0.
a. Deflection for the given load case where force is applied to the end of a beam, where
the opposite end is fixated.
Deflection = 𝛿 =
𝐹𝐿3
3𝐸𝐼
where I =
𝑏ℎ 3
12
Therefore, for the given case, the following parameters, when substituted into the
equation provides:
𝑏=ℎ=𝑡
∶ 𝐼=
𝑇4
𝑇4
12
b. Moment 𝐼 = 12 ∶ 𝐷𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛 (𝛿) =
𝛿=
4𝐹𝐿3
𝐸𝑇 4
4𝐹𝐿3
𝐸𝑇 4
∶ 𝛿=
Rearranging the equation provides
4𝐹𝐿3 1
( ) Therefore
𝑇4 𝐸
1
𝐸
𝑀=( )
c.
Figure 1. Metal
Tungsten alloys possess the highest young’s modulus among metals and thus should provide the least
deflection. This is followed by Medium carbon steel, which is closely followed by lo and high carbon
steel respectively.
Figure 2. Ceramics
Withing carbides, Tungsten Carbides (WC) have the highest young’s modulus and are followed by Silicon
carbides and Aluminum nitrides.
Figure 3. Wood
Wood on the other has very low young’s modulus, therefore has higher deflections. Within the category
of wood, Hard oak possesses the highest modulus followed by bamboo and then softwood.
Figure 4: Metal , Wood & Ceramics
It is evident the ceramics have the highest young’s modulus and thus would show a lower deflection
than the other materials. Ceramics are followed by metals, with tungsten alloys leading the category.
Woods on the other hand have the least young’s modulus and therefore are more susceptible to larger
deflections.
1.
a. Deflection for the given load case where force is across the length of the beam, where
the opposite end is fixated
Deflection = 𝛿 =
𝐹𝐿3
8𝐸𝐼
where I =
𝑏ℎ 3
12
Therefore, for the given case, the following parameters, when substituted into the
equation provides:
𝑏=ℎ=𝑇
∶ 𝐼=
b. Deflection = 𝛿 =
3𝐹𝐿3
2𝐸𝑇 4
𝑇4
12
∶ 𝛿=
12𝐹𝐿3
8𝐸𝑇 4
− (𝑎)
3𝐹𝐿3
= 2𝐸𝑇 4
𝐹 = 𝐴. 𝑔. 𝜌 − (𝑏)
Substituting (b) in (a)
𝛿=
3. 𝐴. 𝑔. 𝜌𝐿3
3. 𝑔. 𝜌. 𝐿3
=
2𝐸𝑇 4
2𝐸𝑇 2
⋮
𝜌
𝑀=( )
𝐸
𝑇ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒 𝑖𝑡 𝑖𝑠 𝑛𝑒𝑒𝑑𝑒𝑑 𝑡𝑜 𝑚𝑖𝑛𝑖𝑚𝑖𝑠𝑒 𝑀 𝑓𝑜𝑟 𝑐ℎ𝑜𝑜𝑠𝑖𝑛𝑔 𝑡ℎ𝑒 𝑚𝑎𝑡𝑒𝑟𝑖𝑎𝑙 𝑤𝑖𝑡ℎ 𝑔𝑖𝑣𝑒𝑛 𝑐𝑟𝑖𝑡𝑒𝑟𝑖𝑎
c.
Figure 5. Ceramics
It can be observed that ceramics such as SiC, AlC & Alumina possess a low Density/E ratio, and therefore
provide a lower deflection for the given loading condition.
Figure 6. metals
Within metals it can be observed that Berylliun, followed by Berylium aluminuma alloy & Ruthenium
possess the last Desnity/E ratio and therefore would possess the least defelction, although, the force
needs to be taken into consideration as these can be brittle.
Figure 7. Wood
Withing the category of woods, Sande followed by Spruce, Fir and Mersawa possess the lease Desnity/E
ratio and hence would show the least deflection,
Figure 8. Wood, Metal & Ceramics
Comparing the three Materials, it is evident that ceramics possess the least density/E ratio and
therefore show the least deflection. It must also be considered that it is load dependent and that
ceramics are brittle. Metals follow ceramics, and possess an intermedia D/E ratio, that is inbetween
woods and ceramics.
2.
3𝐹𝐿3
a. Deflection = 𝛿 = 2𝐸𝑇 4
𝑚
𝜌=
𝐴𝐿
𝐴 = 𝑇2
∶
𝑚 = 𝜌𝐴𝐿 ∶
𝐴 = 𝑇2 =
3𝐹𝐿3
3𝐹𝜌2 𝐿5
3𝐹𝜌2 𝐿5
𝛿=
=
∶
𝑚
=
(
)
2𝐸𝐴2
2𝐸𝑚2
2𝐸𝛿
0.5
𝑚
𝜌𝐿
𝜌
⋮ 𝑀 = ( 0.5 )
𝐸
Or
𝐹 = 𝐴. 𝑔. 𝜌
0.5
0.5
1
3𝐹𝜌2 𝐿5
3(𝑔𝐴𝜌)𝜌2 𝐿5
3(𝑚)𝜌2 𝐿4
𝜌2
2
𝑚=(
) =(
) 𝑚 =(
) ⋮𝑀=( )
2𝐸𝛿
2𝐸𝛿
2𝐸𝛿
𝐸
𝑇ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒 𝑖𝑡 𝑖𝑠 𝑛𝑒𝑒𝑑𝑒𝑑 𝑡𝑜 𝒎𝒊𝒏𝒊𝒎𝒊𝒔𝒆 𝑀 𝑓𝑜𝑟 𝑐ℎ𝑜𝑜𝑠𝑖𝑛𝑔 𝑡ℎ𝑒 𝑙𝑖𝑔ℎ𝑡𝑠𝑡 𝑚𝑎𝑡𝑒𝑟𝑖𝑎𝑙 𝑤𝑖𝑡ℎ 𝑔𝑖𝑣𝑒𝑛 𝑐𝑟𝑖𝑡𝑒𝑟𝑖𝑎
Figure 9. Metals
Figure 10. Metals ( Method 2)
In the case of metals, Magnesium alloys, followed by aluminum and titanium alloys possess the least M
ratio and there fore are optimum for choosing a lightweight metal among its group.
Figure 11. Ceramics
Figure 12. Ceramics (method 2)
Ceramics such as BC, SiC, AlN and SiN, possess the lowest M ratio and therefore are ideal light ceramics
for use for the given load configuration.
Figure 13. Woods
Figure 14. Wood ( Method 2 )
Woods such as bamboo, softwood, hardwood, possess the lowest M ratio and therefore are suitable for
use for the given load configuration.
Figure 15. Wood, Metals and Ceramics.
Figure 16. Wood, Metals and Ceramics ( Method 2)
It can be observed that the ceramics such as BC, SiC, are ideal light materials for the given load
configuration. It is followed by Bamboo and other ceramics. The most optimum metals are Magnesium
alloys.
3.
a. Deflection for the given load case where force is applied to the end of a beam, where
the opposite end is fixated.
Deflection = 𝛿 =
𝐹𝐿3
3𝐸𝐼
where I =
𝑏ℎ 3
12
Therefore, for the given case, the following parameters, when substituted into the
equation provides:
𝑇4
4𝐹𝐿3
𝑏=ℎ=𝑡
∶
𝐼=
∶ 𝛿=
12
𝐸𝑇 4
𝐴 = 𝑇2 =
: 𝑚 = 𝜌𝐴𝐿
𝛿=
4𝐹𝐿3 (𝜌𝐿)2
𝐸𝑚2
=
4𝐹𝐿3 (𝜌𝐿)2
𝐸𝑚2
𝑚
𝜌𝐿
0.5
4𝐹𝐿5 (𝜌)2
)
Therefore 𝑀
𝐸𝛿
∶ 𝑚 =(
𝛿
= 𝐸 0.5
𝑇ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒 𝑖𝑡 𝑖𝑠 𝑛𝑒𝑒𝑑𝑒𝑑 𝑡𝑜 𝒎𝒊𝒏𝒊𝒎𝒊𝒔𝒆 𝑀 𝑓𝑜𝑟 𝑐ℎ𝑜𝑜𝑠𝑖𝑛𝑔 𝑡ℎ𝑒 𝑙𝑖𝑔ℎ𝑡𝑒𝑠𝑡 𝑚𝑎𝑡𝑒𝑟𝑖𝑎𝑙 𝑤𝑖𝑡ℎ 𝑔𝑖𝑣𝑒𝑛 𝑐𝑟𝑖𝑡𝑒𝑟𝑖𝑎
Figure 17. Wood
Woods such as bamboo, softwood, hardwood, possess the lowest M ratio and therefore are
suitable for use for the given load configuration.
Ceramics such as BC, SiC, AlN and SiN, possess the lowest M ratio and therefore are ideal light ceramics
for use for the given load configuration.
In the case of metals, Magnesium alloys, followed by aluminum and titanium alloys possess the least M
ratio and there fore are optimum for choosing a lightweight metal among its group.
It can be observed that the ceramics such as BC, SiC, are ideal light materials for the given load
configuration. It is followed by Bamboo and other ceramics. The most optimum metals are Magnesium
alloys.