Stresses in Beams B the end of this lesson, ou should be able to: Calculate the maimum stress in a beam undergoing a bending moment 1 Outline Curvature Normal Strain Normal Stress Neutral is Moment of nertia Shear Stress Combined ial Loading Organization Last week ou learned how to calculate: Shear Force Bending Moments This week, ou will use: Those calculations New equations To calculate the stress within beams undergoing bending 1
Pure Bending (Constant Moment): (Shear force must be 0: V dm/d ) Non-Uniform Bending: 5 Curvature in Pure Bending Radius of Curvature: 1 Curvature: κ 6
Curvature in Pure Bending From triangle O m 1 m : dθ ds 1 dθ κ ds 7 Curvature in Pure Bending For small deflections, dsd: 1 dθ κ d + Positive Curvature 8 Longitudinal Strain Top Compresses Bottom Stretches Neutral is: No strain Length of line ef: L d 1 ( ) d dθ L1 d d θ 9
Longitudinal Strain Original length of line ef: d Elongation δ d L 1 d Strain: δ ε d ε κ 10 Comments Strain is maimum on top/bottom surface Stress varies linearl with distance from neutral surface, regardless of shape of object Strain produces stress, since there is no where for material to go 11 Eample Length L 16 ft. Height h 1 ft. Longitudinal strain on bottom surface is 0.0015 Distance from bottom surface to neutral surface is 0.5 ft. What is radius of curvature, curvature κ, and deflection δ of the beam? 1
ε κ Solutions: 0.5 ft 00 ft ε 0.0015 1 κ 0.005 ft 1 O'C O'C cosθ δ O ' C ' O ' C 1 cosθ ( ) ssume B L, for nearl flat curve: L 1 L sinθ θ sin 1.16 ( ) δ 00 ft 1 cos1.16 0.960in 1 Comments on Eample Not useful for finding deflection Limited to pure bending Useful for showing relative magnitudes of, L, and δ. We will talk about useful was to find deflection later ( weeks from now?) 1 Normal Stress in Beams E Eε Eκ 15 5
How to find Neutral is? Resultant Force in X direction 0: d 0 Eκ d 0 d 0 E and κ are constants Centroid! Neutral is is located at Centroid (When there is no aial force) 16 How to find Moment-Curvature Relationship? Resultant Moment in direction Bending Moment M: dm d M d κ E d κ E d M κ E d Moment of nertia of the cross section area 1 M κ E 17 nertia is confusing! nertia (cceleration) m (mass) Dimensions: m Polar Moment of nertia (Torque) P d Dimensions: L F ma τ r ma T P Moment of nertia rea d Dimensions: L M κ E Bending Properties Mass dm Dimensions: ml M α Rotational cceleration 18 6
Sign Convention 19 Fleure Formula From Previous Slides: Eκ M κ E M Observations: Stress is proportional to bending moment M Stress is inversel proportional to moment of nertia Ι Stress varies linearl with distance from neutral ais 0 Section Moduli Section Moduli S is Moment of nertia at surface M S 1 7
Moment of nertia (rea) & Section Moduli 6 S b h 1 S 6 Moment of nertia is more sensitive to height than width Disclaimers These equations are onl for regions of: Pure Bending (no shear loads) Prismatic beams Homogeneous Linearl Elastic ε κ 1 M κ E E Eκ Non-uniform bending produces warping. Normal stresses are not altered b warping We can use theor of pure bending to calculate normal stresses M rec circle 6 1 Eample steel wire is bent around a clindrical drum. d mm R 0 0.5m E 00 GPa Proportional limit p1 100 MPa Determine the bending moment M and maimum bending stress in the wire. 8
Useful Equations: ε κ 1 M κ E E Eκ circle M 6 Solution rec 1 d R0 + circle 6 1 M κ E E M Eπ d 6 Eπ d M 5.01Nm R ( ) 0 + d R0 + d E < p1 100 MPa E d Ed 797MPa R d R0 + d 5 0 + Eample Determine the maimum tensile and compressive stress in the beam due to bending 6 Solution Useful Equations: ε κ 1 M κ E E Eκ circle M 6 Non-uniform bending Strateg: Using Bending-Moment Diagram to calculate maimum Moment Calculate Reaction Forces Draw Shear-Force Diagram Draw Bending-Moment Diagram rec 1 Use -M/Ι to calculate normal stresses 7 9
Solution 1) Reactions: M ( 1.5 k / ft )( ft )( 11 ft ) ( 1 k )( 9 ft ) + B ( ft ) 0 B 1.1k F ( 1.5 k / ft)( ft) ( 1k ) + B 0.59k ) Diagrams: Useful Equations: M circle 6 rec 1 ) Ma M: 8 Solution ) Maimum Stress M ma 151.6 k ft rec M ( 8.8 /1)( 7 /1) 0.696 ft 1 1 Useful Equations: M circle 6 rec 1 Centroid Middle (area is smmetrical) ( 151.6kft )( 1.5 in /1) c 5 k / ft.696 ft ( 151.6kft )( 1.5 in /1) t 5 k / ft.696 ft 9 Eample Determine the maimum tensile and compressive stresses in the beam due to the uniform load Cross section 0 10
Solution Reaction Forces:.6 kn, B 10.8 kn Maimum Positive M: M.05 knm Maimum Negative M: M -.6 knm 1 Solution Find Neutral is: Calculate Centroid Y 1 6 mm, 1 1 mm Y 0 mm, 960 mm Y 0 mm, 960 mm ( 6mm)( 1mm ) + ( 0mm)( 960mm ) 1 + ( 960 ) c 18.8mm i i 1 i mm mm Solution (continued) Calculate Moment of nertia Parallel is Theorem: + d z c Seg 1 Seg, c ( b t ) t 97mm ( b t ) t 1mm 1 th 51000mm 1 th 960mm d c 1 -t/ 1.8mm c 1 -h/ -1.5mm z 555600 mm 956600 mm z 1 + z + z.69 10 mm 6 11
Solution (continued) Calculate Maimum Stress Maimum positive stress.05 knm ( mm) M.05kNm 18.8 c 15.MPa 6.69 10 mm ( ) M.05kNm 18.8mm 80mm t 50.5MPa 6.69 10 mm Maimum negative stress -.6 knm ( ) M.6kNm 18.8mm 80mm c 89.8MPa 6.69 10 mm ( mm) M.6kNm 18.8 t 6.9MPa 6.69 10 mm c -15. MPa Eplanation of Solution t 6.9 MPa t 50.5 MPa c -89.8 MPa Maimum Compressive stress -89.8 MPa, Maimum Tensile stress 50.5 MPa 5 Shear Stress in Beams Pure Bending does not produce Shear Non-uniform Bending produces shear Use Fleure formula to calculate normal stress Use the following concepts to calculate shear stress 6 1
Shear Stress Normal stresses are: 0 at neutral ais Maimum at surface Shear stresses are: Maimum at neutral ais 0 at surface 7 Shear Stress in Rectangular Beams VQ τ Q d b Vh V τ ma 8 Q is the integral of the cross sectional area above the level at which shear stress is being evaluated. Y is with respect to the neutral ais 8 Shear Stress in Circular Beams Can onl calculate shear stress easil at the neutral ais V V τ ma π r Hollow Shaft: τ V V r + r r + r 1 1 ma π r r + r1 9 1
Shear Stress in Rectangular Beams VQ τ Q d b Vh V τ ma 8 0 Eample Determine the normal stress and shear stress at point C, which is located 1 in below the top of the beam and 8 in from the right hand support. Show these stresses on a sketch of a stress element at point C. 1 Solution 1) Determine shear force V C and bending moment M C at point C: M C 17,90 lb-in V C -1600 lb ( 1in )( in) ) Moment of nertia: 5.in 1 1 M ( 1790lb in)( 1in ) ) Normal Stress at C: C 60 psi 5.in ) Shear Stress at C: centroid ( )( ) ( 1600lb)( 1.5in ) Q d 1.5in 1in 1.5in VQ τ 50 psi b in in Convention: ( 5. )( 1 ) 1
ial Loading Stock Beams Length / 10 *height Can Superimpose ial and Bending Moments Neutral is changes Non-normal Loads: N M Eccentric Loads P Pe + Tpical Eam Question Calculate the Shear stress at an point on the bar below as a function of length L, moment M1, height h, thickness t, Young s modulus E, and Poisson s ratio ν: Hint: You should be able to solve it b the time ou are done reading this sentence 15