Expression 27: "M" Subscript, 1 , Baseline left parenthesis, "f" Subscript, 1 , Baseline , "m" , "x" , right parenthesis equals negative 1 half "W" "x" squared minus left parenthesis, "m" minus StartFraction, "L" Over 2 , EndFraction , right parenthesis left parenthesis, negative StartFraction, "F" Subscript, "a" , Baseline Over 2 , EndFraction minus "f" Subscript, 1 , Baseline , right parenthesis plus 1 eighth "W" "L" squaredM1f1,m,x=−12Wx2−m−L2−Fa2−f1+18WL2
27
Expression 28: "K" left parenthesis, "f" Subscript, 1 , Baseline , "m" , "x" , right parenthesis equals StartFraction, 1 Over "E" "I" , EndFraction left brace, StartAbsoluteValue, "x" , EndAbsoluteValue less than "m" : "M" Subscript, 1 , Baseline left parenthesis, "f" Subscript, 1 , Baseline , "m" , "x" , right parenthesis , "M" Subscript, 2 , Baseline left parenthesis, "f" Subscript, 1 , Baseline , "m" , "x" , right parenthesis , right brace left brace, StartAbsoluteValue, "x" , EndAbsoluteValue less than or equal to StartFraction, "L" Over 2 , EndFraction , right braceKf1,m,x=1EIx<m:M1f1,m,x,M2f1,m,xx≤L2
28
Slope
29
Expression 30: "T" Subscript, 1 , Baseline left parenthesis, "f" Subscript, 1 , Baseline , "m" , "x" , right parenthesis equals negative 1 sixth "W" "x" cubed minus left parenthesis, "m" minus StartFraction, "L" Over 2 , EndFraction , right parenthesis left parenthesis, negative StartFraction, "F" Subscript, "a" , Baseline Over 2 , EndFraction minus "f" Subscript, 1 , Baseline , right parenthesis "x" plus 1 eighth "W" "L" squared "x"T1f1,m,x=−16Wx3−m−L2−Fa2−f1x+18WL2x
30
Expression 31: "T" Subscript, 2 , Baseline left parenthesis, "f" Subscript, 1 , Baseline , "m" , "x" , right parenthesis equals negative 1 sixth "W" "x" cubed minus left parenthesis, 1 half sign left parenthesis, "x" , right parenthesis "x" squared minus StartFraction, "L" Over 2 , EndFraction "x" , right parenthesis left parenthesis, negative StartFraction, "F" Subscript, "a" , Baseline Over 2 , EndFraction minus "f" Subscript, 1 , Baseline , right parenthesis plus 1 eighth "W" "L" squared "x" minus signum left parenthesis, "x" , right parenthesis left parenthesis, 1 half "m" squared , right parenthesis left parenthesis, negative StartFraction, "F" Subscript, "a" , Baseline Over 2 , EndFraction minus "f" Subscript, 1 , Baseline , right parenthesisT2f1,m,x=−16Wx3−12signxx2−L2x−Fa2−f1+18WL2x−signx12m2−Fa2−f1
31
Expression 32: "T" left parenthesis, "f" Subscript, 1 , Baseline , "m" , "x" , right parenthesis equals StartFraction, 1 Over "E" "I" , EndFraction left brace, StartAbsoluteValue, "x" , EndAbsoluteValue less than "m" : "T" Subscript, 1 , Baseline left parenthesis, "f" Subscript, 1 , Baseline , "m" , "x" , right parenthesis , "T" Subscript, 2 , Baseline left parenthesis, "f" Subscript, 1 , Baseline , "m" , "x" , right parenthesis , right brace left brace, StartAbsoluteValue, "x" , EndAbsoluteValue less than or equal to StartFraction, "L" Over 2 , EndFraction , right braceTf1,m,x=1EIx<m:T1f1,m,x,T2f1,m,xx≤L2
32
Deflection
33
Expression 34: Start integral from negative StartFraction, "L" Over 2 , EndFraction to "x" , end integral, "T" left parenthesis, "F" Subscript, 1 , Baseline left parenthesis, "M" , right parenthesis , "M" , "s" , right parenthesis "d" "s"∫x−L2TF1M,M,sds
34
Expression 35: "U" Subscript, 2 , Baseline left parenthesis, "f" Subscript, 1 , Baseline , "m" , "x" , right parenthesis equals negative StartFraction, 1 Over 24 , EndFraction "W" "x" to the 4th power minus left parenthesis, 1 sixth sign left parenthesis, "x" , right parenthesis "x" cubed minus StartFraction, "L" Over 4 , EndFraction "x" squared , right parenthesis left parenthesis, negative StartFraction, "F" Subscript, "a" , Baseline Over 2 , EndFraction minus "f" Subscript, 1 , Baseline , right parenthesis plus StartFraction, 1 Over 16 , EndFraction "W" "L" squared "x" squared minus signum left parenthesis, "x" , right parenthesis left parenthesis, 1 half "m" squared , right parenthesis left parenthesis, negative StartFraction, "F" Subscript, "a" , Baseline Over 2 , EndFraction minus "f" Subscript, 1 , Baseline , right parenthesis "x" minus StartFraction, 5 Over 384 , EndFraction "W" "L" to the 4th power plus left parenthesis, negative StartFraction, 1 Over 24 , EndFraction "L" cubed plus 1 quarter "m" squared "L" , right parenthesis left parenthesis, negative StartFraction, "F" Subscript, "a" , Baseline Over 2 , EndFraction minus "f" Subscript, 1 , Baseline , right parenthesisU2f1,m,x=−124Wx4−16signxx3−L4x2−Fa2−f1+116WL2x2−signx12m2−Fa2−f1x−5384WL4+−124L3+14m2L−Fa2−f1
35
Zero deflection at M
36
Expression 37: "U" Subscript, 2 , Baseline left parenthesis, "F" Subscript, 1 , Baseline left parenthesis, "M" , right parenthesis , "M" , "M" , right parenthesisU2F1M,M,M
equals=
2.2 7 3 7 3 6 7 5 4 4 times 10 to the negative 13th power2.2737367544×10−13
37
Expression 38: left parenthesis, negative 2 thirds "M" cubed plus 1 half "M" squared "L" minus StartFraction, 1 Over 24 , EndFraction "L" cubed , right parenthesis left parenthesis, negative StartFraction, "F" Subscript, "a" , Baseline Over 2 , EndFraction minus "F" Subscript, 1 , Baseline left parenthesis, "M" , right parenthesis , right parenthesis plus "W" left parenthesis, StartFraction, 1 Over 16 , EndFraction "L" squared "M" squared minus StartFraction, 5 Over 384 , EndFraction "L" to the 4th power minus StartFraction, 1 Over 24 , EndFraction "M" to the 4th power , right parenthesis−23M3+12M2L−124L3−Fa2−F1M+W116L2M2−5384L4−124M4
equals=
00
38
Expression 39: "F" Subscript, 1 , Baseline left parenthesis, "m" , right parenthesis equals StartFraction, StartNestedFraction, 1 NestedOver 16 , EndNestedFraction "L" squared "m" squared minus StartNestedFraction, 5 NestedOver 384 , EndNestedFraction "L" to the 4th power minus StartNestedFraction, 1 NestedOver 24 , EndNestedFraction "m" to the 4th power Over negative 2 thirds "m" cubed plus 1 half "m" squared "L" minus StartNestedFraction, 1 NestedOver 24 , EndNestedFraction "L" cubed , EndFraction "W" minus StartFraction, "F" Subscript, "a" , Baseline Over 2 , EndFractionF1m=116L2m2−5384L4−124m4−23m3+12m2L−124L3W−Fa2
39
Finish U
40
Expression 41: "U" Subscript, 1 , Baseline left parenthesis, "f" Subscript, 1 , Baseline , "m" , "x" , right parenthesis equals negative StartFraction, 1 Over 24 , EndFraction "W" "x" to the 4th power minus 1 half left parenthesis, "m" minus StartFraction, "L" Over 2 , EndFraction , right parenthesis left parenthesis, negative StartFraction, "F" Subscript, "a" , Baseline Over 2 , EndFraction minus "f" Subscript, 1 , Baseline , right parenthesis "x" squared plus StartFraction, 1 Over 16 , EndFraction "W" "L" squared "x" squared plus StartFraction, 1 Over 24 , EndFraction "W" "m" to the 4th power plus left parenthesis, 1 half "m" cubed minus 1 quarter "m" squared "L" , right parenthesis left parenthesis, negative StartFraction, "F" Subscript, "a" , Baseline Over 2 , EndFraction minus "f" Subscript, 1 , Baseline , right parenthesis minus StartFraction, 1 Over 16 , EndFraction "W" "L" squared "m" squaredU1f1,m,x=−124Wx4−12m−L2−Fa2−f1x2+116WL2x2+124Wm4+12m3−14m2L−Fa2−f1−116WL2m2
41
Expression 42: "U" left parenthesis, "f" Subscript, 1 , Baseline , "m" , "x" , right parenthesis equals StartFraction, 1 Over "E" "I" , EndFraction left brace, StartAbsoluteValue, "x" , EndAbsoluteValue less than "m" : "U" Subscript, 1 , Baseline left parenthesis, "f" Subscript, 1 , Baseline , "m" , "x" , right parenthesis , "U" Subscript, 2 , Baseline left parenthesis, "f" Subscript, 1 , Baseline , "m" , "x" , right parenthesis , right brace left brace, StartAbsoluteValue, "x" , EndAbsoluteValue less than or equal to StartFraction, "L" Over 2 , EndFraction , right braceUf1,m,x=1EIx<m:U1f1,m,x,U2f1,m,xx≤L2
42
Shear and tensile/compressive stresses, in MPa, don't look like an issue in our case
43
Expression 44: 10 to the negative 6th power times StartFraction, 3 Over 2 "b" "h" , EndFraction "S" left parenthesis, "F" Subscript, 1 , Baseline left parenthesis, "M" , right parenthesis , "M" , "x" , right parenthesis10−6·32bhSF1M,M,x
44
Expression 45: negative 10 to the negative 6th power times StartFraction, "h" Over 2 "I" , EndFraction "K" left parenthesis, "F" Subscript, 1 , Baseline left parenthesis, "M" , right parenthesis , "M" , "x" , right parenthesis−10−6·h2IKF1M,M,x
45
Deflection
46
Expression 47: "T" left parenthesis, "F" Subscript, 1 , Baseline left parenthesis, "M" , right parenthesis , "M" , "x" , right parenthesisTF1M,M,x
47
Expression 48: "U" left parenthesis, "F" Subscript, 1 , Baseline left parenthesis, "M" , right parenthesis , "M" , "x" , right parenthesisUF1M,M,x
48
Expression 49:
49
Optimize for minimum deflection
50
Expression 51: "N" equals 1000N=1000
1010
10001000
51
Expression 52: "X" equals "L" times left parenthesis, negative 0.5 plus 0.5 times StartFraction, left parenthesis, left bracket, 1... "N" , right bracket minus 0.5 , right parenthesis Over "N" , EndFraction , right parenthesisX=L·−0.5+0.5·1...N−0.5N
equals=
negative 2.1 6 3 9 1 7 5−2.1639175
negative 2.1 6 1 7 5 2 5−2.1617525
negative 2.1 5 9 5 8 7 5−2.1595875
negative 2.1 5 7 4 2 2 5−2.1574225
negative 2.1 5 5 2 5 7 5−2.1552575
negative 2.1 5 3 0 9 2 5−2.1530925
negative 2.1 5 0 9 2 7 5−2.1509275
negative 2.1 4 8 7 6 2 5−2.1487625
negative 2.1 4 6 5 9 7 5−2.1465975
negative 2.1 4 4 4 3 2 5−2.1444325
negative 2.1 4 2 2 6 7 5−2.1422675
negative 2.1 4 0 1 0 2 5−2.1401025
negative 2.1 3 7 9 3 7 5−2.1379375
negative 2.1 3 5 7 7 2 5−2.1357725
negative 2.1 3 3 6 0 7 5−2.1336075
1000 element list
52
Expression 53: "U" left parenthesis, "F" Subscript, 1 , Baseline left parenthesis, "M" , right parenthesis , "M" , "X" , right parenthesis tilde 0UF1M,M,X~0
53
Expression 54: StartFraction, 2 "M" Over "L" , EndFraction2ML
