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7/28/2019 Priliminary concept of Structural Analysis
1/22
Lecture 4: PRELIMINARY CONCEPTS OF
STRUCTURAL ANALYSIS
Introduction
In this class we will focus on the structural analysis of framed structures. We will learn
about the flexibility method first, and then learn how to use the primary analytical tools
associated with the stiffness method. Framed structures consist of components with
lengths that are significantly larger than cross-sectional areas. Both analytical methods
are applicable to structures of all types, but the stiffness method dominates, and the
structural analysis of machine components that fall outside the definition of framed
structures are treated in another course. We will concentrate on :
Beams
Plane trusses
Space trusses
Plane frames
Grids
Space frames
Loads on these elements consist of concentrated forces, distributed loads and/or couples.
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Lecture 4: PRELIMINARY CONCEPTS OF
STRUCTURAL ANALYSIS
Continuous Beam
Loads on a beam are applied in a plane containing an axis of symmetry
Beams have one or more points of support referred to as reactions but in this course they
will be more often referred to as nodes. NodesA, B, andCrepresent reactions. NodeD
identifies a location on the beam (the free end) where we wish to extract information.
Beams deflect in the plane of the loads. Internal forces consist of shear forces, bending
moments, torques (take CVE 513), and axial loads
Shear Moment Axial load
V M A Actions
y x Displacements (translations, rotations)
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Lecture 4: PRELIMINARY CONCEPTS OF
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All structural components are in same plane. Forces act in the plane of structure.
External forces and reactions to those forces are considered to act only at the nodes and
result in forces in the members which are either tensile or compressive forces. Thus all
members are two force members.
Loads acting on members are replaced by statically equivalent forces at the joints. Sothe momentM1, the distributed loadw and the forceP4 would have to be replaced by
equivalent joint loads to conduct an analysis.
Joints are assume hinged, so no bending moments are transmitted through a joint and
absolutely no twisting moments can be applied to the truss (consider a gusset plate).
Plane TrussTruss stabilizing mechanical floor
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Lecture 4: PRELIMINARY CONCEPTS OF
STRUCTURAL ANALYSIS
Space Truss
Forces and structural elements are no longer
confined to a plane. A space frame truss is athree-dimensional framework of members
pinned at their ends. A tetrahedron shape is the
simplest space truss, consisting of six members
which meet at four joints. Large planar
structures may be composed from tetrahedronswith common edges. Space trusses are
employed in the base structures of large free-
standing power line pylons
As in planar trusses only axial tensile orcompressive forces can be developed.
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Lecture 4: PRELIMINARY CONCEPTS OF
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Grid
Elements can intersect at rigid or flexible connections
All forces are normal to the plane of the structure. Typically used to support roofs with
no internal column support (think of indoor sports arenas).
All couples have their vectors in the plane of the grid. Torques can be sustained.
Each member is assumed to have two axes of symmetry so that bending and torsion can
occur independently of one another (see unsymmetrical bending in CVE 513)
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Lecture 4: PRELIMINARY CONCEPTS OF
STRUCTURAL ANALYSIS
Plane Frame
Joints are no longer required to be hinges. They can be rigid, or they can sustain rotation.
Forces and deflection are contained in the plane X-Y.
All couples have moment vectors parallel to Z-axis.
Internal resultants consist of bending moments, shearing forces and axial forces.
Joints may transfer moment
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Lecture 4: PRELIMINARY CONCEPTS OF
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Space Frame
Most general type of framed structure.
No restrictions on location of joints, directions of members, or directions of loads.
Members are assumed to have two axes of symmetry for the same reason grids have
two axes of symmetry.
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Lecture 4: PRELIMINARY CONCEPTS OF
STRUCTURAL ANALYSIS
Displacements Translations and Rotations
When a structure is subjected to loads it deforms and as a consequence points in the
original configuration displace to new positions (the mathematics describing this
process are discussed in detail in CVE 513 and CVE 604)
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Lecture 4: PRELIMINARY CONCEPTS OF
STRUCTURAL ANALYSIS
Actions And Displacements
The terms action and displacement are used to describe two fundamental concepts in
engineering mechanics. An action is most commonly a single force or a moment.
An action may also be a combination of forces, moments, or distributed loads. We will talk
about this more when we discuss the concept of equivalent joint loads.
Out of necessity forces, moments and distributed loads must be related to corresponding
displacements at their point of application (and elsewhere) in a unique manner. We need anotation that allows for this correspondence.
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Lecture 4: PRELIMINARY CONCEPTS OF
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Consider the following notation and the subscripts in the figure below:
The letterA is used to denote actions -this includes concentrated forces and
couples. Internal forces and moments at reactions are also considered actions.
The letterD is used to denote displacements -this includes translations and
rotations.
Consider the beam shown below subjected to several actions producing several
displacements:
At each node there are three possible displacements for this two dimensional structure:
two translations and a rotation.
Clearly three actions are identified as
well as three displacements.
Intuitively the actions and
displacements are associated with
nodes located at the points ofapplication ofA1,A2, andA3. A2 and
A3 are applied at the same node.
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Lecture 4: PRELIMINARY CONCEPTS OF
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If we can determine the quantitiesD11 throughD33 then by superposition each displacement
can be written as follows:
3332313
2322212
1312111
DDDD
DDDD
DDDD
Each action may
contribute to each
displacement identified.
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Lecture 4: PRELIMINARY CONCEPTS OF
STRUCTURAL ANALYSIS
Equilibrium
The objectives of any structural analysis is the determination of reactions at supports and
internal actions (bending moments, shearing forces, etc.). A correct solution for any of thesequantities must satisfy the equations of equilibrium:
In the stiffness method of analysis the equilibrium conditions at the joints of the structure are
the basic equations that are solved.
000 ZYX MMM
000 ZYX FFF
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Lecture 4: PRELIMINARY CONCEPTS OF
STRUCTURAL ANALYSIS
Compatibility
The continuity of the displacements throughout the structure must be satisfied in a correct
structural analysis. This is sometimes referred to as conditions of geometry.
As an example, compatibility conditions must be satisfied at all points of support. If a
horizontal roller support is present then the vertical displacement must be zero at that support.
We always impose compatibility at a joint. If two structural elements frame into a joint then
there displacements and rotations at the connection must be the same or consistent with each
other.
We apply a much more rigorous mathematical definition in CVE 604 for compatibility. It is
simply noted here that strain is a function of displacement. There are 6 components of strain
and only 3 components of displacement at a point in a three dimensional analysis. A
compatible displacement field will produce an appropriate state of strain at a point.
Flexibility methods use equations that express the compatibility of the displacements.
Understanding this issue as it applies to structural analyses give the student a better feel as
to how a structure behaves and an ability to judge the correctness of a solution.
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Lecture 4: PRELIMINARY CONCEPTS OF
STRUCTURAL ANALYSIS
Static And Kinematic Indeterminacy
There are two types of indeterminacy to consider depending on whether actions or
displacements are of interest. When actions are the unknowns which is typical for theflexibility method, then static indeterminacy is of paramount interest. From your early
undergraduate education this meant that there were an excess of unknowns relative to the
number of equations of static equilibrium
The beam in (a) isstatically
indeterminate to
the first degree.
The truss in (c) is
statically
indeterminate to
second degree.
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Lecture 4: PRELIMINARY CONCEPTS OF
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One of these four unknown is referred to as a static redundant. The number of static
redundant represents the degree of static indeterminacy of the structures
3
4
)(
NESE
NUA
RMRHUAActionsUnknown BAAA
Let
NUA = Number of unknown actions
NESE = Number of equations of static equilibrium
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Lecture 4: PRELIMINARY CONCEPTS OF
STRUCTURAL ANALYSIS
A distinction may also be made between external and internal indeterminacy. The
beam in the previous slide is externally statically indeterminate to the first degree.
The truss below is determinate from the standpoint that we could calculate thereactions given the loads applied. However, we would be unable to find the internal
forces in the cross members. The truss is internally indeterminate to the second
degree.
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Lecture 4: PRELIMINARY CONCEPTS OF
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Two Dimensional Trusses
Degree of static indeterminacy = (b + r) - (2j)
b = number of members
r = number of reactions
j = number of joints (this includes
the joints at the reactions)
Criteria In Determining Static Indeterminacy
Two Dimensional Beams
Degree of static indeterminacy = r - (c + 3)
r = number of reactions
c = number of internal conditions (c = 1 for a hinge; c = 2 for a roller; andc = 0 for a
structure with no geometric instability)
Three Dimensional Trusses
Degree of static indeterminacy = (b + r) - (3j)
b = number of members
r = number of reactions
j = number of joints (this includes the
joints at the reactions)
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Lecture 4: PRELIMINARY CONCEPTS OF
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Two Dimensional Frames
Degree of static indeterminacy = (3b + r) - 3j
b = number of members
r = number of reactions
j = number of jointsc = number of internal conditions
Three Dimensional Frames
Degree of indeterminacy = (6b + r) - 6j
b = Number of members
r = Number of reactions
j = Number of joints
c = Number of internal conditions
Criteria In Determining Static Indeterminacy (continued)
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Lecture 4: PRELIMINARY CONCEPTS OF
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STRUCTURAL ANALYSIS
For the stiffness method the displacements at the joints are unknown quantities. Thus
kinematic indeterminacy is important here. When a structure is subjected to loads each joint
may undergo translations and/or rotations. At supports some displacements will be known,
others will not. The number of unknown joint displacements corresponds to the kinematicindeterminacy of structure. Reconsider the beams and the truss from the previous slide.
The beam in (a) is
kinematically
indeterminate to the
second degree.
The beam in (b) is
kinematically
determinate. All jointdisplacements are
known, i.e., they are all
zero (displacements
and rotations).
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Lecture 4: PRELIMINARY CONCEPTS OF
STRUCTURAL ANALYSIS
Consider the beam in Figure (a). At joint A the beam is fixed and cannot undergo any joint
displacement. However at joint B the beam is free to translate in the horizontal direction
and rotate in the plane of the beam. Thus the beam is kinematically indeterminate to the
second degree.
The truss in in (c) can undergo two displacements at each joint. Although rotations can take
place at each joint, since moments cannot be sustained at truss joints, rotations have no
physical significance in this problem. The truss is kinematically indeterminate to the ninth
degree.
Often structural members are very stiff in the axial direction. Thus very little axial
displacement will take place. Removing the axial load or deformation from the system ofunknowns can reduce the degree of indeterminacy of the structure.
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Lecture 4: PRELIMINARY CONCEPTS OF
STRUCTURAL ANALYSIS
Mobile Structures
When the number of reactive forces is greater than the number of equations of static
equilibrium for the entire structure taken as a free body, the structure is statically
indeterminate
However a problem can appear to be statically determinate when it is not. Consider the
beam above. This is a planar problem. Thus in general there are three equations of
statics available namely
But the summation of forces in the x-direction is not applicable, and the structure is
mobile.
000 ZYX MFF
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