Simple trussesSimple trusses

A truss structure is composed of slender members A truss structure is composed of slender members joined together at their end pointsjoined together at their end points

Members are commonly wooden struts or metal barsMembers are commonly wooden struts or metal bars Joint connections are formed by bolting or welding the Joint connections are formed by bolting or welding the

ends of the members to a common plate (gusset ends of the members to a common plate (gusset plate) or by simply passing a large bolt or pin through plate) or by simply passing a large bolt or pin through each of the memberseach of the members

Planar trusses lie in a single plane (often seen Planar trusses lie in a single plane (often seen supporting roofs and bridges), 2-D analysis of forces supporting roofs and bridges), 2-D analysis of forces appropriateappropriate

Method of jointsMethod of joints

If a truss is in equilibrium, then each of its joints must If a truss is in equilibrium, then each of its joints must also be in equilibriumalso be in equilibrium

The method of joints consists of satisfying the The method of joints consists of satisfying the equilibrium conditions for the forces exerted “on the equilibrium conditions for the forces exerted “on the pin” at each joint of the trusspin” at each joint of the truss

Truss members are all straight two-force members Truss members are all straight two-force members lying in the same planelying in the same plane– The force system acting at each pin is coplanar and The force system acting at each pin is coplanar and

concurrent (intersecting)concurrent (intersecting)– Rotational or moment equilibrium is automatically Rotational or moment equilibrium is automatically

satisfied at the joint, only need to satisfy ∑ Fsatisfied at the joint, only need to satisfy ∑ Fxx = 0, ∑ F = 0, ∑ Fyy = = 00

Method of joints – Procedure for Method of joints – Procedure for analysisanalysis

Draw the free-body diagram of a joint having at least Draw the free-body diagram of a joint having at least one known force and at most two unknown forces one known force and at most two unknown forces (may need to first determine external reactions at the (may need to first determine external reactions at the truss supports)truss supports)

Establish the sense of the unknown forcesEstablish the sense of the unknown forces– Always assume the unknown member forces acting on Always assume the unknown member forces acting on

the joint’s free-body diagram to be in tension (pulling on the joint’s free-body diagram to be in tension (pulling on the “pin”)the “pin”)

– Assume what is believed to be the correct sense of an Assume what is believed to be the correct sense of an unknown member forceunknown member force

– In both cases a negative value indicates that the sense In both cases a negative value indicates that the sense chosen must be reversedchosen must be reversed

Method of joints – Procedure for Method of joints – Procedure for analysisanalysis(continued)(continued)

Orient the x and y axes such that the forces can be Orient the x and y axes such that the forces can be easily resolved into their x and y componentseasily resolved into their x and y components

Apply ∑ FApply ∑ Fxx = 0 and ∑ F = 0 and ∑ Fyy = 0 and solve for the unknown = 0 and solve for the unknown member forces and verify their correct sensemember forces and verify their correct sense

Continue to analyze each of the other joints, choosing Continue to analyze each of the other joints, choosing ones having at most two unknowns and at least one ones having at most two unknowns and at least one known forceknown force

Members in compression “push” on the joint and Members in compression “push” on the joint and members in tension “pull” on the jointmembers in tension “pull” on the joint

Mechanics of Materials and building codes are used to Mechanics of Materials and building codes are used to size the members once the forces are knownsize the members once the forces are known

Zero-force membersZero-force members

Truss analysis using the method of joints is greatly Truss analysis using the method of joints is greatly simplified if one is able to determine those members which simplified if one is able to determine those members which support no loading (zero-force members)support no loading (zero-force members)

These zero-force members are used to increase stability of These zero-force members are used to increase stability of the truss during construction and to provide support if the the truss during construction and to provide support if the applied loading is changedapplied loading is changed

If only two members form a truss joint and no external load If only two members form a truss joint and no external load or support reaction is applied to the joint, the members or support reaction is applied to the joint, the members must be zero-force members, must be zero-force members, SHOWSHOW

If three members form a truss for which two of the members If three members form a truss for which two of the members are collinear, the third member is a zero-force member are collinear, the third member is a zero-force member provided no external force or support reaction is applied, provided no external force or support reaction is applied, SHOWSHOW

EXAMPLES (pg 276)EXAMPLES (pg 276)

Method of sectionsMethod of sections

Based on the principle that if a body is in equilibrium, then any Based on the principle that if a body is in equilibrium, then any part of the body is also in equilibriumpart of the body is also in equilibrium

Procedure for analysisProcedure for analysis– Section or “cut” the truss through the members where the forces Section or “cut” the truss through the members where the forces

are to be determinedare to be determined– Before isolating the appropriate section, it may be necessary to Before isolating the appropriate section, it may be necessary to

determine the truss’s external reactions (then 3 equs. of equilibrium determine the truss’s external reactions (then 3 equs. of equilibrium can be used to solve for unknown member forces in the section)can be used to solve for unknown member forces in the section)

– Draw the free-body diagram of that part of the sectioned truss that Draw the free-body diagram of that part of the sectioned truss that has the least number of forces acting on ithas the least number of forces acting on it

– Establish the sense of the unknown member forcesEstablish the sense of the unknown member forces– Apply 3 equs. of equilibrium trying to avoid equations that need to Apply 3 equs. of equilibrium trying to avoid equations that need to

be solved simultaneouslybe solved simultaneously Moments should be summed about a point that lies at the intersection of Moments should be summed about a point that lies at the intersection of

the lines of action of two unknown forcesthe lines of action of two unknown forces If two unknown forces are parallel – sum forces perpendicular to the If two unknown forces are parallel – sum forces perpendicular to the

direction of these unknownsdirection of these unknowns

EXAMPLES (pg 287)EXAMPLES (pg 287)

Frames and machinesFrames and machines

Structures are often composed of pin-connected Structures are often composed of pin-connected multiforce membersmultiforce members

Frames are generally stationary and are used to Frames are generally stationary and are used to support loadssupport loads

Machines contain moving parts and are designed to Machines contain moving parts and are designed to transmit and alter the effect of forcestransmit and alter the effect of forces

Can apply the equations of equilibrium to each Can apply the equations of equilibrium to each member of the frame or machine to determine the member of the frame or machine to determine the forces acting at the joints and supports (assuming the forces acting at the joints and supports (assuming the frame or machine is properly constrained and contains frame or machine is properly constrained and contains no more supports or members than are necessary to no more supports or members than are necessary to prevent collapse) prevent collapse)

Frames and machines – procedure Frames and machines – procedure for analysisfor analysis

Construct applicable free-body diagramsConstruct applicable free-body diagrams– Draw an outline of the shapeDraw an outline of the shape– Show all forces or couple moments that act on the partShow all forces or couple moments that act on the part– Indicate dimensions needed for determining momentsIndicate dimensions needed for determining moments– Identify all two-force members in the structureIdentify all two-force members in the structure

All loadings are applied at the jointAll loadings are applied at the joint Members are joined together by smooth pinsMembers are joined together by smooth pins Members have two equal but opposite forces acting at their Members have two equal but opposite forces acting at their

points of applicationpoints of application The line of action of the forces are along the axis of the The line of action of the forces are along the axis of the

membersmembers

– Forces common to any two contacting members act with Forces common to any two contacting members act with equal magnitudes but opposite sense on the respective equal magnitudes but opposite sense on the respective membersmembers

Apply the equations of equilibriumApply the equations of equilibrium EXAMPLES (pg 313)EXAMPLES (pg 313)