determinate and indeterminate structures

STRUCTURAL ANALYSIS-II
v shiva
Assistant Professor,
Dept. of Civil Engineering,
Anurag Engineering College, Kodada
Mobile: 8328188940
03-08-2020
vempati shiva

UNIT – I (TOPIC - I)
What is Structure?
Structural analysis?
Types of connections(supports)
Equations of equilibrium
Determinate and indeterminate structures
Degree of static indeterminacy
Internal static indeterminacy
External static indeterminacy
Degree of static indeterminacy for:
i. Beams
ii. Frames(2-D and 3-D)
iii. Trusses(2-D and 3-D)
Kinematic indeterminacy
03-08-2020 vempati shiva

• By the end of the session:
• Student will know the difference between determinate and
indeterminate structure.
• student will be able to calculate static and kinematic indeterminacy in
a structure.
• student will understand the requirement of compatibility equations that
are to be formulated in order to solve unknowns.

STRUCTURE
• A structure consists of a series of connected parts used to support a
load.
• Connected parts are called structural elements which include beams,
columns and tie rods.
• The combination of structural elements and the materials from which
they are composed is referred to as structural system.

STRUCTURALANALYSIS
• Structural analysis may be defined as the prediction of performance of a
given structure under stipulated loads or other external effects, such as
support movements and temperature changes.
• Performance characteristics include axial forces, shear forces, and bending
moments, deflections, and support reactions.
• These unknown quantities may be obtained by formulating a suitable
number of independent equations. These equations can be obtained from the
following relations which govern the behaviour of structure:
1. Equilibrium
2. Stress – Strain relation
3. kinematics

CONNECTIONS
• Structural members are jointed together by rigid(fixed) or/and
flexible(hinged) connections.
• A rigid connection or joint prevents relative translations and rotations
of the member ends connected to it (Original angles between members
intersecting at rigid joint are maintained after the deformation also).
Rigid joints capable of transmitting forces as well as moments also.
• A hinged joint prevents only relative translations of member ends.
These are capable of transmitting only forces.

EQUATIONS OF EQUILIBRIUM
• A structure initially at rest and remains at rest when subjected to a system of forces
and couples is said to be in a state of static equilibrium.
• The conditions of zero resultant force and zero resultant couple can be expressed
as:
• For planar structure:
• ΣFx = 0
• ΣFy = 0
• ΣMz = 0
• When a planar structure subjected to concurrent coplanar force system, the above
requirements for equilibrium reduces to
• ΣFx = 0
• ΣFy = 0

• For space structure:
• ΣFx = 0
• ΣFy = 0
• ΣFz = 0
• ΣMx = 0
• ΣMy = 0
• ΣMz = 0

INDETERMINATE STRUCTURE
• If the unknown forces in a structure are more than the equilibrium
equations then such structure is referred to as indeterminate structure.
• Indeterminacy can be of static indeterminacy or kinematic
indeterminacy.

INDETERMINACY
STATIC INDETERMINACY
EXTERNAL STATIC INDETERMINACY INTERNAL STATIC INDETERMINACY
KINEMATIC INDETERMINACY

External static indeterminacy(Dse):
• When an all the external reactions cannot be evaluated from the static
equilibrium equations alone, then such structure is referred to as
externally indeterminate.
Internal static indeterminacy(Dsi):
• When an all the internal forces cannot be evaluated from the static
equilibrium equations alone, then such structure is referred to as
internally indeterminate.
Degree of Static indeterminacy(Ds) = Dse + Dsi

Need of degree of indeterminacy:
• Degree of indeterminacy = Total no. of unknown forces – Total no. of
equilibrium equations
• Degree of indeterminacy gives us the additional
equations(compatibility equations) required to find unknown forces.
• Thus the Degree of indeterminacy is equal to the number of additional
equations(other than static equilibrium equations) required to solve the
unknown forces in that structure.

BEAMS
Find the degree of static indeterminacy for the following beams:
Number of reactions(re) = 3
Number of internal forces(i) = 0
Total number of unknown forces(re + i) = 3 + 0 = 3
Number of parts(n) = 1
Total number of equations of equilibrium available = 3n = 3 X 1 = 3
Degree of static indeterminacy = (re + i) – 3n
= (3 + 0) – 3
= 0 (Determinate)03-08-2020 vempati shiva

Degree of static indeterminacy = (r + i) – 3n
= (4 + 0) – 3
= 1 (Indeterminate)

= (6 + 0) – 3
= 3 (Indeterminate)

= (4 + 2) – 6
= 0 (determinate)

= (4 + 1) – 6
= -1 (Determinate but UNSTABLE)03-08-2020 vempati shiva

STABILITY
Structural stability is the major concern of the structural designer. To
ensure the stability, a structure must have enough support reaction
along with proper arrangement of members. The overall stability of
the structure can be divided into:
• External stability
• Internal stability

External stability
• For stability of structures there should
be no rigid body movement of
structure due to loading. So it should
have proper supports to restrain
translation and rotation motion.
There should be min. 3 no. of externally
independent support reactions.
All reactions should not be parallel
All reactions should not be linearly
concurrent otherwise rotational
unstability will setup.

Internal stability
• For internal stability no part of
the structure can move rigidly
relative to the other part so that
geometry of the structure is
preserved, however small
elastic deformations are
permitted.
• To preserve geometry enough
number of members and their
adequate arrangement is
required.
• For geometric stability there
should not be any condition of
mechanism. Mechanism is
formed when there are three
collinear hinges.03-08-2020 vempati shiva

FRAMES

FRAMES
• Frames always forms closed loops. Each time when you cut the loop it releases
three unknown forces(axial force, shear force and moment).
• If there are ‘c’ closed loops then 3c is the internal static indeterminacy.
For 2-D Frames:
External static indeterminacy(Dse) = re – 3
Internal static indeterminacy(Dsi) = 3c – rr
re = number of external reactions
c = number of closed loops
rr = number of reactions released.(in case of any internal hinges)
rr = Σ(m’ – 1)
m’ = number of members connecting the hinge.

For 3-D Frames:
External static indeterminacy(Dse) = re – 6
Internal static indeterminacy(Dsi) = 6c – rr
re = number of external reactions
c = number of closed loops
rr = number of reactions released.(in case of any internal hinges)
rr = Σ3(m’ – 1)
m’ = number of members connecting the hinge.

• Find the degree of static indeterminacy for
the frames shown:
• No of external reactions = 3;
• No. of static equilibrium equations = 3
• External static indeterminacy(Dse) = 3 – 3 = 0;
• No. of closed loops(c) = 0;
• No. of unknowns in a closed loop = 3c.
• Internal static indeterminacy(Dsi) = 3c – rr.
= 3X0 – 0 = 0;
Ds = Dse + Dsi;
∴ degree of static indeterminacy(Ds) = 0 + 0 = 0
= 0 (Determinate structure)
Alternatively:
To make the given frame into a closed loop, I need to add one rotational
restraint at support A and one horizontal restraint and one rotational
restraint at support D. so total 3 restraints I need to add to make it into a
closed loop.
As we know every closed loop has 3 unknown forces, now the above
frame has 3 unknown forces. But as we have added 3 restraints to make it
into closed loop , subtract those 3 restraints from 3 unknown forces which
comes out to be 0 and it is a determinate structure.

the frames shown:
• No. of internal releases(rr) = 0;
= 3X0 – 0 = 0;
Ds = Dse + Dsi;
= 3 (Indeterminate structure)03-08-2020 vempati shiva

• Find the degree of static indeterminacy for the
frames shown:
• No. of internal releases(rr) = 0;
= 3X4 – 0 = 12;
Ds = Dse + Dsi;
= 18 (Indeterminate structure)

• Find the degree of static indeterminacy for the
frames shown:
• No. of internal releases(rr) = Σ(m’ – 1)
• = Σ(2’ – 1) = 1
= 3X1 – 1 = 2;
Ds = Dse + Dsi;

the Plane frames shown:
• External static indeterminacy(Dse) = 12 – 3 =
9;
• No. of internal releases(rr) = Σ(m’ – 1) ;
• = (3 – 1) + (3 – 1) = 4 ;
= 3X3 – 4 = 5;
Ds = Dse + Dsi;
= 14 (Indeterminate structure)03-08-2020 vempati shiva

Find the degree of degree of indeterminacy for the
space FRAMES shown:
• No of external reactions = 4X6 = 24;
• No. of internal releases(rr) = Σ3(m’ – 1) = 0 ;
= 6X1 – 0 = 6;
Ds = Dse + Dsi;

Find the degree of degree of indeterminacy for the space FRAMES
shown:
• No of external reactions = 6X6 = 36;
• External static indeterminacy(Dse) = 36 – 6 =
30;
• No. of internal releases(rr) = Σ3(m’ – 1) = 0 ;
= 6X16 – 0 = 96;
Ds = Dse + Dsi;
∴ degree of static indeterminacy(Ds) = 30 + 96
= 126

Find the degree of degree of indeterminacy for the space
FRAMES shown:
• No of external reactions = 2X6 + 2X3 = 18;
• No. of internal releases(rr) = Σ3(m’ – 1) = 3(2-1)=3 ;
= 6X1 – 3 = 3;
Ds = Dse + Dsi;

TRUSSES
• Trusses are most common type of structure used in
constructing building roofs, bridges and towers etc.
• A truss can be constructed by straight slender members
joined together at their end by bolting, riveting or welding.
Classification of trusses:
• Plane trusses(2-D)
• Space trusses(3-D)

• External static indeterminacy(Dse) = re – 3
• Internal static indeterminacy(Dsi) = m – (2j – 3)
re = no. of external reactions;
m = no. of members;
j = no. of joints;
• Degree of static indeterminacy(Ds) = Dse + Dsi
Example-1:
re = 3;
Dse = 3 – 3 = 0;
∴external static indeterminacy = 0
m = 3;
j = 3;
Dsi = m – (2j - 3);
Dsi = 3 – (2X3 – 3);
Dsi = 0;
∴internal static indeterminacy = 0;
Ds = Dse + Dsi
Ds = 0;
∴Degree of static indeterminacy = 0
Alternatively:
Ds = Dsi + Dse;
Ds = re – 3 + m – (2j - 3);
Ds = re – 3 + m – 2j + 3;
Ds = m + re -2j;
Ds = 3 + 3 – 2X3;
Ds = 6 – 6;
Ds = 0;

Example-2:
re = 6;
Dse = 6 – 3 = 3;
m = 3;
j = 4;
Dsi = m – (2j - 3);
Dsi = 3 – (2X4 – 3);
Dsi = -2;
∴internal static indeterminacy = -2;
Ds = Dse + Dsi
Ds = 3 – 2;
Alternatively:
Ds = Dsi + Dse;
Ds = re – 3 + m – (2j - 3);
Ds = re – 3 + m – 2j + 3;
Ds = m + re -2j;
Ds = 3 + 3 – 2X3;
Ds = 6 – 6;
Ds = 0;03-08-2020 vempati shiva

Example-2:
re = 4;
Dse = 4 – 3 = 1;
m = 23;
j = 13;
Dsi = m – (2j - 3);
Dsi = 23 – (2X13 – 3);
Dsi = 0;
∴internal static indeterminacy = 0;
Ds = Dse + Dsi
Ds = 1 – 0;

DEGREE OF KINEMATIC INDETERMINACY
or
DEGREE OF FREEDOM(Dk)
Degree of kinematic indeterminacy refers to the total number of independent available
degree of freedom at all joints. The degree of kinematic indeterminacy may be defined as
the total number of unrestraint displacement components at all joints.
S.NO TYPE OF JOINT POSSIBLE DEGREE OF FREEDOM
1. 2 -D Truss joint Two degree of freedoms are available
1. ∆x 2. ∆y
2. 3 –D Truss joint Three degree of freedoms are available
1. ∆x 2. ∆y 3. ∆z
3. 2 –D Rigid joint Three degree of freedoms are available
1. ∆x 2. ∆y 3. θz
4. 3 –D Rigid joint six degree of freedoms are available
1. ∆x 2. ∆y 3. ∆z
2. 4. θx 5. θy 6. θz

• Plane truss (2–D truss)
• Dk = 2j – re
• space truss (3–D truss)
• Dk = 3j – re
• Rigid jointed Plane frame (2–D frame)
• Dk = 3j - re – m”
• Rigid jointed space frame (3–D frame)
• Dk = 6j - re – m”
• Dk = degree of freedom;
• re = no. of external reactions;
• J = no. of joints;
• M” = no. of axially rigid members;

Find the degree of kinematic indeterminacy for the following BEAMS:
If the beam is axially flexible:
• No. of external reactions = 3;
• No. of joints = 2;
• No. of axially rigid members = 0;
• Degree of freedom (Dk) = 3j – re – m”;
Dk = 3X2 – 3 – 0 ;
Dk = 3
If the beam is axially rigid:
• Degree of freedom (Dk) = 2j – re – m” ;
Dk = 2X3 – 3 – 1 ;
Dk = 2

Find the degree of kinematic indeterminacy for the following BEAMS:
If the beam is axially flexible:
• Degree of freedom (Dk) = 3j – re – m”;
Dk = 3X2 – 4 – 0 ;
Dk = 2
If the beam is axially rigid:
As the beam is already axially restrained by reactions.
Dk will be same as the previous case.
∴ Dk = 2

Find the degree of kinematic indeterminacy for the following FRAMES:
• No. of released reactions rr = Σ(m’ – 1);
rr = 2 – 1 = 1
• Degree of freedom (Dk) = 3j – re – m” + rr;
Dk = 3X4 – 5 – 0 + 1 ;
Dk = 8

Find the degree of kinematic indeterminacy for
the FRAME shown, consider all beams are
axially rigid:
• No. of released reactions rr = Σ(m’ – 1) = 0 ;
Degree of freedom (Dk) = 3j – re – m” + rr;
Dk = 3X9 – 7 – 4 + 0 ;
Dk = 16

Find the degree of kinematic indeterminacy for the following TRUSSES:
• Degree of freedom (Dk) = 2j – re
Dk = 2X3 – 3 ;
Dk = 3

Find the degree of kinematic indeterminacy for the following TRUSSES:
• No. of external reactions, re = 3;
• No. of joints, j = 4;
• Degree of freedom (Dk) = 2j – re
Dk = 2X4 – 3 ;
Dk = 5

Find the degree of Static indeterminacy and kinematic
indeterminacy for the following plane structures:

determinate and indeterminate structures

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