Concept of Reinforced Cement Concrete Design Based on Working Stress Method (WSM) Suraj Kr Ray B. Tech (civil)

Transcription

1 Concept of Reinforced Cement Concrete Design Suraj Kr Ray B. Tech (civil)

2 Concept of Reinforced Cement Concrete Design (RCC) Concrete: Concrete is a substance which obtained by mixing Cement, Sand, Aggregates & water in a suitable proportion. And it can be cast in any desired shape & size for bear external loads. Reinforced Cement Concrete (RCC): Concrete is strong in Compression but weak in tension. Tensile Strength of concrete can be increase by providing steel inside concrete according to requirement. So, it can be define as: Reinforced Cement Concrete is a combined substance in which concrete forms a layer surrounding Steel and also, concrete and steel both are hardly bonded to each other is termed as Reinforced Cement Concrete (RCC). Advantage and Disadvantage of Reinforced Cement Concrete (RCC) Advantage of RCC: It has high compressive strength as compare of simple Concrete. It has better fire resistance capacity than steel. It needs less maintenance as compare of steel. It can be cast in any required shape at site too. By using steel in concrete cross-sectional dimension of structure can be reduced. In some structures like pile, dams, etc it is more economical. Disadvantage of RCC: It needs mixing, casting and curing, all of which affect the final strength of concrete. The cost of the forms used to cast concrete is relatively high. It has low compressive strength as compared to steel. In columns/beams of multi-storey buildings Cracks develop in concrete due to shrinkage and the application of live loads. Page 1 of 22

3 Singly Reinforced Beam Based on Working Stress Method Structure of Concrete containing Reinforcements (RCC Structures): Slab: A concrete slab is common element of structures/ buildings. It presents as horizontally in form of roof or floors. They may carry gravity load as well as lateral loads. Depth of slabs is very small as compare of its length and width. Beams: A member of structure in which length is greater than other two Dimensions (breadth, depth) is termed as Beam. Column: A member of structure in which Depth is greater than other two Dimensions (breadth, length) is termed as Column. Frames: Walls: A frame is structural member consists of Slab, Beams and columns. Walls are vertical plate elements resisting gravity as well as lateral loads e.g. retaining walls, basement walls etc. Page 2 of 22

4 Concept of Reinforced Cement Concrete Design (RCC) Loads working on structures: Members of structure must be design for support specific loads. The loads which used to occur on structural member are divided into three categories, 1. Dead loads 2. Live loads 3. Environmental loads Dead loads: A load which is constant in magnitude and fixed in location throughout the lifetime of the structure is termed as Dead loads. E.g. load of beams, columns, walls, slab, etc Live loads: A load which is get changing in magnitude and varies in location with certain time is termed as Live loads. Environmental loads: Consists mainly of snow loads, wind pressure and suction, earthquake loads (i.e inertial forces) caused by earthquake motions. Soil pressure on subsurface portion of structures, loads from possible ponding of rainwater on flat surfaces and forces caused by temperature differences. Like live loads, environmental loads at any given time are uncertain both in magnitude and distribution. Reinforced Cement Concrete Design Methods Design of is an art to provide a safe, economical, serviceable & functional structure. According to art of design and uses of materials there are following methods: 1. Ultimate/Strength design Method 2. Working stress Design Method 3. Limit State Design Method Page 3 of 22

5 Singly Reinforced Beam Based on Working Stress Method WORKING STRESS METHOD OF DESIGN Such type of system of design in which we consider the elastic limit in all calculation is termed as Working Stress Method of design. In short it is called WSM design method. The base of WSM is the concept of modular ratio, which is the ratio of young s modulus of steel by young s modulus of concrete and it is adopted as m =. ASSUMPTIONS FOR THE WORKING STRESS METHOD OF DESIGN:- Plane section remains plane before and after bending. Concrete in tension zone is ignored during the analysis of beam. Obey Hook s law so the stress-strain relationship of steel and concrete will be straight line under working load. Steel and concrete form a composite structure. SOME SYMBOLES USED IN RCC :- cbc = Allowable stress in concrete in bending in compression in direct stress. st = Allowable stress in steel in bending in tension in direct stress. sc = Allowable stress in steel in bending in compression in direct stress. y = Characteristic strength of steel. ck = Characteristic strength of concrete. A st = Area of steel in tension zone. A sc = Area of steel in compression zone. m = Modular ratio (m = 280/3 cbc) ECC = effective concrete cover CCC = clear concrete cover Page 4 of 22

6 Concept of Reinforced Cement Concrete Design (RCC) Methods of R.C.C. design Working Stress Method Limit Stress Method 1. The stress in an element is obtained from 1. The stress is obtained from design load The working load and compare with and compare with design strength. Permissible stresses. (Load = working load considered.) (Load = Working load x Load factor) 2. This method follows linear stress-strain 2. In this method, it follows linear Behaviour of both materials. Relationship (one of the major diff.). 3. Modular ratio can be used to determine 3. The ultimate stresses of materials allowable stresses. itself are used as allowable stress. 4. Material capabilities are under estimated 4. The material capabilities are not under to large extent. Factor of safety are used estimated as much as they are in WSM. in working stress method. Partially factor of safety are used. Allowable stress in conc., cbc = ck/fos cbc ck/partial FOS Allowable stress in steel, st = y/fos st = y/partial FOS 5. Its result gives an uneconomical section. 5. This method of design gives economical section. Page 5 of 22

7 Singly Reinforced Beam Based on Working Stress Method Types of RCC sections Balanced Section Unbalanced section Under Reinforced Over Reinforced Page 6 of 22

8 Concept of Reinforced Cement Concrete Design (RCC) Balanced sections Under reinforced sections Over reinforced sections Such type of section In which concrete and Steel attain its Permissible strength is termed as balanced section or critical section or economical section. Such types of section in which steel attain its permissible stress but concrete attain stress lower than its permissible stress. Such type of section in which concrete attain its permissible strength but steel remain below to its permissible strength. This type sections occurs when amount of provided steel is neither less nor more than the steel required for a critical section. In this type of sections critical Neutral axis and Actual neutral axis are same line. This type of sections occurs when area of provided steel is less than the area of steel required for balanced section. Actual Neutral axis remains above than the critical neutral axis. This type sections occurs when area of provided steel is more than the area of steel required for balanced section. Actual neutral axis remains below than critical neutral axis. Actual neutral axis Actual Critical NA NA Critical Actual NA NA Stress diagram Stress diagram Stress diagram cbc cbc cbc st/m st/m cbc< cbc st/m st< st Moment of resistence, MR=bn c cbc = st A st (d n/3) = Qbd 2 Moment of resistence, MR=bn cbc = st A st (d n/3) Moment of resistence, MR=bn cbc = st A st (d n/3) Page 7 of 22

9 Singly Reinforced Beam Based on Working Stress Method SOME FORMULAS AND THEIR DERIVATIONS:- Neutral axis:- Neutral axis is an imaginary axis about which the moment of area of tension zone and compression zone is equal. In fig; b= Breadth of cross-section n Compression zone d= Effective depth of cross-section d n= Depth of neutral axis NA Ast= steel area provided to that section Now, Total area of compression zone= bn Tension zone b Total area of tension zone= Ast (in tension zone concrete also present but it is not considered) Taking moment of area of compression zone about neutral axis: =Area of compression zone x distance of C.G. (compression zone) from NA = (b.n).n/2 b Again taking moment of tension zone about NA = Area of tension zone x distance of C.G. (tension zone) from NA = ma st x (d-n).. From assumption of R.C.C. bn(n/2) = mast (d-n) Page 8 of 22

10 Concept of Reinforced Cement Concrete Design (RCC) MOMENT OF RESISTANCE: When a beam bends under loads, its top fibres become shorter than its actual size and bottom fibres become larger than its actual size. Due to this bending max m compressive stress occur on the outer most top fibre and max m tensile stress occur on most outer bottom layer to prevent beam from this bending. There should be provided some additional strength to the beam. Such type of moment which works against the Bending Moment (B.M.) is termed as MOMENT OF RESISTANCE (M.R.). M.R. = Resultant compressive force x leaver arm = Resultant tensile force x leaver arm (Leaver arm is the distance between two points one of them is resultant tensile force and another one is resultant compressive force) Mathematically it can be represent as MR = = σst Ast This formula is for balanced section in case of unbalanced section In over reinforced section st will replace with st in under reinforced section cbc will replace with cbc for balanced section M.R. = Qbd 2 here Q = J = (1- ) (J is known as co-efficient of lever arm) K = or (here r = and m =.) % Steel = p = Page 9 of 22

11 Singly Reinforced Beam Based on Working Stress Method Balanced or Critical or Economical section:- A section of RCC in which most distant concrete fibre in compression and steel in tension attain its permissible stresses simultaneously, is termed as Balanced section. Since in this type of section, used concrete and steel both get fully utilized that s why it is also known as Economical or Critical section. Bending stress:- Bending stress is the longitudinal stress that is introduced at a point in a body subjected to loads (perpendicular to the length) that cause it to bend. In RCC beam compression zone get compressed and tension zone get elongate as a result of bending stress. Compression zone Tension zone Beam Before Bending Beam after Bending Neutral Axis (nc or n):- Neutral axis is the axis through a beam where the stress is zero; that is there is neither compression nor tension. Page 10 of 22

12 Concept of Reinforced Cement Concrete Design (RCC) Value of Neutral axis (nc) for Balanced section:- = = = 1+ = Let =r (. is constant for any grade of steel & concrete) = = = ) d Let = k (k =co-efficient of Neutral axis).. (For a critical section) Page 11 of 22

13 Singly Reinforced Beam Based on Working Stress Method Moment of Resistance (M.R.) = Qbd 2 for balanced/critical section:- MR = b n c cbc = b k d cbc (. n c = kd) = b k d cbc = b d 2 [ j k cbc ] (let = j = co-efficient of lever arm for a critical section) MR = Q b d 2 ( Q = j k cbc ) Percentage of steel for a Critical section (p):- Taking moments of area of concrete and steel about Neutral axis: Area of concrete x distance of CG from NA = Area of steel x distance of CG from NA b n c x = m A st x ( d - n c ) b = m A st x ( d - n c ) b = m A st x ( d - kd ) b = m A st x d( 1 - k) Page 12 of 22

14 Concept of Reinforced Cement Concrete Design (RCC) b = m A st x d( 1 - k) b = m A st x ( 1 - k) = p = x 100 = TYPES OF PROBLEMS IN SINGLY REINFORCED BEAM: 1. To find Moment of resistance of singly reinforced beam. 2. To find stress in steel and concrete if M.R. is given. 3. To design a beam for given conditions. For solving problems Step 1: Step 2: Write down all given values under a title Given Data. Determine actual depth of Neutral axis (i.e. n) for given section by taking Moment of Compression zone & Tension zone about NA. Step 3: Step 4: b = m A st x ( d - n ) Calculate Critical NA (i.e. n c). For this we can use this relationship n c = k d or, = (this relationship came from similar triangle of stress diagram) Compare values of n and n c If n < n c ; section is under reinforced use this formula MR=bn cbc = st A st (d n/3) If n > n c ; section is over reinforced use this formula MR=bn cbc = st A st (d n/3) Page 13 of 22

15 Singly Reinforced Beam Based on Working Stress Method Example 1: The cross-section of a singly reinforced concrete beam is 300 mm wide and 500 mm depth to the centre of the tension reinforcement which consist of four bars of 16 mm diameter. If the stress in concrete and steel are not exceed 7 N/mm 2 and 140 N/mm 2 respectively, determine the moment of resistance of the beam. Take m = Sol: Given Data: b= 300mm = wide of beam d= 500mm = depth of beam N= 4 = no. of bars Ø= 16mm = dia of bars A st= no. of bars x d 2 = 4 x 16 2 = mm 2 = 7 N/mm 2 = 140 N/mm 2 m= To find Actual Natural axis (n):- b = m A st x ( d - n ) 300 x = x 804 x (500 n) 150 n 2 = n 150 n n =0 n = mm ( another value of n ( mm) is not possible so ignored) Page 14 of 22

16 Concept of Reinforced Cement Concrete Design (RCC) To find critical Natural axis (n c):- n c = kd =0.4 x 500 = 200mm ( k = = = = ) Comparison between n and n c:- n = mm < n c = 200 mm So, it is the case of under-reinforced To find MR:- MR = st A st = 140 x 804 x = N-mm.. MR = 50.4 KN-m Example 2 :- The moment of resistance of a rectangular singly reinforced beam of width b mm and effective depth d mm is 0.9bd 2 N-mm. If the stress in the outside fibbers of concrete and steel do not exceed 5 N/mm 2 and 140 N/mm 2 respectively and the modular ratio equals 18, determine the ratio of depth of natural axis from the extreme compression fibres to the effective depth of the beam and the ratio of the area of the tensile steel to the effective area of the beam. sol n :- Page 15 of 22

17 Singly Reinforced Beam Based on Working Stress Method Given Data:- b = b mm = width of beam d = d mm = effective depth of beam MR= 0.9bd 2 N-mm 5 N/mm N/mm 2 m= 18 To find:- 1. =? 2. =? For Balanced section: MR = Qbd 2 = 0.87 bd 2 m = r = K = j = 1 Q = j k 5 N/mm N/mm But given MR = 0.90 bd 2 Given MR ( = 0.90 bd 2 ) > MR for Balanced section (0.87 bd 2 ).. Given section is Over reinforced So Concrete will attain its permissible stress earlier than steel Now, MR = 0.9 bd 2 = 0.90 d 2 = 1.08 d 2 = 3dn n = 1.08 = 3n n 2 (Let ) Page 16 of 22

18 Concept of Reinforced Cement Concrete Design (RCC).... = = = Now from stress diagram: BOC AOD.. Since section is Over reinforced So, concrete will reach at its permissible stress Which is 2.. [. ( ].. 2 i.e. For equilibrium Resultant Compression Force must be equal to Resultant Tension Force ( = = Page 17 of 22

19 Singly Reinforced Beam Based on Working Stress Method Example 3 :- Find the percentage of tensile reinforcement necessary for a singly reinforced balanced rectangular section if the permissible stress in concrete and steel are c and t Newton/mm 2 respectively and modular ratio is m. sol n :- Given Data: 2 2 m = m for equilibrium; Total compression force = Total tension force i.e. (1) for balanced section; n c = k d = = = = Putting this value of n c in eq n (1).. Required percentage of Reinforcement = Page 18 of 22

20 Concept of Reinforced Cement Concrete Design (RCC) Example 4 :- A reinforcement concrete beam 300mm wide by 600mm total depth has a span of 8 metres. Find the necessary tension reinforcement at the centre of the span to enable the beam to carry a load of 6000N/m in addition to its own weight; sol n : Consider cover below the steel centre= 40mm Weight of beam = 25000N/m 3 Permissible stress in concrete = 7N/mm 2 Permissible stress in steel = 230N/mm 2 Modular ratio = Self weight of beam:- Weight of beam = 25000N/m 3.. Weight of unit length of beam = x 1 x b x d = x 1 x x N/m = 4500 N/m Total load resist by beam = U. D.L. + self weight of beam W = = N/m Max bending moment (B.M.) for a UDL = BM max = = N-m = 84 x 10 6 N-mm For M 20 & Fe 415 ; j = 0.9 Moment or resistance ( MR ) = Stress Force x Lever arm MR = st A st MR = st A st (. n c = kd) MR = st A st x jd ( ) Page 19 of 22

21 Singly Reinforced Beam Based on Working Stress Method Example 5 :- MR = st A st x jd.. A st = 2 Design a rectangular beam section for a simply supported beam of 7m span and having a UDL of 35KN/m run by working stress method of design. Use M 20 & Fe 415 steel rods. Sol n : Given Data: 7m (length (or span) of the beam) UDL = 35 KN/m For M 20 & Fe 415 m = r = K = j = 1 Q = j k 7 N/mm N/mm Max m Bending Moment for given UDL = = = KNm = x 10 6 Nm Since this is a Design Problem & during design always consider a balanced section. So; for a Balanced section: MR = Q b d 2 = 0.9 b d 2 (Since Breadth and depth of Beam is not defined so it depends upon the designer that what value he want to take A/C to experience.) For a balanced section,.. MR = 0.9 x we assumed, b = x 10 6 = 0.9 x.. b = = mm So we take; b = 390 mm d = 780 mm Area of steel = = 2 Page 20 of 22

22 Suppose provide 20 mm bars Area of one bar = =314 mm 2 No of Bars required = So providing 5 bars of 20 mm diameter Area of steel provided = 5 x 314 = 1570 mm 2 2 Hence ok Check for the designed beam: From assumption: Moment of area of compression zone about NA = Moment of area of tension zone about NA.. Concept of Reinforced Cement Concrete Design (RCC) n 2 = (13.33 x 1570 x 780) (13.33 x 1570 x n) For balanced section: n c = kd = 0.22 x 780 = mm n n c It is clear that concrete will get its permissible strength before steel so; MR = = = KNm > KNm Hence ok. Page 21 of 22