Strength of materials - Chapter 7: Pure bending - Nguyễn Sỹ Lâm

Nội dung text: Strength of materials - Chapter 7: Pure bending - Nguyễn Sỹ Lâm

1. CHAPTER 7: PURE BENDING 7.0 Introduction 7.1 Bending deformation 7.2 Stress due to pure bending 7.3 Composite section 7.4 Stress concentration 7.5 Plastic analysis
2. 7.0 INTRODUCTION OTHER LOADING TYPES • Eccentric Loading: Axial loading which does not pass through section centroid produces internal forces equivalent to an axial force and a couple • Transverse Loading: Concentrated or distributed transverse load produces internal forces equivalent to a shear force and a couple • Principle of Superposition: The normal stress due to pure bending may be combined with the normal stress due to axial loading and shear stress due to shear loading to find the complete state of stress.
3. 7.1 BENDING DEFORMATIONS BENDING DEFORMATIONS Beam with a plane of symmetry in pure bending: • member remains symmetric • bends uniformly to form a circular arc • cross-sectional plane passes through arc center and remains planar • length of top decreases and length of bottom increases • a neutral surface must exist that is parallel to the upper and lower surfaces and for which the length does not change • stresses and strains are negative (compressive) above the neutral plane and positive (tension) below it
4. 7.2 STRESS DUE TO PURE BENDING STRESS DUE TO BENDING • For a linearly elastic material, y  E E x x c m y  (stress varies linearly) c m • For static equilibrium, • For static equilibrium, y Fx 0  x dA  m dA y c M y x dA y  m dA c  m 0 y dA   I c M m y2 dA m c c First moment with respect to neutral Mc M  plane is zero. Therefore, the neutral m I S surface must pass through the y section centroid. Substituting   x c m My  x I
5. 7.2 STRESS DUE TO PURE BENDING DEFORMATIONS IN A TRANSVERSE CROSS SECTION • Deformation due to bending moment M is quantified by the curvature of the neutral surface 1   1 Mc m m c Ec Ec I M EI • Although cross sectional planes remain planar when subjected to bending moments, in-plane deformations are nonzero, y y     y x z x • Expansion above the neutral surface and contraction below it cause an in-plane curvature, 1  anticlastic curvature
6. 7.2 STRESS DUE TO PURE BENDING EXAMPLE 7.01 SOLUTION: • Based on the cross section geometry, calculate the location of the section centroid and moment of inertia.  yA 2 Y Ix  I Ad  A • Apply the elastic flexural formula to find the maximum tensile and compressive stresses. Mc  m I A cast-iron machine part is acted upon by a 3 kN-m couple. Knowing E = 165 • Calculate the curvature GPa and neglecting the effects of fillets 1 M (đường gờ cong), determine (a) the EI maximum tensile and compressive stresses, (b) the radius of curvature.
7. 7.2 STRESS DUE TO PURE BENDING EXAMPLE 7.01 • Apply the elastic flexural formula to find the maximum tensile and compressive stresses. Mc  m I M cA 3 kNm 0.022m  76.0 MPa  A A I 868 10 9 mm4 M cB 3 kNm 0.038m  B  B 131.3 MPa I 868 10 9 mm4 • Calculate the curvature 1 M EI 3 kNm 1 3 -1 20.95 10 m 165 GPa 868 10-9 m4 47.7 m
8. 7.2 STRESS DUE TO PURE BENDING TECHNICAL EXPRESSION M x  x y I x x tension tension ymax cmax compression compression ymax cmax y M x tension  c M x max I max  ctension   x max I max tension For safety: x M x compression  min cmax  min  compression I x
9. 7.3 COMPOSITE SECTION EXAMPLE 7.02 SOLUTION: • Transform the bar to an equivalent cross section made entirely of brass • Evaluate the cross sectional properties of the transformed section • Calculate the maximum stress in the transformed section. This is the correct maximum stress for the brass pieces of the bar. Bar is made from bonded pieces of 6 steel (Es = 29x10 psi) and brass • Determine the maximum stress in the 6 (Eb = 15x10 psi). Determine the steel portion of the bar by multiplying maximum stress in the steel and the maximum stress for the transformed brass when a moment of 40 kip*in section by the ratio of the moduli of is applied. elasticity.
10. 7.3 COMPOSITE SECTION REINFORCED CONCRETE BEAMS • Concrete beams subjected to bending moments are reinforced by steel rods. • The steel rods carry the entire tensile load below the neutral surface. The upper part of the concrete beam carries the compressive load. • In the transformed section, the cross sectional area of the steel, As, is replaced by the equivalent area nAs where n = Es/Ec. • To determine the location of the neutral axis, x bx n A d x 0 2 s 1 b x2 n A x n A d 0 2 s s • The normal stress in the concrete and steel My  x I  c  x  s n x
11. 7.3 COMPOSITE SECTION REINFORCED CONCRETE BEAMS – EXAMPLE 7.03 SOLUTION: • Transform to a section made entirely of concrete. E 29 106 psi n s 8.06 6 Ec 3.6 10 psi 5 2 2 nAs 8.06 2 in 4.95in 4 8 • Evaluate the geometric properties of the transformed section. x 12x 4.95 4 x 0 x 1.450in 2 I 1 12in 1.45in 3 4.95in 2 2.55in 2 44.4in 4 3 • Calculate the maximum stresses. Mc1 40kip in 1.45in  c c 1.306ksi I 44.4in 4 Mc2 40kip in 2.55in  s n 8.06  s 18.52ksi I 44.4in 4
12. 7.5 PLASTIC ANALYSIS PLASTIC DEFORMATIONS • For any member subjected to pure bending y   strain varies linearly across the section x c m • If the member is made of a linearly elastic material, the neutral axis passes through the section centroid My and  x I • For a material with a nonlinear stress-strain curve, the neutral axis location is found by satisfying Fx  x dA 0 M y x dA • For a member with vertical and horizontal planes of symmetry and a material with the same tensile and compressive stress-strain relationship, the neutral axis is located at the section centroid and the stress- strain relationship may be used to map the strain distribution from the stress distribution.
13. 7.5 PLASTIC ANALYSIS MEMBERS MADE OF AN ELASTOPLASTIC MATERIAL • Rectangular beam made of an elastoplastic material Mc    x Y m I I   M  maximum elastic moment m Y Y c Y • If the moment is increased beyond the maximum elastic moment, plastic zones develop around an elastic core. y2 M 3 M 1 1 Y y elastic core half - thickness 2 Y 3 2 Y c • In the limit as the moment is increased further, the elastic core thickness goes to zero, corresponding to a fully plastic deformation. M 3 M plastic moment p 2 Y M k p shape factor (depends only on cross section shape) MY
14. 7.5 PLASTIC ANALYSIS RESIDUAL STRESSES • Plastic zones develop in a member made of an elastoplastic material if the bending moment is large enough. • Since the linear relation between normal stress and strain applies at all points during the unloading phase, it may be handled by assuming the member to be fully elastic. • Residual stresses are obtained by applying the principle of superposition to combine the stresses due to loading with a moment M (elastoplastic deformation) and unloading with a moment -M (elastic deformation). • The final value of stress at a point will not, in general, be zero.
15. 7.5 PLASTIC ANALYSIS EXAMPLE 7.04, 7.05 • Thickness of elastic core: y2 M 3 M 1 1 Y 2 Y 3 2 c y2 36.8kNm 3 28.8kNm 1 1 Y 2 3 2 c y y Y Y 0.666 2y 80mm c 60mm Y • Radius of curvature: • Maximum elastic moment:  240 106 Pa  Y Y 9 I 2 E 200 10 Pa 2 bc2 2 50 10 3m 60 10 3m c 3 3 1.2 10 3 6 3 120 10 m yY Y I 6 3 MY Y 120 10 m 240MPa c y 40 10 3m Y 28.8 kN m 3 33.3m Y 1.2 10