Clubes de convergencia de la pobreza multidimensional en el Perú: 2004-2018 1
4. Conclusiones y recomendaciones
Although a one-metre wide strip of the slab is considered as a beam of width b = 1000 mm for the analysis/design 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 due to the Poisson effect. Similarly, the part below the NA is subjected to a lateral contraction. Hence, after bending, the cross section will strictly not be rectangular, but nearly‡ trapezoidal, as shown (greatly exaggerated) in Fig. 4.23(d).
In the case of a one-way slab, for a design strip such as shown in Fig. 4.23(c, e), such lateral displacements (and hence strains) are prevented by the remainder of the slab on either side (except at the two edges). In other words, in order for the rectangular section to remain rectangular even after bending (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. 4.23(e). This is known as the ‘plain strain’ condition [Ref. 4.1]. These lateral stresses give rise to secondary moments in the transverse direction as shown in Fig. 4.23(e).
Hence, even a one-way slab will need (‘secondary’) reinforcements in the transverse direction to resist these secondary moments. Furthermore, bending moments in the transverse direction are generated locally when the slab is subject to concentrated loads. Also, shrinkage and temperature effects introduce secondary stresses which require transverse reinforcement.
† When concentrated loads act on a one-way slab, the simplified procedure given in Cl. 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’ (‘anticlastic’) curvature in the transverse direction. Thus the top surface will be curved rather than straight [see Ref. 4.1].
EXAMPLE 4.17
Determine (a) the allowable moment (at service loads) and (b) the ultimate moment of resistance of a 150 mm thick slab, reinforced with 10 mm φ bars at 200 mm spacing located at an effective depth of 125 mm. Assume M 20 concrete and Fe 415 steel.
The neutral axis depth kd is obtained by considering moments of areas in the transformed−cracked section [Eq. 4.12], and considering b = 1000 mm
( )
1000× kd 2 2 = 13.33 × 393 × (125 – kd)
Solving, kd= 31.33 mm <kbd =0.289 × 125 = 36.1 mm Hence, the section is ‘under−reinforced (WSM)’.
⇒
fst =σ = 230 MPa stApplying Eq. 4.65, or using analysis aids [Table A.2(a)],
M bd
uR
2 = 0 87 415 0 314
100 1 415
20 0 314 . × × . ×⎛ − × 100
⎝⎜ ⎞
⎠⎟
. = 1.060 MPa
⇒
MuR =1060. ×1000×1252 = 16.56 × 106 Nmm/m = 16.6 kNm/m.REVIEW QUESTIONS
4.1 What is the fundamental assumption in flexural theory? Is it valid at the ultimate state?
4.2 Explain the concept of ‘transformed section’, as applied to the analysis of reinforced concrete beams under service loads.
4.3 Why does the 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 in 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 (WSM) and limit state method (LSM). Discuss the difference in meaning.
4.7 Why is it undesirable to design over-reinforced sections in (a) WSM, (b) LSM?
4.8 The concept of locating the neutral axis as a centroidal axis (in a reinforced concrete beam section under flexure) is applied in WSM, but not in LSM.
Why?
4.9 Why is it uneconomical to use high strength steel as compression reinforcement in design by WSM?
4.10 Justify the Code specification for the limiting neutral axis depth in LSM.
4.11 “The ultimate moment 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 Compute and plot the ratio
all uR
M
M for a given singly reinforced beam section
for values of pt in the range 0.0 to 2.0, considering combinations of (i) M 20 and Fe 250 and (ii) M 25 and Fe 415. (Refer Figs 4.13 and 4.19). Comment on the graphs generated, in terms of the safety underlying beam sections that are designed in accordance with WSM.
4.13 Define “effective flange width”.
4.14 What are the various factors that influence the effective flange width in a T-beam? To what extent are these factors accommodated in the empirical formula given in the Code?
4.15 Is it correct to model the interior beams in a continuous beam-supported slab system as T-beams for determining their flexural strength at all sections?
4.16 Discuss the variation of the ultimate moment of resistance of a 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 T-beam sections (at the ultimate limit state), given that it lies outside the flange.
4.18 Given percentages of tension steel (pt) and compression steel (pc) of a doubly reinforced section, how is it possible to decide whether the beam is under-reinforced or over-under-reinforced (at the ultimate limit state)?
4.19 Show that the procedure for analysing the flexural strength of reinforced concrete slabs is similar to that of beams.
4.20 What are the significant differences between the behaviour in bending of a beam of rectangular section and a strip of a very wide one-way slab?
4.21 Why 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 any additional requirement to be satisfied for ‘safety’?
4.23 If a balanced singly reinforced beam section is experimentally tested to failure, what is the ratio of actual moment capacity to predicted capacity (as per Code) likely to be? (Hint: to estimate actual strength, no safety factors should be applied; also, there is no effect of sustained loading).
PROBLEMS
4.1 A beam has a rectangular section as shown in Fig. 4.24. Assuming M 20 concrete and Fe 250 steel,
(a) compute the stresses in concrete and steel under a service load moment of 125 kNm. Check the calculations using the flexure formula.
[Ans. : 4.84 MPa; 499.0 MPa]
(b) determine the allowable moment capacity of the section under service loads. Also determine the corresponding stresses induced in concrete and steel.
[Ans. : 164 kNm; 6.35 MPa; 130 MPa]
4.2 Determine the allowable moment capacity of the beam section [Fig. 4.24] of Problem 4.1, as well as the corresponding stresses in concrete and steel (under service loads), considering
(i) M 20 concrete and Fe 415 steel;
[Ans. : 181 kNm; 7.00 MPa; 143 MPa]
(ii) M 25 concrete and Fe 250 steel.
[Ans. : 164 kNm; 6.35 MPa; 130 MPa]
350
700
2 – 28 φ
2 – 25 φ
Fig. 4.24 Problems 4.1 – 4.3
30 clear cover
4.3 Determine the ultimate moment of resistance of the beam section [Fig. 4.24] of Problem 4.1, considering
(i) M 20 concrete and Fe 250 steel;
[Ans. : 278 kNm]
(ii) M 20 concrete and Fe 415 steel;
[Ans. : 420 kNm]
(iii) M 25 concrete and Fe 250 steel;
[Ans. : 285 kNm]
(iv) M 25 concrete and Fe 415 steel.
[Ans. : 440 kNm]
Compare the various results, and state whether or not, in each case, the beam section complies with the Code requirements for flexure.
700
300
4 – 25 φ
Fig. 4.25 Problems 4.4 – 4.5
655
4.4 A beam carries a uniformly distributed service load (including self-weight) of 38 kN/m on a simply supported span of 7.0 m. 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;
[Ans. : 10.4 MPa; 209 MPa]
(b) the allowable service load (in kN/m) that the beam can carry (as per the
Code). [Ans. : 25.5 kN/m]
4.5 Determine the ultimate moment of resistance of the beam section [Fig. 4.25] of Problem 4.4. Hence, compute the effective load factor (i.e., ultimate load/service load), considering the service load of 38 kN/m cited in Problem 4.4.
[Ans. : 366 kNm; 1.57]
4.6 The cross-sectional dimensions of a T- 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 kNm;
[Ans. : 4.30 MPa; 92.9 MPa]
(b) the allowable moment capacity of the section at service loads.
[Ans. : 244 kNm]
1300 100
325 7 – 28 φ
420
325
7 – 25 φ 1000
100
Fig. 4.26 Problems 4.6 – 4.7 Fig. 4.27 Problems 4.8 – 4.9 d = 420 500
4.7 Determine the ultimate moment of resistance of the T - beam section [Fig. 4.26]
of Problem 4.6.
[Ans. : 509 kNm]
4.8 Assuming M 25 concrete and Fe 415 steel, compute the ultimate moment of resistance of the L - beam section shown in Fig. 4.27.
[Ans. : 447 kNm]
4.9 Determine the ultimate moment of resistance of the L- section [Fig. 4.27] of Problem 4.8, considering Fe 250 grade steel (in lieu of Fe 415).
[Ans. : 288 kNm]
4.10 A doubly reinforced beam 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 kNm;
[Ans. : 11.7 MPa; 170 MPa; 218 MPa]
(b) the allowable service load moment capacity of section.
[Ans. : 74.6 kNm]
4.11 Determine the ultimate moment of resistance of the beam section [Fig. 4.28] of Problem 4.10.
[Ans. : 201 kNm]
250
400
30 clear
30 clear 3 – 22 φ 3 – 28 φ
M 20 concrete Fe 415 steel
Fig. 4.28 Problems 4.10 – 4.12
4.12 Repeat Problem 4.11, considering the compression bars to comprise 3 – 20 φ (instead of 3 – 22 φ, as shown in Fig. 4.28).
[Ans. : 196 kNm]
4.13 Determine (a) the allowable moment (at service loads) and (b) the ultimate moment of resistance of a 100 mm thick slab, reinforced with 8 mm φ bars at 200 mm spacing located at an effective depth of 75 mm. Assume M 20 concrete and Fe 415 steel.
[Ans. : (a) 4.21 kN/m;
(b) 6.33 kN/m]
4.14 A simply supported one–way slab has an effective span of 3.5 metres. It is 150 mm thick, and is reinforced with 10 mm φ bars @ 200 mm spacing located at an effective depth of 125 mm. Assuming M 20 concrete and Fe 415 steel, determine the superimposed service load (in kN/m2) that the slab can safely carry (i) according to WSM , and (ii) according to LSM (assuming a load factor of 1.5).
[Ans. : (i) 3.01 kN/m2; (ii) 3.46 kN/m2] REFERENCES
4.1 Timoshenko, S.P. and Goodier, J.N., Theory of Elasticity, Second edition, McGraw-Hill, 1951.
4.2 Hognestad, E., Hanson, N.W. and McHenry,D., Concrete Stress Distribution in Ultimate Strength Design, Journal ACI, Vol. 52, Dec. 1955, pp 455−479 4.3 Popov, E.P., Mechanics of Solids, Prentice-Hall Inc., Englewood Cliffs, New
Jersey, 1976.
4.4 Rüsch, H, Researches Towards a General Flexural Theory for Structural Concrete, Journal ACI, Vol. 57, July 1960, pp 1–28.
4.5 — Explanatory Handbook on Indian Standard Code of Practice for Plain and Reinforced Concrete (IS 456:1978), Special Publication SP:24, Bureau of Indian Standards, New Delhi, 1983.