N e e d f o r P l a s t i c D e f o r m a t i o n s . C o n c e p t o f C o n t r o l l e d P l a s t i c D e f o r m a t i o n As already stated, in structures designed according to modern codes plastic defor-mations are expected under severe earthquakes. Acceptance of post-elastic response to high intensity seismic motions is the fundamental concept of up-to-date structural seismic design.
The need for development of plastic deformations during earthquakes is due to eco-nomic reasons. Design of structures for responding elastically to any seismic intensity involves resource consumption (financial and technical as well), unacceptable even for most wealthy societies.
The possibility to accept post-elastic structural and non-structural deformations results from the following:
• Transitory character of seismic actions (they have a duration of several seconds to several minutes).
• Reversal character of seismic inertial forces.
• Low frequency of high intensity earthquakes.
• Ability of reinforced concrete structures to develop substantial post-elastic defor-mations especially when certain specific measures are observed.
• Reparability, in most cases, of damage involved by development of post-elastic deformations.
The most significant progress of recent researches is, perhaps, the development of methods and practical procedures for controlling the seismic post-elastic response.
This means that, nowadays, a conception and design tools are available for imposing, with high degree of credibility, location of plastic deformation (within the structure) and for limiting the extent of those deformations. The location of plastic defor-mation can be chosen in naturally ductile and easy-to-repair elements (Paulay, T., Bachmann, H. & Moser, K. 1990).
The post-elastic controlled seismic response is associated with the concept of energy dissipating mechanism.
The seismic energy input absorbed by structures is dissipated partially through damp-ing and partially through the work of plastic moments (“plastic energy dissipation’’), until the static equilibrium is reached.
The capacity of the whole structure to dissipate energy through plastic deformations depends upon the ductility of plastic zones and upon their number and location. When the post-elastic response of a structure involves a great number of plastic zones, located in elements with high ductility (like beams), the overall energy dissipating capacity of the structure is substantial. The structures that exhibit a limited number of plastic zones, located within elements with low ductility, possess a limited capacity to dissipate energy through plastic deformations and its ability to survive strong earthquakes is questionable.
Location of plastic zones within a structure depends upon the ratio between effective internal forces due to progressive loading and critical sections’ resistance. If cer-tain critical zones, considered to be suitable for developing plastic deformations, are appropriately designed and detailed while all other zones are endowed with enough capacity to resist elastically, the overall structure response to seismic motion will be controlled and not random.
M e c h a n i s m s f o r P l a s t i c E n e r g y D i s s i p a t i o n
The following discussion is more easily understood if we accept that the plastic deformations are concentrated in plastic hinges (see chapter 2).
Depending upon the stiffness of different frame members and resistance capacity of critical sections, different distributions of plastic hinges occur during strong earth-quakes. The structure fails when enough plastic hinges occur to transform it in a kinematical mechanism. Different mechanism types can be encountered according to the plastic hinge distribution. Basically, there are two limit cases (Fig. 6.6):
(a) Story mechanism – the plastic hinges are located at the ends of columns of a single story (“soft and weak story’’) (Fig. 6.6a)
(b) Beam mechanism – the plastic hinges are located at the ends of all beams and at the columns’ bottom (Fig. 6.6b).
The story mechanism involves a limited number plastic hinges located in columns, which are elements with low ductility (because of the compressive axial force). On the
Figure 6.6 Types of dissipating mechanisms a) Story mechanism, b) Beam mechanism
other hand, the columns are vital members for the overall structure stability i.e. column failure can lead to building collapse. Since this mechanism involves a small number of plastic hinges, the amount of energy dissipated by each plastic hinge is substantial (total energy is shared within a limited number of hinges). Consequently, large plastic rotation demands are required for these hinges. It follows that the story mechanism is very unfavorable.
In contrast with story mechanism, the beam mechanism involves a great number of plastic hinges so that the amount of energy to be dissipated by each plastic hinge is small. Consequently, plastic rotations required within each plastic hinges are small so that the ductility requirement can be easily fulfilled. Moreover, the beams subjected to bending (without axial force) have naturally good ductile behavior without supple-mentary constructive measures. If a beam fails, the overall structure stability is not endangered.
The conclusion, which follows from above considerations, is that the beam mecha-nism is a favorable one. Accordingly, procedures should be defined and implemented in order to ensure a structural post-elastic behavior as close as possible to that model.
For this purpose a practical method has been developed (Paulay, 1980) called method of capacity design.
Hinges that occur in beams rather than in columns characterize the favorable dis-sipating mechanism. Accordingly, that mechanism could be defined through the brief statement “weak beams-strong columns’’.
The method of capacity design fulfills the requirement of weak beams-strong columns by appropriately adapting the analysis based on equivalent seismic forces (current seismic design method). The basic judgment is the following: plastic deforma-tions will develop in predicted secdeforma-tions if we design these secdeforma-tions to the internal forces resulted from conventional analysis while all other critical sections are proportioned and designed to the highest magnitude of internal forces which can be developed by the structure during earthquakes (Fig. 6.7). Consequently, all members of the structure, except the plastic zones, will behave elastically.
The overall plastic response of the structure will mobilize only imposed plastic zones, appropriately designed and detailed. According to this approach, the internal design forces within members supposed to behave elastically are considerably higher than those determined by the conventional analysis.
In other words, the structure should not be designed to provide “equal capacity’’
to all its components but an advantageous capacities hierarchy should be observed.
Accordingly, chosen elements will meet earlier plastic deformations while others (having higher capacity) will respond elastically to the external input.
The key of the procedure is to assess, as accurately as possible, maximum capacities which can be developed by sections and elements. This means that, the actual resistance of a material, section or element can be substantially higher than that used for current design. We say that the structural components possess over-strength due to different factors. Three main over-strength factors can be identified:
(a) Material over-strength – the actual resistance of materials (steel, concrete, masonry) is normally higher that its design value
(b) Over-strength due to constructive provisions which oblige sometimes to appro-priately increase the elements’ sizes, reinforcement amount, etc. always on the safe side
Figure 6.7 Potential plastic zones of a multistory frame
(c) System over-strength quantifies the capacity of a part of the structure or of the whole structure to develop higher resistance that required by design demand.
For example, for a reinforced concrete beam, part of a frame, we can define:
• Reinforcement over-strength factor λ0= actual average yield strength/design strength fyd≈ 1.25
• Plastic hinge over-strength factor 0=M0
ME=Asλ0fyd(d− a/2) ME
where: MEis the bending moment resulted from elastic analysis under seismic forces, As– reinforcement actual area and other notation are shown in Figure 6.8.
Beam over-strength factor (i.e. system over-strength factor of sub-frame involving the continuous beam with n potential plastic hinges):
ψ0=
n 1Mo,i
n 1ME,i =
n
1(0ME)i
n 1ME,i
(6.2)
S t e p s o f C a p a c i t y D e s i g n M e t h o d f o r F r a m e s
The method of capacity design for frame structures involves the following steps:
1. Structural analysis under gravity loads and conventional (equivalent) seismic forces (elastic analysis). Internal forces (bending moments, shear forces, axial
forces) due to the two types of actions (gravity and seismic equivalent forces) are thus determined: MG, VG, NG, respectively ME, VE, NE.
2. Choose potential plastic zones (desired). For frames, according to above consid-erations, plastic zones are expected to develop at beams’ ends and at the columns’
base sections.
3. Design all potential plastic zones for bending, using the internal forces above determined. Detail all plastic zones for bending taking into account the rules for ensuring development of substantial plastic deformations without premature failure due to adverse phenomena (see subchapter 6.7).
4. Calculate the flexural capacity of beam ends (plastic hinges) determined with the effective reinforcement amount, taking into account the over-strength of reinforcement (see Fig. 6.8).
5. Determine the shear force in beams, associated to the flexural capacity of plastic zones determined with over-strength (see Fig. 6.10). This is the maximum shear force which can be developed within the beam.
6. Design and detail of plastic zones for shear using conservative procedures provided by codes and the shear force above determined.
7. Determine the maximum magnitude of internal forces in columns. For this pur-pose, calculate the magnitude of internal forces in columns associated with moment capacity of beams with over-strength and, thereafter, use appropriate correction factors for taking into account all other adverse phenomena that can be developed during earthquake: dynamic magnification of internal forces, effect of higher vibration modes, spatial effects (plastic hinges developed at all beam ends adjacent to the column).
Accordingly, the maximum internal forces in columns will be:
– Bending moments:
M= Mscol×
0Mbeamsr
Mbeamss × KM (6.3)
where KMis a coefficient which take into account the adverse phenomena listed above.
Figure 6.8 Bending resistance with over-strength a) Section, b) Strain, c) Stress
– Axial forces:
N =
Vas (6.4)
8. Design and detailing of columns as common elastic elements.
C o m m e n ts
1. It can be noted that, when capacity design method is implemented, the only internal forces determined through elastic (conventional) analysis directly used in design process (with their primary magnitudes) are the bending moments in potential plastic zones of the structure. The rest of internal forces initially deter-mined are then adjusted (magnified) in order to prevent plasticization in undesired elements.
2. The capacity design method has a much wider range of use than the case of frames. Basically, it consists in adjustment of internal forces magnitudes, deter-mined through conventional elastic analysis with equivalent seismic forces, so that the plastic deformations (which mean actually structural damage) are developed only in controlled easy-to-repair zones while the rest of the structure respond elastically to the seismic input. This objective is obtained through a suitable hier-archy of members’ capacity. Although the procedure does not explicitly quantify the post-elastic behavior, it ensures a conservative control of post-elastic struc-ture behavior. The advantage is that this task is accomplished through a simple approach, familiar to structural designers.
6.6 Drift Control of Frames Subjected to Seismic Actions