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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 supported

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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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