RC II Robel

RCS II Plastic Moment Redistribution Chapter I

AAiT, Department of Civil Engineering . Page 1

CHAPTER I

Plastic Moment Redistribution

1.1. Introduction

It is known that an indeterminate beam or frame normally will not fail when the ultimate

moment capacity of just one critical section is reached. After formation of plastic hinges at the

more highly stressed sections, substantial redistribution of moments occurs at the critical

sections as loads are further increased before collapse of the structure takes place.

Redistribution of moments permits the designer to modify, within limits, the moment diagrams

for which the members are to be designed. This enables the designer to reduce the congestion

of reinforcement, which often occurs in high moment areas, such as at the junction of girders

with columns.

Method of analysis allowed in EBCS -2

i. Elastic, optionally followed by inelastic (plastic) moment redistribution

ii. Plastic analysis

iii. Non Linear analysis

1.2. Moment curvature relationship

Although it is not needed explicitly in ordinary design, the relation between moment applied to

a given beam section and the resulting curvature, through the full range of loading to failure, is

important to the study of member ductility, understanding the development of plastic hinges,

and accounting for the redistribution of elastic moments that occur in most RC structures

before collapse.

Curvature is defined as the angle change per unit length at any given location along the axis of a

member subjected to loads as seen in figure 1.2-1.

Figure 1.2-1 Curvature



From similarity of triangles,

Moment of Inertia of Transformed Section

When a beam made of two materials is loaded, the different values of E for the two materials

lead to different stress distribution since one materials is stiffer and accepts more stress for a

given strain than the other .

However, the elastic beam theory can be used if the beam is hypothetically transformed to

either an all steel beam or an all concrete beam, customarily the later. This is done by replacing

the area of the steel with an equivalent area of concrete having centroid at the level of the

centroid of the steel. The replaced concrete will experience the same force and strain as the

steel.



Figure 1.2-2 Transformed Sections

With the above transformed sections and the idealized stress-strain relationships for steel and

concrete figure 1.2-3 (b) and (c) the usual assumptions regarding perfect bond and plane

sections, it is possible to calculate the relation between M and for a typical under-reinforced

concrete beam section, subject to flexural cracking as follows.

Figure 1.2-3 Under-reinforced concrete beam section, subject to flexural cracking

In the limit case of figure 1.2-3b

Where, lut is the moment of inertia of the un-cracked transformed section.



Figure 1.2-4 Moment-Curvature relationship of reinforced beam

These values (cr , Mcr) provide information needed to plot point 1of the M- graph of figure 1.2-4.

When the tensile cracking occurs at the section, the stiffness is immediately reduced, and curvature increases to point 2 with no increase in moment. In the limit case, the concrete strain just reaches the proportional limit as shown in figure figure 1.2-3 (c) and the steel is below the yield strain.

(el , Mel) provides point 3 on the graph and the curvature at point 2 can be found from the ratio Mcr/Mel.

Once the proportional limit is exceeded, the concrete is well into the inelastic range, although the steel has not yet yielded. The NA depth, C1 is less than the depth a = Kd and is changing with increasing load as the shape of the concrete stress distribution and the steel stress changes.

It is now convenient to adopt a numerical solution to find the concrete compressive force 'C' and the location of its centroid for any arbitrarily selected value of maximum concrete strain c in the range el< c cu.

The compressive strain diagram is divided into an arbitrary number of steps and the corresponding stress for each strain read from the stress-strain curve concrete. The stepwise representation of the actual continuous stress block is integrated numerically to find C, and its point of application is located taking moments of the concrete forces about the top of the



section. The basic equilibrium requirement, C =T, can be used to find the correct location of the NA, for the particular compressive strain selected, following an iterative procedure.

Alternative to numerical integration, formulae for determining the total compressive force as

stated in EBCS 2-1995 can be used and are given below.

i. cm 2 and N.A. within the section

ii. cm 2 and N.A. within the section

iii. cm 2 and N.A. outside the section

Then the total compressive force will be,

1.3. Plastic hinges and collapse mechanisms

If a short segment of a reinforced concrete beam is subjected to a bending moment, continued

plastic rotation is assumed to occur after the calculated ultimate moment Mu is reached, with

no change in applied moment. The beam behaves as if there were a hinge at that point.

However, the hinge will not be friction free, but will have a constant resistance to rotation.

If such a plastic hinge forms in a determinate structure, as shown in figure 1.3-1, an

uncontrolled deflection takes place and the structure will collapse. The resulting system is

referred to as a mechanism. This implies that a statically determinate system requires the

formation of only one plastic hinge in order to become a mechanism.



Figure 1.3-1

In the case of indeterminate structures, stability may be maintained even though hinges have

formed at several cross sections. The formation of such hinges in indeterminate structures

permits a redistribution of moments within the beam or frame.

For illustration let us see the behavior of an indeterminate beam of figure 1.3-2. It will be

assumed for simplicity that the beam is symmetrically reinforced, so that the negative bending

capacity is the same as the positive. Let the load P be increased gradually until the elastic

moment at the fixed support, 3PL/16 is just equal to the plastic moment capacity of the section,

Mu. This load is

At this load the positive moment under the load is

PL, as shown in figure 1.3-2.

Figure 1.3-2

The beam still responds elastically everywhere but at the left support. At that point the actual

fixed support can be replaced for purpose of analysis with a plastic hinge offering a known

resisting moment Mu, which makes the beam statically determinate.



The load can be increased further until the moment under the load also becomes equal to Mu,

at which load the second hinge forms. The structure is converted into a mechanism, as shown

in figure 1.3-2 c, and collapse occurs. The moment diagram at collapse is shown in figure 1.3-2d.

The magnitude of the load causing collapse is easily calculated from the geometry of figure 1.3-

2d.

From which

By comparison of equation 1.2 and 1.1, it is evident that an increase of 12.5% is possible

beyond the load which caused the formation of the first plastic hinge, before the beam will

actually collapse. Due to the formation of plastic hinges, a redistribution of moments has

occurred such that, at failure, the ratio between positive moment and negative moment is

equal to that assumed in reinforcing the structure.

1.4. Rotation Requirement

It may be evident that there is a direct relation between the amount of redistribution desired

and the amount of inelastic rotation at the critical sections of a beam required to produce the

desired redistribution. In general, the greater the modification of the elastic-moment ratio, the

greater the required rotation capacity to accomplish that change. To illustrate, if the beam of

figure 1.2-2a had been reinforced according to the elastic-moment diagram of figure 1.2-2.b, no

inelastic-rotation capacity at all would be required. The beam would, at least in theory, yield

simultaneously at the left support and at mid-span. On the other hand, if the reinforcement at

the left support had been deliberately reduced (and the mid-span reinforcement

correspondingly increased), inelastic rotation at the support would be required before the

strength at mid-span could be realized.

Reinforced concrete members with bending are designed to have certain ductility, which

ensures that the member is capable of undergoing a certain amount of rotation after yielding of

the tension steel reinforcement and before crushing of the concrete in compression.

Generally, the amount of redistribution depends on

Hinge sections must be able to undergo necessary inelastic deformation. Since the

inelastic rotational capacity is a function of reinforcement ratio as in figure 1.4-1, this

implies an upper limit on the reinforcement,

Hinges should not occur at service load since wide cracks develop at hinge location, and



Equilibrium must be maintained.

Figure 1.4-1 Moment-curvature diagram

To ensure that designs remain under-reinforced (ductile), EBCS-2 recommends that the ratio

x/d, at sections of largest moment, does not exceed the values given by the following equations

as functions of percent plastic moment redistribution.

Where

For example, for 20% redistribution

In moment redistribution usually it is the maximum support moments, which are (adjusted)

reduced so that economizing in reinforcing steel and also reducing congestion of bars at the

column.

Requirements for applying moment - redistribution are:-



Equilibrium between internal and external forces must be maintained; hence it is

necessary to recalculate the span moments.

Maximum redistribution is 30% ... (

Redistribution kx * kz

*

0 1.0 0.450 0.295 0.814

10 0.9 0.368 0.252 0.840

20 0.8 0.288 0.205 0.880

30 0.7 0.208 0.143 0.914

Table 1.4-1 Moment Redistribution Design Factors

Design procedure using table No. 1a & 1b (with moment redistribution)

Steps

Calculate

a) If , where km* is the value of km shown shaded in general design table No. 1a,

corresponding to %age moment redistribution, section is singly reinforced.

Enter the general design table 1a using km and concrete grade.

Read ks from general design table No. 1a corresponding to steel grade.

Evaluate

b) If , the section has to be doubly reinforced.

Calculate kmkm*

Read ks and ks' corresponding to kmkm* and steel grade from table No. 1b and No.

1a respectively.

Assume d2=d" and read (correction factor) from table No. 1a using kmkm* and

d2/d.

Read ' corresponding to d2/d and %age moment redistribution from table No. 1b.

Calculate



Design procedure using general design chart (with moment redistribution)

Calculate

a) If , section is singly reinforced.

Evaluate Z from chart using

Evaluate

b) If , section is doubly reinforced.

Evaluate Z* from chart using

Evaluate

Calculate

RCS II Continuous Beams, One-Way Solid And Ribbed Slabs Chapter II


CHAPTER II

CONTINUOUS BEAMS AND ONE-WAY RIBBED SLABS

2.1. Introduction

Continuous beams, one-way slabs and continuous one-way ribbed slabs are indeterminate

structures for which live load variation has to be considered. This is because dead load is always

there but live load might vary during the life time of these structures.

One-way slabs transmit their load mainly in one direction (i.e., the direction. of span). A 1m

strip is taken in the direction of span and treated similar to continuous beams.

Elastic analysis such as slope-deflection, moment distribution and matrix method or plastic

analysis or approximate method such as the use of moment coefficient or such methods as

portal or cantilever can be used.

2.2. Analysis and design of continuous beams

The three major stages in the design of a continuous beam are design for flexure, design for

shear, and design of longitudinal reinforcement details. In addition, it is necessary to consider

deflections and crack control and, in some cases, torsion. When the area supported by a beam

exceeds 37m2, it is usually possible to use a reduced live load in calculating the moments and

shears in the beam.

Figure 2.2-1 One-way slab and continuous beam

The largest moment in continuous beams or one-way slabs or frames occur when some spans

are loaded and the others are not. Influence lines are used to determine which spans should be

loaded and which spans should not be to find the maximum load effect.

Figure 2.2-2a shows influence line for moment at B. The loading pattern that will give the

largest positive moment at consists of load on all spans having positive influence ordinates.



Such loading is shown in figure 2.2-2b and is called alternate span loading or checkerboard

loading.

The maximum negative moment at C results from loading all spans having negative influence

ordinate as shown in figure 2.2-2d and is referred as an adjacent span loading.

Figure 2.2-2 Influence line for moment and loading patterns

Similarly, loading for maximum shear may be obtained by loading spans with positive shear

influence ordinate and are shown in figure 2.2-3.



Figure 2.2-3 Influence line for shear

In ACI code, it is required to design continues beam and one-way slabs to be design for the

following loading patterns,

1. Factored load on all spans with factored partition load and factored live load on two

adjacent spans and no live load on any other span. This will give the maximum negative

moment and maximum shear at the support between the two loaded spans. This

loading case is repeated for each interior support.

2. Factored dead load on all spans with factored partition load and a factored live load on

alternate spans. This will give maximum positive moment at the middle of the loaded

span, minimum positive moments (which could even be negative) at the middle of the

unloaded spans, and maximum negative moment at the exterior support.

3. Factored dead and live load on all spans. Although this condition represents the

maximum vertical loading possible, it is unlikely to cause the maximum reaction, shear

forces, or bending moment for continuous beams.

After obtaining the maximum load effects of continuous beams, the design of continuous

beams is carried out as discussed in reinforced concrete structures I course for no moment

redistribution case and as in chapter I of this course encase of moment redistribution. For

convenience, design steps of no redistribution by using the general design table (km - ks table) is

recalled below. Note that charts can also be used for design given in EBCS 2-1995 Part 2.



Steps for design using design table (no moment redistribution)

1. Evaluate Km

2. Enter the general design table No 1.a using Km and concrete grade.

a. If Km Km*, the value of Km show shaded in design Table No 1.a, then the section is

singly reinforced.

- Enter the design table No 1.a using Km and concrete grade

- Read Ks from the table corresponding to the steel grade and Km

- Evaluate As

b. If Km Km*, then the section should be doubly reinforced.

- Evaluate Km / Km * and d/d

- Read Ks, Ks, and from the same table corresponding to Km / Km *, d/d and

concrete grade

- Evaluate

Design using general design chart

1. Calculate

2. Enter the general design chart,

If , section is singly reinforced.

Evaluate Z from by reading value of from chart using

Evaluate

If , section is doubly reinforced.

Evaluate Z from chart using

Evaluate

Calculate



2.3. Analysis and design of one-way slabs

Slabs are flat plates used to provide useful horizontal surfaces mainly for roofs and floors of

buildings, parking lots, airfields, roadway etc.

Classification: Beam supported slabs may be classified as:

1. One-way slabs - main reinforcement in each element runs in one direction only. (Ly/Lx

>2). There are two types one way solid slabs and one way ribbed slabs.

2. Two-way slabs - main reinforcement runs in both directions where ratio of long to short

span is less than two. (Ly/Lx 2)

Others types of slab include flat slab, flat plates, two way ribbed or grid slabs etc.

One-ways slabs are considered as rectangular beams of comparatively large ratio of width to

depth and ratio of longer span to width (short span) is greater than two.

When Ly/Lx > 2, about 90% or more of the total load is carried by the short span, i.e., bending

takes place in the direction of the shorter span.

Figure 2.3-1 One-way slab



Analysis and design is than carried out by assuming a beam of unit width with a depth equal to

the thickness of the slab, continuous over the supporting beam and span equal to the distance

between supports (in the short direction or strip A and B) as shown in figure 2.3-2. The strip

may be analyzed in the same way as singly reinforced rectangular sections. Near the ends the

panel adjacent to the girders, some load is resisted by bending in the longitudinal strips (strips

C) and less by the transvers strips (strip A). But for design purpose the effect is ignored and is

indirectly accounted by extending top reinforcements into the top of the slabs on each side of

the girders across the ends of the panel.

Figure 2.3-2 One-way and two-way slab action

The load per unit area on the slab would be the load per unit length on this imaginary beam of

unit width. As the loads being transmitted to the supporting beams, all reinforcement shall be

placed at right angles to these beams. However some additional bars may be placed in the

other direction to carry temperature and shrinkage stresses.

Generally the design consists of selecting a slab thickness for deflection requirements and

flexural design is carried out by considering the slab as series of rectangular beams side by side.

Remarks:

The following minimum thicknesses shall be adopted in design:

a) 60 mm for slabs not exposed to concentrated loads (e.g inaccessible roofs)

b) 80 mm for slabs exposed mainly to distributed loads.

c) 100 mm for slabs exposed to light moving concentrated loads (e.g slabs accessible to

light motor vehicles)

d) 120 mm for slabs exposed to heavy dynamic moving loads (eg. slabs accessible to heavy

vehicles)

Unless conditions warrant some change, cover to reinforcement is 15 mm.

The ratio of the secondary reinforcement to the main reinforcement shall be at least

equal to 0.2.

The geometrical ratio of main reinforcement in a slab shall not be less than



The spacing between main bars for slabs shall not exceed the smaller of 2h or 350 mm.

The spacing between secondary bars shall not exceed 400 mm.

2.4. Analysis and design of one-way ribbed slabs

Long-span floors for relatively light live loads can be constructed as a series of closely spaced,

cast-in-place T-beams (or joists or ribs) with a cross section as shown in figure 2.4-1. The joists

span one way between beams. Most often, removable metal forms referred to as fillers or pans

are used to form the joists. Occasionally, joist floors are built by using clay-tile fillers, which

serve as forms for the concrete in the ribs that are left in place to serve as the celling.

Figure 2.4-1 Typical ribbed slab cross-section

General Requirements:

Because joists are closely spaced, thickness of slab (topping),

{

Ribs shall not be less than 70 mm in width

Ribs shall have a depth, excluding any topping, of not more than 4 times the minimum

width of the rib.

Rib spacing shall not exceed 1.0 m

The topping shall be provided with a reinforcement mesh providing in each direction a

cross sectional area not less than 0.001 of the section of the slab.

If the rib spacing exceeds 1.0 m, the topping shall be designed as a slab resting on ribs

considering load concentrations, if any.

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 floor thickness.

Procedure for design of ribbed slabs

1. Thickness of toppings and ribs assumed based on minimum requirement.

2. Loads may be computed on the basis of centerline of the spacing of joists.

3. The joists are analyzed as regular continuous or T -beams supported by girders.

4. 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.

5. Determine flexural reinforcement and consider minimum provision in the final solution.

6. Provide the topping or slab with reinforcement as per temp and shrinkage requirement.

7. Design the girder as a beam.

RCS II Two-way Slabs Chapter III

AAiT, Department of Civil Engineering .

Page 19

CHAPTER III

TWO-WAY SLABS

3.1. Introduction

Slabs with the ratio of the longer to the shorter span, between 1 & 2 transfer their load in

two orthogonal directions. i.e. some portion of the load in the short direction and the

remaining portion of the load in the long direction. These slabs are called two-way slabs and

they deflect into a dish shaped curvature. This means that they have curvature in both

directions and because moments are proportional to curvature, there are moments in both

directions, which require reinforcement in the tension zone.

3.2. Analysis and design of two way beam supported slabs

For the slab shown in figure 3.1.-1, if beams are incorporated within the depth of the slab

itself the slab carries load in two directions. The load at A may be thought of as being carried

from A to B and C by one strip of slab, and from B to D and E, and so on, by other slab strips.

Because the slab must transmit loads in two directions it is referred to as two way slab.

Figure 3.2-1 Two-way slab

Consider the simply supported panel under uniform load w.

Figure 3.2-2

Let wx and wy be load in the x and y direction in which,



Page 20

Where kx and ky are load distributing factors in the short and long directions respectively.

Because the imaginary strips actually are part of the same slab, their deflections at the

intersection point must be the same. Equating the center deflections of the strips in the

short (x) and long (y) directions gives

Thus

Analysis using Table coefficients [EBCS 2-1995]

The coefficients kx & ky as obtained using the previous discussion are approximate because

the actual behavior of a slab is more complex than the two intersecting strips. The outer

strips not only bend, but also twist. The twisting results in torsional moments and stress

pronounced near the corners.

Moments for individual panels with edge simply supported or fully fixed may be computed

from:-

Where: Mi : is the design initial moment per unit width at the point of reference.

i : coefficient given in Table A-1 (EBCS 2-1995) as a function of Ly/Lx ratio and

support condition

pd: design uniform load

Lx, Ly : shorter and longer span of the panel respectively



Page 21

Figure 3.2-3 Notations of critical moments

The subscripts have the following meaning.

s Support

f - field (span)

y, x - directions in the long & short span, respectively.

Division of slabs into middle and edge strips is illustrated in Fig. A-4.

The maximum design moments calculated as above apply only to the middle strips and no

redistribution shall be made.

Reinforcement in an edge strip, parallel to the edge, need not be less than minimum areas

of tension reinforcement.

Figure 3.2-4 Division of slab into middle strip and edge strip

Moment adjustment

For each support over which the slab is continuous, there will be two adjacent support

moments. The difference may be distributed between the panels at either side of support to

equalize their moments as in moment distribution method for frames.



Page 22

There are two alternatives: -

a. When Ms 0.2 Ms,large

The average of initial moments may be used.

b. When Ms 0.2 Ms,large

Apply moment distribution only to adjacent spans.

Steps to be followed

1. Support and span moments are first evaluated for individual panels using

coefficients from Table A-1 .

2. The unbalanced moment is distributed using the moment distribution method.

3. When the support moment is decreased, the span moments Mxf and Myf are then

increased to allow for the changes of support moments (equilibrium). This increase

is computed as: -

Where cx and cy are coefficient from Table A-2 (EBCS-2)

Flexural reinforcement

The ratio of the secondary reinforcement to the main reinforcement shall be at least

equal to 0.2.

The geometrical ratio of main reinforcement in a slab shall not be less than

The spacing between main bars shall not exceed the smaller of 2h or 350 mm

The spacing between secondary bars shall not exceed 400 mm.

Load on beams

The design uniform loads on beams supporting solid slabs may be computed using: -

Where and are load transfer coefficient given in Table A-3 (EBCS-2)

The shear force carried by concrete in slab can be taken as the one given for beams.



Page 23

3.3. Analysis and Design of Flat Slabs

3.3.1. Introduction

Concrete two-way slabs may in some cases be supported by relatively shallow, flexible

beams, or directly by columns without the use of beams or girders. Such slabs are generally

referred as column supported two-way slabs. Beams may also be used where the slab is

interrupted as around stair, walls or at discontinuous edges.

In practice column supported two-way slabs take various forms:

Flat Plates: they are flat slabs with flat soffit. Such slabs have uniform thickness supported

on columns. They are used for relatively light loads, as experienced in apartments or similar

buildings. Flat plats are most economical for spans from 4.5m to 6m (see Fig. 3.3.1-1a).

Flat Slabs: they are slab systems with the load transfer to the column is accomplished by

thickening the slab near the column, using drop panels and/or by flaring the top of the

column to form a column capital. They may be used for heavy industrial loads and for spans

of 6m to 9m (see Fig. 3.3.1-1c)

Waffle Slabs: they are two-way joist systems with reduced self-weights. They are used for

spans from 7.5m to 12m. (Note: for large spans, the thickness required to transmit the

vertical loads to the columns exceeds that required for bending. As a result the concrete at

the middle of the panel is not efficiently used. To lighten the slab, reduce the slab moments,

and save material, the slab at mid span can be replaced by intersecting ribs. Near the

columns the full depth is retained to transmit loads from the slab to the columns (see Fig.

3.3.1-1b)

In this chapter, consideration will be given to flat slabs with or without drop panels or

column capitals.



Page 24

Figure 3.3.1-1 Types of two way column supported slabs

For analysis and design purpose the panel in flat slab is divided in to column strips and

middle strips as shown below. (EBSC 2)

Figure 3.3.1-2 Division of panels in Flat slabs



Page 25

A column strip is a design strip with a width on each side of a column centerline equal to

0.25 Lx or if drops with dimension not less than Lx/3 are used, a width equal to the drop

dimension. A middle strip is a design strip bounded by two column strips.

The drop panels are rectangular (may be square) and influence the distribution of moments

in the slab. The smaller dimension of the drop is at least one third of the smaller dimension

of the surrounding panels, Lx/3 and the drop may be 25 to 50 percent thicker than the rest

of the slab.

3.3.2. Load Transfer in Flat Slabs

Consider the following column supported two way slabs. If a surface load w is applied (see

Fig. 3.3.2-1a), it is shared between imaginary slab strips la in the short direction and lb in the

longer direction. Note that the portion of the load that is carried by the long strips lb is

delivered to the beams B1 which in turn carried in the short direction plus that directly

carried in the short direction by the slab strips la, sums up to 100 percent of the load applied

to the panel. The same is true in the other direction.

A similar situation is obtained in the flat plate floor (see Fig. 3.3.2-1b) where broad strips of

the slab centered on the column lines in each direction serve the same function as the

beams. Therefore; for column supported construction, 100 percent of the applied load must

be carried in each direction, jointly by the slab and its supporting beams.

Figure 3.3.2-1 Column Supported two-way slabs (a) with beams (b) without beams



Page 26

3.3.3. Moments in Flat Slab Floors

Consider the flat slab floor supported by columns at A, B, C, and D as shown in Fig. 3.3.3-1a.

Figure 3.3.3-1 Moment Variation (a) critical-moment section (b) moment variation along a span (c) moment variation along the width of critical section

Longitudinal Distributions of Moments

For the determination of moment in the direction of span l1, the slab may be considered as a

broad, flat beam of width l2.

The load, P2 = wl2 per m length of span.

From the requirement of statics:

In the longitudinal direction (see fig. 3.3.3-1b)

In the perpendicular direction

From the above static moment in each direction, the moment in the long direction is larger

than those in the short direction unlike the situation for the slab with stiff edge beams.



Page 27

Lateral Distributions of Moments

The moments across the width of critical sections such as AB or EF are not constant as

shown qualitatively (see fig. 3.3.3-1c). For design purpose, moments may be considered

constant within the bounds of a middle strip or column strip, unless beams are present in

column lines.

3.3.4. Practical Analysis of Flat Slab Floors

The two methods for the analysis of flat slabs are:

a) Direct Design Method

b) Equivalent Frame Method

Generally, for both methods of analysis, the negative moments greater than those at a

distance hc/2 from the center-line of the column may be ignored provided the moment Mo

obtained as the sum of the maximum positive design moment and the average of the

negative design moments in anyone span of the slab for the whole panel width is such that:

(

)

Where L1 is the panel length parallel to span, measured from centers of columns.

L2 is the panel width, measured from centers of columns

hc is the effective diameter of a column or column head (see below)

When the above condition is not satisfied, the negative design moments shall be increased.

The effective diameter of a column or column head hc is the diameter of a circle whose area

equals the cross-sectional area of the column or, if column heads are used, the area of the

column head based on the effective dimensions as defined below. In no case shall hc be

taken as greater than one-quarter of the shortest span framing in to the column.

The effective dimensions of a column head for use in calculation of hc are limited according

to the depth of the head. In any direction, the effective dimension of a head Lh shall be

taken as the lesser of the actual dimension Lho or Lh,max, where Lh,max is given by:

For a flared head, the actual dimension Lho is that measured to the center of the reinforcing

steel (see Fig. 3.3.4-1)



Page 28

Figure 3.3.4-1 Types of Column Head

3.3.5. Direct Design Method as per EBCS 2, 1995

According to the EBCS 2 specification, the direct design method of analysis is subjected to

the following restrictions.

Design is based on the single load case of all spans loaded with the maximum design

ultimate load.

There are at least three rows of panels of approximately equal span in the direction

being considered.

Successive span length in each direction shall not differ by more than one-third of

the longer span.

Maximum offsets of columns from either axis between center lines of successive

columns shall not exceed 10% of the span (in the direction of the offset).

Longitudinal Distribution

The distribution of design span and support moments depends on the relative stiffness of

the different sections which in turn depends on the restraint provided for the slab by the

supports. Accordingly, the distribution factors are given in the following table.

Outer support Near center of first span

First interior support

Center of interior

span

Interior support

Column Wall

Moment -0.040FL -0.020FL 0.083FL -0.063FL 0.071FL -0.055FL

Shear 0.45F 0.40F - 0.60F - 0.50F

Total Column moments

0.040FL - - 0.022FL - 0.022FL

Table 3.3.5-1 Bending Moment and Shear Force Coefficients for Flat slabs of Three or More Equal Spans.



Page 29

NOTE:

F is the total design ultimate load on the strip of slab between adjacent columns

considered.

L is the effective span = L1-2hc/3

The limitations of Section A.4.3.1(2) of EBCS 2, need not be checked

The moments shall not be redistributed

Lateral Distribution

The design moment obtained from the above (or equivalent frame analysis) shall be divided

b/n the column and middle strips according to the following table.

Apportionment been column and middle strip expressed as percentages of the total negative or positive design moment

Column Strip (%) Middle Strip (%)

Negative 75 25

Positive 55 45

Table 3.3.5-2 Distribution of Design Moments in Panels of Flat Slabs

NOTE: For the case where the width of the column strip is taken as equal to that of the drop

and the middle strip is thereby increased in width, the design moments to be resisted by the

middle strip shall be increased in proportion to its increased width. The design moments to

be resisted by the column strip may be decreased by an amount such that the total positive

and the total negative design moments resisted by the column strip and middle strip

together are unchanged.

3.3.6. Equivalent Frame Method

The direct design method is applicable when the proposed structures satisfy the restrictions

on geometry and loading. If the structure does not satisfy the criteria, the more general

method of elastic analysis is the equivalent frame method.

In the equivalent frame method, the structure is divided in to continuous frames centered

on the column lines on either side of the columns, extending both longitudinally and

transversely. Each frame is composed of a broad continuous beam and a row of columns.



Page 30

Figure 3.3.6-1 Building idealization for equivalent frame analysis

Equivalent Frame Method as per EBCS 2, 1995

According to the EBCS 2 specification, Equivalent Frame Method of analysis is treated as

follows:

(1) The width of slab used to define the effective stiffness of the slab will depend upon

the aspect ratio of the panels and the type of loading, but the following provisions

may be applied in the absence of more accurate methods:

In the case of vertical loading, the full width of the Panel, and

For lateral loading, half the width of the panel may be used to calculate the

stiffness of the slab.

(2) The moment of inertia of any section of slab or column used in calculating the

relative stiffness of members may be assumed to be that of the cross section of the

concrete alone.

(3) Moments and forces within a system of flat slab panels may be obtained from

analysis of the structure under the single load case of maximum design load on all

spans or panels simultaneously, provided:

The ratio of the characteristic imposed load to the characteristic dead load does

not exceed 1.25.

The characteristic imposed load does not exceed 5.0 kN/m2 excluding partitions.

(4) Where it is not appropriate to analyze for the single load case of maximum design

load on all spans, it will be sufficient to consider following arrangement of vertical

loads:



Page 31

All spans loaded with the maximum design ultimate load, and

Alternate spans with the maximum design ultimate load and all other spans

loaded with the minimum design ultimate load (1.0Gk).

(5) Each frame may be analyzed in its entirety by any elastic method. Alternatively, for

vertical loads only, each strip of floor and roof may be analyzed as a separate frame

with the columns above and below fixed in position and direction at their

extremities. In either case, the analysis shall be carried out for the "appropriate

design ultimate loads on each span calculated for a strip of slab of width equal to the

distance between center lines of the panels on each side of the columns.

Equivalent Frame Method as per ACI Code

According to the ACI Code specification, the Equivalent Frame method was developed with

the assumption that the analysis would be done using the moment distribution method.

A. Basis of Analysis

The equivalent Frame method was developed with the assumption that the analysis would

be done using the moment distribution method. For vertical loading, each floor with its

columns may be analyzed separately by assuming the columns to be fixed at the floors

above and below.

B. Moment of Inertia of Slab Beam

The slab beam includes the portion of then slab bounded by panel centerlines on each side

of the columns, together with column line beams or drop panels (if used).

The moment of inertia used for analysis may be based on the concrete cross-section,

neglecting reinforcement, but variations in cross section along the member axis should be

accounted for (see below).



Page 32

Figure 3.3.6-2 EI values for slab with drop



Page 33

Figure 3.3.6-3 EI values for slab and beam

C. The equivalent Column

In the equivalent frame method of analysis, the columns are considered to be attached to

the continuous slab beam by torsional members transverse to the direction of the span for

which moments are being found. Torsional deformation of these transverse supporting

members reduces the effective flexural stiffness provided by the actual column at the

support.



Page 34

Figure 3.3.6-4 Frame action and twisting of edge member

The above effects can be considered by replacing the actual beam and columns with an

equivalent column having the following stiffness:

Where: Kec = Flexural stiffness of equivalent column



Page 35

Kc = flexural stiffness of actual column

Kt = torsional stiffness of edge beam

The torsional Stiffness Kt can be calculated by:

(

)

Where: Ecs = modulus of elasticity of slab concrete

c2 = size of rectangular column, capital, or bracket in the direction of l2.

C = cross sectional constant (roughly equivalent to polar moment of inertia)

The torsional constant C can be calculated by:

(

)

Where: x is the shorter side of a rectangle and y is the longer side.

C is calculated by sub-dividing the cross section of torsional members in to component

rectangles and the sub-division is to maximize the value of C.

The torsional members according to ACI Code are as follows:



Page 36

Figure 3.3.6-5 Torsional members

D. Arrangement of Live Load for Analysis

(1) If the un-factored live load does not exceed 0.75 times the un-factored dead load, it

is not necessary to consider pattern loadings, and only the case of full factored live

load and dead load on all spans need to be analyzed.

(2) If the un-factored live load exceeds 0.75 times the un-factored dead load the

following pattern loadings need to be considered.

a. For maximum positive moment, factored dead load on all spans and 0.75

times the full factored live load on the panel in question and on alternate

panels.

b. For maximum negative moment at an interior support, factored dead load on

all panels and 0.75 times the full factored live load on the two adjacent

panels.

The final design moments shall not be less than for the case of full factored dead and live

load on all panels.



Page 37

3.3.7. Shear in Flat Slabs, as per EBCS 2

The concrete section (thickness of the slab) must be adequate to sustain the shear force,

since stirrups are not convenient.

Two types of shear are considered

a) Beam type Shear: Diagonal tension Failure and critical section is considered at d

distance from the face of the column or capital and Vc is the same expression given

earlier for beams or solid slabs.

b) Punching Shear: perimeter shear which occurs in slabs without beams around

columns. It is characterized by formation of a truncated punching cone or pyramid

around concentrated loads or reactions. The outline of the critical section is shown in

Fig. below.

Figure 3.3.7-1 Critical section remote from a free edge

The shear force to be resisted can be calculated as the total design load on the area

bounded by the panel centerlines around the column less the load applied with in the area

defined by the critical shear perimeter.

The punching shear resistance without shear reinforcement is:

( )



Page 38

CHAPTER IV

COLUMNS

4.1. Introduction

A column is a vertical structural member transmitting axial compression loads with or

without moments. The cross sectional dimensions of a column are generally considerably

less than its height. Column support mainly vertical loads from the floors and roof and

transmit these loads to the foundation

In construction, the reinforcement and concrete for the beam and slabs in a floor are place

once the concrete has hardened; the reinforcement and concrete for the columns over that

floor are placed followed by the next higher floor.

Columns may be classified based on the following criteria:

a) Classification on the basis of geometry; rectangular, square, circular, L-shaped, T-

shaped, etc. depending on the structural or architectural requirements.

b) Classification on the basis of composition; composite columns, in-filled columns, etc.

c) Classification on the basis of lateral reinforcement; tied columns, spiral columns.

d) Classification on the basis of manner by which lateral stability is provided to the

structure as a whole; braced columns, un-braced columns.

e) Classification on the basis of sensitivity to second order effect due to lateral

displacements; sway columns, non-sway columns.

f) Classification on the basis of degree of slenderness; short column, slender column.

g) Classification on the basis of loading: axially loaded column, columns under uni-axial

bending, columns under biaxial bending.

Composite/In-filled Columns

a) Composite Columns: Columns in which steel structural members are encased in a

concrete. Main reinforcement bars positioned with ties or spirals are placed around the

structural member.

b) In-filled Columns: Columns having steel pipes filled with plain concrete or lightly

reinforced concrete.

Figure 4.1-1 Composite Columns and in-filled columns



Page 39

Tied/Spiral Columns

a) Tied Columns: Columns where main (longitudinal) reinforcements are held in position by

separate ties spaced at equal intervals along the length. Tied columns may be, square,

rectangular, L-shaped, circular or any other required shape. And over 95% of all columns

in buildings in non-seismic regions are tied columns.

Figure 4.1-2 Tied Columns

b) Spiral Columns: Columns which are usually circular in cross section and longitudinal bars

are wrapped by a closely spaced spiral.

Figure 4.1-3 Spiral Columns

Behavior of Tied and Spiral columns

The load deflection diagrams (see Fig. 4.1-4) show the behavior of tied and spiral columns

subjected to axial load.

Figure 4.1-4 Load deflection behavior of tied and spiral columns



Page 40

The initial parts of these diagrams are similar. As the maximum load is reached vertical

cracks and crushing develops in the concrete shell outside the ties or spirals, and this

concrete spalls off. When this happens in a tied column, the capacity of the core that

remains is less than the load and the concrete core crushes and the reinforcement buckles

outward between the ties. This occurs suddenly, without warning, in a brittle manner.

When the shell spalls off in spiral columns, the column doesnt fail immediately because the

strength of the core has been enhanced by the tri axial stress resulting from the

confinement of the core by the spiral reinforcement. As a result the column can undergo

large deformations before collapses (yielding of spirals). Such failure is more ductile and

gives warning to the impending failure.

Accordingly, ductility in columns can be ensured by providing spirals or closely spaced ties.

4.2. Classification of Compression Members

4.2.1. Braced/Un-braced Columns a) Un-braced columns

An un-braced structure is one in which frames action is used to resist horizontal loads. In

such a structure, the horizontal loads are transmitted to the foundations through bending

action in the beams and columns. The moments in the columns due to this bending can

substantially reduce their axial (vertical) load carrying capacity. Un-braced structures are

generally quit flexible and allow horizontal displacement (see Fig. 4.2.1-1). When this

displacement is sufficiently large to influence significantly the column moments, the

structure is termed a sway frame.

Figure 4.2.1-1 Sway Frame/ Un-braced columns

b) Braced columns:

Although, fully non sway structures are difficult to achieve in practice, EBCS-2 or EC-2 allows

a structure to be classified as non-sway if it is braced against lateral loads using substantial



Page 41

bracing members such as shear walls, elevators, stairwell shafts, diagonal bracings or a

combination of these (See Fig. 4.2.1-2). A column with in such a non-sway structure is

considered to be braced and the second order moment on such column, P-, is negligible.

This may be assumed to be the case if the frame attracts not more than 10% of the

horizontal loads.

Figure 4.2.1-2 Non-sway Frame / Braced columns

4.2.2. Short/Slender Columns a) Short columns

They are columns with low slenderness ratio and their strengths are governed by the

strength of the materials and the geometry of the cross section.

b) Slender columns

They are columns with high slenderness ratio and their strength may be significantly

reduced by lateral deflection.

When an unbalanced moment or as moment due to eccentric loading is applied to a

column, the member responds by bending as shown in Fig. below. If the deflection at the

center of the member is, , then at the center there is a force P and a total moment of M +

P. The second order bending component, P, is due to the extra eccentricity of the axial

load which results from the deflection. If the column is short is small and this second order

moment is negligible. If on the other hand, the column is long and slender, is large and P

must be calculated and added to the applied moment M.



Page 42

Figure 4.2.2-1 Forces in slender column

4.3. Classification of Columns on the Basis of Loading

4.3.1. Axially loaded columns They are columns subjected to axial or concentric load without moments. They occur rarely.

When concentric axial load acts on a short column, its ultimate capacity may be obtained,

recognizing the nonlinear response of both materials, from:

Where Ag is gross concrete area Ast is total reinforcement area

When concentric axial load acts on a long column (

, its ultimate capacity may be

obtained from:

4.3.2. Column under uni-axial bending

Almost all compression members in concrete structures are subjected to moments in

addition to axial loads. These may be due to the load not being centered on the column or

may result from the column resisting a portion of the unbalanced moments at the end of

the beams supported by columns.



Page 43

Figure 4.3.2-1 Equivalent eccentricity of column load

When a member is subjected to combined axial compression Pd and moment Md, it is more

convenient to replace the axial load and the moment with an equivalent Pd applied at

eccentricity ed as shown below.

Interaction diagram

The presence of bending in axially loaded members can reduce the axial load capacity of the

member

To illustrate conceptually the interaction between moment and axial load in a column, an

idealized homogenous and elastic column with a compressive strength, fcu, equal to its

tensile strength, ftu, will be considered. For such a column failure would occurs in a

compression when the maximum stresses reached fcu as given by:

Dividing both sides by fcu gives:



Page 44

Figure 4.3.2-2 Interaction Chart for an elastic column

The maximum axial load the column could support is obtained when M = 0, and is Pmax =

fcuA.

Similarly the maximum moment that can be supported occurs when P=0 and is Mmax = fcuI/C.

Substituting Pmax and Mmax gives:

This is known as interaction equations because it shows the interaction of or relationship

between P and M at failure. It is plotted as line AB (see Fig.). A similar equation for a tensile

load, P, governed by ftu, gives line BC in the figure. The plot is referred to as an interaction

diagram.

Points on the lines represent combination of P and M corresponding to the resistance of the

section. A point inside the diagram such as E represents a combination of P and M that will

not cause failure. Load combinations falling on the line or outside the line, such as point F

will equal or exceed the resistance of the section and hence will cause failure.

Interaction Diagrams for Reinforced concrete Columns



Page 45

Since reinforced concrete is not elastic and has a tensile strength that is lower than its

compressive strength, the general shape of the diagram resembles Fig. 4.3.2-3

Figure 4.3.2-3 Interaction diagram for column in combined bending and axial load

Balanced condition: For a given cross section the design axial force Pb acts at one specific

eccentricity eb to cause failure by simultaneous yielding of tension steel and crushing of

concrete (see Fig. 4.3.2- 3)

Tension failure controls: For a very large eccentricity of the axial force Pn, the failure is

triggered by yielding of the tension steel. The horizontal axis corresponds to an infinite

value of e, i.e. pure bending at moment capacity Mo (see Fig. 4.3.2-3)

Compression failure controls: For a very small eccentricity of the axial force Pn, the failure is

governed by concrete compression. The vertical axis corresponds to e = 0 and Po is the

capacity of the column if concentrically loaded (see Fig. 4.3.2-3)

Interaction diagrams for columns are generally computed by assuming a series of strain

distributions, each corresponding to a particular point on the interaction diagram, and

computing the corresponding values of P and M (strain compatibility analysis).

The calculation process can be illustrated as follow for one particular strain distribution.



Page 46

Figure 4.3.2-4 Stress-Strain relationship for column

In the actual design, interaction charts prepared for uniaxial bending can be used. The

procedure involves:

Assume a cross section, d and evaluate d/h to choose appropriate chart

Compute:

o Normal force ratio:

o Moment ratios:

Enter the chart and pick (the mechanical steel ratio), if the coordinate (, ) lies

within the families of curves. If the coordinate (, ) lies outside the chart, the cross

section is small and a new trail need to be made.

Compute

Check Atot satisfies the maximum and minimum provisions

Determine the distribution of bars in accordance with the charts requirement

4.3.3. Column under bi-axial bending

There are situations in which axial compression is accompanied by simultaneous bending

about both principal axes of the section. This is the case in corner columns, interior or edge

columns with irregular column layout. For such columns, the determination of failure load is

extremely laborious and making manual computation difficult.

Consider the Rc column section shown under axial force P acting with eccentricities ex and

ey, such that ex = My/p, ey = Mx/P from centroidal axes (Fig. 4.3.3-1c).

In Fig. Fig. 4.3.3-1a the section is subjected to bending about the y axis only with eccentricity

ex. The corresponding strength interaction curve is shown as Case (a) (see Fig. 4.3.3-1d).

Such a curve can be established by the usual methods for uni-axial bending. Similarly, in Fig.

4.1-16b the section is subjected to bending about the x axis only with eccentricity ey. The

corresponding strength interaction curve is shown as Case (b) (see Fig. 4.3.3-1d). For case

(c), which combines x and y axis bending, the orientation of the resultant eccentricity is

defined by the angle



Page 47

Bending for this case is about an axis defined by the angle with respect to the x-axis. For

other values of , similar curves are obtained to define the failure surface for axial load plus

bi-axial bending.

Any combination of Pu, Mux, and Muy falling outside the surface would represent failure.

Note that the failure surface can be described either by a set of curves defined by radial

planes passing through the Pn axis or by a set of curves defined by horizontal plane

intersections, each for a constant Pn, defining the load contours (see Fig. 4.3.3-1).

Figure 4.3.3-1 Interaction diagram for compression plus bi-axial bending

Computation commences with the successive choice of neutral axis distance c for each value

of q. Then using the strain compatibility and stress-strain relationship, bar forces and the

concrete compressive resultant can be determined. Then Pn, Mnx, and Mny (a point on the

interaction surface) can be determined using the equation of equilibrium

Since the determination of the neutral axis requires several trials, the procedure using the

above expressions is tedious. Thus, the following simple approximate methods are widely

used.

1. Load contour method: It is an approximation on load versus moment interaction

surface (see Fig. 4.3.3-1). Accordingly, the general non-dimensional interaction

equation of family of load contours is given by:

(

)

(

)



Page 48

(

)

where: Mdx = Pdey Mdy = Pdex

Mdxo = Mdx when Mdy = 0 (design capacity under uni-axial bending about x) Mdyo = Mdy when Mdx = 0 (design capacity under uni-axial bending about y)

2. Reciprocal method/Breslers equation: It is an approximation of bowl shaped failure

surface by the following reciprocal load interaction equation.

Where: Pd = design (ultimate) load capacity of the section with eccentricities edy and edx

Pdxo = ultimate load capacity of the section for uni axial bending with edx only (edy = 0)

Pdyo = ultimate load capacity of the section for uni axial bending with edy only (edx = 0) Pdo = concentric axial load capacity (edx = edy = 0) However interaction charts prepared for biaxial bending can be used for actual design. The

procedure involves:

Select cross section dimensions h and b and also h and b

Calculate h/h and b/b and select suitable chart

Compute:

Normal force ratio:

Moment ratios: and

Select suitable chart which satisfy and ratio:

Enter the chart to obtain

Compute

Check Atot satisfies the maximum and minimum provisions

Determine the distribution of bars in accordance with the charts requirement



Page 49

4.4. Analysis of columns according to EBCS 2 (short and slender)

Classification of Frames

A frame may be classified as non-sway for a given load case if the critical load ratio for that

load case satisfies the criterion:

Where: Nsd is the design value of the total vertical load Ncr is its critical value for failure in a sway mode

In Beam-and-column type plane frames in building structures with beams connecting each

column at each story level may be classified as non-sway for a given load case, when first-

order theory is used, the horizontal displacements in each story due to the design loads

(both horizontal and vertical), plus the initial sway imperfection satisfy the following

criteria.

Where: is the horizontal displacement at the top of the story, relative to the bottom of the story

L is the story height H is the total horizontal reaction at the bottom of the story N is the total vertical reaction at the bottom of the story,

For frame structures, the effects of imperfections may be allowed for in frame analysis by

means of an equivalent geometric imperfection in the form of an initial sway imperfection

(assuming that the structure is inclined to the vertical at an angle) determined by:

a. For single story frames or for structures loaded mainly at the top

b. For other types of frames

Where the effects of imperfections are smaller than the effects of design horizontal actions,

their influence may be ignored. Imperfections need net be considered in accidental

combinations of actions.

The displacement in the above equation shall be determined using stiffness values for

beams and columns corresponding to the ultimate limit state. As an approximation,



Page 50

displacements calculated using moment of inertia of the gross section may be multiplied by

the ratio of the gross column stiffness Ig to the effective column stiffness Ie (see the

following section) to obtain .

All frames including sway frames shall also be checked for adequate resistance to failure in

non-sway modes

Determination of story buckling Load Ncr

Unless more accurate methods are used, the buckling load of a story may be assumed to be

equal to that of the substitute beam-column frame defined in Fig. and may be determined

as:

Where: EIe is the effective stiffness of the substitute column designed using the equivalent reinforcement area. Le is the effective length. It may be determined using the stiffness properties of the gross concrete section for both beams and columns of the substitute frame (see Fig. 4.4-1c )

In lieu of a more accurate determination, the effective stiffness of a column EIe may be

taken as:

Where: Ec = 1100fcd Es is the modulus of elasticity of steel Ic, Is, are the moments of inertia of the concrete and reinforcement sections, respectively, of the substitute column, with respect to the centroid of the concrete section (see Fig. 4.4-1c)

or alternatively

Where: Mb is the balanced moment capacity of the substitute column 1/rb is the curvature at balanced load and may be taken as

(

)

The equivalent reinforcement areas, As, tot, in the substitute column to be used for

calculating Is and Mb may be obtained by designing the substitute column at each floor level

to carry the story design axial load and amplified sway moment at the critical section. The



Page 51

equivalent column dimensions of the substitute column may be taken as shown in Fig,

below, in the case of rectangular columns. Circular columns may be replaced by square

columns of the same cross-sectional area. In the above, concrete cover and bar

arrangement in the substitute columns shall be taken to be the same as those of the actual

columns.

The amplified sway moment, to be used for the design of the substitute column, may be

found iteratively taking the first-order design moment in the substitute column as an initial

value.

In lieu of more accurate determination, the first-order design moment, Mdl, at the critical

section of the substitute column may be determined using:

Where: 1 and 2 are defined before and shall not exceed 10.

Figure 4.4-1 Substitute Multi-Story Beam-Column Frame

Slenderness Ratio



Page 52

The significance of P (i.e. whether a column is short or slender) is defined by a slenderness

ratio.

In EBCS 2, the slenderness ratio is defined as follows:

a) For isolated columns, the slenderness ratio is defined by:

where: Le is the effective buckling length i is the minimum radius of gyration. The radius of gyration is equal to

Where: I is the second moment of area of the section A is cross sectional area

b) For multistory sway frames comprising rectangular sub frames, the following expression

may be used to calculate the slenderness ratio of the columns in the same story.

where: A is the sum of the cross-sectional areas of all the columns of the story Kl is the total lateral stiffness of the columns of the story (story rigidity), with modulus of elasticity taken as unity L is the story height

Limits of Slenderness

The slenderness ratio of concrete columns shall not exceed 140

Second order moment in a column can be ignored if

a) For sway frames, the greater of

{

b) For non-sway frames

Where: M1 and M2 are the first-order (calculated) moments at the ends, M2 being

always positive and greater in magnitude than M1, and M1 being positive if member is

bent in single curvature and negative if bent in double curvature



Page 53

Effective Length of Columns

Effective buckling length is the length between points of inflection of columns and it is the

length which is effective against buckling. The greater the effective length, the more likely

the column is to be buckle.

The effective length of the column, Le, can be determined from Fig 4.4-2, alignment charts

(see Fig. 4.4-3), or using approximate equations.

i. Figure for idealized condition is used when the support conditions of the column can be

closely represented by those shown in the figure below.

Figure 4.4-2 Effective length factors for centrally loaded columns with various idealized conditions



Page 54

ii. The alignment chart (see Fig. 4.4-3) is used for members that are parts of a framework.

Figure 4.4-3 Alignment Charts/Nomo graph for effective length of columns in continuous frames

iii. Approximate equations. The effect of end restrained is quantified by the two end

restrain factors 1 and 2

Figure 4.4-4 Model for computing stiffness coefficients



Page 55

Where Ecm is modulus of elasticity of concrete Lcol is column height

Lb is span of the beam Icol, Ib are moment of inertia of the column and beam respectively

is factor taking in to account the condition of restraint of the beam at the opposite end

= 1.0 opposite end elastically or rigidly restrained = 0.5 opposite end free to rotate = 0 for cantilever beam

Note that: if the end of the column is fixed, the theoretical value of is 0, but an value of

1 is recommended for use. On the other hand, if the end of the member is pinned, the

theoretical value of is infinity, but an value of 10 is recommended for use. The rationale

behind the foregoing recommendations is that no support in reality can be truly fixed or

pinned.

The following approximate equations can be used provided that the values of 1 and 2

dont exceed 10 (see EBCS 2).

(a) Non-sway mode

(b) In Sway mode

Or Conservatively,

Where 1 and 2 are as defined above and m is defined as:

Note that: for flats slab construction, an equivalent beam shall be taken as having the width

and thickness of the slab forming the column strip.

Design of columns, EBSC-2 1995

General



Page 56

The internal forces and moments may generally be determined by elastic global analysis

using either first order theory or second order theory.

a) First-order theory, using the initial geometry of the structure, may be used in the

following cases

Non-sway frames

Braced frames

Design methods which make indirect allowances for second-order effects.

b) Second-order theory, taking into account the influence of the deformation of the

structure, may be used in all cases.

A. Design of Non-sway Frames

Individual non-sway compression members shall be considered to be isolated elements and

be designed accordingly.

Design of Isolated Columns

For buildings, a design method may be used which assumes the compression members to be

isolated. The additional eccentricity induced in the column by its deflection is then

calculated as a function of slenderness ratio and curvature at the critical section

Total eccentricity

1. The total eccentricity to be used for the design of columns of constant cross-section at

the critical section is given by:

Where: ee is equivalent constant first-order eccentricity of the design axial load ea is the additional eccentricity allowance for imperfections. For isolated columns:

e2 is the second-order eccentricity First order equivalent eccentricity

i. For first-order eccentricity e0 is equal at both ends of a column

ii. For first-order moments varying linearly along the length, the equivalent

eccentricity is the higher of the following two values:



Page 57

Where: e01 and e02 are the first-order eccentricities at the ends, e02 being positive and

greater in magnitude than e01. e01 is positive if the column bents in single curvature and

negative if the column bends in double curvature.

Figure 4.4-5

iii. For different eccentrics at the ends, (2) above, the critical end section shall be

checked for first order moments:

Second order eccentricity

i. The second-order eccentricity e2 of an isolated column may be obtained as

Where: Le is the effective buckling length of the column k1= /20 - 0.75 for 15 35 k1= 1.0 for >35 l/r is the curvature at the critical section.

ii. The curvature is approximated by:

(

)

Where: d is the effective column dimension in the plane of buckling k2 =Md /Mb

Md is the design moment at the critical section including second-order effects

Mb is the balanced moment capacity of the column. iii. The appropriate value of k2 may be found iteratively taking an initial value

corresponding to first-order actions.



Page 58

Design of Sway Frames

The second order effects in the sway mode can be accounted using either of the following

two methods:

a. Second-order elastic global analysis: When this analysis is used, the resulting forces and

moment may directly be used for member design.

b. Amplified Sway Moments Method: In this method, the sway moments found by a first-

order analysis shall be increased by multiplying them by the moment magnification

factor:

Where: Nsd is the design value of the total vertical load Ncr is its critical value for failure in a sway mode. The amplified sway moments method shall not be used when the critical load ratio

Sway moments are those associated with the horizontal translation of the

top of story relative to the bottom of that story. They arise from horizontal loading and may also arise from vertical loading if either the structure or the loading is asymmetrical.

As an alternative to determining

direct, the following approximation may be used

in beam and-column type frames

In the presence of torsional eccentricity in any floor of a structure, unless more accurate

methods are used, the sway moments due to torsion should be increased by multiplying

them by the larger moment magnification factor s, obtained for the two orthogonal

directions of the lateral loads acting on the structure.

Effect of Creep

Creep effects may be ignored if the increase in the first-order bending moments due to

creep deformation and longitudinal force does not exceed 10%.

The effect of creep can be accounted by:

a) For isolated columns in non-sway structures, creep may be allowed for by multiplying

the curvature for short-term loads( see the expression of curvature in second order

eccentricity) by (1 + d), where d, is the ratio of dead load design moment to total

design moment, always taken as positive.

b) For sway frames, the effective column stiffness may be divided by (1 + d), where d, is

as defined above.

Slender columns bent about the major axis



Page 59

A slender column bent about the major axis may be treated as bi-axially loaded with initial

eccentricity ea acting about the minor axis

Biaxial Bending of Columns

a) Small Ratios of Relative Eccentricity

Columns of rectangular cross-section which are subjected to biaxial bending may be

checked separately for uni-axial bending in each respective direction provided the relative

eccentricities are such that k 0.2; where k denotes the ratio of the smaller relative

eccentricity to the larger relative eccentricity.

The relative eccentricity, for a given direction, is defined as the ratio of the total eccentricity,

allowing for initial eccentricity and second-order effects in that direction, to the column

width in the same direction.

b) Approximate Method

Columns of rectangular cross-section which are subjected to biaxial bending may be

checked separately for uni-axial bending in each respective.

If the above condition is not satisfied, the following approximate method of calculation can

be used, in the absence of more accurate methods.

For this approximate method, one-fourth of the total reinforcement must either be

distributed along each face of the column or at each corner. The column shall be designed

for uni-axial bending with the following equivalent uni-axial eccentricity of load, eeq along

the axis parallel to the larger relative eccentricity:

Where: etot denotes the total eccentricity in the direction of the larger relative eccentricity k denotes the relative eccentricity ratio as defined in above.

may be obtained from the following table as a function of the relative normal force

0

0.2

0.4

0.6

0.8

1.0 0.6

0.8

0.9

0.7

0.6

0.5

Detailing

The minimum lateral dimension of a column shall be at least 150 mm.

Longitudinal Reinforcement

The area of longitudinal reinforcement shall neither be less than 0.008Ac nor more

than 0.08AC. The upper limit shall be observed even where bars overlap.



Page 60

For columns with a larger cross-section than required by considerations of loading, a

reduced effective area not less than one-half die total area may be used to

determine minimum reinforcement and design strength

The minimum number of longitudinal reinforcing bars shall be 6 for bars in a circular

arrangement and 4 for bars in a rectangular arrangement

The diameter of longitudinal bars shall not be less than 12 mm

Lateral Reinforcement

The diameter of ties or spirals shall not be less than 6 mm or one quarter of the

diameter of the longitudinal bars.

The center-to-center spacing of lateral reinforcement shall not exceed:

12 times the minimum diameter of longitudinal bars. least dimension of column 300 mm

Ties shall be arranged such that every bar or group of bars placed in a corner and

alternate longitudinal bar shall have lateral support provided by the corner of a tie

with an included angle of not more than 1350 and no bar shall be further than 150

mm clear on each side along the tie from such a laterally supported bar( see Fig. )

Up to five longitudinal bars in each corner may be secured against lateral buckling by

means of the main ties. The center-to-center distance between the outermost of

these bars and the corner bar shall not exceed 15 times the diameter of the tie (see

Fig.)

Spirals or circular ties may be used for longitudinal bars located around the

perimeter of a circle. The pitch of spirals shall not exceed 100 mm.

Figure 4.4-6 a) Measurement between laterally Supported column bars (b) Requirements for main and intermediate ties

CHAPTER IPlastic Moment Redistribution1.1. Introduction1.2. Moment curvature relationship1.3. Plastic hinges and collapse mechanisms1.4. Rotation RequirementCHAPTER IICONTINUOUS BEAMS AND ONE-WAY RIBBED SLABS2.2.1. Introduction2.2. Analysis and design of continuous beams2.3. Analysis and design of one-way slabs2.4. Analysis and design of one-way ribbed slabsCHAPTER IIITWO-WAY SLABS1.2.3.3.1. Introduction3.2. Analysis and design of two way beam supported slabs3.3.1.3.2.3.3. Analysis and Design of Flat Slabs3.3.1. Introduction3.3.2. Load Transfer in Flat Slabs3.3.3. Moments in Flat Slab Floors3.3.4. Practical Analysis of Flat Slab Floors3.3.5. Direct Design Method as per EBCS 2, 19953.3.6. Equivalent Frame Method3.3.7. Shear in Flat Slabs, as per EBCS 2CHAPTER IVCOLUMNS4.4.1. Introduction4.2. Classification of Compression Members4.2.1. Braced/Un-braced Columns4.2.2. Short/Slender Columns4.3. Classification of Columns on the Basis of Loading4.3.1. Axially loaded columns4.4.1.4.2.4.3.4.3.1.4.3.2. Column under uni-axial bending4.3.3. Column under bi-axial bending4.4. Analysis of columns according to EBCS 2 (short and slender)

RC II Robel

Documents

moment of inertia

moment diagrams

steel beam

moment curvature relationship

concrete figure

given beam section

plastic analysis

indeterminate beam