Continuity in RC beams and frames-AFM (Complete).pdf

CONTINUITY IN
REINFORCED CONCRETE
BEAMS AND FRAMES
1
Course Instructors:
Prof. Dr. Asad-ur-Rehman Khan /
Dr. Aslam Faqeer Mohammad

INFLUENCE LINES
2
Influence lines are important in the design of structures
that resist large live loads.
If a structure is subjected to a live or moving load, the
variation in shear and moment is best described using
influence lines.
An influence line represents the variation of the
reaction, shear, moment, or deflection at a specific point
in a member as a concentrated force moves over the
member

INFLUENCE LINES
3
Once the influence line is drawn, the location of the
live load which will cause the greatest influence on the
structure can be found very quickly.
Therefore, influence lines are important in the design
of a structure where the loads move along the span
(bridges, cranes, conveyors, etc.).

INFLUENCE LINES
4
Although the procedure for constructing an influence
line is rather simple, it is important to remember the
difference between constructing an influence line and
constructing a shear or moment diagram
Influence lines represent the effect of a moving load
only at a specified point on a member.
Whereas shear and moment diagrams represent the
effect of fixed loads at all points along the member.

INFLUENCE LINES
5
Procedure for determining the influence line at a point
𝑷 for any function (reaction, shear, or moment).
1. Place a unit load (a load whose magnitude is equal to
one) at a point, 𝑥, along the member.
2. Use the equations of equilibrium to find the value of
the reaction, shear, or moment at a specific point 𝑃 due
the concentrated load as a function of 𝑥.
3. Plot the values of the reaction, shear, or moment for
the member.

INFLUENCE LINES
6
Construct the influence line for the vertical reaction
at A of the beam shown in figure
When the unit load is placed
a variable distance x from A,
the reaction 𝐴𝑦 as a function
of x can be determined from
summing the moment about
𝐵.

INFLUENCE LINES
8
Construct the influence line for the shear at point
C of the beam in figure
Here two equations have to be determined since there
are two segments for the influence line due to the
discontinuity of shear at C.

INFLUENCE LINES
10
Construct the influence line for the moment at
point C of the beam in figure
Here two equations have to be determined since there
are two segments for the influence line due to the
discontinuity of shear at C.

INFLUENCE LINES
12
Since beams or girders are usually major load–carrying
members in large structures, it is important to draw
influence lines for reaction, shear, and moment at
specified points.
Once an influence line has been drawn, it is possible to
locate the live loads on the beam so that the maximum
value of the reaction, shear, or moment is produced.
This is very important in the design procedure.

QUALITATIVE INFLUENCE LINES
13
In 1886, Heinrich Müller-Breslau develop a method
for rapidly constructing the shape of an influence line.
The Muller-Breslau principle states:
The influence line for a function (reaction, shear,
moment) is to the same scale as the deflected shape of
the beam when the beam is acted on by the function.
To draw the deflected shape properly, the ability of the
beam to resist the applied function must be removed.

14
For example, consider the following simply supported
beam.
Let’s try to find the shape of the influence line for the
vertical reaction at A.

15
Remove the ability to resist movement in the vertical
direction at A by using a guided roller

16
Consider the following simply supported beam.
Let’s try to find the
shape of the influence
line for the shear at
the mid-point (point
C). Remove the ability
to resist shear at point
C.

17
Consider the following simply supported beam.
Let’s try to find the
shape of the influence
line for the moment at
the mid-point (point
C). Remove the ability
to resist moment at
point C.

INDETERMINATE STRUCTURES
20
A unique need in concrete design (especially when
one considers that reinforcement may be terminated at
various locations along spans) is to place loading for
maximum effect. Therefore, we would like to study
ways to place live loading in various portions of an
indeterminate structure to create maximum positive
moment, maximum negative moment, maximum axial
loading in columns, etc.

21
Let’s Consider the 5-span continuous beam shown
below:
Let’s suppose we would like to determine the loading
pattern that will cause maximum positive moment
within span AC (i.e. at point B).
A
B D F H K
C E G J L

22
One technique that will allow us to do this is the
Mueller-Breslau approach. This approach will allow
us to develop qualitative influence lines for
indeterminate structures by examining the deflected
shape of the structure when subjected to loading
representing the internal force we are interested in. We
want to place loading on the 5-span beam so that
maximum positive moment is created within span AC.

23
To do this, simply impart a deformation in the
direction of this internal force (i.e. positive moment)
and draw the resulting deflected shape of the
continuous beam.

24
Any location on the deflected shape that is on the
same-side as the “zero-line” should be loaded and any
part “under” the zero-line should not be loaded. This
leads to the pattern loading shown below.
A
B D F H K
C E G J L

25
Now, let’s examine the loading required to create
maximum negative moment over the support at point
C on the beam. To find this loading pattern, we simply
deform the beam in the same manner as negative
moment at C would. This is shown below.
A
B D F H K
C E G J L

26
The loading pattern that will deform the beam in the
same manner is that which will cause maximum
negative moment at C. This loading pattern is as
shown below

27
The Mueller-Breslau principle can also be applied to
transverse shear within the system. For example, to
create maximum transverse shear in a span, we simply
deform the beam in a manner corresponding to
positive shear as shown in the example below

28
Further Examples

29
The same principles apply to building frameworks.
Sketching the deflected shape is a bit more difficult,
however. If we would like to establish the loading
pattern to generate the maximum positive moment in
span CD of the building shown below, we deform the
framework in the same manner as positive moment
within CD will as shown below. Extending the
deformed shape of the structure after positive bending
is induced in span CD gives indication as to where
live loading is to be placed. The checkerboard pattern
that results is shown in the figure below.

31
The checkerboard pattern that results is shown in the
figure below.

32
We should also be
concerned with creating
maximum moment and
maximum axial loading in
columns within the
framework. The deflected
shape shown to generate
single curvature bending
in columns CC’ and DD’.

33
The loading that accentuates this behaviour is shown
below.

34
As we will see in the design of column, both reverse
curvature and single curvature column bending will have
their own unique effects on assessing a column’s
strength. Reverse curvature bending can actually
improve the axial load carrying capacity when compared
to the same moment magnitudes causing single-curvature
bending.
If one would like to cause large magnitude reverse
curvature bending in columns CC’ and DD’, the loading
pattern shown below should be implemented.

Simplifications Allowed in Frame Analysis
36
If we were to consider all the loading possibilities that
create maximum effect within the members of a
structural system, our lives as designers would be rather
complicated. Fortunately, the ACI 318 provisions have
set out to simplify the process, while maintaining
accuracy in computations. These simplified provisions
have based upon many structural analyses for buildings
of usual configuration with usual loading.

37
The first aspect to the ACI simplifications is the creation
of a beam and column subassembly that can be utilized
for analysis. This subassemblage first defines a floor
level for consideration. The columns adjacent to the
beam defining this floor level are assumed to have their
far ends fixed if an intermediate floor is considered.
If floor level includes “basement” columns, or columns
connected to foundation components, the end conditions
assumed for these columns should be reflective of the
foundation conditions.

38
Once the subassembly has been defined, the engineer
can then turn attention to the loading. The arrangement
of live loading allowed by ACI 8.9 is as follows:

39
These loading arrangements are shown below

40
The rather unique aspect regarding loading in concrete
design, is that shear force and bending moment
envelopes are used. One loading condition will provide
maximum positive moment within a span and another
loading condition will cause maximum negative moment
at the supports. As a result, we will be superimposing
different moment diagrams for different loading
conditions onto a single diagram that will be the basis
for the design of our member, including reinforcement
sizing, bar cut-offs, etc.

41
A shear force and moment envelope diagram using the
three previous loading configurations is shown below

42
Our previous discussion has been limited to beam
design. However, we also have to design reinforced
concrete columns. There are ACI 318 provisions that
aid the designer in ensuring that columns are subjected
to the worse-case loading, while keeping procedures
manageable. ACI 8.8 provides the provisions and
guidance.

Modeling or Idealization of the Structure
43
The use of statics, moment distribution, slope
deflection, or matrix methods to analyze reinforced
concrete structures all depend upon an analytical
model. This model must be capable of capturing all
pertinent and important behavioral characteristics of the
structure and give accurate representations for the
internal forces within, and deflections of, the structure.
Modeling structures can sometimes seem to be equal
parts art and science. However, there are some
“standard” techniques that we may use to model
structures within the spirit of the ACI 318 provisions.

44
Effective Span Length
Reinforced concrete members have finite dimension to
them. For example, it is not uncommon to have
columns that are 24 inches square and therefore, these
columns will remove 2 feet from the centerline-to-
centerline span for any beam they support. Therefore,
understanding the effective span of members within the
context of analytical modeling is very important.

45
We can begin our discussion by considering a
subassemblage from a reinforced concrete framework
shown on the next sheet. As indicated, the members
DO NOT have a zero thickness (sticklike) appearance
in real life. Therefore, let’s take a look at what effect
beam and column dimensions have on the moment
diagrams that result from analysis and this will help us
in understanding what the effective length is in the
analysis of reinforced concrete structures.

46
𝑀𝑏1
𝑀𝑐1
𝑀𝑐2
𝑀𝑏2 𝑀𝑏3

47
The moments in the columns, corresponding to the
centerlines of the beams, are shown as 𝑀𝑐1 and 𝑀𝑐2 ,
respectively. There are three moments of inter est in the
beam member. These are the moments at the centerline
of the columns and the mid-span (i.e. maximum
positive moment) in the beams; 𝑀𝑏1, 𝑀𝑏2 and 𝑀𝑏3.

48
A frame analysis using the centerline-to-centerline
dimensions of the members will result in moments at
the ends of the members that are not realistic. The
moments referenced previously were taken from the
frame analysis and are indicated at the centerlines of
the joints. ACI 318 recommends that the moments at
the face of the members be used.

49
Therefore, there will be a significant reduction in
negative moment in the beams and a moderate
reduction in the columns when the face-moments are
considered. These design moments are indicated in the
figure above as well. The column moments are often
taken simply to be those at the centreline of the beam
members because the column moment diagram gradient
is often “shallow”.

50
Therefore, there will be a significant reduction in
negative moment in the beams and a moderate
reduction in the columns when the face-moments are
considered. These design moments are indicated in the
figure above as well. The column moments are often
taken simply to be those at the centreline of the beam
members because the column moment diagram gradient
is often “shallow”.

51
Moments of Inertia
As per previous discussion related to cracking in
reinforced concrete members should lead to an
appreciation of the difficulty in assigning moments of
inertia for members within a concrete structure. First of
all, the members may be T-beams or columns bent in
single or reverse curvature. Furthermore, the extent of
cracking will depend upon the magnitude of the loading
and the ratio of 𝑀𝑐𝑟/𝑀𝑎 at any given section along the
member’s spans.

52
Moments of Inertia

53
Moments of Inertia
As always, the ACI 318 provisions come riding in to the
rescue. There are a couple of different recommendations
for the moments of inertia of the members within
concrete frameworks. The commentary of ACI 8.6.1
contains the following recommendations:
For braced frames, relative values of stiffness are
important. Two usual assumptions are to use gross 𝑬𝑰
values for all members or, to use half the gross 𝑬𝑰 of the
beam stem for beams and the gross 𝑬𝑰 for the columns

54
Moments of Inertia
The second set of recommendations (for frames allowed
to sway) comes from ACI 10.11.1

55
Moments of Inertia
The second set of recommendations (for frames allowed
to sway) comes from ACI 10.11.1

56
Moments of Inertia
One should always stop and consider what the ACI 318
provisions are “saying” when these types of provisions
are read. First of all, the moment of inertia of the
members in the analytical model are important.
Furthermore, cracking at the ultimate and service
loading conditions are very important to consider in the
analysis. Therefore, ACI 318 appears to be subscribing
to the following philosophy in the analysis of reinforced
concrete members:

57
Moments of Inertia
1. Beams will suffer from considerably more flexural
cracking at the ultimate loading condition, and
therefore, the moment of inertia is assumed to be 1/2
that of the columns.
2. The columns will have a beneficial compression
force in them at the ultimate loading condition and
therefore cracking will likely be less than that in the
beams.

58
Moments of Inertia
3. The moment of inertia of the columns is taken to be
twice that of the beams, and therefore, the columns
will be carrying more loading due to their increased
relative stiffness and also will be relied upon to a
larger extent in the design of the framework
4. In essence, the columns will be designed for larger
forces, and therefore, they will be stronger than they
likely will need to be, which will result in load-path
redundancy.

59
Support Conditions
Most framework and continuous beam analysis will
begin with a transformation of a three-dimensional
structure to a two-dimensional model. The 2D model is
often much simpler (analytically), but modeling the
support conditions can still be a little cumbersome.
Furthermore, there are situations where the framing
members will add rotational restraint to the two-
dimensional model that is difficult to quantify, but
necessary to ensure analytical accuracy.

60
Support Conditions
The support conditions leading to “three-dimensional
behavior” in two-dimensional analytical models are very
important. Examples of these conditions are listed
below:
1. If a beam is cast monolithically with a wall and the
wall runs parallel to the beam member, the beam end
may be considered as fixed.

61
Support Conditions
2. If the wall is relatively thin, or perpendicular to the
monolithically-cast beam, the beam end may be
considered as pinned.

62
Support Conditions
3. When a masonry foundation wall, or other very
slender wall (flexurally) not cast monolithically with the
beam is used for support, it is likely too slender to
contribute to rotational restraint and therefore, zero
flexural restraint may be assumed.

63
Support Conditions
4. A similarly difficult situation arises with the
foundation at columns. The magnitude of support
restraint unfortunately is heavily dependent upon the
type of foundation, and the soil upon which it rests. A
fixed support does not exist unless the slope of the
column during loading maintains its angle (e.g. zero
slope before and after). This is likely to be approached
only in situations of pile caps or solid rock (pile cap
situation shown below).

64
Support Conditions

65
Support Conditions

66
Support Conditions

67
Support Conditions

68
Support Conditions
In the case of typical footings, the support condition is
likely less than “rigid”. A conservative approach would
be to assume the foundation as pinned during the frame
analysis and detail the foundation and column/footing
connection accordingly. The second condition would be
to assume that the base is fixed and design the footing
and column/footing connection for the moment that
arises from the fixed base condition.

69
Support Conditions
The third would be to estimate the stiffness of the soil
and footing and work these two estimates into a
rotational restraint condition at the base of the column
that can be utilized in a frame analysis.
The first two are very quick and the third requires a good
deal of engineering experience.

70
Support Conditions
5. A very difficult situation
arises when beam members
are cast monolithically with
supporting girders and then
girders are then cast
monolithically with
supporting columns. An
example of such a framing
system is shown

71
Support Conditions
If we were to determine the
bending moment diagram
for the typical beam above,
one certainly must consider
the torsional stiffness of the
longitudinal girder
contributing to the flexural
restraint at the discontinuous
end of the beam.

ACI 318 Moment Coefficient Method
72
The most useful method for carrying out continuous
beam analysis, in the absence of software or the
moment-distribution technique, is the ACI Coefficient
Method. When the limiting assumptions of the method
are satisfied, the technique can very quickly provide
good structural analysis solutions for continuous beam
and slab systems.
The method located in ACI 8.3.3 is founded upon the
two fundamental relationships shown below:
𝑀𝑢 = 𝐶𝑚𝑜𝑚𝑤𝑢𝑙𝑛
2
𝑉
𝑢 = 𝐶𝑠ℎ𝑒𝑎𝑟𝑤𝑢𝑙𝑛/2

73
where:
𝐶𝑚𝑜𝑚 is an ACI 318 moment coefficient that depends
upon end conditions and moment location;
𝐶𝑠ℎ𝑒𝑎𝑟 is an ACI 318 shear coefficient;
𝑤𝑢 is the factored uniformly distributed loading
considered; and
𝑙𝑛 is the clear span of the beam if positive moment is
considered and the average of adjacent clear spans if
negative moment is considered.

74

75
It is VERY IMPORTANT to realize that analysis using
the coefficient method must adhere to the following
restrictions/assumptions:
1. There are two or more spans.
2. The spans are approximately equal with the
difference in spans being no greater that 20% (of the
shorter span).
3. The loading is uniformly distributed.
4. The unit live loading does not exceed three times the
unit dead load.
5. The members are prismatic (cross-section constant
throughout spans).

76

77
DESIGN LIMITATIONS ACCORDING TO ACI CODE
The following limitations are specified by the ACI Code:
1.A typical imaginary strip 1 ft (or 1m) wide is assumed.
2. The minimum thickness of one-way slabs using grade 60 steel according to the ACI Code, for
solid slabs and for beams or ribbed one-way slabs should be equal to the following:
• For simply supported spans: solid slabs, h = L/20 (ribbed slabs, h = L/16).
• For one-end continuous spans: solid slabs, h = L/24 (ribbed slabs, h = L/18.5).
• For both-end continuous spans: solid slabs, h = L/28 (ribbed slabs, h = L/21).
• For cantilever spans: solid slabs, h = L/10 (ribbed slabs, h = L/8).
• For fy other than 60 ksi, these values shall be multiplied by (0.4 + 0.01fy), where fy is in ksi. This
minimum thickness should be used unless computation of deflection indicates a lesser thickness
can be used without adverse effects.
3. Deflection is to be checked when the slab supports are attached to construction likely to be
damaged by large deflections. Deflection limits are set by the ACI Code,

78
DESIGN LIMITATIONS ACCORDING TO ACI CODE
4. It is preferable to choose slab depth to the nearest 1/2in. (or 10 mm).
5. Shear should be checked, although it does not usually control.
6. Concrete cover in slabs shall not be less than 3/4in. (20 mm) at surfaces not exposed to
weather or ground. In this case, d = h − 3/4in. − (half-bar diameter).
7. In structural slabs of uniform thickness, the minimum amount of reinforcement in the direction of
the span shall not be less than that required for shrinkage and temperature reinforcement (ACI Code,
Section 7.12).
8. The principal reinforcement shall be spaced not farther apart than three times the slab thickness
nor more than 18 in. (ACI Code, Section 7.6.5).
9. Straight-bar systems may be used in both tops and bottoms of continuous slabs. An alternative
bar system of straight and bent (trussed) bars placed alternately may also be used.
10. In addition to main reinforcement, steel bars at right angles to the main must be provided.
This additional steel is called secondary, distribution, shrinkage, or temperature reinforcement.

79
To analyze a continuous beam system using the
method, one needs to first isolate the continuous
member within the framing system and determine the
factored loading to be used. The engineer must then
decide if the ends of the exterior spans are
unrestrained, or are cast monolithically with a spandrel
girder or column.
Once these have been determined, ACI 8.3.3 provides
the following shear and moment coefficients (ACI
318):

80

81

82
As have discussed in the past, our goal in analysis of
continuous (indeterminate) systems is to determine
positive and negative moment envelopes and transverse
shear envelopes that can be used for design. The use of
the ACI 318 coefficient method makes things a little
cumbersome, but the approach is certainly of great use
to designers in concrete.

84
Example 1: Example Demonstrating Use of ACI 318
Coefficients to Generate Moment and Shear Envelopes.
Consider a typical 4th-floor four-span continuous beam
with uniformly distributed dead loading equal to 1.5 klf
(including the beam self-weight) and uniformly
distributed live loading equal to 2.0 klf. Assume that all
spans are equal and the centreline to centreline
distances of 18-inch square columns is 25 feet. The
story height (above and below) the beam considered is
14 feet. The beam cross-section can be taken as 18-
inches wide by 30-inches high.

85
Use the ACI 318 coefficient approach to determine the
bending moment diagram envelope for the system.
Solution:
1. Create a sketch of the system considered.
2. Validate that the assumptions for use of the
coefficient method are met.
3. Compute the factored loading and clear spans
4. Determine Maximums in the Positive Moment
Regions
5. Determine Maximums in the Negative Moment
Regions

86

87

88

89

90

91

92
Example (Nadim Hassoun)
The slab–beam floor system shown in Fig. below carries a uniform
live load of 130 psf and a dead load that consists of the slab’s own
weight plus 80 psf. Using the ACI moment coefficients, design a
typical interior continuous beam and draw detailed sections. Use fc=
4ksi, fy= 60 ksi, beam width(b) = 12 in., 12 × 12-in. columns, and a
slab thickness of 5.0 in.

93

94

95

96

97

98
The slab–beam floor system shown in Fig. carries a uniformly distributed
dead load (excluding weight of slab and beam) of 40 psf and a live load of
100 psf. Using the ACI Code coefficients, design

Continuity in RC beams and frames-AFM (Complete).pdf

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