BENDING STRESS

Bending stress is calculated from Bending Moment and Area Moment.

Smartboard Notes:

Bending-Stress.pdf

Bending-Stress.one

The 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 mm⁴ (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 mm⁴

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 = WL³/(48EI)

What will be the deflection at this stress?

= 33437 x 10200³ / (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 = WL³/(48EI)

So W = 48EID/L³

= 48 x 200000 x 86.4E6 x 15 / 10200³

= 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)