1. ☻ 1.0 Axial Forces 2.0 Bending of Beams M M
Now we consider the elastic deformation of beams (bars) under bending loads. M M

2. Application to a Bar Fn Mt Ft Mn Normal Force: Bending Moment: S.B.
Shear Force:
K.J. Ft
Torque or Twisting Moment: Mn

3. Examples of Devices under Bending Loading:
Atrium Structure
Excavator
Yacht
Car Chassis

4. 2.0 Bending of Beams sx 2.1 Revision – Bending Moments
(Refer: B,C & A – Sec’s 6.1,6.2)
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)

5. RECALL… 2.1 Revision – Bending Moments Q (SFD) (BMD) M
(Refer: B, C & A – Chapter 6)
Last year Jason Ingham introduced Shear Force and Bending Moment Diagrams. 3m
12 kN Q
(SFD) M
(BMD)

6. P Mxz RAy RBy A B Mxz Consider the simply supported beam below:
(Refer: B, C&A – Sections 1.14, 1.15, 1.16, 6.1) x y
Radius of Curvature, R P
Deflected Shape A B
RBy
RAy
Mxz
Mxz
What stresses are generated within, due to bending?

7. Recall: Axial Deformation WP A B
RBy
Mxz
RAy
Bending
Recall: Axial Deformation W u
Load (W)
Extension (u)
Bending Moment (Mxz)
Curvature (1/R)
Axial Stiffness
Flexural Stiffness

8. sx is NOT UNIFORM through the section depth
Axial Stress Due to Bending:
Mxz=Bending Moment y
sx (Tension)
sx (Compression)
Mxz x
sx=0
Beam
Unlike stress generated by axial loads, due to bending:
sx is NOT UNIFORM through the section depth
sx DEPENDS ON:
(i) Bending Moment, Mxz
(ii) Geometry of Cross-section

9. Mxz Qxy Qxy=Shear Force Sign Conventions: y Mxz=Bending Moment x
-ve sx
Mxz
Qxy
+ve sx
+VE (POSITIVE)
“Happy” Beam is +VE
“Sad” Beam is -VE

10. P Mxz=P.L RAy=P P.L Mxz P Qxy P Mxz Mxz Qxy Qxy
Example 1: Bending Moment Diagrams P y
Mxz=P.L A B x
RAy=P L
P.L
Qxy
Mxz x P P
Mxz
Qxy
Mxz
Qxy
Q & M are POSITIVE

11. Shear Force Diagram (SFD) Bending Moment Diagram (BMD)P y L
P.L x
Mxz
Qxy A B x P
Qxy
+ve P
Shear Force Diagram (SFD)
Mxz
Bending Moment Diagram (BMD)
-ve
-P.L
To find sx and deflections, need to know Mxz.

12. P P A A C B Qxy Mxz a b y Example 2: Macaulay’s Notation a x x
Where can only be +VE or ZERO.

13. P A C B Mxz BMD: y a b x x (i) When 1 (ii) When 2 Eq. 1 Eq. 2 +ve A B1
(ii) When 2
Mxz
BMD:
Eq. 1
Eq. 2
+ve A B C

14. wL2 Mxz= 2 RAy=wL wL2 Mxz wx 2 Qxy wL
Distributed Load w per unit length
Example 3: Distributed Load y x
Mxz=
wL2 2 A B x
RAy=wL L wx
wL2 2
Qxy
Mxz wL

15. Mxz
BMD: L x
-ve
-wL2 2

16. Summary – Is anything Necessary for Revision
Generating Bending Moment Diagrams is a key skill you must revise. From these we will determine:
Stress Distributions within beams,
and the resulting Deflections
Apart from the revision problems on Sheet 4, you might try these sources:
B, C & A
Worked Examples, pg
Problems, 6.1 to 6.8, pg 173
Jason Ingham’s problem sheets: