# BENDING STRESS

Bending stress is calculated from Bending Moment and Area Moment.

Smartboard Notes: Bending-Stress.pdf Bending-Stress.oneThe formula for determining the bending stress in a beam
under simple bending is^{}:

where;

σ is the bending stress (Ivonaff uses f_{b})

*M*- the moment about the neutral axis

*y*- the distance to the neutral axis

*I*

_{x}- the second moment of area about the neutral axis

*x*

*b*- the width of the section being analyzed

*h*- the depth of the section being analyzed

Vertical Bending Ixx |
Lateral Bending Iyy |
Torsion J |

**Example:** Lifting a House

Central support point

Universal beam spanning 10.2 m

Along the beam.

#### Find Stress in this beam.

Span = 10.2m

Flange thickness 10mm, overall height 305mm, overall width 168mm.

Identify the beam: **310UB/40.4**

So Ixx = 86.4E6 mm^{4} (or 86400000)

What is the maximum deflection of the beam over this span when stress is 150MPa? (Point load = worst case)

Bending Stress:

From this we can determine the maximum value of M that can be handled;

Bending stress: σ = 150 MPa

Distance to the neutral
axis y = 304/2 = 152 mm

Second
moment of area* I _{x}_{}* = 86.4E6 mm

^{4}

M = σ *I _{x }*/ y

= 150 x 86.4E6 / 152

= 85263158 Nmm (85263.158 Nm)

Now find W from the deflection formula in the following table:

Pick the 5th row - simply supported with a load at the centre;

Max Bending Moment M = WL/4

So W = 4M/L

= 4 x 85263158 / 10200

= 33437 N (3411 kg or 3.4 tonnes)

This is the load that produces a stress of 150MPa. This is a Safety Factor of about 2. (Assuming 300MPa max)

**Deflection **

Deflection D = WL^{3}/(48EI)

What will be the deflection at this stress?

= 33437 x 10200^{3} / (48 x 200000 x 86.4E6)

= 42.78 mm

So this is deflection when the stress is 150MPa.

Measure the current deflection;

There is about 15mm deflection.

This is about 1/3 of the calculated deflection,

Stress is about 1/3 of the 150MPa = 50MPa

Load of about 1/3 of 4.3 tonnes = 1.5 tonnes.

#### Determine actual stress and load from the deflection:

Deflection D = 15mm

From equation: D = WL^{3}/(48EI)

So W = 48EID/L^{3}

= 48 x 200000 x 86.4E6 x 15 / 10200^{3}

= 11724 N (1196 kg or 1.2 tonnes)

Calculate Bending Moment:

M = WL/4

= 11724 x 10200 / 4

= 29896194 Nmm

Calculate Bending Stress:

= 29896194 x 152 / 86.4E6

= 52.595 MPa (S.F. = 4.75 on 250 MPa steel)

#### Questions:

Assignment: Do all questions

Questions 29:1-29:11 (Chapter 29.3: Shear force and Bending
Moment)

OPTIONAL: Questions 30:1-30:9 (Chapter 30: Deflection of Beams)