find second moment of area for cases 4-7
Plane Trusses
It is defined as, two or more elements like beams or any two or more force members, which when assembled together, behaves like a complete structure or as a single structure. They generally consist of two force member which means any component structure where the force is applied only at two points. The point of contact of joints of truss are known as nodes. They are generally made up of triangular patterns. Nodes are the points where all the external forces and the reactionary forces due to them act and shows whether the force is tensile or compressive. There are various characteristics of trusses and are characterized as Simple truss, planar truss or the Space Frame truss.
Equilibrium Equations
If a body is said to be at rest or moving with a uniform velocity, the body is in equilibrium condition. This means that all the forces are balanced in the body. It can be understood with the help of Newton's first law of motion which states that the resultant force on a system is null, where the system remains to be at rest or moves at uniform motion. It is when the rate of the forward reaction is equal to the rate of the backward reaction.
Force Systems
When a body comes in interaction with other bodies, they exert various forces on each other. Any system is under the influence of some kind of force. For example, laptop kept on table exerts force on the table and table exerts equal force on it, hence the system is in balance or equilibrium. When two or more materials interact then more than one force act at a time, hence it is called as force systems.
find second moment of area for cases 4-7
![CASE BEAM
1.
2.
3.
4.
5.
6.
7.
Labe
STEEL
TOSS
ALUMINIUM
ALUMINIUM
→→
ALUMINIUM
+11
46
7
+1+
ALUMINIUM
H
RIMU
44
RIMU
b
(mm)
16
40
40
51
51
40
60
d
(mm)
16
40
40
25.4
25.4
60
40
W
(mm)
na
m
3.4
3.4
na
na
t
(mm)
na
m
3
3.4
3.4
na
na
Deflection
indicator
reading (h)
(Divisions)
4.84
0.93
2.71
4.17
4.20
0.29
1.03
Note the deflection gauge(DTI) is calibrated so that 1 division equals 0.01 mm
The Young Modulus values [E] for the three materials
Steel
Hearn uses the figure of 207 GPa for E for low carbon steel and this is a typical value used in
many texts (1).
Aluminium Alloy
Hearn uses the figure of 69 GPa for an E value for Aluminium alloys and again this is a typical
value used in texts (1).
Rimu (also called Red Pine) Dacrydium Cupressinum
Use the figure of 22 GPa for an E value for Rimu. A New Zealand native tree species, Rimu is
a medium density hardwood. Density is around 560 kg/m³ but will vary considerably (2). Like
all timbers, the material properties for Rimu are hard to source and the variation for each
property is much greater than that for metals and plastics. One source for the material
properties for Rimu lists eight different strength groups and they have E values ranging from
7.9 GPa up to 21.5 GPa (2).
In addition the timber is an anisotropic material where the properties vary with the direction.
For example the strength across the grain of timber is much weaker than the strength along
the grain. Ashby records an E value parallel to the grain for Pine of 10 GPa, but when
perpendicular to the grain it is only 1.0 GPa (a tenth as much) (3). Also when timber is milled,
various other cuts are possible, see the figure below.
SLATEX
HIGH QUALITY
TALE
QUARTER
Man
cook
CHALITIES OF
WM
LIW TO UM GALITY
See 2) on the previous page
The diagram below represents the beam flexed to a radius 'R' between two supports which
are placed a distance 'L' apart.
Applying Pythagoras' Theorem:
As 'h' is small compared to 'R' and 'L' then
L²
42Rh
Radius of Curvature,
(L)² =R²-(R-h)² =R²-R² +2Rh-h²
(2)
beam bottom flange
= 2Rh-h²
L
And as is very small, it may be neglected to give
L²
R=
8h
so R =
L²h
8h 2
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