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Chapter 6 - Analysis of Structures. Sections 6.9 - 6.11. Sections 6.9 - 6.11 - Frames and Machines. Frames differ from trusses because they contain at least one multiforce member. A multiforce has at least 3 forces acting on it. Remember that a truss is made entirely of 2 force members! - PowerPoint PPT Presentation
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Chapter 6 - Analysis of Structures
Sections 6.9 - 6.11
Sections 6.9 - 6.11 - Frames and Machines
Frames differ from trusses because they contain at least one multiforce member.
A multiforce has at least 3 forces acting on it. Remember that a truss is made entirely of 2 force members!
In Fig 6.20, the frame shown contains two multiforce members: AD and CF.
Sections 6.9 - 6.11 - Frames and Machines cont.
With any multiforce member it can no longer be assumed that all forces follow the axis line of the member. This makes the analysis more difficult.
The procedure for analyzing the forces in a frame is usually to first solve for the reactions by considering the frame as a single rigid body.
Sections 6.9 - 6.11 - Frames and Machines cont.
Once you have the external reactions you must separate the members and examine each with a free body diagram.
In the free body diagram, show all forces acting on the pins in each member, all external support reactions and all external applied loads.
Sections 6.9 - 6.11 - Frames and Machines cont.
When you solve for an unknown force acting on a pin, you can apply an equal and opposite force acting on the same pin in a connected member. (See Fig 6.20)
For a multiforce member you have three equilibrium equations to work with.– Usually MP, Fx, Fy but you may use more
than one moment equation.
Sections 6.9 - 6.11 - Frames and Machines cont.
To determine whether a frame is
statically determinate, break it into
members and count the unknowns.
If there are more unknowns than there
are independent equations, the frame is
indeterminate.
Sample Problem 6.4Determine the force in DE and the force at pin C.
160 mm
100 mm
480 kN
Ay
Ax
B
B
MA = 0 = B x 160 - 480 x 100
B = (480 x 100)/160 = 300 N
Fx = 0 = 300 N - Ax
Ax = 300 N
Fy = 0 = - Ay - 480
Ay = - 480 N
Sample Problem 6.4 cont.
170
150
80
100 mm
80 mm
220 mm
Ax = 300 N
Ay = 480 N
Cx
Cy
DE
MC = 0 = 300 N x 220 mm + (150/170)DE x 80 mm + (80/170)DE x 100 mm
0 = 300 x 220 + 70.6 DE + 47 DE
DE = -66,000/117.6 = -561.2 N (Compression)
Sample Problem 6.4 cont.At pin C:
Fx = 0 = - Cx - 300 N - (150/170)(561)
Cx = 795 N
Fy = 0 = - Cy + 480 N - (80/170)(DE)
Cy = 216 N
•Now these are the forces exerted at C on member ACE.
•However for member BCD these forces would have to be turned around.
480
B
Cy = 216 N
Cx = 795 N
Sample Problem 6.5
A
B
CD
E F
3.6 m
4.8 m
2.7 m
2.7 m
Determine the components of the forces acting on each member of the frame shown.
Sample Problem 6.5 cont.
A
B
CD
E F
3.6 m
4.8 m
Ex
Ey F
2400 N
Free Body: Entire Frame
Since the external reactions involve only three unknowns, we compute the reactions by considering the free-body diagram of the entire frame.
Sample Problem 6.5 cont.
+ ME = 0: -(2400 N)(3.6 m) + F(4.8 m) = 0
F = + 1800 N F = 1800 N
+ Fy = 0: -2400 N + 1800 N + Ey = 0
Ey = + 600 N Ey = 600 N
+ Fx = 0: Ex = 0
Members. The frame is now dismembered; since only two members are connected at each joint, equal and opposite components are shown on each member at each joint.
Sample Problem 6.5 cont.
+ MB = 0: -(2400 N)(3.6 m) + Cy(2.4 m) = 0
Cy = + 3600 N
+ MC = 0: -(2400 N)(1.2 m) + By(2.4 m) = 0
By = + 1200 N
+ Fx = 0: -Bx + Cx = 0
We note that neither Bx nor Cx can be obtained by considering only member BCD. The positive values obtained for By and Cy indicate that the force components By and Cy are directed as assumed.
Free Body: Member BCD B C D
1.2 m2.4 mBy
Bx Cx
Cy
2400 N
Sample Problem 6.5 cont.Free Body: Member ABE
+ MB = 0: Bx(2.7 m) = 0 Bx = 0
+ Fx = 0: +Bx - Ax = 0 Ax = 0
+ Fy = 0: -Ay + By + 600 N = 0
-Ay + 1200 N + 600 N = 0 Ay = +1800 NA
B
E
2.7 m
2.7 m
600 N
Ax
Ay
By
Bx
Free Body: Member BCD
Returning now to member BCD, we write
+ Fx = 0: -Bx + Cx = 0; 0 + Cx = 0
Cx = 0
Sample Problem 6.5 cont.Free Body: Member ACF (check)
All unknown components have now been found; to check the results, we verify that member ACF is in equilibrium.
+ MC = (1800 N)(2.4 m) - Ay(2.4 m) - Ax(2.7 m)
= (1800 N)(2.4 m) - (1800 N)(2,4 m) - 0 = 0
0 = 0 (checks)A
F
C2.4 m
1800 N
Ax
Ay
Cy
Cx
Sample Problem 6.6Determine the forces acting on the vertical members of the frame.
600 lb
F
Fy
Fx
Ex
Ey
10'
6'
E
(1) Since this frame has 2 pin supports, it is technically indeterminate.
However you can still solve for Ey and Fy, which will get you started.
ME = 0 = - 600 x 10' + Fy x 6'
Fy = 6000/6 = 1000 lb (1)
Fy = 0 = 1000 + Ey
Ey = -1000 lb (1)
Sample Problem 6.6 cont.
Now write a FBD for AE.
Fy = 0 = -1000 + (2.5/6.5)CD + (2.5/6.5)AB
ME = 0 = (6/6.5)AB x 10' -(6/6.5)CD x 2.5 - 600 x 10
CD = (1000 - .385)AB/.385
CD = 2597 - AB
6.5
6
2.5
2.5
6
6.5
600 lb
Ey = 1000 lb
Ex
CD
AB
2.5
7.5
(2)
Sample Problem 6.6 cont.0 = 9.23 AB - 2.31 CD - 6000
0 = 9.23 AB - 2.31(2597 - AB) - 6000
0 = 11.54 AB - 12,000
AB = 12000/11.54 = 1040 lb (force as drawn in FBD (2))
CD = 2597 - 1040 = 1557 lb (force as drawn in FBD (2))
and Fx = 0 = 600 lb - (6/6.5)AB + (6/6.5)CD - Ex
0 = 600 - 960 + 1437 - Ex
Ex = 1077 lb (2)
Sample Problem 6.6 cont.Now for BF
2.5 6.5
6
6
2.5 6.5
2.5
5.0
D
B
F Fx
Fy
AB
CD
Fy = 0 = -((6/6.5)AB x 7.5) + ((6/6.5)CD x 5.0)
6.92 AB = 4.62 CD
AB = (4.62/6.92)CD = .67 CD
Fy = 0 = -(2.5/6.5)AB - (2.5/6.5)CD +1000
1000 = .385 AB + .385 CD
1000 = .258 CD + .385 CD
CD = 1556 lb
AB = 1040 lbFx = 0 = (6/6.5)AB - (6/6.5)CD + Fx
Fx = 480 lb