Tos Exam 2.docx

Equilibrium - A structure is considered to be in equilibrium if, initially at rest, it remains at rest when subjected to a system of forces and couples.

External Forces • External forces are the actions of other bodies on the structure under consideration.  Applied forces - usually referred to as loads (e.g., live loads and wind loads), have a tendency to move the structure and are usually known in the analysis. • Reaction forces - or reactions, are the forces exerted by supports on the structure and have a tendency to prevent its motion and keep it in equilibrium. The reactions are usually among the unknowns to be determined by the analysis. The state of equilibrium or motion of the structure as a whole is governed solely by the external forces acting on it.

Internal Forces • Internal forces - are the forces and couples exerted on a member or portion of the structure by the rest of the structure. Principle of superposition - The principle of superposition forms the basis for much of the theory of structural analysis. It may be stated as follows: The total displacement or internal loadings (stress) at a point in a structure subjected to several external loadings can be determined by adding together the displacements or internal loadings (stress) caused by each of the external loads acting separately. Two requirements must be imposed for the principle of superposition to apply: 1. The material must behave in a linear-elastic manner, so that Hooke’s law is valid, and therefore the load will be proportional to displacement. 2. The geometry of the structure must not undergo significant change when the loads are applied.

Plane and Space Trusses Truss - A truss is a structure composed of slender members joined together at their end points. The joint connections are usually formed by bolting or welding the ends of the members to a common plate, called a gusset plate. Common Types of Trusses Plane Trusses -If all the members of a truss and the applied loads lie in a single plane, the truss is called a plane truss. Plane trusses are commonly used for supporting decks of bridges and roofs of buildings. Coplanar Trusses Simple Trusses - The simplest framework that is rigid or stable is a triangle. Consequently, a simple truss is constructed by starting with a basic triangular element. Compound Trusses - A compound truss is formed by connecting two or more simple trusses together. Quite often this type of truss is used to support loads acting over a large span, since it is cheaper to construct a somewhat lighter compound truss than to use a heavier single simple truss. Complex Trusses - A complex truss is one that cannot be classified as being either simple or compound. Space Trusses - are analyzed as three-dimensional bodies subjected to three-dimensional force systems.

Determinacy and Stability Determinacy statically determinate - When all the forces in a structure can be determined strictly from these equations. statically indeterminate - Structures having more unknown forces than available equilibrium equations. compatibility equations – the additional equations needed to solve for the unknown reactions are obtained by relating the applied loads and reactions to the displacement or slope at different points on the structure.

Beams and Frames The approach that can be used for determining the static instability, determinacy, and indeterminacy of internally unstable structures is as follows 𝒓 + 𝒇 < 𝟑𝒏 Statically unstable 𝒓 + 𝒇 = 𝟑𝒏 Statically determinate 𝒓 + 𝒇 > 𝟑𝒏 Statically indeterminate For indeterminate structures 𝒊 = 𝒓 + 𝒇 − 𝟑𝒏 where: 𝒓 = Support reactions 𝒇 = Internal forces 𝒏 = Members or portions 𝒊 = Degree of external indeterminacy

Truss Plane Truss The approach that can be used for determining the static instability, determinacy, and indeterminacy of internally unstable structures are as follows 𝒓 + 𝒃 < 𝟐𝒋 Statically unstable 𝒓 + 𝒃 = 𝟐𝒋 Statically determinate 𝒓 + 𝒃 > 𝟐𝒋 Statically indeterminate where: 𝒓 = Support reactions 𝒃 = Truss members 𝒋 = Joints Space Truss Realizing that in three dimensions there are three equations of equilibrium available for each joint then for a space truss 𝒓 + 𝒃 < 𝟑𝒋 Statically unstable 𝒓 + 𝒃 = 𝟑𝒋 Statically determinate 𝒓 + 𝒃 > 𝟑𝒋 Statically indeterminate where: 𝒓 = Support reactions 𝒃 = Truss members 𝒋 = Joints

Stability Partial Constraints - In some cases a structure or one of its members may have fewer reactive forces than equations of equilibrium that must be satisfied. Improper Constraints - In some cases there may be as many unknown forces as there are equations of equilibrium; however, instability or movement of a structure or its members can develop because of improper constraining by the supports. ** Another way in which improper constraining leads to instability occurs when the reactive forces are all parallel. Internal loadings Method of sections - Internal load at a specified point in a member can be determined Sign Convention - An easy way to remember sign convention is to isolate a small segment of the member and note that positive normal force tends to elongate the segment, positive shear tends to rotate the segment clockwise, and positive bending moment tends to bend the segment concave upward,