8.2 ANALYSIS OF CONTINUOUS BEAMS AND FRAMES.
In reinforced concrete structures, as much of the concrete as is practical is placed in one
single operation. Reinforcing steel is not terminated at the ends of a member but is
extended through the joints into adjacent members. At construction joints, special care is
taken to bond the new concrete to the old by carefully cleaning the latter, by extending the
reinforcement through the joint, and by other means. As a result, reinforced concrete
structures usually represent monolithic, or continuous, units. A load applied at one location
causes deformation and stress at all other locations. Even in precast concrete construction,
which resembles steel construction in that individual members are brought to the job site
and joined in the field, connections are often designed to provide for the transfer of moment
as well as shear and axial load, producing at least partial continuity.
For structural analysis of continuous beam or rib to obtain the shear and moment diagrams,
it shall be permitted according to ACI code, 8.11.2, to assume that the arrangement of live
load is limited to combinations of:
a. Factored dead load on all spans with full factored live load on two adjacent
spans; and
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b. Factored dead load on all spans with full factored live load on alternate spans.
Span 1 Span2 Span 3
Load Case 1: ACI-8.11.2-a
LL
DL
Load Case 2: ACI-8.11.2-a
LL
DL
Load Case 3: ACI-8.11.2-b
LL LL
DL
Load Case 4: ACI-8.11.2-b
LL
DL
2
From each case we get the Maximum moment:
• Maximum negative moment from load cases 1+2 (ACI-8.11.2-a)
• Maximum positive moment from load cases 3+4 (ACI-8.11.2-b)
• Envelope moment diagram from all possible load cases.
Moment Diagram from
Load Case 1
Moment Diagram from
Load Case 2
Moment Diagram from
Load Case 3
Moment Diagram from
Load Case 4
3
Moment Diagrams of all Load cases
Envelope Moment Diagram from all Load cases
8.3 ANALYSIS AND DESIGN OF ONE-WAY SOLID SLABS. ACI CODE LIMITATIONS.
If the concrete slab is cast in one uniform thickness without any type of voids, it can be
referred to as a solid slab. In a one-way stab nearly all the loading is transferred in the short
direction, and the slab may be treated as a beam. A unit strip of slab, usually 1 m at right
angles to the supporting girders, is considered a rectangular beam. The beam has a unit
width with a depth equal to the thickness of the slab and a span length equal to the distance
between the supports. A one-way slab thus consists of a series of rectangular beams placed
side by side.
If the slab is one span only and rests freely on its supports, the maximum positive moment
for a uniformly distributed load of is , where is the span length
between the supports. If the same slab is built monolithically with the supporting beams or
is continuous over several supports, the positive and negative moments are calculated by
structural analysis or by moment coefficients as for continuous beams. The ACI Code,
Section 8.3, permits the use of moment and shear coefficients in the case of two or more
approximately equal spans.
The maximum positive and negative moments and shears are computed from the following
expressions:
4
( )
( )
where and are moment and shear coefficients given in table below and figure (page
193).
For all positive midspan moments, all shears and the negative moment at exterior
supports, , is for the span under consideration. For the negative moment at interior
supports, , shall be taken as ( ) as defined in the figure above.
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The conditions under which the moment coefficients for continuous beams and slabs should
be used can be summarized as follows:
1. Spans are approximately equal: Longer span (shorter span).
2. Loads are uniformly distributed.
3. The ratio (live load/dead load) is less than or equal to .
4. For slabs with spans less than or equal to , negative bending moment at
face of all supports is ( )
5. For an unrestrained discontinuous end, the coefficient is at end support and
( ) at midspan.
6. Shear force at C is and at the face of all other support is ( ) .
7. The members are prismatic.
When these conditions are not satisfied, structural analysis is required. In structural analysis,
the negative bending moments at the centers of the supports are calculated. The value that
may be considered in the design is the negative moment at the face of the support, ACI
8.9.2, 8.9.3.
The following limitations are specified by the ACI code:
Atypical imaginary strip 1m wide is assumed.
The minimum thickness of one-way slabs using grade 420 steel can be defined
according to the ACI Code, 9.5.2.1, Table 9.5a, for solid slabs and for beams or ribbed
one-way slabs .
ACI 9.5.2.1 – Minimum thickness stipulated in Table 9.5(a) shall apply for one-way
construction not supporting or attached to partitions or other construction likely to be
damaged by large deflections, unless computation of deflection indicates a lesser thickness
can be used without adverse effects.
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Solution:
Minimum thickness (deflection requirements).
( )
( )
Take slab thickness
Assume bar diameter for main reinforcement.
Loads calculation:
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Reinforced Concrete II Dr. Nasr Abboushi
Check whether thickness is adequate for shear:
× ×
√ √
- for shear.
×
The thickness of the slab is adequate enough.
Even, if for solid slabs, the thickness of the slab will be enough.
Factored moments at sections A, B, C, D, E:
For the negative moment at interior supports, , shall be taken as ( ).
Here ( )
( )
Location
C ( )
D ( )
Slab Design for the positive moments:
Midspan section B:
( √ ) ( √ )
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Use then
Take .
Step ( ) is the smallest of:
1. ×
2.
( ) ( )
( ) ( )
–
Midspan section E:
( √ ) ( √ )
Use then
Take .
–
Slab Design for the negative moments:
Note that the second support has two values of moments by analysis, at section C and
section D. In construction, the provided reinforcement will be the same bar diameters on
opposite sides of the support, so the design may be done for the maximum moment of the
two moments at C and D (Only one design for Support section C).
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Support section C:
Assume bar diameter for main reinforcement.
( √ ) ( √ )
Use then
Take .
–
Support section D (interior D supports):
Assume bar diameter for main reinforcement.
( √ ) ( √ )
Use then
Take .
–
Support section A:
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Assume bar diameter for main reinforcement.
( √ ) ( √ )
Provide
Use then
Take .
–
Temperature and shrinkage reinforcement.
( )
Take .
Step ( ) is the smallest of:
1. ×
2.
–
Required Provided
Location Reinforcement
A 3.3 ( )
B 3.3 ( )
C and first
3.3 ( )
interior D
Interior D 3.3 ( )
E 3.3 ( )
Temperature and shrinkage
reinforcement ( )
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