CH 2 Continous Beams & Ribbed Slabs

8/20/2019 CH 2 Continous Beams & Ribbed Slabs

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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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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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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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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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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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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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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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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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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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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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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


• 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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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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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)

CH 2 Continous Beams & Ribbed Slabs

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