Chapter 3. Load and Stress Analysis

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1 Chapter 3 Load and Stress Analysis

2 2 Shear Force and Bending Moments in Beams Internal shear force V & bending moment M must ensure equilibrium Fig. 3 2

3 Sign Conventions for Bending and Shear Fig. 3 3

4 4 Distributed Load on Beam Distributed load q(x) = load intensity Units of force per unit length Fig. 3 4

5 5 Relationships between Load, Shear, and Bending

6 6 Cartesian Stress Components Plane stress occurs = stresses on one surface are zero Fig. 3 8

7 7 Plane-Stress Transformation Equations Fig. 3 9

8 8 Principal Stresses for Plane Stress principal directions principal stresses Zero shear stresses at principal surfaces Third principal stress = zero for plane stress

9 9 Extreme-value Shear Stresses for Plane Stress Max shear stresses: on surfaces that are ±45º from principal directions Two extreme-value shear stresses:

10 10 Mohr s Circle Diagram Relation between x-y stresses and principal stresses Relationship is a circle with center at C = (s, t) = [(s x + s y )/2, 0] R s x s 2 y 2 t 2 xy

11 Mohr s Circle Diagram 11 Fig. 3 10

12 12 Elastic Strain For a stress element undergoing s x, s y, and s z, simultaneously,

13 13 Elastic Strain Hooke s law for shear: Shear strain g = change in a right angle of a stress element when subjected to pure shear stress. G = shear modulus of elasticity For a linear, isotropic, homogeneous material,

14 14 Uniformly Distributed Stresses For tension and compression, For direct shear (no bending present),

15 15 Normal Stresses for Beams in Bending Straight beam in positive bending x axis = neutral axis xz plane = neutral plane Neutral axis is coincident with centroidal axis of the cross section Fig. 3 13

16 16 Normal Stresses for Beams in Bending Bending stress varies linearly with distance from neutral axis, y Fig. 3 14

17 17 Transverse Shear Stress (TSS) Fig TSS is always accompanied with bending stress

18 18 Transverse Shear Stress in a Rectangular Beam

19 Torsion Angle of twist for a solid round bar 19 Fig. 3 21

20 20 Stress Concentration Localized increase of stress near discontinuities K t = Theoretical (Geometric) Stress Concentration Factor

21 21 Theoretical Stress Concentration Factor A-15 and A-16 Peterson s Factors Stress-Concentration

22 22 Stress Concentration for Static and Ductile Conditions With static loads and ductile materials Highest stressed fibers yield (cold work) Load is shared with next fibers Cold working is localized Overall part does not see damage unless ultimate strength is exceeded Stress concentration effect is commonly ignored for static loads on ductile materials

23 23 Stresses in Pressurized Cylinders Fig Tangential and radial stresses

24 24 Stresses in Pressurized Cylinders Special case of zero outside pressure, p o = 0

25 25 Stresses in Pressurized Cylinders If ends are closed, then longitudinal stresses also exist

26 26 Thin-Walled Vessels Cylindrical pressure vessel with wall thickness 1/10 or less of the radius Radial stress is quite small compared to tangential stress Average tangential stress

27 27 Thin-Walled Vessels Maximum tangential stress Longitudinal stress (if ends are closed)

28 28 Curved Beams in Bending Fig. 3 34

29 29 Curved Beams in Bending Location of neutral axis Stress distribution

30 30 Curved Beams in Bending Stress at inner and outer surfaces

31 31 Example 3-15 Plot the distribution of stresses across section A A of the crane hook shown in Fig. 3 35a. The cross section is rectangular, with b = 0.75 in and h = 4 in, and the load is F = 5000 lbf.

32 32 Example 3-15 Fig. 3 35

33 33 Formulas for Sections of Curved Beams (Table 3-4)

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