Department of Mechanical Engineering
ENGR 0135
Chapter 6-2
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Equilibrium equations in 2D
Two-force membersStatically determinate problemsFrames and MachinesStatically indeterminate problems
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Equilibrium in two dimension (xy-plane)
Basic equations– Two force balance equations + 1 moment balance equation– The moment balance is about any point in the plane
Alternative set of equations– Maximum number of independent equations = 3– Alternatively,
» 1 force balance + 2 moment balance equations» 3 moment balance equations
0kC
0jiR
==
=+=
∑∑∑
z
yx
M
FF 000 === ∑∑∑ zyx MFF
Scalar equationsVector equations
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Two-Force Members
= frequently used in 2D (and 3D) structures= requirements:
– The ends of the member are pins– Member weight is neglected – The external forces act at the ends– The equilibrium of the member requires these force to be equal,
opposite, and collinear passes the ends of the member
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Two-force membersThe shape of the member determines the internal force
Straight member Only axial (normal) internal
force
Curved members May have normal, lateral
(shearing) and moment
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Statically determinate structure
The support reactions can be determined from the equilibrium equations
Number of unknowns < 3
Find the reaction (support) forces
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2D problem - example Given:
– Geometry– Forces– Mass of W = 100 kg
Assumption:– Member weight is neglected
Questions:– Reactions at the supports A
(pin) and B (roller)
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Solving the problem Construct the FBD
– Remove the supports– Replace them with reaction
forces– Identify the unknowns: Bx, Ax,
Ay
– Replace the weight with a vertical downward force
Utilize the force and moment balance equations
Solve the unknowns
000 === ∑∑∑ zyx MFF
Roller
Smooth pin
This structure is called a TRUSS (consists only of 2-force members)
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Many ways on how to solve the 2D determinate problems
Given– Structure geometry– Weight of the ladder AB (W) = 250 lb– Notice: point D = intersection of the lines of
action of reaction forces at A and C
Questions:– Determine the support reactions
» Contact at C normal to the surface» Cable tension along the cable, in
tension» Contact at A normal to surface
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Many ways on how to solve the 2D determinate problems
Method 1– Use 2 force balance equations and 1
moment equation about A
Method 2– Use 3 moment equations about D, C, and
A (why about these points?)
Method 3– Use moment equation about D and two
force balance equations
000 === ∑∑∑ zyx MFF
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Problems involving curved 2-force members
Pins
FBD
Four unknonws:Ax, Ay, Bx, By
Note: Member AC is a 2-force member.The reaction force must pass both ends.
3 unknowns: A, Bx, By
C
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Problems involving distributed load
First, find the resultant force and its directionand location of the distributed load
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Problems involving pulleys
Pulleys change the direction of the cable tension
It is assumed to be frictionless It does not change the
magnitude
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Problems involving pulleys
Given:– Weight of the person– Frictionless pulleys
Questions:– The force must be exerted on the rope
to support himself
In the text:– Uses two FBDs– Two force balance equations (both in
vertical direction)
WTT =+ 21
02 21 =−TTWT =13
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Alternative solution
W
TT
T
WT =3
They are coming from the same cable the same magnitude