emech

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Equilibrium Equations
• Six scalar equations are required to express the
conditions for the equilibrium of a rigid body in the
general three dimensional case.
   
   
000
000
zyx
zyx
MMM
FFF
• These equations can be solved for no more than 6
unknowns which generally represent reactions at supports
or connections.
• The scalar equations are conveniently obtained by applying the
vector forms of the conditions for equilibrium,
     00 FrMF O

Engineers Mechanics- 3D equilibrium
Reactions at Supports and Connections for a Three-
Dimensional Structure
Engineers Mechanics- 3D equilibrium
Ball and Socket
Roller on Bridge

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Reactions at Supports and Connections for a Three-
Dimensional Structure
Engineers Mechanics- 3D equilibrium
Universal Joint
Bearing
Engineers Mechanics- 3D-equilibrium
Problem 1 :
SOLUTION KEY:
A and B are hinged.
Staticallyindeterminate
Take moment about AB to
solve for FCD
A window is temporarily held open in the 50o position shown by a wooden
prop CD until a crank-type opening mechanism can be installed. If a=0.8m
and b=1.2 m and the mass of the window is 50 kg with mass center at its
geometric center, determine the compressive force FCD in the prop

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Engineers Mechanics- 3D-equilibrium
Problem 1 :
SOLUTION:
The co-ordinates of point D:
(0.8*sin (50), -0.8*cos(50),0)
= (0.613,-0.514,0)
FCD = FCD [ 0.828 i + 0.386 j + 0.406 k ]
Engineers Mechanics- 3D-equilibrium
Problem 1:
1 - 6
Summation of moments about Z axis passing through lineAB:

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Engineers Mechanics- 3D-equilibrium
Problem 2:
SOLUTION KEY:
A rectangular sign over a store has a mass of 100kg, with the center of mass in
the center of the rectangle. The support against the wall at point C may be
treated as a ball-and-socket joint. At corner D support is provided in the y-
direction only. Calculate the tensions T1 and T2 in the supporting wires, the total
force supported at C, and the lateral force R
6 unknowns
Staticallydeterminate
No force contribute to the
moment about x axis except
Cy
Engineers Mechanics- 3D-equilibrium
Problem 2:
1 - 8
SOLUTION:
∑MX = 0;
∑MAB = 0 ;

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Engineers Mechanics- 3D-equilibrium
Problem 2:
1 - 9
On solving, we get-
Engineers Mechanics- 3D-equilibrium
Problem 2:

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Engineers Mechanics- 3D-equilibrium
Problem 3:
The two bars AB and OD, pinned together at C, form the diagonals of a
horizontal square AOBD. The ends A and O are attached to a vertical wall
by ball and socket joint, point B is supported by a cable BE, and a vertical
load P is applied at D. Find the components of the reactions at A and O,
and tension in the cable
SOLUTION KEY:
total unknown reactions = 12
total equations = 12 (from 2 rigid
bodies)
Therefore, the problem is statically
determinate.
The components of reactions can
be solved by taking force and
moment equilibrium.
Engineers Mechanics- 3D-equilibrium
Problem 3:
∑MOX = -AY.a + Pa = 0
=> AY = P
∑MOY = 0
=>AX.a = 0
=> AX = 0
SOLUTION:

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Engineers Mechanics- 3D-equilibrium
Problem 3:
∑FX= 0
=> AZ + OZ = 0
=> AZ = -OZ = -P
AX = 0
AY = PAZ = -POX
= POY = -POZ = P
AX = 0
AY = P
AZ = -P
OX = P
OY = -P
OZ = P
Engineers Mechanics- 3D-equilibrium
Problem 4:
Three identical steel balls, each of mass m, are placed in the cylindrical
ring which rests on a horitontal surface and whose height is slightly
greater than the radius of the balls. The diameter of the ring is such that
the balls are virtually touching one another. A fourth identical ball is then
placed on top of the three balls. Determine the force P exerted by the ring
on each of the three lower balls.
SOLUTION KEY:
Consider the equilibrium of the top
ball. Three equal contact reactions
balances the weight
Take the equilibrium of one of the
lower balls to find the force
exerted by the ring

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Engineers Mechanics- 3D-equilibrium
Problem 4:
2 − (
√3
) = 2 2⁄
Equilibrium of top ball
= 3 cos − = 0
3
2 2 3⁄
2
= => = √6⁄
Lower ball
= 0; − sin = 0
=
√6
2 √3⁄
2
=
3√2
AO =
AO =
cos 30
=
2
√3
OC = 22 − (
2
√3
)2 = 2 2 3⁄
Engineers Mechanics- 3D-equilibrium
Problem 5:
A uniform 0.5mx0.75m steel plate ABCD has a mass of 40 kg and is attached
to ball-and-socket joints at A and B. Knowing that the plate leans against a
frictionless vertical wall at D, determine (a) the location of D, (b) the reaction
at D.
SOLUTION KEY:
Find the location of point D from
the geometry of the problem
Take moment of all the forces
about axis AB and set it zero to
find the unknown reaction at D.
Vector mechanics could be easier!!

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Engineers Mechanics- 3D-equilibrium
Problem 5:
/ / 0D A B A r r
(a) The location of D follows from
the geometry of the problem
Denoting the coordinates of D by (0, y, z):
   / 0.1 m 0.7 mD A y z    r i j k
   / 0.3 m 0.4 mB A  r i k
Thus, / / 0.03 0.4 0.28 0D A B A z     r r
Or 0.625 m.z 
   2 22
/ 0.1 m 0.625 m 0.7 m 0.75 mD A y     r
0.73951 my 
Engineers Mechanics- 3D-equilibrium
Problem 5:
   2 2
0.3 0.4
0.6 0.8
0.3 0.4
AB
AB
AB

   
 
i k
i k


     / 0.1 m 0.73951 m 0.075 mD A    r i j k
     / 0.4 m 0.73951 m 0.625 m 0.3 mD B     r i j k
D DNN i
      2
40 kg 9.81 m/s 392.4 Nmg     W j j j
Note:
rG/B = 0.5 rD/B

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   / /0: 0AB AB D A D AB G BM       r N r W 
Engineers Mechanics- 3D-equilibrium
Problem 5:
0.6 0 0.8 0.6 0 0.8
0.1 0.73951 0.075 0.2 0.36976 0.1625 0
0 0 0 392.4 0DN
 
    

0.59161 24.525 0DN   41.455 NDN 
 41.455 ND N i 