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The linear Miner rule sums consumed cycle fractions from multiple constant-amplitude fatigue blocks.
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Formula
D dimensionless; ni and Ni in cycles
| Symbol | Meaning | Unit |
|---|---|---|
| Cumulative linear damage sum | dimensionless | |
| Applied cycles in block i | load cycles | |
| Cycles to failure at constant level i from a fatigue curve | load cycles |
D=\sum_i\frac{n_i}{N_i}Linear accumulation treats each block as the fraction ni/Ni consumed; D≈1 is often used as a model failure threshold. Sequence effects, mean stress, scatter and load interaction are not captured, so real design needs appropriate fatigue data and safety rules. Each Ni must correspond to the material, surface, geometry and load ratio.
Linear accumulation treats each block as the fraction ni/Ni consumed; D≈1 is often used as a model failure threshold. Sequence effects, mean stress, scatter and load interaction are not captured, so real design needs appropriate fatigue data and safety rules. Each Ni must correspond to the material, surface, geometry and load ratio.
The linear Miner rule sums consumed cycle fractions from multiple constant-amplitude fatigue blocks.
Base equation for the target quantity
Preserve signs, boundary conditions and units when rearranging.
Work through the numerical example for “Miner rule for cumulative fatigue damage” and check its unit and magnitude.
Blocks 2,000/10,000 and 30,000/100,000 cycles give D=0.20+0.30=0.50; this is modeled damage, not guaranteed remaining life.
Substituting a related quantity that shares the same symbol.
Use variable definitions and check: D dimensionless; ni and Ni in cycles.
Formula cluster
The linear Miner rule sums consumed cycle fractions from multiple constant-amplitude fatigue blocks.
Blocks 2,000/10,000 and 30,000/100,000 cycles give D=0.20+0.30=0.50; this is modeled damage, not guaranteed remaining life.
Compare variable-load histories using a simple linear fatigue model.
Here is how to work through a typical Miner rule for cumulative fatigue damage (D = Σ ni/Ni) task step by step:
Task
Work through the numerical example for “Miner rule for cumulative fatigue damage” and check its unit and magnitude.
Solution path
Blocks 2,000/10,000 and 30,000/100,000 cycles give D=0.20+0.30=0.50; this is modeled damage, not guaranteed remaining life.
The ratio estimates the modeled life fraction consumed by a load block. If 2,000 cycles are applied and the relevant fatigue curve predicts 10,000, the fraction is 0.2. Add fractions across blocks; the model ignores interactions. The linear sum is an engineering approximation and depends on fatigue lives Ni from suitable data for each stress block. Application: Compare variable-load histories using a simple linear fatigue model. The numerical example Blocks 2,000/10,000 and 30,000/100,000 cycles give D=0.20+0.30=0.50; this is modeled damage, not guaranteed remaining life anchors the idea in a concrete measurement case.
No. D=1 is a common convention in the linear rule, not an exact physical constant. Test data scatter, and load sequence can affect crack formation. Limits and safety requirements vary with component, material and failure consequences. The linear sum is an engineering approximation and depends on fatigue lives Ni from suitable data for each stress block. The corresponding units are D dimensionless; ni and Ni in cycles. Blocks 2,000/10,000 and 30,000/100,000 cycles give D=0.20+0.30=0.50; this is modeled damage, not guaranteed remaining life. This checks that the reference quantities are compatible.
Ni must come from a fatigue curve or validated model for the relevant stress amplitude. Surface condition, notches, mean stress, material condition and environment can change life substantially. A generic table value without these details can misestimate damage and is not a component qualification. The linear sum is an engineering approximation and depends on fatigue lives Ni from suitable data for each stress block. Scope: Linear accumulation treats each block as the fraction ni/Ni consumed; D≈1 is often used as a model failure threshold. Sequence effects, mean stress, scatter and load interaction are not captured, so real design needs appropriate fatigue data and safety rules. Each Ni must correspond to the material, surface, geometry and load ratio. A practical application is Compare variable-load histories using a simple linear fatigue model.
Write the quantity and unit from the variable list next to every value: D dimensionless; ni and Ni in cycles. Substitute only after converting to a consistent unit system, and state sign or reference direction explicitly. The example Blocks 2,000/10,000 and 30,000/100,000 cycles give D=0.20+0.30=0.50; this is modeled damage, not guaranteed remaining life. is a calculation check: result, unit and order of magnitude must fit the described situation. Round after intermediate calculations so small contributions are not lost prematurely. The linear sum is an engineering approximation and depends on fatigue lives Ni from suitable data for each stress block.
The background states this starting condition: Linear accumulation treats each block as the fraction ni/Ni consumed; D≈1 is often used as a model failure. That limits when the result is a valid model prediction. Compare the problem’s boundary conditions with this assumption before substituting values; real components, material properties or loads can introduce additional effects. The cited reference supports the relation used here but does not remove the need to check whether the specific case fits the model. The linear sum is an engineering approximation and depends on fatigue lives Ni from suitable data for each stress block.
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Curated exam set for "Miner rule for cumulative fatigue damage":
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Which formula describes Miner rule for cumulative fatigue damage?
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How do you rearrange Miner rule for cumulative fatigue damage for Base equation for the target quantity?
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Which common mistake happens with Miner rule for cumulative fatigue damage?
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