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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.
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FAQs on Structural Analysis Formulas for Civil Engineering Exam - Structural Analysis - Civil Engineering (CE)
| 1. What are the basic principles of structural analysis in civil engineering? |  |
Ans. Structural analysis in civil engineering is based on the principles of equilibrium, compatibility, and stiffness. Equilibrium ensures that the forces and moments acting on a structure are balanced, while compatibility ensures that the structure deforms in a compatible manner. Stiffness refers to the resistance of a structure to deformation under applied loads.
| 2. How is structural analysis used in the design of civil engineering structures? | |
Ans. Structural analysis plays a crucial role in the design of civil engineering structures. It helps engineers determine the internal forces and deformations within a structure under various loading conditions. By analyzing these internal forces, engineers can ensure that the structure is safe, efficient, and capable of withstanding the intended loads and environmental conditions.
| 3. What are the different methods of structural analysis commonly used in civil engineering? | |
Ans. The commonly used methods of structural analysis in civil engineering include the method of joints, method of sections, and the finite element method (FEM). The method of joints is used to analyze trusses by considering the equilibrium of forces at each joint. The method of sections is used to analyze the internal forces in a structure by cutting it into sections and applying equilibrium equations. FEM is a numerical method used to analyze complex structures by dividing them into smaller elements interconnected at nodes.
| 4. How does structural analysis help in assessing the safety of civil engineering structures? | |
Ans. Structural analysis helps assess the safety of civil engineering structures by determining the internal forces and deformations within the structure. Engineers compare these internal forces with the strength and capacity of the materials used in the structure. By ensuring that the internal forces are within the acceptable limits, engineers can ensure that the structure will not fail under the applied loads and will remain stable and safe.
| 5. Can structural analysis be used to predict the behavior of civil engineering structures during earthquakes? | |
Ans. Yes, structural analysis is an important tool for predicting the behavior of civil engineering structures during earthquakes. Engineers can model the structure using appropriate analysis methods and simulate the seismic forces acting on the structure. By considering the dynamic behavior of the structure under earthquake loads, engineers can evaluate its response, identify potential weaknesses, and implement design measures to enhance its seismic performance.
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Structural Analysis Formulas for Civil Engineering Exam Notes
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The content of Structural Analysis Formulas for Civil Engineering Exam is prepared as per the latest Civil Engineering (CE) syllabus.
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