Chapter 4

1

Reinforced concrete frame
s
and frame analysis
Reinforced concrete frames generally consist of slabs, beams, girders, columns and
foundation ‘elements.’
Slabs may be 1
–
way or 2
–
way systems.
Example building alternates #1 & #2 use 1

way
slabs.
One
–
way slab
s are typically supported by beams pe
rpendicular to the shorter
span.
Beams may be directly supported by the columns or may be girder supported.
Girders (if present) are typically supported by columns and are loaded by beam end
reactions (concentrated lo
ads from beams).
Although exceptions exist
,
slab loading is typically considered to be due to u
niform area
dead and live loads on floors and uniform dead and snow loads on roofs.
Uniform loading on slabs produces uniformly distributed line loads on
suppo
rting
beams.
Frames may be braced (non
–
sway) or unbraced (sway frames).
If a frame is unbraced the
frame must resist
,
in additional to gravity load, lateral loads due to wind or earthquake
forces. Additional moments and shears are produced by lateral for
ces in the beams and
columns associated with the lateral bracing system. Analysis and design of unbraced
frames is beyond the scope of this course.
Braced (non
–
sway) frames include elements
explicitly designed to resist the lateral forces.
Such elements
include shear walls, elevator cores, and stair walls
(not shown on our
example building drawing)
. Thus the slabs
,
beams
,
columns and footings are typically
designed to resist only gravity loadings.
The analysis and design examples in this course consider
only braced frames under
gravity loading. The examples are for buildings of moderate size (floor area) and height.
B
uildings of less than 15,000
ft
2
floor area and three to four stories in height make up
approximately 90% of the current reinforced concr
ete building inventory.
In braced frames with ‘regular’ column spacing the
interior
columns support mainly axial
loading. Moments are small and column design is typically for axial compression only.
If should be noted, however, that the exterior (wall)
columns
are
subject
to significant
(uniaxial) bending moments even when only gravity loading is considered.
Corner columns are typically subject to biaxial bending moment as well as axial lo
ad
.
In
braced frames columns can be considered ‘short’ (non
–
slend
er) if
the
column unbraced
length (clear height between flo
o
r elements) is
≤
12 times
the least column dimension.
In
this course we will only design non
–
slender (short) columns with and without bending
moment in
addition
to axial load.
Footings will be designed
for
vertical axial load
centered on the footing.
This loading is mo
stly due to the axial in the column.
Chapter 4

2

Analysis of reinforced concrete frames: introductory discussion
Full three dimensional analysis of reinforced concrete frames
are
typically done by
computer.
The stiffness method (includes finite element analysis when
appropriate) is by
far most
popular analysis method when approximate methods are considered to be adequate.
Generally these computer analysis programs require initial member size
s
or stiffnesses a
s
input
,
thus an
initial design
is required even if final
analysis will be don
e
by computer.
What is desired for strength design is the maximum moments, shears, and axial force
resultants in the slabs, beams and column sections. This allows sizing of the critical
cross
–
sections.
Fortunately, the ACI code provi
sions provide relatively simple approximate methods for
analysis of braced frames subject to gravity loadings.
This allows sizing of the important
cross
–
sections and preliminary (often final) selection
of reinforcement.
One very useful simplification is
to ‘slice’ the three
–
dimensional frames along the column lines. (The
‘slice’ shown goes into and out of the page half way to
the next two
–
dimensional slice).
An explicit code provision allows each level in the
two
–
dimensional ‘slice’ to be consider
ed
for analysis as
beams (with slabs as flanges) supported by columns
which are
fixed
at the floor levels above and below the
level under consideration.
The ‘model’ shown is allowed by the ACI code for a typical floor.
In a typical ‘regular’ frame many of
the models will be similar
,
greatly simplifying the
original analysis problem.
Although the problem has been simplified the issue of
loading patterns
still complicates
the analysis as discussed in the next subsection.
Chapter 4

3

Loading ‘patterns’ required by ‘m
odel’ codes
Dead loads are assumed to always exist at every level of the structure.
Live loads are, however, variable and may or may not occur on some spans.
The ACI
Code require
s
consideration of live load patterns at each level of a continuous
s
truct
ure. The required loading
patt
erns
are:
1.
Dead load (DL)
on all spans
plus full live load
(LL) on all spans
2.
Dead load (DL)
on all spans
plus full live load
(LL) on
adjacent
spans
3.
Dead load (DL)
on all spans
plus full live load
(LL) on
alternate
spans
Note there two alternate span load cases for the three
span example structure shown.
Although full shear and moment diagrams for each load
case may be useful if is often only necessary
to
obtain
load re
sultants at ‘critical sections’.
C
ritical sections
are discussed in the next subsection.
Note: For strength design of R.C. frames:
If D and L are service loads on the floor:
DL = 1.2 D and LL = 1.6 L
Thus . . .
DL+LL = 1.2 D + 1.6 L
LL
DL
LL
DL
LL
DL
LL
DL
LL
DL
DL
LL
DL
DL
LL
DL
DL
LL
DL
DL
Chapter 4

4

Critical sections for flexure and shear in one
–
floor frame models
Critical sections (CS) along framing lines are those locations at which maximum values
of either positive moments, negative moments, or shear resultants occur.
The critical section for positive moment is typically near the center line of each span
under n
ormal loading conditions. For uniform loading patterns the critical section for
positive moment is
assumed
to be at the span center line.
The critical section
s
for negative moments are at the
faces
of support (ends of slab or
beam clear spans). These lo
cations are also the critical sections for shear.
For a framing line of several spans beams are typically sized for the
maximum moment
occurring at any critical section
along the framing line.
The five critical sections for a symmetrical three
–
span,
ACI
single floor model are
illustrated below:
Note: The information and concepts presented on the following page are useful if the
structure requires actual detailed analysis.
I would like you to learn this material.
After ‘trial’ member cross

secti
ons for the slabs, beams and columns have been selected
detailed analysis can be done by ‘hand’ computations (moment distribution method) or
by a plane

frame analysis program.
An alternative to detailed analysis is ‘approximate analysis’ which is explicit
ly allowed
by the ACI Code for many common structural and loading situations.
We will u
se approximate analysis to deter
mine the moments and shears at the critical
sections (CS’s) of our frame.
After a short ‘time out’ during which I will attempt to
explai
n the basis of th
e approximate method
I wi
ll present
,
and you will learn
,
the
approximate ACI analysis method.
Personal Note: Due to the complexity of detailed analysis which involves many
assumptions regarding cracking of the members which greatly effect
s member stiffness
) I
have to believe
that the results from approximate analysis are just a accurate as ‘exact’
analysis in many cases!
C
L
Floor model
x = critical section
2
5
4
3
1
Chapter 4

5

Loading patterns which produce maximum moments and shears at critical section
of a framing line
(Framing line = conti
nuous series of spans from end

end of structure).
Note: The examples illustrated below are for a three span framing line with equal (or
nearly equal) column spacing. Beam line loads are assumed to be uniformly distributed
and of equal (or nearly equal) v
alue.
DL = dead load (always on all spans)
LL = live load (placed on all adjacent or alternate spans
)
Important critical sections shown circled as
.
Load case 1:
DL and LL on all spans produces maximum vertical load on columns
but
does not produce maximum moment or shear resultants at any critical section in the b
e
am
framing line!
Load case 2:
DL on all spans, LL on alternate spans (as
shown).
This case produces maximum negative
moment and shear at critical section 1 and maxim
um
positive moment at 2.
Load case 3:
DL or all spans
,
LL on adjacent spans (as
shown). This case produces maximum negative
moments and shears at critical sections 3 and 4.
Load case 4:
DL on all spans and LL on center span.
This load case produces ma
ximum positive moment at
critical section 5.
Comments: For equal spans the design moment at
critical section 3 is typically used to ‘size’ the beam
cross
–
section
. Other moments at critical sections
are
smaller
. These
sections typically have reduced ten
sion
reinforcement area than critical section
3
.
The actual moment values at each critical section depend on
the span, loadings, ratio of dead load to live load and beam to
column stiffness ratio
s
.
Our ‘simple model’
has been
complicated by the requireme
nt
to consider loading patterns!
Fortunately, the ACI code
allows
,
in many common situations
,
analysis by ACI
coefficients
. This is the subject of the next subsection.
2
1
LL
DL
DL
LL
DL
4
3
LL
DL
LL
DL
DL
DL
LL
DL
DL
5
L
n
L
‘clear’ span
††
–
††††
獰sn
C
L
C
L
Chapter 4

6

‘
Short Timeout’ Page:
For reference consider the three span floor structure present
ed on the previous page. Also
look at the figures on the last page of this chapter for moment diagrams discussed below.
Suppose
that all of the beams of the structure were connected to the columns using only
‘pin’ connections (simply supported). If these
beams were uniformly loaded the end
moments would be zero and the maximum moment at the centerline would = wL
2
/8.
This fact can be derived using
statics
alone, the value is a called the ‘static moment’.
If this were the actual system I would reinforce the
beam using the appropriate A
s
+
at the
centerline of the beam for M
+
=wL
2
/8.
Now suppose that the all of the beams were rigidly connected to rigid columns
(fixed

fixed end support) and uniformly loaded. An elastic analysis results in end moments of
wL
2
/12
and moment at the beam centerlines of wL
2
/24.
If this were the actual system I would reinforce the beam using the appropriate A
s
+
at the
centerline of the beam for M
+
=wL
2
/24 and the ends of the beam using the appropriate A
s

for M

=wL
2
/12.
Note that wL
2
/24 + wL
2
/12 = wL
2
/8 (the ‘static moment
’)!
Neither of the support c
onditions supposed above is correct
.
Beams are not connected to
columns by ‘pin
s
’ and columns are not rigid
, they are flexible
members.
The actual
situation is most likely some
what
betwe
en the
two
extreme
solutions given above.
For now consider the interior (middle) span of our symmetric example floor structure. It
is a FACT that whatever the actual solution is M
+
 + M

 = wL
2
/8!
( x=absolute value ).
Example: M
+
= M

= wL
2
/16
s
atisfies the static moment requirement.
If I were to
reinforce this beam using equal A
s
+
and A
s

reinforcement appropriate for wL
2
/16 the
beam
would be ‘safe’ (strong enough, at with respect
to
moment capacity).
All the above solutions ‘work’! All are ‘s
afe’. So who cares about a more exact analysis?
Well,
serviceability
also matters
.
Getting the analysis as close a possible to the actual
structural situation will provide better deflection estimates and control of cracking. These
are service load issues.
Getting column moment estimates a closely as possible is also
good practice.
Chapter 4

7

Frame analysis using ACI coefficients
The previous subsections have discussed back ground material related to:
Reinforced concrete frames and their typical components.
Simp
lifications typically used to analyze actual three
–
dimensional frames using
single
–
level models.
The complications associated with various (code required) ‘loading patterns.’
Now that we have ‘covered
’
the determination
of the
critical load patterns for
obtaining
the maximum values of negative moments, positive moments an
d shears for members of
continuou
s frames
you
should be pleased to learn that in many cases we can actually
avoid
these considerations for typical
one
–
way slab systems
and for
beams
subj
ect to
uniform load (typical of DL and LL on building roofs and floors).
The ACI code allows the use of certain specified
coefficients
for analysis of slabs and
beams by which maximum anticipated moments and shears can be computed at member
ends and cente
rlines.
The ACI coefficients can
only
be used if certain specified conditions exist. These
conditions are given in the next subsection.
To compute design values for positive
M
u
and negative
M
u
only the applied line loading
(
w
u
,
k
/
ft
) and the clear span
(
L
n
,
ft
) must be known (or estimated).
Typically
for floors
we compute the factored line loading (
w
u
) using tributary widths and
factored dead plus (reduced) live loadings (adding some allowance for estimated stem
weight of beams).
Clear span is computed
from centerline to centerline span reduced by (typically
estimated) actual width of supports.
When the ACI coefficients (
x
) are used:
Design m
oments are computed as:
x
L
w
M
n
u
u
2
Design s
hears are
computed as:
x
L
w
V
n
u
u
(except
for 15% inc. @ 1
st
int. support)
1/
x
= applicable ACI coeff
icient, ‘x’ varies with location of CS, support
conditionbs and type of resultant under consideration (M or V).
Chapter 4

8

Requirements for analysis by ACI coefficients for design moments and shears
For
members
which form part of continuous frame
, moments and shears for design are
permitted to be determined by ‘ACI coefficient
s
’ when the following conditions are meet:
There are two or more spans
.
This condition is typical in building frames. It is true
for all frames which we
encounter in this course.
Loads are uniformly distributed
.
This condition allows ACI coefficients for analysis of our slabs and beams.
Girders have significant concentrated ‘point’ loads and the girder frames can not
directly
use
coefficients for analysis.
Unfactored live load must not exceed three times unfactored dead load
This condition is not typ
ically limiting. Consider the extreme case of
basic
(unreduced) live load of 100
psf
and a
thin slab of about
the minimum =
4".
DL
=
(4/12)(150) = 50
psf
.
LL/DL = 100/50 = 2 < 3 (OK)
Members are prismatic
(constant cross
–
section)
For economic reasons ‘haunched’ members are rarely used.
Spans are approximate
ly
as equal. Longer of two adjacent spans must not be
greater than the sh
orter span by more than 20 percent.
In our
frames and in many other cases spans in one direction qualify
(In our
example frames all
slab and beam
spans are
very nearly
equal
)
.
Values of the ACI coefficients
Rather than specifying the values of numerous
ACI coefficients ‘in writing’ a much more
efficient method is to use figures.
Each of the various ACI coefficients are shown on the graphical analysis aid provided by
an A
ppendix
‘A’ page
.
Review that page.
Note that support conditions and location of th
e critical section under consideration
determine the coefficient to be used.
Note that one
–
way slab
s
with clear span less than 10
ft
have a separate set of coefficients
for negative moment (end moments).
Also be aware that in moderate height buildings th
e column size does not generally
qualify the column as ‘stiff’!
Chapter 4

9

Observe that, for beam lines with equal spans
,
the maximum moment is the negative
moment value given by
10
/
2
n
u
u
L
w
M
.
This negative
M
u
value is typically used to
‘size’ the concre
te cross
–
section for the entire beam ‘line.’
Where the coefficients produce smaller
M
u
values the same cross
–
section is used but less
tension reinforcement area is required.
Observe that t
he design shear resultant is specified as
2
/
n
u
u
L
w
V
at all locations
except for the beam end at the first interior support for
end spans.
At this location
2
/
15
.
1
n
u
u
L
w
V
(Note the 15% increase at these special end span locations).
P
ractical design issue #1:
slab, beam, column and footing ‘marks
’.
In a typical frame there are a variety of slabs, beams, columns and footings (basic
member ‘classes’).
Although many members of a certain ‘class’ may be ‘similar’, design loads (moments
shears axial forces, etc.) will vary. This will require us to va
ry the reinforcement even if
the members of a certain class have
the same
concrete cross
–
section.
For
e
ase of reference in design we typically assign identification
symbols
to elements of
each class (see engineering marks following the Example Building pl
an).
For example we might identify interior beams as B1, B2, etc. and spandrel beams (beams
along outside walls) as SB1, SB2, etc.
Footings supporting interior columns might be marked as F1, footing support
ing
wall
columns, F2, and footing supporting cor
ner column, F3, etc.
For slabs marks we might use S1, S2, etc.
Slabs, beams and footing which are exactly
the same (including reinforcement) are given
the same ‘mark’.
For columns we choose ‘column lines’ in one direction and label them as A, B, C, D, e
tc.
We then label the ‘column lines’ in the other (typically perpendicular) direction as 1, 2, 3,
4, etc.
Each column is then
uniquely
identified as A1, A2, A3, B1, B2, B3, etc.
Chapter 4

10

Practical design issue #2:
class examples and homework
Examples will be
for deformed bar reinforcement of grade 60,
f
y
= 60,000
psi
.
Bars will be straight except for standard hooks, stirrups and column ‘ties’.
Concrete will be normal weight (150
psf
with reinforcement) with
f
c
' = 4,000
psi
.
with the
exception of
f
c
' = 3,000
psi
for footings.
For efficient formwork our designs will conform with the following specifications:
All slabs will be of the same thickness and designed to avoid deflection
computations
.
All beams
and girders (and beams & joists)
will be of the same d
epth
.
All interior beams will be of the same width
with interior girders
twice
as wide.
All spandrel beams and girders will be of the same width (if practical).
All columns will have the same concrete cro
ss
–
section at all locations and from
level to le
vel. All columns will be standard
sizes.
Maximum flexural bar size shall be #10.
Minimu
m flexural bar size shall be #4 in slabs and #5 in beams.
Minimum clear cover for flexural bars in slabs
& joists
shall be ¾" (code).
Maximum stirrups and the bar si
zes shall be #4, with minimum clear cover of 1½
inches (code).
All footings supporting interior columns will be the same size and thickness.
All footings supporting exterior columns (non
–
corner) will be the same size and
thickness.
All footings supporti
ng corner columns will be of the same size and thickness.
Minimum clear cover to steel in footings shall be 3"
(code)
.
Maximum course aggregate size
i
s specified as ¾".
Chapter 4

11

Practical design issue #3:
Static moment concepts
Consider a simply supported b
eam under
uniform load.
The moment diagram shows that the
maximum moment (determined by static
s
)
is:
8
2
max
wL
M
Next consider a beam fixed at both ends
under uniform load.
Elastic analysis gives the maximum
positive moment as
wL
2
/24 (at c
enter line
of span) and maximum negative moment
as
wL
2
/12 (at supports).
Note the sum of
wL
2
/24 +
wL
2
/12 =
wL
2
/8
as in the simply supported beam under
uniform load.
Next consider a beam supported by elastic
(flexible) columns.
Analysis reveals that th
e
end moments
depend on the ratio
of the beam to column
stiffness.
The sum of the maximum positive moment (at center line) plus maximum negative
moment (at supports)
will still be
the static moment,
M
o
=
wL
2
/8.
Now suppose that you have considered the ‘sys
tem’
shown
in the flexible column
example as best representing your reinforced concrete beam design problem.
You select the positive moment steel and the negative moment steel for
M
max
=
wL
2
/16.
The beam is sa
f
e (for flexural loading) if
w
does not excee
d the factored design load
w
u
.
Let’s suppose now that the actual system behavior is much more like the fixed ended
beam case (columns are far stiffer than anticipated originally).
Is your design (for flexural reinforcement) still safe (strong enough)?
wL
2
/8
w
wL
2
/8
w
wL
2
/
24
wL
2
/
12
wL
2
/8
w
wL
2
/
16
wL
2
/
16
Chapter 4

12

T
he answer is YES if ‘certain conditions’ are met.
As we increase the load w towards
w
u
, a
t
same point we will exceed
your
design end
moment of
wL
2
/16 and the negative moment steel will yield. At this point plastic hinges
will form at the ends of the span
.
For additional loading the span acts as a simple beam system and accumulates positive
moment at the span centerline. Since we have provided reinforcement
for
the full static
moment,
M
o
=
wL
2
/8, the span will not fail in flexure until
w
reaches the fact
ored design
moment
w
u
unless the ‘hinges’ fail before w
u
is reached.
The major point illustrate
d
by the above discussion is that in continuous structures
redistribution
of resisting moments can occur if sufficient rotational ductility
at the
‘hinges’
is a
vailable.
With respect to the previous discussion several very important points should be clarified.
1.
The steel must yield for an effective plastic hinge to form. Thus, the steel must be
adequately ‘anchored’ in the concrete.
2.
After th
e steel yields as
t
he ‘hinge’ rotate
s
compressive strains in the concrete will
increase rapidly. The concrete must not fracture or ‘spall off’ when subjected to
the required hinge rotations associated with load level
w
u
.
In short,
the reinforced concrete section must have
adequate flexural
ductility
(hinge rotational capacity).
3.
The previous discussion seems to imply that the ‘exact’ analysis is relatively
unimportant. The system will not ‘fail’ (in flexure) even if the analysis is quite
wrong (if enough flexural ductility
has been provided).
The issue of ‘failure’, however, involves more than flexural strength of the beam system.
Serviceability issues include deflection and crack control issues. If excessive rotations
occur at plastic hinges under
actually expected serv
ice loads
deflections at the beam
center line increase and large cracks open at the hinges. Thus a ‘reasonable’ analysis is
highly desirable from a serviceability point of view.
Exterior
and
corner column moments also depend strongly
on the actual beam e
nd
moments, thus a ‘reasonable’ analysis is important for the design of the
se
column
elements subject to axial loading plus bending.
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