Transcript Title

SM1-05: Statics 5:Statically determined bar structures
Trusses
STATICALLY DETERMINED PLANE
BAR STURCTURES
(TRUSSES)
M.Chrzanowski: Strength of Materials
1/10
SM1-05: Statics 5:Statically determined bar structures
Trusses
Formal definition:
A frame is a plane (2D) set of straight bars connected at hinged joints (corners) loaded at hinges
by concentrated forces
The simplest truss consists of three
bars connected in hinges.
If the only loading will be forces acting at hinges
then only cross sectional will be normal forces
which can be found considering equilibrium of
hinges.
Unstable!
Frame
M.Chrzanowski: Strength of Materials
Truss
2/10
SM1-05: Statics 5:Statically determined bar structures
Trusses
Motivation to use trusses is quite different
Trusses are aimed to span large areas with a light but durable structures
Frame
Truss
M.Chrzanowski: Strength of Materials
3/10
SM1-05: Statics 5:Statically determined bar structures
M.Chrzanowski: Strength of Materials
Trusses
4/10
SM1-05: Statics 5:Statically determined bar structures
Trusses
Under the assumptions:
if the structure consists of straight bars connected and loaded at hinges
its elements have to bear the normal cross-sectional forces only
Strut
Tie
The structure has to be kinematically stable!
M.Chrzanowski: Strength of Materials
5/10
SM1-05: Statics 5:Statically determined bar structures
Trusses
Too many joints,
too few bars!
w – number of joints
p – number of bars
Too many bars,
too few joints
ν = 2w – p – 3
Structure has to be kinematically stable,
but can be statically determine or in-determine!
Kinematics
2w = number of equations
p +3 = number of unknowns
M.Chrzanowski: Strength of Materials
Statics
ν>0
ν=0
unstable
stable
undefined
determine
ν<0
stable
In-determine
6/10
SM1-05: Statics 5:Statically determined bar structures
Examples of kinematically and statically
determination
w = 10, p = 17
ν = 2·10 – 17 – 3 = 0
Trusses
ν = 2w – p – 3
Internally and externally
determined
Kinematically stable
Internally determined
w = 10, p = 17
ν = 2·10 – 17 – 4 = -1
Externally indetermined
Kinematically stable
w = 10, p =16
ν = 2·10 – 16 – 3 = 1
Externally determined
Internally indetermined
Kinematically stable
M.Chrzanowski: Strength of Materials
7/10
SM1-05: Statics 5:Statically determined bar structures
Trusses
Normal forces in truss elements are to be found from the fundamental axiom that
if the whole structure is in equilibrium then any part of it is in equilibrium, too.
Such a part of a structure can be obtained by cutting off the truss through three
bars not converging in a point (A), or through two bars converging in a node. (B).
In the former case we have three equations of equilibrium, in the latter – two.
B
A node can be cut off; then we have two
equations of equilibrium (C) , or
any bar with one equation of equilibrium (D).
Y
X = 0
A
Y = 0
MK = 0
B
X = 0
C
X = 0
Y = 0
X
X
D
X = 0
Y = 0
M.Chrzanowski: Strength of Materials
8/10
SM1-05: Statics 5:Statically determined bar structures
Trusses
Certain bars are required solely for the purpose of keeping a truss kinematically
stable. The cross-sectional forces in these bars vanish; one can call them „0-bars”.
A
A
B
C
C
There are three cases in which we can easily spot „0-bars”:
A. When only two bars converge in a node which is free of
loading
B. When a node connects two bars but the loading acts
along of any of these bars
C. When unloaded node connects three bars, two of them
being co-linear
M.Chrzanowski: Strength of Materials
9/10
SM1-05: Statics 5:Statically determined bar structures
How does
truss work?
P=2
A
5
7
All forces in kN
6
8
9
11
12
1
2
10
13
3
4
R=1
R=1
B7
5
6
8
9
11
1
C
7
11
1
12
2
10
13
3
5
6
8
9
12
2
Trusses
4
M.Chrzanowski: Strength of Materials
Frame A
1
2
2
1
-1
-1
-1,41
-1,41
-1,41
-1,41
1
0
1
frame B
1
1
1
1
-2
-2
-1,41
1,41
1,41
-1,41
0
-2
0
frame C
1
2
1
1
-1
-2
-1,41
-1,41
1,41
-1,41
1
-1
0
10
Bars under tension
4
Bars under compression
„0-bars”
13
3
Bar
1
2
3
4
5
6
7
8
9
10
11
12
13
10/10
SM1-05: Statics 5:Statically determined bar structures
Trusses
stop
M.Chrzanowski: Strength of Materials
11/10