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EAST WEST INSTITUTE OF TECHNOLOGYDEPARTMENT OF CIVIL ENGINEERINGDESIGN OF CONCRETE CYLINDRICAL SHELL ROOF By, GURURAJA B II SEMESTER , M TECH STRUCTURAL ENGINEERING PLAN AREA CROPPED

CONTENTS

1. INTRODUCTION 2. THEORY OF SHELLS 3. CLASSIFICATION 4. SCOPE AND OBJECTIVE 5. DIMENSIONS 6. EQUILIBRIUM EQUATIONS 7. STRAIN DISPLACEMENT EQUATIONS 8. STRESS- STRAIN RELATIONS 9. FORCE- DISPLACEMENT RELATIONS 10. DESIGN ASSUMPTIONS 11. RECOMMENDATIONS OF IS 2210:1988 12. ASCE METHOD OF ANALYSIS 13. MEMBRANE FORCES 14. BOUNDARY CONDITIONS 15. MEMBRANE ANALYSIS 16. BENDING ANALYSIS 17. RESULTANT STRESSES 18. STATIC CHECKS 19. DESIGN AND DETAILING 20. DRAWBACKS OF METHOD 21. CONCLUSION. 22. APPENDIX 23. REFERENCES THEORY OF SHELLS Thin shell: Curved structure in space in which the thickness is small compared to the radius and other dimensions. Generatrix, Directrix: A curve which moves parallel to itself over another stationary curve to generate a surface is called generatrix and the stationary curve the directrix.Gauss curvature: Product of two principal radii of curvature at any point on the surface of the shell.

Synclastic surface: A surface for which gauss curvature is positive.Developable surface: A surface for which gauss curvature is zero.Anticlastic surface: A surface for which gauss curvature is negative.Traverse : Structures provided to support the shell & preserve the geometry of the structure.

CLASSIFICATION OF SHELLSBased on thicknessThick shells d/R > 1/20 Thin shells d/R < 1/20Based on Gaussian CurvatureSynclasticAnticlasticBased on generatrixShells of Translation Shells of revolution

Cylindrical shells:

These are formed by a straight line generator moving along a specified directrix. The common types of cylindrical shells are the following with straight line generators :Circular cylindrical shells-Circular arc directrixCatenary cylindrical shells-Catenary directrixParabolic cylindrical shells-Parabola directrixElliptic cylindrical shells-Semi ellipse

THEORIES OF CYLINDRICAL SHELLSFlugges theoryGibsons theoryDKJs theoryDischingers theoryAas Jakobsons theoryHolands theoryVlassovs theoryLundgrens theoryASCE manual methodFinsterwalder s theorySchorers theorySCOPE/ OBJECTIVE

The study presents the introduction to ASCE method of design and analysis of shells according to manual 31.The study is initiated by a practical case study and design to cover a open theatre of a collage campus of area approaximately 25 x 14 m 0n plan.Objective is to present the accuracy of the method and applicability limitations to practical design solution.This paper also considers the criteria of IS 2210: 1988.And the main objective is to present an introduction to elastic design and analysis of RC structures.

MEMBRANE STRESS

MEMBRANE FORCES & MOMENTS

Equilibrium Equations(Tx/ x)+(1/R)(S/ )+X = 0

(1/R)*(T/ )+ (S/ x) (1/ R)*(Mx/ x)+ Y =0

(2Mx/ x2)+(1/R2)*(2Mx/ 2)+ (1/R)*(2Mx/ x )+ (1/R)*(2Mx/ x )+(T/R)+ Z = 0

(Mx /R)+(Tx- Tx) = 0 STRESS STRAIN RELATIONS

x = 1/Eh(Nx - Ny)y = 1/Eh(Ny -Nx)xy = Nxy/Gh E= Young's modulus = Poison's ratio G = Shear modulus h = Thickness of shell.

STRAIN DISPLACEMENT RELATIONSHIP Strains in middle surface can be found by knowing displacements in each co-ordinate direction. x = (1/A)*(u/x) + (v/AB)* (A/y) + (w/R1)

Y = (1/B)*(v/y) + (u/AB)* (B/y) + (w/R2)

xy = (1/A)*(v/x) + (1/B)*(u/y) (u/AB)*(A/y) (v/AB)*(B/x)

FORCE DISPLACEMENT RELATIONSHIPS

ASCE METHOD It makes use of the tables to find out resultant stresses of manual no 31.Determinate stresses analysis on shells Membrane forces determination.2. Indeterminate analysis to suit practical boundary conditions of shell. Bending/Correction analysis.

GENERAL ASSUMPTIONS 1.Stress in Z direction z is considered negligible 2. Points on line normal to middle surface before deformation remain on line normal to the middle surface even after deformation. 3. All displacements are small , hence change in geometry of shell will not affect the static equilibrium of shell. 4. material is homogeneous, isotropic and linear elastic. 5. All the stress along the thickness is constant.BOUNDARY CONDITIONSFOR CONSIDERED SHELL Cylindrical simply supported shell without edgebeam.HenceAt x = 0 and x = L, Tx = 0, T = 0At x = L/2 , S = 0Shear along the longitudinal edges are 0.

Geometric properties of RC ShellDESCRIPTIONVALUESk35 DEGREEEL17 mR24.41 mt0.10 mCONCRETE MIX AND STEEL GRADEM 25 AND Fe 415DEAD LOAD3.704 KN/sqmtrsFINISHES0.47 KN/sqmtrsTOTAL LOAD4.174 KN/sqmtrsYOUNGS MODULUS (E)25 X 10^6 KN/sqmtrsPOISSONS RATIO ()0.20 PARAMETER SELECTION FACTORSSHELL TYPE : r/L RADIUS r : should not lie in working platformTHICKNESS : min according to ASCE & IS 2210CHORD WIDTH : based on plan area and type of shellDIRECTRIX : based on crown , central height requirementSPAN : based on orientation and type of shell.

SHELL DIMENSIONSChord width B = 28 mRadius r =24.41 mDirectrix = circularRise h = 4.42 mHeight of crown at higher ground level = 3.2+4.42 = 7.62 mk = 35 degree Curvilinear length of shell = 29.822 mSpan L = 17 m

SYDNEY OPERA, 1973 PRESTRESSED CONCRETE

L,OCEANOGRAPHIC, VALENCEA,STEELFRC

Fedala water tank: hyperbolical-concrete silo, WATER STORAGE in MORROCCO

Dimension checks 1.r/L = 1.435 > 0.6 ( type 2 shell) short shell2. h > 1/10 B ; 4.42 > (0.1 * 28) = 2.8 ( safe)3. Radius = 24.41m > 7.62+3.71 =11.33m (acoustic design safe)4. Minimum thickness = 50mm and 75mm t = 100mm hence ok5. B max for short shell = 122 m 6. Semicentral angle k < 40 degree ( to neglect wind load)Load Combination

1.Dead load+ Live load 2.Dead load + Live load+ Wind load. 3.Dead load+ Live load + Earth quake load 4.Dead load+ Live load +Wind load +Earthquake load.

Live load acc to IS 875 for curved slab of > 10 deg = 0.75 kn/m2Equivalent dead load = Pd = 0.704 kn/m2Dead load = 3 kn/m2Finishes = 0.49 kn/m2Total design load UDL on curve surface = 4.174 KN/sqmtr

.FOURIER SINUSOIDAL LOADDesign load is considered as a Sinusoidal expressed as Fourier load as P for n=1,3,5.....We consider n=1 and 3 onlyFor n=1 and 3,P=(4/)*3 =3.82 KN/m2.

Equivalent dead loadConverted Live Load parallel to chord width to curved surfacePd=PL (sin k)/ k Pd=Equivalent Dead LoadPL= Live Loadk= Semi Central Angle.Pd =( 0.75*(sin 35)/( 35 radians))=0.704 KN/m2Now Total Design Load = Dead Load +Pd Design load, P = 3+0.704 +0.49 = 4.174KN/m2

MEMBRANE ANALYSISFrom Fourier loading and simply supported shells.Tx at x = 0 is 0 and max at x = L/2T at x= 0 is 0 and max at x = L/2S at x = L/2 is and max at x = 0 at endsCharacteristic of directorix =K= (s/(r*t*L2)^(1/4)) K = S/5.135 = (s/r)*(180/) s = distance of key points from edge to crown. s is taken at each one m on the curve length from springing till Crown. Into 15 parts and 16 points.And for K values are calculated .

MEMBRANE ANALYSISTO FIND DETERMINATE MEMBRANE FORCESTx = Nx = force in x directionTx = P*r*(L/r)2* (co-efficient Tx)*sin (x/L)T = N = force in tangential or lateral direction T = P *r*(co-efficient T)* sin (x/L)S = lateral shear force along the spanS = P*r*( L/r)*(co-efficient S)* cos (x/L)For shell considered with Fourier loading with n=1Loads varying as 0 to maximum at centre of span.Refer Table 1B of ASCE manual 31Parameters = k s = distance on curve from springing , longitudinal edge.

CALCULATION OF MEMBRANE FORCES FROM TABLE 1B MANUAL 31MEMBRANE FORCES AND DISPLACEMENTS V H Tx S T0.6710.469-9.25682-29.2383-83.36520000.750.433-9.7587-25.4734-88.14930000.82140.383-10.2048-21.5482-92.22080000.8830.321-10.6509-17.4629-95.57990000.9330.251-10.9297-13.2173-98.73530000.9690.17-11.1305-8.97176-100.2620000000.99240.087-11.2588-4.44583-101.38200010-11.29780-101.789CORRECTION ANALYSISTo suit the practical boundary condition of the shellApply correction opposite loads which create bending moment and forces in x, directions hence bending analysis. To find bending stresses and add to membrane stresses. No edge member to take longitudinal load at springings hence make the S and T = 0 at = 0 . Hence equations are at = 0 . T = 0 S = 0 Tx is taken care at edges by longitudinal tensionCALCULATION OF CORRECTION LOADS To satisfy above boundary condition apply TL and SL at either ends to satisfy static equilibrium of forces.TL = P*r*(co-efficient of T at s = 0) = 4.174*24.41* 0.819 = 83.445 KN/m2SL = P*r*(L/r)*(co-efficient of S at s = 0) = 4.174*24.41*(17/24.41)*0.365 = 25.899KN/m2 Find the Tx , T ,S and M due to TL & SL Table 3A for type 2 shells Parameters = K= s/5.135 and r*t/L2K = 0,0.1,0.2,0.4,0.8.. r*t/L2 = (24.41*0.1)/172 = 0.00845

CORRECTION CO-EFFICIENTSrt/L2 = 0.008k=s/5.135Tx CO TLTx CO SLT CO TLT CO SLS CO TLS CO SLM CO TLM CO SL016.2255.811001000.1940625.3822.990.7850.1036-1.9450.1910.13830.03790.388123-0.771.1770.3670.0988-2.316-0.187-0.03150.0410.5821850.776247-4.278-0.414-0.313-0.004-1.094-0.263-0.6617-0.4220.9703091.164371.3584321.552494-0.131-0.199-0.24-0.05670.7190.084-0.7409-0.1361.7465551.9406172.1346792.3287412.5228022.7168642.89346-0.0860.0220.050.0132-0.178-0.0250.12710.0227CORRECTION BENDING STRESSES S k=s/5.135 Tx OF TL T OF TL S OF TLM OF TL000-1353.9-83.4450012.34723511.11896-449.101-65.5043162.3005-1.1540424.69446922.2379264.25265-30.6243193.25860.26285237.04170433.35688000049.38893944.47584356.977726.1182991.288835.521556511.7361755.59480000614.0834166.713760000716.4306477.832720000818.7778888.9516810.931320.0268-59.9976.18244921.12511100.070600001023.47235111.189600001125.81958122.308600001228.16682133.427500001330.51405144.546500001432.86129155.6654000014.9134.99727165.78377.17627-4.1722514.85321-1.06059CORRECTION ANALYSIS BENDING STRESSES S k=s/5.135 T OF SL Tx OF SL S OF SL M 0F SL0000-150.473-25.899012.34723511.11896-2.6821-77.438-4.94671-0.0981624.69446922.23792-2.55783-30.48314.843113-0.1061937.04170433.35688000049.38893944.475840.10355610.722196.8114371.092938511.7361755.59480000614.0834166.713760000716.4306477.832720000818.7778888.951681.4679065.153901-2.175520.352226921.12511100.070600001023.47235111.189600001125.81958122.308600001228.16682133.427500001330.51405144.546500001432.86129155.6654000014.9134.99727165.7837-0.34173-0.569780.647475-0.05879STRESS RESULTANTSSk=s/5.135 Tx RESULT T RESUL S RESULT M RESULT000-1495.11-0.079813.3393250157.29611.11896-526.539-68.1864157.3538-1.25222114.59222.2379243.5282354.96713223.57510.1566663171.88833.3568800004229.18444.47584377.9047118.4427119.64856.6144935286.4855.594800006343.77666.7137610.6509295.5798717.4628907401.07277.8327200008458.36888.9516827.01494120.23-48.95516.5346669515.664100.0706000010572.96111.189611.13049100.26228.97176011630.256122.3086000012687.552133.4275000013744.848144.546511.25875101.38184.445828014802.144155.6654000014.91854.2834165.783717.9042897.2750215.50069-1.11938VARIATION OF T v/s sVARIATION OF S v/s sVARIATION OF Tx v/s Tx v/s sVARIATION OF M V/S sVARIATION OF M & T V/S sRESULTANT STRESS over sRESULTANT STRESSES AT X=0 FOR S & X= L/2Tx RESULTT RESULS RESULTM RESULT-1495.11-0.079813.3393250-526.539-68.1864157.3538-1.2522015643.5282354.96713223.57510.15666585377.9047118.4427119.64856.6144934510.6509295.5798717.46289027.01494120.23-48.95516.5346664511.13049100.26228.97176011.25875101.38184.445828017.9042897.2750215.50069-1.11937668RESULTANT STRESS AT X= L/4 Tx RESULTT RESULS RESULTM RESULT-1057.96-4.5E-051E-050-372.322-48.216132.12188-0.8853829.9938938.926861.794020.110767266.399783.8145638.348234.6768396.67411367.6502610.93793018.2228785.0832-6.427034.6203676.97462370.964335.61948807.05499271.756822.784657011.7509868.853023.717641-0.79146VARIATION OF SHEAR AT X=0 AND X=L/4REINFORCEMENTS

Longitudinal reinforcement for TxThis is designed by considering max allowable stress in steel Transverse reinforcement for T & MTension moments & tensile lateral force is taken care by these bars D Diagonal reinforcement for shear S & this also should take care of the principal tension caused due to Tx, T & S Principal tension is given by N1,2=(Tx+T )/2 sqrt{((Tx+T )/2)2 +S2

Direction of principal tension is given by Tan2=2S/(Tx+T )

DESIGN OF REINFORCEMENTDESIGN VALUESAT X=0 FOR S & X= L/2Tx=377.9047 at s=4mT=120.23 at s=8mM=6.61449345 at s=4mS= 223.5751 at s=3mAT X= L/4Tx=266.3997 at s=4mT=85.0832 at s=8mM=4.676839 at s=4mS= 61.79402 at s=3m

CALCULATIONSAllowable stress in steel is taken as fs=140 N/mm2 Longitudinal reinforcement Ast=Tx(max)/fs=2699.785mm2Transverse reinforcement for moment Ast= M (max)* 100/(fs*0.87*d)=72.40mm2 Diagonal steel as simply supported hence at X=0, Tx & T = 0 hence principle tension = S = 223.5751 Kn/mm2 Ast= S/fs =1596.928 mm2 DetailingDESCRIPTION

DISTANCE (m)

DESIGN FORCEAst mm2

DIAMETER mm

SPACING mm(C/C)LONGITUDINAL BARSFROM s= 6 to 14.91m sp = 600 mm X= L/2 to L/4377.90 KN/m2699.785 16 75 X= L/4 to L/8266.3997KN/m1349.8916150X= L/8 to 094.47 KN/m674.9516300TRANSVERSE BARS s=0 to 3m tops=3 to 14.91m bottom X= L/2 to L/4

120.23KN/m858.785

1090X= L/4 to L/8

85.0832 KN/m429.3910180DIAGONAL STEEL @ 45 TO Tx BARS X= 0 to L/4223.57KN/m1596.92816125X= L/4 to L/2111.785 KN/m798.46416250REINFORCEMENT IN PLAN

SECTIONAL VIEW OF REINFORCEMENT

DRAW BACKSThis method gives the force coefficients at fixed key points .hence stress values at any required points on the shell cannot be analyzed it gives only variations.It gives an approximate analysis of the bending stress upon corrective loads on boundary conditions.It does not give rectification for random distribution values.CONCLUSIONAnalysis done for the Fourier loading 1st n value gives acceptable design forces throughout the section of the shell.For more accurate results same analysis for n=3 has to be done.For the validation of this method further model analysis of the proposed design & detailing has to be done.

REFERENCESVarghese P C, Limit state design of reinforced concrete,PHI Learning,New Delhi,2005.Chandrasekaran K,Analysis of thin concrete shells, Tata McGraw-Hill,NewDelhi,1986. Ramaswamy G S, Design & construction of concrete shell roof, Tata McGraw-Hill,NewYork,1968.ASCE Manual 31IS 2210-1988IS 875-1998 PART-1 AND PART-2 THANK YOUSheet1Sk=s/5.135S/Rk - CO EFF TxCO EFF SCO EFF T000035-0.166-0.365-0.81910.19406171160.04096681692.347234739932.652765260120.38812342320.08193363384.694469479730.3055305203-0.175-0.318-0.86630.58218513490.12290045067.041704219627.958295780440.77624684650.16386726759.388938959425.6110610406-0.183-0.269-0.90650.97030855810.204834084411.736173699323.263826300761.16437026970.245800901314.083408439220.9165915608-0.191-0.218-0.93971.35843198140.286767718116.43064317918.56935682181.5524936930.32773453518.777877918916.2221220811-0.196-0.165-0.9791.74655540460.368701351921.125112658713.8748873413101.94061711620.409668168823.472347398611.5276526014-0.1996-0.112-0.985112.13467882790.450634985725.81958213859.1804178615122.32874053950.491601802528.16681687836.8331831217132.52280225110.532568619430.51405161824.4859483818-0.2019-0.0555-0.996142.71686396270.573535436332.8612863582.13871364214.912.89346012030.610815239734.99726997130.0027300287-0.20260-1