REINFORCED CONCRETE DESIGN
T- beams and L-beams
4.8 ANALYSIS O F SLABS A S RECTANGULAR BEAMS
Slabs under flexure behave in much the same way as beams. A slab of thickness subject to a bending moment uniformly distributed over its width [Fig.
may be treated as a wide shallow beam for the purpose of analysis and design.
TYPICAL STRIP
slab
cross section design strip
offered b
Induced
rectangular beam bending design strip, subject to bending
4.23 Analysis of slabs
IN FLEXURE 161 In such slabs, the reinforcing bars are usually spaced over the width of the slab. For computations are generally based on a typical one-metre wide strip of the slab considered as a beam [Fig. with b mm.
The loads are generally uniformly distributedt and expressed in units of
If is the centre-to-centre spacing of bars in mm, then the number of bars i n the metre wide strip is given by Accordingly, denoting as the cross-sectional area of one bar (equal to 2 the area of tensile steel expressed in units of
is given by
In practice, reinforced concrete slabs are generally and singly reinforced. In the example to follow, the analysis of a typical slab is undertaken to
moment resisting capacity at working loads as well as at the
4.8.1 T r a n s v e r s e M o m e n t s in One- way S l a b s
Although a one-metre wide strip of the slab is considered as a beam of width b for the for flexural strength, there is a difference which the student will do well to bear in mind. As a beam bends (sags), the portion of the section above the neutral axis is under compression and hence subjected to a lateral expansion to the Poisson effect. Similarly, the part below the NA i s subjected to a lateral contraction. Hence, after bending, the cross section will strictly not be rectangular, but nearlyt trapezoidal, as shown (greatly exaggerated) in Fig.
In the case of a one-way slab, for a design strip such as shown in Fig. such lateral displacements (and hence strains) are prevented by the remainder of the slab on either side (except at the two edges). In words, in order for the rectangular section to remain rectangular even after binding (as a slice of a long cylindrically bent surface, with no transverse curvature, should be), the remainder of the slab restrains the lateral displacements and strains, by inducing lateral stresses on the design strip as shown in Fig. is known as the 'plain strain' condition [Ref. These lateral stresses give rise to secondary moments in the transverse direction as shown in Fig.
When concentrated loads on a one-way slab, the simplified procedure given in 24 3.2 of the Code may be adopted.
'To be exact, just as the beam undergoes a 'sagging' curvature along the span, there will be a 'hogging' curvature in the transverse direction. Thus the top surface will be curved than straight [see Ref. 4.11.
REINFORCED CONCRETE DESIGN
EXAMPLE 4.17
Determine (a) the allowable moment (at service loads) and the moment of of a 150 mm thick slab, reinforced 10 bars at 200
located at an effective depth of 125 Assume M 20 and Fe 415 steel.
SOLUTION transformed-cracked section and considering b
13.33 x 393 x (125
Applying 4.65, or using analysis aids [Table 0.314 415 0.314
4.3 Why does Code specify an effectively higher modular ratio for compression reinforcement, as compared to tension reinforcement?
4.4 Justify the assumption that concrete resists no flexural tensile stress. i n reinforced concrete beams.
4.5 Describe the moment-curvature relationship for reinforced concrete beams.
What are the possible modes of failure?
4.6 The term 'balanced section' is used in both working stress method and state method Discuss the difference in meaning.
4.7 Why is it undesirable to design sections in (a) WSM, (b) LSM?
4.8 The of locating neutral axis as a (in a
concrete flexure) is applied in WSM, but not in LSM.
Why?
4.9 Why is it lo use high strength steel as compression reinforcement in design by WSM?
4.10 Justify the Code for the limiting axis depth in LSM.
4.11 ultimate of resistance of a singly reinforced beam section can be calculated either in terms of the concrete compressive strength or the steel tensile strength", Is this statement justified in all cases?
4.12 and plot the - for a singly beam section
....
for values of in range 0.0 to 2.0, considering combinations of M 20 and Fe 250 and M 25 and 415. (Refer Figs 4.13 and 4.19). on the system as T-beams for determining their flexural strength at all sections?
4.16 Discuss the variation of ultimate moment of resistance singly reinforced beam of given rectangular cross-section and material properties with the area of tension steel.
4.17 Explain how the neutral axis is located in sections (at the ultimate limit state), given that it lies outside the flange.
164 REINFORCED CONCRETE
4.18 Given percentages of tension steel (p,) and compression steel of a doubly reinforced section, how is it possible to the beam is reinforced or (at the ultimate limit
4.19 Show that the procedure for flexural of slabs is to that of
4.20 What are the significant differences the in bending of a beam of and strip of a very wide one-way slab?
4.21 is it necessary to provide transverse reinforcement in a one-way slab?
4.22 "A reinforced concrete beam can be considered to be safe in flexure if its ultimate moment of resistance (as per Code) at any section exceeds the factored moment due to the loads at that section". Explain the meaning of safety as implied in this statement. Does the Code call for additional to be satisfied for 'safety'?
4.23 If a balanced singly reinforced is experimentally tested to failure, what is ratio of actual moment capacity to predicted capacity (as per Code) likely to be? to estimate actual no safety factors should be applied: also, is no effect of loading).
PROBLEMS
4.1 A beam has a rectangular section as shown in Fig. 4.24. M 20 concrete and Fe 250 steel,
(a) compute the stresses in concrete and steel under a service load moment of 125 !dim. Check the calculations using the flexure formula.
: 4.84 99.0 MPal (b) determine the capacity of the section under loads.
Also determine the corresponding stresses induced in concrete and steel.
: !dim, 6.35 130 MPal 4.2 Determine the allowable moment capacity of the beam section [Fig. of Problem4.1, as well as the corresponding stresses in concrete and steel (under service loads), considering
M 20 concrete and Fe 415 steel;
: 181 143 MPal
M 25 concrete and Fe 250 steel.
: 165.7 6.84 130 MPal
IN FLEXURE 165
Fig. 4.24 Problems 4.1
-
4.34.3 Determine the ultimate moment of resistance of the beam section [Fig. of Problem 4.1, considering
M 20 concrete and 250 steel:
[Ans. : 278 M 20 concrete and Fe 415 steel:
: 420 (iii) M 25 concrete and Fe 250 steel;
: 285 M 25 concrete and Fe 415 steel.
: 440 Compare the various results, and state whether or not, in each case, the beam section complies with the Code requirements for flexure.
4.25 Problems 4.4
-
4.54.4 A beam carries a uniformly distributed service load (including of 38 on a simply supported span of 7.0 The cross-section of the beam is shown in Fig. 4.25. Assuming M 20 concrete and Fe 415 steel, compute (a) the stresses developed in concrete and steel at applied service loads;
: 10.4 209 MPal
REINFORCED CONCRETE DESIGN
(b) the service load (in that the beam can carry (as per the
Code). : 25.5
4.5 Detennine the moment of resistance of the beam section [Fig. of Problem4.4. Hence, compute the effective load factor ultimate load), considering the service load of 38 cited in Problem 4.4.
: 366
4.6 The cross-sectional dimensions of a beam are given in Fig. 4.26. Assuming M 20 concrete and Fe 415 steel, compute :
(a) the stresses in concrete and steel under a service load moment of 150
: 4.30 92.9 (b) the moment capacity of the at service loads.
: 244
Fig. 4.26 Problems 4.6
-
4.7 4.27 Problems 4.8-
4.94.7 Determine the ultimate moment of resistance of the T beam section [Fig.
of Problem 4.6.
: 509 4.8 M 25 concrete and Fe 415 steel, compute the ultimate moment of
of the L beam section shown in Fig. 4.27.
: 447 4.9 Determine the ultimate moment of resistance of the section [Fig. of
Problem 4.8, considering Fe 250 grade (in lieu of Fe
: 288 4.10 A doubly reinforced section is shown in Fig. 4.28. Assuming M 20
concrete and Fe 415 steel, compute
(a) the stresses in concrete and steel under a service load moment of 125 kiim;
: 11.7 218
the allowable service load moment capacity of section.
74.6
BEHAVIOUR IN FLEXURE 167
4.1 1 the moment of of beam section of
Fig. 4.28 Problems 4.10
-
4.124.12 Repeat Problem4.11, considering the compression bars to comprise 3 - 20 (instead of 3
-
22 as shown in Fig. 4.28).: 196 4.13 Determine (a) the allowable moment (at service loads) and the ultimate
moment of resistance of a 100 mm thick slab, reinforced with 8 mm bars at 200 spacing located at an effective depth of 75 Assume M 20 concrete and Fe 415 steel.
: ( a ) 4.21 6.33 4.14 A simply supported slab has an effective span of 3.5 metres. It is 150
thick, and is reinforced with mm bars 200 mm spacing located at an effective depth of 125 Assuming M 20 concrete and Fe 415 steel.
determine the superimposed service load (in that the slab can safely to WSM , and according to (assuming a load factor of 1.5).
REFERENCES
4.1 Timoshcnko, and Goodier, Theory of Second edition, McGraw-Hill, 195
4.2 Hognestad, E., Hanson, N.W. and Concrete Stress
in ACI, Vol. 52, pp 455-479
4.3 E.P., of Solids, Prentice-Hall Inc., Englewood Cliffs, New Jersey, 1976.
4.4 H, Researches Towards a General Theory for