Civil engineering mechanics & structural foundations of Splitting Tensile Strength.
Direct uniaxial tension testing of concrete is notoriously difficult to execute because gripping specimens without inducing severe stress concentrations, eccentric bending, and premature end shearing is virtually impossible in routine testing. To circumvent these challenges, Fernando Carneiro and Akazawa independently developed the splitting tensile test (commonly termed the Brazilian test, standardized under ASTM C496 and BS EN 12390-6). In this test, a standard concrete cylinder (150 mm diameter by 300 mm length) is placed horizontally between the flat platens of a compression machine, with narrow sacrificial plywood bearing strips interposed along the top and bottom diametral generator lines. When a compressive line load P is applied, the elasticity solution derived by Hertz and Timoshenko shows that a state of nearly uniform transverse tensile stress develops across approximately 80% of the vertical diametral plane. Once this transverse tensile stress reaches the tensile capacity of the concrete, a clean vertical crack splits the cylinder into two halves. In this laboratory, splitting tensile tests are performed on standard concrete cylinders to evaluate the splitting tensile strength f_ct and calibrate the Brazilian split conversion factor.
1. Theory of Elasticity for Diametral Line Compression
The mathematical foundation of the Brazilian splitting test originates from the classical two-dimensional theory of elasticity solved by Heinrich Hertz and Stephen Timoshenko. Consider a solid circular cylinder of diameter $D = 2R$ and length $L$ subjected to opposing concentrated compressive line loads $P$ acting along diametral generator lines in the $x\text{-}y$ plane ($y$-axis aligned with loading direction, $x$-axis horizontal).
By superimposing two semi-infinite Flamant line-load stress fields onto a uniform radial boundary correction, the complete two-dimensional stress tensor within the circular disk is established. Along the vertical diametral loading axis ($x = 0$, $-R \le y \le R$):
\[ \sigma_x(0, y) = \frac{2 P}{\pi L D} = \text{constant} \] \[ \sigma_y(0, y) = - \frac{2 P}{\pi L} \left[ \frac{2}{D - 2y} + \frac{2}{D + 2y} - \frac{1}{D} \right] \]Remarkably, the horizontal transverse normal stress $\sigma_x$ along the vertical centerline is completely uniform, positive (tensile), and independent of vertical position $y$ across the central section of the disk.
2. The Brazilian Splitting Strength Formula
Because brittle concrete is vastly weaker in tension than in compression ($f_t \ll f_c$), failure initiates by tensile cleavage along the vertical diametral plane when the uniform tensile stress $\sigma_x$ reaches the tensile strength $f_{ct}$. Equating $\sigma_x$ at peak load $P$ to splitting tensile strength $f_{ct}$:
\[ f_{ct} = \frac{2 P}{\pi L D} \]For standard cylinder dimensions ($D = 150\text{ mm}$, $L = 300\text{ mm}$):
\[ \frac{f_{ct}}{P} = \frac{2}{\pi \times 300\text{ mm} \times 150\text{ mm}} = \frac{2}{141,371.67\text{ mm}^2} = 1.41471 \times 10^{-5}\text{ mm}^{-2} = 0.0141471\text{ MPa/kN} \]3. Cushioning Role of Sacrificial Plywood Strips
Concentrated line loads acting directly on brittle concrete would induce localized crushing and plastic pulverization at the platen contact. Interposing soft plywood bearing strips distributes the contact load over a finite contact band of width $2b \approx 12\text{ to }16\text{ mm}$. Finite element analysis confirms that contact strip width $2b / D \le 0.10$ reduces vertical compressive peak stresses directly under the platen to safe levels without altering the uniform tensile stress state across the central $80\%$ of the vertical diametral plane.
Viva voce questions & analytical theory
How does the Brazilian splitting test produce tensile failure using a compressive machine?
Theory of elasticity demonstrates that diametral line compression generates uniform horizontal tensile principal stresses sigma_x = 2 P / (pi L D) across the vertical centerline; concrete fails in tension because its tensile strength is only ~10% of compressive strength.
Why are soft plywood bearing strips placed between the cylinder and steel platens?
Plywood strips crush plastically under load, distributing the contact force over a finite narrow width (about D/10), which suppresses extreme contact shear stress concentrations and prevents premature crushing near the platens.
What is the theoretical distribution of horizontal transverse stress along the vertical diameter?
Horizontal transverse stress sigma_x is remarkably uniform and tensile over the central 75% to 80% of the diameter, switching to high compressive stresses only in the immediate vicinity (within ~0.1 D) of the bearing strips.
How does splitting tensile strength f_ct compare quantitatively to direct tensile strength f_t?
Splitting tensile strength f_ct is typically 5% to 15% higher than direct uniaxial tensile strength f_t due to biaxial compression-tension stress states near the platens and minor friction restraint along bearing strips.
Why is the splitting test widely preferred over direct tension testing in civil engineering practice?
Direct tension testing requires gluing high-strength end plates or gripping specimens, which almost always induces parasitic eccentricity, premature grip pullout, or bending; the Brazilian test uses standard compression cylinders with zero special preparation.
What empirical relationship links splitting tensile strength to compressive strength in CEB-FIP / ACI codes?
CEB-FIP Model Code and ACI 318 model splitting tensile strength as f_ct = 0.50 to 0.56 sqrt(f'c) in MPa, reflecting square-root scaling with matrix density.
Experimental precautions and laboratory safety
- Ensure diametral generator line marked on the cylinder is perfectly vertical and aligned with the center of the loading platens.
- Always use fresh, unused plywood bearing strips for each test; never reuse compressed or warped strips.
- Avoid eccentric platen contact that could cause premature localized corner splitting or out-of-plane torsional shear.
- Keep specimen moist right up to the moment of testing to prevent surface drying microcracks from degrading tensile capacity.
Theoretical assumptions & boundary condition limits
- The civil engineering physical models apply within the specified calibrated operational window (100.0 to 250.0 kN).
- Assumes homogeneous, isotropic continuum material behavior without unmodeled micro-fracture anomalies.
- Continuous sensor readings reflect ordinary least-squares regression across successive observations.
- Extreme inelastic yielding or cyclic degradation requires non-linear constitutive continuum damage mechanics.