Engineering Mechanics : Trusses and Frames

When the truss members lie essentially in a single plane, that type of truss is called a plane truss. When the members of the truss lie in three-dimension, the truss is called a space truss. 

CLASSIFICATION OF TRUSS

• EFFICIENT OR PERFECT TRUSS- If the number of members in the truss is just sufficient to prevent distortion of its shape when loaded externally, it called a perfect truss. A perfect truss satisfies the equation m = 2j-3, where  j is the number of joints and m is the number of members.

• DEFICIENT OR COLLASIBLE TRUSS- When the number of members is less then 2j - 3, it is a deficient truss. It is an imperfect truss. It is also known as collapsible truss as under the action of collapsible forces, the truss tends to collapse.

• REDUNDANT TRUSS- An imperfect truss in which the number of members is more than 2j-3 is called the redundant truss.

Why to provide redundant member

• To maintain alignment of two members during construction
• To increase stability during construction
• To maintain stability during loading (Ex: to prevent buckling of compression members)
• To provide support if the applied loading is changed
• To act as backup members in case some members fail or require strengthening

2 force member- A member of the truss is a two-force member. Here, the forces are collinear and therefore a member of the truss would either be in tension or compression.

ASSUMPTIONS OF TRUSS ANALYSIS

• Members are linked by their ends by smooth frictionless pins. The weight of members is negligible.
• Loads and reactions are applied to truss at joints only.
• The line connecting the joint centres present at the end of the member coincides with the centroidal axis of each member.
• All members function as two-force members ie, they are subjected to either tension or compression.

It is physically impossible for all these conditions to be satisfied exactly in an actual truss, and therefore a truss in which these idealized conditions are assumed is called an ideal truss.

EQUATION OF EQUILIBRIUM- When a balance of force and moment is maintained, a structure or one of its members are in equilibrium. Force and moment equation of equilibrium must  be satisfied for this.

ΣF_X= 0  ΣF_Y = 0 and ΣM = 0

Whenever these equations are applied, it is first necessary to draw a free body diagram of the truss or its members. For the selected member, only its outlined shapes must be drawn free from its supports and surroundings.

ANALYTICAL METHOD OF TRUSS ANALYSIS

• METHOD OF JOINTS- For analyzing or designing a truss, the force in each of its members must be obtained. If we were to consider a free body diagram of the entire truss, then the forces in the members would be internal forces, and they could not be obtained from an equilibrium analysis. Instead if we consider the equilibrium of a joint of the truss then member force becomes an external force on the joints free body diagram, and the equations of equilibrium can be applied to obtain its magnitude. Hence, method of joints is based upon this.

At each joint, the forces in the members meet at the joint and these forces at the joint, if any, constitute a system of concurrent forces. Two independent equations of equilibrium can therefore be formed at each joint  i.e. ΣF_H = 0 And ΣF_V = 0 

When three members are meeting at a joint where no load acts and out of these three, two are collinear, then the force in third member will be zero, as shown below.

• METHOD OF SECTIONS- When few members of the truss are to be analyzed for forces, we can do it by the method of sections. If a body is in equilibrium then its any part is also in equilibrium, and this forms the basis of the method of sections.

When two pairs of collinear members are joined as shown in figure, the forces in each pair must be equal and opposite.

In the method of sections, a section is of the truss is cut, such that not more than three unknown forces are required to be computed. Free body diagram is drawn for either of its two parts.

Three equilibrium equations i.e. ΣF_X = 0 ,ΣF_Y = 0 and ΣM = 0 for a plane are used to find out three unknown forces. The force in almost any desired member may be found directly from an analysis of a section which has cut that member is a basic advantage of the method of sections method. Firstly, the reactions are needed to be found.

Structural Analysis: Space Truss

• 6 bars joined at their ends to form the edges of a tetrahedron as the basic non-collapsible unit.
• 3 additional concurrent bars whose ends are attached to three joints on the existing structure are required to add a new rigid unit to extend the structure.
• If center lines of joined members intersect at a point. If two force members assumption is justified i.e. each member under Compression or Tension. Such a truss is called a simple space truss.

Structural Analysis: Space Truss

• For a truss with “m” number of two force members, and maximum 6 unknown support
  reactions : Total Unknowns = m + 6
• Therefore:
  m + 6 = 3j: Statically Determinate Internally
  m + 6 > 3j: Statically Indeterminate Internally
  m + 6 < 3j: Unstable Truss

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