Page 170 - Mechanical Behavior of Materials

Section 4.9 Bending and Torsion Tests 171

highest bending moment occurs at midspan and is M = PL/4. Equation 4.33 then gives
3L
σ fb = 2 P f (4.34)
8tc
where P f is the fracture force in the bending test and σ fb is the calculated fracture stress. This
is usually identified as the bend strength or the flexural strength, with the quaint term modulus of
rupture in bending also being used. Values for some ceramics are given in Table 4.7.
Such values of σ fb should always be identified as being from a bending test. This is because
they may not agree precisely with values from tension tests, primarily due to departure of the
stress–strain curve from linearity. Note that brittle materials are usually stronger in compression
than in tension, so the maximum tension stress is the cause of failure in the beam. Corrections for
nonlinearity in the stress–strain curve could be made on the basis of methods presented later in
Chapter 13, but this is virtually never done.
Yield strengths in bending are also sometimes evaluated. Equation 4.34 is used, but with P f
replaced by a load P i , corresponding to a strain offset or other means of identifying the beginning of
yielding. Such σ o values are less likely than σ fb to be affected by nonlinear stress–strain behavior,
but agreement with values from tension tests is affected by the previously noted insensitivity of the
test to the beginning of yielding.
The elastic modulus may also be obtained from a bending test. For example, for three-point
bending, as in Fig. 4.40(a), linear-elastic analysis gives the maximum deflection at midspan. From
Fig. A.4(a), this is
PL 3
v = (4.35)
48EI
The value of E may then be calculated from the slope dP/dv of the initial linear portion of the load
versus deflection curve:
L 3 dP L 3 dP
E = = (4.36)
48I dv 32tc 3 dv
Elastic moduli derived from bending are generally reasonably close to those from tension
or compression tests of the same material, but several possible causes of discrepancy exist:
(1) Local elastic or plastic deformations at the supports and/or points of load application may not
be small compared with the beam deflection. (2) In relatively short beams, significant deformations
due to shear stress may occur that are not considered by the ideal beam theory used. (3) The
material may have differing elastic moduli in tension and compression, so that an intermediate
value is obtained from the bending test. Hence, values of E from bending need to be identified
as such.
For four-point bending, or for other modes of loading or shapes of cross section, Eqs. 4.34 to
4.36 need to be replaced by the analogous relationships from Appendix A that apply.

4.9.2 Heat-Deflection Test
In this test used for polymers, small beams having rectangular cross sections are loaded in three-
point bending with the use of a special apparatus described in ASTM Standard No. D648. Beams
2c = 13 mm deep, and t = 3 to 13 mm thick, are loaded over a span of L = 100 mm. A force