Unsymmetrical Bending of Beams
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- Horatio Booker
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1 Unsmmetrical Bending of Beams
2 ntroduction Beam structural member takes transverse loads Cross-sectional dimensions much smaller than length Beam width same range of thickness/depth Thin & thick beams f l 15 t thin beam Thin beam Euler Bernoulli s beam Thick beam Timoshenko beam rd_mech@ahoo.co.in
3 ntroduction n thin beams deformation due to shear negligible Thick beams shear deformation considered Beam one dimensional structural member length is ver high than lateral dimensions Various parameters > function of single independent variable 3 rd_mech@ahoo.co.in
4 Sign convention Following sign convention is followed ve moments o Orientation of beam Beam bending in and planes ψ ve rotations ψ ψ 4 rd_mech@ahoo.co.in
5 Sign convention Shear force and bending moments F F Positive shear force and bending moments Conve upward ve direction F F 5 rd_mech@ahoo.co.in
6 Unsmmetrical bending Load applied in the plane of smmetr P Load applied at some orientation θ P G G Smmetrical cross-sectional load applied in the plane of smmetr - Bending takes place in plane Bending takes place in both planes and 6 rd_mech@ahoo.co.in
7 Unsmmetrical bending Cross-section is unsmmetrical P P Bending takes place in both planes 7 rd_mech@ahoo.co.in
8 Euler Bernoulli s beam theor Basic assumptions Length is much higher than lateral dimensions l 15 t Plane cross section remains plane before and after bending Stresses in lateral directions B B negligible Thin beam strain variation is linear across cross-section Hookean material linear elastic B 8 rd_mech@ahoo.co.in
9 Bending of arbitrar cross section beam n arbitrar cross-section beam oriented along -direction ε N G σ d css coincides with centroid dn, σ Eε a E b c ( a b c) Equilibrium of the section requires σ d F > N Eternal moments E( a b c)d ntegrate this for total force in - direction 9 rd_mech@ahoo.co.in
10 Bending of arbitrar cross section beam This equals to eternal force in direction N ( )d N σ d E a b N N Ea d Eb d Ec Ec > c N E c d Origin of co-ordinate sstem is selected at centroid - Coefficients of Ea and Eb constants vanish 10 rd_mech@ahoo.co.in
11 Bending of arbitrar cross section beam oment wrt ais d ( σ d ) d E ε d N G σ d ntegrating over cross-sectional area > Eb E ( a b c) Ea d Ec d d d 11 rd_mech@ahoo.co.in
12 Bending of arbitrar cross section beam oment about ais Ea d Eb d Ec > Ea Eb - (1) d oment about ais ( a b c) E d Ea d Eb d Ec > Ea Eb - () a and b unknowns solve simultaneous algebraic equations (1) and () d 1 rd_mech@ahoo.co.in
13 Bending of arbitrar cross section beam Solving (1) and () a and b values are as following a (, b ) E( ) E Strain is ε E ( ) ( E ) E This represents a straight line N 13 rd_mech@ahoo.co.in
14 Bending of arbitrar cross section beam Bending stress multipl aial strain (bending strain) with Young s modulus (E) Neutral ais /plane no stress/strain > 0 ( ) ( ) > slope > m N N ( ) 0 14 rd_mech@ahoo.co.in
15 Bending of arbitrar cross section beam Equation of neutral ais N ( ) Neutral ais doesn t pass through origin centroid of cross-section f no aial load > N 0 G N α slope, m N. equation m Passes through centroid tanα 15 rd_mech@ahoo.co.in
16 Bending of arbitrar cross section beam Neutral ais (N. ) Equation of neutral ais completel depends on geometr and loading moments and aial load ial load offset N.. does not pass through centroid of cross-section Orientation (slope) purel depends on moments and geometr Orientation not decided b aial load ndependent of material properties - isotropic Simplifing equation if co-ordinate sstem coincides with principal aes 16 rd_mech@ahoo.co.in
17 Bending of arbitrar cross section beam Bending in a single plane - no aial load > N 0 G σ d 0 > > N ( ) Equation of N. depends on area moment of inertia 17 rd_mech@ahoo.co.in
18 Principal aes rea moment of inertia geometric propert d, d o d p (, ) θ Product moment of inertia d Rotate o css to a new css o at an angle θ o θ p (, ) 18 rd_mech@ahoo.co.in
19 Transformation Principal aes sin θ cos θ cos θ sin θ cos θ cos d θ d sin o ( cosθ sin θ ) θ sin d θ p (, ) θ sin θ sin θ d - (1) d 19 rd_mech@ahoo.co.in
20 Transformation Principal aes ( sinθ cosθ ) d sin cos d sin θ d cos θ d sinθ θ θ sinθ - d () 0 rd_mech@ahoo.co.in
21 Principal aes Product moment of inertia a propert defined wrt a set of perpendicular aes ling in the same plane of area d > ( )( cosθ sinθ sinθ cosθ ) d > cosθ sinθ d d ( cos θ sin θ ) d 1 rd_mech@ahoo.co.in
22 Product. Principal aes 1 sin θ Product. can be ve or ve depends on selection of co-ordinate sstem ( ) cos θ - (3) Variation of, and wrt θ harmonic When product. changes sign it passes through ero also Principal aes is a css, where product. is ero rd_mech@ahoo.co.in
23 Principal aes Variation of, and aimum value of inimum value of 0 θ 3 rd_mech@ahoo.co.in
24 Principal aes Product. 0, area. maimum and minimum Orientation corresponding to ma. & min. principal aes > θ, θ 90 0 From (3), 0 1 sin > θ ( ) tanθ ( ) cos θ rea moment of inertia a second order tensor 0 4 rd_mech@ahoo.co.in
25 Smmetric sections Product moment of inertia Principal aes d Co-ordinate sstem is selected smmetricall, is positive and negative. - o Summation over the area vanishes ero. YoZ principal coordinate sstem. Yo Z another co-ordinate sstem. Product moment of inertia doesn t vanish. One ais of smmetr principal ais O 5 rd_mech@ahoo.co.in
26 Bending of arbitrar cross section beam Co-ordinate sstem is aligned with principal css product. > 0 N G σ d ε E( ) E( ) E N ε E E E Simplified epression gain, N. equation can be obtained b making strain ero N 6 N rd_mech@ahoo.co.in
27 Bending of arbitrar cross section beam When no aial load acts N 0 equation of N. N N > G Straight line passing through centroid f, 0 > Equation of N. > 0 This is ais 7 rd_mech@ahoo.co.in
28 ial deformation ial deformation in bending w B B ψ o ψ v B View - V ψ V 8 rd_mech@ahoo.co.in
29 ial deformation ial deformation due to aial load, N > u o ial deformation due to bending Bending in plane > -ψ Bending in plane > -ψ ial deformation due to all loads u( ) u ψ ( ) ψ ( ) o -ve signs rotations are opposite to moments 9 rd_mech@ahoo.co.in
30 ial deformation n Euler-Bernoulli s theor shears strains negligible o u u ψ ψ v u γ 0 w u γ 0 rd_mech@ahoo.co.in 30 v v ψ ψ γ > > 0 0 w w ψ ψ γ > > 0 0
31 ial deformation ial deformation in terms of deflection gradient v w u u v w u u o rd_mech@ahoo.co.in 31,, strain, v a w b u c a b c v w u u o o ε ε
32 Curvatures Curvature gradient of slope b a v w o E > E v w > E E c u N E > E u oment curvature relationships o N 3 rd_mech@ahoo.co.in
33 Shear stress distribution Shear stress in a solid section F b rea τ d b Solid section F b * F b force due to bending stress τ - shear acting on plane b d b local width on which shear to be estimated 33 rd_mech@ahoo.co.in
34 Shear stress distribution Force due to bending stress on area Bending force at d F b σ d F * b F b F b d σ d σ d Shear force acting on plane b d F τ bd s 34 rd_mech@ahoo.co.in
35 Equilibrium > F b Shear stress distribution σ F * b F b d > F d b F b F τb F b σ b F s 0 τ bd 0 τb d > τb σ F b * d b area, which is above the plane on which shear to be found τ F b d rea 35 rd_mech@ahoo.co.in
36 Shear stress distribution Bending stress σ Eε σ 1 1 N f a beam is subjected to pure bending no variation of bending moment along length σ 0, 0, 0 > τ 0 36 rd_mech@ahoo.co.in
37 Shear stress distribution Shear stress eists when transverse loads q are applied q V* V V N G σ d V > V * d Force equilibrium of infinitesimal element V V * qd V 0 d qd V 0 > V q - (1) 37 rd_mech@ahoo.co.in
38 Shear stress distribution oment equilibrium q d * V * d qd d V V d d V - () 0 q Higher order terms are neglected From (1) and () V ( d) q 0 > V* V * d q 38 rd_mech@ahoo.co.in
39 Shear stress distribution When transverse loads are acting Using equation (), V V V V rd_mech@ahoo.co.in 39 V V d b σ σ σ τ 1 1 d V V b τ d d, V V b τ
40 Shear stress distribution Transverse loading in plane - V τ b V τ b - V V - - V 0 V G Resembles shear flow, q Shear flow at a given horiontal section depends on moment of area above that section wrt css passing through centroid This shear flow is not constant. Varies from ero to maimum 40 rd_mech@ahoo.co.in
41 Shear stress distribution Thin walled beam subjected to transverse loads F b d τ t F * b Equilibrium of infinitesimal element gives stress distribution in the wall 41 rd_mech@ahoo.co.in
42 Shear stress distribution Shear stress in thin walled beam τ t V V Shear flow in the wall of thin beam - Sum of the shear forces in the cross-section eternall applied shear force > vertical equilibrium of the section - q Shear flow is not constant varies from point to point along the wall Shear is purel because of transverse loads, which cause bending > Bending shear Shear due to pure torsion > Torsion - shear 4 rd_mech@ahoo.co.in
43 Shear center Equilibrium of beam cross-section requires Forces and moment equilibrium Twisting will not take place if moment due to shear developed in the wall is equal to moment due to eternal transverse load Variable location of application of eternal load Shear flow is a function of eternal transversal load. oments of forces taken wrt an point in space gives point of application of eternal load shear center Shear center independent of eternal load depends onl on geometr q V 43 rd_mech@ahoo.co.in
44 Shear center Shear center (S. C) ma or ma not pass through centroid of beam cross-section When there is no aial load, N. passes through centroid Loads acting in both planes q e Q q e e Q e S.C 44 rd_mech@ahoo.co.in
45 Shear center Some specific cross-sections F 1 F 3 e o G h Q F Equilibrium in direction > Q F 3 Equilibrium in direction > F 1 - F 0 oment of forces wrt o Q e h h F1 F1 0 > e F Q 1 h 45 rd_mech@ahoo.co.in
46 Shear center When load acts at S.C, it produces onl bending no twisting takes place Q e G F 1 F a Vertical equilibrium > F 1 F Q Horiontal equilibrium sum of forces in horiontal web 0 b Taking moments wrt G F 1a F b Q e 0 > e F1a Q F b Epress F 1 and F in terms of transverse load, Q e will be a function of geometric parameters 46 rd_mech@ahoo.co.in
47 Shear center Smmetric cross-section F 1 F 1 o Q Since cross-section is smmetric, shear distribution in each leg is smmetric wrt plane Taking moments about O h G F 3 F F F h Fh 0 - no moment Eternal load should pass through ais of smmetr shear center lies in the plane smmetr n smmetric cross-sections shear center lies in the plane of smmetr 47 rd_mech@ahoo.co.in
48 Shear center Load acting at some angle Q α Q Q cosα G SC G SC SC G Q Q sinαα Singl smmetric Point of application of load has to through S.C no twisting takes place 48 rd_mech@ahoo.co.in
49 Shear center Doubl smmetric beam Q G Smmetric in plane S.C lies on - ais G Q Smmetric in plane S.C lies on - ais n a doubl smmetric beam shear center lies at centroid 49 rd_mech@ahoo.co.in
50 Shear center Location of shear center for some cross-sections 50
51 51
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