Analysis Of Trusses
Mariam Obregon
One of the most common structures, especially for lightweight construction over long spans, is a truss. A truss consists of a number of long struts or bars (slender members) joined at their ends. The individual pieces are called members and the locations where they meet are called joints. Fig. 6.1 shows examples of trusses.
There are two main types of trusses, spatial and planar trusses (Fig. 6.1). A planar truss, being the topic of this chapter, is a truss with all its members lying in a plane. A common type of planar trusses is a simple truss. A simple truss consists of rigid triangular units in a way that the members of any unit do not cross members of other triangular units. Figure 6.2 shows examples of simple trusses.
To better understand how a simple truss is created, consider the structure shown in Fig. 6.3a. This structure consists of bars joined at their ends using frictionless pins. If a force acts on the structure, the structure deforms and collapses (Fig. 63b). However, the structure becomes stable (rigid) if a diagonal member preventing the deformation is added as shown in Fig. 6.3b. This structure is now a simple truss; it consists of (non-crossing) triangle units (Fig. 6.3c). Each triangle is a rigid unit assuming that the bars are rigid.
The main purpose of a structural analysis on a truss is to determine the internal forces of the members. The member forces are needed for designing the members and joints. To analyze a truss, two simplifying assumptions can be used. These assumptions, idealizing a real truss in practice, are as follows.
This assumption is almost true in practice; a physical truss performs optimally when the loads are applied at its joints. To achieve this, loads are transferred to the joints by beams or other structural members.
For example,consider a bridge with its deck connected to trusses through floor beams (Fig. 6.5a). The loads (weight of vehicles for example) on the deck are supported by the floor beams connected (at their ends) to the joints of the trusses. Figure 6.5b shows the FBD of the deck and beams of the bridge and the support reactions from the truss joints. Fig. 6.5c shows the action and reaction forces between the ends of the beams and the joints of the trusses. Finally, each truss is loaded as demonstrated in Fig. 6.5d.
The weight of the members of a truss are usually negligible in analysis as the weight of a member is much smaller than the member force. However, if the weights of a member are to be considered, a vertical force (in the direction of the gravity) being equal to half of the weight of the member is applied at each end of the member.
The above assumptions result in a truss member acting as a two-force member. To prove this, a truss loaded with arbitrary loads is considered and one of its member () is arbitrarily chosen (Fig. 6.7a). Isolating the member , we draw its FBD as shown in Fig. 6.7b.
Because each end of the member is a pin (hinge) connection, only a force (no moment) acts at each end. An end force can be decomposed along two directions being the member axis, , and an axis perpendicular to the member as shown in Fig.6.7c. The (planar) equations of equilibrium for the member are,
As shown in Fig. 6.8b, the force of a truss member is either a tensile (T) or a compressive (C) force acting at each end. The tensile force tends to elongate the member by pulling on it. The compressive force, on the other hand, tends to shorten the member by pushing on it.
In truss analysis problems, member forces are unknown. There are two methods to solve for these forces, being the method of joints, and the method of sections. Both of these tactics will be expanded upon later in this chapter.
Truss bridges such as the High Level Bridge in Edmonton (Fig. 6.9a) are designed to carry heavy loads and span fairly long distances (longest span of 88 m in the case of the High Level Bridge). However, as mentioned earlier, construction and maintenance costs of the connections between members makes them less cost effective compared to other types of bridges (e.g. girder bridges) for modern applications. For this reason, most truss bridges you see are decades old.
Tower cranes (Fig. 6.9b) are made of trusses for various reasons but the primary reason is to make the crane as light as possible. This makes it easier to construct, deconstruct, and ship cranes from one site to the next on trucks.
Trusses are commonly used to support roofs, particularly those in buildings that require long spans (i.e. distances between supports) like those in athletic facilities (Fig. 6.9c) and airports. Unlike with bridges, roof trusses are protected against the elements so maintenance costs are lower. This makes trusses cost effective for long spans in modern low-rise buildings.
Truss members are also commonly used in steel buildings to resist lateral loads (sideways forces that come from wind and earthquakes). A prominent example of trusses being used as a lateral load resisting system is The Bow skyscraper in Calgary (Fig. 6.9d) which was the tallest building in western Canada when it was completed in 2012. To limit how much The Bow sways when the wind is blowing (which would make occupants uncomfortable), large trusses were added on the building which makes for a dramatic structural (as well as architectural) feature. Most buildings are more modest (e.g. DICE at the University of Alberta) with their lateral load resisting systems but also use truss members to prevent them from swaying excessively.
Trusses do not need to be made of steel. Though wood is much weaker than steel it is lightweight, cheap, and easy to work with (i.e. it can be assembled with hand tools). Wooden trusses similar to the one shown in Figure 6.9e are commonly used to support roofs in houses. The triangular shape of these trusses also makes it easier to construct sloped roofs.
Each FBD is drawn from the perspective of the joint as an isolated body (particle); thus, all forces (even internal member forces) are external forces to the joints. The force of a member is labeled by referring to the labels of the ends (joints) of the member (e.g. or as in Fig. 10). The same label is used in the FBD of the joints.
As already shown in Fig 6.6b and Fig 6.10, a joint is considered to be a point at which member forces are concurrent. Therefore, in truss analysis, a joint can be modeled as a particle and the particle equilibrium equations (the scalar formulation) from Section 5.1 can be applied here,
Since there are two equations of equilibrium at each joint, two unknown member forces at most can be determined per joint. The line of actions of the forces at each joint are already known (they are along the members as explained in the previous section). Therefore, the equations are used to determine the magnitudes and sense of direction (i.e. tension or compression) . The method of joints is illustrated by the following example.
1- Draw the FBD of joint A. Isolating joint A from its surroundings (Fig. 6.11b), we consider the external load and the member forces at the joint. The member forces are assumed as tensile forces (Fig 6.11b) and therefore pointing away from the joint. Since the number of unknown member forces is two, they can be solved for using the equations of equilibrium of joint A.
4- Identify the true sense of directions (tension or compression). Any negative solution indicates that the sense of direction of the force was incorrectly assumed. As we initially assumed a tensile force for a member, any negative value (of a magnitude) implies that the force should be a compressive force and the true direction in the FBD should be the opposite. In this example, and are negative, therefore, they are compressive forces and we write,
To avoid potential confusion, we need to be consistent in assuming the direction of unknown forces, such as all members in tension. In this way, we can identify the true senses quickly, negative values mean compression, positive means tension.
Joint has two unknown support reactions in addition to the unknown member force , therefore the two equations of equilibrium cannot be solved for three unknowns. However, there are two unknowns, a support reaction and the unknown member force at joint . Choosing node , we write,
As it can be observed, the support reaction is also determined. We may choose not to solve for the support reactions if not needed. In this problem though, the support reactions can be obtained by writing the equations of equilibrium for the whole truss (try it yourself).
Zero-force members. Zero-force members are the members that do not carry any member forces. In some cases, due to the external loading situations and/or support designs, a member can behave as a zero-force member. In the first two cases demonstrated below, the zero force members can be easily identified by observation.
1- Two non-collinear members forming a joint are zero-force members if the joint is not externally loaded or not connected to a support. Figure 6.14a shows a general case of this type of zero-force members, i.e. members and .
To prove that members and are zero-force members, the FBD of joint is considered as shown in Fig. 6.14b. Setting a Cartesian coordinate system with the x axis being in the direction of and therefore the y axis perpendicular to the x axis as shown, we can write the equilibrium equations as,
2- A member connected to two collinear members at the same joint is a zero-force member if the joint is not externally loaded or has support reactions. َMember shown in Fig 6.16a is a general case of this kind of zero-force member.
To prove that member is a zero-force member, the FBD of joint is considered as shown in Fig. 6.16b. Setting a Cartesian coordinate system with the x axis being along the line of action of the collinear member forces and and the y axis perpendicular to the x axis (Fig. 6.16b), we can write the equilibrium equations as,
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