Lesson #01 Truss Analysis I

1. Lesson #01Truss Analysis I • 1. Lesson Objectives • At the conclusion of this lesson, you should be able to do the following: •  Define the term truss. •  Explain the assumptions necessary for truss analysis. •  Calculate the internal forces in truss members using: • Method of Joints • Method of Sections

2. 4. Key Definitions • Truss - A structure composed of straight, slender members joined together at their end points and loaded only at the joints. • Member - A component of a structure. In a truss, members are typically made of metal or wood. • Joint - A connection at which two or more members are attached together. In older trusses, joints are typically made from heavy metal pins or from assemblies of metal plates connected together with rivets or bolts. In modern trusses, joints are usually made from metal plates connected together by bolts or welds.

3. Definition and Examples This beautiful structure, the Glen Canyon Dam Bridge over the Colorado River in the U.S., uses an important and very efficient type of structural system called a truss.

4. A truss is a structure composed of straight, slender members joined together at their end points and loaded only at the joints. From this definition, you can infer that truss structures are composed of two types of components—members and joints. The members are usually made of metal or wood, and they are slender, meaning that their length is significantly greater than their width or thickness. Members are connected together at joints, which typically use bolts, rivets, or welds to attach the ends of the members to each other. The result is a rigid framework that is capable of carrying loads very efficiently. Trusses can span long distances and generally have low weight in comparison with the loads they carry. As a result of this load-carrying efficiency, trusses are used in many different practical applications. A few examples are provided below. • The long-span highway bridge pictured below is called a deck truss, because the deck of the bridge—the concrete road surface—rests on top of the main trusses.

10. Assumptions • The principal objective of this lesson is to calculate the internal forces in truss members. Before we can accomplish this objective, however, we must make three important assumptions: • (1) Truss members are connected at their ends only. • (2) Truss members are connected by frictionless pins. • (3) Trusses are loaded only at the joints. • Although these assumptions do not precisely reflect the way that actual trusses are built, they provide reasonably accurate analysis results for most properly configured truss designs. • In CE-301, we will also assume that the weight of the truss members themselves may be neglected in the analysis. This assumption is made primarily for the sake of simplicity in an introductory course. The assumption of negligible self-weight might be reasonably accurate for some kinds of trusses—lightweight short-span roof trusses, for example. But this assumption would not be accurate for long-span trusses like the highway bridges pictured above. The self-weight of such structures would typically be very large in proportion of the total load the bridges carry and thus could not be safely neglected.

11. When the self-weight of a truss is considered in an analysis, the weight of each individual member is divided in half, and each half of the weight is applied to the joint at each end of the member. This procedure ensures that assumption (3) above is not violated. • Given that all members in a truss have frictionless pins at each end, have loads applied only at the joints, and have no applied couple moments, we can conclude that all members in an idealized truss must be two-force members. This observation greatly facilitates the analysis of trusses, because it leads to the conclusion that the internal force in each truss member is oriented along the longitudinal axis of the member. Thus a truss member will always be subjected to either tension or compression (as defined in Lesson 7).

12. Method of Joints • There are two methods for calculating internal forces in truss members—the Method of Joints and the Method of Sections. Both methods derive from a simple but powerful concept: if a structure is in equilibrium, then every part of that structure must also be in equilibrium. • The general procedure for the Method of Joints is as follows: • (1) Isolate a single joint, and draw a free body diagram of it. Replace every member connected to the joint with an unknown force drawn at the same orientation as the member. It is advisable to assume that all unknown member forces are in tension. The free body diagram must also include all external loads and reactions applied at the joint. • (2) Write the relevant equations of equilibrium for the joint and, if possible, solve for the unknown member forces. Because the unknown forces are assumed to be in tension on the diagram, a positive result means that the member is in tension, while a negative result indicates compression. Because the free body diagram of the joint represents a concurrent force system, there are only two independent equations of equilibrium available at each joint. Thus, in applying the Method of Joints, it is always advisable to select joints for which only two (or fewer) unknown forces are exposed.

14. We can now proceed to joint B. Here there are three connected members (AB, BD, and BC), and so the free body diagram of this joint might appear to have three unknown forces. However, the force FAB is already known from the analysis of joint A; thus, FBCand FBD are the only unknown forces at joint B. Once again, they can be calculated using ΣFx=0 and ΣFy=0. • At this point, there does not appear to be a joint for which the free body diagram will have only two unknown forces. For example, joint C will have three unknowns—FCD, FCF, and FCE. Joint D will also have three unknowns—FCD, FDF, and the unknown reaction caused by the roller support. One approach to solving this problem is to write the equilibrium equations for all the joints in the truss, and then solve the system of equations simultaneously. However, in this case, it is considerably easier to draw a free body diagram of the truss as a whole and then solve for the unknown reaction at joint D. Once this reaction is known, then FCDand FDF can be calculated from the equations of equilibrium for joint D. And once FCDis known, joint C can be analyzed to determine FCF and FCE. The remainder of the analysis would proceed in this manner, stepping from joint to joint, until all of the member forces have been calculated. • Example Problems 6-18 and 6-19 provide good illustrations of the Method of Joints. It is important to note that, even though both of these solutions begin with a calculation of the external reactions of the truss, it is not always necessary to solve for reactions in a truss analysis.

15. Method of Sections • The Method of Sections uses essentially the same procedure as the Method of Joints, except that a section of the truss is isolated, rather than just a single joint. The general procedure is as follows: • (1) Isolate a section of the truss by cutting through members. Draw a free body diagram of the section, including all external loads and reactions that are applied to the isolated section. Every member that was cut in order to isolate the section should be replaced with an unknown force drawn at the same orientation as the member. Again, it is advisable to assume that all unknown member forces are in tension. • (2) Write the relevant equations of equilibrium for the section and, if possible, solve for the unknown member forces. In most cases, the free body diagram of a section will be a non-concurrent force system, resulting in three independent equations of equilibrium being available to solve for the unknown forces. Thus, in applying the Method of Sections, it is always advisable to make the section cut such that three (or fewer) unknown forces are exposed. • The Method of Sections is particularly useful when a small number of specific member forces need to be calculated in the middle of a truss. For example, if we only needed to calculate FGE and FHE in the truss below, the Method of Sections would be far more efficient than the Method of Joints. To calculate FGE and FHE, we would first make the section cut shown, then isolate the left-hand side of the truss and draw a free body diagram of the section. Note that the cut must pass through members GE, HE, and HF, in order to fully isolate the truss. This cut exposes three unknown forces, FGE, FHE, and FHF. When the unknown reactions at joint K are included, there are five unknown forces to solve for. The force system is non-concurrent, and so there are three equilibrium equations available. Once again, the problem can only be solved by first drawing a free body diagram of the entire structure and solving for the reactions Kx and Ky. Once these two forces are known, FHE can be calculated from ΣFy=0, and FGE from ΣMH=0.

17. Example Problems 6-20 and 6-21 provide good illustrations of the Method of Sections. Note that Problem 6-21 is solved without calculating the reactions of the structure. • Most students find the most challenging aspect of truss analysis to be the selection of appropriate joint cuts or section cuts. This skill is best acquired through practice; however, you may also find the following guidance helpful: • (1) Whether making joint cuts or section cuts, always cut the member(s) of interest, in order to expose the unknown force(s) you are required to solve for. • (2) The cut must completely isolate a portion of the structure—either a joint or a section. In other words, you must always cut all the way through the truss. • (3) Attempt to make cuts in a manner that exposes only as many internal member forces as there are equilibrium equations available to solve. For the Method of Joints, attempt to expose no more than two unknown forces with each cut. For the Method of Sections, expose no more than three unknowns.

19. End of lesson Question