Mechanics in Energy Resources Engineering - Chapter 5 Stresses in Beams (Basic topics)

Size: px

Start display at page:

Download "Mechanics in Energy Resources Engineering - Chapter 5 Stresses in Beams (Basic topics)"

Miranda Davis
5 years ago
Views:

1 Week 7, 14 March Mechanics in Energy Resources Engineering - Chapter 5 Stresses in Beams (Basic topics) Ki-Bok Min, PhD Assistant Professor Energy Resources Engineering i Seoul National University

2 Shear Forces and Bending Moments Preview Introduction Types of Beams, Loads, and Reactions Shear Forces and Bending Moments Relationships Between Loads, Shear Forces and Bending Moments Shear-Force and Bending-Moment Diagrams

3 Shear-Force and Bending-Moment Diagrams Concentrated Load Shear Force Diagram, Pb V L (0 x a ) Pa V L (0 x a ) Bending Moment Diagram Pbx M L ( a x L ) Pa M ( L x ) L ( a x L ) Slope dv/dx =0 q=0 Area for x < a increase in M Area for a<x<b decrease in M Slope dm/dx = V

4 Shear-Force and Bending-Moment Diagrams Uniform Load From Moment Equilibrium, R A R B ql 2 From Free Body Diagram, ql V RA qx qx 2 x qlx qx M RAxqx Slope of V? Slope of M? 2

Shear-Force and Bending-Moment Diagrams Several

P P P A B 1 2 3 From Free Body Diagram, V R M R

1 A 1 1 1 2 1 V R P B 3 M R Lx P Lb x a xa V R

5 Shear-Force and Bending-Moment Diagrams Several Concentrated Loads From Moment Equilibrium, R R P P P A B From Free Body Diagram, V R M R x (0 xa) A A V R P M R x P ( x a ) ( a x a ) A 1 A V R P B 3 M R Lx P Lb x a xa V R ( ) ( ) ( ) B B M R ( Lx) ( a xl) B 3

6 Saint-Venant Principle Effect of material property? Point Load Point Load Case 1 E = 2E5 MPa Case 2 E = 2 MPa

7 Outline Introduction Pure Bending and Nonuniform Bending Curvature of Beam Longitudinal Strains in Beams Normal Stress in Beams Design of Beams for Bending Stresses Nonprismatic Beams Shear Stresses in Beams of Rectangular Cross Section Shear Stresses in Beams of Circular Cross Section Shear Stresses in the Webs of Beams with Flanges

8 Introduction Chapter 4 Shear forces (V) & Bending Moments (M). How about stresses and strains associated with V & M? Assumption: Beams are symmetric about the xy plane. y-axis is an axis of symmetry of the cross section All loads act in this same plane known as the All loads act in this same plane, known as the plane of bending

Occurs only in regions with zero shear force Nonuniform bending Flexure in

9 Pure Bending and Nonuniform Bending Pure Bending: Flexure of a beam under a constant bending moment. Occurs only in regions with zero shear force Nonuniform bending Flexure in the presence of shear forces Bending moment changes Simple beam AB loaded by two couples M 1 Constant bending moment & shear force 0

10 Pure Bending and Nonuniform Bending Other examples nonuniform bending Pure bending nonuniform bending

11 Curvature of a Beam dfiiti definition Strains and stresses due to lateral load are directly related to the curvature of deflection curve. Two points m 1 & m 2 on the deflection curve Center of curvature Radius of curvature Center of curvature Radius of curvature Curvature (κ, 곡률, 曲率 ): reciprocal of the radius of curvature Measure of how sharply a beam is bent 1 1

12 Curvature of a Beam From the geometry of triangle O m 1 m 2, d ds By rearranging, 1 d ds Center of curvature Radius of curvature Under the assumption of small deflections deflection curve is nearly flat 1 d dx

13 Curvature of a Beam sign convention Sign convention of curvature (+) : beam is bent concave upward ( 위로오목 ) (-): beam is bent concave downward ( 아래로오목 )

14 Longitudinal Strains in Beams Basic assumption: Cross section of a beam in pure bending remain plane (There can be deformation in the plane itself) Upper part: shorten compression Lower part: elongate tension

15 Longitudinal Strains in Beams Neutral surface: no longitudinal strain Neutral axis: neutral surface s intersection with cross- sectional plane At neutral surface: dx d Length L1 of line ef after bending y L1 ( y ) d dx dx

16 Longitudinal Strains in Beams Longitudinal strain y x yy Strain-curvature relation Longitudinal strain is proportional to the curvature & distance y from the neutral surface (regardless of the material) Longitudinal stress expected Transverse strains due to Poisson s ratio does not induce transverse stress, why?

17 Normal Stress in Beams From Hooke s Law, Ey x Ex Ey Stresses are compression above the neutral surface with positive curvature Still not practical. Why? Determine y & relationship between κ (curvature) y p ( ) and M (Bending Moment)

18 Normal Stress in Beams Resultant of the normal stresses Resultant force in x direction is zero The resultant moment is equal to the bending moment M

Normal Stress in Beams Location of Neutral Axis

cross section A da x A A yda EydA 0 0 First moment

Y axis is also axis of symmetry The origin O of

19 Normal Stress in Beams Location of Neutral Axis Because there is no resultant force acting on the cross section A da x A A yda EydA 0 0 First moment of the area of the cross section evaluated with respect to z-axis is zero. z-axis must pass through the centroid. Y axis is also axis of symmetry The origin O of coordinates is located at the The origin O of coordinates is located at the centroid of the cross sectional area

20 Normal Stress in Beams Moment-Curvature t Relationship Elemental moment dm yda x 2 2 M Ey da E y da A M EI Moment-Curvature t Equation 1 M EI A I A 2 yda Flexural rigidity: a measure of the resistance of a beam to bending

21 Normal Stress in Beams Flexure Formula ( 굽힘공식 ) Finally, bending stress due to bending moment is: x Bending stress My I Maximum tensile and compressive bending stresses occur at points located farthest from the neutral axis. Mc M I S Mc M I S 2 S 1 I c S2 1 I c 2

Doubly symmetric shapes: when c 1 = c 2 = c Maximum tensile and maximum

22 Normal Stress in Beams Flexure Formula ( 굽힘공식 ) Section modulus: combines properties into a single quantity. Doubly symmetric shapes: when c 1 = c 2 = c Maximum tensile and maximum compressive stresses are equal numerically Mc M 1 2 I S A beam of rectangular cross section I bh 12 3 S bh 6 2

23 Example 5-3 Maximum tensile and compressive stress in the beam due to bending?

24 Summary Introduction Pure Bending and Nonuniform Bending Curvature of Beam Longitudinal Strains in Beams Normal Stress in Beams Design of Beams for Bending Stresses Nonprismatic Beams Shear Stresses in Beams of Rectangular Cross Section Shear Stresses in Beams of Circular Cross Section Shear Stresses in the Webs of Beams with Flanges

[8] Bending and Shear Loading of Beams

[8] Bending and Shear Loading of Beams Page 1 of 28 [8] Bending and Shear Loading of Beams [8.1] Bending of Beams (will not be covered in class) [8.2] Bending Strain and Stress [8.3] Shear in Straight