a28c1ff5-e7b3-4941-938a-acac115e571a 278 2 5 decimal

&[DATE] &[TIME] - &[FILENAME]

Oppdragsnr.:

Oppdragsnavn:

Beregningsnr.:

Emne: Ståltemperaturer ved brann

Utarbeidet:

Kontrollert:

Nidaros Byggkompetanse AS

TOPPTEKST

NS-EN 1991-1-2:2002+NA:2008 & NS-EN 1993-1-2:2005+NA:2009

INNDATA

Profilfaktor for boks ratio.Am.b_V 70102^* :

Profilfaktor ratio.Am_V 107102^* :

Skyggeeffekt k.sh 0.9 ratio.Am.b_V ratio.Am_V / * : 0.589

Densitet stål ρ.a 7850 :

k.sh ratio.Am_V * 102^/ 63

Varmeoverføringskoeffisient α.c 25 :

Konfigurasjonsfaktor Φ 1.0 :

Bygningsdelens overflateemissivitet ε.m 1.0 :

Brannens emissivitet ε.f 1.0 :

Stephan Boltzmann konstant σ 5.67 108 -^* :

Standard temperatur-tidkurve t θ.g 203458 t * 1 + log10 * + :

SYSTEM AV DIFFERENSIALLIGNINGER

θ.a c.a 425 7.73 101^/ θ.a * + 1.69 103^/ θ.a 2^* - 2.22 106^/ θ.a 3^* + 66613002738 θ.a - / + 54517820 θ.a 731 - / + 65041 mat :

t θ.a D k.sh θ.a c.a 1 el ρ.a * / ratio.Am_V * α.c t θ.g θ.a - (* Φ ε.m * ε.f * σ * t θ.g (4^θ.a 4^- (* + (* k.sh θ.a c.a 2 el ρ.a * / ratio.Am_V * α.c t θ.g θ.a - (* Φ ε.m * ε.f * σ * t θ.g (4^θ.a 4^- (* + (* k.sh θ.a c.a 3 el ρ.a * / ratio.Am_V * α.c t θ.g θ.a - (* Φ ε.m * ε.f * σ * t θ.g (4^θ.a 4^- (* + (* k.sh θ.a c.a 4 el ρ.a * / ratio.Am_V * α.c t θ.g θ.a - (* Φ ε.m * ε.f * σ * t θ.g (4^θ.a 4^- (* + (* 41 mat :

Oppdragsnr.:

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Beregningsnr.:

Emne: Ståltemperaturer ved brann

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Nidaros Byggkompetanse AS

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INITIALBETINGELSER OG LØSNING AV DIFFERENSIALLIGNINGER

θ.a.init 20 :

θ.a.12 600 :

θ.a.23 735 :

θ.a.34 900 :

steps 5 :

t.init 0 :

t.12 21.70135 :

t.23 31.80617 :

t.34 50.57102 :

t.final 120 :

res θ.a.init t.init t.12 steps t.12 t.init - (* t θ.a D 1 el dn_GearsBDF θ.a.12 t.12 t.23 steps t.23 t.12 - (* t θ.a D 2 el dn_GearsBDF θ.a.23 t.23 t.34 steps t.34 t.23 - (* t θ.a D 3 el dn_GearsBDF θ.a.34 t.34 t.final steps t.final t.34 - (* t θ.a D 4 el dn_GearsBDF 41 mat :

θ.a t.12 res 1 el 2 col res 1 el 1 col θ.a ainterp :

θ.a.12 t.12 21.701352

θ.a t.23 res 2 el 2 col res 2 el 1 col θ.a ainterp :

θ.a.23 t.23 31.806172

θ.a t.34 res 3 el 2 col res 3 el 1 col θ.a ainterp :

θ.a.34 t.34 50.571027

check t.12 θ.a.12 t.12 - abs 1105 -^* < t12 er tilstrekkelig itterert For stort avvik på t12 if t.23 θ.a.23 t.23 - abs 1105 -^* < t23 er tilstrekkelig itterert For stort avvik på t23 if t.34 θ.a.34 t.34 - abs 1105 -^* < t34 er tilstrekkelig itterert For stort avvik på t34 if 31 mat : t12 er tilstrekkelig itterert t23 er tilstrekkelig itterert t34 er tilstrekkelig itterert 31 mat

FJERNING AV DUPLIKATVERDIER I OVERGANGER OG SAMMENSTILLING NY ENHETLIG LØSNING

res.stripped res 1 el 1 res 1 el rows 1 - 12 submatrix res 2 el 1 res 2 el rows 1 - 12 submatrix res 3 el 1 res 3 el rows 1 - 12 submatrix res 4 el 41 mat :

res.stacked res.stripped 1 el res.stripped 2 el res.stripped 3 el res 4 el stack :

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FUNKSJONSUTTRKK VED INTERPOLERING AV ENHETLIG LØSNING

θ.a.cr t res.stacked 2 col res.stacked 1 col θ.a.cr ainterp :

t.req θ.a res.stacked 1 col res.stacked 2 col t.req ainterp :

θ.a.cr 650 :

-->

θ.a.cr t 24.25

t.req 30 :

-->

t.req θ.a 726.02

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res.stacked x θ.g x t.init t.final lele / t.req t.req θ.a t.req 0 2 2 mat 0 t.req θ.a t.req t.req θ.a 2 2 mat t.req t.req θ.a 1 2 mat 0 θ.a.cr θ.a.cr t θ.a.cr 2 2 mat θ.a.cr t θ.a.cr θ.a.cr t 0 2 2 mat θ.a.cr t θ.a.cr 1 2 mat 8 1 sys

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THE FOLLOWING ARE BITS AND PIECES IRRELEVANT TO SOLVING MY SPECIFIC PROBLEM, BUT QUITE INTERESTING NONE THE LESS

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taking the interpolated function θ.a(t) and recasting it with increments specied by the user for tabulation

start step finish x function inc U start finish step range : i 1 U rows range f i el U i el function eval : for U f augment 31 line :

res.recast t.init 1 t.final t θ.a inc :

res.recast 0201 30.36 2 49.11 3724 97.54 5 124.96 62 mat

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using built-in solve() to solve θ.a(t) for both θ.a and t

time before temperature reaches critical. θ.a.cr is obtained from μ.0 indicated by circle in plot

θ.a.cr 1030 :

θ.a t t θ.a θ.a.cr - 0 ≡ t 0120 solve : 108.7

-->

temperature at time when it is OK for stucture to fail indicated by cross in plot

t.req 120 :

t θ.a t.req θ.a θ.a - 0 ≡ θ.a 01200 solve : 1045.39

-->

res.stacked x θ.g x t.init t.final lele / t.req t θ.a t.req 0 2 2 mat 0 t θ.a t.req t θ.a 2 2 mat 0 θ.a.cr θ.a t θ.a.cr 2 2 mat θ.a t θ.a.cr θ.a t 0 2 2 mat t.req t θ.a x 20 red θ.a t θ.a.cr o 20 blue 2 5 mat 7 1 sys

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inversing θ.g(t) and θ.a(t) and plotting them

t θ.g 203458 t * 1 + log10 * + :

t θ.a res.stacked 1 col res.stacked 2 col t ainterp :

L H N Lo Hi inverse dx H L - N / : i 1 N range θ i el L dx - (dx i * + : for j 1 θ rows range t.g j el t θ.g θ - (j el 0 ≡ t Lo Hi solve : t.a j el t θ.a θ - (j el 0 ≡ t Lo Hi solve : 21 line for θ t.g t.a augment 41 line :

Ω 20120505 inverse 1001045300120 inverse 21 mat :

Ω 1 el 1 col Ω 1 el 2 col augment Ω 1 el 1 col Ω 1 el 3 col augment Ω 2 el 1 col Ω 2 el 2 col augment Ω 2 el 1 col Ω 2 el 3 col augment 4 1 sys

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correlation between specific heat and temperature

θ.s c.s 425 7.73 101^/ θ.s * + 1.69 103^/ θ.s 2^* - 2.22 106^/ θ.s 3^* + θ.s 600 ≤ 66613002738 θ.s - / + 600 θ.s < (θ.s 735 ≤ (& 54517820 θ.s 731 - / + 735 θ.s < (θ.s 900 ≤ (& 650 cases :

x c 10 x ≤ (x 1200 ≤ (& cases :

x c.s x c *

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correlation between critical temperature and utilization

θ.a.cr 39.19 1 0.9674 μ.0 3.833^* / 1 - ln * 482 + :

original equation

μ.0 1 e θ.a.cr 482 - 39.19 /^1 + / 0.9674 / 3.833 nthroot :

equation solved symbolically by hand for μ.0

--------------------------

μ.0 θ.a.cr 39.19 1 0.9674 μ.0 3.833^* / 1 - ln * 482 + :

defining function

θ.a.cr 496 :

given

μ.0 μ.0 θ.a.cr θ.a.cr - 0 ≡ μ.0 01 solve : 0.8

using solve() to solve the equation for μ.0

μ.0 0.8 :

given

using solve() to solve the equation for θ.a.cr

θ.a.cr μ.0 θ.a.cr θ.a.cr - 0 ≡ θ.a.cr 01200 solve : 496.05

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