Bending stress is calculated from Bending Moment and Area Moment.Smartboard Notes: Bending-Stress.pdf Bending-Stress.one
where;σ is the bending stress (Ivonaff uses fb)
M - the moment about the neutral axis
y - the distance to the neutral axis
Ix - 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
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)
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 Ix = 86.4E6 mm4
M = σ Ix / 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 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)
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)