Structural Analysis Formulas for Civil Engineering Exam | Structural Analysis - Civil Engineering (CE) PDF Download

 Page 1


Short Notes on Structural Analysis 
Static Indeterminacy 
• If a structure cannot be analyzed for external and internal reactions using static equilibrium 
conditions alone then such a structure is called indeterminate structure 
External static indeterminacy: 
• It is related with the support system of the structure and it is equal to number of external 
reaction components in addition to number of static equilibrium equations. 
Internal static indeterminacy: 
• It refers to the geometric stability of the structure. If after knowing the external reactions it is 
not possible to determine all internal forces/internal reactions using static equilibrium equations 
alone then the structure is said to be internally indeterminate. 
Kinematic Indeterminacy 
• It the number of unknown displacement components are greater than the number of 
compatibility equations, for these structures additional equations based on equilibrium must be 
written in order to obtain sufficient number of equations for the determination of all the 
unknown displacement components. 
•  The number of these additional equations necessary is known as degree of kinematic 
indeterminacy or degree of freedom of the structure. 
 
Three Hinged Arches 
 
 
(i) Three Hinged Parabolic Arch of Span L and rise 'h' carrying a UDL ovr the whole span 
 0
S
D ? 
 
2
2
0
8
2
C
XA
BM
wl
H
h
wx
M V x Hy
?
?
? ? ?
 
where, H = Horizontal thrust 
 VA = Vertical reaction at 
2
wl
A ? 
 
2
2
A
wx
Vx
??
? ?
? ?? ?
?
?
? ?
??
 Simply supported beam moment i.e., moment caused by vertical 
reactions. 
 Hy = H-moment 
 DS = Degree of static indeterminacy 
 BMC = Bending Moment at C. 
Page 2


Short Notes on Structural Analysis 
Static Indeterminacy 
• If a structure cannot be analyzed for external and internal reactions using static equilibrium 
conditions alone then such a structure is called indeterminate structure 
External static indeterminacy: 
• It is related with the support system of the structure and it is equal to number of external 
reaction components in addition to number of static equilibrium equations. 
Internal static indeterminacy: 
• It refers to the geometric stability of the structure. If after knowing the external reactions it is 
not possible to determine all internal forces/internal reactions using static equilibrium equations 
alone then the structure is said to be internally indeterminate. 
Kinematic Indeterminacy 
• It the number of unknown displacement components are greater than the number of 
compatibility equations, for these structures additional equations based on equilibrium must be 
written in order to obtain sufficient number of equations for the determination of all the 
unknown displacement components. 
•  The number of these additional equations necessary is known as degree of kinematic 
indeterminacy or degree of freedom of the structure. 
 
Three Hinged Arches 
 
 
(i) Three Hinged Parabolic Arch of Span L and rise 'h' carrying a UDL ovr the whole span 
 0
S
D ? 
 
2
2
0
8
2
C
XA
BM
wl
H
h
wx
M V x Hy
?
?
? ? ?
 
where, H = Horizontal thrust 
 VA = Vertical reaction at 
2
wl
A ? 
 
2
2
A
wx
Vx
??
? ?
? ?? ?
?
?
? ?
??
 Simply supported beam moment i.e., moment caused by vertical 
reactions. 
 Hy = H-moment 
 DS = Degree of static indeterminacy 
 BMC = Bending Moment at C. 
(ii) Three Hinged Semicircular Arch of Radius R carrying a UDL over the whole span. 
 
2
wR
H ? 
 
2
2
[sin sin ]
2
X
wR
M ??
?
?? 
  
 
2
max
8
wR
M
?
? 
 0
C
BM ? 
 Point of contraflexure = 0     
(iii) Three Hinged Parabolic Arch Having Abutments at Different Levels  
 (a) When it is subjected to UDL over whole span. 
  
 
2
1 2
2( )
AB
wl
HH
h h
??
?
 
 
1
1
1 2
lh
l
h h
?
?
 
 
2
2
1 2
lh
l
h h
?
?
 
 0
C
BM ? 
(b) When it is subjected to concentrated load W at crown 
  
 
? ?
2
1 2
wl
H
h h
?
?
 
(iii) Three Hinged Semicircular Arch Carrying Concentrated Load W at Crown  
 
2
A B
W
HV V ? ?? 
 
Temperature Effect on Three Hinged Arches 
 
(i) 
22
4
4
lh
hT
h
?
??
?
? ?
? ?? ?
?
?
? ?
??
 
Where,     h ? = free rise in crown height  
      l = length of arch 
   h = rise of arch 
 a = coefficient of thermal expansion 
 T= rise in temperature in 
0
C 
(ii) 
1
H
h
? 
 Where, H = horizontal thrust 
Page 3


Short Notes on Structural Analysis 
Static Indeterminacy 
• If a structure cannot be analyzed for external and internal reactions using static equilibrium 
conditions alone then such a structure is called indeterminate structure 
External static indeterminacy: 
• It is related with the support system of the structure and it is equal to number of external 
reaction components in addition to number of static equilibrium equations. 
Internal static indeterminacy: 
• It refers to the geometric stability of the structure. If after knowing the external reactions it is 
not possible to determine all internal forces/internal reactions using static equilibrium equations 
alone then the structure is said to be internally indeterminate. 
Kinematic Indeterminacy 
• It the number of unknown displacement components are greater than the number of 
compatibility equations, for these structures additional equations based on equilibrium must be 
written in order to obtain sufficient number of equations for the determination of all the 
unknown displacement components. 
•  The number of these additional equations necessary is known as degree of kinematic 
indeterminacy or degree of freedom of the structure. 
 
Three Hinged Arches 
 
 
(i) Three Hinged Parabolic Arch of Span L and rise 'h' carrying a UDL ovr the whole span 
 0
S
D ? 
 
2
2
0
8
2
C
XA
BM
wl
H
h
wx
M V x Hy
?
?
? ? ?
 
where, H = Horizontal thrust 
 VA = Vertical reaction at 
2
wl
A ? 
 
2
2
A
wx
Vx
??
? ?
? ?? ?
?
?
? ?
??
 Simply supported beam moment i.e., moment caused by vertical 
reactions. 
 Hy = H-moment 
 DS = Degree of static indeterminacy 
 BMC = Bending Moment at C. 
(ii) Three Hinged Semicircular Arch of Radius R carrying a UDL over the whole span. 
 
2
wR
H ? 
 
2
2
[sin sin ]
2
X
wR
M ??
?
?? 
  
 
2
max
8
wR
M
?
? 
 0
C
BM ? 
 Point of contraflexure = 0     
(iii) Three Hinged Parabolic Arch Having Abutments at Different Levels  
 (a) When it is subjected to UDL over whole span. 
  
 
2
1 2
2( )
AB
wl
HH
h h
??
?
 
 
1
1
1 2
lh
l
h h
?
?
 
 
2
2
1 2
lh
l
h h
?
?
 
 0
C
BM ? 
(b) When it is subjected to concentrated load W at crown 
  
 
? ?
2
1 2
wl
H
h h
?
?
 
(iii) Three Hinged Semicircular Arch Carrying Concentrated Load W at Crown  
 
2
A B
W
HV V ? ?? 
 
Temperature Effect on Three Hinged Arches 
 
(i) 
22
4
4
lh
hT
h
?
??
?
? ?
? ?? ?
?
?
? ?
??
 
Where,     h ? = free rise in crown height  
      l = length of arch 
   h = rise of arch 
 a = coefficient of thermal expansion 
 T= rise in temperature in 
0
C 
(ii) 
1
H
h
? 
 Where, H = horizontal thrust 
  and  h = rise of arch 
(iii) % Decrease in horizontal thrust 100
h
h
?
?? 
 
Two Hinged Arches 
 
2
ds
My
El
H
y ds
El
?
?
?
 
DS = 1  
Where, M = Simply support Beam moment caused by vertical force.  
(i) Two hinged semicircular arch of radius R carrying a concentrated load 'w' at the town. 
w
H
?
? 
 
(ii) Two hinged semicircular arch of radius R carrying a load w at a section, the radius vector 
corresponding to which makes an angle a with the horizontal. 
 
2
sin
w
H ?
?
? 
(iii) A two hinged semicircular arch of radius R carrying a UDL w per unit length over the 
whole span. 
 
 
4
3
wR
H
?
? ? 
(iv) A two hinged semicircular arch of radius R carrying a distributed load uniformly varying 
from zero at the left end to w per unit run at the right end. 
 
2
3
wR
H
?
?? 
(v) A two hinged parabolic arch carries a UDL of w per unit run on entire span. If the span off 
the arch is L and its rise is h. 
 
2
8
wl
H
h
? 
(vi) When half of the parabolic arch is loaded by UDL, then the horizontal reaction at support 
is given by 
 
2
16
wl
H
h
? 
(vii) When two hinged parabolic arch carries varying UDL, from zero to w the horizontal thrust 
is given by 
 
2
16
wl
H
h
? 
Page 4


Short Notes on Structural Analysis 
Static Indeterminacy 
• If a structure cannot be analyzed for external and internal reactions using static equilibrium 
conditions alone then such a structure is called indeterminate structure 
External static indeterminacy: 
• It is related with the support system of the structure and it is equal to number of external 
reaction components in addition to number of static equilibrium equations. 
Internal static indeterminacy: 
• It refers to the geometric stability of the structure. If after knowing the external reactions it is 
not possible to determine all internal forces/internal reactions using static equilibrium equations 
alone then the structure is said to be internally indeterminate. 
Kinematic Indeterminacy 
• It the number of unknown displacement components are greater than the number of 
compatibility equations, for these structures additional equations based on equilibrium must be 
written in order to obtain sufficient number of equations for the determination of all the 
unknown displacement components. 
•  The number of these additional equations necessary is known as degree of kinematic 
indeterminacy or degree of freedom of the structure. 
 
Three Hinged Arches 
 
 
(i) Three Hinged Parabolic Arch of Span L and rise 'h' carrying a UDL ovr the whole span 
 0
S
D ? 
 
2
2
0
8
2
C
XA
BM
wl
H
h
wx
M V x Hy
?
?
? ? ?
 
where, H = Horizontal thrust 
 VA = Vertical reaction at 
2
wl
A ? 
 
2
2
A
wx
Vx
??
? ?
? ?? ?
?
?
? ?
??
 Simply supported beam moment i.e., moment caused by vertical 
reactions. 
 Hy = H-moment 
 DS = Degree of static indeterminacy 
 BMC = Bending Moment at C. 
(ii) Three Hinged Semicircular Arch of Radius R carrying a UDL over the whole span. 
 
2
wR
H ? 
 
2
2
[sin sin ]
2
X
wR
M ??
?
?? 
  
 
2
max
8
wR
M
?
? 
 0
C
BM ? 
 Point of contraflexure = 0     
(iii) Three Hinged Parabolic Arch Having Abutments at Different Levels  
 (a) When it is subjected to UDL over whole span. 
  
 
2
1 2
2( )
AB
wl
HH
h h
??
?
 
 
1
1
1 2
lh
l
h h
?
?
 
 
2
2
1 2
lh
l
h h
?
?
 
 0
C
BM ? 
(b) When it is subjected to concentrated load W at crown 
  
 
? ?
2
1 2
wl
H
h h
?
?
 
(iii) Three Hinged Semicircular Arch Carrying Concentrated Load W at Crown  
 
2
A B
W
HV V ? ?? 
 
Temperature Effect on Three Hinged Arches 
 
(i) 
22
4
4
lh
hT
h
?
??
?
? ?
? ?? ?
?
?
? ?
??
 
Where,     h ? = free rise in crown height  
      l = length of arch 
   h = rise of arch 
 a = coefficient of thermal expansion 
 T= rise in temperature in 
0
C 
(ii) 
1
H
h
? 
 Where, H = horizontal thrust 
  and  h = rise of arch 
(iii) % Decrease in horizontal thrust 100
h
h
?
?? 
 
Two Hinged Arches 
 
2
ds
My
El
H
y ds
El
?
?
?
 
DS = 1  
Where, M = Simply support Beam moment caused by vertical force.  
(i) Two hinged semicircular arch of radius R carrying a concentrated load 'w' at the town. 
w
H
?
? 
 
(ii) Two hinged semicircular arch of radius R carrying a load w at a section, the radius vector 
corresponding to which makes an angle a with the horizontal. 
 
2
sin
w
H ?
?
? 
(iii) A two hinged semicircular arch of radius R carrying a UDL w per unit length over the 
whole span. 
 
 
4
3
wR
H
?
? ? 
(iv) A two hinged semicircular arch of radius R carrying a distributed load uniformly varying 
from zero at the left end to w per unit run at the right end. 
 
2
3
wR
H
?
?? 
(v) A two hinged parabolic arch carries a UDL of w per unit run on entire span. If the span off 
the arch is L and its rise is h. 
 
2
8
wl
H
h
? 
(vi) When half of the parabolic arch is loaded by UDL, then the horizontal reaction at support 
is given by 
 
2
16
wl
H
h
? 
(vii) When two hinged parabolic arch carries varying UDL, from zero to w the horizontal thrust 
is given by 
 
2
16
wl
H
h
? 
(viii) A two  hinged parabolic arch of span l and rise h carries a concentrated load w at the 
crown. 
25
128
wl
H
h
? 
 
 
Temperature Effect on Two Hinged Arches 
 
2
lT
H
y ds
El
?
?
?
 
(i) 
2
4El T
H
R
?
?
?  
where H = Horizontal thrust for two hinged semicircular arch due to rise in temperature by 
T 
0
C. 
(ii) 
0
2
15
8
El T
H
h
?
? 
where l0 = Moment of inertia of the arch at crown. 
 H = Horizontal thrust for two hinged parabolic arch due to rise in temperature T 
0
C. 
 
Reaction Locus for a Two Hinged Arch 
(a) Two Hinged Semicircular Arch 
Reaction locus is straight line parallel to the line joining abutments and height at 
2
R ?
 
  
(b) Two Hinged Parabolic Arch 
 
2
2 2
1.6hL
y PE
L Lx x
??
??
 
 
Eddy's Theorem 
 
X
My ? 
where,  MX = BM at any section 
 y = distance between given arch linear arch 
 
Trusses: 
Degree of Static Indeterminacy   
(i) 2
Se
D mr j ? ?? where, DS = Degree of static indeterminacy m = Number of members, 
re = Total external reactions,  
j = Total number of joints  
(ii)  DS = 0 ?Truss is determinate 
 If Dse = +1 & Dsi = –1 then DS = 0 at specified point. 
(iii) DS > 0 ? Truss is indeterminate or dedundant. 
Page 5


Short Notes on Structural Analysis 
Static Indeterminacy 
• If a structure cannot be analyzed for external and internal reactions using static equilibrium 
conditions alone then such a structure is called indeterminate structure 
External static indeterminacy: 
• It is related with the support system of the structure and it is equal to number of external 
reaction components in addition to number of static equilibrium equations. 
Internal static indeterminacy: 
• It refers to the geometric stability of the structure. If after knowing the external reactions it is 
not possible to determine all internal forces/internal reactions using static equilibrium equations 
alone then the structure is said to be internally indeterminate. 
Kinematic Indeterminacy 
• It the number of unknown displacement components are greater than the number of 
compatibility equations, for these structures additional equations based on equilibrium must be 
written in order to obtain sufficient number of equations for the determination of all the 
unknown displacement components. 
•  The number of these additional equations necessary is known as degree of kinematic 
indeterminacy or degree of freedom of the structure. 
 
Three Hinged Arches 
 
 
(i) Three Hinged Parabolic Arch of Span L and rise 'h' carrying a UDL ovr the whole span 
 0
S
D ? 
 
2
2
0
8
2
C
XA
BM
wl
H
h
wx
M V x Hy
?
?
? ? ?
 
where, H = Horizontal thrust 
 VA = Vertical reaction at 
2
wl
A ? 
 
2
2
A
wx
Vx
??
? ?
? ?? ?
?
?
? ?
??
 Simply supported beam moment i.e., moment caused by vertical 
reactions. 
 Hy = H-moment 
 DS = Degree of static indeterminacy 
 BMC = Bending Moment at C. 
(ii) Three Hinged Semicircular Arch of Radius R carrying a UDL over the whole span. 
 
2
wR
H ? 
 
2
2
[sin sin ]
2
X
wR
M ??
?
?? 
  
 
2
max
8
wR
M
?
? 
 0
C
BM ? 
 Point of contraflexure = 0     
(iii) Three Hinged Parabolic Arch Having Abutments at Different Levels  
 (a) When it is subjected to UDL over whole span. 
  
 
2
1 2
2( )
AB
wl
HH
h h
??
?
 
 
1
1
1 2
lh
l
h h
?
?
 
 
2
2
1 2
lh
l
h h
?
?
 
 0
C
BM ? 
(b) When it is subjected to concentrated load W at crown 
  
 
? ?
2
1 2
wl
H
h h
?
?
 
(iii) Three Hinged Semicircular Arch Carrying Concentrated Load W at Crown  
 
2
A B
W
HV V ? ?? 
 
Temperature Effect on Three Hinged Arches 
 
(i) 
22
4
4
lh
hT
h
?
??
?
? ?
? ?? ?
?
?
? ?
??
 
Where,     h ? = free rise in crown height  
      l = length of arch 
   h = rise of arch 
 a = coefficient of thermal expansion 
 T= rise in temperature in 
0
C 
(ii) 
1
H
h
? 
 Where, H = horizontal thrust 
  and  h = rise of arch 
(iii) % Decrease in horizontal thrust 100
h
h
?
?? 
 
Two Hinged Arches 
 
2
ds
My
El
H
y ds
El
?
?
?
 
DS = 1  
Where, M = Simply support Beam moment caused by vertical force.  
(i) Two hinged semicircular arch of radius R carrying a concentrated load 'w' at the town. 
w
H
?
? 
 
(ii) Two hinged semicircular arch of radius R carrying a load w at a section, the radius vector 
corresponding to which makes an angle a with the horizontal. 
 
2
sin
w
H ?
?
? 
(iii) A two hinged semicircular arch of radius R carrying a UDL w per unit length over the 
whole span. 
 
 
4
3
wR
H
?
? ? 
(iv) A two hinged semicircular arch of radius R carrying a distributed load uniformly varying 
from zero at the left end to w per unit run at the right end. 
 
2
3
wR
H
?
?? 
(v) A two hinged parabolic arch carries a UDL of w per unit run on entire span. If the span off 
the arch is L and its rise is h. 
 
2
8
wl
H
h
? 
(vi) When half of the parabolic arch is loaded by UDL, then the horizontal reaction at support 
is given by 
 
2
16
wl
H
h
? 
(vii) When two hinged parabolic arch carries varying UDL, from zero to w the horizontal thrust 
is given by 
 
2
16
wl
H
h
? 
(viii) A two  hinged parabolic arch of span l and rise h carries a concentrated load w at the 
crown. 
25
128
wl
H
h
? 
 
 
Temperature Effect on Two Hinged Arches 
 
2
lT
H
y ds
El
?
?
?
 
(i) 
2
4El T
H
R
?
?
?  
where H = Horizontal thrust for two hinged semicircular arch due to rise in temperature by 
T 
0
C. 
(ii) 
0
2
15
8
El T
H
h
?
? 
where l0 = Moment of inertia of the arch at crown. 
 H = Horizontal thrust for two hinged parabolic arch due to rise in temperature T 
0
C. 
 
Reaction Locus for a Two Hinged Arch 
(a) Two Hinged Semicircular Arch 
Reaction locus is straight line parallel to the line joining abutments and height at 
2
R ?
 
  
(b) Two Hinged Parabolic Arch 
 
2
2 2
1.6hL
y PE
L Lx x
??
??
 
 
Eddy's Theorem 
 
X
My ? 
where,  MX = BM at any section 
 y = distance between given arch linear arch 
 
Trusses: 
Degree of Static Indeterminacy   
(i) 2
Se
D mr j ? ?? where, DS = Degree of static indeterminacy m = Number of members, 
re = Total external reactions,  
j = Total number of joints  
(ii)  DS = 0 ?Truss is determinate 
 If Dse = +1 & Dsi = –1 then DS = 0 at specified point. 
(iii) DS > 0 ? Truss is indeterminate or dedundant. 
 
Truss Member Carrying Zero forces 
(i)  M1, M2, M3 meet at a joint 
 M1 & M2 are collinear 
 ?M3 carries zero force 
 where M1, M2, M3 
 represents member. 
 
(ii) M1 & M2 are non collinear and Fext= 0 
 
12
& MM ? carries zero force. 
 
Indeterminate Truss 
(i)  Final force in the truss member 
 S = P + kX and 
2
PkL
AE
X
kL
AE
??
?
?
 
 sign convn ? +ve for tension, –ve  for compression 
where,  
S = Final force in the truss member 
K = Force in the member when unit load is applied in the redundant member 
L = Length of the member 
A = Area of the member 
E = Modulus of elasticity 
P = Force in the member when truss become determinate after removing one of the member. 
P = Zero for redundant member. 
 
Lack of Fit in Truss 
U
X
?
??
?
 where, 
2
2
QL
U
AE
?? 
Q = Force induce in the member due to that member which is '' ? too short or '' ? too long is 
pulled by force 'X'. 
 
Deflection of Truss 
 
C
PL
y kL T
AE
?
??
?? ?? ?
??
??
 
Where, yC = Deflection of truss due to effect of loading & temp. both. 
If effect of temperature is neglected then 
 
C
PkL
y
AE
?
? 
? ? Coefficient of thermal expansion 
T = Change in temperature 
T = +ve it temperature is increased 
T = -ve it temperature is decreased 
P & K have same meaning as mentioned above.