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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 methodslengths 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 trussesp• Plane frames• Grids• Space frames
Loads on these elements consist of concentrated forces, distributed loads and/or couples.
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. Nodes A, B, and C represent reactions. Node Didentifies a location on the beam (the free end) where we wish to extract informationidentifies 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
h i l l dShear Moment Axial loadV M A Actionsy θ x Displacements (translations, rotations)
Lecture 4: PRELIMINARY CONCEPTS OF STRUCTURAL ANALYSIS
Plane Truss Truss stabilizing mechanical floor
All structural components are in same plane. Forces act in the plane of structure. E t l f d ti t th f id d t t l t th d dExternal 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. So the moment M1, the distributed load w and the force P4 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).
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 a three-dimensional framework of members pinned at their ends. A tetrahedron shape is thepinned 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 tetrahedrons
ith d S twith common edges. Space trusses are employed in the base structures of large free-standing power line pylons
As in planar trusses only axial tensile or compressive forces can be developed.
Lecture 4: PRELIMINARY CONCEPTS OF 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 withAll 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)
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 YForces 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
Lecture 4: PRELIMINARY CONCEPTS OF STRUCTURAL ANALYSIS
Space Frame
Most general type of framed structure. g yp
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 y y gtwo axes of symmetry.
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 j q poriginal configuration displace to new positions (the mathematics describing this process are discussed in detail in CVE 513 and CVE 604)
Lecture 4: PRELIMINARY CONCEPTS OF STRUCTURAL ANALYSIS
Actions And DisplacementsThe terms “action” and “displacement” are used to describe two fundamental concepts in engineering mechanics An action is most commonly a single force or a momentengineering 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 talkAn 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 a notation that allows for this correspondence.
Lecture 4: PRELIMINARY CONCEPTS OF STRUCTURAL ANALYSIS
Consider the following notation and the subscripts in the figure below:
The letter A is used to denote actions - this includes concentrated forces and fcouples. Internal forces and moments at reactions are also considered actions.
The letter D is used to denote displacements - this includes translations and rotationsrotations.
Consider the beam shown below subjected to several actions producing several displacements:
Clearly three actions are identified asClearly three actions are identified as well as three displacements. Intuitively the actions and displacements are associated with nodes located at the points of application of A1, A2, and A3. A2 and A3 are applied at the same node.
At each node there are three possible displacements for this two dimensional structure: two translations and a rotation.
Lecture 4: PRELIMINARY CONCEPTS OF STRUCTURAL ANALYSIS
Each action may contribute to each displacement identified.
If we can determine the quantities D11 through D33 then by superposition each displacement can be written as follows:
2322212
1312111
DDDD
DDDD
++=
++=
3332313
2322212
DDDD
DDDD
++=
++
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 these quantities must satisfy the equations of equilibrium:
∑∑∑ === 000 MMM
∑∑∑ === 000 ZYX FFF
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
q
Lecture 4: PRELIMINARY CONCEPTS OF STRUCTURAL ANALYSIS
Compatibility
The continuity of the displacements throughout the structure must be satisfied in a correct y p gstructural 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 supporthorizontal 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 strainsimply 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.
l ibili h d i h h ibili f h di lFlexibility 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.
Lecture 4: PRELIMINARY CONCEPTS OF STRUCTURAL ANALYSIS
Static And Kinematic Indeterminacy
There are two types of indeterminacy to consider depending on whether actions orThere 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 the flexibility 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) is staticallystatically indeterminate to the first degree.
The truss in (c) isThe truss in (c) is statically indeterminate to second degree.
Lecture 4: PRELIMINARY CONCEPTS OF STRUCTURAL ANALYSIS
LetNUA = Number of unknown actionsNESE = Number of equations of static equilibrium
[ ])( = RMRHUAActionsUnknown BAAA[ ]
34
)(
==
NESENUA
RMRHUAActionsUnknown BAAA
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
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 degreebeam in the previous slide is externally statically indeterminate to the first degree. The truss below is determinate from the standpoint that we could calculate the reactions 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.
Lecture 4: PRELIMINARY CONCEPTS OF STRUCTURAL ANALYSIS
Criteria In Determining Static IndeterminacyTwo Dimensional Beams
Degree of static indeterminacy = r - (c + 3)
b f tir = number of reactionsc = number of internal conditions (c = 1 for a hinge; c = 2 for a roller; and c = 0 for a
structure with no geometric instability)
Two Dimensional Trusses
Degree of static indeterminacy = (b + r) - (2j)
Three Dimensional Trusses
Degree of static indeterminacy = (b + r) - (3j)Degree of static indeterminacy (b + r) (2j)
b = number of membersr = number of reactionsj b f j i t (thi i l d
g f y ( ) ( j)
b = number of membersr = number of reactionsj b f j i t (thi i l d thj = number of joints (this includes
the joints at the reactions)j = number of joints (this includes the
joints at the reactions)
Lecture 4: PRELIMINARY CONCEPTS OF STRUCTURAL ANALYSIS
Two Dimensional Frames
Criteria In Determining Static Indeterminacy (continued)
Two Dimensional Frames
Degree of static indeterminacy = (b + r) - (2j + c)
b = number of membersr = number of reactionsj = number of jointsc = number of internal conditions
Three Dimensional Framesc number of internal conditions
Degree of indeterminacy = (b + r) - (3j + c)
b = Number of membersr = Number of reactionsr = Number of reactionsj = Number of jointsc = Number of internal conditions
Lecture 4: PRELIMINARY CONCEPTS OF 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
d l i d/ i A di l ill b kmay undergo translations and/or rotations. At supports some displacements will be known, others will not. The number of unknown joint displacements corresponds to the kinematic indeterminacy 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 joint displacements are known, i.e., they are all zero (displacementszero (displacements and rotations).
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
d dsecond 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 p j , j ,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 of unknowns can reduce the degree of indeterminacy of the structure.
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 namelystatics available namely
But the summation of forces in the x-direction is not applicable, and the structure is bil
∑∑∑ === 000 ZYX MFF
mobile.