Lecture - 02

Determinacy, Indeterminacy & stability: Part-I

Equilibrium of Structures

A structure is considered to be in equilibrium when

It is initially at rest. [is not moving or accelerating in any direction]
It remains at rest under the action of forces and couples.

If a structure is in equilibrium:

All its members and components are also in equilibrium.
Each part must maintain a balance of force and moment.
Each and every force acting upon a body or part of the body is resisted by either another equal and opposite force or set of forces whose net result is zero.

teeter -totter

tug of war rope

In real-world applications:

Structures are typically arranged in three-dimensional configurations.
For a space structure subjected to 3 dimensional forces and couples:

Resultant force must be zero.
Resultant couple (moment) must also be zero.

The conditions of zero resultant force and zero resultant couple can be expressed as

These six equations are called the equations of equilibrium of space structures and are the necessary and sufficient conditions for equilibrium. The first three equations ensure that there is no resultant force acting on the structure, and the last three equations express the fact that there is no resultant couple acting on the structure.

For a plane structure lying in the xy plane and subjected to a coplanar system of forces and couples, the necessary and sufficient conditions for equilibrium can be expressed as

· These three are known as the equations of equilibrium for plane structures.

ü First Equation – ∑Fx=0

· States that the algebraic sum of all x-components of forces is zero.

· Implies no net force in the horizontal direction.

ü Second Equation – ∑Fy=0

· States that the algebraic sum of all y-components of forces is zero.

· Implies no net force in the vertical direction.

ü Third Equation – ∑Mz=0

· States that the algebraic sum of all moments about any point in the plane (including moments from couples) is zero.

· Implies no net rotational effect (couple) on the structure.

The first two equations together ensure the resultant force acting on the structure is zero.
The third equation ensures the resultant moment (or couple) acting on the structure is zero.

· All three equations must be satisfied simultaneously for the structure to be in a state of complete static equilibrium.

Concurrent Force Systems

Equilibrium of a Structure under a Concurrent Force System

ü A concurrent force system is one where the lines of action of all forces intersect at a single point.

ü When a structure is in equilibrium under a concurrent force system:

· The moment equilibrium equations are automatically satisfied.

· Only the force equilibrium equations need to be considered.

· [Moment = Force x Perpendicular distance from the point (Lever arm)]

· [In concurrent force system, all force act through a common point, the perpendicular distance from that point to each force’s line of action is zero, M= Fx0 = 0]

· Similarly, for a plane structure subjected to a concurrent coplanar force system, the equilibrium equations can be expressed as

Two-Force and Three-Force Structures

Two-Force Members:

A structure (or member) in equilibrium under only two forces must satisfy the following:

The forces must be equal in magnitude.
The forces must be opposite in direction.
The forces must be collinear (act along the same straight line).
Example: Standard link or rod in a truss, Tie or strut connecting two joints.

Three-Force Members:

A structure in equilibrium under only three forces must satisfy:

The forces must be either concurrent (their lines of action intersect at a common point) or
The forces must be parallel.
Example: A beam supported at two points with a point loading somewhere between.

FREE-BODY DIAGRAM

A free-body diagram (FBD) is a very useful sketch of the structure showing all forces (including couples) applied to it and having all supports replaced with their corresponding reactions.

External and Internal Forces

External Forces in Structural Analysis

External forces are the actions exerted by other bodies on the structure under consideration.
For analysis purposes, external forces are commonly classified into two types:

1. Applied Forces (Loads):

Also referred to as loads (e.g., live loads, dead loads, wind loads).
These forces:

Tend to move or deform the structure.
Are typically known quantities in structural analysis.

2. Reaction Forces (Reactions):

Forces exerted by supports or connections on the structure.
These forces:

Tend to oppose motion and maintain equilibrium.
Are usually unknowns to be determined through analysis.

· The state of equilibrium or motion of a structure depends entirely on the external forces acting on it (i.e., both applied and reaction forces).

Internal Forces in Structures

Internal forces are the forces and couples that:

Develop within a member or portion of a structure.
Are exerted by the rest of the structure on that member or portion.
Structural elements must resist internal forces to remain stable.
Internal forces are generated within a structure when external loads (e.g., dead load, snow load, wind load) are applied.
These internal forces are essential for maintaining the equilibrium and structural integrity.
If structural elements fail to resist internal forces, they will break or collapse.

Characteristics of Internal Forces:

They are responsible for holding the structure together.
According to Newton’s Third Law:

Internal forces always occur in equal and opposite pairs.
One part of the structure exerts an internal force, and the adjacent part reacts with an equal but opposite force.

Equilibrium Implication:

Because internal force pairs cancel each other, they:

Do not appear in the equations of equilibrium for the entire structure.
Are only considered when analyzing individual members or portions of the structure.

ü Internal forces are usually among the unknowns.

ü They are determined by:

o Isolating a member or section of the structure.

o Applying the equations of equilibrium to that part.

Types of Supports for Plane Structures

Supports are structural elements used to:

Attach structures to the ground or other bodies.
Restrict movement caused by applied loads.

· Function of Supports:

· Applied loads tend to move the structure.

· Supports oppose this movement by exerting reaction forces or couples.

· These reactions neutralize the effect of loads and maintain equilibrium.

Characteristics of Supports

Static Determinacy, Indeterminacy, and Instability

Internal Stability of Structures

A structure is said to be internally stable (or rigid) if it can maintain its shape and resist deformation when detached from its external supports. In other words, an internally stable structure remains a rigid body and does not undergo unintended movement or collapse under load.

Internally Stable Structures:

Maintain their shape without external support.
Can resist small disturbances without significant displacement.
Are typically formed by rigid connections (e.g., triangulated frameworks).
Examples: Triangular frames, properly braced trusses.

Consider a straight beam resting on supports.

If the supports are removed, the beam may fall due to the applied loads but:

It will not change its shape — it remains straight.
It remains a rigid body.

Therefore, the beam is internally stable, even though it is externally unstable without support.

Internally Unstable (Non-Rigid) Structure:

A structure is considered internally unstable (or nonrigid) if it cannot maintain its shape and may undergo large displacements even under small disturbances when not supported externally.

Internally Unstable Structures:

Cannot maintain their shape when detached from supports.
Undergo large displacements with even small external disturbances.
Usually contain hinged or unbraced joints that allow parts to rotate freely.
May collapse under its own weight when detached from supports.
Examples:

1. A structure with two rigid members (like AB and BC) connected by a hinge at B — which allows rotation and doesn't resist shape change.

2. Consider a frame made of two rigid segments, ABC and CDE, connected at hinge C. When the supports are removed:

· The two parts can rotate freely around the hinge at C.

· The frame changes shape — a sign of internal instability.

· So, even though the members themselves are rigid, the structure as a whole is not.

Static Determinacy of Internally Stable Structures

Statically Determinate Externally:

A structure is externally statically determinate if all support reactions can be found using the equations of equilibrium.
since there are only three equilibrium equations, they cannot be used to determine more than three reactions.
Thus, a plane structure that is statically determinate externally must be supported by exactly three reactions.
The term ‘Externally’: clarify that the member is statically determinate with regard to the calculation of the reactions.

Some examples of externally statically determinate plane structures:

Statically Indeterminate Externally:

If a structure has more than three support reactions, then:

The three equilibrium equations are not sufficient to solve for all reactions.
Such a structure is called externally statically indeterminate.

The extra reactions (beyond what’s needed for equilibrium) are called external redundant.
The number of these redundant is the degree of external indeterminacy.

· if a structure has r reactions (r>3) , then the degree of external indeterminacy can be written as

i_e = r - 3

Some examples of externally statically indeterminate plane structures:

Statically Unstable Externally

When a structure has fewer than three support reactions:

It lacks sufficient constraints to resist all types of planar movement.
The structure cannot remain in equilibrium under general loading conditions.
Such structures are referred to as externally statically unstable.

Consider a truss supported by only two rollers.

These two rollers provide only two vertical reactions.

Although these reactions:

Can prevent vertical movement.
Can resist rotation to some extent.

They cannot prevent horizontal movement (no horizontal reaction).

Hence, the truss is not fully constrained and can slide horizontally → making it statically unstable externally.

The conditions of static instability, determinacy, and indeterminacy of plane internally stable structures can be summarized as follows:

· r < 3 the structure is statically unstable externally

· r = 3 the structure is statically determinate externally

· r > 3 the structure is statically indeterminate externally

where r=5 number of reactions.

Geometrically Unstable Structures

Sometimes, a structure may be supported by three reactions (r = 3) — which is the correct number for external static determinacy — yet still be unstable.
This happens due to an improper arrangement of supports and is called: Geometric (or External) Instability

Important Conditions:

1. If r < 3 → The structure is definitely unstable

· This is both necessary and sufficient to conclude instability.

2. If r = 3 or r > 3 →

· This is necessary for stability or indeterminacy but not sufficient.

· You must also check the geometry of the reaction arrangement.

Geometrically Unstable Cases (Even When r = 3):

Case 1: All Reactions are Parallel

A truss is supported by three vertical reactions (all parallel).
Though r = 3, the structure can translate horizontally.
There's no resistance to horizontal movement → unstable.

Case 2: All Reactions are Concurrent (Intersect at One Point)

A beam is supported by three reactions whose lines of action meet at a single point (say, Point A).
The structure cannot resist rotation about Point A.
The moment equation ∑M_A=0 becomes ineffective.
Hence, the structure is geometrically unstable, despite having r = 3.

Here,

The beam is supported at three points: A, B, and C.
The support reactions R_A, R_B,R_Care shown as non-parallel.
All three lines of action intersect at point D above the beam.
Loads:

Uniform load w over span AB
Point load P acting diagonally at C

Although the reactions are not parallel, they are concurrent (all intersect at point D).
This configuration fails to resist rotation about point D.
So, the structure cannot satisfy moment equilibrium about that point.

Structural Analysis and Design I : Determinacy, Indeterminacy and stability: Part-I