bridge.org
... ...
@@ -23,3 +23,86 @@ but it can be worse ?
23 23
24 24
What about if placed on soil?
25 25
[[id:1d521a37-86e1-4b38-9fca-f6bdd5cd96bf][boussinesq solutions]]?
26
+
27
+* simple beam-component strength estimate
28
+:PROPERTIES:
29
+:ID: 93f81d43-4314-43e8-a1ea-124378c14f71
30
+:END:
31
+consider center point-loading of a beam with height 50mm, width 75mm, length 725mm.
32
+
33
+suppose the steel reinforcement is 2 rods with a diameter of 12.7mm, 20mm from the bottom of the beam
34
+
35
+** pure concrete beam
36
+:PROPERTIES:
37
+:ID: 4c193c21-60d6-45d9-a925-bd12653667df
38
+:END:
39
+based on 3MPa modulus of rupture,
40
+the [[id:57a8c77a-c98f-4a1c-ac57-8a4b356118dc][splitting tension test calculation]]
41
+gives a maximum supported force of
42
+#+begin_src julia
43
+ uconvert(u"N",3u"MPa"*2*75u"mm"*(50u"mm")^2/(3*725u"mm"))
44
+#+end_src
45
+
46
+#+RESULTS:
47
+#+begin_results
48
+517.2413793103449 N
49
+#+end_results
50
+
51
+** with rebar, assuming cracking
52
+:PROPERTIES:
53
+:ID: 309c3353-64a7-41c7-a45e-48e86acd458c
54
+:END:
55
+Let $y=0$ be the side with rebar,
56
+and let $y=y_0$ be the neutral surface,
57
+
58
+Let the concrete crush strength be 30MPa,
59
+the streel's tensile strength be 200MPa,
60
+
61
+Assume $n=8$ for the steel's modulus over the concrete
62
+#+begin_src julia
63
+ @syms y y0
64
+ cIcrc(u) =integrate(u,(y,y0,50),(x,0,75))
65
+ cIcrs(u) =0.5*8*(3.14*6.35^2)*(subs(u,x=>25,y=>18)+subs(u,x=>50,y=>18))
66
+ y0v=cIcrc(y-y0)+cIcrs(y-y0) |> simplify |> (u->solve(u,y0)[1])
67
+#+end_src
68
+
69
+#+RESULTS:
70
+#+begin_results
71
+31.1520370152924
72
+#+end_results
73
+
74
+#+begin_src julia
75
+ Iv=cIcrc((y-y0)^2)+ cIcrs((y-y0)^2) |> subs(y0=>y0v)
76
+#+end_src
77
+
78
+#+RESULTS:
79
+#+begin_results
80
+342599.125813200
81
+#+end_results
82
+
83
+for a 30MPa crush strength concrete,
84
+by [[id:0c49d881-25e0-4e93-b1f9-a8e8cc3ad365][elastic flexure formulas]]
85
+
86
+#+begin_src julia
87
+crushload=uconvert( u"N" , 30u"MPa"*(Iv*1u"mm^4")/((75-y0v)*1u"mm")*4/725u"mm")
88
+#+end_src
89
+
90
+#+RESULTS:
91
+#+begin_results
92
+1293.24279495813 N
93
+#+end_results
94
+
95
+the stress in the steel is
96
+#+begin_src julia
97
+ uconvert(u"MPa", (crushload*725u"mm"/4)*8*(y0v-18)*1u"mm"/(Iv*1u"mm^4"))
98
+#+end_src
99
+
100
+#+RESULTS:
101
+#+begin_results
102
+71.9871270820728 MPa
103
+#+end_results
104
+
105
+So we put in way more steel than required,
106
+since it has a yield strength >250MPa.
107
+
108
+This tells me that *the bridge would benefit from using fibre reinforcement* to improve the crush strength.