Chapter 5 Elastic Strain, Deflection, and Stability 1. Elastic Stress-Strain Relationship

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1 Chapter 5 Elastic Strain, Deflection, and Stability Elastic Stress-Strain Relationship A stress in the x-direction causes a strain in the x-direction by σ x also causes a strain in the y-direction & z-direction by Chapter 5 Elastic Strain, Deflection, and Stability 1

2 E Resulting Strain Each Direction Stress x y z σ x σ y σ z Chapter 5 Elastic Strain, Deflection, and Stability 2

3 Adding the columns to obtain the total strain in each direction ε x ε y ε z Shear strain γ xy, γ yz, γ zx Note: shear strain on a given plane is by the shear stresses on other planes. Generalized Hooke s Law Only elastic constants are needed for an material. G Chapter 5 Elastic Strain, Deflection, and Stability 3

4 Only two are independent elastic constant Pure shear stress Mohr s circle σ ε τ γ/2 ε 1 ε 2 from G τ/γ ε 3 γ Chapter 5 Elastic Strain, Deflection, and Stability 4

5 Example: 1. The stress that develops in the y-direction. 2. The strain in the z-direction. 3. The strain in the x-direction. 4. The stiffness E σ z / ε z in the z-direciton. Is this equal to E? ε y & σ x Chapter 5 Elastic Strain, Deflection, and Stability 5

6 1. ε y 0 σ y 2. ε z 3. ε x 4. E σ z / ε z Chapter 5 Elastic Strain, Deflection, and Stability 6

7 Volumetric Strain & Hydrostatic Stress Volume changes associated with. Shear strains cause only dv Since VLWH dv V ε v V dv ν 0. 5 ε ν > 0. 5 tensile stress decrease volume V Chapter 5 Elastic Strain, Deflection, and Stability 7

8 Hydrostatic stresses Invariant σ h σ v Volumetric strain hydrostatic stress Constant modulus B Chapter 5 Elastic Strain, Deflection, and Stability 8

9 Castigliano s Method Useful in computing elastic deflection and redundant reactions Deflection Figure 5.15 General load deflection curve for elastic range U U stored elastic energy is equal to times. du du Deflection, In general case, Chapter 5 Elastic Strain, Deflection, and Stability 9

10 Axial Loading Case U U δ U Sample problem 5.4 Chapter 5 Elastic Strain, Deflection, and Stability 10

11 Sample problem 5.4 con t. 1. M V Q M Px 2 valid only x 0 x L 2 2. U 3. U δ P Chapter 5 Elastic Strain, Deflection, and Stability 11

12 Problem 5.15 (page233) What are the angular and linear displacements of point A of Figure 5.15? Known: Figure P.15 is given. Find: Calculate the angular and linear displacements of point A. Chapter 5 Elastic Strain, Deflection, and Stability 12

13 Problem 5.19 (page 234) Figure 5.19 shows a steel shaft supported by self-aligning bearings and subjected to a uniformly distributed load. Using Castigliano s method, determine the required diameter d to limit the deflection to 0.2mm. Known: A steel shaft supported by self-aligning bearings is subjected to a uniformly distributed load. Find: Using Castigliano s Method, determine the required diameter, d, to limit the deflection to 0.2mm. Assumption: 1. The steel shaft remains in the elastic region. 2. The transverse shear deflection is negligible. Analysis: Chapter 5 Elastic Strain, Deflection, and Stability 13

14 Problem 5.23(page 235) In order to reduce the deflection of the I-beam cantilever shown, a support is to be added at S. (a). What vertical force at S is needed to reduce the deflection at this point to zero? (b). What force is needed to cause an upward deflection at S of 5mm? (c). What can you say about the effect of these forces at S on the bending stresses at the point of beam attachment? Assumptions: 1. The beam remains elastic. 2. Transverse shear deflection is negligible. Analysis: Chapter 5 Elastic Strain, Deflection, and Stability 14

15 Redundant Reactions by Castigliano s Method Reduntant reaction: force or moment that is for equilibrium. As magnitude of a redundant reaction is varied, changes, But remains. Castigliano s theorem states that the associated with any reaction that can be varied without upsetting equilibrium. The deflection. Chapter 5 Elastic Strain, Deflection, and Stability 15

16 Sample Problem 5.9 Figure 5.22 Find: Determine the tension in the guy wire Assumption: Analysis: At point a Chapter 5 Elastic Strain, Deflection, and Stability 16

17 M Bending energy below point a 2 3 M u dy 0 2EI The horizontal deflection at point a δ 0 u F F Chapter 5 Elastic Strain, Deflection, and Stability 17

18 Euler Column Buckling Figure 5.24 B0 Q ρ cr xl, y0 Asin ρl 0 2 d y 2 dx M EI ρ 2 I Aρ S cr or S E cr Chapter 5 Elastic Strain, Deflection, and Stability 18

19 Le / p 10 S cr 0. 1 E Fig5.25 Log-log plot of Euler Eq (dimensionless, hence applies to all materials within their elastic range). Fig5.26 Euler column buckling curves illustrated for two values of E and S y. Chapter 5 Elastic Strain, Deflection, and Stability 19

20 Figure 5.27 Equivalent column lengths for various end conditions Figure 5.28 Euler and Johnson column curves illustrated for two valuses of E and S y Chapter 5 Elastic Strain, Deflection, and Stability 20

21 Secant formula for the loading, taking the into account. S cr Pcr A S y L 1+ ( ec )sec ( e 2 ) ρ ρ 4AE Where c denotes the distance from the neutral bending plane to the extreme fiber. P cr Chapter 5 Elastic Strain, Deflection, and Stability 21

22 Draft paper 1/2 Chapter 5 Elastic Strain, Deflection, and Stability 22

23 Draft paper 2/2 Chapter 5 Elastic Strain, Deflection, and Stability 23