Slenderness Effects in Reinforced Concrete Beams. Paper by P. Revathi and Devdas Menon/AUTHORS' CLOSURE

ACI Structural Journal, May/Jun 2008 by Solanki, Himat

(ProQuest: ... denotes formulae omitted.)

Discussion by Himat Solanki

Professional Engineer, Building Dept., Sarasota County Government, Sarasota, FL.

The authors have presented an interesting observation on reinforced concrete slender beams. The discusser would like to offer the following comments:

1. The warping effect is not significant in rectangular slender beams when beams are simply supported. When end conditions are not simple, the warping effect would have some effect on the stability of beams. For example

M^sub bcr^ = C^sub b^[{π^sup 2Ely^/(k^sub y^l)^sup 2^}{(π^sup 2^EI^sub w^ /(k^sub z^l)^sup 2^) + GJ}]^sup 1/2^ (18)

where23

2. In Reference 13, there is no case for two loads at the middle-third location condition. Based on the discusser?s simple linear interpolation from the uniformly distributed load and centrally concentrated load for a simply supported slender beam, coefficient C^sub l^ [asymptotically =] 3.26 instead of the authors? C^sub l^ [asymptotically =] 3.33 was found. The C^sub l^ [asymptotically =] 3.26 value is consistent with Stiglat.24 The difference is not significant, but some cases, such as in nonrectangular section, may have some impact;

3. The authors have mentioned B^sub eff^ and K^sub eff^. Are these values at fully cracked stage? (B^sub eff^ and K^sub eff^ notations confuse the discusser because the effective value is normally the value between the noncracked and the fully cracked sections);

4. Based on Eq. (4) through (8), the R value appears to be for the noncracked section, but it was considered along with cracked values (B^sub eff^ and K^sub eff^) in Eq. (16). The R value could be expressed as

R = ζ^sub o^[function of]'^sub c^α^sub o^(1 ζ^sub o^β^sub o^h) (19)

where ζ^sub o^ is the depth coefficient of compressive force from the actual concrete stress block at its centroid = C/ζbh[function of]'^sub c^; β^sub o^ is the distance coefficient of compressive force to the top of the beam; ζ is the coefficient of depth with respect to neutral axis; ζh is the depth of the neutral axis of the beam; C is the concrete compressive force from actual concrete stress block at its centroid; and [function of]'^sub c^ is the concrete cylinder compressive strength at 28 days;

5. It appears that the authors have not considered the selfweight of the beams. The discusser believes that the self-weight of the beams does contribute a significant moment, that is, uniformly distributed load condition, and it can be expressed as

M = 3.54/l[B^sub eff^K^sub eff^]^sup 1/2^sup (20a)

6. Based on the numerous previous published studies, for the rectangular beam with under-reinforced section at its fully cracked stage, the average Bcracked and Kcracked values could be approximated to 0.3Bcracked and 0.03Kcracked. Based on this concept, the authors? Eq. (1) could be modified to

M^sub bcr^ = 3.33/l[(0.03B^sub cracked^)(0.03K^sub cracked^)]^sup 1/2^ (20b)

Assuming G [asymptotically =] 0.43E^sub c^ (Poisson' ratio v [asymptotically =] 1/6)

M^sub bcr^ = 3.33/l[(0.009)(0.43)(E^sub c^)^sup 2^]^sup 1/2^ = 0.2072E^sub c^/l (20c)

...

... (20d)

Based on the aforementioned equation, M^sub bcr^ is an independent of the cross section of the beams, as compared with the authors' Eq. (16); and

7. Rafla25 suggested that the I-type beam can be analyzed as an equivalent rectangular beam when the ratio of the width of the flange to the width of the web is less than or equal to 3, and it can be expressed as

b^sup 3^^sub equivalent^ = 6/h(I^sub y^I^sub t^)^sup 1/2^ (21)

where I^sub y^ is the moment of inertia about the y-axis and I^sub t^ is the torsional moment of inertia.

REFERENCES

23. Toyama, K., et al., "Inelastic Lateral Buckling Behavior of H-Shaped Steel Beams-Column," Summaries of Technical Papers of Annual Meeting of the Architectural Institute of Japan, July 2006, pp. 831-832. (in Japanese)

24. Stiglat, K., "Zur Näherungsberechnung der Kipplasten von Stahlbeton und Spanbeton trägern ?ber Vergleichsschlankheiten," Beton und Stahbetonbau, V. 86, No. 10, Oct. 1991, pp. 237-240.

25. Rafla, K., "Beitrag zur frage Kippstabilität aufgehangter Balken," dissertation, TH Brauschweig, Germany, 1968.

AUTHORS? CLOSURE

The authors thank the discussers for expressing interest in the research work and for their valuable comments.

Response to discussion by Subramanian

It is true that slender concrete beams are perhaps not commonly used in practice, and often they are connected integrally to slabs. When they are encountered in practice (as in precast slab panels that need to be transported, balcony parapets, facia projections, and side walls of tanks), however, they need to be appropriately designed. The current design provisions (that allow slender beams with slenderness ratios Ld/b2 up to 250) are clearly inadequate for this purpose. There is definitely a need for experimental and theoretical studies in this direction, and the two papers published recently by the authors reflect attempts in this direction. Wall thicknesses in practice can be as low as 80 mm (3.15 in.) (for precast units, with minimal covers), and not 200 mm (8 in.), as visualized by the discusser. The specimen dimensions adopted for testing by the authors (as well as by previous researchers worldwide), covering slenderness ratios Ld/b2 in the range of 150 to 350 (with spans limited to 6 m [6.6 yd]), could be accommodated with slab thicknesses of 80 and 100 mm (3.15 and 4 in.). Indeed, the results can be extrapolated to cover slender beams of larger thicknesses. Full-scale prototype experiments on slender beams with thicknesses as much as 200 to 400 mm (8 to 16 in.), as suggested by the discusser, would involve specimens with spans and depths that are practically not feasible under normal laboratory conditions.