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Survivability under environmental stresses and faults

Model civil engineering system survival against uncertain loads, thermal exposure, wear, interruption and shared faults, with twenty examples.

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1. Load-strength margin

Definitions & inputs. C capacity,L demand,g=C−L margin,μ and σ means and standard deviations.

  1. Define success before assigning any probability.

    Ps=P(C>L)=P(g>0)P_s=P(C>L)=P(g>0)

  2. Propagate the difference including dependence.

    μg=μC−μL,σg2=σC2+σL2−2Cov⁡(C,L)\mu_g=\mu_C-\mu_L,\quad\sigma_g^2=\sigma_C^2+\sigma_L^2-2\operatorname{Cov}(C,L)

  3. A Gaussian margin converts standardized separation into survival probability.

    Ps=Φ(μg/σg)P_s=\Phi(\mu_g/\sigma_g)

Interpretation. Tail extrapolation and an incorrect capacity model can dominate the result.

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2. Hazards and common causes

Definitions & inputs. H discrete environmental scenario,Ps survival conditional on H,pcc shared disabling-event probability.

  1. Average conditional survival over the actual exposure distribution.

    Ps=∑hP(S∣H=h)P(H=h)P_s=\sum_hP(S|H=h)P(H=h)

  2. A parallel pair is defeated by the shared event or both independent residual failures.

    Ps=(1−pcc)[1−(1−r)2]P_s=(1-p_{cc})[1-(1-r)^2]

  3. Multiply conditional probabilities, not unconditional ones when dependence exists.

    P(S1∩S2)=P(S1)P(S2∣S1)P(S_1\cap S_2)=P(S_1)P(S_2|S_1)

Interpretation. Redundancy benefits depend on separation of shared power, environment, software and other common causes.

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3. Accumulated exposure

Definitions & inputs. ni cycles at level i,Ni fitted cycles to failure,μ attenuation coefficient,x thickness,D dose.

  1. Miner’s index accumulates fractions of fitted fatigue life.

    Df=∑ini/NiD_f=\sum_i n_i/N_i

  2. Independent removal per unit path gives exponential attenuation.

    I=I0e−μxI=I_0e^{-\mu x}

  3. Solve the attenuation law for one-half transmission.

    x1/2=ln⁡2/μx_{1/2}=\ln2/\mu

Interpretation. Miner’s D=1 is an approximate criterion; radiation buildup and spectral changes are excluded from the narrow-beam law.

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4. Continuity and recovery

Definitions & inputs. E usable backup energy,P load,td detection,ts switching,tr recovery duration.

  1. Stored usable energy supports a constant power load for a finite time.

    thold=E/Pt_{hold}=E/P

  2. Sequential detection and switching create a service interruption.

    tgap=td+tst_{gap}=t_d+t_s

  3. Integrate fractional lost function q(t) over recovery.

    Rloss=∫0tr[1−q(t)]dtR_{loss}=\int_0^{t_r}[1-q(t)]dt

Interpretation. A system can survive physically while its service is unavailable; state both acceptance criteria.

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Graphical worked example

Single-energy attenuation coefficient μ=0.2 cm⁻¹; scattering buildup is omitted. X axis: Shield thickness (cm). Y axis: Narrow-beam transmission (dimensionless).
Single-energy attenuation coefficient μ=0.2 cm⁻¹; scattering buildup is omitted. Related worked calculation · Download SVG · Plot data

Twenty worked examples

Open a problem to see its defined inputs, assumptions, equation, numerical substitution, result, and interpretation. Values are illustrative analytical exercises.

Example 01. Capacity margin

Definitions & inputs. Capacity 150 N,demand 100 N.

  1. Choose the governing model and isolate the requested quantity.

    M=C−LM=C-L

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    M=150−100M=150-100

  3. Evaluate the expression; the result uses the units shown.

    Result=50 N\mathrm{Result}=50\ {\rm N}

Interpretation. Positive margin only addresses the defined load and failure mode.

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Example 02. Capacity-to-demand ratio

Definitions & inputs. Capacity 150 N,demand 100 N.

  1. Choose the governing model and isolate the requested quantity.

    n=C/Ln=C/L

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    n=150/100n=150/100

  3. Evaluate the expression; the result uses the units shown.

    Result=1.5 \mathrm{Result}=1.5\ {}

Interpretation. A ratio is not a survival probability.

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Example 03. Normalized reserve

Definitions & inputs. Capacity 150 N,demand 100 N.

  1. Choose the governing model and isolate the requested quantity.

    m=(C−L)/Lm=(C-L)/L

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    m=(150−100)/100m=(150-100)/100

  3. Evaluate the expression; the result uses the units shown.

    Result=0.5 \mathrm{Result}=0.5\ {}

Interpretation. This convention expresses reserve relative to demand.

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Example 04. Independent margin deviation

Definitions & inputs. σC=3 N,σL=4 N.

  1. Choose the governing model and isolate the requested quantity.

    σg=σC2+σL2\sigma_g=\sqrt{\sigma_C^2+\sigma_L^2}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    σg=9+16\sigma_g=\sqrt{9+16}

  3. Evaluate the expression; the result uses the units shown.

    Result=5 N\mathrm{Result}=5\ {\rm N}

Interpretation. Correlated loads and capacity require a covariance term.

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Example 05. Standardized margin

Definitions & inputs. μC=120 N,μL=100 N,σg=10 N.

  1. Choose the governing model and isolate the requested quantity.

    β=(μC−μL)/σg\beta=(\mu_C-\mu_L)/\sigma_g

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    β=(120−100)/10\beta=(120-100)/10

  3. Evaluate the expression; the result uses the units shown.

    Result=2 \mathrm{Result}=2\ {}

Interpretation. β is a reliability index under this normal-margin model.

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Example 06. Gaussian survival

Definitions & inputs. Margin mean 20 N,standard deviation 10 N.

  1. Choose the governing model and isolate the requested quantity.

    Ps=Φ(2)P_s=\Phi(2)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    Ps=12[1+erf⁡(2/2)]P_s=\tfrac12[1+\operatorname{erf}(2/\sqrt2)]

  3. Evaluate the expression; the result uses the units shown.

    Result=0.9772499 \mathrm{Result}=0.9772499\ {}

Interpretation. The result depends on normal tails and the validity of the threshold model.

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Example 07. Scenario-weighted survival

Definitions & inputs. Mild scenario probability .8 with survival .99; severe probability .2 with survival .8.

  1. Choose the governing model and isolate the requested quantity.

    Ps=∑hP(S∣h)P(h)P_s=\sum_hP(S|h)P(h)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    Ps=0.8(0.99)+0.2(0.8)P_s=0.8(0.99)+0.2(0.8)

  3. Evaluate the expression; the result uses the units shown.

    Result=0.952 \mathrm{Result}=0.952\ {}

Interpretation. The two scenarios are exhaustive and mutually exclusive.

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Example 08. Shared-fault parallel pair

Definitions & inputs. Shared disabling probability .02; conditional unit reliability .9 independent otherwise.

  1. Choose the governing model and isolate the requested quantity.

    Ps=(1−pcc)[1−(1−r)2]P_s=(1-p_{cc})[1-(1-r)^2]

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    Ps=0.98(1−0.12)P_s=0.98(1-0.1^2)

  3. Evaluate the expression; the result uses the units shown.

    Result=0.9702 \mathrm{Result}=0.9702\ {}

Interpretation. The shared event prevents the independent-pair result of .99.

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Example 09. Sequential-stage survival

Definitions & inputs. Stage1 survival .95; conditional stage2 survival given stage1 .9.

  1. Choose the governing model and isolate the requested quantity.

    Ps=P(S1)P(S2∣S1)P_s=P(S_1)P(S_2|S_1)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    Ps=0.95(0.9)P_s=0.95(0.9)

  3. Evaluate the expression; the result uses the units shown.

    Result=0.855 \mathrm{Result}=0.855\ {}

Interpretation. Conditional input handles dependence between stages.

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Example 10. Narrow-beam transmission

Definitions & inputs. μ=.2 cm⁻¹,thickness 5 cm.

  1. Choose the governing model and isolate the requested quantity.

    I/I0=e−μxI/I_0=e^{-\mu x}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    I/I0=e−1I/I_0=e^{-1}

  3. Evaluate the expression; the result uses the units shown.

    Result=0.3678794 \mathrm{Result}=0.3678794\ {}

Interpretation. Single energy, narrow beam and no buildup are assumed.

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Example 11. Half-value thickness

Definitions & inputs. μ=.2 cm⁻¹.

  1. Choose the governing model and isolate the requested quantity.

    x1/2=ln⁡2/μx_{1/2}=\ln2/\mu

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    x1/2=ln⁡2/0.2x_{1/2}=\ln2/0.2

  3. Evaluate the expression; the result uses the units shown.

    Result=3.465736 cm\mathrm{Result}=3.465736\ {\rm cm}

Interpretation. Material and photon energy determine the attenuation coefficient.

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Example 12. Exposure time to dose threshold

Definitions & inputs. Defined device limit 100 Gy,constant absorbed dose rate .5 Gy/h.

  1. Choose the governing model and isolate the requested quantity.

    t=Dlimit/D˙t=D_{limit}/\dot D

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    t=100/0.5t=100/0.5

  3. Evaluate the expression; the result uses the units shown.

    Result=200 h\mathrm{Result}=200\ {\rm h}

Interpretation. This hypothetical device threshold is not a human exposure guideline.

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Example 13. Cumulative fatigue index

Definitions & inputs. 1000 cycles at N1=10000;2000 at N2=20000.

  1. Choose the governing model and isolate the requested quantity.

    D=n1/N1+n2/N2D=n_1/N_1+n_2/N_2

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    D=1000/10000+2000/20000D=1000/10000+2000/20000

  3. Evaluate the expression; the result uses the units shown.

    Result=0.2 \mathrm{Result}=0.2\ {}

Interpretation. Linear damage ignores sequence effects and scatter in fatigue life.

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Example 14. Remaining illustrative fatigue cycles

Definitions & inputs. Current D=.2;future cycles at fitted N=10000 until D=1.

  1. Choose the governing model and isolate the requested quantity.

    n=(1−D)Nn=(1-D)N

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    n=0.8(10000)n=0.8(10000)

  3. Evaluate the expression; the result uses the units shown.

    Result=8000 cycles\mathrm{Result}=8000\ {\rm cycles}

Interpretation. This is a Miner-rule estimate, not a guaranteed remaining life.

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Example 15. Adiabatic thermal buffer

Definitions & inputs. Heat capacity 1000 J/K,allowed rise 20 K,net heat input 100 W.

  1. Choose the governing model and isolate the requested quantity.

    t=CΔT/Pt=C\Delta T/P

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    t=1000(20)/100t=1000(20)/100

  3. Evaluate the expression; the result uses the units shown.

    Result=200 s\mathrm{Result}=200\ {\rm s}

Interpretation. No heat rejection occurs in this bound.

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Example 16. Cold-start temperature margin

Definitions & inputs. Predicted minimum −15°C,qualified minimum −20°C.

  1. Choose the governing model and isolate the requested quantity.

    M=Tpred−TminM=T_{pred}-T_{min}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    M=−15−(−20)M=-15-(-20)

  3. Evaluate the expression; the result uses the units shown.

    Result=5 K\mathrm{Result}=5\ {\rm K}

Interpretation. The relevant qualification applies to the actual startup condition.

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Example 17. Backup-energy hold time

Definitions & inputs. Usable energy 500 Wh,critical load 100 W.

  1. Choose the governing model and isolate the requested quantity.

    t=E/Pt=E/P

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    t=500/100t=500/100

  3. Evaluate the expression; the result uses the units shown.

    Result=5 h\mathrm{Result}=5\ {\rm h}

Interpretation. Conversion losses are already included in usable energy.

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Example 18. Detection plus switchover gap

Definitions & inputs. Detection 20 ms,subsequent switch 5 ms.

  1. Choose the governing model and isolate the requested quantity.

    tgap=td+tst_{gap}=t_d+t_s

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    tgap=20+5t_{gap}=20+5

  3. Evaluate the expression; the result uses the units shown.

    Result=25 ms\mathrm{Result}=25\ {\rm ms}

Interpretation. The function must tolerate the full interruption.

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Example 19. Linear recovery lost-service time

Definitions & inputs. Function rises linearly from zero to full in 4 h.

  1. Choose the governing model and isolate the requested quantity.

    Rloss=∫0tr(1−t/tr)dt=tr/2R_{loss}=\int_0^{t_r}(1-t/t_r)dt=t_r/2

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    Rloss=4/2R_{loss}=4/2

  3. Evaluate the expression; the result uses the units shown.

    Result=2 h\mathrm{Result}=2\ {\rm h}

Interpretation. This equals two hours of complete loss spread over a four-hour recovery.

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Example 20. Two independent exposure intervals

Definitions & inputs. Survival .98 in each interval with the stated independence.

  1. Choose the governing model and isolate the requested quantity.

    Ps=r1r2P_s=r_1r_2

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    Ps=0.982P_s=0.98^2

  3. Evaluate the expression; the result uses the units shown.

    Result=0.9604 \mathrm{Result}=0.9604\ {}

Interpretation. Aging and accumulated damage often make independence inappropriate.

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Symbols and units

Each derivation and problem defines its own symbols and inputs. Symbols may be reused with different meanings in other subjects. Keep units consistent, retain sufficient precision during calculation, and apply the stated validity limits.