CIE3109 : Structural Mechanics 4

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CI3109 CI3109 : Structural echanics 4 13-14 Unsmmetrical and/or inhomogeneous cross sections Introduction General theor for extension and bending Unsmmetrical cross sections example : curvature

1 CI3109 CI3109 : Structural echanics Unsmmetrical and/or inhomogeneous cross sections Introduction General theor for extension and bending Unsmmetrical cross sections example : curvature versus loading example : normal stress distribution Deformation 15 Inhomogeneous cross sections Refinement of the theor xamples Core and Shear Stresses Core of the cross section Shear stresses in a non-smmetrical cross sections Shear centre in a thin walled non-smmetrical cross section Ir J.W. Welleman bladnr 1

2 QUSTION What is the stress and strain distribution in a cross section due to extension and bending in case of a unsmmetrical and/or non-homogeneous cross section? concrete 1 steel 2 a b c 1 Ir J.W. Welleman bladnr 2

3 BNDING ND XTNSION F ϕ F ϕ x ϕ F u x F x u u x RLTION BTWN XTRNL LODING ND DISPLCNTS - Structural level - Cross sectional level RRK The definition of moments differs from the international standard! Ir J.W. Welleman bladnr 3

4 SSUPTIONS Cross section bar axis axis of smmetr 1 FIBR ODL Fibres 2 SLL ROTTIONS OF TH CROSS SCTIONS 3 UNIXIL STRSS SITUTION IN TH FIBRS 4 LINR LSTIC TRIL Hooke s law : σ Fibre Cross section Ir J.W. Welleman bladnr 4

5 SOLUTION PTH o Determine the strain distribution due to the displacements of the cross section o Determine the stress distribution caused b these strains o Determine the resulting forces in the cross section Load Stress Strain Displacement Fq NV uϕ quilibrium Constitutive Kinematic equations equations equations Ir J.W. Welleman bladnr 5

6 KINTIC QUTIONS RLTION BTWN STRINS ND DISPLCNTS Fibre Cross section ϕ u x x-axis u x x-axis u ϕ u ϕ ϕ -axis ϕ d dx u -axis ϕ d u dx Ir J.W. Welleman bladnr 6

7 DISPLCNT u in fibres // to x-direction u x u φ φ or x u x u u u x STRIN & CURVTUR u x x u u x u x-axis -axis x-axis -axis strain strain 2 du d u x ; ; 2 dx dx d u dx 2 2 Ir J.W. Welleman bladnr 7

8 CURVTUR o Curvature is first order tensor o is curvature in x--plane o is curvature in x--plane o curvature in x-k plane is denoted with k RRK The definition of the curvature differs from the international standard! tanα 2 2 k k k Ir J.W. Welleman bladnr 8

9 STRIN DISTRIBUTION SSNTILS: o strain in fibre which coincides with the x-axis o slope of strain diagram in -direc o slope of strain diagram in -direc Ir J.W. Welleman bladnr 9

10 NUTRL LIN : ZRO STRINS IN TH CROSS SCTION k is perpendicular to neutral line k 0 Ir J.W. Welleman bladnr 10

11 Ir J.W. Welleman bladnr 11 CONSTITUTIV RLTION : STRIN ND STRSS o Relation between strain and stress in a fibre [ ] σ σ QUILIBRIU RLTION : FRO STRSS TO FORCS o Relation between stresses and resulting cross sectional forces like N V and

12 Ir J.W. Welleman bladnr 12 QUILIBRIU d d d d d d N σ σ σ

13 Ir J.W. Welleman bladnr 13 LBORT THS RLTIONS 2 2 S d d d S d d d S S d d d N RSULT : DOUBL-LTTR-SYBOL : S S S S N RRK : RBITRRY POSITION OF TH COORDINT SYST

14 NORL FORC CNTR NC NC NC NC NC S S d d d d NC NC d S d S NC NC d THUS : S NC NC S Ir J.W. Welleman bladnr 14

15 SPCIL CS : ORIGIN OF TH COORDINT SYST IN TH NORL FORC CNTR NC N extension bending definition NC : S S 0 Ir J.W. Welleman bladnr 15

16 RSULT o BNDING ND XTNSION R UNCOUPLD o ONT IS a 1 e ORDR TNSOR 2 tanα m 2 RRK The definition of the moment differs from the international standard! oment is acting in the x-m plane Ir J.W. Welleman bladnr 16

17 Ir J.W. Welleman bladnr 17 SURY tension compression CURVTUR k N.L. x-axis N.L. -axis -axis α k ONT m x-axis -axis -axis α m ONT tension compression N.C. N.C. N σ

18 XPL 1 NC n.a. 30 deg 2a n.a. a Determine the position of the loading plane m-m for the specified position of the neutral line which makes an angle of 30 o degrees with the -axis. Ir J.W. Welleman bladnr 18

19 XPL 2 C triangular cross section is loaded b pure bending onl. The cross section is homogeneous and the oungs modulus is. The normal stress in point and C is equal to 10 N/mm 2. 6a NC B a 20 mm a Calculate the magnitute and direction of the resulting bending moment b Compute the stress in point B 3a Use the next sheet with some formulae Ir J.W. Welleman bladnr 19

20 ONT OF INRTI FORULS FOR TRINGULR CROSS SCTION I I I bh b 3 3 h bh 3 tanα 1 36 bh 3 tanα 2 Ir J.W. Welleman bladnr 20

THEME IS FIRST OCCURANCE OF YIELDING THE LIMIT?

CIE309 : PLASTICITY THEME IS FIRST OCCURANCE OF YIELDING THE LIMIT? M M - N N + + σ = σ = + f f BENDING EXTENSION Ir J.W. Welleman page nr 0 kn Normal conditions during the life time WHAT HAPPENS DUE TO