The attempt to optimize the weight of a truss bridge given multiple constraints. Needed to withstand a given load amount while still reducing weight.

1. TrussBridge​ ​Optimization
May​ ​2,​ ​2015
By​ ​Ralph​ ​Gemade,​ ​Alphonse​ ​Mugisha,​ ​Mathew​ ​Quirong,​ ​Johann​ ​Ortiz-Franco

2. Abstract:
In​ ​this​ ​project,​ ​we​ ​were​ ​instructed​ ​to​ ​make​ ​three​ ​iterations​ ​of​ ​a​ ​truss​ ​bridge​ ​given
multiple​ ​constraints​ ​to​ ​work​ ​with.​ ​The​ ​goal​ ​was​ ​to​ ​build​ ​the​ ​truss​ ​bridge​ ​and​ ​improve​ ​upon​ ​it,
each​ ​time​ ​reducing​ ​the​ ​weight​ ​of​ ​the​ ​bridge​ ​while​ ​it​ ​still​ ​met​ ​the​ ​required​ ​standards.​ ​Upon
completing​ ​our​ ​fully​ ​optimized​ ​truss​ ​bridge,​ ​it​ ​weighed​ ​15.28kN.​ ​This​ ​was​ ​a​ ​significant
improvement​ ​from​ ​our​ ​initial​ ​truss​ ​which​ ​weighed​ ​28.35kN.
Introduction:
The​ ​goal​ ​of​ ​this​ ​project​ ​is​ ​for​ ​our​ ​group​ ​to​ ​design​ ​and​ ​optimize​ ​a​ ​bridge​ ​consisting​ ​of​ ​6
modules,​ ​each​ ​spanning​ ​52​ ​meters​ ​in​ ​length.​ ​It​ ​must​ ​fit​ ​within​ ​a​ ​9​ ​meter​ ​height​ ​and​ ​support​ ​a
deck​ ​at​ ​a​ ​height​ ​of​ ​4.5​ ​meters​ ​above​ ​the​ ​pier.​ ​With​ ​the​ ​use​ ​of​ ​Dr.​ ​Frame,​ ​we​ ​were​ ​able​ ​to​ ​design
half​ ​of​ ​a​ ​module​ ​as​ ​each​ ​module​ ​is​ ​vertically​ ​symmetrical​ ​at​ ​the​ ​center​ ​point​ ​between​ ​two​ ​piers
and​ ​again​ ​along​ ​the​ ​middle​ ​of​ ​the​ ​pier​ ​itself.​ ​To​ ​accounts​ ​for​ ​the​ ​presence​ ​of​ ​the​ ​rest​ ​of​ ​the
structure,​ ​we​ ​utilized​ ​the​ ​software​ ​by​ ​positioning​ ​rollers​ ​along​ ​those​ ​lines​ ​of​ ​symmetry​ ​in​ ​order
to​ ​constrain​ ​the​ ​movement​ ​of​ ​the​ ​structure​ ​as​ ​if​ ​the​ ​rest​ ​of​ ​the​ ​structure​ ​was​ ​present.​ ​In​ ​designing
the​ ​bridge,we​ ​had​ ​multiple​ ​constraints​ ​that​ ​we​ ​had​ ​to​ ​follow,​ ​and​ ​these​ ​included​ ​the​ ​geometrical
constraints​ ​of​ ​the​ ​bridge​ ​as​ ​well​ ​as​ ​the​ ​type​ ​of​ ​material​ ​we​ ​could​ ​use​ ​in​ ​the​ ​construction​ ​of​ ​our
truss​ ​bridge​ ​and​ ​the​ ​support​ ​conditions​ ​of​ ​the​ ​bridge.​ ​One​ ​of​ ​these​ ​constraints​ ​was​ ​the​ ​joints
resting​ ​on​ ​the​ ​pier​ ​that​ ​had​ ​to​ ​be​ ​pin​ ​connected.​ ​This​ ​means​ ​that​ ​the​ ​joints​ ​are​ ​not​ ​allowed​ ​to
translate​ ​in​ ​either​ ​vertical​ ​or​ ​horizontal​ ​direction;​ ​they​ ​can​ ​only​ ​rotate.​ ​Members​ ​along​ ​the
symmetry​ ​lines​ ​​ ​must​ ​be​ ​free​ ​to​ ​move​ ​​ ​along​ ​the​ ​line​ ​of​ ​symmetry​ ​and​ ​constrained​ ​in​ ​the
direction​ ​perpendicular​ ​to​ ​the​ ​symmetry​ ​line.​ ​Not​ ​only​ ​this,​ ​but​ ​the​ ​members​ ​can​ ​be​ ​no​ ​more
than​ ​10m​ ​in​ ​length.​ ​The​ ​material​ ​used​ ​in​ ​the​ ​truss​ ​are​ ​straight​ ​bars​ ​of​ ​pine​ ​timber​ ​and​ ​the​ ​cross
sections​ ​available​ ​for​ ​construction​ ​of​ ​the​ ​bridge​ ​are,​ ​5x5​ ​cm,​ ​10x10cm,​ ​20x20​ ​cm,​ ​30x30​ ​cm,
and​ ​40x40cm.​ ​We’re​ ​allowed​ ​to​ ​use​ ​different​ ​cross​ ​section​ ​dimensions​ ​in​ ​the​ ​same​ ​truss​ ​design.
The​ ​mechanical​ ​properties​ ​of​ ​pine​ ​are,​ ​elastic​ ​modulus,​ ​E=​ ​10​ ​GPa,​ ​strength​ ​in​ ​tension,​ ​σst​ ​=100
MPa,​ ​strength​ ​in​ ​compression,​ ​σcr​ ​=27​ ​MPa,​ ​and​ ​density​ ​of​ ​the​ ​wood,​ ​ρ​ ​=​ ​350​ ​kg/m​3​
.
For​ ​the​ ​live​ ​load​ ​acting​ ​on​ ​the​ ​truss​ ​due​ ​to​ ​traffic​ ​and​ ​the​ ​deck​ ​weight​ ​itself,​ ​we​ ​have
4.9kN​ ​for​ ​every​ ​meter​ ​of​ ​the​ ​road​ ​deck.​ ​Thus​ ​we​ ​have​ ​that​ ​the​ ​total​ ​live​ ​is​ ​127.4​ ​kN.​ ​Apart​ ​from

3. all​ ​of​ ​this,​ ​the​ ​imperial​ ​building​ ​code​ ​requires​ ​that​ ​as​ ​we​ ​construct​ ​our​ ​bridge,​ ​we​ ​use​ ​a​ ​factor​ ​of
safety​ ​of
5​ ​to​ ​ensure​ ​that​ ​the​ ​structure​ ​is​ ​very​ ​safe​ ​for​ ​use​ ​by​ ​the​ ​public.With​ ​this​ ​safety​ ​factor​ ​we​ ​have
that​ ​allowed​ ​stress​ ​in​ ​tension​ ​is​ ​20MPa​ ​and​ ​allowed​ ​stress​ ​in​ ​compression​ ​is​ ​5.4MPa.
Truss​ ​Design:
In​ ​designing​ ​our​ ​truss,​ ​we​ ​began​ ​with​ ​a​ ​simple​ ​design​ ​that​ ​would​ ​not​ ​fail​ ​under​ ​the
different​ ​tests​ ​that​ ​it​ ​had​ ​to​ ​pass,​ ​which​ ​meant​ ​that​ ​the​ ​stresses​ ​in​ ​the​ ​elements​ ​had​ ​to​ ​be​ ​less​ ​that
what​ ​was​ ​allowed,​ ​and​ ​this​ ​was​ ​5.4MPa​ ​for​ ​the​ ​elements​ ​in​ ​compression​ ​and​ ​20MPa​ ​for​ ​those​ ​in
tension.​ ​The​ ​objective​ ​of​ ​the​ ​initial​ ​design​ ​was​ ​to​ ​find​ ​a​ ​truss​ ​that​ ​worked​ ​the​ ​different
constraints​ ​given.​ ​Once​ ​the​ ​truss​ ​design​ ​was​ ​determined,​ ​the​ ​goal​ ​from​ ​there​ ​was​ ​to​ ​optimize​ ​it​ ​to
the​ ​best​ ​of​ ​our​ ​ability.​ ​The​ ​initial​ ​step​ ​taken​ ​toward​ ​optimization​ ​of​ ​our​ ​truss​ ​was​ ​to​ ​remove​ ​the
members​ ​which​ ​we​ ​considered​ ​unnecessary.​ ​These​ ​included​ ​members​ ​that​ ​carried​ ​very​ ​little
force​ ​without​ ​helping​ ​to​ ​reduce​ ​the​ ​force​ ​of​ ​the​ ​other​ ​members.​ ​Although​ ​this​ ​reduced​ ​the​ ​weight
of​ ​the​ ​truss​ ​by​ ​a​ ​good​ ​amount,​ ​more​ ​changes​ ​could​ ​be​ ​made​ ​to​ ​reduce​ ​the​ ​weight​ ​even​ ​further.
To​ ​do​ ​so,​ ​we​ ​began​ ​by​ ​decreasing​ ​the​ ​cross​ ​sectional​ ​areas​ ​of​ ​the​ ​members​ ​that​ ​carried​ ​little
forces​ ​as​ ​well​ ​as​ ​those​ ​in​ ​tension​ ​for​ ​they​ ​had​ ​a​ ​higher​ ​allowable​ ​stress​ ​and​ ​making​ ​sure​ ​that​ ​the
stress​ ​in​ ​these​ ​members​ ​was​ ​less​ ​than​ ​the​ ​allowable​ ​stress.​ ​For​ ​this​ ​reason​ ​the​ ​truss​ ​could​ ​be​ ​light
and​ ​still​ ​be​ ​able​ ​to​ ​carry​ ​the​ ​desired​ ​loads.
Figure​ ​1:​ ​Truss​ ​Geometry

4. ​ ​Figure​ ​2:​ ​Live​ ​Loads
Figure​ ​3:​ ​Dead​ ​Loads

5. Element Side​ ​[m] area​ ​[m^2]
2nd​ ​area
mom.​ ​[m^4] length​ ​[m]
Dead​ ​Load
[kN]
Live​ ​Load
[kN]
AB 0.3 0.09 0.000675 4.50 1.3906
AC 0.3 0.09 0.000675 6.50 2.0086 31.85
AD 0.3 0.09 0.000675 7.90 2.4412
CD 0.3 0.09 0.000675 4.50 1.3906
CF 0.3 0.09 0.000675 6.50 2.0086 31.85
CG 0.3 0.09 0.000675 7.90 2.4412
EG 0.4 0.16 0.002133333 4.20 2.3073
FG 0.3 0.09 0.000675 4.50 1.3906
FH 0.3 0.09 0.000675 6.50 2.0086 31.85
FI 0.3 0.09 0.000675 7.00 2.1631
GI 0.3 0.09 0.000675 6.80 2.1013
HI 0.3 0.09 0.000675 2.50 0.7725
HJ 0.3 0.09 0.000675 6.50 2.0086 31.85
HK 0.3 0.09 0.000675 7.00 2.1631
IK 0.3 0.09 0.000675 6.50 2.0086
JK 0.3 0.09 0.000675 2.50 0.7725
Table​ ​1:​ ​Element​ ​cross​ ​sectional​ ​area,​ ​dead​ ​loads,​ ​and​ ​live​ ​loads