CHAPTER TWO

DESIGN OF SOLID SLABS AND

INTRODUCTION TO YIELD LINE ANALYSIS OF SLABS

2.1 Two-way Edge Supported Slabs For one-way slab:

slab supported on two opposite edges only only one plane of bending exists, and the load is transferred to those two supports.

If a slab is supported on all the four edges: (Fig. 2.1) the load is transferred on to the four supports bending takes place along both spans (Fig 2.2). bending moments and deflection are considerably

reduced as compared to one-way slab. If the corners are fixed at the wall support,

bending moment and deflection are further reduced; but special torsion reinforcement is needed for torsion resistance.

.

.

One-way slabs carry load in one direction.

Two-way slabs carry load in two directions

a) One-way Slab b) Two-way slab

c) Two-way slab 3D

Fig2.1 Types of Structural Slabs

Fig.2.2 Bending of center strips of two-way slabs on simple edge supports

In two-way square slab, the two-way action is equal in each direction.

Due to this reinforcement is also provided in both directions.

In long narrow slabs, (the ratio Ly/Lx >2 ), the two-way action is effectively reduced to one-way action in the direction of short span.

The CODE table coefficients for moments and shear are given depending on aspect ratios, and support conditions of slab panel.

However, analysis using approximate theories which provide satisfactory result for some cases (the Rankine-Grashoff’s method) can be used, particularly for simply supported two way slab.

2.2 Rankine-Grashoff’s Approximate Method This is suitable for analysis of two-way simply

supported slab if corners are not held (fig 2.2 below).

Assume that load is shared between strips of unit width running in the two directions as shown below (interdependent in action).

At their common intersection point, their deflections are equal.

wx and wy are values of the share of loads obtained from compatibility of equal deflections of strips at the center of the slab.

04/28/2023

Fig 2.2 Moments and moment variations

Assuming slab strips as simply supported beam subjected to uniform load, the maximum deflection and maximum bending moment of slab strips are obtained as shown above.

Therefore, the bending moment per unit width in both directions are given by substituting wx and wy into equations of maximum bending moment of slab strip as,

From these two equations of moment, it can be seen that a larger share of moment goes along the shorter span.

Values of bending moment coefficients & aspect ratio are given table below.

2.3 Analysis of Two-way Rectangular Slab using Code’s

Coefficients (EBCS-2) EBCS-2/95 section A.3.3 provide moment coefficient table

for analysis of rectangular slab panels subjected to uniformly distributed load with provision for torsion at the corners depending on aspect ratios and support conditions of slab.

1.0 1.1 1.2 1.3 1.4 1.5 1.75 2.0 2.5 3.0

0.0625 0.074 0.084 0.093 0.099 0.104 0.113 0.118 0.122 0.124

0.0625 0.061 0.059 0.055 0.051 0.046 0.037 0.029 0.020 0.014

xy ll

x

y

xy ll

Code Method:can also be used for analysis of slab subjected to concentrated load in addition to a uniform load by treating concentrated load as equivalent-uniform load provided that the sum of the non-uniform load on a panel does not exceed 20% of the total load.

assumes unyielding supports of slab.

Unyielding (fixed) supports of slab may be ensured by proportioning supports of slab with depth larger than or equal to 2.5 times thickness of slab.

Maximum moments for individual slab panels with edges either simply supported (discontinuous) or fully fixed (continuous) are given by,

04/28/2023

Notation used for different critical moments and edge numbers are as shown below.

Figure 2.3 Code Method of Slab Analysis

Subscripts used for moments and moment coefficient have the following meaning.

s--support -ve moment f--field or span +ve moment x--direction of shorter span y--direction of longer span

Therefore, the maximum support and span moments per unit width for two-way system are given by the following equations:

Moment coefficient table given by EBCS-2/95 provide moment coefficients for nine separate slab panels with different possible support conditions as shown below.

Figure 2.4: Possible two-way rectangular slab panel with different support

For slab panel with support condition different from those given above, interpolate linearly between the neighboring supports condition of slab panels.

The maximum design load for all slab panels, in strength limit state method is given

For purpose of design of slab and provision of reinforcement, the slab panel is divided into middle and edge strips as shown below.

LDwd 6.13.1

For intermediate support in continuous slab, there will thus be two different support moments (from left and right sides).

The difference may be distributed between the slab panels on either side of the support to equalize their moments, as in the moment distribution method for frames.

Table 2:Bending moment coefficients for rectangular panels supported on four sides with provision for torsion at corner

Two methods of differing accuracy are specified by EBCS-2 to distribute the intermediate support moments: method I and method II (section A.3.3.2).

Method I: - Dimensioning in this method is carried out either for:

initial moment directly, or average of initial moments at the support

This method may be used: When the difference between initial support

moments are less than 20% of the larger moment, and

For internal structures where live load does not exceed 2.5 times the dead load (qk <= 2.5 gk)or for external structures 0.8 times dead load (qk <= 0.8 gk).

Method II:- Used when conditions given in method I are not met, Other more accurate method is also possible The unbalanced support moment is locally distributed at

each edge without iteration using the moment distribution method depending on the relative stiffness of the adjacent slab panels.

The relative stiffness of each slab panels shall be taken proportional to its gross moment of inertia divided by the smaller span.

If the support moment is decreased while carrying out moment distribution of unbalanced support moment, the span moments Mxf & Myf are then increased to allow for the change of support moments.

This increase is calculated as being equal to the change of the support moment multiplied by the factor given in table 3.

If a support moment is increased, no adjustment shall be made to the span moments.

Table 3 Factors for adjusting span moments mxf and myf

At corners of discontinuous corners of two-way slab, special torsion reinforcement is required at top (why?) along diagonal and at bottom (why?) perpendicular to the diagonal of rectangle extends for the length about lx/5 from corner as shown below.

Alternatively, mesh reinforcement may be provided at top and bottom of the corner of the rectangle.

2.4 Loads on Supporting Beams and Maximum Shear-force of Two-way slab

The load on two-way slab transferred to the supporting beams may be assumed as the load within tributary area of slab bounded by the intersection of line from the corners with the median line of the panel parallel to the long side as shown below.

The lines that divide the load on to the supporting beam correspond to the assumed crack-lines of yield-line theory of slab.

According to EBCS-2/95, the design loads on supporting beam and the design shear-force of two-way slab subjected to a uniformly distributed load considering torsion at corners may be determined using the following equation.

where are shear-force coefficient given by the code as a function of aspect ratio, and supporting condition of slab panel (refer table 4)

service or factored uniform design load depending on method of design

vi

dw

The design load on supporting beam is assumed to be distributed over a length of 0.75 times the span length of beam as shown below.

Table 4 Shear-force coefficients for uniformly loaded rectangular panels supported on four sides with provision for torsion at corner

2.5 Design of section of Solid Slabs Designed as a singly reinforced section without

shear reinforcement Flexural reinforcement of slabs is applied in the same

way as singly reinforced rectangular beams with clear cover about 15mm for mild exposure condition or 25mm for moderate exposure condition.

Design Procedure: Determine slab-thickness from deflection

requirement.

D = d + c + Ø Check adequacy of thickness for both flexure

and shear.

Thickness of slab is adequate for shear if the design shear stress developed in slab is less than or equal to the shear strength provided by the slab.

Design shear strength of concrete slab in the ultimate limit state method that prevents diagonal tension failure according to EBCS-2/95 shall be taken as:

If thickness of slab is adequate for flexure and for shear, then required flexural reinforcement are determined using ultimate limit state for flexure as:

The required area of tension steel in ULS for flexure is determined using:

Limitation of Flexural Reinforcement of slabs (EBCS-2/95)

Secondary reinforcement is the area of steel corresponding to the minimum reinforcement ratio of main reinforcement.

2.6 Yield Line Theory for Slabs2.6.1 Introduction: Rectangular one way or two way slabs under normal

uniform loading can be analyzed and then designed using coefficients.

For irregular shapes, varied support conditions, presence of openings, varied loading and more complex conditions, the yield line theory is found to be useful.

The yield line theory is an ultimate load method of analysis of slab, i.e. the BM at the verge of collapse is used as the basis for design.

At collapse loads, an under reinforced slab begins to crack with the reinforcement yielding at points of high moment.

The crack lines or the yield lines propagate with the increase in deflection until the slab is broken into a number of segments.

A yield line is a line in the plane of the slab across which reinforcing bars have yielded and about which excessive deformation (plastic rotation) under constant limit moment (ultimate moment) continues to occur leading to failure.

The yield line method developed by Johansen is applicable to collapse by yielding of under-reinforced concrete slab.

It is based on the Upper bound theorem & According to this theorem, for any assumed collapse mechanism, if the collapse load is calculated by equating the energy dissipation at the plastic ‘hinges’ to the work done by the external load, then the load so calculated is equal to or greater than the true collapse load.

Fig.2.9 shows some yield line patterns. Rectangular one way or two way slabs under normal uniform loading can be analyzed and then designed using coefficients obtained from Tables published for this purpose.

In the one-way continuous slab shown in Fig.2.9 (a), straight yield lines form with a sagging yield line at the bottom of the slab near mid-span and hogging yield lines over the supports.

The yield line patterns for a square and a rectangular simply supported two-way slab subjected to a uniform load are shown in Fig.2.9 (b) and Fig.2.9 (c) respectively.

The deformed shape of the square slab is an inverted pyramid and that of the rectangular slab is an inverted roof shape.

Fig 2.9 (a) Continuous one-way slab; (b) square slab; (c) rectangular slab.