Priliminary concept of Structural Analysis

7/28/2019 Priliminary concept of Structural Analysis

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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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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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STRUCTURAL ANALYSIS

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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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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STRUCTURAL ANALYSIS

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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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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STRUCTURAL ANALYSIS

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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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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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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STRUCTURAL ANALYSIS

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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STRUCTURAL ANALYSIS

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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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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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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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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STRUCTURAL ANALYSIS

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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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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STRUCTURAL ANALYSIS

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)


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)



j = number of joints (this includes the

joints at the reactions)


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STRUCTURAL ANALYSIS

Two Dimensional Frames

Degree of static indeterminacy = (3b + r) - 3j



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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STRUCTURAL ANALYSIS


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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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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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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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Priliminary concept of Structural Analysis