ANALYSIS OF STATICALLY DETERMINATE STRUCTURES
Support Connections
Support Connections
Equations of Equilibrium
It may be recalled from statics that a structure or one of its
members is in equilibrium when it maintains a balance of force
and moment.
Whenever these equations are applied, it is necessary to draw a
free-body diagram (FBD) of the structure or its members. If a
member is selected, it must be isolated from its supports and
surroundings and its outlined shape drawn. All the forces and
couple moments must be shown that act on the member. In this
regard, the types of reactions at the supports can be
determined using Table 2-1. also, recall that forces common to
two members act with equal magnitudes but opposite
directions on the respective FBD of the members.
Determinacy and Stability
Before starting the force analysis of a structure, it is necessary
to establish the determinacy and stability of the structure.
Determinacy
When all the forces in a structure can be determined strictly
from the equations of equilibrium, the structure is referred to as
statically determinate. Structures having more unknown forces
than available equilibrium equations are called statically
indeterminate.
Determinacy and Stability
Determinacy
In particular, if a structure is statically indeterminate, the
additional equations needed to solve for the unknown reactions
are obtained by relating the applied loads and reactions to the
displacement or slope at different points on the structure.
These equations, which are referred to as compatibility
equations, must be equal in number to the degree of
indeterminacy of the structure. Compatibility equations involve
the geometric and physical properties of the structure.
Determinacy and Stability
Determinacy
Example 1
Classify each of the beams shown in Fig. 2-19a through 2-19d
as statically determinate or statically indeterminate.
Determinacy and Stability
Determinacy
Example 2
Classify each of the pin-connected structures shown in Fig. 2-
20a through 2-20d as statically determinate or statically
indeterminate.
Determinacy and Stability
Determinacy
Example 3
Classify each of the frames shown in Fig. 2-21a through 2-21c
as statically determinate or statically indeterminate.
Determinacy and Stability
Stability
To ensure the equilibrium of a structure or
its members, it is not only necessary to
satisfy the equations of equilibrium, but
the members must also be properly held
or constrained by their supports. Two
situations may occur where the conditions
for proper constraint have not been met.
Partial Constraints. In some cases a
structure or one of its members may have
fewer reactive forces than equations of
equilibrium that must be satisfied.
Determinacy and Stability
Stability
Improper Constraints. In some cases there may be as many
unknown forces as there are equations of equilibrium; however,
instability or movement of a structure or its members can
develop because of improper constraining by the supports.
Determinacy and Stability
Stability
Improper Constraints.
In general, a structure will be geometrically unstable – that is, it will
move slightly of collapse – if there are fewer reactive forces than
equations of equilibrium; or if there are enough reactions, instability
will occur if the lines of action of the reactive forces intersect at a
common point or are parallel to one another.
Determinacy and Stability
Stability
If the structure consists of several members or components,
local instability of one or several of these members can
generally be determined by inspection.
If the structure is unstable, it does not matter if it is statically
determinate or indeterminate. In all cases such types of
structures must be avoided in practice.
Determinacy and Stability
Stability
Example 1
Classify each of the structures shown in Fig. 2-25a through 2-
25b as stable or unstable.
Determinacy and Stability
Problem Set 1
Classify each of the structures as statically determinate,
statically indeterminate, or unstable. If indeterminate, specify
the degree of indeterminacy.
Determinacy and Stability
Problem Set 2
Classify each of the frames as statically determinate or
statically indeterminate. If indeterminate, specify the degree of
indeterminacy.
Determinacy and Stability
Problem Set 3
Classify each of the structures as statically determinate,
statically indeterminate, stable, or unstable. If indeterminate,
specify the degree of indeterminacy.
Determinacy and Stability
Problem Set 4
Classify each of the structures as statically determinate,
statically indeterminate, stable, or unstable. If indeterminate,
specify the degree of indeterminacy.
Application of the Equations of Equilibrium
To illustrate the method of force analysis, consider the three-
member frame shown in Fig. 2-26a.
Application of the Equations of Equilibrium
Application of the Equations of Equilibrium
Method of Force Analysis
Example 1
Determine the reactions on the beam shown in Fig. 2-28a.
Application of the Equations of Equilibrium
Method of Force Analysis
Example 2
Determine the reactions on the beam shown in Fig. 2-29a.
Application of the Equations of Equilibrium
Method of Force Analysis
Example 3
Determine the reactions on the beam shown in Fig. 2-30a.
Assume A is a pin and the support at B is a roller (smooth
surface).
Application of the Equations of Equilibrium
Method of Force Analysis
Example 4
The compound beam in Fig. 2-31a is fixed at A. Determine the
reactions at A, B, and C. Assume that the connection at B is a
pin and C is a roller.
Application of the Equations of Equilibrium
Method of Force Analysis
Example 5
Determine the horizontal and vertical components of reaction at
the pins A, B, and C of the two-member frame shown in Fig. 2-
32a.
Application of the Equations of Equilibrium
Method of Force Analysis
Example 6
The side of the building in Fig.
2-33a is subjected to a wind
loading that creates a uniform
normal pressure of 15 kPa on
the windward side and a
suction pressure of 5 kPa on the
leeward side. Determine the
horizontal and vertical
components of reaction at the
pin connections A, B, and C of
the supporting gabble arch.
Application of the Equations of Equilibrium
Method of Force Analysis
Example 7
Determine the reactions on the beam.
Application of the Equations of Equilibrium
Method of Force Analysis
Example 8
Determine the horizontal and vertical components at A, B, and
C. Assume the frame is pin connected at these points. The
joints at D and E are fixed connected.
Application of the Equations of Equilibrium
Method of Force Analysis
Example 9
Determine the reactions at the supports A, B, D, and F.
Application of the Equations of Equilibrium