Theory of Structural
Chapter (2)
Fundamental Principles
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Chapter Outline
2.1
2.2
2.3
2.4
2.5
2.6
Idealized Structure
Connections
Supports and Reactions
principle of Superposition
Equation of Equilibrium
Determinacy and Stability
2.6.1 Determinacy and Stability of Beams
2.6.2 Determinacy and Stability of Trusses
2.6.3 Determinacy and Stability of Frames
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2.1 Idealized Structure
Generally, structures are complex and must be idealized or simplified
into a form that can be analyzed.
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2.1 Idealized Structure
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2.1 Idealized Structure
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2.1 Idealized Structure
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Example:
The floor of a classroom is to be supported by the bar joists shown in Figure.
Each joist is 15 ft long and they are spaced 2.5 ft on centers. The floor itself is
to be made from lightweight concrete that is 4 in. thick. Neglect the weight of
the joists and the corrugated metal deck, and determine the load that acts
along each joist.
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SOLUTION
The dead load on the floor is due to the weight of the concrete slab.
From Table 1.3 for 4 in. of lightweight concrete
(4)(8 lb/ft2) = 32 lb/ft2
From Table 1.4,
the live load for a classroom is 40 lb/ft2
Thus the total
floor load is 32 lb/ft2 + 40 lb/ft2 = 72 lb/ft2
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For the floor system,
L1= 2.5 ft and L2 = 15 ft.
Since L2/L1 > 2
the concrete slab is treated as a one-way slab.
The tributary area for each joist is shown in Figure b. Therefore the uniform
load along its length is
This loading and the end reactions on each joist are shown in Fig. c
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Example:
The concrete girders shown in the photo of the passenger car parking garage
span 30 ft and are 15 ft on center. If the floor slab is 5 in. thick and made of
reinforced stone concrete, and the specified live load is 50 lb/ft2 (see Table
1.4), determine the distributed load the floor system transmits to each interior
girder.
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SOLUTION
Here, L2 = 30 ft and L1 = 15 ft, so that L2 / L1 = 2.
We have a two way slab.
From Table 1.2, for reinforced stone concrete, the specific weight of the
concrete is 150 lb/ft3 .
Thus the design floor loading is
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A trapezoidal distributed loading is transmitted to each interior girder AB from
each of its two sides ➀ and ➁.
The maximum intensity of each of these
distributed loadings is
So that on the girder this intensity becomes
Note: For design, consideration
should also be given to the weight of
the girder.
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2.2 Connections
 Two types of connections are commonly used to joint members
of structures:
 Rigid connections
 Flexible [or Hinged] connections.
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2.2 Connections
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2.3 Supports and Reactions
 If a support prevents translation of a body in a given direction.
A force is developed on the body in that direction. There are
three types of supports:
 Roller: Prevent displacement in only one direction and allow
rotation.
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2.3 Supports and Reactions
Hinge (Pin):
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2.3 Supports and Reactions
Fixed: prevents rotation and displacement in all directions.
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2.3 Supports and Reactions
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2.3 Supports and Reactions
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2.3 Supports and Reactions
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Real Life Supports
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Real Life Supports
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2.4 principle of Superposition
The total displacement or internal loadings (stress) at a point in a
structure subjected to several external loadings can be determined by
adding together the displacements or internal loadings (stress) caused
by each of the external loads acting separately.
Requirements:
1. The material must behave in a linear-elastic
manner, so that Hooke’s law is valid, and therefore
the load will be proportional to displacement.
2. The geometry of the structure must not
undergo significant change when the loads are
applied, i.e., small displacement theory applies.
Large displacements will significantly change the
position and orientation of the loads.
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2.5 Equation of Equilibrium
For structures idealized in three directions
For most structures idealized in two directions
The ideal cut section internal loadings
V= Shear Force
N= Normal Force
M= Bending Moment
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2.6 Determinacy and Stability
A structure is Statically Determinate when the
equilibrium equations can be utilized to determine
all forces (all support reactions and internal forces)
at any locations in the structure.
A structure is statically indeterminate when there
are more unknown forces than available equilibrium
equations. 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.
The Degree of Static Indeterminate is a number
of static unknowns (support reactions and / or
internal forces that exceeds the number of independent
equilibrium equations and static conditions available.
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2.6 Determinacy and Stability
A Statically Stable Structure is a structure that can resist any
actions without the development (mechanism) on the entire
structure or within any parts of the structure.
A Statically Unstable Structure is a structure that exhibits the
rigid body displacement (mechanism) for the entire structure or
within any parts of the structure when subjected to the
particular action.
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2.6 Determinacy and Stability
An externally, Statically Unstable Structure is a statically
unstable structure with the development of the rigid body
displacement on the entire structure.
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2.6 Determinacy and Stability
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2.6 Determinacy and Stability
An internally, Statically Unstable Structure is a statically unstable
structure with the development of the mechanism only on certain
parts of components of the structure.
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2.6.1 Determinacy and Stability of Beams
In the entire beam structure, there are only three independent
equations (equations of equilibrium). However, if the internal
hinges are provided, there will be an additional equilibrium
equation (moment at hinge=0) for the hinge. Hence, the following
rule can be used for determining the determinacy of beams:
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2.6.1 Determinacy and Stability of Beams
Examples:
r= 3
nc= 0
Statically Determinate
Stable
r= 5
nc= 0
Statically Indeterminate to 2nd
Stable
r= 5
nc= 1
Statically Indeterminate to 1st
Stable
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2.6.1 Determinacy and Stability of Beams
Examples:
r= 6
nc= 1
Statically Indeterminate to 2nd
Stable
r= 5
nc= 3
Unstable
r= 4
nc= 2
Unstable
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2.6.1 Determinacy and Stability of Beams
Examples:
r=
, nc=
r=
, nc=
r=
, nc=
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2.6.2 Determinacy and Stability of Trusses
Suppose that m is the total number of members in a truss and j the
total number of joints. Then the following relationship may be
written for determinacy of trusses with total number of reaction r.
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2.6.2 Determinacy and Stability of Trusses
Examples:
m= 6 , r= 3 , j= 4
m+r = 9
2j = 8
Statically Indeterminate to 1st
Stable
m= 12 , r= 3 , j= 7
m+r =15
2j = 14
Statically Indeterminate to 1st
Stable
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2.6.2 Determinacy and Stability of Trusses
Examples:
m= 22 , r= 4 , j= 13
m+r =26
2j = 26
Statically Determinate
Stable
m= 16 , r= 4 , j= 10
m+r =20
2j = 20
Statically Determinate
Unstable
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2.6.2 Determinacy and Stability of Trusses
Examples:
m= , r=
m+r
2j
m= , r=
m+r
2j
, j=
, j=
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2.6.3 Determinacy and Stability of Frames
A frame structure is a structure that consists of frame members being
connected by frame rigid joints. This type of structures can resist both
transverse and longitudinal loadings. For a given statically stable frame
structure that consists of j joints and m members, the following relationship
may be written for determinacy of frames with total number of reaction r:
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2.6.3 Determinacy and Stability of Frames
Examples:
m= 6 , r= 4 , j= 6 , nc = 0
3m+r = 22
3j+nc = 18
Statically Indeterminate to 4th
Stable
m= 3 , r= 4
3m+r = 13
3j+nc = 14
Unstable
, j= 4 , nc = 2
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2.6.3 Determinacy and Stability of Frames
Examples:
m= 3 , r= 3
3m+r = 12
3j+nc = 13
Unstable
, j= 4 , nc = 1
m= 8 , r= 6 , j= 8 , nc = 0
3m+r = 30
3j+nc = 24
Statically Indeterminate to 6th
Stable
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2.6.3 Determinacy and Stability of Frames
Example:
m= , r= , j=
3m+r
3j+nc
, nc
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