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CHAPTER II
CONTINUOUS BEAMS ANDONE-WAY RIBBED SLABS
ABRHAM E.
SOPHONYAS A.
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Introduction
Live load might vary in structures during service.
Live load variation has to be considered for design of:
Continuous beams,
one-way slabs &
continuous one-way ribbed slabs
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Pattern Loadings for Live Load
The largest moments in a continuous beam or frame
occur when some spans are loaded with variable loads(live loads) and others not.
Diagrams, referred to as influence lines, often are used
to determine which spans should and should not be
loaded.
An influence line is a graph of the variation in the
moment, shear, etc. at one particular point in a beam
due to a unit load that moves across the beam Fig (a) is an influence line for the moment at point C in
the two span beam shown in Fig (b).
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Pattern Loadings for Live Load
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The horizontal axis refers tothe position of a unit load
(1 kN) on the beam, and
The vertical ordinates are
the moment at C due to theunit load acting at the point
in question.
The derivation of the
ordinates at B, C, and E
called (influence ordinates)
is illustrated in Fig (c) to (e).
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Pattern Loadings for Live Load
If a concentrated load P kN acted at point E, the
moment at C would be P times the influence ordinateat E, or M = -0.9P kNm.
If a uniform load w acted on the span A-D, the moment
at C would be w times the area of the influence
diagram from A to D.
Fog (a) shows that a load placed anywhere b/n A and D
will cause positive moment at point C, whereas a load
placed anywhere b/n D and F will cause a negativemoment at C.
Thus, to get the maximum positive moment at C, we
must load span A-D only.27-Mar-12 5
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Pattern Loadings for Live Load
Two principal methods are used to calculate
influence lines.
In the 1st, unit load is placed successively at evenly
spaced points along the span, and the moment (or
shear) is calculated at the point for which theinfluence line is being drawn as shown above.
The 2nd procedure, known as Mueller Breslau
Principle is based on the principle of virtual work
The PVW states that the total work done during avirtual displacement of a structure is zero if the
structure is in equilibrium.
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Pattern Loadings for Live Load
The use of the M-BP to compute an influence line for
moment at C is illustrated in Fig (f).
Thus the deflected shape of the structure caused by
the unit virtual rotation, C =1, has the same shape &
is proportional to the influence line for moment at C. The M-BP is used as a qualitative guide to the shape
of influence lines to determine where to load a
structure to cause maximum moments or shears at
various points.
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Pattern Loadings for Live Load
27-Mar-12 8Fig. Influence lines for moments & loading patterns
Insert a fictitious hinge
at the section under
consideration,
Introduce a rotation
therein in a directioncorresponding to the
moment desired.
The resulting deflected
shape, due to a unitrotation, gives the
desired influence line.
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Pattern Loadings for Live Load
27-Mar-12 9Fig. Influence lines for shear
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Fig. Influence lines and gravity load patterns for a plane frame
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Observations:
Maximum ‘positive’ moment in a span occurs when live
loads are placed on that span & every other alternate
span
The maximum ‘negative’ moment at a support section
occurs when live loads are placed on the span (BC) inwhich the support section is located as well as the
adjoining span CD, and also on every alternate span
thereafter,
The influence of loads on spans far removed from thesections under consideration is relatively small.
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2 Moment Redistribution with Pattern Loadings
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2 Moment Redistribution with Pattern Loadings
Case (a): maximum positive moment in exterior spans
Case (b): maximum positive moment in interior span Case (c): maximum negative moment over the interior
support.
Assume that a 10% adjustment of maximum negative
and positive moments is permitted throughout.
An overall reduction in design moments through theentire three-span beam may be possible.
Case (a): Adjusting the maximum positive momentupward by 10%, one obtains a positive moment of 98kNm, which results in an upward adjustment of thesupport moment to 104 kNm.
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2 Moment Redistribution with Pattern Loadings
Case (b): By a similar redistribution of moments, areduced middle-span moment of 64 kNm is
accompanied by an increase in the support moment
from 78 to 86 kNm.
Case (c): First interior support moment for loadingcase (c) is decreased by 10% to 121 kNm.
To limit the increase in the controlling span moment
of the interior span, the right interior support
moment is not decreased.
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2 Moment Redistribution with Pattern Loadings
The positive moments in the left exterior span and inthe interior span corresponding to the modified
moment at the left interior support are 90 and 57kNm respectively.
Observations:
The reduction obtained for the span moments incases (a) and (b) was achieved at the expense ofincreasing the moment at the first support.
However the increased support moment in each case
was less than the moment for which that supportwould have to be designed based on the loading c,which produced the maximum moment.
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2 Moment Redistribution with Pattern Loadings
Similarly, the reduction in support moment in case(c) was taken at the expense of an increase in span
moments in the two adjacent spans.
However, in each case the increased span moments
were less than the maximum span momentsobtained for other loading conditions
The final design moments at all critical sections are
underlined.
The net result is a reduction in design moments over
the entire beam
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98
2 Moment Redistribution with Pattern Loadings
• Draw the envelope BMD
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9864 kNm
121 121
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Analysis and Design of One-way Ribbed Slabs
• Hollow block floors proved economic for spans of morethan 5 m with light or moderate live loads, such as
hospitals, offices or residential buildings.
• They are not suitable for structures having heavy liveloads such as warehouses or parking garages.
•
The joists span one way between beams.
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Fig Typical ribbed slab cross-section
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Arrangement of Ribs in Plan
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Arrangement of Ribs in Plan
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The designer has to make up his mind regarding
the option he prefers.Some designers opt to run the ribs in a direction
that leads to smaller moments and shears in thesupporting beams which means much morereinforcement in the ribs.
Other designers opt to run the ribs in the shorterdirection which leads to much more
reinforcement in the supporting beams.The later option leads to more economical
design.
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Arrangement of Ribs in Plan
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Advantages of Ribbed Slabs
The main advantage of using hollow blocks is the
reduction in weight by removing the part of the
concrete below the neutral axis.
Additional advantages are:
1. Ease of construction.
2. Hollow blocks make it possible to have smooth
ceiling which is often required for architectural
considerations.3. Provides good sound and temperature insulation
properties.
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General Requirements:
Thickness of slab (topping):
t ≥ max (40mm, 110
∗ ℎ )
Width of ribs shall not be less than 70 mm.
Depth of ribs , excluding any topping, shall not be more
than 4 times the minimum width of the rib. Rib spacing shall not exceed 1.0 m
Minimum mesh reinforcement area of 0.001 times
section of slab shall be provided for the topping. If the rib spacing exceeds 1.0 m, the topping shall be
designed as a slab resting on ribs considering load
concentrations, if any.27-Mar-12 25
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Transverse ribs shall be provided if the span of the
ribbed slab exceeds 6.0 m.
When transverse ribs are provided, the center-to-center distance shall not exceed 20 times the overall
depth of the ribbed slab.
The transverse ribs shall be designed for at least half
the values of maximum moments and shear force in
the longitudinal ribs.
The girder supporting the joist may be rectangular or
T-beam with the flange thickness equal to the floorthickness.
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Design Procedures
Thickness of toppings and ribs assumed based on
minimum requirement. Loads may be computed on the basis of centerline of
the spacing of joists.
The joists are analyzed as regular continuous or T -
beams supported by girders.
Shear reinforcement shall not be provided in the
narrow web of joist thus a check for the section
capacity against shear is carried out. The shear capacity may be approximated as 1.1 Vc of
regular rectangular sections.
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Determine flexural reinforcement and consider
minimum provision in the final solution.
Provide the topping or slab with reinforcement as
per temperature and shrinkage requirement.
Design the girder as a beam.
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• Design the
floor slabsystem.
• Design the
girders.
• Live load =
4kN/m2
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Exercise (Group Work)