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Chapter Objectives Chapter Outline To find forces in Truss by
Method of joints Method of sections To analyse internal forces in a frame and machine composed of pin-connected members Chapter Outline Simple Trusses The Method of Joints Zero-Force Members The Method of Sections Frames and Machines
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6.1 Simple Trusses Truss is composed of connected members where joints are allowed to rotate (no bending moment) and the members ideally support either compression or tension only Planar Trusses Use in roof or a long span bridge The weight of the roof is transferred by a truss structure only joints at purlin
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6.1 Simple Trusses Planar Trusses
Analyse internal forces in members in 2D For bridges and roofs, 2D planar trusses are used frequently for ease of design and construction
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6.1 Simple Trusses Assumptions
External forces acting at the joints only - Neglect the weights of members except for detail designs Joints are pin connection and no friction - Joints are connected with all lines of actions of all member and lines of members share a common point
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6.1 Simple Trusses Simple Truss Truss is rigid
Truss is composed of triangles
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6.2 The Method of Joints All external forces, e.g. support reactions, can be found from equilibrium To analyse truss, forces in all members must be found Internal forces in members are either tension or compression Internal forces at joints share a common point and lie in the same Apply ∑Fx = 0 and ∑Fy = 0 at each joint to find internal forces
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6.2 The Method of Joints Procedure for Analysis
Consider FBD at the joints with not more than 2 unknown force If there are is FBD as mentioned above, find forces from support reactions first Find all angles using geometry in x, y plane Apply ∑Fx = 0 & ∑Fy = 0 Find unknown force
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Example 6.1 Determine the force in each member of the truss and indicate whether the members are in tension or compression.
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Solution 2 unknown member forces at joint B
2 uknown member forces and 1 unknown reaction force at joint C 2 unknown member forces and 2 unknown reaction forces at point A For Joint B,
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Solution For Joint C, For Joint A,
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Solution FBD of each pin shows the effect of all the connected members and external forces applied to the pin FBD of each member shows only the effect of the end pins on the member
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6.3 Zero-Force Members ใช้ช่วยในการวิเคราะห์ด้วยวิธีจุดต่อ
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Example 6.4 Using the method of joints, determine all the zero-force members of the Fink roof truss. Assume all joints are pin connected.
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Solution For Joint G, GC is a zero-force member. For Joint D,
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Solution For Joint F, For Joint B and H
FHC satisfy ∑Fy = 0 and therefore HC is not a zero-force member.
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6.4 The Method of Sections Can be used to find internal forces
For an object in equilibrium, all members in the object must also be in equilibrium To find internal forces in members, cut the truss and separate into two parts and display all forces acting on the forces in member, both internal and external forces
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6.4 The Method of Sections Consider a truss below, the truss is cut along a-a For members that are cut, internal forces are equal in magnitude but in opposite direction based on Newton’s third
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6.4 The Method of Sections Procedure for Analysis Free-Body Diagram
Choose where to cut a truss Find all support reactions Apply equilibrium equations to find internal forces along the cut Use FBD that is simple, i.e. not many forces Find angles for unknown forces or as necessary Apply equilibrium equations ∑Fx = 0 & ∑Fy = 0 & ∑Mz = 0
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Example 6.5 Determine the force in members GE, GC, and BC of the truss. Indicate whether the members are in tension or compression.
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Solution Choose section a-a since it cuts through the three members
Draw FBD of the entire truss
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Solution Draw FBD for the section portion
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6.6 Frames and Machines Composed of many parts connected by pin joints
Apply equilibrium equations for each member to find internal forces FBD Separate each part by sketching external forces for each part Display all known forces and moments Use unknown for unknown forces and moments Find distance as nd angles if known Apply equilibrium equationsใช้สมการสมดุล Assume the directions of unknown forces sketch FBD
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Example 6.9 For the frame, draw the free-body diagram of (a) each member, (b) the pin at B and (c) the two members connected together.
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Solution Part (a) BA and BC are not two-force member
AB is subjected to the resultant forces from the pins
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Solution Part (b) Pin at B is subjected to two forces, force of the member BC and AB on the pin For equilibrium, forces and respective components must be equal but opposite Bx and By shown equal and opposite on members AB
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Solution Part (c) Bx and By are not shown as they form equal but opposite internal forces Unknown force at A and C must act in the same sense Couple moment M is used to find reactions at A and C
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