CASE 1. 3. BEAM STEEL IE ALUMINIUM Hel ALLI MILIM (mm) 16 40 40 (mm) 16 40 40 (mm) na 3 3 (mm) na 3 3 Deflection indicator reading (h) (Divisions) 4.84 0.93 2.71 Steel Hearn uses th many texts (1 Aluminium Al Hearn uses th value used in Rimu (also ca Use the figure a medium des all timbers, th
CASE 1. 3. BEAM STEEL IE ALUMINIUM Hel ALLI MILIM (mm) 16 40 40 (mm) 16 40 40 (mm) na 3 3 (mm) na 3 3 Deflection indicator reading (h) (Divisions) 4.84 0.93 2.71 Steel Hearn uses th many texts (1 Aluminium Al Hearn uses th value used in Rimu (also ca Use the figure a medium des all timbers, th
Chapter2: Loads On Structures
Section: Chapter Questions
Problem 1P
Related questions
Question
Find second moment of area mm^4 for each case
![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
B](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb39c1794-81d5-4842-bec5-958b9b2a6559%2F4929576f-5259-4a3e-b319-7aee0cf18b70%2Fwa0qsf8_processed.png&w=3840&q=75)
Transcribed Image Text: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
B
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