n > 0.5 – elastic-brittle behavior
Consequently, in order to ensure ductile behavior of column plastic potential zone, for each of above cases the following measures have to be taken:
(a) For n= 0.0–0.2 – no special measures are required except those related with preventing the buckling of longitudinal reinforcement, i.e. stabilizing stirrups (spaced at less than 6 dmin)
(b) For n= 0.3–0.5 – the ductility is improved if the concrete is confined so that it develops large limit strains. Thus, within potential plastic zones of columns, with n= 0.3–0.5, hoops or spiral transverse reinforcement have to be provided.
Figure 6.12 Shear arm and aspect ratio of the columns
(c) n > 0.5 has to be avoided for columns with potential plastic zones, by increasing their cross section.
Within potential plastic zone no lapped splices are allowed. Accordingly, the con-nection of longitudinal bars has to be made outside of the potential plastic zone or special connectors have to be used.
The shear behavior of the columns is governed by the magnitude of the shear arm:
ash= M
Vh (6.9)
Since throughout the column height the bending moment varies linearly and the shear force is constant, the dimensionless magnitude of the shear arm ash/h (h is the cross section height) is equivalent to the aspect ratio:
λ=l0
h
where l0 is the distance between the section with maximum moment and that with zero moment (Fig. 6.12). (For very rigid beams in respect with columns l0= H/2).
According to the aspect ratio, the shear behavior of the columns evidences three distinct patterns (see Fig. 2.28):
• For short columns (λ <≈2.5) the failure is governed by shear. It has a brittle character and occurs through a critical inclined crack. No effective way exists to ensure a ductile failure of these elements so that no post-elastic deformations are allowed in short columns.
Figure 6.13 Shear forces due to seismic action within beam-column joints
• Middle-length columns (λ = about 2.5 to 5) show a semi-ductile flexure-shear failure. This means that, for these columns, the failure starts by the yielding of lon-gitudinal reinforcement within the zones with maximum bending moments and is completed by a shear inclined critical crack. If the premature failure due to shear is prevented, the column is able to develop substantial (flexural) deformations.
Accordingly, stirrups (size and spacing) have to be calculated with the formula used for the beams, for the shear force associated with the plastic moments at the column ends. The stirrups will be extended over the whole column height.
• Long columns (λ > about 5) are predominantly subjected to bending (with axial force) and, consequently, have a “natural’’ ductile behavior, with plastic hinges at both ends.
Of course, this implies that the column slenderness is not critical from the point of view of the buckling.