Load Scaling, Linearity & Multi-Level Static Testing

Why Load Level Matters in Correlation

A strain gauge reading is the product of two things: the structural response (how much strain the structure produces per unit load) and the load magnitude (how much load was applied). A correlation that compares only the absolute strain at a single load level conflates these two — a 10% discrepancy might be a 10% response error with the correct load, or a correct response with a 10% load error, or any combination. Load scaling and linearity checking separate these two contributions and make the correlation more precise and more diagnostic. They also confirm the fundamental assumption of linear static correlation: that the structure is behaving linearly and elastically, so that the FE model's linear assumption is valid.

Why It Matters

Without linearity confirmation and load normalisation, a correlation can be misleading in two directions. First, it can be falsely reassuring: if the test load is lower than intended and the strain is correspondingly lower, the absolute strain might happen to match the FE prediction at the intended load — a coincidence of two errors. Second, it can be falsely alarming: if the structure is slightly nonlinear at the test load, the strain will not match the linear FE prediction, and the discrepancy will be attributed to model error when it is actually nonlinearity. Linearity checking and load normalisation prevent both misinterpretations and ensure the comparison is between like quantities.

A single-load correlation conflates structural response and load magnitude. A load ladder separates them — and confirms the linear assumption on which the comparison rests.

Linearity and Proportional Response

In a linear elastic structure, the strain at any point is directly proportional to the applied load. Double the load and the strain doubles; halve the load and the strain halves. The plot of strain against load is a straight line through the origin. This is the behaviour the linear static FE model assumes. If the test structure behaves this way, the linear model is valid for comparison. If it does not — if the plot curves, if the strain does not return to zero on unloading, if the slope changes with load level — the structure is nonlinear and the linear model is not valid. The linearity check is therefore the gatekeeper of linear static correlation: it must be passed before any strain comparison is interpreted.

Proportional response (linear elastic):

ε(F)  =  k · F

where:
ε(F)  =  strain at load F
k     =  proportionality constant (strain per unit load)
F     =  applied load

The load–strain plot is a straight line through the origin.
The slope k is the structural response, independent of load.

If ε(F) is NOT proportional to F, the response is nonlinear:
yielding, joint slip, contact opening, large displacement.

The Load Ladder

The load ladder is the practical implementation of the linearity check. Instead of applying a single load and recording the strain, the test applies a sequence of increasing load levels — typically 20%, 40%, 60%, 80% and 100% of the maximum test load — and records strain at each level. The strain is plotted against load for each gauge. If the points fall on a straight line through the origin, the response is linear. If they curve, the response is nonlinear. If they are linear but offset from the origin, there is a preload or zero offset. If the unloading path differs from the loading path, there is hysteresis. The load ladder is cheap — it adds only a few minutes to the test — and it is the single most valuable diagnostic in static correlation testing.

Load ladder and linearity check:

  Strain
   ↑
   |        · 100% load
   |      /
   |    /  · 80%
   |  /  /
   |/  /  · 60%
   |  /
   | /  · 40%
   |/
   |· 20%
   |_____________→ Load
   0

  Points on straight line through origin → linear elastic ✓
  Points curving upward                   → yielding or softening
  Points on line but offset from origin   → preload or zero offset
  Different unload vs load path           → hysteresis (joint slip)

Load Scaling for FE Comparison

If the FE model is run at a reference load that differs from the test load, the FE strain must be scaled to the test load before comparison. In a linear structure, this is a simple proportionality: the FE strain at the test load equals the FE strain at the reference load multiplied by the ratio of test load to reference load. This is valid only for linear response — for nonlinear response, the FE model must be run at the test load directly. Load scaling is also used when the test applies multiple loads: if the FE model has separate load cases for each, the results are scaled by the load ratios and superposed. Again, this is valid only for linear response.

Load scaling (linear response):

ε_pred  =  ε_ref  ×  (F_test / F_ref)

For multiple loads (superposition, linear only):

ε_pred  =  Σ  ε_ref,i  ×  (F_test,i / F_ref,i)

where:
ε_pred     =  FE strain predicted at the test load
ε_ref      =  FE strain at the reference load
F_test     =  actual test load
F_ref      =  FE reference load
F_test,i   =  test load for load case i
F_ref,i    =  FE reference load for load case i

VALID ONLY FOR LINEAR ELASTIC RESPONSE.

CONSTRAINT: Load scaling and superposition are valid only for linear elastic response. If the structure is nonlinear, run the FE model directly at the test load — do not scale a linear result.

Load-Normalised Strain (microstrain per kN)

The most robust single-number comparison is the load-normalised strain: the strain per unit load, in microstrain per kN. The test normalised strain is the slope of the load–strain line from the load ladder. The FE normalised strain is the FE strain divided by the FE load. Comparing these two values isolates the structural response from the load magnitude — if the test load was 48 kN and the FE load was 50 kN, the absolute strains differ by 4% even if the response is identical, but the normalised strains are directly comparable. The normalised strain is also independent of the load level, so comparisons can be made across different load levels and different test runs. For a well-behaved linear structure, the normalised strain is a constant property of the structure at that gauge location.

Detecting Nonlinearity

When the load–strain plot is not a straight line through the origin, the structure is behaving nonlinearly, and the source of nonlinearity should be identified before the correlation is interpreted. Different nonlinearity patterns point to different causes. A curve that bends downward (strain increases less than proportionally with load) suggests material yielding — the structure is softening as stress exceeds yield. A curve that bends upward suggests joint slip or contact closure — the structure is stiffening as gaps close. An offset from the origin without curvature suggests a preload or zero datum problem, not nonlinearity. Hysteresis (different unload and load paths) suggests joint slip, friction or material micro-yield. Each pattern has a different implication for whether the linear FE model is valid and what should be done about it.

• Downward curvature (softening) → material yielding; reduce load to elastic range or run nonlinear FE
• Upward curvature (stiffening) → joint slip or contact closure; reduce load or model the nonlinearity
• Linear but offset from origin → preload or zero datum error, not nonlinearity; reconcile the datum
• Hysteresis (unload ≠ load path) → joint slip, friction or micro-yield; investigate the joint behaviour
• Sudden jump in strain → slip, fracture or gauge debond; investigate immediately

Repeatability and the Load Ladder

The load ladder also provides the repeatability check. After reaching the maximum load, the structure is unloaded and the ladder is repeated. If the second ladder follows the same path as the first, the gauge and the structure are repeatable. If the second ladder is offset, the structure has changed — possibly due to joint bedding-in, residual set or gauge drift. If the zero does not return, there is a permanent set or a zero shift. The repeatability check is important because a non-repeating gauge cannot be reliably correlated — the measurement is not stable. The load ladder, run twice, provides linearity, normalisation and repeatability in a single test sequence, making it the most efficient and informative static test protocol.

VERIFICATION: Run the load ladder twice — load, unload, reload. A repeating ladder confirms linearity and measurement stability. A non-repeating ladder flags joint bedding-in, permanent set or gauge problems before the correlation is attempted.

What Nonlinearity Means for the Correlation

When the load ladder reveals nonlinearity, the engineer faces a decision about how to proceed with the correlation. The options depend on the intended use of the model and the source of the nonlinearity. If the nonlinearity is due to material yielding and the model is intended for elastic stress analysis, the correlation should be performed at load levels below the onset of nonlinearity — the linear range is the valid envelope, and the correlation confirms the model within that envelope. If the nonlinearity is due to joint slip or contact and the model is intended for service load prediction, the engineer must decide whether the service loads are in the linear range or the nonlinear range. If the service loads are in the linear range, correlate there and note the nonlinearity as a finding. If the service loads are in the nonlinear range, a linear static model is not adequate, and a nonlinear analysis — with nonlinear material properties, contact, or joint models — must be built and correlated. Attempting to force a linear model to match nonlinear test data by adjusting parameters is not correlation — it is curve-fitting that produces a model valid at one load level and wrong at every other. The honest response to nonlinearity is either to restrict the correlation to the linear envelope or to build the right model for the nonlinear regime.

CONSIDERATION: When nonlinearity is detected, either restrict the correlation to the linear load range and state the envelope, or build a nonlinear FE model. Do not adjust a linear model to match nonlinear data — the result is valid at one load level only.

Key Takeaways

• In a linear elastic structure, strain is proportional to load — the load–strain plot is a straight line through the origin
• The load ladder (multiple load levels) is the cheapest and most valuable diagnostic in static correlation testing
• Load scaling allows FE strain at a reference load to be predicted at the test load — valid only for linear response
• Load-normalised strain (με/kN) isolates the structural response from the load magnitude — the most robust comparison
• Non-proportional response flags nonlinearity — yielding, joint slip, contact — and invalidates linear FE comparison until resolved
• Run the ladder twice to check repeatability as well as linearity

Key takeaways

• In a linear elastic structure, strain is proportional to load — the load–strain plot should be a straight line through the origin, and this is the first check of valid correlation.
• Load scaling allows FE strain at a reference load to be predicted at any other load by simple proportionality — but only if the response is linear.
• Load-normalised strain (microstrain per kN) strips out load magnitude and isolates the structural response — the most robust single-number comparison.
• A load ladder of multiple levels exposes nonlinearity, preload offsets and hysteresis that a single endpoint measurement would miss entirely.
• Non-proportional response flags yielding, joint slip or contact — a linear FE model is not valid for comparison until the response is confirmed linear.