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Page 1 Short Notes on Concrete Structures Working Stress Method Modular Ratio S C E m E = o m = Modular ratio o E S = Modulus of elasticity of steel o E C = Modulus of elasticity of concrete Equivalent Area of Concrete CS AmA = o A C = Area of concrete o AS = Area of steel Critical Depth of Neutral Axis (X C ) C mc X d tmc æö ÷ ç = ÷ ç ÷ ÷ ç èø + Here, D = Overall depth d = Efffective depth cbc s = c = permissible stress in concrete st s = t = permissible stress in steel Actual depth of Neutral axis (X a ) 2 () 2 a st a BX mA d x =- Special case : (i) when ac XX = for balanced section (ii) when ac XX > for over reinforced section (iii) when ac XX < for under reinforced section Doubly Reinforced Rectangular Section Critical depth of Neutral axis, (X C ) C mc Xd tmc =· + Actual depth of Neutral axis, (X a ) 2 (1.5 1) ( ) ( ) 2 a SC a c st a bX mAX d mAdx +- - = - Page 2 Short Notes on Concrete Structures Working Stress Method Modular Ratio S C E m E = o m = Modular ratio o E S = Modulus of elasticity of steel o E C = Modulus of elasticity of concrete Equivalent Area of Concrete CS AmA = o A C = Area of concrete o AS = Area of steel Critical Depth of Neutral Axis (X C ) C mc X d tmc æö ÷ ç = ÷ ç ÷ ÷ ç èø + Here, D = Overall depth d = Efffective depth cbc s = c = permissible stress in concrete st s = t = permissible stress in steel Actual depth of Neutral axis (X a ) 2 () 2 a st a BX mA d x =- Special case : (i) when ac XX = for balanced section (ii) when ac XX > for over reinforced section (iii) when ac XX < for under reinforced section Doubly Reinforced Rectangular Section Critical depth of Neutral axis, (X C ) C mc Xd tmc =· + Actual depth of Neutral axis, (X a ) 2 (1.5 1) ( ) ( ) 2 a SC a c st a bX mAX d mAdx +- - = - Singly Reinforced T-Section Effective width of flange ? For beam casted monolithic with slab 0 12 6 6 22 wf f w l bd B Minimum or ll b ìæö ï ÷ ï ç ÷ ++ ï ç ÷ ç ï ÷ ç èø ï ï ï ï = í ï ï ï ï ++ ï ï ï ï î ? For isolated T-beam 0 0 4 fw l Bb l B =+ æö ÷ ç + ÷ ç ÷ ç ÷ èø l 0 = Distance between points of zero moments in the beam B = Total width of flange b w = Width of web Critical depth of Neutral axis (X c ) C mc Xd tmc æö ÷ ç = ÷ ç ÷ ÷ ç èø + ? When Neutral axis is in flange area o Actual depth of Neutral axis 2 () 2 a st a BX mA d X =- Here, X a = Actual depth of Neutral axis ? Moment of resistance (M r ) When Neutral axis is in web area ? For actual depth of neutral axis Page 3 Short Notes on Concrete Structures Working Stress Method Modular Ratio S C E m E = o m = Modular ratio o E S = Modulus of elasticity of steel o E C = Modulus of elasticity of concrete Equivalent Area of Concrete CS AmA = o A C = Area of concrete o AS = Area of steel Critical Depth of Neutral Axis (X C ) C mc X d tmc æö ÷ ç = ÷ ç ÷ ÷ ç èø + Here, D = Overall depth d = Efffective depth cbc s = c = permissible stress in concrete st s = t = permissible stress in steel Actual depth of Neutral axis (X a ) 2 () 2 a st a BX mA d x =- Special case : (i) when ac XX = for balanced section (ii) when ac XX > for over reinforced section (iii) when ac XX < for under reinforced section Doubly Reinforced Rectangular Section Critical depth of Neutral axis, (X C ) C mc Xd tmc =· + Actual depth of Neutral axis, (X a ) 2 (1.5 1) ( ) ( ) 2 a SC a c st a bX mAX d mAdx +- - = - Singly Reinforced T-Section Effective width of flange ? For beam casted monolithic with slab 0 12 6 6 22 wf f w l bd B Minimum or ll b ìæö ï ÷ ï ç ÷ ++ ï ç ÷ ç ï ÷ ç èø ï ï ï ï = í ï ï ï ï ++ ï ï ï ï î ? For isolated T-beam 0 0 4 fw l Bb l B =+ æö ÷ ç + ÷ ç ÷ ç ÷ èø l 0 = Distance between points of zero moments in the beam B = Total width of flange b w = Width of web Critical depth of Neutral axis (X c ) C mc Xd tmc æö ÷ ç = ÷ ç ÷ ÷ ç èø + ? When Neutral axis is in flange area o Actual depth of Neutral axis 2 () 2 a st a BX mA d X =- Here, X a = Actual depth of Neutral axis ? Moment of resistance (M r ) When Neutral axis is in web area ? For actual depth of neutral axis ? Moment of resistance (M r ) Limit State Method Design stress strain curve at ultimate state ? Design value of strength o For concrete 0.67 0.45 1.5 ck dck mc f f ff g == = mc g = Partial factor of safety for concrete = 1.5 f d = design value of strength o For steel 0.87 1.15 y dy f ff == Singly Reinforced Beam ? Limiting depth of neutral axis (x u, lim ) ,lim 700 0.87 1100 u y x d f =´ + ? Actual depth of neutral axis (X u ) 0.87 0.36 yst u ck fA CT X fb =? = ? Lever arm = d – 0.42 X u ? Ultimate moment of resistance 0.36 ( 0.42 ) ucku u MfbXd X =- 0.87 ( 0.42 ) uyst u MfAd X =- Special cases 1. Under-reinforced section : X u < X u,lim 0.36 ( 0.42 ) ucku u MfbXd X =- 0.87 ( 0.42 ) uyst u MfAd X =- 2. Balanced section: X u = X u,lim ,lim ,lim 0.36 ( 0.42 ) ucku u MfbXd X =- ,lim 0.87 ( 0.42 ) uyst u MfAd X =- 3. Over reinforced section : X u > X u,lim X u limited to X u,lim Moment of resistance limited to (M u,lim ) Doubly Reinforced Section Page 4 Short Notes on Concrete Structures Working Stress Method Modular Ratio S C E m E = o m = Modular ratio o E S = Modulus of elasticity of steel o E C = Modulus of elasticity of concrete Equivalent Area of Concrete CS AmA = o A C = Area of concrete o AS = Area of steel Critical Depth of Neutral Axis (X C ) C mc X d tmc æö ÷ ç = ÷ ç ÷ ÷ ç èø + Here, D = Overall depth d = Efffective depth cbc s = c = permissible stress in concrete st s = t = permissible stress in steel Actual depth of Neutral axis (X a ) 2 () 2 a st a BX mA d x =- Special case : (i) when ac XX = for balanced section (ii) when ac XX > for over reinforced section (iii) when ac XX < for under reinforced section Doubly Reinforced Rectangular Section Critical depth of Neutral axis, (X C ) C mc Xd tmc =· + Actual depth of Neutral axis, (X a ) 2 (1.5 1) ( ) ( ) 2 a SC a c st a bX mAX d mAdx +- - = - Singly Reinforced T-Section Effective width of flange ? For beam casted monolithic with slab 0 12 6 6 22 wf f w l bd B Minimum or ll b ìæö ï ÷ ï ç ÷ ++ ï ç ÷ ç ï ÷ ç èø ï ï ï ï = í ï ï ï ï ++ ï ï ï ï î ? For isolated T-beam 0 0 4 fw l Bb l B =+ æö ÷ ç + ÷ ç ÷ ç ÷ èø l 0 = Distance between points of zero moments in the beam B = Total width of flange b w = Width of web Critical depth of Neutral axis (X c ) C mc Xd tmc æö ÷ ç = ÷ ç ÷ ÷ ç èø + ? When Neutral axis is in flange area o Actual depth of Neutral axis 2 () 2 a st a BX mA d X =- Here, X a = Actual depth of Neutral axis ? Moment of resistance (M r ) When Neutral axis is in web area ? For actual depth of neutral axis ? Moment of resistance (M r ) Limit State Method Design stress strain curve at ultimate state ? Design value of strength o For concrete 0.67 0.45 1.5 ck dck mc f f ff g == = mc g = Partial factor of safety for concrete = 1.5 f d = design value of strength o For steel 0.87 1.15 y dy f ff == Singly Reinforced Beam ? Limiting depth of neutral axis (x u, lim ) ,lim 700 0.87 1100 u y x d f =´ + ? Actual depth of neutral axis (X u ) 0.87 0.36 yst u ck fA CT X fb =? = ? Lever arm = d – 0.42 X u ? Ultimate moment of resistance 0.36 ( 0.42 ) ucku u MfbXd X =- 0.87 ( 0.42 ) uyst u MfAd X =- Special cases 1. Under-reinforced section : X u < X u,lim 0.36 ( 0.42 ) ucku u MfbXd X =- 0.87 ( 0.42 ) uyst u MfAd X =- 2. Balanced section: X u = X u,lim ,lim ,lim 0.36 ( 0.42 ) ucku u MfbXd X =- ,lim 0.87 ( 0.42 ) uyst u MfAd X =- 3. Over reinforced section : X u > X u,lim X u limited to X u,lim Moment of resistance limited to (M u,lim ) Doubly Reinforced Section ? Limiting depth of neutral axis ,lim 700 0.87 1100 u y Xd f =´ + ? For actual depth of neutral axis (X u ) 12 0.36 ( 0.45 ) 0.87 ck u sc ck sc y st CT C C T fbX f f A fA =? + = ? +- = ? Ultimate moment of resistance 0.36 ( 0.42 ) ( 0.45 ) ( ) ucku u sc cksc C MfbXd X f fAdd =- +- - f SC = stress in compression T-Beam ? Limiting depth of neutral axis ,lim 700 0.87 1100 u y Xd f =´ + ? Singly reinforced T-Beam o When NA is in flange area X u < D f o 0.87 0.36 yst uf ck f fA XD fb =< o Ultimate moment of resistance 0.36 ( 0.42 ) 0.87 ( 0.42 ) uckfu u u yst u MfbXd XorM fAd X =- = - o When NA is in web area X u > D f ? X u > D f and 3 7 fu DX < ? For actual depth of neutral axis 0.36 0.45 ( ) 0.87 ck w u ck f w f y st fbx f b b D fA +- = ? Ultimate moment of resistance 0.36 ( 0.42 ) 0.45 ( ) 2 f uckwu u ckfwf D Mfbxd x fbbDd æö ÷ ç =- + - - ÷ ç ÷ ç ÷ èø 12 0.87 ( 0.42 ) 0.87 2 f u y st u y st D MfAd x fAd æö ÷ ç =- + - ÷ ç ÷ ç ÷ èø 12 0.36 0.45 ( ) , 0.87 0.87 ck w u ck f w f st st yy fbx f b b D AA ff - == ? When Xu > Df and 3 7 fu DX > Page 5 Short Notes on Concrete Structures Working Stress Method Modular Ratio S C E m E = o m = Modular ratio o E S = Modulus of elasticity of steel o E C = Modulus of elasticity of concrete Equivalent Area of Concrete CS AmA = o A C = Area of concrete o AS = Area of steel Critical Depth of Neutral Axis (X C ) C mc X d tmc æö ÷ ç = ÷ ç ÷ ÷ ç èø + Here, D = Overall depth d = Efffective depth cbc s = c = permissible stress in concrete st s = t = permissible stress in steel Actual depth of Neutral axis (X a ) 2 () 2 a st a BX mA d x =- Special case : (i) when ac XX = for balanced section (ii) when ac XX > for over reinforced section (iii) when ac XX < for under reinforced section Doubly Reinforced Rectangular Section Critical depth of Neutral axis, (X C ) C mc Xd tmc =· + Actual depth of Neutral axis, (X a ) 2 (1.5 1) ( ) ( ) 2 a SC a c st a bX mAX d mAdx +- - = - Singly Reinforced T-Section Effective width of flange ? For beam casted monolithic with slab 0 12 6 6 22 wf f w l bd B Minimum or ll b ìæö ï ÷ ï ç ÷ ++ ï ç ÷ ç ï ÷ ç èø ï ï ï ï = í ï ï ï ï ++ ï ï ï ï î ? For isolated T-beam 0 0 4 fw l Bb l B =+ æö ÷ ç + ÷ ç ÷ ç ÷ èø l 0 = Distance between points of zero moments in the beam B = Total width of flange b w = Width of web Critical depth of Neutral axis (X c ) C mc Xd tmc æö ÷ ç = ÷ ç ÷ ÷ ç èø + ? When Neutral axis is in flange area o Actual depth of Neutral axis 2 () 2 a st a BX mA d X =- Here, X a = Actual depth of Neutral axis ? Moment of resistance (M r ) When Neutral axis is in web area ? For actual depth of neutral axis ? Moment of resistance (M r ) Limit State Method Design stress strain curve at ultimate state ? Design value of strength o For concrete 0.67 0.45 1.5 ck dck mc f f ff g == = mc g = Partial factor of safety for concrete = 1.5 f d = design value of strength o For steel 0.87 1.15 y dy f ff == Singly Reinforced Beam ? Limiting depth of neutral axis (x u, lim ) ,lim 700 0.87 1100 u y x d f =´ + ? Actual depth of neutral axis (X u ) 0.87 0.36 yst u ck fA CT X fb =? = ? Lever arm = d – 0.42 X u ? Ultimate moment of resistance 0.36 ( 0.42 ) ucku u MfbXd X =- 0.87 ( 0.42 ) uyst u MfAd X =- Special cases 1. Under-reinforced section : X u < X u,lim 0.36 ( 0.42 ) ucku u MfbXd X =- 0.87 ( 0.42 ) uyst u MfAd X =- 2. Balanced section: X u = X u,lim ,lim ,lim 0.36 ( 0.42 ) ucku u MfbXd X =- ,lim 0.87 ( 0.42 ) uyst u MfAd X =- 3. Over reinforced section : X u > X u,lim X u limited to X u,lim Moment of resistance limited to (M u,lim ) Doubly Reinforced Section ? Limiting depth of neutral axis ,lim 700 0.87 1100 u y Xd f =´ + ? For actual depth of neutral axis (X u ) 12 0.36 ( 0.45 ) 0.87 ck u sc ck sc y st CT C C T fbX f f A fA =? + = ? +- = ? Ultimate moment of resistance 0.36 ( 0.42 ) ( 0.45 ) ( ) ucku u sc cksc C MfbXd X f fAdd =- +- - f SC = stress in compression T-Beam ? Limiting depth of neutral axis ,lim 700 0.87 1100 u y Xd f =´ + ? Singly reinforced T-Beam o When NA is in flange area X u < D f o 0.87 0.36 yst uf ck f fA XD fb =< o Ultimate moment of resistance 0.36 ( 0.42 ) 0.87 ( 0.42 ) uckfu u u yst u MfbXd XorM fAd X =- = - o When NA is in web area X u > D f ? X u > D f and 3 7 fu DX < ? For actual depth of neutral axis 0.36 0.45 ( ) 0.87 ck w u ck f w f y st fbx f b b D fA +- = ? Ultimate moment of resistance 0.36 ( 0.42 ) 0.45 ( ) 2 f uckwu u ckfwf D Mfbxd x fbbDd æö ÷ ç =- + - - ÷ ç ÷ ç ÷ èø 12 0.87 ( 0.42 ) 0.87 2 f u y st u y st D MfAd x fAd æö ÷ ç =- + - ÷ ç ÷ ç ÷ èø 12 0.36 0.45 ( ) , 0.87 0.87 ck w u ck f w f st st yy fbx f b b D AA ff - == ? When Xu > Df and 3 7 fu DX > 0.15 0.65 fu ff yX DD =+ < ? For actual depth of neutral axis 12 0.36 0.45 ( ) 0.87 0.87 ck w u ck f w f y st y st fbX f b b y fA fA +- = + 0.36 0.45 ( ) 0.87 ck w u ck f w f y st fbX f b b y fA +- = Design Beams and Slabs and Columns Effective span Simply supported beams and slabs ( l eff ) ? 0 0 minimum eff lw l ld ì + ï ï = í ï + ï î Here, l 0 = clear span w = width of support d = depth of beam or slab For continuous beam ? If width of support 1 12 < of clear span 0 0 minimum eff lw l ld ì + ï ï = í ï + ï î ? If width of support 1 12 > of clear span o When one end fixed other end continuous or both end continuous. l eff = l 0 o When one end continuous and other end simply supportedRead More
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FAQs on Concrete Structure Formulas for Civil Engineering Exam - RCC & Prestressed Concrete - Civil Engineering (CE)
| 1. What are the formulas commonly used in designing concrete structures for civil engineering exams? |  |
Ans. Some of the commonly used formulas in designing concrete structures for civil engineering exams include:
- The formula for calculating the required area of steel reinforcement, which is given by As = (0.85 * fy * bd) / fy.
- The formula for calculating the moment of inertia of a rectangular section, which is given by I = (bd^3) / 12.
- The formula for calculating the section modulus of a rectangular section, which is given by S = (bd^2) / 6.
- The formula for calculating the compressive strength of concrete, which is given by f'c = 0.85 * fc.
- The formula for calculating the bending moment in a simply supported beam, which is given by M = (wL^2) / 8.
| 2. How do I calculate the required area of steel reinforcement in a concrete structure? | |
Ans. The required area of steel reinforcement in a concrete structure can be calculated using the formula As = (0.85 * fy * bd) / fy, where As is the required area of steel reinforcement, fy is the yield strength of the steel, b is the width of the concrete section, and d is the effective depth of the section.
| 3. What is the formula for calculating the moment of inertia of a rectangular section in concrete structures? | |
Ans. The formula for calculating the moment of inertia of a rectangular section in concrete structures is given by I = (bd^3) / 12, where I is the moment of inertia, b is the width of the section, and d is the depth of the section.
| 4. How can I calculate the compressive strength of concrete? | |
Ans. The compressive strength of concrete can be calculated using the formula f'c = 0.85 * fc, where f'c is the compressive strength of concrete and fc is the specified compressive strength of concrete.
| 5. How do I calculate the bending moment in a simply supported beam in concrete structures? | |
Ans. The bending moment in a simply supported beam in concrete structures can be calculated using the formula M = (wL^2) / 8, where M is the bending moment, w is the uniformly distributed load, and L is the span length of the beam.
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