CHAPTER_1 Statically Indeterminate Structures
1.1 Types of indeterminate structures
Those structures, whose reaction components and internal stresses cannot be fully
analyzed by using the equations of static equilibrium, are statically indeterminate
structures. For eg. fixed beams, continuous beams, two hinged arches etc.
Three basic types of indeterminate structures are as follows:
1) Discrete (skeletal) structures: ⇒Members are discrete and connected by joints.
⇒Represented by simultaneous linear equations.
⇒ Types:
 Pin joined
 Rigid joined
Fig: Rigid joined
Fig: Pin joined
2) Continuum Structures:⇒Members have surface with continuous joints.
⇒Represented by partial differential equations.
⇒ Types:
 Surface structures (slab, shell etc)
 Solid structure (Space dome, etc)
3) Discrete and continuum structures:⇒Combination of above two.
 Bridges with slabs
1.2 Static indeterminacy and methods of determination for various types
of structures
Static indeterminate structures are those which cannot be analyzed with the help of static
equilibrium alone. For the analysis of these structures, it becomes necessary to consider
the deformation of structure because the equations of statics are not sufficient for the
solution of problems.
In the case of statically indeterminate structure, the number of unknowns is greater than
the number of independent equation derived from the condition of static equilibrium. The
number of these additional equations, necessary for the solution of problems is known as
degree of static indeterminacy or degree of redundancy.
Types of static indeterminacy
a. External Indeterminacy (DSE)
 DSE is related to the support system of structure.
 If the reaction components are more than the available static equilibrium
equation, then the system is externally indeterminate.
For beam
DSE= r-(3+C)
Where, r= No. of independent external reaction component
C= Number of special cases (C=1 for internal hinge & C=2 for sliding
roller)
For frame and truss
Plane structure: DSE= r-3
Space structure: DSE= r-6
Where, r= No. of independent external reaction component
b. Internal Indeterminacy (DSI)
 DSI is related to the member force of the structure.
 In the beam, there is no internal indeterminacy.
For Frame(Rigid jointed)
Plane structure: DSI= 3*No. of cuts to have open configuration
Space structure: DSI= 6*No. of cuts to have open configuration
For Truss(Pin-jointed frame)
Plane structure: DSI = m-(2j-3)
Space structure: DSI = m-(3j-6)
Where,
m= No. of members
j = No. of joints
For hybrid/Skeletal structure
 Some of the joint is pin-jointed and some is rigid jointed.
Plane structure:DSI= 3*No. of cuts to have open configuration- ∑(𝑚 − 1)
Space structure:DSI= 6*No. of cuts to have open configuration- ∑ 3(𝑚 − 1)
Where, 𝑚′= Number of members meeting at hinge joint.
Total degree of Static indeterminacy (DSI)=DSE+ DSI
Alternatively,
For Frame(Rigid jointed)
Plane structure: DSI = (3m+r)-3j
Space structure: DSI=(6m+r)-6j
For Truss (pin-jointed frame)
Plane structure: DSI=(m+r)-2j
Space structure: DSI=(m+r)-3j
For hybrid/Skeletal structure
 Some of the joint is pin-jointed and some is rigid jointed.
Plane structure: DSI =(3m+r)-(3j+C)
Space structure: DSI= (6m+r)-(6j+C)
Where,
m= No. of members
j = No. of joints
C= no. of special conditions=∑(𝑚 − 1) ; for plane & ∑ 3(𝑚 − 1);for space
𝑚′= Number of members meeting at hinge joint.
1.3 Kinematic indeterminacy and methods of determination for various
types of structures
A structure is said to be kinematic indeterminate if the displacement components of joint
cannot be determined by the compatibility equation alone.
The number of additional equation necessary for the determination of all the independent
components is known as degree of kinematic indeterminacy or degree of freedom (𝐷 ).
Plane truss:
𝐷 = 2𝑗 − 𝑟
Space truss:
𝐷 = 3𝑗 − 𝑟
Plane frame:
𝐷 = 3𝑗 − 𝑟 , member extensible
Space frame:
𝐷 = 6𝑗 − 𝑟 , member extensible
Plane frame:
𝐷 = 3𝑗 − (𝑟 + 𝑚) , member inextensible
Space frame:
True for
unbraced
frame
only
𝐷 = 6𝑗 − (𝑟 + 𝑚) , member inextensible
Where, j= Number of joints
r=Number of compatibility equation or number of constraints imposed by support
condition
m= number of members
Fig: Unbraced frame
Fig: Braced frame
Hybrid/skeletal structure:
 Degree of kinematic indeterminacy of hybrid structure are found out by considering
members are inextensible
𝐷 = 3𝑗 − (𝑟 + 𝑚) +
(𝑚 − 1) + 3𝑖
Where,
i= Number of internal hinges not at joint.
𝑚 = Number of member at internal hinges at joint.