STATICALLY DETERMINED PLANE BAR STURCTURES (FRAMES, ARCHES)

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STATICALLY DETERMINED PLANE BAR STURCTURES
(FRAMES, ARCHES)
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Frames
Formal definition A frame is a plane (2D) set
of beams connected at stiff and/or hinged joints
(corners)
HINGED JOINT
STIFF JOINT
Joints have to be in the equilibrium!
!?!
?
?
!?!
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Frames
What is the difference between beams and
frames? Why do we need to make frames?
Hey, you!
Beam or frame?
Beam
Frame
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Frames
Equilibrium equations
MA1 0
MA2 0

For n hinged joints (if any!) at A1, A2 An points

MAn 0

kinematic stability of a structure (c.f.
Theoretical Mechanics)
Examples of unstable structures
Centre of instability
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Frames
After we determine reactions and check stability
we can deal with a frame as a set of individual
beams, applying all techniques which have been
demonstrated for beams. But, besides of diagrams
of M and Q we have to make diagrams of N, too.
Some problems can be encountered with sloping
members
W
W q?x
q?s
qq?x/?sq cos?
qs qcos? qcos2?
?x/?s cos?
ns qsin? qsin ? cos?
s
q
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Frames
Example Diagrams of M, Q, N for a simple frame
sin?0,6 cos ?0,8
1,5 kN/m
3
3
1
2 kN
1
3kN
0,33
2 kN
2 kN
1 kN
M kNm
2
2
2
1,2
Q kN
2
1,6
0,6
2
0,8
2
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Frames
Checking the equilibrium at a joint
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Arches
Formal definition An arch is a plane (2D) set
of curved beams connected at stiff and/or hinged
joints
FRAME
Stones and other brittle materials do not sustain
an extension
ARCH
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Arches
C
Mc 0
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Arches
y
C
Parabolic arch
Parabolic arch
C
Semi-circular arch
Semi-circular arch
xC
r
yC
h
?
x
To determine reactions we only need to know
position and magnitude of loads and position of
the hinge and supports
r, ?C
xC , yC
But to determine the cross-sectional forces we do
need the equation describing shape of the arch
including coordinates of any point and its
tangent.
y a bx cx2
x rcos?
a,b,c from for x 0 y 0
(in polar coordinates)
for x 2l y 0
for x l y h
(Symetric arch)
? arctg dy/dx
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Arches
Example parabolic arch under concentrated force
P
C
A
B
HA
HB
VB
RB
RA
VA
VA
?MA 0
Pl - VB 2l 0
?Y 0
VB P/2
Mc 0
?X 0
HA/VA HB/VB l/h
HA HB VA l/h
Higher the ratio l/h (i.e. lower the ratio h /
l) higher the value of horizontal reaction H
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Arches
?
M RA?
RB
RB
??RA
NCHA
QA
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HA(P/2)(l/h)
NH
Bar axis
At C ?0
At A ??0
For (l/h)ltlt1 steep arch
QH
NV
QV
VAP/2
For (l/h)gtgt1 shallow arch
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NCHA - (P/2) (l/h)
RB
NAgtP/2
P/2
P/2
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tension
Example semi-circular arch under horizontal
force
?0,3r
0,7P
P/2
P/2
P/2
P/2
P/2
P/2
P/2
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Quantitative comparison of frame, quasi-arch and
arch
P
P
?
r
r
r
P
P
r
r
r
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Frame
r
Pr/2
Pr
Pr/2
P
P/2
P
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Quasi-arch
tension
Pr/2
0,55 Pr
P/2
P
P/2
P/2
P
1,1P
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Comparison
P/2
P/2
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