6.1 Chapter 6: Structural Analysis Chapter Objectives • To show how to determine the forces in the members of a truss using the method of joints and the method of sections. • To analyze the forces acting on the members of frames and machines composed of pin-connected members. In this chapter, we shall consider problems dealing with the equilibrium of structures made of several connected parts. • These problems call for the determination of the following forces. 1. External forces acting on the structure (reactions). 2. “Internal forces” – forces that hold together the various parts of the structure. Newton’s Third Law states, “The forces of action and reaction between bodies in contact have the same magnitude, same line of action, and opposite sense.” In this chapter, we shall consider three broad categories of engineering structures. 1. Trusses. • Trusses consist of straight members connected at joints located at the ends of the members. • Members of a truss are “two-force” members. 2. Frames. • Frames always contain at least one “multi-force member.” 3. Machines. • Machines are designed to transmit and modify forces, and are structures containing moving parts. • Machines, like frames, always contain at least one “multi-force member.” 6.1 Simple Truss A truss is one major type of engineering structures and is used in bridges and buildings. • A truss consists of straight members connected at joints. • Truss members are connected at their ends only. • No member is continuous through a joint.
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6.1
Chapter 6: Structural Analysis
Chapter Objectives
• To show how to determine the forces in the members of a truss using the method
of joints and the method of sections.
• To analyze the forces acting on the members of frames and machines composed of
pin-connected members.
In this chapter, we shall consider problems dealing with the equilibrium of structures
made of several connected parts.
• These problems call for the determination of the following forces.
1. External forces acting on the structure (reactions).
2. “Internal forces” – forces that hold together the various parts of the
structure.
Newton’s Third Law states, “The forces of action and reaction between bodies in
contact have the same magnitude, same line of action, and opposite sense.”
In this chapter, we shall consider three broad categories of engineering structures.
1. Trusses.
• Trusses consist of straight members connected at joints located at the ends of
the members.
• Members of a truss are “two-force” members.
2. Frames.
• Frames always contain at least one “multi-force member.”
3. Machines.
• Machines are designed to transmit and modify forces, and are structures
containing moving parts.
• Machines, like frames, always contain at least one “multi-force member.”
6.1 Simple Truss
A truss is one major type of engineering structures and is used in bridges and
buildings.
• A truss consists of straight members connected at joints.
• Truss members are connected at their ends only.
• No member is continuous through a joint.
6.2
Truss Frame
The members of a truss are slender and can support little lateral load.
• All loads must be applied at the joints and not along the members themselves.
• In the case of bridge trusses, the dead
loads and traffic loads from the deck are
first carried by “stringers” which in turn
transmit the loads to “floor beams”, and
then the loads are finally transmitted to
the joints of the supporting side trusses
(ref. Fig. 6-2 in textbook).
Assumptions for Design
1. All loadings are applied at the joints.
2. The members are joined together by smooth pins.
Because of these two assumptions, each truss member acts as a two-force
member.
• The member is either in tension or compression.
Although the members are actually joined together by means of riveted, bolted, or
welded connections, the members are considered to be pinned together.
• The forces acting at each end of the member reduce to a single force and no
couple.
• Each member is treated as a two-force member.
Actual connection (Gusset Plate)
The moments that are created at
the ends of the members are small
and considered insignificant.
6.3
Simple Truss
The triangle ABC is considered the “basic truss.”
• The truss is said to be a “rigid truss” meaning
that the truss is stable and will not collapse.
• The only possible deformation involves small
changes in the length of its members.
A large truss may be obtained by successively adding two members, attaching them
to separate existing joints, and connecting them at a new joint.
• A truss constructed in this manner is called a “simple truss.”
• In a simple truss, the total number of members is related to the total number
of joints by the following equation.
m = 2 n – 3
where
n = the total number of joints
m = the total number of members
6.2 The Method of Joints
A truss may be considered as a group of pins and two-force members.
• We can dismember a truss and draw a free-body diagram for each pin.
• Since the entire truss is in equilibrium, each pin must be in equilibrium.
Consider the truss shown below.
6.4
Each pin represents a concurrent force system.
• At each pin we can write only two equations of equilibrium (a.k.a. equations of
statics).
∑ Fx = 0 and ∑ Fy = 0
In the case of a simple truss, the number of members and the number of pins
(joints) are related by the following equation.
m = 2 n – 3
where
n = the number of pins (joints)
m = number of members
and the number of unknowns that can be determined by the pins is
m + 3 (i.e. 2 n = m + 3).
• Thus, the forces in all the members plus the two components of the reaction at
A (i.e. Ax and Ay) and the vertical reaction at C (i.e. Cy) may be found using the
free-body diagrams of the pins.
• Typically, the entire truss is treated as a rigid body to determine the reactions
at the supports.
The “Method of Joints” solution is good if the entire truss is to be analyzed (i.e. all
the forces in each member is required).
• The solution must begin where there are only two unknown forces, usually at one