L2. Bending of beams: Normal stresses, PNA: , CCSM: chap 9.1-2

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1 L2. Bending of beams: ormal stresses, P: , CCSM: chap àcoordinate system üship coordinate system übeam coordinate system üstress resultats - Section forces üstress - strain tensors (recall structure) àormal stresses in beams üuniaxial bar problem (1D stress problem) übeam problem at bending Kinematical relations Stress - strain relations (Hooke's law) Generalized stress - strain relation Evaluation of normal stresses due to bending Example: L - beam (non-symmetric) Equilibrium Governing Material equations and Computational for bending problem Mechanics Group 1

2 àcoordinate system üship coordinate system z : heave HhävningL φ: roll HrullningL θ: pitch HstampningL übeam coordinate system 17 (1) (2) (3) ote! c=centroid (neutral axis) coincides with center of gravity for homogeneous material. Sectional forces (stress resultants): üstress resultats - Section forces Consider sectional forces, cf. Fig., due to bending defined as = σ x M y = σ x z, T = τ xz 18 M z = σ x H yl, V = τ xy M x = Material and Computational Mechanics Group Hτ xy He z zl τ xz He y yll 2

3 à ormal stresses in beams ü Uniaxial bar problem (1D stress problem) Equilibrium Kinematics Elastic material behavior ü Beam problem at bending ssumptions i) plane cross-section remains plane after deformation ii) Shear deformations are neglected iii) Elastic material behavior assumed (Hooke's law) Material and Computational Mechanics Group 3

4 Kinematical relations ssumption i) fl "common" in-plane displacements for the cross section u y, zd = v@xd u y, zd = w@xd ssumption ii) fl express out-plane shear strains (from kinematics of 3D solid) Integration γ xy = u x y + u y x = u x y + v x = 0 u x y = v γ xz = u x z + u z x = u x z + w x = 0 u x z = w u y, zd = v x y + g@x, zd u x z = w x = g z ote! h@xd = u 0,0D = u@xd g@x, zd = w x z + h@xd u y, zd = v x y + g@x, zd = v x y w x z +u@xd Material and Computational Mechanics Group 4

5 Kinematical relations Express normal strain ε x = u x x = u x 2 v x 2 y 2 w x 2 z =ε 0 +κ z y +κ y z Generalized strains : ε 0 = u x ; κ y = 2 w x 2 ; κ Z = 2 v x 2 ote! Plane bending about y-axis ( 2 v x 2 = 0) fl ε x = u x x =ε 0 κ y z Material and Computational Mechanics Group 5

6 Stress - strain relations (Hooke's law) Consider linear elasticity fl In particular σ x = E ε x = E Hε 0 +κ z y +κ y zl Generalized stress - strain relation = Ÿ E Hε 0 +κ z y +κ y zl M z = Ÿ E Hε 0 y +κ z y 2 +κ y yzl M y = Ÿ E Hε 0 z +κ z yz +κ y z 2 L = i y M j z z = k M y { i E j k 1 y z y y 2 yz z y yz z i j z 2 { k ε 0 κ z κ y y z { Consider special case: i) homogeneous beam fl E@x, y, zd = E ii) Centroid as origin of coordinate system fl E = E; y = y = 0 Reduced generalized stress strain relation: i y M j z k M y { i z = E j k I yz 0 y i I yz z j I y { k ε 0 y κ z z i j κ y { k ε 0 y κ z z = 1 ED κ y { i j k D I y I yz 0 I yz y i z j { k M z M y y z { 2 with D = I y I yz and Material = y 2 and ; Computational I y = z 2 ; Mechanics I yz = yz Group 6

7 Evaluation of normal stresses due to bending From Hooke's law and inverted stress - strain relation σ x = E Hε 0 +κ z y +κ y zl = E@1, y, zd i j k ε 0 κ z κ y y z =...= { σ x = + z HM z I yz + M y L y HM y I yz + M z I y L D ote! Bending stress evaluation requires: 1) Determination of cross section properties: centroid c,, I y, I yz fl Consider Steiners theorem for (composed) thinwalled cross sections rea deviation inertia moment I yz : Cross sections ) Determination of M z, M y and???? fl equilibrium considerations Material and Computational Mechanics Group 7

8 Example: L - beam (non-symmetric) data = 8b 2 300, b 1 400, t 1 10, t 2 10< 1) Evaluation: Centroid, consider conditions y = 0 c y = b 2 Hb 2 2 t 1 L t 2 b 1 t Hb 1 t 1 + b 2 t 2 L = mm z = 0 c z = b 1 2 t 1 b 2 t b 1 t b 2 t 2 = mm 2) Evaluation: rea moment of inertia, Steiner's theorem yields fl = y 2 = mm 4 I y = z 2 = mm 4 I yz = yz Material = and Computational 10 7 mm 4 Mechanics Group 8

9 Equilibrium In view of fig. we obtain w.r.t to action in the x-direction: 20 +H +dl+q x dx = 0 d dx +q x = 0 In view of fig. we obtain w.r.t to action in the xz plane: T +HT +dtl+q z dx = 0 dt dx +q z = 0 d2 M y dx 2 + q z = 0 M y +HM y + dm y L HT + dtl dx + q z dx 1 2 dx = 0 dm y dx T = 0 In view of fig. we obtain w.r.t to action in the xy plane: V +HV +dvl+q y dx = 0 dv dx +q y = 0 d2 M z dx 2 +q y = 0 M z +HM z + dm z L+HV + dvl dx + q y dx 1 2 dx = 0 dm z dx + V = 0 Material and Computational Mechanics Group 9

10 Governing equations for bending problem ormal action: + q x = 0; = E u HE u L + q x = 0 Bending action, from constitutive relation and kinematics we have J M z M y = E J I yz I yz I y J κ z κ y ; κ z = v ; κ y = w fl M z + q y = 0; M z = E v EI yz w HE v + EI yz w L + q y = 0 M y + q z = 0; M y = EI yz v EI y w HEI yz v + EI y w L + q z = 0 Example: Simply supported beam 16 Material and Computational Mechanics Group 10

11 Governing equations for bending problem ote! the relation x-x represent a coupled set of equations, that may typically be solved by direct integration and identification of boundary conditons for both v and w! Integration yields HE v +EI yz w L + q y = 0 E v EI yz w = q y x+ b x3 + b x2 +b 3 x +b 4 HEI yz v +EI y w L + q z = 0 EI yz v EI y w = q z x+ a x3 + a x2 +a 3 x +a 4 J v w = 1 ED J I y I yz i ŸŸŸŸq z x + b x 3 + b x 2 +b 3 x+b 4 y j k ŸŸŸŸq y x +a x 3 + a x2 + a 3 x +a z 4 { I yz Identification 8 boundary conditions for the displacements J v finally yields the total solution. w ote! In practice, solution often established via introduction of principal coordinate system, defined by I yz = 0 fl Uncoupled set of equations ote! Unified handling in terms of structural element stiffness relations, cf. CCSM, chap Material and Computational Mechanics Group 11

12 Thanks for today! Material and Computational Mechanics Group 12

13 ü Stress - strain tensors (recall structure) Consider stress tensor w.r.t. Cartesian basis fl Components σ = i σ x τ xy j k τ xz τ xy σ y τ yz τ xz τ yz σ z y z { ote! traction t w.r.t. arbitrary surface (with orienation n) fl t = σ n i j k t x t y t z y i z = j { k σ x τ xy τ xz τ xy σ y τ yz τ xz τ yz σ z y i z j { k n x n y n z y z { ssocited with stress tensor fl Strain tensor ε in component form ε = 1 2 i j k 2 ε x γ xy γ xz γ xy 2 ε y γ yz γ xz y γ yz z 2 ε z { with ε x = u x x ; ε y = u y y ; ε z = u z z γ xy = u x y Material x and ; γ Computational xz = u x + u z Mechanics Group z + u y z x ; γ yz = u y + u z y 13

14 Moments of inertia - Steiners theorem Consider definition of area moments of inertia: = y 2 ; I y = z 2 ; I yz = yz Establish integration over subregions = y 2 = 8y = y i + a yi < = Hy i + a yi L i 2 = i i=1 I y = z 2 = 8z = z i + a zi < = Hz i + a zi L i 2 = i i=1 i=1 i=1 i=1 i=1 2 Hi + a yi i L 2 HI yi + a zi i L I yz = yz = i i=1 i=1 Hz i + a zi L Hy i + a yi L = HI yzi + a yi a zi i L i i=1 ote! Cross section built up by thin rectangular parts i = btfl I i = bt3 12 Jor I i = tb3 12 ; I yzi := 0 Material and Computational Mechanics Group 14

15 Material and Computational Mechanics Group 15

16 Material and Computational Mechanics Group 16

17 ü Example simply supported beam: Evaluation midspan deflections ote! From elementary loading case: = 5 H2 LL4 q 384 " EI " What is "EI" in the present case? fl Consider coupled problem J M z M y = E J I yz I yz I y J κ z κ y In present case we have (from equilibrium considerations) M z + q y = M z = 0 M z = 0duetoM = M LD = 0 J M z = 0 M y = E J I yz I yz I y J κ z κ y κ z + I yz κ y = 0 M y = E i k We thus obtain ji y Iyz 2 y z κ y { p z = w@ld = From the conditions 5 H2 LL 4 q 384 E II y I yz 2 M = 5 ql4 24 E II y I yz 2 M κ z = I yz κ y ; κ z = v ; κ y = w v = I yz w v = I yz w + a 1 x + a 2 Synchronous boundary conditions: 8v@0D = w@0d = 0, v@2 LD = w@2 LD = 0 < a 1 = a 2 = 0 v = I yz w whereby p y = I yz w@ld = 5 ql 4 I yz 24 E HI y I yz 2 L Material and Computational Mechanics Group ote! Sign of I yz determine the sign of the lateral displacement p y! 17

18 Material and Computational Mechanics Group 18

19 Material and Computational Mechanics Group 19

20 Material and Computational Mechanics Group 20

21 Material and Computational Mechanics Group 21

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