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

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

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

4. Geometry of Deformation:
Hookes Law: 1
and

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

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

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

8. - Applied Bending Moment - N.m
- Property of Cross-Sectional Area
- m4
- Stress due to Mxz
- N/m2 or Pa
- Distance from the Neutral Axis
- m
- Young’s Modulus of Beam Material
- N/m2 or Pa
- Radius of Curvature due to Mxz
- m y y’ y’ x z NA o
Neutral Axis

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

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

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

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

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

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

15. Maximum Stress: y
89.6
40.4
Mxz x NA
(N/m2 or Pa)

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

17. Neutral Axis Position, y
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.
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.