Engineering Mechanics of Solids by Popov Chapter 1: Stress, Axial Loads and Safety Concepts

1. Chapter 1
Stress, Axial Loads & Safety Concepts
Mechanics of Solids

2. • This subject involves analytical methods for determining the strength,
stiffness(deformation characteristics) and stability of material.
Part A
General concept: Stress
• Internal resistance of a body = the nature of forces setup within a body
to balance the effect of the externally applied forces.
• Sectioning of a body = The externally applied forces to one side of an
arbitrary cut must be balanced by the internal forces developed at the
cut.
2

3. • D’Alembert principle: If the force so computed is applied to the body
at its mass center in a direction opposite to the acceleration, the
dynamic problem is reduced to one of statics.
Fig. 1: Sectioned body: (a) free body with some internal forces, (b) enlarged view with components of ∆𝑃
• Stress = components of intensity of force per unit area.
3

4. • First subscript of τ indicates that the plane perpendicular to x axis is
considered, the second subscript designates the direction of the
stress component.
• Force perpendicular to the section is called the ‘normal stress’ at a
point. There can be Tensile or Compressive forces.
• SI Unit = N / 𝑚2 or Pa
• 1 in = 25 mm, 1 lbf = 4.4 N, 1 psi = 7 kpa
4

5. Stress Tensor
• Stress Transformation = the process of stresses changing from one set
of coordinate axis to another is termed stress transformation.
• Tensor = The state of a stress at a point which can be defined by 3
components on each of the 3 mutually perpendicular axis.
Fig. 2: (a) General state of stress acting on an infinitesimal element in the initial coordinate system.
(b) General state of stress acting on an infinitesimal element defined in a rotated system of coordinate axes.
All stresses have positive sense
5

6. • 3 normal stresses: 𝜏 𝑥𝑥 = 𝜎𝑥, 𝜏 𝑦𝑦 = 𝜎 𝑦, 𝜏 𝑧𝑧 = 𝜎𝑧
• 6 shearing stresses: 𝜏 𝑥𝑦, 𝜏 𝑦𝑥, 𝜏 𝑦𝑧, 𝜏 𝑧𝑦, 𝜏 𝑧𝑥, 𝜏 𝑥𝑧
• Force vector, P = 𝑃𝑋, 𝑃𝑌, 𝑃𝑍
• Matrix Representation of a stress Tensor:
6

7. Fig. 3: Elements in pure shear
• Simplifying above equation: 𝜏 𝑥𝑦 = 𝜏 𝑦𝑥
• Similarly, 𝜏 𝑦𝑧 = 𝜏 𝑧𝑦, 𝜏 𝑧𝑥 = 𝜏 𝑥𝑧. Stress Tensor is symmetric.
• Shear stresses always occurs in two pairs.
7

8. • Two- dimensional stress is referred to as plane stress.
Fig. 4: Elements in plane stress
• There can be Uniaxial, Biaxial or Tri-axial strain.
8

9. Differential equations of equilibrium
• An infinitesimal element of a body must be in equilibrium.
• For the 2D case, the system of stresses acting on an infinitesimal
element (dx)(dy)(1) is shown in figure:
• The possibility of an increment in stresses from one face of the
element to another is accounted for.
• Rate of change of 𝜎𝑥 in X – direction is
𝜕𝜎 𝑥
𝜕𝑥
and a step of dx is made,
the increment is
𝜕𝜎 𝑥
𝜕𝑥
dx.
9

10. Fig. 5: Infinitesimal element with stress and body forces
10

11. • For 3 dimensional case,
• For 2D case, 3 unknowns: 𝜎𝑥, 𝜎 𝑦, 𝜏 𝑥𝑦 and 2 equations
• For 3D case, 6 stresses and 3 equations –Internally Indeterminate
• Finite element method – small finite elements rather than
infinitesimal volume.
11

12. Part B: Stress Analysis of axially loaded bars
Fig. 6: Stresses on inclined sections in axially loaded bars
• The applied force, the reaction, as well as the equilibrating force P at the
section acts through the centroid of the bar section.
12

13. Fig. 7: sectioning of a prismatic bar on mutually perpendicular planes
13

14. • Force P can be resolved to 2 components: 1) Normal component – P ×
cos θ, 2) Shear Component – P × sin θ.
• The area of the inclined cross section – A / cos θ.
• Normal Stress: 𝜎 𝜃 and Shear Stress: 𝜏 𝜃.
• Maximum Normal stress at θ = 0°.
• For, θ = 90° No stresses act along the top and bottom free boundaries
of the bar.
• 𝜏 𝑚𝑎𝑥 at θ = ±45°
14

15. • Maximum Normal stress equation is only valid for axially loaded
compression members that are rather chunky, i.e., to short blocks.
• Bearing Stress - Sometimes, compressive stresses arise when one
body is supported by another. If the resultant of applied forces
coincides with centroid of the contact area between two bodies.
15

16. • In real materials, there is no homogeneity of stress distribution, for
that average stress represents the amount of stress.
Fig. 8: (a) Stress irregularity in material due to inhomogeneity,
(b) variation of tensile stress across a plate in a rolling operation, (c) residual stresses in a rolled plate.
• Residual stresses- In casting, high internal stresses due to uneven
cooling.
16

17. • Shear Stresses:
Fig. 9: Loading condition causing shear stresses between interfaces of glued blocks
• L 𝑢ders lines: glide or slip planes on a polished surface of a specimen.
17

18. • Average Shear Stress:
• Bearing Stress in Fig. 10 (a), 𝜎𝑏 = P/ td. t = plate thickness, d = bolt
diameter.
Fig. 10: Loading conditions causing shear and bearing stresses in bolts
18

19. • For Fig. 10(e) bearing stress for middle plane, 𝜎1 = P/ t1d and outer
plane, 𝜎2 = P/ 2t2d.
• Fig. 10(a) - Single Shear - shear force on one surface
• Fig. 10(e) - Double shear - shear force on two surfaces
Fig. 11: Loading condition causing critical shear in two planes of fillet weld
• Maximum shear stress in welded joint occurs in the planes a-a and b-
b.
19

20. Analysis for normal and shear stresses
• For the equilibrium of body in space, conditions for static equilibrium:
• In a planar problem, such as X-Y plane, relation 𝐹𝑍 = 0, 𝑀 𝑋 = 0, 𝑀 𝑌 = 0 while still
valid are trivial.
Fig. 11: Identical beams with identical loading having different support conditions

21. • fig. 11(a) Statically Determinate - If the equations of statics suffice for
determining the external reactions and internal stress resultants
• fig. 11(b) & (c) Indeterminate – number of independent equations of
statics is insufficient to solve for the reactions.
• The reactions that can be removed and leaving a stable system
statistically determinate are ‘Superfluous’ or ‘Redundant’.
• fig. 11(b) any one of the vertical reactions can be removed and the
structural system remains stable and tractable.
• fig. 11(c) any two reactions can be dispensed with for the beam.
• Depending on the number of redundant internal forces or reactions,
the system is said to be Indeterminate to the first degree fig. 11(b), or
second degree fig. 11(c).
21

22. Part C: Deterministic and Probabilistic design
bases
22

23. Deterministic Design of Members: Axially loaded bars
• Ultimate load: Force necessary to cause rupture.
• Ultimate Strength: Ultimate load divided by original cross sectional area
of the specimen.
• Fatigue Strength: Number of cycles required to break the specimen at a
particular stress under the application of a fluctuating load.
• Corresponding curve: S-N diagram (Stress - Number)
• At low stress an infinitely large number of reversals of stress can take
place before the material fractures. The limiting stress at which this
occurs is called the ‘endurance limit’ of the material.
• This limit being dependent on stress, is measured in ksi or Mpa.

24. Fig. 12: Fatigue strength of 18-8 SS at various temperatures (reciprocating beam stress)
24

25. • Deterministic approach: base on past experience and number of
experiments.
• Probabilistic approach: after identification of the main parameters in
a given stress analysis problem, their statistical variability is assessed.
To estimate structural safety.
• In the design of advanced aircraft, offshore structures, buildings and
bridges.
• Factor of Safety:
• In the aircraft Industry:
margin of Safety =
25

26. • In an alternative approach ‘Ultimate load’ is obtained by multiplying
the working loads by a suitably chosen ‘load factors’.
• Allowable stress design approach (ASD): The required net area A of a
member,
26

27. Probabilistic Basis for Structural Design
• In this approach, variability in material properties, fabrication-size
tolerances, as well as uncertainties in loading and even design
approximation, can be appraised on a statistical basis.
• Probabilistic approach has the advantage in consistency in the factor of
safety, even for complex structural assemblies.
• The test results are termed as “Population Samples”.
• Sample Mean (average), 𝑋
• Sample Variance, S2
where,Xi = ith sample
27

28. Experimental Evidence
Fig. 13: (a) Histogram of maximum compression strength for Western Hemlock (wood)
(b) Frequency diagram of compression yield strength of ASTM grades A7 and A36 steels.
28

29. • Standard deviation, S = square root of variance.
• Coefficient of Variation, V =
𝑆
𝑋
.
• The expected sample value is the 𝑋; S is a measure of dispersion
(scatter) of data, V is dimensionless.
Fig. 14: Normal Probability density function of Z.
29

30. Theoretical Basis
• These bell shaped curves of probability density functions(PDFs) are
based on normal or Gaussian distribution.
• Mean:
• Variance:
• The constant
1
2𝜋
is selected so that the normalized frequency
diagram encloses a unit area.
30

31. Fig. 15: Examples of probabilities of outcomes at different amounts of standard deviation from the mean
• fig. 15(a) - the probability of occurrence of an outcome between one
standard deviation on the either side of mean is 68.27 %.
• Fig. 15(b) – between 2 standard deviations: 95.45 %
31

32. • The areas enclosed under the curve tails that are three standard
deviations from the mean are only 0.135% of the total outcomes.
• The small number of outcomes likely to take place under fz(z) several
standard deviations away from the mean is of the utmost importance
in appraising structural safety.
Practical Formulations
• Statistically determined resistance PDF fR(r), and a corresponding load
effect PDF fQ(q) define the behavior of the same critical parameter
such as a force, stress or deflection.
• It is assumed that load effect has larger standard deviation from the
mean, than that for the member resistance.
32

33. • Conventional factor of safety= Rn / Qn
• For R > Q, no failure can occur. That is safe design.
• For R < Q, a failure would take place.
Fig. 16: PDF for the two main random variables (load and resistance)
Fig. 17: Probabilistic definition of safe and unsafe structural design
33

34. • The probability of failure, pf Is given by the area under the tail of the
curve to the left of the origin.
Fig. 18: (a) Normal and (b) lognormal probability density functions
• A possible magnitude of pf may be surmised from fig. 18(b). A
member will survive in all instances to the right of the origin.
34

35. • 𝛽𝜎 𝑅−𝑄 = 𝜇 𝑅−𝑄, where 𝛽 = safety index and 𝜎 𝑅−𝑄= standard
deviation.
• Variance of a linear function of two independent normal variables,
𝜎 𝑅−𝑄
2
, is the sum of the variances of its part.
• A larger 𝛽 results in fewer failures and thus, a more conservative
design. Here, R is the resistance and Q is load effect.
• When the distributions of R and Q are skewed and the lognormal
distribution rather than the normal is appropriate. Concept of ‘factor
of safety’ (R / Q) can be used.
35

36. • Where, 𝜇 𝑅 and 𝜇 𝑄 are the mean values for respective functions, and
𝛿 𝑅 and 𝛿 𝑄 are coefficients of variation.
• For routine applications, a β on the order of 3 is considered
appropriate.
• Uncertainties in design variables can be explicitly included by using
the coefficients of the variation in the design parameters, resulting in
more consistent reliability of structures and machines.
• This approach is also suitable for the serviceability limit states, control
of maximum deflections or limitations on undesirable vibrations.
• Lognormal distribution for a random variable R,
𝜉 𝑅
2
= ln 1 + 𝛿 𝑅
2
𝜆 𝑅 =
ln 𝜇 𝑅
1 + 𝛿 𝑅
2
36

37. Bolted and Riveted Connections
• A connection design approach based on preventing slippage between
the faying surfaces.
• Ductile deformations and/or slip between the faying surfaces permits
an equal redistribution of the applied force before the ultimate
capacity of a connection is reached.
Fig. 19: Assumed action for a bolted or a riveted connection
37

38. • In single lap joints, the connections are in single shear, and the plates
near the connector tend to bend to maintain the axial force
concentric.
Fig. 20: Bending of plates commonly neglected in lap joints
• When connectors are arranged as shown in Fig. 21(a), determining
the net section in tension poses no difficulty.
• If the rows for the bolt holes are closely spaced and staggered, as
shown in Fig. 21(b), a zigzag tear maybe more likely to occur than a
tear across a normal section b-b.
• It is also necessary to have a sufficient edge distance to prevent a
shear failure across the c-c planes shown in Fig. 21(c)
38

39. Fig. 21: possible modes of failures in bolted joints (connections): (a) net section, (b) zig-zag tear, and
(c) tear out due to insufficient edge(end) distance along the lines c-c
• Failure in bearing is approximated on the basis of an average bearing
stress acting over the projected area of the connector’s shank onto a
plate, i.e., on area td.
• The frictional resistance between the at the connectors has been
neglected.
39

40. • If the clamping forces developed by the connector is both sufficiently
large and reliable, the capacity of the joint can be determined on the
basis of the friction force between the faying surfaces.
• With the use of high-strength bolts with the yield strength on the
order of 100 ksi (700 MPa), this is an acceptable method in structural
steel design.
• The required tightening of such bolts is usually specified to be about
70% of their tensile strength.
Fig. 22: (a), (b) Illustration of a bearing failure, and (c) assumed stress distribution
40

41. Welded Joints
• Butt welds – Fig.23 and fillet welds – fig. 24 are particularly common.
Fig. 23: Complete penetration butt welds. (a) single V-groove weld, (b) double V-groove weld
Fig. 24: an example of a fillet weld
41

42. • The strength of butt welds is simply found by multiplying the cross-
section area of the thinner plate being connected by the allowable
stress for welds.
• Allowable strength – certain percentage of the strength of the original
solid plate of the parent material.
• For ordinary purpose 20% reduction in the allowable stresses for the
weld compared to the solid plate maybe used.
• Fillet welds are designated by the size of the legs, Fig. 24(b) which are
usually made of equal width ‘w’.
• Throat – the smallest dimension across a weld - 2 × 𝑤
42