4. TRUSSES

Cables and trusses are 2D or 3D structures composed of an assemblage of simple one-dimensional components which transfer only axial forces along their axis. Cables can carry only tensile forces, trusses can carry tensile and compressive forces. Cables tend to be flexible, and hence, they tend to oscillate and therefore must be stiffened. Trusses are extensively used for bridges, long span roofs, electric towers, and space structures.

This chunk introduces the concept of trusses, explaining that they are structural frameworks made of simple components that mostly transfer forces along their length (axial forces). Cables are only able to hold tension and are flexible, which can lead to oscillation. Trusses, on the other hand, can handle both tension and compression, making them suitable for various heavy-duty constructions such as bridges and roofs.

Imagine a hammock (representing a cable) that only supports you when it is pulled tight, versus a bridge (representing a truss) that can support not only your weight (tension) but also the weight of a truck (compression) as it moves across, showing how trusses are robust in various applications.

For trusses, it is assumed that 1. Bars are pin-connected. 2. Joints are frictionless hinges. 3. Loads are applied at the joints only.

This chunk outlines the basic assumptions made when analyzing trusses. First, the bars that make up the truss are assumed to be pin-connected, which allows them to rotate freely at the joints. Second, the joints are treated as frictionless hinges, meaning they do not resist rotation or movement. Lastly, it's assumed that external loads are only applied at these joints, simplifying the calculation of forces.

Think of a mobile hanging from the ceiling where each part can freely rotate at the joints (like pin connections), allowing the mobile to sway gently without resistance. This illustrates how the assumptions about connection types in trusses help simplify calculations in engineering.

A truss would typically be composed of triangular elements with the bars on the upper chord under compression and those along the lower chord under tension. Depending on the orientation of the diagonals, they can be under either tension or compression.

In truss structures, the design typically consists of triangular shapes which help distribute forces evenly. The upper bars or chords experience compression, meaning they are being 'pushed together', while the lower chords are under tension, 'pulling apart'. The diagonal bars can either be under tension or compression depending on their orientation and load direction, which is key in maintaining the integrity of the truss.

Consider a set of ropes supporting a hanging piece of art. The ropes at the top (upper chord) want to push down or compress, while those at the bottom (lower chord) are stretched or in tension, serving as an example of how trusses balance forces akin to a simple hanging art installation.

In a Pratt truss, the diagonal members are under tension, while in a Howe truss, they are in compression. The Pratt design is an excellent choice for steel as its slender members under tension are not prone to buckling. The vertical members are less likely to buckle because they are shorter. On the other hand, the Howe truss is often preferred for heavy timber trusses.

This chunk describes different types of trusses. The Pratt truss has diagonal members that deal with tension forces, making it ideal for lightweight structures like steel frameworks that resist buckling. Conversely, the Howe truss has diagonal members that handle compression, making it suitable for heavier materials like timber. The design choice affects how forces are managed in construction.

Think of a bicycle brake system. The cables tightening to stop the bike represent tension (like in a Pratt truss), while the metal rods that compress to apply brakes are akin to the Howe truss. The way forces are managed differently in each component reflects the design choices made in building trusses.

Trusses are statically determinate when all the bar forces can be determined from the equations of statics alone. Otherwise, the truss is statically indeterminate. A truss may be statically/externally determinate or indeterminate with respect to the reactions (more than 3 or 6 reactions in 2D or 3D problems respectively).

This section clarifies the conditions under which a truss can be statically determined or not. A statically determinate truss allows all internal forces to be found using basic static equations, while an indeterminate truss does not, making the calculations more complex. The number of external reactions involved also determines stability and whether it’s externally or internally determined.

Consider a simple puzzle. If you have enough pieces to solve it (statically determinate), you can complete it without guessing. However, if there are extra pieces or missing edges (statically indeterminate), it becomes challenging, similar to how knowing the reaction forces in a truss helps or hinders analysis.

For 2D trusses, the external equations of equilibrium which can be used to determine the reactions are ΣF_x = 0, ΣF_y = 0 and ΣM = 0. For 3D trusses, the available equations are ΣF_x = 0, ΣF_y = 0, ΣF_z = 0 and ΣM_x = 0, ΣM_y = 0, ΣM_z = 0.

This chunk explains the mathematical foundation for analyzing trusses. For 2D structures, two force equations along the X and Y axes, and one moment equation are essential for finding reactions. For 3D structures, three force equations and three moment equations are necessary. Knowing how to apply these equations allows engineers to analyze complex structures accurately.

Imagine balancing a seesaw. You need to know how much weight is on each side (similar to ΣF), and you need to understand how far they are from the pivot (like ΣM) to ensure balance. This balancing act mirrors how engineers use equilibrium equations to maintain stability in trusses.

In truss analysis, there is no sign convention. A member is assumed to be under tension (or compression). If after analysis, the force is found to be negative, then this would imply that the wrong assumption was made, and that the member should have been under compression (or tension).

This section highlights how assumptions play a role in truss analysis. Initially, engineers assume whether a member is in tension or compression without a strict sign convention. After performing calculations, if a negative value emerges, it indicates the assumption was incorrect, requiring adjustments to how forces are viewed.

Think of a game of 'guess what’s in the box'. If you believe it’s a soft toy (tension) but it turns out to be a rock (compression), your initial guess was wrong. Similarly, in truss analysis, incorrect assumptions lead to a reevaluation of forces acting within the structure.

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