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2 .2 Stresses in Beams

s x. s x. P. x. M xz. M xz. P 1. P 2. 2 .0 Bending of Beams. ☻. 2 .1 Revision – Bending Moments. (Refer: B,C & A – Sec’s 6.3-6.6 ). 2 .2 Stresses in Beams. 2 .3 Combined Bending and Axial Loading. (Refer: B,C & A – Sec’s 6.11, 6.12 ). 2 . 4 Deflections in Beams.

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2 .2 Stresses in Beams

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  1. sx sx P x Mxz Mxz P1 P2 2.0 Bending of Beams ☻ 2.1 Revision – Bending Moments (Refer: B,C & A –Sec’s 6.3-6.6) 2.2 Stresses in Beams 2.3 Combined Bending and Axial Loading (Refer: B,C & A –Sec’s 6.11, 6.12) 2.4Deflections in Beams (Refer: B,C & A –Sec’s 7.1-7.4) 2.5Buckling (Refer: B,C & A –Sec’s 10.1, 10.2)

  2. y Mxz Mxz y C D y’ y’ x z NA A B dx Neutral Axis dq R Mxz Mxz sx=0 on the Neutral Axis. In general we must find the position of the Neutral Axis. y’ C’ D’ A’ B’ 2.2Stresses in Beams (Refer: B, C & A–Sec 6.3, 6.4, 6.5, 6.6) 2.2.1 The Engineering Beam Theory Compression No Stress Tension

  3. Mxz Mxz C D y’ A B dq R Mxz Mxz y’ C’ D’ A’ B’ Plane surfaces remain plane Beam material is elastic and only Assumptions

  4. and 1 Geometry of Deformation: Hookes Law:

  5. dx y Linear Distribution of sx x (Eqn ) 1 y’ NA 0 Neutral Axis -ve +ve sx Note: y E is a Material Property x is Curvature Mxz Mxz dx 1

  6. Let y But dA Mxz sx y’ x z First Moment of Area Area, A y “Neutral Axis” coincides with the XZ plane through the centroid. Centroid y’ y’ x z NA Neutral Axis Equilibrium: Then y’ is measured from the centroidal axis of the beam cross-section.

  7. y dA Mxz sx y’ x z Area, A as 1 Let 2 & 1 2 Equilibrium: =The 2nd Moment of Area about Z-axis THE SIMPLE BEAM THEORY:

  8. - Applied Bending Moment - N.m - m4 - Property of Cross-Sectional Area - N/m2or Pa - Stress due to Mxz - m - Distance from the Neutral Axis - N/m2or Pa - Young’s Modulus of Beam Material - m - Radius of Curvature due to Mxz y y’ y’ x z NA o Neutral Axis

  9. y dA sx y’ y’ is measured from the Centroidal or Neutral Axis, z. o x Iz is the 2nd Moment of Area about the Centroidal or Neutral Axis, z. Mxz z y Area, A dA y Centroidal Axis dA y’ z z o o y’ n (Definition) 2.2.2 Properties of Area (Refer: B, C & A–Appendix A, p598-601) Position of Centroidal or Neutral Axis:

  10. y 200 10 z Centroidal Axis o 125 120 60 n 20 Example: (Dimensions in mm)

  11. y Definition: dA z’ y’ z o Example: y dy y’ z o 2nd Moment of Area:

  12. Definition: n Example: y dy y’ z o n The Parallel Axis Theorem: y z o

  13. 30.4 89.6 200 10 3 20 30.4 35.4 2 89.6 1 20 y Example: (Dimensions in mm) 200 10 • What is Iz? • What is maximum sx? o z 120 20

  14. y Example: (Dimensions in mm) 200 10 • What is Iz? • What is maximum sx? 30.4 o z 120 89.6 200 10 20 3 20 30.4 35.4 2 89.6 1 20

  15. Mxz 40.4 89.6 Maximum Stress: y x NA (N/m2or Pa)

  16. dA z’ R y’ The Polar 2nd Moment of Area (About the X-axis) Example: y dR From Symmetry, R z o The Perpendicular Axis Theorem: y z o

  17. Calculating properties of beam cross sections is a necessary part of the analysis. • Neutral Axis Position, y • 2nd Moments of Area, Iy, Iz, Jx Properties of Areas are discussed in Appendix A of B, C & A. 2.2.3 Summary The Engineering Beam Theory determines the axial stress distribution generated across the section of a beam. It is applicable to long, slender load carrying devices.

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