Design of Circular Beams

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1 The Islamic University of Gaza Department of Civil Engineering Design of Special Reinforced Concrete Structures Design of Circular Beams Dr. Mohammed Arafa 1

2 Circular Beams They are most frequently used in circular reservoirs, spherical dome, curved balconies etc. Circular beams loaded and supported loads normal to their plans. The center of gravity of loads does not coincide with the centerline axis of the member. Dr. Mohammed Arafa

3 Circular Beams Circular beam are subjected to torsional moments in addition to shear and bending moment The torsional moment causes overturning of the beams unless the end supported of the beams are properly restrained. Dr. Mohammed Arafa 3

4 Semi-circular beam simply supported on three equally spaced supports Dr. Mohammed Arafa 4

5 Semi-circular beam supported on three supports Supports Reactions Let R A = R C = V 1 R B = V Taking the moment of the reaction about the line passing through B and parallel to AC M 0 V 1R wr R WR V 1 R V V F y 0 WR wr Rw Dr. Mohammed Arafa 5

6 S.F and B.M at Point P at angle from support C Dr. Mohammed Arafa 6

7 Semi-circular beam supported on three supports Taking the bending moment at point P at angle from the support say, C equal: R sin / M V 1R sin wr sin / / wr M R sin wr sin / M wr sin sin / Dr. Mohammed Arafa 7

8 Semi-circular beam supported on three supports Maximum negative moment occurs at point B M M at M max. B ve B ( ) 0.49wR Maximum Positive moment dm Max. M ve where ( 0) d max dm wr cos sin / cos / 0 d tan 9 43' M wR Dr. Mohammed Arafa 8

9 Semi-circular beam supported on three supports Torsional Moment at point P R sin / T V 1 R R cos wr R cos / / T wr cos sin dt The section of maximum torsion 0 d dt 0 at 59 6' d T 0.104wR max. Dr. Mohammed Arafa 9

10 Circular beams loaded uniformly and supported on symmetrically placed columns A B Dr. Mohammed Arafa 10

11 Circular beams at symmetrically placed columns Dr. Mohammed Arafa 11

12 Circular beams at symmetrically placed columns Consider portion AB of beam between two consecutive columns A and B be with angle The load on portion AB of beam =wr The center of gravity of the load will lie at a distance R sin( /) ( /) from the center Let M 0 be the bending moment and V 0 be the shear at the supports Dr. Mohammed Arafa 1

13 Circular beams at symmetrically placed columns From geometry V 0 wr / The moment M 0 at the support can be resolved along the line AB and at right angle to it. The component along line AB is M 0 sin( /) and 0 at right angle of it. Taking the moment of all forces about line AB M cos( /) Dr. Mohammed Arafa 13

14 Circular beams at symmetrically placed columns Taking the moment of all forces about line AB R sin / M 0 sin / wr R cos / / M M 0 0 wr wr sin / cos / sin / / sin / 1 cot / Dr. Mohammed Arafa 14

15 S.F and B.M. at point P, at angle from the support Load between points A and P is wr Shearing force at point P wr V P wr wr Dr. Mohammed Arafa 15

16 Shear force and bending moment at point P The direction of vector representing bending moment at point P will act along line PO R sin / M V 0R sin M 0cos wr sin / / wr M wr wr M sin 1 cot / cos sin / wr sin cos cot / cos sin / since cos 1sin / M 1 sin cot / cos wr Dr. Mohammed Arafa 16

17 Shear force and bending moment at point P The direction of vector representing torsion at point P will act at right angle to line PO wr R sin / T M sin R R cos wr R cos / 0 / since R R cos R 1 cos R sin / T wr 1 cot / sin wr sin / wr 1 sin / cos / 1 T wr 1 cot / sin wr sin / wr 1 sin T T sin cot / sin sin / sin wr cot / sin sin / wr since sin / 1cos T cos sin cot / sin Dr. Mohammed Arafa 17 wr

18 Shear force and bending moment at point P To obtain maximum value of torsional moment wr R sin / T M 0 sin R R cos wr R cos / / dt wr 1 sin sin cot / cos 0 d M dt 0 d 1 sin sin cot / cos 0 wr i.e. point of maximum torsion will be point of zero moment Dr. Mohammed Arafa 18

19 Circular beams loaded uniformly and supported on symmetrically n placed columns # of Supports R V K K` K`` Angle for Max Torsion 4 P/4 P/ P/5 P/ P/6 P/ P/7 P/ P/8 P/ P/9 P/ P/10 P/ P/1 P/ P M M T max rw support Midspan KwR K '' wr K ' wr T max M max(+ve) Dr. Mohammed Arafa T max M max(-ve) V max ` / M max(-ve) V max O 19

Example 1 Design a semi-circular beam supported on three-equally spaced columns. The centers of the columns are on a circular curve of diameter 8m.

20 Example 1 Design a semi-circular beam supported on three-equally spaced columns. The centers of the columns are on a circular curve of diameter 8m. The beam is support a uniformly distributed factored load of 5.14 t/m in addition to its own weight. f 350 kg / cm and f 400 kg / cm ' c y min L r 6.8m h L / / cm use Beam 40 70cm o. w. of the beam =0.4(0.7)(.5)(1.) = 0.84 t/m Total load= =5.98 t/m Dr. Mohammed Arafa 0

21 Example 1 M max( ve ) 0.49wR 0.49(5.98)(4) t. m M max( ve ) wR (5.98)(4) t. m T max 0.104wR 0.104(5.98)(4) 9.97 t. m Reactions wr 5.98(4) R A 13.65ton R wr (5.98)(4) 47.84ton B wr Shear at point P: V wr Dr. Mohammed Arafa 1

22 Example 1 Shear Force Diagram Bending Moment Diagram Dr. Mohammed Arafa

23 Example 1 Design for Reinforcement d= =63.95 A A ve s ( ve ) ve s ( ve ) cm cm min min Dr. Mohammed Arafa 3

24 Example 1 Design for Shear V 3.9 ton ( T 0.0at middle support) V max c (40)(63.95) 19.0ton V S 6.53 ton 0.75 AV (63.95) S AV cm / cm S Dr. Mohammed Arafa 4

25 Example 1 Design for Torsion if T u 0.65 p cp f A ' c cp cp cp 0h 0 0 h neglect torsion A bh, p b h, A x y, p x y and A 0.85x y The following equation has to be satisfied to check for ductlity V u Tu p h V c '.1 f c bw d 1.7Aoh bw d Reinforcement AT Tu S f A ys f A A p A p ' AT c cp AT l h and l,min h S f y S Dr. Mohammed Arafa 5

26 Example 1 Design for Torsion At section of maximum torsion is located at =59.43 T=9.97 t.m wr V u wr (4)( ) 11.6ton 360 M 0.0 V (40)(63.95) 19.0ton V No shear reinf.required c f ' ' 0.65 f c Acp c p cp 1.3 t. m T i.e torsion must be considered u u Dr. Mohammed Arafa 6

27 Example 1 Design for Torsion: Ductility Check The following equation has to be satisfied 0 0 V u Tu p h Vc '.1 f c bwd 1.7Aoh bwd x y p x y cm A h oh cm 3 5 V u Tu p h kg / cm bwd 1.7Aoh V c 19.0 bd w '.1 fc kg / cm i.e section dimension are adequate for preventing brittle failure due to combined shear stresses. Dr. Mohammed Arafa 7

28 Example 1 Design for Torsion 5 AT Tu S f A A S T min ys cm / 3.5b w f 400 ys cm / AV AT cm / cm S S A 1.13 Use (closed stirrups) with cm S 11.5 cm cm / cm Dr. Mohammed Arafa 8

29 Example 1 Design for Torsion AT Al ph cm S l,min s, ve total s, ve total ' 1.3 f c Acp A T h f y S 400 A p cm Al 5.33cm 3 Use 0 top and 414skin reinforcement A A cm use 618mm cm use80mm Dr. Mohammed Arafa 9

30 Example 1 Design for Torsion Dr. Mohammed Arafa 30

31 Internal Forces in Circular beams Using polar coordinate dv w ds rd ds dv wr d dm ds dt dr dm ds dt V dr T r T V r as r = constant A B P dm T Vr d Dr. Mohammed Arafa 31

32 Internal Forces in Circular beams Using polar coordinate The component of the moment M along ds in the radial direction must be equal to the difference of the torsional moments along the same element dt Md dt M d Mds r This means that the torsional moment is maximum at point of zero bending moment Dr. Mohammed Arafa 3

33 Internal Forces in Circular beams Using polar coordinate Differentiating the equation dm T Vr d d M dt dv r d d d d d M M wr d Vr d r w The solution of this deferential equation gives M A sin B coswr Dr. Mohammed Arafa 33

34 Internal Forces in Circular beams Using polar coordinate M A sin B coswr The constants can be determined from conditions at supports The internal forces are in this manner statically indeterminate. In many cases, they can be determined from the conditions of equilibrium alone because the statically indeterminate values are either known or equal to zero. Dr. Mohammed Arafa 34

35 Circular beams loaded uniformly and supported on symmetrically n placed columns n rw R n 0 0 n rw V n The bending moment and shearing force V at any section at an angle from the centerline between two successive supports are given by: M r w n T r w n V cos 1 sin0 sin sin0 rw Dr. Mohammed Arafa 35

36 SEMICIRCULAR BEAM FIXED AT END SUPPORTS If a semicircular beam supports a concrete slab, as shown the ratio of the length to the width of the slab is r/r =, and the slab is considered a one-way slab. The beam will be subjected to a distributed load, which causes torsional moments in addition to the bending moments and shearing forces. The load on the curved beam will be proportional to its distance from the support AB Dr. Mohammed Arafa 36

37 SEMICIRCULAR BEAM FIXED AT END SUPPORTS Load on the curved beam per unit length wr sin w' Shear Force VA VB wr 0. 39wr 8 Bending Moment at support 3 3 wr M A MB 3 Torsional Moment at support T A 0.11wr Dr. Mohammed Arafa 37

38 SEMICIRCULAR BEAM FIXED AT END SUPPORTS Bending Moment at section N 1 cos M N wr sin T A M M N 1 cos wr sin Midspan wr at = Torsional Moment at section sin 1 cos 1 sin cos T N wr T A T N T Midspan 0.11sin 3 1 wr cos 1 sin 0.11cos at = N Dr. Mohammed Arafa 38

39 BMD TMD Dr. Mohammed Arafa 39

40 Fixed-end Semicircular Beam Under Uniform Loading Dr. Mohammed Arafa 40

41 Fixed-end Semicircular Beam Under Uniform Loading Dr. Mohammed Arafa 41

42 Fixed End Semicircular Beam with Uniform Loading Load on the curved beam per unit length Shear Force V V 1.57wr A B Bending Moment at support M M wr A B Torsional Moment at support T A 0. 3wr w Dr. Mohammed Arafa 4

43 Fixed End Semicircular Beam with Uniform Loading Bending Moment at section N M N wr T A M M N wr Midspan sin 1 sin 4 sin wr = Torsional Moment at section N cos T N wr cos T A T T N 4 cos 0 = wr Midspan Dr. Mohammed Arafa 43

44 Fixed End Circular Beam under Uniform Loading Dr. Mohammed Arafa 44

45 Fixed End Circular Beam under Uniform Loading Dr. Mohammed Arafa 45

46 Fixed End Semicircular Beam with Uniform Loading Load on the curved beam per unit length w Bending Moment at Centerline wr M C K K K K K K 4 C cos Torsional Moment at Centerline T C 0 Bending Moment at Section N M M wr N Torsional Moment T M sin wr N C 1 cos at Section N sin Dr. Mohammed Arafa 46

47 Fixed End Semicircular Beam with Uniform Loading Bending Moment at Support A M M wr A C cos 1 cos Torsional Moment at Support A T M sin wr A C sin Dr. Mohammed Arafa 47

48 Values of and J The value of G/E for concrete may be assumed to be equal to The Value of J 1 4 For circcular section J r For square section of side x J 0.141x For rectangular section of short side x and long side J K ' xy 3 4 y Dr. Mohammed Arafa 48

49 Values of and J For rectangular section of short side x and long side y Dr. Mohammed Arafa 49

50 Problems P1- Design a circular beam that supported on six equally spaced columns, and its centerline lies on a circle 7m in diameter. The beam carries a uniform dead load of 1 t/m` and live load of ' 6kg/m. Use f 300 kg / cm and f 400 kg / cm c P. Find the values of factored shear, moment and torsion at critical sections for the fixed end semicircular beam, if the diameter of the circle is 10 m. The beam is apart of floor slab that caries a uniform dead load (including its own) weight of 800 kg/m and live load of 00kg/m. P3. Find the values of factored shear, moment and torsion at critical sections for the 60 o V-shape beam, with a=8m. The beam carries a uniform dead load (including its own) weight of 10 t/m` and live load of 4t/m`. Assume the ratio of long to short side of the rectangular section is. y 50

Design of Circular Beams

The Islamic University of Gaza Department of Civil Engineering Design of Special Reinforced Concrete Structures Design of Circular Beams ENGC 6353 Dr. Mohammed Arafa 1 Circular Beams They are most frequently