Theory of Structures (Stability and determinacy).pptx

CEng 3203-Theory of Structures
Chapter One
Introduction to indeterminate
structure
10/19/2024 Stability & Determinacy of structures 1

Outlines
Introduction
Determinacy of structures for different types of
beams, portal frames and truss
Stability of structures
Structural loads
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Stability & Determinacy of structures

Learning outcomes of this chapter
Describe the stability and determinacy.
For analysis of indeterminate structures, adopt an
appropriate structural analysis method.
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4
3

Introduction
What is a structures?
A structure refers to a system of connected parts
used to support loads.
The fundamental purpose of a structure is to
transmit loads from the point of application to the
point of support and through the foundations to the
ground.
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Introduction
Before going into the analysis of any structure, it
is necessary to identify its statically type
(classification) i.e. whether it is determinate or
indeterminate, stable or unstable.
An unstable arrangement of supports and structural
members should be avoided.
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Determinacy of structures
What does mean determinacy of structures?
The determinacy of structures refers to the degree to which the
structural system is fixed.
Why it is needed to fix structures?
To determine the loads it must withstand, materials used to
construct it.
Indeterminacy is introduced in structures on account of
functional requirements, limitations on types of framing and
need for stiffness.
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Based on response of the structures to the external
applied loads in the form of reactions or
displacements, indeterminacy of the structures
maybe classified into two general categories.
a) Static indeterminacy due to support reactions
b) Kinematic indeterminacy due to joints
displacement
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a) Static indeterminacy
Depending upon determinacy, structures maybe
classified into two general categories.
i. Statically determinate
ii. Statically indeterminate
 When all support reactions in a structure can be
determined from equilibrium equations solely, the
structure is referred to as statically determinate.
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 When all support reactions in a structure cannot be determined
from equilibrium equations solely (i.e. more unknown support
reactions than available equilibrium equations), the structure is
referred to as statically indeterminate.
In addition to more unknown support reactions, a structure
is statically indeterminate when it possesses more
members than necessary for stability (and equilibrium).
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These excess members or support reactions are
called redundant or degree of static indeterminacy.
The degree of static indeterminacy is the number of
unknown support reactions in excess of the
available equilibrium equations.
To analysis indeterminate structures, we need ways
of establishing additional equations.
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These additional equations may be derived from
compatibility of deformation or from conditions of
symmetry and known as compatibility equations.
Compatibility equations must be equal to the
number of static degree indeterminacy.
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External and Internal Static Indeterminacy
 The static indeterminacy of a structure can be external (with
respect to support reactions or restraints) or internal (with
respect to member forces or internal forces).
 When all support reactions in a structure cannot be
determined from equilibrium equations solely (i.e. more
unknown support reactions than available equilibrium
equations), the structure is referred to as statically
indeterminate externally.
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A structure is statically indeterminate internally when
it is not possible to determine all internal forces by
using the equations of static equilibrium.
For the great majority of structures, the question of
whether or not they are statically indeterminate
internally can be decided by inspection.
For certain structures this is not so, and for these types
rules have to be established.
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Total degree of static indeterminacy of structures can be
given by the following equation:
Degree of static indeterminacy of structures externally
Degree of static indeterminacy of structures internally
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Where:
is the number of internal forces for each member
is the number of members
is the number of unknown reactions
is the number of equations of equilibrium at each joint
is the number of joints of the structures
is the number of internal releases (internal hinges and/or
internal rollers)
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If a structure is supported by more than three reactions,
then all the reactions cannot be determined from the three
equations of equilibrium.
Such structures are termed statically indeterminate
externally ().
The reactions in excess of those necessary for equilibrium are
called external redundants, and the number of external
redundants is referred to as the degree of external
indeterminacy.
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If a structure is supported by fewer than three
support reactions, the reactions are not sufficient to
prevent all possible movements of the structure in
its plane.
Such a structure cannot remain in equilibrium
under a general system of loads and is therefore,
referred to as statically unstable externally ().
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Determinacy of Structures-Beams
A beam is a structural element that is capable of
withstanding load primarily by resisting bending.
Beams generally carry vertical loadings (in plane
moments and forces normal to the axis of the
beam).
Thus, for beams , , and the total degree of static
indeterminacy for a beam can be calculated as:
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Total degree of static indeterminacy for a beam is
the degree of external static indeterminacy as no
internal force in any member of a beam remains
unknown, if beam is statically determinate
externally.
Thus for beams, and
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Note: In the determination of , components of the
support reactions in the direction parallel to the
beam axis must be ignored since there is no internal
axial force in any bean members.
Stability of a beam is depends on external supports
only and its determinacy relates on the number of
available and conditional equations.
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Let is the available number of reaction
components, is the minimum number of reaction
components required for stability usually , and n is
the number of special/ conditional equation.
If
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Note: is not a sufficient condition for stability.
For a coplanar structure there are at most three
equilibrium equations for each part, so that if there is a
total of parts and force and moment reaction
components, we have:
o , unstable
o , statically determinate
o , statically indeterminate
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Examples (Determine total degree of static
indeterminacy , degree of external and internal
indeterminacy of the following beams)
Total degree of static indeterminacy for beam
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𝑎¿
𝐹𝐵𝐷

Where:
is the number of members
is the number of joints of the structures
is the number of internal releases (internal hinges
and/or shear releases)
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Degree of external indeterminacy of a beam
For beams
Degree of internal indeterminacy of a beam
For a beams
 A beam is statically indeterminate to the second
degree.
OR
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 is number force and moment reaction components
is total number of parts
Degree of static indeterminacy of a beam
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 is total number of parts
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𝑏¿ 𝐹𝐵𝐷

Where:
internal rollers)
 A beam is statically indeterminate to the first degree.
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 is total number of parts
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Where:
internal rollers)
 A beam is statically determinate.
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A beam is unstable.
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𝑑¿ 𝐹𝐵𝐷

A beam is unstable.
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𝑒 ¿ 𝐹𝐵𝐷

Determinacy of Structures-Trusses
A simple truss can be made by combining three bars
to form a triangle.
Stability depends partly on external supports and
partly on the arrangement of members or bars.
Three reaction components are required for external
stability and determinacy of a plane truss without
condition equations.
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For trusses (pin jointed frames, ) and thus total
degree of static indeterminacy .
External degree static of indeterminacy
The external statically classification of the truss
depends on the total number of reaction
components, and their arrangement.
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 where is the number of equilibrium equations and
is the number of conditional equations. For trusses .
Let is the available number of reaction components,
is the minimum number of reaction components
required for stability usually , and n is the number of
special/ conditional equation.
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If
The condition for is necessary but not sufficient
conditions for statically classification because the
arrangement of the reaction components may render
the truss unstable.
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Internal degree static of indeterminacy .
Let is the number of members required to form an
internally statically determinate truss that connects
joints and has reaction components required for
external stability.
If is the actual number of bar forces in the truss, then
the following criteria hold true for internal
classification:
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If
Consider the trusses shown below. The truss shown
in fig. (a) is stable whereas the truss shown in fig.
(b) is unstable since the geometric arrangement of
the members is not maintained.
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Determinacy of Structures-Frames
 A frame is a structure of continuous members being connected by
rigidly joints.
 This type of structures ca resist both transverse and longitudinal
loadings.
 For frames () and thus total degree of static indeterminacy .
 Stability depends partly on external supports and partly on
moment resisting joints.
 External degree static of indeterminacy where is the number of
equilibrium equations and is the number of conditional equations.
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The number of conditional equation introduced by
a hinge joint is equal to the number of members at
the joint minus one.
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Let is the available number of reaction components,
is the minimum number of reaction components
required for stability usually , and n is the number of
special/ conditional equation.
If
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Internal degree static of indeterminacy .
Let is the number of members required to form an
internally statically determinate frame that connects
joints and has reaction components required for
external stability.
If is the actual number of bar forces in the frame
is the number of unknowns
is the number of available equations
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Then the following criteria
hold true for internal
classification:
 If
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Stability of structures
The resistance offered by a structure to
undesirable movement like sliding, collapsing,
overturning, etc. is called stability.
A statically stable structure is a structure that can
resist any actions without the development of the
rigid body movement (mechanism) on the entire
structure or within any parts of the structure.
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For the static stability of a structure, proper and
sufficient number of constraints must be provided
to the entire structure in terms of supports so as to
prevent the rigid body movement (translation and
rotation) in the structure.
This means stability depends upon the support
conditions and arrangements of members.
10/19/2024 47

In structural analysis, a structure is said to be stable
when it can support any possible system of applied
loads.
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.
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Based on this concept of static stability, a statically
unstable structure is a structure that is not statically
stable.
In the other word, a structure that exhibits the rigid
body movement (mechanism) for the entire
structure or within any parts of the structure when
subjected to a particular action.
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Stability can be divided into two as external and internal.
External stability
A structure in which there are insufficient number of
constraints or improper constraints to prevent motion
from taking place is called an unstable entire structure.
This is external instability.
What matters is not only the number of support reactions
but also their arrangement.
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Structures for which the numbers of reaction
components are greater than or equal to the number
of available equilibrium equations but that are
unstable due to arrangement of these reaction
components are said to be geometrically unstable.
When the reaction elements are three or more like
supports that are either parallel or concurrent, they
are not sufficient to maintain static equilibrium.
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For a structure to be externally stable, the following
conditions must be met:
a) Non-Parallel reactions
The reactions at the supports should not be parallel to
each other.
If the reactions are parallel, even if there is sufficient
numbers of reaction, it cannot be restrain rigid body
movement which leading to external instability.
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b) Non-Concurrent reactions
 The reactions at the supports should not pass through the same
point.
 If the reactions are concurrent, it indicates that the forces at the
supports are acting in a single line of action and they cannot
prevent rotation of the structure about their concurrent point
which can result in external instability.
 By satisfying these conditions of external stability, a structure can
maintain its overall stability and resist external loads and forces.
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10/19/2024 54
Parallel supports
Beam
Truss

For this case of parallel supports, even if we have
sufficient number of supports, due to their improper
arrangement, they will offer no resistance to
horizontal movement, thus making the arrangement
unstable.
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10/19/2024 56
Concurrent supports

Internal stability
A structure is considered to be internally stable or
rigid, if it maintains its shape and remains a rigid body
when isolated from the supports.
Conversely, a structure is termed internally unstable (or
non-rigid) if it cannot maintain its shape and may
undergo large displacements under small disturbances
when not supported externally.
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This internal instability is caused by too many
internal releases (i.e. internal hinges) or improper
arrangement of members.
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Examples of internally stable structures
Note that each of the structures shown forms a rigid body and each can
maintain its shape under loads.

Each structure is composed of two rigid parts, AB and
BC connected by a hinged joint B which cannot prevent
the rotation of one part with respect to the other.
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Examples of internally unstable structures

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Internally unstable truss due to improper arrangement
of members
Stable truss

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Internally unstable structures due to improper
arrangement of members

Generally, to investigate the static stability of a
structure, first find degree of static indeterminacy .
If the degree of static indeterminacy of a given
structure is less than zero, i.e. , then the structure is
statically unstable.
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For degree of static indeterminacy , investigate the
structure for the development of rigid body motion
(mechanism) within the structure under a particular
action.
If there is any mechanism, then the structure is
statically unstable.
If no mechanism, then the structure is statically stable.
10/19/2024 63

Mechanism of the entire structure is caused due to
insufficient number of constraints or improper
constraints (parallel and concurrent constraints).
While the mechanism in any portion of the structure
is caused by too many internal releases or improper
arrangement of members.
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Investigate the static stability of the following
structure
Since , the structure is unstable internally.
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𝑺𝑰=𝒃+𝒓 −𝟐 𝒋=𝟏𝟏+𝟒−𝟐∗𝟖=−𝟏

However, by investigating all parts of this structure,
there exists a pattern of rigid body motion
(mechanism) as shown in the figure.
10/19/2024 66

Therefore, the structure is statically unstable. Thus,
static stability of the structure cannot be concluded
from .
10/19/2024 67

Since , thus, static stability of the structure cannot be
concluded from .
However, by investigating all parts of this structure, there
is no development of rigid body motion within any parts
of the structure. Therefore, the structure is statically stable.
10/19/2024 68

Structural loads
What is a load?
A load is a weight or source of pressure due to
materials or persons acting on a structure during
their lifetime.
Accurate estimation of the magnitudes of these
loads is a very important aspect of the structural
analysis process.
10/19/2024 69

Structural loads
The loads that act on common civil engineering
structures can be grouped according to their nature
and source into three classes:
a) Dead loads (Permanent loads),
b) Live loads (Variable loads) and
c) Environmental loads
10/19/2024 70

Structural loads
a) Dead loads
Dead loads are those that act on the structure as a
result of the weight of the structure itself and of the
components of the system that are permanent
fixtures.
As a result, dead loads are characterized as having
fixed magnitudes and positions.
10/19/2024 71

Structural loads
Examples of dead loads are the weights of the
structural members themselves such as beams and
columns, the weights of roof surfaces, floor slabs,
ceilings or permanent partitions walls and so on.
How we can determine dead loads?
Dead loads associated with the structure can be
determined if the materials and sizes of the various
components are known.
10/19/2024 72

Structural loads
The determination of the characteristic values of self-
weight and of the dimensions and densities shall be in
accordance with ES EN 1990:2015 ‘General actions’
Part 1-1 section 4.1.2.
Nominal dimensions should be those as shown on the
drawings.
 where A is members cross-section and is a unit weight
of composed materials
10/19/2024 73

Structural loads
10/19/2024 74

Structural loads
b) Live loads
Live loads also known as imposed loads are those that
may vary over time and often result from the
occupancy of the structure.
Typically live loads may include people, movable
objects like furniture, vehicles and so on.
Live loads are variable as they depend on usage and
capacity.
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Structural loads
How we can determine live loads?
The magnitude of live loads are varying on the structure
and its impossible to obtain accurate values.
Because of this, design codes such as our very nation’s
‘ES EN 1991:2015 Part 1-1 ‘General Actions’ section
6.3 provide equivalent loads for various building
categories and their specific usages.
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Structural loads
10/19/2024 77
Imposed
Loads
on
Buildings

Structural loads
10/19/2024 78

Structural loads
The imposed loads are modelled by uniformly
distributed loads, line loads or concentrated loads or
combinations of these loads.
10/19/2024 79

Structural loads
Values of imposed actions
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Structural loads
10/19/2024 81

Structural loads
10/19/2024 82

Structural loads
Imposed Loads on Bridges
Live loads on bridges consist primarily of vehicles,
which vary in weight, size, and frequency of
occurrence (cars, trucks, buses).
Live loads are often concentrated at specific points
where vehicles are located on the bridge.
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Structural loads
10/19/2024 84
Imposed Loads on Bridges

Structural loads
10/19/2024 85
Weight
shown
are
for
each
one
axle
=
2
wheels

Structural loads
Differences between live loads on the buildings and live
loads on the bridges
The differences between live loads on buildings and
live loads on bridges are primarily related to their
functions, design considerations and usage patterns.
Live loads in buildings vary based on the occupancy
type (residential, commercial, etc.), furniture,
equipment, and people.
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Structural loads
Live loads are often assumed to be uniformly
distributed across floors, although concentrated
loads (like heavy furniture) may be considered.
Live loads in buildings are typically considered to
be sustained for longer periods, as buildings are
generally static environments with continuous
occupancy.
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Structural loads
Live loads on buildings may not account for significant
dynamic effects, as the movement of occupants is generally
slower and less impactful compared to vehicles on a bridge.
Live loads on bridges are often concentrated at specific
points where vehicles are located on the bridge.
Live loads on bridges are typically transient, as vehicles are
not stationary; they pass over the structure relatively
quickly.
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Structural loads
 Bridges must account for dynamic impact factors due to the
acceleration and deceleration of vehicles, as well as potential
collisions, which can increase the effective live load.
In summary:
 Live loads on buildings are primarily influenced by occupancy
and usage patterns, with a focus on static and sustained loads.
 Live loads on bridges are characterized by dynamic vehicle
traffic, requiring considerations for concentrated loads and
dynamic effects.
10/19/2024 89

Structural loads
c) Environmental loads
Environmental loads refer to the forces and stresses
exerted on a structure due to natural environmental
factors.
These loads can significantly impact the design and
performance of civil engineering structures, requiring
careful consideration to ensure safety and functionality.
10/19/2024 90

Structural loads
Environmental loads include:
i. Wind Loads
ii. Seismic Loads
iii. Soil Loads
iv. Snow Loads
v. Temperature Effects
vi. Rain Loads
10/19/2024 91

Structural loads
i. Wind Loads
Wind loads are the environmental load cases caused
by a moving (blowing) air mass against an
obstructing structure and hence subsequently
converting the kinetic energy of the wind into a
potential energy of pressure.
This wind pressures can be either positive or negative.
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Structural loads
10/19/2024 93

Structural loads
10/19/2024 94

Structural loads
Positive wind pressures: are loads having pushing
effects (towards the structure).
Negative wind pressures (suctions): are those loads
having pulling effects (away from the structure).
These loadings caused by wind are expected to affect
three perspective faces of a given structure.
These are the windward, leeward and sides of a
structure
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Structural loads
Windward faces
Those in the upwind direction or from which the
wind is coming.
Windward faces are always loaded with positive
wind pressures.
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Structural loads
Leeward faces
Those which are away (downward or downwind)
from the upcoming wind.
Leeward faces are always loaded with negative
wind pressures (suctions).
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Structural loads
Side faces
Those faces parallel to the direction of the wind.
Side walls can be loaded with positive or negative
pressures depending on the geometric plan shape of
the structure.
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Structural loads
Wind Loads-Analysis
Exposure extent of the structure is dependent on
terrain roughness and topography of its location,
expresses the vulnerability of the structure to wind
actions.
Because of this, design codes such as our very nation’s
‘ES EN 1991:2015 ‘General Actions’ Part 1-4 section
5.2 provide characteristics wind loads value.
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Structural loads
ii. Earthquake Loads
10/19/2024 100

Structural loads
 An earthquake is a sudden motion in the earth’s upper crust
(Lithosphere) caused by abrupt releases of accumulated strains
energy in rocks below, volcanic activity, landslides and collapse
of underground caves.
The ground surface moves both in horizontal and vertical
directions during an earthquake.
The magnitude of the vertical ground motion is usually
small and does not have a significant effect on most
structures.
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Structural loads
It is the horizontal component of ground motion that
causes structural damage and must be considered in
design of structures located in earthquake prone
areas.
The seismic base shear force for each horizontal
direction in which the building is analyzed shall be
determined by ES EN 1998:2015 section 4.3.3.2.
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Quiz (5%)
1) The preconditions for checking of the determinacy
of a given structure is its stability. (True/False) (1
mark)
2) Investigate the external and internal stability and
determine the total degree of static indeterminacy,
external and internal indeterminacy of the
following framed structures. (3 marks)
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Quiz (5%)
3) What are the causes of the external and internal
instability of structures? (1 mark)
10/19/2024 104
𝑐 ¿
𝑎¿ 𝑏¿

10/19/2024 105
END OF THE CHAPTER

Theory of Structures (Stability and determinacy).pptx

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