Flexural Modulus of Plastics: Formula, Testing and Design Use

Flexural modulus, also called bending modulus, measures how strongly a material resists bending within a defined range of deformation. It is commonly reported in MPa or GPa and determined using a three-point bending test. For a rectangular specimen, the initial linear bending response can be calculated using Ef = L³m/(4bd³), where m is the load versus deflection slope.

The reported value depends on specimen geometry, loading rate, material condition, and how deflection is measured. ASTM D790 and ISO 178 use related testing principles but do not define every modulus calculation in the same way. Neither flexural strength nor the stiffness of a finished part can be determined from flexural modulus alone.

1. What Is Flexural Modulus?

Flexural modulus, also termed the flexural modulus of elasticity, quantifies resistance to bending within a stated strain interval. The flexural modulus of plastics is not necessarily identical to tensile Young’s modulus. When geometry and loading match, a larger modulus generally produces less elastic deflection.

In a three-point bending test, a rectangular specimen rests on two supports while a loading nose applies force at the center. The upper surface experiences compression, the lower surface experiences tension, and the stress varies through the specimen thickness.

Flexural properties vary with resin grade, molding direction, reinforcement, conditioning and test procedure. A coupon measurement does not establish the stiffness of a ribbed or notched part.

Compare values only when the material identity, strain interval, specimen direction and test method are compatible. A reported 3,000 MPa modulus alone cannot tell you the displacement of a bracket.

1.1. Flexural Modulus vs Flexural Strength

Flexural modulus measures resistance to bending deformation. Flexural strength is a stress endpoint, not a stiffness value.

The modulus comes from the slope defined by the test method and its specified strain range.

Flexural strength is the applicable maximum nominal bending stress. It need not be the stress at fracture.

Two materials may have the same initial modulus but different yield or break behavior. They can deflect similarly at a small load and fail at different loads.

Likewise, a stronger plastic may have a lower modulus and deflect more in service.

A cover or instrument bracket may stay below its strength limit yet move far enough to lose alignment.

ASTM D790-25 also distinguishes maximum flexural stress, stress at yield, stress at break, and stress at a specified strain. These values should not be treated as interchangeable. Flexural strength vs tensile strength is a separate comparison because the loading states are different.

1.2. Flexural Modulus vs Young’s Modulus

The comparison of flexural modulus vs Young’s modulus starts with loading direction. Young’s modulus describes the relationship between axial stress and axial strain during uniaxial loading. Flexural modulus describes bending response, which involves tension on one side of the specimen and compression on the other.

For an isotropic, homogeneous, linear elastic solid, bending and tensile moduli can be close when beam assumptions hold. Anisotropy, tension and compression asymmetry, and viscoelasticity remove that guarantee.

Temperature, moisture, testing rate, section thickness and processing orientation can change the reported response.

Reinforced and printed polymers may have different stiffness along and across their principal directions.

Test methods also matter. ASTM D790-25 allows an initial tangent modulus and a chord modulus calculated over specified stress and strain points. ISO 178:2019 defines its flexural modulus using the stress difference between 0.05% and 0.25% flexural strain. These results are not automatically equivalent.

Tensile behavior is a different question; the elastic modulus and stiffness discussion addresses uniaxial loading and part geometry.

When reviewing a material datasheet, check whether the reported number comes from tensile testing or bending testing before using it in a structural calculation. Flexural modulus vs elastic modulus and flexural modulus vs tensile modulus are useful questions, but the measurement methods and assumptions must be named.

2. Flexural Modulus Formula and Calculation

The flexural modulus formula for a rectangular, simply supported specimen with a central point load follows small-deflection beam theory. In ASTM D790-25, the initial tangent expression is Ef = L³m/(4bd³). The flexural modulus equation is not a universal correction for shear, material nonlinearity, contact indentation, or fixture movement.

Variable	Meaning	Unit
Ef	Initial tangent bending modulus	MPa (N/mm²)
L	Measured distance between support centers	mm
m	Initial linear load versus midspan deflection slope	N/mm
b	Specimen width	mm
d	Specimen thickness	mm

For the modulus of elasticity in bending, thickness enters the denominator as d³, a 1% thickness input error changes calculated modulus by approximately 3% with other inputs fixed. Use the specified specimen measurement method.

2.1. Flexural Stress and Strain Equations

For a conventional three point bending formula under the relevant small-deflection assumptions, the nominal flexural stress formula is σf = 3PL/(2bd²). The flexural strain formula is εf = 6Dd/L², where P is the central load in newtons and D is the midspan deflection in millimeters. Use dimensionless strain in the formulas, not a percentage entered as a whole number.

This gives stress and modulus in MPa when force is in N and dimensions are in mm. Under ISO 178:2019, the flexural modulus comes from the specified stress and strain interval rather than automatically from the steepest portion of the curve. ASTM D790-25 also describes a large-deflection stress correction under its stated applicability conditions.

2.2. Worked Calculation Example

For this flexural modulus calculation, the case geometry (L = 64.0 mm, b = 10.0 mm, d = 4.0 mm), an interval slope of approximately 29.3 N/mm gives:

Ef = (64.0³ × 29.3)/(4 × 10.0 × 4.0³) = 3,000 MPa (rounded to three significant figures).

This slope must come from specimen midspan deflection with appropriate compliance treatment. ISO 178 determines modulus over its defined strain interval; the equivalent beam formula does not turn that interval into an ASTM initial-tangent measurement.

2.3. Tangent, Chord and Secant Modulus

Modulus	How it is obtained	When it matters
Initial tangent modulus	Slope of the initial straight portion of the stress and strain curve	Describes early bending response under the specified method
Chord modulus	(σ2 − σ1)/(ε2 − ε1) between declared points	Allows a stated interval to be compared
Secant modulus	σ/ε from the selected curve point and origin	Depends on the chosen point; not the initial tangent by default

A nonlinear flexural stress strain curve can produce different numbers from the same specimen. State the interval or point rather than printing only “flexural modulus.” Seating artifacts may be corrected where the applicable method permits, but genuine material curvature must remain in the result.

3. How Is Flexural Modulus Tested?

A three point bending test loads a rectangular coupon at midspan between two supports. Force and midspan deflection are converted to nominal stress and strain. A flexural modulus test requires the chosen modulus interval and deflection method to be reported.

A 3 point bend test and a four point bending test have different load distributions. The equations below apply to the stated three-point setup, not both fixtures.

3.1. Specimen Geometry, Span and Fixture Setup

For common molded plastic specimens, ASTM D790-25 gives a preferred length of 127 mm, width of 12.7 mm and thickness of 3.2 mm, with a nominal span to depth ratio of 16:1. The 127 mm specimen length is not the support span. ISO 178:2019 prefers an 80 × 10 × 4 mm specimen with a 64 mm span under its usual 16:1 setup. Material-specific exceptions can apply.

Measure each flexural test specimen as required by the applicable method. Highly anisotropic laminates may need a larger support span to reduce transverse shear. Contact radii and indentation can influence the load versus deflection record.

3.2. Speed, Conditioning and Measurement Errors

Flexural testing of viscoelastic plastics is sensitive to speed and environment. ASTM D790-25 Procedure A uses a nominal outer-fiber strain rate of 0.01 min⁻¹; its Procedure B uses 0.10 min⁻¹ under the conditions stated by the method. ISO 178:2019 uses approximately 1% per minute for the modulus segment, with a common 2 mm/min setting for the preferred specimen and separate options for the later test segment.

Record temperature, humidity, conditioning and the actual flexural strain rate. The recorded machine crosshead movement is not necessarily the true center deflection: seating, contact indentation and frame compliance can enter the displacement. ISO 178:2019 identifies a Type III correction route; ASTM D790-25 describes measurement types and compliance considerations. Short, thick specimens may also show shear deformation that is not accounted for by elementary bending theory.

When transverse shear affects the measured deflection, shear modulus in polymers supplies the complementary material parameter.

4. ASTM D790 vs ISO 178: Key Differences

Both standards use three-point loading, but their reported values are not interchangeable by default. Check material identity, direction, specimen geometry, rate, conditioning, deflection measurement and modulus definition.

Decision field	ASTM D790-25	ISO 178:2019	Engineering consequence
Preferred common specimen	127 × 12.7 × 3.2 mm	80 × 10 × 4 mm	Different geometry and preparation can affect response
Usual support span	Nominal 16 × specimen depth	(16 ± 1) × specimen thickness	Use measured span in calculations
Modulus definition	Initial tangent or specified chord modulus	Stress difference from 0.05% to 0.25% strain divided by 0.0020	Do not pool unlike modulus definitions
Test speed	Procedure A and conditional B	Modulus speed followed by Method A or B sequence	Record rate and procedure separately
Deflection	Identify measurement method and compliance handling	Use qualified measurement or specified Type III correction	Machine travel is not always specimen deflection
Environment	Specified conditioning and test conditions, typically via ASTM D618	Material standard or ISO 291 conditions	Compare conditioned specimens, not names alone

4.1. ASTM D790-25: Procedure and Modulus Definition

ASTM D790-25 calculates initial tangent flexural modulus from the straight initial part of the force and displacement curve: EB = L³m/(4bd³). A chord modulus uses the stress difference over a declared strain interval. A secant value is also curve-point dependent. These labels belong in the result, alongside Procedure A or B, specimen direction, span, loading speed and measurement type (ASTM D790-25, Sections 10, 12 and 13).

The method commonly ends a determination at rupture or the relevant 5% outer-fiber strain boundary. That boundary does not mean every polymer has fractured or yielded. A stress reported at 5% strain should not be renamed “stress at break” without a break.

4.2. ISO 178:2019: The 0.05% to 0.25% Modulus Interval

ISO 178:2019 evaluates Ef = (σf2 − σf1)/(0.0025 − 0.0005), using the two specified nominal flexural strain points or permitted interval treatment. For the preferred 4 mm thick specimen and 64 mm span, the corresponding deflections are approximately 0.0853 and 0.4267 mm.

This small displacement window explains why deflection measurement accuracy matters. It also explains why an ISO 178 chord-like interval measurement is not automatically identical to an ASTM initial tangent value, particularly when the curve is nonlinear. The ISO standard itself cautions against directly translating flexural test values into engineering design properties without checking material behavior and constraints (ISO 178:2019, Sections 3, 5 and 9).

5. Flexural Modulus in Plastics and 3D Printing

Plastic flexural modulus reflects not only resin chemistry but also manufacturing history. Fiber orientation, molded skin structure, voids and residual stresses can alter the response measured in bending.

5.1. Molding Direction, Thickness and Material Conditioning

Melt flow can align reinforcement or polymer structure. Skin and core cooling histories differ. Moisture-sensitive grades respond to conditioning, and viscoelastic grades respond to test speed. Record the loading direction relative to the molded orientation.

Do not compare ABS flexural modulus, PEEK flexural modulus, polycarbonate flexural modulus, polypropylene flexural modulus or PLA flexural modulus as fixed universal numbers without grade and test conditions. A high flexural modulus may favor dimensional stability under a specified short load but may not suit a living hinge or compliant latch. Plastic anisotropy and injection molding material properties belong in the selection decision.

Grade-specific molding tradeoffs appear in the injection molding materials guide, beyond the flexural response discussed here.

5.2. 3D Printing Orientation and Anisotropy

A 3D printing flexural modulus value should identify build direction, raster, layer height, infill and specimen preparation. Otherwise, XY and Z figures may describe different structures rather than a reliable material comparison.

3D printing anisotropy also complicates the difference between stiffness and failure. The question behind 3D printing orientation strength is often whether an interlayer interface becomes the weak path under bending, not whether one printed direction always has the highest modulus. 3D printing flexural strength and modulus are distinct outcomes and require distinct test descriptions.

For a separate analysis of axial loading and failure, see tensile stress in 3D printing

Build orientation should match the expected bending loads. Our additive manufacturing and 3D printing capabilities provide relevant process options.

6. Flexural Modulus vs Flexural Stiffness in Design

Flexural modulus is a specimen-derived material response. Bending stiffness of a beam depends on both modulus and section geometry. For a rectangular section, I = bd³/12 and bending rigidity is EI.

For a simply supported beam under a central point load, the small-deflection beam deflection formula is δ = FL³/(48EI). This flexural stiffness formula describes a specific beam and boundary condition, not a general plastic part with ribs, holes, bosses, contact zones or complex constraints. In a nonlinear or time-dependent material, the correct effective modulus may change with time, temperature or strain level.

At unchanged width, doubling thickness increases I eightfold under the rectangular beam model. The part may still be limited by contacts, notches, supports or local stress.

Under a sustained load, plastic creep design limits become relevant because a short-time modulus does not predict long-term movement.

7. Using Flexural Modulus for Material Selection and Part Design

Set the permitted displacement, applied load, operating temperature, duration and actual supports first. Compare supplier data on a consistent test basis; then test or calculate the finished plastic part design against its functional limit.

7.1. Engineering Case: Test Data and Part Deflection

EPOC CRAFTER’s engineering case 工程算例-弯模-018 concerns an injection molded instrument guide plate made from unfilled Delrin 500P NC010 POM homopolymer. The plate is modeled as a rectangular, simply supported member loaded through a central rounded nose. The material batch is anonymized. Testing and the short-time structural interpretation are given for a room-temperature environment around 23°C.

Five ISO 178:2019 specimens were evaluated along the molded flow direction using a 10.0 mm width, 4.0 mm thickness and 64.0 mm support span. The modulus window was 0.05% to 0.25% nominal surface strain. Midspan deflection was independently measured with machine compliance accounted for. The five reported moduli were 2,950, 3,010, 2,990, 3,030 and 3,020 MPa.

Case item	Value	Engineering meaning
Material	Delrin 500P NC010; unfilled, injection molded	Specific grade and manufacturing direction matter
ISO 178:2019 flexural modulus	Mean 3,000 MPa; n = 5; sample SD 31.6 MPa	Short-time bending response in the tested interval
Plate section	b = 10.0 mm, d = 4.0 mm; span 64.0 mm	Rectangular simply supported idealization
Applied central force	30.0 N	Short-duration load for the beam calculation
Bending rigidity	EI = 160,000 N·mm²	Combines modulus with cross-sectional geometry
Calculated midspan displacement	1.024 mm	Small-deflection linear beam result
Specified displacement criterion	No more than 1.20 mm	Calculated displacement lies below criterion
Nominal outer-fiber bending stress	18.0 MPa	Does not represent localized contact stress

The calculation uses I = 10.0 × 4.0³/12 = 53.333 mm⁴. With E = 3,000 MPa, EI = 160,000 N·mm². Hence δ = 30.0 × 64.0³/(48 × 160,000) = 1.024 mm, giving 29.30 N/mm under the adopted elastic beam model. The corresponding load for a 1.20 mm displacement is about 35.2 N under the same model.

The short-time linear beam result is 1.024 mm against a 1.20 mm displacement limit. Its predicted nominal stress is 18.0 MPa. The 30 N equivalent nominal surface strain is approximately 0.60%, beyond the 0.25% upper limit used to determine ISO 178 modulus; that extrapolation, contact indentation, shear response and time-dependent behavior limit the prediction. The calculation does not establish a separate strength or long-term pass result.

For the molded plate, fiber or melt-flow direction and section thickness belong in the specification for rapid injection molding.

7.2. Supplier Flexural Data Checklist

A material stiffness or flexural modulus figure is useful only with its test context. Record the following fields before ranking candidate materials:

Check	What to record	Decision consequence
Material identity	Commercial grade, filled or unfilled state, processing history	Reject comparisons based on resin-family name alone
Testing method	ASTM D790-25 or ISO 178:2019; method/procedure	Avoid mixing unlike calculations
Geometry and direction	Width, thickness, support span, molded or printed orientation	Identify potential geometry and anisotropy effects
Modulus definition	Initial tangent, chord interval or other declared definition	Check whether quoted values describe the same strain range
Conditioning and rate	Temperature, humidity, conditioning duration, loading speed	Recognize rate- and moisture-dependent behavior
Displacement measurement	Midspan instrument or qualified compliance correction	Avoid treating machine travel as specimen deformation
Statistics	Specimen count, mean and dispersion	Assess whether differences exceed observed scatter
Part checks	Load, support, contact, displacement limit, operating time	Do not turn a coupon modulus into a structural pass claim

Before requesting comparative flexural testing, narrow candidate grades using engineering materials and properties alongside the expected service environment.

Part-level support and fit behavior can then be checked with rapid prototyping services before committing to production tooling.

8. FAQ

8.1. What units are used for flexural modulus?

The usual flexural modulus units are MPa and GPa. One GPa equals 1,000 MPa because strain is dimensionless. Compare unit systems before transferring a number to an analysis.

8.2. How to Calculate Flexural Modulus from a Three-Point Test?

For a rectangular three-point beam, Ef = L³m/(4bd³) relates slope and geometry under the stated beam assumptions. Under ISO 178:2019, determine the stress versus strain slope from 0.05% to 0.25% flexural strain; ASTM D790-25 distinguishes initial tangent and chord results.

8.3. Can Flexural Modulus Replace Young’s Modulus?

No. Flexural modulus versus Young’s modulus is a comparison of bending and axial test responses. They may agree in an ideal linear elastic isotropic case but cannot automatically be exchanged for polymers or printed parts.

8.4. Are ASTM D790 and ISO 178 Results Directly Comparable?

No. Both use three-point bending, but different specimen dimensions, speed histories, deflection methods and modulus definitions may produce different numbers. Compare matched conditions or keep the results separate.

Specify the method, grade, orientation and conditioning alongside a flexural modulus. Use the beam equations only with defensible geometry, boundaries and material behavior. For the POM guide plate, the next meaningful check is part displacement and local contact behavior under the actual service load.