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-sectionalsectional 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.
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 D 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 θ x Displacements (translations, rotations)
Plane Truss Truss stabilizing mechanical floor All structural components are in same plane. Forces act in the plane of structure. External forces and reactions to those forces are considered d 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 M 1, the distributed load w and the force P 4 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).
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 the simplest space truss, consisting of six members which meet at four joints. Large planar structures may be composed from tetrahedrons with common edges. Space trusses are employed in the base structures of large freestanding power line pylons As in planar trusses only axial tensile or compressive forces can be developed.
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)
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
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 y y g two axes of symmetry.
Displacements Translations and Rotations When a structure is subjected to loads it deforms and as a consequence points in the j q p original configuration displace to new positions (the mathematics describing this process are discussed in detail in CVE 513 and CVE 604)
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 a notation that allows for this correspondence.
Consider the following notation and the subscripts in the figure below: The letter A is used to denote actions - this includes concentrated forces and couples. Internal forces and moments at reactions are also considered actions. The letter D is used to denote displacements - this includes translations and rotations. Consider the beam shown below subjected to several actions producing several displacements: Clearly 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 A 1, A 2, and A 3. A 2 and A 3 are applied at the same node. At each node there are three possible displacements for this two dimensional structure: two translations and a rotation.
Each action may contribute to each displacement identified. If we can determine the quantities D 11 through D 33 then by superposition each displacement can be written as follows: D = D + D + D 1 11 12 13 D 2 = D 21 + D 22 + D 23 D 3 = D 31 + D 32 + D 33
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: FX 0 FY = 0 FZ = = 0 M X 0 M Y = 0 M Z = = 0 In the stiffness method of analysis the equilibrium conditions at the joints of the structure are the basic equations that are solved.
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. l ibili h d i h h ibili f h di l 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.
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 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 statically indeterminate to the first degree. The truss in (c) is The truss in (c) is statically indeterminate to second degree.
Let NUA = Number of unknown actions NESE = Number of equations of static equilibrium Unknown Actions [ H R M R ] ( UA ) = R NUA = 4 NESE = 3 A A A B 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
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 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.
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; and c = 0 for a structure with no geometric instability) Two Dimensional Trusses Degree of static indeterminacy = (b + r) - (2j) b = number of members r = number of reactions j = number of fjit joints (this includes ld the joints at the reactions) Three Dimensional Trusses Degree of static indeterminacy = (b + r) - (3j) b = number of members r = number of reactions j = number of fjit joints (this includes ld the joints at the reactions)
Two Dimensional Frames Lecture 4: PRELIMINARY CONCEPTS OF Criteria In Determining Static Indeterminacy (continued) Degree of static indeterminacy = (b + r) - (2j + c) b = number of members r = number of reactions j = number of joints c = number of internal conditions Three Dimensional Frames Degree of indeterminacy = (b + r) - (3j + c) b = Number of members r = Number of reactions j = Number of joints c = Number of internal conditions
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 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 (displacements and rotations).
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 ddegree. 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 of unknowns can reduce the degree of indeterminacy of the structure.
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 FX = 0 FY = 0 M Z = 0 But the summation of forces in the x-direction is not applicable, and the structure is mobile.