ASM Metals HandBook Vol. 8 - Mechanical Testing and Evaluation

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Reprinted in part from Mechanical Metallurgy, 3rd ed., McGraw-Hill, New York, 1986, p 275–

295, with

permission

References cited in this section

26. “Standard Methods of Compression Testing of Metallic Materials at Room Temperature,” E-9, Annual

Book of ASTM Standards, ASTM, Philadelphia, PA

27. G. Sines, T. Okada, and S. Mack, Fixture for Accurate Load Path in Axial Compression, Compression

Testing of Homogeneous Materials and Composites, R. Chait and R. Papirno, Ed., STP 808, ASTM,

Philadelphia, PA, 1983, p 97–108

28. P. Dadras and J.F. Thomas, Deformation Inhomogeneities in Upset Forging, Compression Testing of

Homogeneous Materials and Composites, R. Chait and R. Papirno, Ed., STP 808, ASTM, Philadelphia,

PA, 1983, p 24–39

29. M.L. Lovato and M.G. Stout, Met. Trans. A, Vol 23, 1992, p 935–951

Mechanical Behavior Under Tensile and Compressive Loads*

George E. Dieter, University of Maryland

References

1. G.E. Dieter, Introduction to Ductility, in Ductility, American Society for Metals, 1968

2. P.G. Nelson and J. Winlock, ASTM Bull., Vol 156, Jan 1949, p 53

3. D.J. Mack, Trans. AIME, Vol 166, 1946, p 68–85

4. P.E. Armstrong, Measurement of Elastic Constants, in Techniques of Metals Research, Vol V, R.F.

Brunshaw, Ed., Interscience, New York, 1971

5. H.E. Davis, G.E. Troxell, and G.F.W. Hauck, The Testing of Engineering Materials, McGraw-Hill, New

York, 1964, p 33

6. J.R. Low and F. Garofalo, Proc. Soc. Exp. Stress Anal., Vol 4 (No. 2), 1947, p 16–25

7. J.R. Low, Properties of Metals in Materials Engineering, American Society for Metals, 1949

8. J. Datsko, Material Properties and Manufacturing Processes, John Wiley & Sons, New York, 1966, p

18–20

9. W.B. Morrison, Trans. ASM, Vol 59, 1966, p 824

10. D.C. Ludwigson, Metall. Trans., Vol 2, 1971, p 2825–2828

11. H.J. Kleemola and M.A. Nieminen, Metall. Trans., Vol 5, 1974, p 1863–1866

12. C. Adams and J.G. Beese, Trans. ASME, Series H, Vol 96, 1974, p 123–126

13. J.H. Keeler, Trans. ASM, Vol 47, 1955, p 157–192

14. A. Considére, Ann. Ponts Chaussées, Vol 9, 1885, p 574–775

15. G.W. Geil and N.L. Carwile, J. Res. Natl. Bur. Stand., Vol 45, 1950, p 129

16. P.W. Bridgman, Trans. ASM, Vol 32, 1944, p 553

17. J. Aronofsky, J. Appl. Mech., Vol 18, 1951, p 75–84

18. T.A. Trozera, Trans. ASM, Vol 56, 1963, p 280–282

19. E.R. Marshall and M.C. Shaw, Trans. ASM, Vol 44, 1952, p 716

20. W.J. McG. Tegart, Elements of Mechanical Metallurgy, Macmillan, New York, 1966, p 22

21. N.E. Dowling, Mechanical Behavior of Materials, Prentice-Hall, Englewood Cliffs, NJ, 1993, p 165

22. M.J. Barba, Mem. Soc. Ing. Civils, Part 1, 1880, p 682

23. E.G. Kula and N.N. Fahey, Mater. Res. Stand., Vol 1, 1961, p 631

24. G.B. Espey, M.H. Jones, and W.F. Brown, Jr., ASTM Proc., Vol 59, 1959, p 837

25. J.D. Lubahn, Trans. ASME, Vol 79, 1957, p 111–115

26. “Standard Methods of Compression Testing of Metallic Materials at Room Temperature,” E-9, Annual

Book of ASTM Standards, ASTM, Philadelphia, PA

27. G. Sines, T. Okada, and S. Mack, Fixture for Accurate Load Path in Axial Compression, Compression

Testing of Homogeneous Materials and Composites, R. Chait and R. Papirno, Ed., STP 808, ASTM,

Philadelphia, PA, 1983, p 97–108

28. P. Dadras and J.F. Thomas, Deformation Inhomogeneities in Upset Forging, Compression Testing of

Homogeneous Materials and Composites, R. Chait and R. Papirno, Ed., STP 808, ASTM, Philadelphia,

PA, 1983, p 24–39

29. M.L. Lovato and M.G. Stout, Met. Trans. A, Vol 23, 1992, p 935–951

Footnote

Reprinted in part from Mechanical Metallurgy, 3rd ed., McGraw-Hill, New York, 1986, p 275–

295, with

permission

Stress-Strain Behavior in Bending

P. Dadras, Wright State University (retired)

Introduction

A CHARACTERISTIC FEATURE of bending is the inhomogeneous (nouniform) nature of the deformation.

Therefore, in a bent specimen the strain and stress at a given point are dependent on the location of the point

with respect to the neutral axis of the cross-sectional area of the specimen. In cases where the applied bending

moment varies along the length of the specimen (as in three-point bending), the strains and stresses become

dependent on axial location as well. Because of these inhomogeneities, a full appreciation of stress and strain

distributions is of utmost importance in bending analyses and computations. Stress-strain relationships, strain

curvature, and stress-moment equations are discussed in this article. The formulations are for elastic, elastic-

plastic, and fully plastic bending conditions.

Stress-Strain Behavior in Bending

P. Dadras, Wright State University (retired)

Elastic Bending

Elastic analysis of bending deformation can be performed by simple-beam theory (Ref 1), elasticity solutions,

and numerical methods such as the finite-difference and finite-element methods (Ref 2). Generally, numerical

methods are suitable for bending of specimens that are subjected to complex loading patterns and that have

irregular and/or varying cross-sectional areas. Elasticity solutions are useful when accuracies better than ~5%

are desired. Simple-beam theory is used in most testing applications in which plates, strips, bars, and rods are

bent in three-point or four-point bending modes. The basic assumptions of the simple-beam theory for pure

elastic bending (shear force = 0) are: (a) all sections that are initially plane and perpendicular to the axis of the

beam remain plane and perpendicular to it after bending; (b) all longitudinal elements (fibers) bend into

concentric circular arcs (hence, cylindrical bending); and (c) a one-dimensional stress state is assumed, and the

same stress-strain relationship is used for tension and compression.

The first assumption implies a linear distribution for fiber elongations and contractions, as shown in Fig. 1(a).

The resulting strain distributions (Fig. 1b, c) are given by (engineering bending strain):

(Eq 1)

and (true bending strain):

(Eq 2)

where R

is the radius of curvature of the neutral axis. For ε

≤ 0.1, the difference between these two strain

definitions is ≤5%. Therefore, for most elastic bendings, the engineering strain definition is sufficiently

accurate and more convenient. As shown in Fig. 1, the true strain description indicates a nonlinear strain

distribution and a maximum compressive strain in the concave inner fiber that is greater than the maximum

tensile strain in the outermost fiber.

Fig. 1 Distribution of strain determined by the simple-beam theory. (a) Linear

distribution for fiber elongations and contractions. (b) Distribution of engineering strain.

= radius of neutral axis; R

= inner radius; R

= outer

radius

For a linear elastic material:

(Eq 3)

where E is the modulus of elasticity for axial loading (Young's modulus), and ν is Poisson's ratio. From the

third assumption, σ

= σ

= 0. Therefore:

(Eq 4)

When expressions similar to Eq 3 are written for ε

and ε

, the following results are obtained:

(Eq 5)

Also:

= ε

= 0

(Eq 6)

Figure 2 illustrates grid deformations in the longitudinal and cross directions. A transverse curvature, called

anticlastic curvature (Ref 3) develops with a radius of curvature equal to (R

/ν). Experimental evidence

indicates that the actual radius of anticlastic curvature depends on (b

/2R

h), where b is the width and 2h is the

thickness of the beam. For (b

/2R

h) ≤1 (i.e., narrow beams), the R

/ν estimate is sufficiently accurate. For

plates and wide beams (b

/2R

h > 20), the anticlastic deformation is primarily concentrated at the edges.

Fig. 2 Grid deformations in the longitudinal and cross directions of a beam

The location of the neutral axis, which is the line of zero fiber stress in any given section of a member subject

to bending, is determined from the condition of zero axial forces acting on the beam. Therefore:

(Eq 7)

where A is the cross-sectional area. This equation indicates that the first moment of the cross-sectional area

about the neutral axis is zero, which implies that the neutral and the central (centroidal) axes are coincident.

The moment-curvature and moment-stress relationships are found by equating the externally applied bending

moment to the internal bending moment at any cross section:

(Eq 8)

where M is the bending moment.

For the linear stress distribution shown in Fig. 3, the results are:

(Eq 9)

and

(Eq 10)

where I

is the area moment of inertia of the cross section about the z axis, which is coincident on the centroidal

axis.

Fig. 3 Linear stress distributions in a beam

The sign (positive or negative) of the bending moment is found from the following relationship: (sign of the

bending moment) = (sign of the moment vector) × (sign of the outward normal to the section). For example, the

bending moment acting on section ABCD in Fig. 4 is positive, because the moment vector, , is in the positive

z-direction (right-hand rule) and outward normal of the plane, , is also in the positive x-direction.

Fig. 4 Sign convention for the bending moment

References cited in this section

1. J.M. Gere and S.P. Timoshenko, Mechanics of Materials, 4th ed., PWS Publishing Co., 1997

2. A.C. Ugural and S.K. Fenster, Advanced Strength and Applied Elasticity, 3rd ed., Prentice Hall, 1995

3. D. Horrocks and W. Johnson, On Anticlastic Curvature with Special Reference to Plastic Bending: A

Literature Survey and Some Experimental Investigations, Int. J. Mech. Sci., Vol 9, 1967, p 835–861

Stress-Strain Behavior in Bending

P. Dadras, Wright State University (retired)

Noncylindrical Bending

In the previous section it was stated that the simple beam theory considers the effect of bending moments alone

(shear force = 0) and assumes a bent configuration consisting of concentric circular arcs (cylindrical bending).

For cylindrical bending to occur, these conditions must be met:

• Bending must occur under the action of bending moments alone, which implies zero applied shear force.

• The cross-sectional area of the beam must possess at least one axis of symmetry.

• The vector of the applied bending moment must be in the direction of an axis of symmetry.

For asymmetrical beams such as Z-sections and unequal L-sections, the second and third conditions for

cylindrical bending are not satisfied; for unsymmetrical bending of symmetrical beams, the third condition is

not met. These cases are significant in structural design and will not be considered here. Information on

unsymmetrical loading of straight beams can be found in Ref 2.

In a majority of testing applications such as three-point bending, roll bending, and press-brake forming, the

applied bending moment varies along the length of the specimen. Because shear force V = (dM/dx), such

variations in the bending moment imply a nonzero shear force. Therefore, condition 1 for cylindrical bending is

not met. The resulting shear-stress, τ

, which is determined from the equilibrium considerations at a typical

section m-n (Fig. 5a), is:

(Eq 11)

where

(Eq 12)

is the first moment of the shaded area (A

) with respect to the neutral axis (Fig. 5b). The first moment of the

unshaded area with respect to the neutral axis gives the same Q. The distribution of τ

for a rectangular cross

section is shown in Fig. 5(c).

Fig. 5 Distribution of shear stress, τ

, for a rectangular specimen. See text for details.

Reference cited in this section

2. A.C. Ugural and S.K. Fenster, Advanced Strength and Applied Elasticity, 3rd ed., Prentice Hall, 1995

Stress-Strain Behavior in Bending

P. Dadras, Wright State University (retired)

Elastic-Plastic Bending

The limit for elastic bending, which is the onset of elastic-plastic bending, is reached when the maximum fiber

strain (ε

)

max

= (h/R

) becomes equal to 9 (σ

/E), where σ

is the yield strength and E is Young's modulus of the

material. For bending beyond this limit, the beam consists of a central elastic core and two plastically

deforming zones remote from the neutral axis. For accurate analysis of elastic-plastic bending, factors such as

the shift of the neutral axis from the centroidal axis and the effect of radial (transverse) stresses must be

considered. Elastic-perfect plastic (Ref 4) and elastic-linear hardening (Ref 5) analyses are available. However,

in view of analytical and computational difficulties, an approximate method, which is an extension of the

simple-beam theory, is commonly employed. Therefore, the three assumptions stated for elastic bending will be

enforced. The location of the neutral axis is assumed to be fixed at the centroidal axis, as it is for elastic

bending.

For a beam with the stress-strain curve shown in Fig. 6(a), the development of longitudinal strain and stress at

different stages of deformation is shown in Fig. 6(b) and 6(c), respectively. In Fig. 6(b), a linear strain

distribution given by ε

= (-y/R

) was used. When the strain at the outermost fiber exceeds 0.1, it is suggested

that the true strain distribution (Eq 2) be used. For bending to a radius of curvature equal to R

, the strain

distribution and the subsequent stress distribution (from the σ - ε curve or from a known constitutive equation

for the σ - ε dependence) can be found. The moment, M, required to produce R

is:

(Eq 13)

Fig. 6 Stress-strain distributions in a beam. (a) Stress-strain curve. (b) Strain

distribution. (c) Stress distribution. (d) Stress distribution for elastic-perfect plastic

material

The thickness of the elastic core, C (Fig. 6c), is:

(Eq 14)

Therefore, in bending plates of the same material to the same radius of curvature R

, the fractional thickness of

the elastic core C/2h becomes smaller as the thickness increases.

For elastic-plastic bending, a general equation for the relationship between R

and M (analogous to Eq 9 for the

elastic case) does not exist. For the simple case of an elastic-perfect plastic material (Fig. 6d), the following

equation is obtained:

(Eq 15)

The predictions of this equation at large plastic deformations (C/2h ≤ 0.02) are not reliable.

References cited in this section

4. J. Chakrabarty, Theory of Plasticity, McGraw-Hill, 1987

5. P. Dadras, Plane Strain Elastic-Plastic Bending of a Strain-Hardening Curved Beam, Int. J. Mech. Sci.,

in press Nov 2000

Stress-Strain Behavior in Bending

P. Dadras, Wright State University (retired)

Pure Plastic Bending

The fractional thickness of the elastic core decreases as the ratio R

/2h decreases. For R

/2h ≤10, the thickness

of the elastic core in a hot rolled strip of AISI 1020 steel is ≤3.2%. In such cases it is possible to ignore the

elastic core altogether and use the analysis for pure plastic bending.

Most plastic bending analyses are for sheet-type specimens in which b is much greater than 2 h. The width

strain is small for such a geometry and plane-strain bending deformation (ε

= 0) is commonly assumed. Also,

the first two assumptions of the simple-beam theory are still applied. However, the radial (transverse) stresses

and strains induced by higher curvatures are considered, and a plane-in state, rather than a one-dimensional

stress state (the third assumption of the simple-beam theory), is assumed. Also, the neutral axis is not fixed at

the centroid, and thickness variations due to bending sometimes are incorporated into the solution. Figure 7

shows the geometry of deformation and the strain and stress states at different zones.

Fig. 7 Pure plastic bending of strip specimen. (a) Geometry of deformation. R

= inner

radius of curvature; R

= outer radius of curvature; R

= radius of curvature of the

neutral axis; R

= radius of currently unstretched fiber; R

= current radius of curvature

of original center fiber. (b) Strain and stress states in different zones.---, deformed state. ε

= ln r/R

is the circumferential strain. (c) Stress-strain curves and stress-strain states at

various locations. N = stress-strain state at R

; U = stress-strain state at R

; C = stress-

strain state at R

At the onset of the assumed full plastic condition, R

= R

. In the current state, the fiber with radius of

curvature equal to R

is being overtaken by the neutral axis and is experiencing unloading from a compressive

tangential stress field. Accordingly, all fibers below the neutral axis r ≤ R

have been progressively compressed,

while those situated above R

have been consistently stretched during deformation. All fibers in the interval R

≤ r < R

have been overtaken by the neutral axis. Fibers between R

< r < R

have now been stretched beyond

their original length due to reverse loading, while those located in R

≤ r ≤ R

have yet to recover their original

undeformed length.

As expected, a comprehensive analysis accounting for the described fiber movements is very complicated (Ref

6). A compromise solution (Ref 7), which ignores thickness variations and assumes rigid-perfect plastic

material behavior, provides useful approximations for the stress-strain distributions in plastic bending.

When a rigid-perfect plastic material model is used (Fig. 7c), the same stress-strain relationship applies to all

fibers with r ≥ R

. As a result, the distinction among R

, R

, and R

becomes inconsequential. This eliminates

the complicated task of describing the behavior of fibers in reversed loading and the Bauschinger effect.

The state of stress acting on a typical element is shown in Fig. 8, where σ

is the circumferential (tangential)

stress, and σ

is the radial (transverse) stress. The equilibrium equation for plane strain deformation is:

(Eq 16)

Fig. 8 State of stress acting on a typical element in plane-strain bending

The effective or significant stress, , and strain, , for plane strain deformation, using von Mises criterion, are,

respectively:

(Eq 17)

and

(Eq 18)

where ε

= ln r/R

. Substituting (σ

- σ

) from Eq 16 into Eq 15 and putting σ

= 0 at r = R

and r = R

, these

equations for the distribution of σ

are obtained:

(Eq 19)

(Eq 20)

From the expressions for σ

and Eq 16, the following expressions for σ

are determined:

(Eq 21)