moehlesmrf

7/29/2019 MoehleSmrf

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Special Moment Frames1by Jack P. Moehle

Pacific Earthquake Engineering Research Center

University of California, Berkeley

1. Target Yield Mechanism

designlateral

loads

designlateral

loads

Figure 1.1 Target yield mechanism

One of the guiding principles of seismic design

is to spread yielding throughout the structure so

that large inelastic deformations do not

concentrate in isolated locations. In design of a

Special Moment Resisting Frame (SMRF) it is

important to avoid a yielding mechanism

dominated by yielding of the columns in a single

story, as this can result in very large local

demands in the columns. Instead, it is desirable

in a SMRF that yielding be predominantly in thebeams (Figure 1.1). This is a fundamental

objective in the design of a SMRF.

Note that even if the beams are targeted as the

main elements to yield, some column yielding

must be anticipated. For example, yielding at the

foundation seems likely (Figure 1.1). Also, it is

difficult to completely protect the columns from

yielding in other stories, as will be discussed

later.

Given this capacity-design approach of having plastic hinges in the beams, the beams will be

sized for the design seismic loads (usually based on analysis under code-specified loading), theywill be detailed for ductile response, and the rest of the system will be proportioned to reduce the

likelihood of inelastic action away from the beam plastic hinges.

2. Beams

2.1 Design objective

As discussed above, the design objective in a SMRF is to provide a stiff and strong spine of

columns up the height of the building so that concentrations of inelastic action in isolated stories

are avoided. Therefore, the objective is for beams to form flexural plastic hinges at targetedlocations through the height of the frame. The design also should attempt to avoid inelastic

response in shear as well as anchorage or bond failures.

1

1 Prepared for CE 244, Reinforced Concrete Structures, a graduate class taught at the University of

California, Berkeley


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2.2 Design actions on beams

The seismic demands imposed on building frames are a complicated function of the earthquake

as well as the building stiffness, strength, mass, and configuration. Therefore, it is not possible to

state with any accuracy the demands that beams in SMRFs, in general, need to sustain. However,

some approximations of the level of inelasticity can be made.

According to the International Building Code and Uniform Building Code, SMRFs are allowed

to be designed for a force reduction factor ofR = 8, that is, they are allowed to be designed for a

base shear equal to one-eighth of the value obtained from elastic response analysis. Assuming an

average building overstrength ratio of about 2.5 (actual building strength about 2.5 times the

design value) owing to material overstrengths, section oversizing, strain-hardening, and

interactions among structural components and among structural and nonstructural components

that were not considered in design, the effective strength is about one-third of the strength

required for elastic response. Accepting the equal displacement rule, the global displacement

ductility for the building would be approximately 3. Local concentrations of interstory drift

[Moehle, 1992] reasonably could result in local ductility demand about twice the global value, or

equal to 6. For a typical beam span-to-depth ratio of ten, the local rotational ductility demand

within the flexural plastic hinge would be approximately 2.5 times the local displacement

ductility, resulting in an estimate of rotational ductility equal to 15. (See companion paper on

seismic design principles for more detailed discussion of this topic.)

An alternative approach views demands directly in terms of drift and member rotation

demands. Assuming a global drift angle of 0.015 for a design-level event, the local drift ratio

could reasonably be 0.03. The yield curvature for a typical beam is on the order ofy/0.7h, andthe flexural plastic hinge length can be approximated as being equal to h/2. Assuming that the

gravity load results in moments at the beam ends less than half the moment capacity, and

assuming that the columns do not contribute to drift (that is, they are relatively stiff), the

curvature ductility for a given drift ratio can be calculated. For the drift ratio of 0.03, and for

reasonable aspect ratios, the curvature ductility demand for a beam is approximately 20. This

value is reasonably close to the value obtained in the preceding paragraph.

Both approximations assume that the curvature demand is directly related to the building drift.

This is only approximately correct, and even then only in the case of reversing plastic hinges.

This subject is considered in more detail later.

2.3 Beam behavior

2.3.1 Formation of plastic hinges

An objective in design of SMRFs is to restrict most yielding to beams, which are speciallydetailed to resist the imposed actions. To get an understanding of the actions on a beam, consider

the framing shown in Figure 2.1.

Plastic moment strength can be assumed to be equal to the moment that will develop as theframe is deformed to the maximum deformation under the design earthquake loading. ACI 318

Chapter 21 definesMpras being equal to at least 1.25Mn. It is likely that larger moments will

develop in a beam if it responds to the curvatures that are commonly anticipated for a SMRF. As

discussed previously, curvature ductility demands on the order of 15 to 20 can be anticipated in

some SMRFs.

Under gravity loading shown in gray, the beam develops the moment diagram shown. Under

service level gravity loads, it is expected that the moments will be less than half the nominal

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moment strengths. Also, it is unlikely that the full

service load will be present when the earthquake

occurs, so even smaller moments are likely.

If under earthquake loading the frame sways to

the right, the column shears would be as shown by

the black arrows. The moment diagram would shiftunder this loading, shown by the blue, dashed

curve. The first section likely to reach the plastic

moment is the negative-moment section. This

section will begin to develop plastic rotation before

the positive moment section yields.

As the loading continues, the positive-moment

section may yield and develop plastic rotation.

While this is happening, the rotations at the

negative-moment section continue to grow.

2.3.2 Behavior of reversing plastic hinges

As a building sways back and forth during an

earthquake, the motion of the building drives the

beam plastic hinges through displacement or

deformation histories.

Envision the building drift history as shown by

the simple waveform (Figure 2.2). Assume the

frame sways to point I. While this happens, the end

of the beam that has negative moment (top in tension) under gravity loads is deformed as shown

on the moment-curvature plot by the blue curve to point I. The strain distribution is also shown

next to the beam, showing the top in tension, the bottom in compression note that the tensile

strains exceed the compressive strains for a beam with modest amounts of tension longitudinal

reinforcement. The stress-strain histories of the top and bottom bars also are shown below thebeam.

Mpr-

Mpr+

L R

Bottom in tension

y

y

Mpr-

Mpr+

L R

Bottom in tension

y

y

Figure 2.1 Plastic hinge formation

As the building drifts from point I to II (shown in red), the curvature of the beam reverses, as

shown in the moment-curvature diagram. For the beam cross section shown, the cross-sectional

area of the top reinforcement is larger than that of the bottom reinforcement. As a result, the total

tension force developed by the beam bottom reinforcement is insufficient to yield the top

reinforcement in compression. Therefore, unless other effects dominate (such as loss of bond

through a beam-column joint and subsequent pull-through of the top reinforcement) the top beam

reinforcement will be in compression but will not be strained to the yield point in this case the

cracks that opened at state I remain open through the depth of the beam. If shear forces are high,this can lead to sliding along the cracked interface such sliding can lead to a moment-curvature

relation that is pinched, with slackness occurring for low moment and curvature values. Note thestress-strain histories for the longitudinal reinforcement. The top reinforcement is in compression

stress while showing tensile strain. Because the top reinforcement is in tension strain, the bottom

reinforcement is driven to very large tensile strain in order to achieve the curvatures demanded ofthe beam.

As the beam is flexed to position III, the situation reverses.

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I

II

III

time

M

drift

As

As

I II III

fs

s

fs

s

start here

curvature

II

I

IIII

IIIII

IIIIII

I

II

III

time

M

drift

As

As

I II III

fs

s

fs

s

fs

s

start here

curvature

II

I

IIII

IIIII

IIIIII

Figure 2.2 Reversing plastic hinge behavior

2.3.3 Reversing and non-reversing plastic hingesIf the beam is relatively short and/or the gravity loads relative low compared with seismic

design effects, the beam behavior is likely to be as shown on the left of Figure 2.3. As the beam

is deformed by the building response to the earthquake motions, the moments reach the plastic

moment capacities at the beam ends (adjacent to the column face). As the earthquake sway

reverses, the plastic hinges form again at the same locations, and a reversing plastic hinge forms,

as presumed in the previous discussion.

If the span or gravity loads are relatively large compared with earthquake effects, then a less

desirable behavior can result. This is illustrated on the right-hand side of Figure 2.3. As the

beam is deformed by the earthquake, the moments reach the plastic moment capacities in

negative moment at the column face and in positive moment away from the column face. The

deformed shape is shown. Upon reversal, the same situation occurs, but on opposite ends of thebeam. In this case, the sections that had yielded previously do not yield in the opposite direction,

but instead the plastic hinge forms in a different location. Note the deformed shape. As the

deformations continue to reverse, the plastic hinges do not reverse, but instead continue to build

up rotation. This results in progressively increasing rotations of the plastic hinges, so that for along earthquake the rotations can be very large and the vertical movement of the floor can exceed

serviceable values. This type of behavior should be avoided through design. In evaluation of

existing buildings, this type of behavior should be investigated because if it does occur, the

rotations can well exceed values estimated based on static inelastic analysis.

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Mp-

Mp+

Mp-

Mp+

low gravity loadshigh gravity loads

reversingplastic hingereversingplastic hinge

non-reversingplastic hingenon-reversingplastic hingenon-reversingplastic hinge

Figure 2.3 Reversing and non-reversing plastic hinges

It is possible to determine whether non-reversing plastic hinges are likely. As shown in

the two moment diagrams, reversing plastic hinges are expected if the slope of thepositive moment diagram is negative, while non-reversing plastic hinges are anticipated if

the slope is positive this of course

assumes that the moment strength doesnot change appreciably along the span.

For the case of uniformly distributedload, consider equilibrium of the freebody shown in Figure 2.4. Summing

moments about VR:

wu

ln

MpM

p

+

VRVL

wu

ln

MpM

p

+

VRVL

Figure 2.4 Free-body diagram cut through plastic

hin es at end of beam02

2

=+ + nunLpplw

lVMM

Setting the slope of the moment diagram (the shearVL) equal to zero, we find that a

non-reversing plastic hinge is likely if

+ + ppnu MM

lw

2

2

2.3.4 Computed flexural ductility of beam cross sections

Response of beams in bending can be computed readily using computer programs (e.g.,

UCFyber). Most programs consider only monotonic loading response, though some are able to

represent reversed cyclic loading effects. Regardless the program, the output is only as good as

the input, the computation algorithm, and the ability of the user to make sensible interpretations.

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To gain a sense of the important variables, Figure 2.5 presents results computed using

UCFyber. Longitudinal reinforcement was assumed to have typical Grade 60 reinforcement

properties, including strain-hardening. Concrete confinement was considered within the

boundary of perimeter hoops, and followed the Mander relations. All other concrete was

unconfined. Unconfined concrete had maximum strain capacity of 0.005. Confined concrete was

assumed to have maximum compressive strain capacity of 0.02 (a practical limit for such sections

considering reinforcement buckling and subsequent fracture). The sections were flexed so the topwas in tension.

Unconfined sections (Figure 2.5) were unable to reach the curvature ductilities estimated for

SMRFs (approximately 15 to 20). Sections with the confinement were able to reach

approximately the expected curvature ductility demand. ACI 318, Chapter 21, limits the ratio of

area of tension reinforcement to area of compression reinforcement. Sections 1c and 1u represent

cases that approach the ACI limit. For those sections, the available curvature ductility is barely

equal to the required value even with transverse reinforcement.

Beam 1c Beam 2c

24

18No. 3 stirrups

@ 6 inches

Beam 1u

0

2000

4000

6000

8000

10000

12000

14000

0 0.001 0.002 0.003 0.004 0.005

curvature, 1/inch

moment,kip-inch

Beam 1c

Beam 2c

Beam 2u

20y

y

Beam 1u Beam 2u

No. 9 Grade 60 longitudinal bars,fc = 4 ksi

Beam 1c Beam 2c

24

18No. 3 stirrups

@ 6 inches

Beam 1c Beam 2c

24

18No. 3 stirrups

@ 6 inches

Beam 1uBeam 1u

0

2000

4000

6000

8000

10000

12000

14000

0 0.001 0.002 0.003 0.004 0.005

curvature, 1/inch

moment,kip-inch

Beam 1c

Beam 2c

Beam 2uBeam 2u

20y

y

Beam 1u Beam 2u


Beam 1u Beam 2u


Figure 2.5 Computed moment-curvature relations for unconfined and confined concrete sections. Top

in tension.

2.3.5 Shear behavior

Consider a member subjected to a concentrated lateral load as shown in Figure 2.6. Assume

that the 45-degree truss model effectively represents behavior in flexure and shear. Also, assume

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web member

tension stress

fy0web member

tension stress

fy0

Figure 2.6 Truss model

that the tension chord is designed so that

it does not yield, and that the

compression diagonals are sufficiently

strong that they do not crush. Yielding is

controlled by yielding of the web

reinforcement, shown in blue.

In this case, on the first cycle of

inelastic loading, the relation between

reinforcement strain and applied shear

will be nonlinear, as shown qualitatively

in Figure 2.7. Upon unloading, the

transverse reinforcement will unload and

show a residual tension strain. Upon

loading in the opposite direction, the

transverse reinforcement will yield in

tension again. Under repeated loading,

the tension strain in the transverse

reinforcement will tend to increase

progressively. The result is lateral

dilation of the member cross section,

with widening cracks and eventual

breakdown of the integrity of the

concrete core. The ability to resist

transverse loading decreases with

continued loading. V

transverse

steel strain

V

transverse

steel strain

Figure 2.7 Idealized strains in hoop reinforcement ofshear- ieldin member

Reinforced concrete elements behave

very differently depending on whether

the inelastic action is predominantly in

shear versus predominantly in flexure.

The data shown in Figure 2.8 are fromJirsa [1977]. The upper load-

deformation relation is for a member

where shear dominates; the lower

relation is one where flexure dominates.

Inelastic response in shear shows

strength degradation associated with the

shear-yielding mechanism of Figure 2.7.

The flexural mechanism shows stable

response for this configuration.

Figure 2.8 Load-deformation response of shear-yielding

and flexure-yielding members

2.4 Beam design

2.4.1 Longitudinal reinforcement

The ACI Building Code limits the

longitudinal reinforcement ratio to 0.025.

This limit is based on construction

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considerations (it is difficult to place concrete with this much or more reinforcement), shear

considerations (members with this much reinforcement tend to have excessively high shear

stresses), and bond considerations (reinforcement must be anchored in joints, and this becomes

increasingly difficult as the reinforcement ratio increases).

Some codes limit the reinforcement ratio to some fraction of the balanced reinforcement ratio.

Under reversed loading, where stresses are not uniquely related to strains (Figure 2.2), and inmembers with heavy hoop reinforcement (where concrete compression strains are enhanced),

conditions are considerably different from those on which balanced failure computations are

based, so this concept has been abandoned in the seismic provisions of the ACI code.

As noted previously (Figure 2.2), if the cross-sectional areas of top and bottom longitudinal

reinforcement differ significantly, cracks that open when the larger area of reinforcement yields

will remain open on load reversal, unless the bars slip through the joint because of bond failure.

To reduce consequences of this behavior, ACI limits the ratio of top to bottom reinforcement

areas to between 0.5 and 2.0.

The limits on reinforcement ratios also relate to conventional considerations of flexural

ductility capacity. Figure 2.5 shows moment-curvature relations for some beam cross sections.

When the ratio of tension to compression reinforcement areas differs significantly for large

reinforcement ratios, the flexural ductility capacity is reduced. Confinement of the cross section

with moderate transverse reinforcement considerably improves the computed behavior.

Because of uncertainty in the moment requirements in seismic conditions as the frame responds

to horizontal and vertical excitations, ACI 318 requires that the positive and negative moment

strengths along the span be not less than one-fourth the strength provided at the face of the joint.

Lap splices of longitudinal reinforcement are permitted in ACI 318 only if hoop reinforcement

is provided over the lap length. The maximum spacing of the hoops is not to exceed d/4 or 4

inches. Laps are not to be used within joints, within a distance of twice the member depth of

joints, and at locations where analysis indicates flexural yielding. The requirement for close

spacing is based on the understanding that laps are effective only if closely spaced hoop

reinforcement confines the splice after cover concrete spalls. Mechanical splices can be used, but

should be Type 2.

Mp-

Mp+

low gravity loads

Mp-

Mp+

low gravity loads

Figure 2.9 Moment diagrams of a yielding frame

2.4.2 Transverse reinforcement

Current building codes require special

hoops to confine core concrete and restrain

buckling of longitudinal reinforcement in

plastic-hinge regions of beams. The special

reinforcement is specified to extend a

distance equal to twice the member depth

from the center of the yielding region. The

length of actual plastic hinge is likely to

depend on the details of the moments andshears. One could argue that in the positive

moment region the plastic hinge spreads over

an extended length because the moment

diagram is relatively flat (Figure 2.9). On the

other hand, one could also argue that in the

negative moment region the plastic hinge

spread also is large because the plastic

rotations are larger there and because the

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shear forces and resulting tension shift are more significant there. The practical consideration is

that it is difficult to identify the hinging zone with precision, and inexpensive to provide hoops

over a generous length to cover the lack of precision.

Hoop details within the target plastic-hinge region are designed to confine the core concrete so

that it can reach strains well beyond the spalling strain and so that it can resist shear during these

inelastic excursions; the close spacing of hoops also serves to restrain the longitudinalreinforcement from buckling when in compression. Specific details will depend on the

configuration of the cross section. Figures

2.10 and 2.11 show some typical details of

hoops as required by ACI 318-99. To

restrain buckling, longitudinal reinforcement

is to be restrained as required for columns in

nonseismic construction, that is, corner bars

are to be restrained in corners of hoops, at

least alternate bars are to be restrained by

crossties or intermediate hoop legs, and no

unrestrained bar is to be more than 6 inches

from a restrained bar.

Figure 2.11 Examples of overlapping hoops (/ACI

Detail A Detail B

Detail C

Detail ADetail A Detail B

Detail C

Figure 2.10 Beam hoop details

It is common in US practice to use

crossties having 135-degree bends on one

end and 90-degree bends on the other end.

Although the 90-degree bends are known to

be less effective in restraining buckling after

loss of cover concrete (because the hook is

not embedded in core concrete), it has been

found to produce satisfactory behavior in

members with low axial compression, and it

improves contructibility.

Cap ties sometimes are used (detail B in

Figure 2.10) to ease construction. Whenused, the 90-degree hook should be restrained

by an adjacent slab, as shown. Multiple

hoops can be used as shown in detail C, but

these details can be difficult to construct.

New machinery for bending hoops is making

it feasible to construct complex hoop

configurations from a single piece of steel.

These should be used where economical.

Figure 2.12 is a photograph of beam

reinforcement details in a building

constructed in 2000 in Emeryville,California. The beam stood above the floor

slab, so the details were visible for this

photograph.

According to the ACI 318 code, maximum

spacing of hoop reinforcement within the

plastic hinge region is the smallest of (a) d/4,

(b) 8db, where db is the diameter of the

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Figure 2.13 Design shears for girders and columns(ACI 318)

Figure 2.12 Details in a building in Emeryville, CA

longitudinal reinforcement, (c) 24 db,

where db is the diameter of the transverse

reinforcement, and (d) 12 inches. These

requirements relate to the objective of

confining the core concrete so it can resist

shear under deformation reversals and can

be strained to reasonable compressionstrains, as well as providing resistance to

longitudinal bar buckling. Figure 2.5shows computed behavior of cross sections

with transverse reinforcement satisfying

these provisions.

Also, the beam is to be designed to resist

the shear corresponding to development of

Mprat both ends of the member (Figure

2.13). Within the plastic-hinge region,

when the shear due to seismic effects is

equal to or greater than the gravity shear,

the hoop reinforcement is to be designed to

provide Vs assuming Vc = 0. Of course, this

is not to imply that the concrete carries no

shear, a misinterpretation that could

mislead the engineer to think the concrete

section is unimportant when in fact a stout

section will improve section behavior.

Instead, the intent is to increase the amount

of hoop reinforcement to enable the

concrete section to resist shear under

adverse moment and shear reversals that

are anticipated.

Outside the plastic hinge region, stirrups

are to be provided at spacing not to exceed

d/2. Hoops are not required along this

length.

Figure 2.14 shows a beam detail with

lap-spliced longitudinal reinforcement.

The lap splices are placed near midspan,

which is where the stresses are lowest for

many SMRF frames (seismic actions

predominate over gravity actions). Hoops

are closely spaced near the ends

(presuming flexural plastic hinges form atthose locations). Hoops also are spaced

closely along the lap splice to confine the

splice.

Rather than splice the reinforcement as shown in Figure 2.14, sometimes the reinforcement is

curtailed at alternating positions along the span as shown in Figure 2.15. By cutting the bars in

alternate spans, no lap splices are required.

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Alternatively, mechanical splices can be

used. Type 2 splices (capable of

developing the specified bar ultimate

tensile strength strength) can be placed

anywhere along the span, though it is

advisable to remove these from plastic-

hinge regions.

Figure 2.15 Beam longitudinal reinforcement avoiding laps

gravityseismic + gravity

gravityseismic + gravity

Figure 2.14 Beam reinforcement with lap splices

3. Columns


Design of a SMRF aims to achieve a beam-yield mechanism (Figure 1.1) and avoid a story-

yield mechanism in which the columns in a story yield at the bottom and top of their clear length.

Therefore, a capacity-design approach is used to promote flexural yielding in the beams and

avoid flexural or shear yielding in columns.

The capacity-design process begins by identifying where the inelastic action is intended to

occur. For a SMRF, the inelastic action is intended to be predominantly in the form of flexural

yielding of the beams. The building is analyzed under the design loads to determine the required

flexural strengths of the beam plastic hinges. The beam sections are designed so that the reliable

moment strength is at least equal to the design strength, that is, . Once the beam is

proportioned, the plastic moment strengths of the beam can be determined based on the expectedmaterial properties and the selected cross section. ACI 318 uses the strengthM

un MM

prfor this purpose.

This moment is calculated using conventional ACI procedures with reinforcement yield stress

taken equal to 1.25 times the nominal yield stress. The result is effectively the same as 1.25Mn.

Knowing the beam plastic-moment strengths, it is possible to approach the problem of

designing the columns so that they are stronger than the beams. The design problem is

complicated by uncertainty in the seismic loading demands. Some aspects of the design problem

are discussed in the following text.

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3.2 Design actions on columns

3.2.1 Moments

Figure 3.1 shows schematically the beam and column

moments that are obtained from a code analysis of a building

frame under gravity and lateral loads. Beam moments varynonlinearly because of the presence of distributed loads.

Columns moments equilibrate the beam moments. Assuming

no lateral inertial effect from the columns, the column

moment diagrams are linear. The corresponding shears are

constant along the height of a column in a story.

P

Mx

My

design axial load

corresponding Mx and

My moment strengths

Mb4

Mb1 Mb3

Mb2

(a) P-M interaction diagram

(b) Plan view of joint

directio

nof

build

ingsw

ay

P

Mx

My

design axial load


My moment strengths

P

Mx

My

design axial load


My moment strengths

Mb4

Mb1 Mb3

Mb2

Mb4Mb4

Mb1 Mb3

Mb2

(a) P-M interaction diagram

(b) Plan view of joint

directio

nof

build

ingsw

ay

directio

nof

build

ingsw

ay

Figure 3.3 Biaxial loading

Mb2

Vb2

Mb1

Vb1

Mc2

Mc1

Vc2

Vc1

hb

hc

Vb2

Vb1

Vc2

Vc1

Mb2

Vb2

Mb1

Vb1

Mc2

Mc1

Vc2

Vc1

hb

hc

Vb2

Vb1

Vc2

Vc1

Figure 3.2 Planar equilibrium

beam centerline

column centerline

Figure 3.1 Beam and column

moments

The upper portion of Figure 3.2 shows a free-body

diagram of a portion of a SMRF. The beams and

columns have been cut at inflection points along the

spans. The lower portion of Figure 3.2 shows the

free-body diagram of the joint. Equilibrium of the

joint requires the following relation:

( ) ( )

( ) ( )2

2

2121

2121

cbbbb

bcccc

hVVMM

hVVMM

+++=

+++

For simplicity in expression, this relation often is

written as follows:

= bc MM

The latter expression is mathematically correct only ifthe beam and column dimensions are such that the

terms involving the shears in the previous expression

cancel.

Codes commonly require that the sum of the

column moment strengths exceed the sum of the

beam moment strengths at joints. The intent of this

requirement is to avoid formation of a story yieldmechanism. Usually, the requirement is applied

separately along each principal axis of the building.

In three-dimensional SMRFs, loading along a

diagonal can increase the moment demand on the

columns. This is illustrated in Figure 3.3. Figure3.3(a) shows a P-M interaction diagram for biaxial

bending. A horizontal plane is located at the axial

load corresponding to the loading considered. The

plane cuts the P-M interaction surface as shown.

This defines the column moment strength as a

function of the direction of loading. For a circular

column cross section with symmetric reinforcement,

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the moment strength for bending along any direction is constant for a given axial load. For a

square section with symmetric reinforcement, the moment strength for bending along the

diagonal is somewhat less than for bending along any principal direction, the exact relation

depending on the materials and reinforcement.

Figure 3.3(b) shows an interior beam-column joint in plan. Moment vectors indicate the

directions of the moments in the beams for building sway in the direction indicated. In this case,the sum of the moment strengths of the columns above and below the joint, for loading along the

diagonal, must balance the vector sum of the beam moment strengths. For beams of equal

strengths in the two directions, this can be expressed as

( ) ( )314321 22

1bbbbbbbc MMMMMMMM +=+++==

Because the column moment strength is the same (for circular sections) or less (for square

sections) for loading along the diagonal in comparison with loading along a principal axis, it is

clear that the column demand is increased significantly by diagonal loading. The column

moment strength must be increased by at least 40% for diagonal loading as compared with

loading along a principal axis, if it is to be stronger than the adjacent beams.

(T. Kelly, U. Canterbury, 1974)

+ =

first higher combined

(T. Kelly, U. Canterbury, 1974)

+ =

first higher combined

Figure 3.4 Higher-mode effects on column moment

diagrams

Dynamic response of SMRFs furthercomplicates column moment design. As

shown in Figure 3.4, the column moment

diagrams follow a fairly regular pattern for

first-mode loading (for a yielding system,

modes do not exist in the sense defined by

classical linear dynamics, but the concept

of modes is convenient for discussion and

design purposes). Note that even for this

loading the moment diagrams are

somewhat skewed in the bottom story

because of the greater fixity provided by

the foundation than by the first story abovegrade. For lateral force distributions that

occur during dynamic response and that

represent higher-mode loadings, the

moment diagrams are less regular.

Note that in some cases the column may

not be in contraflexure.

Case A Case B

moments

shear

Case A Case B

moments

shear

Figure 3.5 Relation between shear and moment

Note also that at some times (e.g., 2.73

seconds shown), the moments at a joint are

carried almost entirely by either the columnbelow or above the joint. Unless the

column is much stronger than the beam,yielding in the column is likely.

3.2.2 Shears

Shear is equal to the slope of the moment

diagram (Figure 3.5), so the moment

patterns of Figure 3.4 indicate column

shears as well as moments. In cases where

13


14/33

the column moments are relatively large at both ends, a

case that can be identified in some stories at time 2.73 in

Figure 3.4, the shear also is relatively large. This case is

shown as Case B in Figure 3.5. Therefore, the shears

obtained from a first-mode or inverted triangular lateral

loading are likely to underestimate the shears under

dynamic response.

The maximum column shear can be estimated by

assuming formation of plastic moments at both ends, as

shown in Figure 3.6; this shear may be unnecessarily

large, so alternative estimates are often made. When

using this approach, the axial load should be selected to

obtain a conservatively high estimate of the column

plastic-moment strength.

3.2.3 Axial Loads

Axial loads in SMRFs are

the sum of the shears in the

beams framing into the column

plus the self weight of the

column. As illustrated in

Figure 3.7, the axial loads will

vary around the building as a

function of the column location

and instantaneous earthquake

loading. As shown, for

earthquake inertial loading

from the left toward the right,the gravity and earthquake-

induced axial loads will be

additive for the exterior

columns on the right hand side

of the building. The gravity

and earthquake-induced axial

loads will act in opposite directions for the exterior columns on the left-hand side of the building,

and may result in column tension. For interior columns, the axial loads will tend to be dominated

by gravity loads; variations due to earthquake loading will depend on the stiffness and strength of

the beams framing into the column on either side this determines the shear forces in the beams

and therefore determines the axial load variation. For regular buildings with equal spans and

equal-size and strength beams in all spans, the axial load variation due to earthquake effects willbe relatively small for interior columns.

Figure 3.7 Gravity and earthquake axial loads on columns indifferent locations of a SMRF

earthquake and

gravity effects

additive

predominantly

gravity loads

earthquake and

gravity effects

opposite

earthquake and

gravity effects

additive

predominantly

gravity loads

earthquake and

gravity effects

opposite

Mpr

Pu

Vu

Mpr

Pu

Vu

lnn

pru l

MV =

Mpr

Pu

Vu

Mpr

Pu

Vu

lnn

pru l

MV =

Figure 3.6 Upper-bound column

shear

Limit analysis (plastic analysis) procedures can be used to calculate upper-bounds to column

axial loads. For this purpose, a plastic mechanism is assumed for the building. For a SMRF, itmay be reasonable to assume formation of beam flexural plastic hinges over the height of the

building (Figure 3.8). With this assumption, the beam shears are summed over the height and

added to the column weight to obtain the column axial force.

14


15/33

Corner columns tend to have the

largest fluctuation in axial forces

during an earthquake. This occurs

for two reasons. First the corner

columns usually support the

smallest gravity loads. Second, for

displacements along a diagonaldirection, the beam shears are

additive from both directions(Figure 3.9).

Plastic analysis gives an upper-

bound solution for column axial

forces. Actual values may be less

depending on how much inelastic

response develops in the frame and

depending on how it develops.

Figures 3.8 and 3.9 assume that beam flexural plastic hinges

form over the full height of the building. Studies show that

plastic hinging does not always form in this fashion. Figure

3.10 shows calculated results for a relatively flexible 12-story

building subjected to the 1940 El Center earthquake record.

As shown, beam flexural plastic hinges occur in discrete

locations, those locations migrating up and down the frame as

story drift concentrates in different stories at different times

during the dynamic response.

The patterns shown in Figure 3.10 are highly dependent on

the framing configuration, the dynamic characteristics of the

frame, and the characteristics of the ground motion. For

example, it is likely that a near-field ground motion pulse

could impose massive sway in a frame that would result innearly all the beams yielding at the same time. Even in this

case, and even if the columns are made stronger than the

beams, flexural yielding in general will not extend over the

MprMpr

VpVp

Mpr

Vp

Mpr

Vp

VpVp

VpVp

VpVp

VpVp

VpVp

VpVp

P= Vp + WcolumnP= Vp + Wcolumn P= Vp + Wcolumn

MprMpr

VpVp VpVp

Mpr

Vp

Mpr

Vp

Mpr

Vp

Mpr

Vp

Mpr

Vp

VpVp

VpVp

VpVp

VpVp

VpVp

VpVp

VpVp

VpVp

VpVp

VpVp

VpVp

VpVp

P= Vp + WcolumnP= Vp + Wcolumn P= Vp + WcolumnP= Vp + Wcolumn P= Vp + Wcolumn

Figure 3.8 Plastic analysis to obtain upper-bound column

axial forces

Figure 3.9 Corner column axial

force

Figure 3.10 Calculated locations of beam flexural plastic hinges in a 12-story frame (T. Kelly, 1974)

VbeamVbeam

Pcolumn

Building displaced

toward corner

column

15


16/33

full building height instead, flexural plastic hinges will form in the columns at some height that

depends on the framing configuration, the relative column and beam strengths, and the

distribution of lateral inertial loading.

3.3 Column behaviorBehavior of column cross sections in flexure is essentially the same as that of beams. Analysis

generally follows the same assumptions and approach. The only significant difference for a

column section is that the axial force is not necessarily equal to zero.

Figure 3.11 shows a column cross section and associated analysis assumptions. Strain is

assumed to vary linearly across the section, with maximum compression strain in concrete

assumed to be the limiting parameter in some cases, fracture of longitudinal reinforcement is

the critical behavior, but that will not be considered in detail here. Material stresses are assumed

to be uniquely related to strains as discussed for beams, this assumption is not correct for

inelastic reversed cyclic loading.

b

h

d

Atr

s

cs

fs

fs

As

M

P

Ts

Cs

Cc

As

d

sfs

Ts

As

d

s

cs

fs

fs

M

P

Ts

Cs

Ccsfs

Cs

(a) Low axial load

(b) High axial load

b

h

d

Atr

s

cs

fs

fs

As

M

P

Ts

Cs

Cc

As

d

sfs

Ts

As

d

s

cs

fs

fs

M

P

Ts

Cs

Ccsfs

Cs

(a) Low axial load

(b) High axial load

Figure 3.11 Flexural analysis of column cross sections

For relatively small axial force (Figure 3.11a), the depth of compression zone required to

equilibrate the axial force is relatively small. Therefore, the curvature is relatively large when the

maximum compressive strain capacity is reached. For larger axial force, the depth of

compression zone required to equilibrate the axial force increases; therefore, the curvature

capacity decreases. This effect of axial load on moment strength and curvature capacity is shown

in Figure 3.12. The cross section shown is assumed to be unconfined. For unconfined sections,

the flexural ductility decreases rapidly as axial load increases.

16


17/33

Figure 3.13 plots

interaction diagrams for a

column with 24 inch by 24

inch cross section

reinforced with 16 No. 9

bars as longitudinal

reinforcement. Concretehas compressive strength

of 4000 psi, andreinforcement is Grade 60.

The continuous curves are

calculated behavior

assuming the

reinforcement yields at 60ksi without strain-

hardening, and assuming

concrete is unconfined.

Also shown in Figure

3.13 are plots for two other

assumptions. The broken

curve is calculated

response assuming

reinforcement yields at 60

ksi without strain-

hardening, and assuming

confined concrete

behavior. Transverse

reinforcement comprises a

perimeter hoop plus cross

tie of No 3 bars at

longitudinal spacing of 2.5

inches, which satisfies the

confinement requirements

of ACI 318. Confined

concrete stress-strain

relation is according to Manders model. Concrete confinement with transverse reinforcement

increases the strain capacity and compressive stress capacity of the compression zone. This has

the most significant impact on the P-M interaction diagram for axial forces exceeding the

balanced point (the balanced point in fact changes and becomes ill-defined for confined concrete),

as columns with high axial force are compression-controlled. However, the strength gain is

marginal because the cover concrete has spalled, reducing the effective concrete section.

Ultimate curvature capacity is increased for all axial loads.

unconfined nominal

confined nominal

confined strainhardening

0 0.002 0.004 0.006 0.008

Curvature, 1/inch

-2000

-1000

0

1000

2000

4000

0 5000 10000 15000

Moment, kip-inch

AxialLoad,

kip

3000

24

#9 Grade 60

#3 @ 2.5

fc= 4 ksi unconfined nominal

confined nominal


unconfined nominal

confined nominal


0 0.002 0.004 0.006 0.008

Curvature, 1/inch

-2000

-1000

0

1000

2000

4000

0 5000 10000 15000

Moment, kip-inch

AxialLoad,

kip

3000

0 0.002 0.004 0.006 0.008

Curvature, 1/inch

-2000

-1000

0

1000

2000

4000

0 5000 10000 15000

Moment, kip-inch

AxialLoad,

kip

3000

24

#9 Grade 60

#3 @ 2.5

fc= 4 ksi

24

#9 Grade 60

#3 @ 2.5

fc= 4 ksi

Figure 3.13 Calculated interaction diagrams

P P

Moment, MMoment, M Curvature, Curvature,

Balanced Axial

tension-

controlled

compression-

controlled

Lower Axial

Higher Axial

strains

Figure 3.12 Effect of axial load on flexural cross-section behavior

The dashed curve Figure 3.13 is for the same column cross section with confined concrete, but

now assuming more realistic properties for the longitudinal reinforcement, including yield stress

of 67 ksi, and strain-hardening to 110 ksi. This has the most significant effect on calculated

behavior for low axial loads, as the column is tension-limited in this region. Note, however, that

the effect of strain-hardening is realized mostly because the column has confined concrete, which

increases the compression-zone strain capacity, thereby increasing the curvature capacity,

resulting in increased tension reinforcement strain. Ultimate curvature capacity is reduced

somewhat compared with the case of confined concrete with elasto-plastic reinforcement.

17


18/33

3.4 Column design

3.4.1 Moment and Axial Load

According to US design codes, the reliable flexural strength of the column must be at leastequal to the ultimate design moment. This expression is written as follows:

un MM

Mu is the moment demand obtained from the code-required analysis of the building under thespecified code earthquake representation.

Furthermore, ACI 318 requires that the flexural strengths of the columns shall satisfy

Mc (6/5) Mg

Mc= sum of moments at the faces of the joint corresponding to the nominal flexural strengthof the columns framing into that joint. Column flexural strength is to be calculated for the

factored axial force, consistent with the direction of the lateral forces considered, resulting in thelowest flexural strength.

Mg= sum of moments at the faces of the joint corresponding to the nominal flexural strengthsof the girders framing into that joint. In T-beam construction, where the slab is in tension under

moments at the face of the joint, slab reinforcement within an effective slab width defined in 8.10

is to be assumed to contribute to flexural strength if the slab reinforcement is developed at the

critical section for flexure.

Flexural strengths are summed such that the column moments oppose the beam moments. Thecolumn strength requirement needs to be checked for loading along the two principle directions of

the frame, but it is permitted to consider one framing direction at a time.

Note that this is not a joint equilibrium statement, but simply a requirement that the column

moment strength exceed the girder moment strength by a set ratio.

An alternative approach, which is more

conservative, and which may be viewed as

being more consistent with the capacity-

design philosophy, is to amplify the column

design moments obtained from the code

analysis on the basis of the expected flexural

overstrength of the beams. The procedures is

illustrated in Figure 3.14. Specifically, a

flexural overstrength factoro is defined as

beam centerline

column centerline

Mu,beam

Mu,column

Mpr,beam

Mu,column

beam outline

beam centerline

column centerline

Mu,beam

Mu,column

Mpr,beam

Mu,column

beam outline

Figure 3.14 Column design moments

= beamubeampr

M

M

,

,0

where Mpr,beam = the sum of probablemoment strengths of the beams at the joint

and Mu,beam = the sum of the code-requiredstrengths of the beams at the joint (obtained

from application of the code-specified design

loading). The columns are then designed for

18


19/33

momentsMu,column = oMu,column, whereMu,column = the column moment obtained from the code-specified design loading.

Regardless the method used to obtain moments, the axial loads should be checked for

maximum and minimum values, as either may be critical, depending on the column configuration

and the values of axial loads and moments. Figure 3.15 shows two loading cases, one with lateral

load from the left, the other with lateral load from the right. The loadings are used to determinethe axial loadsPmax andPmin in the column shown. To obtainPmax, the lateral load is shown from

left to right, and a high estimate of the gravity load is imposed. The high estimate is obtained

using a factored load combination that maximizes the axial load. In contrast, to obtainPmin, the

lateral load is applied from right to left, and a low estimate of the gravity load is imposed. In this

latter case, use of a low estimate of gravity load results in a lower estimate of the axial load. As

shown in the interaction diagram of Figure 3.15, both axial load cases need to be checked, as

either one may be critical.

P

M

increasinghigh gravity loads

low gravity

loads

Pmax

Pmin

Pmax

Pmin

P

M

increasingP

M

increasinghigh gravity loads

low gravity

loads

Pmax

Pmin

Pmax

Pmin

Figure 3.15 Axial load combinations for design

3.4 2 Transverse reinforcement

Transverse reinforcement serves to confine the concrete core, thereby increasing its

compressive strain capacity for flexural and axial deformations and improving its toughness forresisting shear. Transverse reinforcement also helps to delay buckling of the longitudinal

reinforcement and improves anchorage and splice strength. Aspects of shear strength are coveredin the next section. The emphasis here is on transverse reinforcement for flexural and axial load

enhancement, as well as buckling and anchorage resistance.

Figure 3.16 idealizes the confining behavior of transverse reinforcement. As the core concrete

is strained longitudinally, it expands laterally. The transverse reinforcement acts passively to

restrain this dilation and thereby results in confining stress. To be most effective as confinement

19


20/33

s

Rectangular Hoop

effectively

confined

concrete

unconfined

concrete

Circular Hoops

s

Rectangular Hoop

effectively

confined

concrete

unconfined

concrete

Circular Hoops

Circular Hoops

Figure 3.16 Confinement effectiveness

reinforcement, transverse reinforcement should

be evenly distributed along the length and

around the perimeter. For circular-cross

section columns, the circular hoops or spiral

provides fairly uniform radial confinement

because of the uniform curvature of the hoops

or spirals. Close spacing of the circular hoopsor spirals improves the uniformity of

confinement along the length, and thereforeimproves confinement effectiveness. For

rectangular-cross section columns, the

rectangular hoops provide resistance effectively

only at their corners or where crossties are

placed. Therefore, confinement effectiveness isimproved by placing longitudinal bars

uniformly around the perimeter and providing

hoops and crossties to restrain their outward

movement. As with circular-cross section

columns, confinement is improved by usingrelatively small longitudinal spacing.

Figure 3.17 depicts

confinement effectiveness

relations obtained using the

model proposed by Mander.

The results are for a circular-

cross-section column, a square-

cross-section column with two

crossties in each direction, a

square-cross-section column

with one crosstie in each

direction, and a square-cross-

section column without

crossties. The relation shown

suggests that square-cross-

section columns without

crossties are much less

effective than columns with

crossties.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

s/h

Aeff/Acc

circular

hoop without cross ties

hoop + 1 crosstie

each direction

hoop + 2 crosstieseach direction

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

s/h

Aeff/Acc

circular

hoop without cross ties

hoop + 1 crosstie

each direction

hoop + 2 crosstieseach direction

Figure 3.17 Confinement effectiveness relations for circular- and

square-cross section columns

The specific requirements for transverse reinforcement for confinement depend on the specific

demands imposed on the column. For example, a column framing into a foundation wall may be

required to develop a plastic-hinge rotation equal to 0.01 or more for a design loading. Under this

imposed rotation demand, the compression strain demand, and therefore the required amount ofconfinement reinforcement, will increase with increasing axial load.

US codes for design of new buildings do not require computation of rotation demands. These

demands are difficult to assess, especially in upper stories where the intent is to maintain an

elastic column, but where yielding may occur owing to higher mode and multi-directional loading

effects. Furthermore, axial loads occurring during earthquakes are difficult to determine during

design. Prevailing design practice is to obtain the design axial load from the code-specified

lateral forces; actual axial forces may deviate significantly from these values during an

20


21/33

earthquake. Plastic analysis procedures, which are not required by US codes, likewise may not

produce an accurate estimate of axial loads.

Given the uncertainty in rotation demands and axial force demands, ACI 318 specifies

transverse reinforcement for confinement of critical regions that is independent of the level of

axial force. The quantities required are intended to produce a column core that is capable of

sustaining axial compression approximately equal to the axial compression capacity of thecolumn before spalling of the concrete shell. This procedure has some advantages. It produces a

column that is relatively tough and capable of sustaining axial forces due to unforeseen loadings

that may crush the column; the result is a column highly resistant to brittle axial compression

failure. The design, construction, and inspection processes are considerably simplified by using a

constant amount of confinement up the height of the column.

The required volume ratio of transverse reinforcement for circular cross sections is equal to

yc

y

c

core

g

s fff

f

A

A'

'

12.0145.0

=

Maximum spacing is not to exceed 3 inches.

For rectangular cross sections, the total cross-sectional area of rectangular hoop reinforcement

is not to be less than that required by either of the following two equations

Ash = 0.3(shcfc/fyh)[(Ag /Ach)_1]

Ash= 0.09shcfc/fyh

Transverse reinforcement is to be provided by either single or overlapping hoops (Figure 3.18).

Crossties are to be of the same bar size and spacing as the hoops, and each end of the crosstiemust engage a peripheral longitudinal reinforcing bar. While it is recognized that 90-degree hooks

on crossties are not as effective as 135-degree hooks, they are considered adequate for most

loadings, and ease construction considerably. Some studies suggest that for heavy axial loads or

confining

stress

Atrfy

Atrfy

Atrfy

Ash

= 3Atr

hc

confining

stressA

trf

yA

trf

yA

trf

yA

trf

y

hc

Ash

= 4Atr

confining

stress

Atrfy

Atrfy

Atrfy

Ash

= 3Atr

hc

confining

stress

AtrfyAtrfy

Atrfy

Atrfy

Ash

= 3Atr

hc

hc

confining

stressA

trf

yA

trf

yA

trf

yA

trf

y

hc

Ash

= 4Atr

confining

stressA

trf

yA

trf

yA

trf

yA

trf

y

hc

confining

stressA

trf

yA

trf

yA

trf

yA

trf

yA

trf

yA

trf

yA

trf

yA

trf

y

hc

Ash

= 4Atr

Figure 3.18 Rectangular hoops

21


22/33

unusually large column rotation demands, 135-degree hooks should be used on both ends. Where

90-degree hooks are used, consecutive crossties shall be alternated end for end along the

longitudinal reinforcement. Perimeter hoops are required to have 135-degree hooks.

Figure 3.18 shows howAsh and hc are defined for a rectangular-cross-section column.

sx

hx8 14

4

6

sx

hx8 14

4

6

Figure 3.19 Spacing limits for hoop

reinforcement as function of horizontal

spacing of hoop legs.

As suggested by Figure 3.17, the same confinement

effectiveness can be achieved by differentcombinations of transverse and longitudinal spacings

of hoop legs. ACI 318 recognizes this by specifying

that the maximum longitudinal spacingsx vary as a

function of the maximum horizontal spacing hx

between legs of hoops or crossties around the column

perimeter (Figure 3.19). In addition, longitudinal

spacing of the transverse reinforcement is not to

exceed (a) one-quarter of the minimum member

dimension and (b) six times the diameter of the

longitudinal reinforcement. The former requirements

relates to confinement and to shear resistance

requirements. The latter requirement is intended torestrain buckling of longitudinal reinforcement until

deformations reach relatively large values.

According to ACI 318, crossties or legs of overlapping hoops shall not be spaced more than 14

in. on center in the direction perpendicular to the longitudinal axis of the structural member. This

longstanding limit is thought to be related to the intention to require crossties in significant

columns, such columns perhaps being considered to have dimensions of 18 inches or larger. As

suggested by Figure 3.17, crossties are desirable for confinement in all columns except very smalland structurally inconsequential columns.

Sometimes, architectural treatments result in thick concrete cover over the confined core. Such

cases should be avoided where possible, as loss of unconfined cover can result in significant and

relatively sudden loss of load-resisting capacity. As a minimum, ACI 318 requires that if thethickness of the concrete outside the confining transverse reinforcement exceeds 4 in., additional

transverse reinforcement shall be provided at a spacing not exceeding 12 in. Concrete cover on

the additional reinforcement shall not exceed 4 in.

The transverse reinforcement described in the preceding paragraphs is to be provided over a

length lo from each joint face and on both sides of any section where flexural yielding is likely to

occur as a result of inelastic lateral displacements of the frame (Figure 3.20). The length lo is not

to be less than (a) the depth of the member at the joint face or at the section where flexural

yielding is likely to occur, (b) one-sixth of the clear span of the member, and (c) 18 in.

Some studies suggest that the length of confinement at end regions should vary as a function of

axial load level. Furthermore, as shown in Figure 3.4 and 3.20, moments in columns in the first

story tend to be shifted toward the base, which may result in yielding extending over a greaterlength than in other typical stories. Some consideration should be given, therefore, to increasingthe length lo in the lower stories.

Also shown in Figure 3.20 is a typical detail at the first-floor level, where a slab on grade is

situated on compacted soil or lean concrete. Usually the structural drawings for such details will

include placement of a soft filler material between the column and slab on grade. If the fill is not

placed, if it hardens or becomes contaminated with debris during construction or during the life of

the building, or if the separation between the slab on grade and column is insufficient, the column

22


23/33

can be restrained by the slab on

grade, shifting the inelastic

action upward along the

column height. This problem

may have been a contributing

factor in the critical damage of

the Imperial County ServicesBuilding duringthe 1979

Imperial Valley Californiaearthquake (Figure 3.21).

Figure 3.20 suggests to

measure the length from the

top of the slab on grade, and

extend the confinement downto the top of the footing.

Where transverse

reinforcement as specified

above is not provided

throughout the full length of

the column, the remainder of

the column length is to contain

spiral or hoop reinforcement with center-to-center

spacing not exceeding the smaller of six times the

diameter of the longitudinal column bars or 6 in.

The specification of relatively close spacing

throughout the height is to avoid a sudden

transition in toughness along the length, which

could result in damage outside the heavily confined

region.

Mechanical splices of longitudinal reinforcementare permitted - Type 1 mechanical splices are not to

be used within a distance equal to twice the

member depth from the column or beam face or

from sections where yielding of the reinforcement

is likely to occur as a result of inelastic lateral

displacements; Type 2 mechanical splices are

permitted to be used at any location. Type 1

mechanical splices are those conforming to chapter

12 of ACI 318 (capable of 125% of the specified

yield strength). Type 2 mechanical splices are

capable of developing not less than the specified

tensile strength of the spliced bar, so that they willbe capable of developing significant inelastic strain

without failure.

Lap splices of longitudinal reinforcement are

permitted only within the center half of the member length, are to be designed as tension lap

splices, and are to be enclosed within transverse reinforcement having longitudinal spacing not

exceeding the minimum of (a) one-quarter of the minimum member dimension, (b) six times the

diameter of the longitudinal reinforcement, and (c)sx defined by Figure 3.19. Horizontal spacing

slab on grade

lo

lo

lo

>lo

closely spaced hoops

along lap splice

Moments

slab on grade

lo

lo

lolo

>lo

closely spaced hoops

along lap splice

Moments

Figure 3.21 Imperial County Servicesbuilding 1979 Imperial Valley

earthquake

Figure 3.20 Moments and hoop spacing along column height

23


24/33


25/33

is obtained not just for the maximum and minimum axial loads, but for all axial loads between

those extremes.

In some cases, the shear according to Figure 3.22a is significantly larger than the frame

mechanism (intended to be a beam-yielding mechanism) can sustain, so the approach of the

preceding paragraphs results in an excessively large design shear. Therefore, it is permitted to

design for the shear determined from joint strengths based on the probable moment strengthMprof the transverse members framing into the joint (Figure 3.22b). ACI 318 provides no guidance

on how this shear is to be determined. As illustrated in Figure 3.5, if the moment is distributed

unevenly between the columns above and below a joint, the shear in one column can be

disproportionately larger than the shear in columns adjacent stories.

Vu

Mu,column

Mu,column =

oMu,column

Vu,column

oVu,column

Vu

Mu,column

Mu,column =

oMu,column

Vu,column

oVu,column

Figure 3.24 Column shear design

Figure 3.24 illustrates one approach to satisfying

the preceding requirement. The approach assumes

that the frame mechanism is controlled by

development of flexural plastic hinges in the beams.

A flexural overstrength factoro is calculated asdescribed previously. Correspondingly, the column

moments are increased fromMu,column to oMu,column.Given that the shear is equal to the slope of themoment diagram, the shear is increased oVu,column,where Vu,column is the column shear calculated from the

code-specified loading. The column shear should be

determined from dynamic analysis rather than static

analysis, so that higher-mode effects are

approximately included. As noted previously,

dynamic response results in shears higher than those

that are obtained from a static lateral loading.

The design shear strength will vary

over the height of the column, in part

because the transverse reinforcementvaries, in part because the inelastic

demands vary over the height. This is

illustrated in Figure 3.25. Figure 3.25b

shows shear strength over height for high

axial loads. In this case, the strength is

greater at the ends than along the middle

because of the closer spacing of hoops

along the length lo. Figure 3.25c shows

the shear strength over height for lowaxial load, or axial tension. Shear

strength degrades with inelastic cyclic

loading, and this degradation may be

especially pronounced for tension

loading. According to ACI 318,

transverse reinforcement along the length lo is to be proportioned to resist shear assuming Vc = 0

when both the following conditions occur: (a) The earthquake-induced shear force represents one-

half or more of the maximum required shear strength within those lengths; and (b) The factored

axial compressive force including earthquake effects is less thanAgfc/20.

lo

lo

(a) column elevation (b) shear

strength for

high axial load

(c) shear strength

for low axial load

lo

lo

lolo

lolo

(a) column elevation (b) shear

strength for

high axial load

(c) shear strength

for low axial load

Figure 3.25 Shear strength

25


26/33

4. Beam-Column Joints

The reader is referred to the ACI 352 report in addition to ACI 318.


The design objective for reinforced concrete beam-column joints in SMRFs is for them to bestronger than the adjacent framing members so the majority of inelastic action occurs in the

adjacent members, in particular, the beams. Sometimes this is referred to as keeping the joint

elastic. However, considering yield penetration of beam reinforcement into the joint and slip of

beam and column reinforcement from the joint, fully elastic behavior is not possible.

Nonetheless, the design should aim to reduce these actions to acceptable levels through

appropriate joint proportioning and reinforcement.

4.2 Design actions on joints

In a SMRF it is assumed

that the design actions can be

obtained from statics. Thebeam is assumed to develop

flexural plastic hinges at the

critical sections as shown in

the right-hand side of Figure

4.1. The corresponding

beam shear is obtained from

statics. The beam moments

and shears are applied to a

free-body diagram of the

column as shown in the left-

hand side of Figure 4.1. For

the purpose of determiningthe column shear for joint

design it usually is sufficient

to assume inflection points in

the column at midheight, as

shown. The column shears

can subsequently be obtained

by statics. The column shear

force thus obtained may not

be suitable for design of the

column; however, it usually

is sufficiently accurate for

determining the designactions on the joint. For

determination of the column

shear for column design see

previous discussion.

lnb

wu

Beam Section

Vb1

Mpr,b1Vb2

Mpr,b2

Vb2Vb1

Mpr,b1

Vc1

Vc2

lnb

wu

Beam Section

Beam Section

Vb1

Mpr,b1Vb2

Mpr,b2

Vb2Vb1

Mpr,b1

Vc1

Vc2

Figure 4.1 Design actions on a joint

Vc1

Vc2

Tb2

Tb1

Tc1

Tc2

Vb1Vb2

Cb1= Tb1

Cb2 = Tb2

Cc1

Cc2

Vc1

Vc2

Tb2

Tb1

Tc1

Tc2

Vb1Vb2

Cb1= Tb1

Cb2 = Tb2

Cc1

Cc2

Tb1 Tb2

Vc1

Vjoint

Tb1 Tb2

Vc1Vc1

Vjoint

(a) Free body diagram showing

external actions on joint

(b) Free body diagram to

determine joint shear

Figure 4.2 Free body diagrams of joint

Figure 4.2a shows a free

body diagram of an interior

joint. The beam moments

have been replaced by the

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corresponding internal force resultants (tension in longitudinal reinforcement and compression in

concrete). It is assumed that the beam does not have axial force; therefore, the compression force

resultant on one side of the joint is equal to the tension force resultant acting on the same side of

the joint. For design purposes, ACI 318 assumes that the tension force is equal to 1.25Asfy, where

As is the cross-sectional area of longitudinal reinforcement andfy is the nominal yield stress. The

factor 1.25 is a low estimate of the overstrength for the reinforcement. According to ACI 318,As

need not include the slab reinforcement in tension, though it would be conservative to do so.

The column moment required to equilibrate the beam moments is obtained as the product of the

column shear and half the column clear height (Figure 4.1). The internal forces associated with

the combined moment and axial force (Figure 4.2a) need not be determined as part of the ACI

design procedure; however, it is worth noting that the values are not necessarily equal because the

column may have axial load.

The design horizontal joint shear is obtained by equilibrium of horizontal forces acting on a

free body diagram of the joint as shown in Figure 4.2b.

4.3 Joint behavior

Two simplified views of joint force-resistingmechanisms are shown in Figure 4.3 and 4.4.

Figure 4.3 depicts what is referred to as a joint

truss model. According to this model,

longitudinal reinforcement tension and

compression forces are resisted entirely by

bond with the concrete. The bond produces

diagonal compression forces in the joint

concrete, which in turn are equilibrated by joint

transverse reinforcement and longitudinal

reinforcement.

For example, referring to Figure 4.3, the

tension force in the beam longitudinalreinforcement entering the joint at the upper

right hand corner of the joint results in bond

forces along the bar length. At point 1, for

example, this bond force (shown by green

arrows) results in an inclined compression strut

in the concrete (shown red). To equilibrate this at point 1 there also is required to be bond stress

between the column longitudinal reinforcement and the concrete, as shown. At point 2, the

diagonal compression strut is resisting by horizontal force in the joint horizontal reinforcement

and by bond in the vertical column reinforcement. At point 3, the joint transverse reinforcement

tension force is equilibrated by another diagonal compression strut in concrete along with bond

stress in the column longitudinal reinforcement. At point 4 the diagonal compression strut is

equilibrated by bond in the horizontal beam reinforcement as well as compression in the concrete.

1

23

4

1

23

4

1

23

4

Figure 4.3 Joint truss model

The truss model as described above requires high bond stresses between the longitudinal

reinforcement and joint concrete, as it will be necessary to resist through bond both the

longitudinal reinforcement tension force on one side of the joint plus the longitudinal

reinforcement compression force from the other side of the joint. To resist this bond requires a

large volume of joint horizontal reinforcement. Joint vertical reinforcement may be required to

assist the column vertical bars in resisting the vertical forces in the joint. The ability of a joint to

resist the bond stresses is hampered by a number of factors. At point 1, for example, the joint is

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in tension due to flexural tension action of the adjacent column. Also, the length through the joint

is not sufficient under normal conditions to develop the bar both in tension and compression. The

problem is exacerbated by cyclic loading, which, among other things, deteriorates the bond

capacity.

An alternative model is the diagonal

compression strut model, shown in Figure 4.4.According to this model, bond stress through

the joint is lost under cyclic loading, so the

reinforcement in tension on one side of the

joint is developed through the compression

zone on the opposite side of the joint, both

within the joint and beyond the joint into the

adjacent beam. This may result in the

longitudinal reinforcement being in tension on

both sides of the joint.

For example, consider the beam flexural

tension force at point a in Figure 4.4. Bond

is likely to be substantial only along thecompressed part of the joint near point b.

The bond stress at this point is likely to be

insufficient to fully develop the bar, so the bar

remains in tension as it emerges from the joint

at point c. Thus, the bar at point c is in

tension even though the concrete in the flexural compression zone of the beam at that same point

is in compression. Because both the top bars at c and bottom bars at d are in tension, the

compression force in the concrete at c may be much higher than would be calculated from

conventional flexural analysis. Confinement of the concrete in this region is important so that

likelihood of brittle compressive failure is reduced.

ab

c

d

ab

c

d

Figure 4.4 Diagonal compression strut model

A result of the model shown in Figure 4.4 is that the joint shear is carried mostly through a

diagonal compression strut as shown in yellow in the figure. Transverse reinforcement isrequired to confine the concrete. The transverse reinforcement strengthens the core concrete and

toughens it so that it can undergo inelastic strain without failure. It also reduces dilation and

crack opening under load reversals so that integrity of the concrete to resist compressive stresses

is maintained.

Note that both the truss model of Figure 4.3 and the diagonal compression stress model of

Figure 4.4 require horizontal joint reinforcement, in one case to equilibrate shear forces and

improve bond capacity, and in the other to enable the concrete to support the diagonal

compression strut. The truss model would predict that the joint strength is directly related to the

amount of transverse reinforcement, assuming that is the limiting component of the mechanism.

The diagonal strut model does not relate the strength directly to the amount of transverse

reinforcement.Also, note that both models require vertical reinforcement, in one case to resist vertical

components of shear forces and to provide bond resistance, and in the other case to provide some

vertical confinement. Either the column longitudinal reinforcement must be well distributed

around the perimeter and remain elastic under lateral loading so it can act as the vertical joint

reinforcement, or additional vertical steel must be added in the joint to resist dilation and vertical

shear.

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Figure 4.5 Effect of transverse reinforcement

quantity on joint shear strength

Some alternative models for joint shear

strength (such as strut and tie models) can be

used to avoid vertical or horizontal joint

reinforcement, or both. These are not pursued

here.

Figure 4.5 shows test data on interior jointssubjected to simulated seismic loading, as

compiled by Otani [1991]. Open circles

correspond to cases where the beam failed in

flexure without apparent joint failure. Open

squares correspond to cases where joints failed

before beam yielding was apparent. Solid

triangles correspond to cases beams first yielded,

and then joints failed. The solid triangles follow

a simple trend for low amounts of joint

transverse reinforcement, the joint shear strength

increases with increasing joint transverse

reinforcement, but at a ratio of about 0.4 percent,

the strength no longer increases for increasing

joint transverse reinforcement ratio. This result

has been interpreted as indicating that the truss

model (Figure 4.3) does not model strength

behavior very well, and has led to the use of the

diagonal compression strut model (Figure 4.4) as

the basis for design according to ACI 318.

Figure 4.6 Effect of development length on

behavior of beam-column joints.

One of the outcomes of the ACI 318 design

procedure (actually, the diagonal compression

strut model) is that the beam longitudinal

reinforcement may sustain bond deterioration that

may result in inadequate performance as thereinforcement slips from (or slides through) the

joint. This slippage can result in a behavior that

is slack as the joint rotations pass near the origin.

Figure 4.6 depicts load-deformation behaviors

resulting from joints with different development

lengths through the joint, identified by the ratio

of the column dimension to the beam longitudinal

bar diameter. Clearly the behavior is improved,

with more stable and rounded hysteresis loops, as

the ratio hc/ db increases. Note the slack or

pinched behavior for small values ofhc/db.

Figure 4.5 shows that the strength of a joint canbe well represented by the ratio of the horizontal

joint shear stress to the compressive strength of

the concrete. This is reasonable considering that

failure is by failure of the diagonal compression

strut (Figure 4.4). In US practices, it has been

more traditional to relate the joint shear strength

to the square root of the compressive strength of

29


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Figure 4.7 Behavior of joints conforming

with code details but having shear exceeding

code values

concrete, based on the traditional notion that

shear strength is a tension failure phenomenon

that can be more directly related to the square

root of the compressive strength rather than the

directly to the compressive strength.

Figure 4.7 shows test results for a jointdesigned with transverse reinforcement meeting

the requirements of ACI 318, but with joint shear

stress exceeding the allowable values of ACI 318

(to be described later) [Kurose, 1991]. The ACI

318 strength values are shown by Vu in the

figures. The joint was subjected to loading in

two horizontal directions, the upper load-

deformation relation for loading in one direction,

the lower load-deformation relation for loading in

the orthogonal direction. The joints were able to

reach shear stress values exceeding the design

value in both directions. However, under this

overload the joints failed in shear. The

deterioration in strength is apparent in the figures.

Although the deformation capacity is

considerable, by the 4% deformation cycles the

strength deterioration probably is unacceptably

high. Better behavior is obtained if the joint

shear stresses are at or below acceptable limits.

4.4 Joint design

According to ACI 318, joint design requires (a) provision of adequate joint shear strength, (b)anchorage of reinforcement, (c) provision of adequate joint transverse reinforcement, and (d)

provision of adequate column flexural strength.

4.4.1 Joint shear

The design horizontal joint shear is obtained by equilibrium of horizontal forces acting on a

free body diagram of the joint as shown in Figure 4.2b. The design joint shear is not to exceed

0.85 times the nominal shear strength. The nominal shear strength of the joint is not tobe taken

greater than the forces specified below for normalweight aggregate concrete. For lightweight

aggregate concrete, the nominal shear strength of the joint is not to be taken greater than three-

quarters of the limits given for normalweight concrete.

For joints confined on all four faces 20 cf Aj

For joints confined on three faces or on two opposite faces 15 cf Aj

For others 12 cf Aj

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For the purpose of determining the appropriate joint shear strength, a member that frames into a

face is considered to provide confinement to the joint if at least three-quarters of the face of the

joint is covered by the framing member.

hbfVV jcnu' =

= 0.85

Values of24 20 15

20 15 12

nonseismic

seismic

Type

Note: To qualify for a geometry shown, the beam must

cover at least 75% of the face of the joint.

hbfVV jcnu' =

= 0.85

Values of24 20 15

20 15 12

nonseismic

seismic

Type

Values of24 20 15

20 15 12

nonseismic

seismic

Type

Note: To qualify for a geometry shown, the beam must

cover at least 75% of the face of the joint.

Figure 4.8 Summary of joint shear strength requirements

Figure 4.8 summarizes the joint

shear strength requirements. Note

that joint geometries typical ofconstruction at the roof level have

not been included here. At the

time of this writing, ACI

Committee 352 is developing

recommendations for joint shear

strength for those geometries.

4.4.2 Anchorage of

reinforcement

It is important for beam and

column longitudinalreinforcement to be anchored

adequately so that the joint can

resist the beam and column

moments.

In interior joints, reinforcement

typically extends through the

joint and is anchored in the

adjacent beam span (Figure 4.9).

ACI 318 requires that the column

dimension parallel to the beam

longitudinal reinforcement be not

less than 20 times the diameter of

the largest longitudinal bar for

normalweight concrete. For

lightweight concrete, the

required dimension is 26 times

the bar diameter. This

requirement helps improve performance of the joint by resisting slip of the beam bars through the

joint. Some slip, however, will occur even with this column dimensional requirement. See

Figure 4.6.

bd20

db

(a) Requirement forinterior connections


dhl

Note: Provide at

least ldh, and always

extend beam bars to

near the back of the

joint

bd20

db



dhl

Note: Provide at

least ldh, and always

extend beam bars to

near the back of the

joint

Figure 4.9 Anchorage/development requirements

ACI 352 recommends that the beam depth be not less than 20 times the diameter of the column

longitudinal reinforcement for the same reason. ACI 318 does not include this requirement.

For exterior joints, beam longitudinal reinforcement usually terminates in the joint with a

standard hook. The provided length for a standard 90 deg hook in normalweight aggregate

concrete must be the largest of8db, 6 in., and the length required by the following expression:

( )cbydh fdf = 65/l This expression assumes that the hook is embedded in a confined beam-column joint. The

expression is limited to for bar sizes No. 3 through No. 11. For lightweight aggregate concrete,

moehlesmrf

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