CSE 212: THEORY OF STRUCTURES II
LECTURE 1: INTRODUTION TO STATICALLY INDETERMINATE STRUCTURES
INTRODUTION TO STATICALLY INDETERMINATE STRUCTURES
1.1 General Introduction
This lecture introduces the concept of static indeterminacy of structures. Specifically, it
reviews the concepts of determinate and indeterminate structures, unstable systems,
degree of static indeterminacy (external and internal) and illustrates advantages and
disadvantages of statically indeterminate structures. Methods of analysis applicable to
statically indeterminate structures are also introduced briefly.
Structures are a group of members, such as beams, columns, slabs, foundations,
girders, and trusses that work as a unit to fulfil a purpose. An engineer's duty is to
design structures in a professional, safe, and economical manner in order to fulfil the
purpose for which it was designed in the first place. Structures are classified into either
being statically determinate or statically indeterminate.
Statically determinate structures are structures that can be analysed using static
equations only, (i.e., equilibrium in all directions).
∑ = ; ∑ = and ∑ = On the other hand, statically indeterminate structures can't be analysed using static
equations only; they require other material properties, such as deformations, in order to
analyse them.
When engineers conduct structural analysis, they calculate the reaction forces due to
the external forces applied to the structure as well as internal forces, such as the
bending moment, shear force, and axial force. Structural analysis is necessary for
structural design in order for the structural engineer to choose the proper sizes and
materials so the structure can economically and effectively resist the effects of the
possible external loads applied to it.
Definition: a statically indeterminate structure is a structure of which the number of
unknown reactions or internal forces exceeds the number of equilibrium equations. The
indeterminacy may arise as a result of added supports or members, or by the general
form of the structure.
During analysis of these structures, additional equations come from compatibility
relationships, which ensure continuity of displacements throughout the structure. The
remaining equations are constructed from member constitutive equations, i.e.,
relationships between stresses and strains and the integration of these equations over
the cross section.
Design of an indeterminate structure is carried out in an iterative manner,
whereby the (relative) sizes of the structural members are initially assumed and used to
analyse the structure. Based on the computed results (displacements and internal
member forces), the member sizes are adjusted to meet governing design criteria. This
iteration process continues until the member sizes based on the results of an analysis
are close to those assumed for that analysis.
Another consequence of statically indeterminate structures is that the relative
variation of member sizes influences the magnitudes of the forces that the
LECTURER: DR. B. OMONDI
SECOND SEMESTER: 2018/2019
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CSE 212: THEORY OF STRUCTURES II
LECTURE 1: INTRODUTION TO STATICALLY INDETERMINATE STRUCTURES
member will experience. Stated in another way, stiffness (large member size
and/or high modulus materials) attracts force.
Despite these difficulties with statically indeterminate structures, an overwhelming
majority of structures being built today are statically indeterminate.
1.2 Static Indeterminacy of Structures
The following figures illustrate some example of statically determinate (Figures 1.1a-c)
and indeterminate structures (Figures 1.2a-c).
(b)
(a)
(c)
Figure 1.1: Statically Determinate Structures
(a)
(b)
(c)
Figure 1.2: Statically Indeterminate Structures
For an indeterminate system, some support(s) or internal member(s) can be removed
without disturbing its equilibrium. These additional supports and members are known
as redundants. A determinate system has the exact number of supports and internal
members that it needs to maintain the equilibrium and no redundants. If a system has
less than required number of supports and internal members to maintain equilibrium,
then it is considered unstable.
An indeterminate system is often described with the number of redundants it possesses
and this number is known as its degree of static indeterminacy.
Thus, mathematically:
Degree of static indeterminacy = Total number of unknown (external and internal)
Forces - Number of independent equations of equilibrium
Example using truss structure
Let m be the number of members in the truss system and n be the number of pin (hinge)
joints connecting these members. Therefore, there will be m number of unknown
internal forces (each is a two-force member) and 2 n numbers of independent joint
LECTURER: DR. B. OMONDI
SECOND SEMESTER: 2018/2019
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CSE 212: THEORY OF STRUCTURES II
LECTURE 1: INTRODUTION TO STATICALLY INDETERMINATE STRUCTURES
equilibrium equations (∑ = and ∑ = for each joint, based on its free body
diagram). If the support reactions involve r unknowns, then:
Total number of unknown forces = m + r
Total number of independent equilibrium equations = 2 n
So, degree of static indeterminacy, d = (m + r) - 2 n
(a) Statically determinate: d =0
(b) Internally indeterminate: d = 1
(c) Externally indeterminate: d = 1
Figure 1.3: Truss structures with different degrees of static indeterminacy
1.3 Advantages of Statically Indeterminate Structures
1) Statically indeterminate structures typically result in smaller stresses and greater
stiffness (smaller deflections) as illustrated for the beam in Figure 1.4. It follows
that a smaller beam section would be required in the fixed-fixed beam case,
resulting in savings in material (economical design).
Figure 1.4: Illustration of stiffness of beam with different support
systems
LECTURER: DR. B. OMONDI
SECOND SEMESTER: 2018/2019
3
CSE 212: THEORY OF STRUCTURES II
LECTURE 1: INTRODUTION TO STATICALLY INDETERMINATE STRUCTURES
2) Determinate beam is unstable if middle support is removed or knocked off.
Statically indeterminate structures introduce redundancy, which may insure that
failure in one part of the structure will not result in catastrophic or collapse failure
of the structure (they are stable). This is illustrated in Figure 1.5.
Figure 1.5: Illustration of stability of beams with different support
systems
1.4 Disadvantages of Statically Indeterminate Structures
(a) Statically indeterminate structure is self-strained due to support settlement, which
produces stresses, as illustrated in Figure 1.5.
Figure 1.6: Self-restraining stresses on indeterminate beam due to support settlement
(b) Statically indeterminate structures are also self-strained due to temperature
changes and fabrication errors, as illustrated in Figure 1.7.
Figure 1.7: Self-restraining stresses on indeterminate beam due to temperature changes
(c) Another disadvantage of statically indeterminate structures is that their analysis
requires the calculation of displacements hence, their cross-sectional dimensions
LECTURER: DR. B. OMONDI
SECOND SEMESTER: 2018/2019
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CSE 212: THEORY OF STRUCTURES II
LECTURE 1: INTRODUTION TO STATICALLY INDETERMINATE STRUCTURES
are required at the onset. The design of such structures therefore becomes a matter
of trial and error, whereas the forces in the members of a statically determinate
structure are independent of member size.
1.5 Methods of analysing statically indeterminate structures
Generally, the methods of analysing statically indeterminate structures can be classified
into two: compatibility methods and equilibrium methods.
Compatibility methods as the name suggests, are based on compatibility relationships
throughout the structure. These methods are also referred to as force methods or the
flexibility methods because the unknowns in the governing equations are forces and
the coefficients of these unknowns are flexibility quantities. The final solution must
satisfy both the conditions of compatibility and the requirements of equilibrium. The
final compatibility equations represent a superposition of a set of partial solutions, each
of which satisfies the requirements of equilibrium. Examples relevant to this course
include the Moment-Area Method, Three-Moment Method and related energy
methods
The equilibrium methods are based on the solution of a set of equations that express the
equilibrium requirements for the structure. They are also called displacement
methods or the stiffness methods, because the unknowns in the governing equations
are displacements and the coefficients of these unknowns are the stiffness quantities.
Examples relevant to this course include Slope-Deflection Method, MomentDistribution Method and related Energy Methods.
Each of these methods are discussed separately in the subsequent lecture notes.
LECTURER: DR. B. OMONDI
SECOND SEMESTER: 2018/2019
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