Assumptions in Truss Analysis
Analysis of a Truss is generally a complex process. We need to simplify such a process. To simplify the process, we make some assumptions that simplify the analysis of truss work significantly. Some assumptions are as follows:
- All the joints in the structure can be represented by a pin connection, i.e, those members are free to rotate at the joints.
The members of a truss are rigidly connected using a gusset plate.
2. Loads are only applied at the joints of the truss. Loads are never applied in the middle of the member as all joints are pinned and members cannot carry Bending Moment, they can carry only Axial Loads.
Each member has to be in equilibrium, therefore, the forces acting at each end of the member must be equal and opposite.
Types of Truss
There are various types of trusses. Different designs carry loads in different ways. The members of these trusses are located in the same plane, these are called Planar Trussses and we can analyze them as Two - Dimensional structures. Following are the various types of trusses:-
- Fink Roof Truss
- Fan Roof Truss
- King Post Truss
- Howe Roof Truss
- Pratt Roof Truss
- Modified Queen Roof Truss
- Howe Bridge Truss
- Pratt Bridge Truss
- Warren Bridge Truss
- Parker Bridge Truss
- Baltimore Bridge Truss
- K Bridge Truss
Types of Truss Based on Determinacy
If the number of unknown forces (reactions and internal forces) of a given structure is equal to equilibrium equations, the structure is known as a determinate structure. Determinacy is of two types, one is Internal Indeterminacy & other is External Indeterminacy.
- Statically Determinate Truss
- Statically Indeterminate Truss
Statically Determinate Truss
If it is possible to determine the reactions, forces, and internal forces in members of a truss by applying an equilibrium equation, that truss is known as a Statically Determinate Truss. To design or analyze a truss, it is essential to determine the force in each of its members. The main purpose is to check whether the members can carry the applied loads without failing. A truss is considered statically determinate if all of its support reactions and member forces can be found using only the equations of equilibrium. For a planar truss to be statically determinate, the number of members added to the number of support reactions must not exceed the number of joints multiplied by 2. This condition is the same as that used previously as a stability criterion.
While solving problems on Statically Determinate Trusses, we consider some assumptions like, Members are subjected to axial forces only (Compression or Tension), Shear Force and Bending Moment are neglected, Self Weight of the members is ignored, Members are assumed to be linear, All joints are smooth, frictionless hinges, Loads and Reactions will act directly or indirectly at joints only.
Degree of Static Indeterminacy
- DS = m+re – 2j where, DS = Degree of static indeterminacy m = Number of members, re = Total external reactions, j = Total number of joints
- DS = 0 ⇒ Truss is determinate
If Dse = + 1 & Dsi = –1 then DS = 0 at specified point.
- DS > 0 ⇒ Truss is indeterminate or dedundant.
Statically Indeterminate Truss
If it is not possible to determine the reaction forces and internal forces in the members of the truss by applying equilibrium equations, it is known as a Statically Indeterminate Structure. It is defined as structures that can't be statically analyzed using only equilibrium equations (statics). These structures indicate that there's at least one more unknown force than there are equations of equilibrium, meaning that the sum of forces and moments in each direction is equal to zero. In 2D structures, there are three equations of equilibrium. Statically Indeterminate Structures are analyzed by Force method or Displacement method. In the Force method, redundant forces are treated as unknowns. In the Displacement method, displacements are treated as unknown.
Zero Force Members
Some members don’t carry any load, they are known as Zero Force Members. There are different ways to find these zero-force members. The purpose of Zero Force Members is to provide stability to the structure and to avoid failure because of unexpected loads. Some of the ways are:
- In a pin joint, if the number of members is three and two of the members are in the same line, the force in the third member is zero. ( No load, No reaction at the joint )
- At the pin joint, if the number of members is 2 and they are in different lines, then the force on both members is zero. ( No load, No reaction shall be present at that joint)
Truss Member Carrying Zero forces
(i) M1, M2, M3 meet at a joint
M1 & M2 are collinear
⇒ M3 carries zero force
where M1, M2, M3
(ii) M1 & M2 are non collinear and Fext = 0
⇒ M1 & M2 carries zero force.
Methods of Joints
Method of Joints is used to find the unknown forces acting on members of a truss. Method of Joints is usually the fastest and most convenient way to find out the unknown forces in truss structure. Equilibrium of a joint is considered in the Method of Joints. Following are the steps that need to be followed to use this method.
- Draw a Free Body Diagram of the truss and solve for reaction forces. Use equilibrium equations to calculate the reaction force.
- Draw FBD of each joint and solve the internal forces. (Unknown forces are found using equilibrium equations.) For members in tension, the force will be acting away from the joint, and for members in compression, the force will be acting towards the joint.
Taking equilibrium of the horizontal forces.
∑Fx = 0, HA = 0
Equilibrium of Vertical Forces
∑Fy = 0
FA + FE = 20kN
MC = 0
dFA = dFE
FA = 10kN
FE = 10 kN
To find force in each joint.
Eg. – Joint A
F AB = -10/Sin 60 = F–11.5kN
F AC = F AB Cos cos 60 = – 5.75 kN
Like this we can find out forces in each joint.
Disadvantages of Method of Joints
To find out the forces in any internal member, we have to find out the forces in prior members. Hence, the Method of Joints is a lengthy method. Hence, it is a time taking method. Another disadvantage of this method is that if the number of members is greater than two, then this method is not suitable.
Method of Section
The equilibrium of a structure is considered in the Method of Sections. In this method, we need to pass a section through the member which we have chosen. A section may divide the entire structure into two separate parts. It is preferred that the section must pass through three members. It is a time-effective method as there is no need to find the force in a prior member to find the force in any chosen internal member. A section may be horizontal, inclined, or vertical. Following are the steps that need to be followed to use this method.
- Draw a Free Body Diagram of truss and solve for reaction forces. Use equilibrium equations to calculate the reaction force.
- Cut the truss through the members of interest.
- Apply the equilibrium equations to solve the internal forces.
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