Module 7. Limit State of Serviceability. Version 2 CE IIT, Kharagpur

Transcription

1 Module 7 Limit State of Seviceability

2 Lesson 17 Limit State of Seviceability

3 nstuction Objectives: At the end of this lesson, the student should be able to: explain the need to check fo the limit state of seviceability afte designing the stuctues by limit state of collapse, diffeentiate between shot- and long-tem deflections, state the influencing factos to both shot- and long-tem deflections, select the peliminay dimensions of stuctues to satisfy the equiements as pe S 456, calculate the shot- and long-tem deflections of designed beams ntoduction Stuctues designed by limit state of collapse ae of compaatively smalle sections than those designed employing woking stess method. They, theefoe, must be checked fo deflection and width of cacks. Excessive deflection of a stuctue o pat theeof advesely affects the appeaance and efficiency of the stuctue, finishes o patitions. Excessive cacking of concete also seiously affects the appeaance and duability of the stuctue. Accodingly, cl of S 456 stipulates that the designe should conside all elevant limit states to ensue an adequate degee of safety and seviceability. Clause 35.3 of S 456 efes to the limit state of seviceability compising deflection in cl and cacking in cl Concete is said to be duable when it pefoms satisfactoily in the woking envionment duing its anticipated exposue conditions duing sevice. Clause 8 of S 456 efes to the duability aspects of concete. Stability of the stuctue against ovetuning and sliding (cl. 20 of S 456), and fie esistance (cl. 21 of S 456) ae some of the othe impotance issues to be kept in mind while designing einfoced concete stuctues. This lesson discusses about the diffeent aspects of deflection of beams and the equiements as pe S 456. n addition, lateal stability of beams is also taken up while selecting the peliminay dimensions of beams. Othe equiements, howeve, ae beyond the scope of this lesson Shot- and Long-tem Deflections As evident fom the names, shot-tem deflection efes to the immediate deflection afte casting and application of patial o full sevice loads, while the long-tem deflection occus ove a long peiod of time lagely due to shinkage

4 and ceep of the mateials. The following factos influence the shot-tem deflection of stuctues: (a) magnitude and distibution of live loads, (b) span and type of end suppots, (c) coss-sectional aea of the membes, (d) amount of steel einfocement and the stess developed in the einfocement, (e) chaacteistic stengths of concete and steel, and (f) amount and extent of cacking. The long-tem deflection is almost two to thee times of the shot-tem deflection. The following ae the majo factos influencing the long-tem deflection of the stuctues. (a) humidity and tempeatue anges duing cuing, (b) age of concete at the time of loading, and (c) type and size of aggegates, wate-cement atio, amount of compession einfocement, size of membes etc., which influence the ceep and shinkage of concete Contol of Deflection Clause 23.2 of S 456 stipulates the limiting deflections unde two heads as given below: (a) The maximum final deflection should not nomally exceed span/250 due to all loads including the effects of tempeatues, ceep and shinkage and measued fom the as-cast level of the suppots of floos, oof and all othe hoizontal membes. (b) The maximum deflection should not nomally exceed the lesse of span/350 o 20 mm including the effects of tempeatue, ceep and shinkage occuing afte eection of patitions and the application of finishes. t is essential that both the equiements ae to be fulfilled fo evey stuctue Selection of Peliminay Dimensions The two equiements of the deflection ae checked afte designing the membes. Howeve, the stuctual design has to be evised if it fails to satisfy any one of the two o both the equiements. n ode to avoid this, S 456 ecommends the guidelines to assume the initial dimensions of the membes which will geneally satisfy the deflection limits. Clause stipulates diffeent span to effective depth atios and cl ecommends limiting slendeness of

5 beams, a elation of b and d of the membes, to ensue lateal stability. They ae given below: (A) Fo the deflection equiements Diffeent basic values of span to effective depth atios fo thee diffeent suppot conditions ae pescibed fo spans up to 10 m, which should be modified unde any o all of the fou diffeent situations: (i) fo spans above 10 m, (ii) depending on the amount and the stess of tension steel einfocement, (iii) depending on the amount of compession einfocement, and (iv) fo flanged beams. These ae funished in Table 7.1. (B) Fo lateal stability The lateal stability of beams depends upon the slendeness atio and the suppot conditions. Accodingly cl of S code stipulates the following: (i) Fo simply suppoted and continuous beams, the clea distance between the lateal estaints shall not exceed the lesse of 60b o 250b 2 /d, whee d is the effective depth and b is the beadth of the compession face midway between the lateal estaints. (ii) Fo cantileve beams, the clea distance fom the fee end of the cantileve to the lateal estaint shall not exceed the lesse of 25b o 100b 2 /d. Table 7.1 Span/depth atios and modification factos Sl. tems Cantileve Simply Continuous No. suppoted 1 Basic values of span to effective depth atio fo spans up to 10 m Modification factos fo spans > 10 m Not applicable as deflection calculations ae to be Multiply values of ow 1 by 10/span in metes. 3 Modification factos depending on aea and stess of steel 4 Modification factos depending as aea of compession steel 5 Modification factos fo flanged beams done. Multiply values of ow 1 o 2 with the modification facto fom Fig.4 of S 456. Futhe multiply the ealie espective value with that obtained fom Fig.5 of S 456. (i) Modify values of ow 1 o 2 as pe Fig.6 of S 456. (ii) Futhe modify as pe ow 3 and/o 4 whee

6 einfocement pecentage to be used on aea of section equal to b f d Calculation of Shot-Tem Deflection Clause C-2 of Annex C of S 456 pescibes the steps of calculating the shot-tem deflection. The code ecommends the usual methods fo elastic deflections using the shot-tem modulus of elasticity of concete E c and effective moment of inetia eff given by the following equation: ; but 12. -( M / M )(z/ d )( 1 x/ d )(b w /b) (7.1) eff eff whee moment of inetia of the cacked section, g M cacking moment equal to (f c g )/y t, whee f c is the modulus of uptue of concete, g is the moment of inetia of the goss section about the centoidal axis neglecting the einfocement, and y t is the distance fom centoidal axis of goss section, neglecting the einfocement, to exteme fibe in tension, M maximum moment unde sevice loads, z leve am, x depth of neutal axis, d effective depth, b w beadth of web, and b beadth of compession face. Fo continuous beams, howeve, the values of, g and M ae to be modified by the following equation: (7.2) X + X X e k1 + (1- k 1 ) X o whee X e modified value of X, X 1, X 2 values of X at the suppots,

7 X o k 1 X value of X at mid span, coefficient given in Table 25 of S 456 and in Table 7.2 hee, and value of, g o M as appopiate. Table 7.2 Values of coefficient k 1 k o less k Note: k 2 is given by (M 1 + M 2 )/(M F1 + M F2 ), whee M 1 and M 2 moments, and M F1 and M F2 fixed end moments. suppot Deflection due to Shinkage Clause C-3 of Annex C of S 456 pescibes the method of calculating the deflection due to shinkage α fom the following equation: (7.3) α k 3 ψ l 2 whee k 3 is a constant which is 0.5 fo cantileves, fo simply suppoted membes, fo membes continuous at one end, and fo fully continuous membes; ψ is shinkage cuvatue equal to k 4 ε /D whee ε is the ultimate shinkage stain of concete. Fo ε, cl of S 456 ecommends an appoximate value of in the absence of test data. (7.4) k ( pt - pc ) / pt 1.0, fo 0.25 pt - pc < t c t t c 0.65( p - p ) / p 1.0, fo p - p whee p t 100A st /bd and p c 100A sc /bd, D is the total depth of the section, and l is the length of span.

8 Deflection Due to Ceep Clause C-4 of Annex C of S 456 stipulates the following method of calculating deflection due to ceep. The ceep deflection due to pemanent loads α is obtained fom the following equation: cc( (7.5) α cc ( α -α whee α initial plus ceep deflection due to pemanent loads obtained using an elastic analysis with an effective modulus of elasticity, E ce Ec /( 1+θ ), θ being the ceep coefficient, and α shot-tem deflection due to pemanent loads using E c Numeical Poblems Poblem 1: Figues and 2 pesent the coss-section and the tensile steel of a simply suppoted T-beam of 8 m span using M 20 and Fe 415 subjected to dead

9 load of 9.3 kn/m and imposed loads of 10.7 kn/m at sevice. Calculate the shotand long-tem deflections and check the equiements of S 456. Solution 1: Step 1: Popeties of plain concete section Taking moment of the aea about the bottom of the beam y t (300)(600)(300) + ( )(100)(550) (300)(600) + ( )(100) mm g 300(429.48) (170.52) (70.52) 3 3 (11.384) (10) 9 mm 4 This can also be computed fom SP-16 as explained below: Hee, b f /b w 7.45, D f /D Using these values in chat 88 of SP-16, we get k g k 1 b w D 3 /12 (2.10)(300)(600) 3 /12 (11.384)(10) 9 mm 4 Step 2: Popeties of the cacked section (Fig ) f 0.7 (cl of S 456) 3.13 N/mm 2 c f ck

10 M f c g /y t 3.13(11.384)(10) 9 / knm E s N/mm 2 E c 5000 f ck (cl of S 456) N/mm 2 m E s /E c 8.94 Taking moment of the compessive concete and tensile steel about the neutal axis, we have (Fig ) b f x 2 /2 m A st (d x) gives (2234)(x 2 /2) (8.94)(1383)(550 x) o x x Solving the equation, we get x mm. z leve am d x/ mm 2234(72.68) (1383)( ) (10) 9 mm 4 M wl 2 /8 ( )(8)(8)/8 160 knm eff. (Eq. 7.1) M z x bw (1- ) ( ) M d d b ( ) ( ) ( So, eff 3.106(10) 9 mm 4.. But eff g ) ( ) Step 3: Shot-tem deflection (sec ) E 5000 (cl of S 456) N/mm 2 c f ck Shot-tem deflection (5/384) wl 4 /E c eff (1) (5)(20)(8) 4 (10 12 )/(384)( )(3.106)(10 9 ) mm

11 Step 4: Deflection due to shinkage (sec ) k ( pt - pc )/ pt 0.72(0.84) ψ k D -7 4 ε / (0.6599)(0.0003)/ (10) k (fom sec ) (2) α k 3 ψ l 2 (Eq. 7.3) (0.125)(3.2995)(10) -7 (64)(10 6 ) 2.64 mm Step 5: Deflection due to ceep (sec ) Equation 7.5 eveals that the deflection due to ceep α cc( can be obtained afte calculating α and α. We calculate α in the next step. Step 5a: Calculation of α Assuming the age of concete at loading as 28 days, cl of S 456 gives θ 1.6. So, E cc E c /(1 + θ ) /( ) N/mm 2 and m E s /E cc / Step 5b: Popeties of cacked section

12 Taking moment of compessive concete and tensile steel about the neutal axis (assuming at a distance of x fom the bottom of the flange as shown in Fig ): 2234(100)(50 + x ) (23.255)(1383)(450 - x ) o x mm which gives x mm. Accodingly, z leve am d x/ mm. 2234(100) 3 / (100)(62.92) (1383)( ) (12.92) 3 / (10 9 ) mm 4 M knm (see Step 2) M w pem l 2 /8 9.3(8)(8)/ knm. eff (1.2) - ( ) ( ) ( ) ( 2234 ) Howeve, to satisfy eff g, eff should be equal to g. So, eff g (10 9 ). Fo the value of g please see Step 1. Step 5c: Calculation of α α 5wl 4 /384(E cc )( eff ) 5(9.3)(8) 4 (10) 12 /384( )(11.384)(10 9 ) (3) mm Step 5d: Calculation of α α 5wl 4 /384(E c )( eff ) 5(9.3)(8) 4 (10) 12 /384( )(11.384)(10 9 ) (4) mm Step 5e: Calculation of deflection due to ceep α cc( α - α

13 (5) mm t is impotant to note that the deflection due to ceep α cc( can be obtained even without computing α. The elationship of α cc( and α is given below. α cc( α - α {5wl 4 /384(E c )( eff )} {(E c /E cc ) 1} α (θ ) Hence, the deflection due to ceep, fo this poblem is: α cc( α (θ ) 1.948(1.6) mm Step 6: Checking of the equiements of S 456 The two equiements egading the contol of deflection ae given in sec They ae checked in the following: Step 6a: Checking of the fist equiement The maximum allowable deflection 8000/ mm The actual final deflection due to all loads (see Eq.1 of Step 3) (see Eq.2 of Step 4) o.k (see Eq.5 of Step 5e) mm < 32 mm. Hence, Step 6b: Checking of the second equiement The maximum allowable deflection is the lesse of span/350 o 20 mm. Hee, span/ mm. So, the maximum allowable deflection 20 mm. The actual final deflection (see Eq.4 of Step 5d) (see Eq.2 of Step 4) (see Eq.5 of step 5e) mm < 20 mm. Hence, o.k. Thus, both the equiements of cl.23.2 of S 456 and as given in sec ae satisfied.

14 Pactice Questions and Poblems with Answes Q.1: Why is it essential to check the stuctues, designed by the limit state of collapse, by the limit state of seviceability? A.1: See sec Q.2: Explain shot- and long-tem deflections and the espective influencing factos of them. A.2: See sec Q.3: State the stipulations of S 456 egading the contol of deflection. A.3: See sec Q.4: How would you select the peliminay dimensions of stuctues to satisfy (i) the deflection equiements, and (ii) the lateal stability? A.4: See se A fo (i) and B fo (ii). Q.5: Check the peliminay coss-sectional dimensions of Poblem 1 of sec (Fig ) if they satisfy the equiements of contol of deflection. The spacing of the beam is 3.5 m c/c. Othe data ae the same as those of Poblem 1 of sec A.5: Step 1: Check fo the effective width b f l o /6 + b w + 6D f o spacing of the beam, whicheve is less. Hee, b f (8000/6) (100) 2234 < Hence, b f 2234 mm is o.k. Step 2: Check fo span to effective depth atio (i) As pe ow 1 of Table 7.1, the basic value of span to effective depth atio is 20. (ii) As pe ow 2 of Table 7.1, the modification facto is 1 since the span 8 m < 10 m. (iii) As pe ow 5 of Table 7.1, the modification facto fo the flanged beam is to be obtained fom Fig. 6 of S 456 fo which the atio of web width to flange width

15 300/ Figue 6 of S 456 gives the modification facto as 0.8. So, the evised span to effective depth atio 20(0.8) 16. (iv) Row 3 of Table 7.1 deals with the aea and stess of tensile steel. At the peliminay stage these values ae to be assumed. Howeve, fo this poblem the aea of steel is given as 1383 mm 2 (2-25T T), fo which p t A st (100)/b f d 1383(100)/(2234)(550) f s 0.58 f y (aea of coss-section of steel equied)/(aea of coss-section of steel povided) 0.58(415)(1) (assuming that the povided steel is the same as equied, which is a ae case). Figue 4 of S 456 gives the modification facto as 1.8. So, the evised span to effective depth atio 16(1.8) (v) Row 4 is concening the amount of compession steel. Hee, compession steel is not thee. So, the modification facto 1. Theefoe, the final span to effective depth atio mm. Accodingly, effective depth of the beam 8000/ mm < 550 Hence, the dimensions of the coss-section ae satisfying the equiements Refeences 1. Reinfoced Concete Limit State Design, 6 th Edition, by Ashok K. Jain, Nem Chand & Bos, Rookee, Limit State Design of Reinfoced Concete, 2 nd Edition, by P.C.Vaghese, Pentice-Hall of ndia Pvt. Ltd., New Delhi, Advanced Reinfoced Concete Design, by P.C.Vaghese, Pentice-Hall of ndia Pvt. Ltd., New Delhi, Reinfoced Concete Design, 2 nd Edition, by S.Unnikishna Pillai and Devdas Menon, Tata McGaw-Hill Publishing Company Limited, New Delhi, Limit State Design of Reinfoced Concete Stuctues, by P.Dayaatnam, Oxfod &.B.H. Publishing Company Pvt. Ltd., New Delhi, Reinfoced Concete Design, 1 st Revised Edition, by S.N.Sinha, Tata McGaw-Hill Publishing Company. New Delhi, Reinfoced Concete, 6 th Edition, by S.K.Mallick and A.P.Gupta, Oxfod & BH Publishing Co. Pvt. Ltd. New Delhi, Behaviou, Analysis & Design of Reinfoced Concete Stuctual Elements, by.c.syal and R.K.Ummat, A.H.Wheele & Co. Ltd., Allahabad, Reinfoced Concete Stuctues, 3 d Edition, by.c.syal and A.K.Goel, A.H.Wheele & Co. Ltd., Allahabad, 1992.

16 10. Textbook of R.C.C, by G.S.Bidie and J.S.Bidie, Wiley Easten Limited, New Delhi, Design of Concete Stuctues, 13 th Edition, by Athu H. Nilson, David Dawin and Chales W. Dolan, Tata McGaw-Hill Publishing Company Limited, New Delhi, Concete Technology, by A.M.Neville and J.J.Books, ELBS with Longman, Popeties of Concete, 4 th Edition, 1 st ndian epint, by A.M.Neville, Longman, Reinfoced Concete Designe s Handbook, 10 th Edition, by C.E.Reynolds and J.C.Steedman, E & FN SPON, London, ndian Standad Plain and Reinfoced Concete Code of Pactice (4 th Revision), S 456: 2000, BS, New Delhi. 16. Design Aids fo Reinfoced Concete to S: , BS, New Delhi Test 17 with Solutions Maximum Maks 50, Maximum Time 30 minutes Answe all questions. TQ.1: Explain shot- and long-tem deflections and the espective influencing factos of them. (10 maks) A.TQ.1: See sec

17 TQ.2: Check the peliminay dimensions of a singly einfoced ectangula cantileve beam of span 4 m (Fig ) using M 20 and Fe 415. (15 maks) A.TQ.2: (i) Fom ow 1 of Table 7.1, the basic value of span to effective depth atio is 7. (ii) Modification facto fo ow 2 is 1 as this is a singly einfoced beam. (iii) Assuming p t as 0.6 and aea of steel to be povided is the same as aea of steel equied, f s 0.58(415(1) N/mm 2. Fom Fig. 4 of S 456, the modification facto Hence, the evised span to effective depth atio is 7(1.18) (iv) Modification factos fo ows 4 and 5 ae 1 as thee is no compession steel and this being a ectangula beam. Hence, the peliminay effective depth needed 4000/ mm < 550 mm. Hence, o.k. TQ.3: Detemine the tensile steel of the cantileve beam of TQ 2 (Fig ) subjected to sevice imposed load of 11.5 kn/m using M 20 and Fe 415. Use Sp-16 fo the design. Calculate shot- and long-tem deflections and check the equiements of S 456 egading the deflection. (25 maks) A.TQ.3: Detemination of tensile steel of the beam using SP-16: Dead load of the beam 0.3(0.6)(25) kn/m 4.5 kn/m Sevice imposed loads 11.5 kn/m Total sevice load 16.0 kn/m Factoed load 16(1.5) 24 kn/m M u 24(4)(4)/2 192 knm Fo this beam of total depth 600 mm, let us assume d 550 mm. M u /bd 2 192/(0.3)(0.55)(0.55) kn/m 2 Table 2 of SP-16 gives the coesponding p t (0.015)/ Again, fo M u pe mete un as 192/ knm/m, chat 15 of SP-16 gives p t 0.68 when d 550 mm.

18 With p t 0.683, A st 0.683(300)(500)/ mm 2. Povide 4-20T to have 1256 mm 2. This gives povided p t 0.761%. Calculation of deflection Step 1: Popeties of concete section y t D/2 300 mm, g bd 3 /12 300(600) 3 /12 5.4(10 9 ) mm 4 Step 2: Popeties of cacked section f c (cl of S 456) 3.13 N/mm 2 y t 300 mm M f c g /y t 3.13(5.4)(10 9 )/ (10 7 ) Nmm E s N/mm 2 E c 5000 f ck (cl of S 456) N/mm 2 m E s /E c 8.94 Taking moment of the compessive concete and tensile steel about the neutal axis (Fig ): 300 x 2 /2 (8.94)(1256)(550 x) o x x

19 This gives x mm and z d x/ / mm. 300(168.88) 3 / (1256)( ) (10 9 ) mm 4 M wl 2 /2 20(4)(4)/2 160 knm eff (1.2) - ( ) ( ) (1- ) (1) (10 ) mm 4 This satisfies eff g. So, eff (10 9 ) mm 4. Step 3: Shot-tem deflection (sec ) E c N/mm 2 (cl of S 456) Shot-tem deflection wl 4 /8E c eff 20(4 4 )(10 12 )/8( )(2.1548)(10 9 ) mm (1) So, shot-tem deflection mm Step 4: Deflection due to shinkage (sec ) k (0.761)/ ψ k D -7 4 ε / (0.664)(0.0003)/ (10) k (fom sec ) (2) α k 3 ψ l 2 (0.5)(3.32)(10) -7 (16)(10 6 ) mm Step 5: Deflection due to ceep (sec ) Step 5a: Calculation of α gives Assuming the age of concete at loading as 28 days, cl of S 456

20 θ 1.6 So, E cc E c /(1 + θ ) N/mm 2 m E s /E cc / Step 5b: Popeties of cacked section Fom Fig , taking moment of compessive concete and tensile steel about the neutal axis, we have: 300 x 2 /2 (23.255)(1256)(550 - x) o x x solving we get x mm z d x/ mm 300( ) 3 /3 + (23.255)(1256)( ) (10) 9 mm 4

21 M 5.634( 10 7 ) Nmm (see Step 2) M w pem l 2 /2 4.5(4 2 )/2 36 knm eff ( ) ( ) (1- ) (1) (10 ) mm 4 Since this satisfies eff g, we have, eff (10 9 ) mm 4. Fo the value of g please see Step 1. Step 5c: Calculation of α α (w pem )( l 4 )/(8E cc eff ) 4.5(4) 4 (10) 12 /8( )(3.5888)(10 9 ) (3) mm Step 5d: Calculation of α α (w pem )( l 4 )/(8E c eff ) 4.5(4) 4 (10) 12 /8( )(3.5888)(10 9 ) (4) mm Step 5e: Calculation of deflection due to ceep α cc( α - α (5) mm Moeove: α cc( α (θ ) gives α cc( 1.794(1.6) mm. Step 6: Checking of the two equiements of S 456 Step 6a: Fist equiement Maximum allowable deflection 4000/ mm The actual deflection (Eq.1 of Step 3) (Eq.2 of Step 4)

22 Step 6b: Second equiement (Eq.5 of Step 5e) > Allowable 16 mm. The allowable deflection is lesse of span/350 o 20 mm. Hee, span/ mm is the allowable deflection. The actual deflection (Eq.4 of Step 5d) (Eq.2 of Step 4) (Eq.5 of step 5e) mm < mm. Remaks: Though the second equiement is satisfying, the fist equiement is not satisfying. Howeve the exta deflection is only 2.81 mm, which can be made up by giving cambe instead of evising the section Summay of this Lesson This lesson illustates the impotance of checking the stuctues fo the limit state of seviceability afte designing by the limit state of collapse. The shotand long-tem deflections along with thei espective influencing factos ae explained. The code equiements fo the contol of deflection and the necessay guidelines fo the selection of dimensions of coss-section ae stated. Numeical examples, solved as illustative example and given in the pactice poblem and test will help the students in undestanding the calculations clealy fo thei application in the design poblems.