Understanding the Application of the Equations of Equilibrium Procedure for Analysis 1 Free-Body Diagrams • Disassemble the structure and draw a freebody diagram of each member. • Recall that reactive forces common to two members act with equal magnitudes but opposite directions on the respective free-body diagrams of the members. • All two-force members should be identified. These members, regardless of their shape, have no external loads on them, and therefore their free-body diagrams are represented with equal but opposite collinear forces acting on their ends. • In many cases it is possible to tell by inspection the proper arrowhead sense of direction of an unknown force or couple moment 2 Equations of Equilibrium • Count the total number of unknowns to make sure that an equivalent number of equilibrium equations can be written for solution. • Many times, the solution for the unknowns will be straightforward if the moment equation is applied about a point (©MO = 0) that lies at the intersection of the lines of action of as many unknown forces as possible. • When applying the force equations and ©Fx = 0 ©Fy = 0, orient the x and y axes along lines that will provide the simplest reduction of the forces into their x and y components. • If the solution of the equilibrium equations yields a negative magnitude for an unknown force or couple moment, it indicates that its arrowhead sense of direction is opposite to that which was assumed on the free-body diagram. Sample Problems Determine the Reactions of the Beam Shown: Regular Figure Solution: FBD Sample Problems Determine the Reactions of the Beam Shown: Solution: Sample Problems Determine the reactions on the beam. Assume A is a pin and the support at B is a roller (smooth surface). Solution: Analysis of Statically Determinate Trusses TRUSS A truss is a structure composed of slender members joined together at their end points. Bridge Trusses Roof Trusses are often used as part of an industrial building frame Are often used to support bridges Analysis of Statically Determinate Trusses ASSUMPTIONS FOR THE DESIGN 1 The members are joined together by smooth pins. In cases where bolted or welded joint connections are used, this assumption is generally satisfactory provided the center lines of the joining members are concurrent at a point 2 All loadings are applied at the joints. In most situations, such as for bridge and roof trusses, this assumption is true. Frequently in the force analysis, the weight of the members is neglected, since the force supported by the members is large in comparison with their weight. Classification of Coplanar Trusses SIMPLE TRUSS Simple Trusses are made with Stable Elements such as shown in the figure, commonly, triangular members. For this method of construction, however, it is important to realize that simple trusses do not have to consist entirely of triangles. Analysis of Statically Determinate Trusses COMPOUND TRUSS 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. Analysis of Statically Determinate Trusses COMPLEX TRUSS A complex truss is one that cannot be classified as being either simple or compound. METHOD OF JOINTS If a truss is in equilibrium, then each of its joints must also be in equilibrium. Hence, the method of joints consists of satisfying the equilibrium conditions and for the forces exerted Fx = 0 Fy = 0 on the pin at each joint of the truss. METHOD OF JOINTS Procedure for Analysis METHOD OF JOINTS Determine the force at joint A, G and B of the roof truss shown in the photo. The dimensions and loadings are shown. State whether the members are in tension or compression. Solution: At Joint A At Joint G At Joint B METHOD OF JOINTS Determine the force at the scissor roof truss shown in the photo. The dimensions and loadings are shown. State whether the members are in tension or compression. METHOD OF SECTION If the forces in only a few members of a truss are to be found, the method of sections generally provides the most direct means of obtaining these forces. The method of sections consists of passing an imaginary section through the truss, thus cutting it into two parts. Provided the entire truss is in equilibrium, each of the two parts must also be in equilibrium; and as a result, the three equations of equilibrium may be applied to either one of these two parts to determine the member forces at the “cut section.” METHOD OF SECTION Procedure for Analysis METHOD OF SECTION Determine the force in members GJ and CO of the roof truss shown in the photo. The dimensions and loadings are shown. State whether the members are in tension or compression. The reactions at the supports have been calculated. MEMBER GC MEMBER CF METHOD OF SECTION Determine the force in members GF and GD of the truss shown. State whether the members are in tension or Compression. The reactions at the supports have been calculated. CABLES AND ARCHES Cables subjected to Concentrated Load When a cable of negligible weight supports several concentrated loads, the cable takes the form of several straight-line segments, each of which is subjected to a constant tensile force. CABLES AND ARCHES Example: Determine the tension in each segment of the cable shown. Also, what is the dimension h? Solution: From Figure: @ Point C: @ Point B: CABLES AND ARCHES Cables subjected to Uniformly Distributed Load 𝑻𝑨𝒚 𝑻𝑨 𝑻𝑩𝒚 𝑻𝑨𝒙 h1 𝑻𝑩 A B x L-x L W (Kn/m) h2 𝑻𝑩𝒙 CABLES AND ARCHES Example: Solve for the value of the Tension at A, Slope of the cable at A and Tension at the lowest Point C. The Cable shown supports a horizontal uniform load of 10 Kn/m. 12 A B 30 6 CABLES AND ARCHES The Three-Hinged Arch is subjected to the two concentrated forces as shown, find the Reaction at A, B, C. 20 kN B 25 15 8 kN 6 C A 23 7.6 23 Thank You!