Posted: January 15th, 2015

ENGINEERING MECHANICS – STATICS 1

Mechanics;

ENGINEERING MECHANICS – STATICS 1
765400

Beams and Bending Stresses 2
Dr Peter Hooper
WS311D WS Building

Overview
Beams – Stresses due to Bending
•
• • • • •

Further work on Maximum Bending Stress in a Beam
Building upon work covered last time Calculation of Maximum Bending Stress Calculation of I for complex sections Parallel Axis Theorem Beam selection

Calculation of maximum Bending Stress in a Beam (cont’d)
• For a given beam and a given loading situation we are interested in determining the maximum stress due to bending. This will become part of the design or selection process for that beam, i.e. ensure that the chosen beam will do the job.

•

Essentially we need to ensure that the chosen beam is fit for purpose

•
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To achieve this, the following procedure should be followed:

Calculation of maximum Bending Stress in a Beam (cont’d)
Calculation of maximum Bending Stress in a Beam To achieve this, the following steps should be followed: 1. Determine the maximum bending moment, Mmax for the beam. This will occur at some point in the beam and may require drawing the BM diagram and finding Mmax. In, symmetrical, simply loaded beams Mmax may be determined by inspection.

2. For the beam cross section, determine the position of the neutral axis (i.e. the position of the centroid) and also determine the second moment of area, I
3. Determine y1 and y2 (distance from neutral axis to outermost fibres of beam and hence the value of ymax, (larger value of y1 and y2). 4. Determine smax from the bending equation:
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Calculation of maximum Bending Stress in a Beam
Calculation of I for complex sections

•

With complex and asymmetrical sections we need to calculate the position of the neutral axis for that section and also be able to determine I for that section.
The section may be able to be split up into rectangular and circular sections but the centroid (and hence neutral axis) for these sections will not coincide with the centroid (NA) of the complete section. The calculation of I about the NA requires the parallel axis theorem.

•

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Calculation of maximum Bending Stress in a Beam
Parallel axis theorem If IGG is the second moment of area of a section about an axis G-G through the centroid of that section and IXX is the second moment of area of that same section about another axis X-X parallel to G-G then

Where: –

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Calculation of maximum Bending Stress in a Beam

•

So if a beam cross section is split up into rectangular and circular sections, I for each section can be determined about the NA for the complete section Then simply added together to find I for the complete section.

•
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Calculation of maximum Bending Stress in a Beam

Position of centroid:

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Calculation of maximum Bending Stress in a Beam – Tutorial Example 1
Q1. Determine the second moment of area of the T-section shown about the line X-X through the centroid parallel to the flange face.
20 mm 120 mm

y
20 mm

A
180 mm

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Q1.

Calculation of maximum Bending Stress in a Beam – Tutorial Example 1
20 mm 120 mm

y
20 mm 180 mm

A X

Area of section Moment of Flange area about edge A-A

Moment of Web area about edge A-A

Q1.

Calculation of maximum Bending Stress in a Beam – Tutorial Example 1
20 mm 120 mm

y
20 mm 180 mm

A X

Total Moment is therefore

If we assume the distance of the centroid to be y from face A-A then

Q1.

Calculation of maximum Bending Stress in a Beam – Tutorial Example 1
20 mm 120 mm

y
20 mm 180 mm

A X

Calculation of I X ? X

Distance of centroid of flange from axis X-X

Q1.

Calculation of maximum Bending Stress in a Beam – Tutorial Example 1
20 mm 120 mm

y
20 mm 180 mm

A X

Consider the flange first Second moment of area of flange about axis X-X

Q1.

Calculation of maximum Bending Stress in a Beam – Tutorial Example 1
20 mm 120 mm

y
20 mm 180 mm

A X

Now consider the web

I for the Web about its own of centroid

Distance of web centroid from axis X-X

Q1.

Calculation of maximum Bending Stress in a Beam – Tutorial Example 1
20 mm 120 mm

y
20 mm 180 mm

A X

I web about axis X-X

Q1.

Calculation of maximum Bending Stress in a Beam – Tutorial Example 1
20 mm 120 mm

y
20 mm 180 mm

A X

Therefore total

I of section about axis X-X

Calculation of maximum Bending Stress in a Beam
Using the tabular of method:

Position of centroid:

? Ay ? ? y? ?A

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Beam Selection
Section Modulus

When selecting a beam from a catalogue to perform a certain duty (loading configuration) we are mainly interested in the maximum stress values (as before):

rearranging: –

Where and is called the section modulus. It depends only on the beam cross section. It depends only on the beam cross section.

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Beam Selection
So to select a suitable beam for a given loading situation

1. Determine the maximum bending moment, Mmax for the loading situation for that beam.
This will occur at some point in the beam and may require drawing the BM diagram and finding Mmax. In, symmetrical, simply loaded beams Mmax may be determined by inspection. 2. For a given beam material, smax can be determined from the UTS for that beam material and the design factor of safety, FOS

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Beam Selection
So to select a suitable beam for a given loading situation (cont’d) 3. Calculate the required section modulus

4. Then use tables of beams/beam properties to find a suitable beam:

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Beam Selection
Manufacturer’s Tables These list beam properties for standard size (commonly available) steel beams. Among the properties are second moment of area, I , section modulus, Z etc. These can be used to help select suitable beams using the method above.

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Conclusions
Beams – Stresses due to Bending
•
• • • • • •

Further work on Maximum Bending Stress in a Beam
Building upon work covered last time Calculation of Maximum Bending Stress Calculation of I for complex sections Parallel Axis Theorem Beam selection NEXT TIME: Torsion of Shafts

Further Reading
• Ivanoff, V. (2010) – “Engineering Mechanics” McGraw Hill, (ISBN 007101003-3) Dodd, D.M. and Richardson R. (1993) – “Engineering Mechanics”, Trumps Copy Centre, Auckland University of Technology. Hannah, J. and Hillier, M.J. (1995) – “Mechanical Engineering Science”. (3rd Edition) Pearson Education Ltd, London (ISBN 0-58225632-1)

•

ENGINEERING MECHANICS – STATICS 1
765400

Beams and Bending Stresses 1
Dr Peter Hooper
WS311D WS Building

Overview
Beams – Stresses due to Bending
• The fundamental Bending Equation • Pure bending

• •
• • •

The relationship between curvature and strain Position of the Neutral Axis
Moment of Resistance

Maximum Bending Stress in a Beam Second Moment of Area for: • Rectangular cross-sections • Circular cross-sections • I Beam sections

Pure Bending
Pure Bending Consider a beam in pure bending. No shear forces. But there will be longitudinal forces set up inside the beam over its cross-section. If the beam is sagging then there will be: tensile stresses in the lower half of the beam and compressive stresses in the top half To find the magnitude of these stresses, a number of assumptions are required

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Pure Bending
Assumptions 1. The beam is initially straight. 2. Bending takes place in the plane of the applied bending moment (the plane of the paper). 3. The cross section of the beam is symmetrical about the plane of bending. If not symmetrical it would twist as well as bend. 4. The stresses are uniform across the width. 5. The material of the beam is elastic and obeys Hooke’s Law. 6. The stresses do not exceed the limit of proportionality. 7. The moduli of elasticity in tension and compression are the same. 8. The plane transverse section of the beam remains a plane section after bending. 9. Each layer of the beam is free to carry stress without interference from adjacent layers.
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Pure Bending
Relation between curvature and strain Consider the straight portion of beam ABCD shown (Fig 17.3 H & H).

After bending the beam becomes an arc, with centre at ‘O’.

1- 5

Pure Bending
Relation between curvature and strain Top layers of the beam are in tension Bottom layers are in compression

Therefore SN is the neutral surface (neither in tension or compression)

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Pure Bending
N’-N’ is the neutral axis. The radius of curvature, of the beam is measured from ‘O’ to the neutral surface. Since SN is neither stretched nor compressed it remains the same length after bending, i.e. SN = S’N’. If ? is the angle (in radians) subtended by S’N’ at ‘O’

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Pure Bending
If ? = The angle subtended by S’N’ at ‘O’ (in radians)

Where y = distance from SN to layer of interest EF

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Pure Bending
so

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Pure Bending
So, strain in EF is ? Now at the layer EF

Which gives us

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Pure Bending
So for a given bending moment at a section the value of R is constant E is constant so this implies that the stress s varies linearly with distance from the neutral axis, y

Hence there is a linear distribution of stress from maximum tensile stress (in this case) at the upper surface to 0 at the neutral axis to maximum compressive stress at the lower surface.
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Pure Bending
Position of the Neutral Axis

This can be found from the fact that there is no net longitudinal force at the neutral axis.
Consider a small strip of width, b and thickness, dy: –

(Fig 17.4, H & H)

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Pure Bending
Position of the Neutral Axis

but

Integrating to find the net axial force;

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Pure Bending
Position of the Neutral Axis Now the net longitudinal (axial) force, over the section is zero, so

But so

is the moment of area of

about the neutral axis

is the moment of the whole area about the neutral axis and this can only be zero if the neutral axis passes through the centroid of the beam cross section.

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Pure Bending
Position of the Neutral Axis Note: 1. For a beam with symmetrical cross section, y1 = y2 and so max. tensile stress = max. compressive stress. 2. For non-symmetrical cross sections, the numerically largest bending stress will occur at the outer layer most distant from the neutral axis (and may be tensile or compressive).

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Pure Bending – Tutorial Example
Calculate the maximum stress in a coil of steel rod of 4mm diameter due to coiling onto a drum of 2m diameter. Assume E = 200 GN/m2

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Pure Bending
Moment of Resistance

Consider the diagram above. The total moment M at the section will be equal and opposite to the moment generated inside the beam by the axial forces about the neutral axis (the moment of resistance). The moment generated by the differential force dF about the neutral axis is

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Pure Bending
Moment of Resistance From the analysis above

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Pure Bending
Moment of Resistance The term is the second moment of area of the section about the neutral axis and is given the symbol I

so And so Which is usually written as

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Pure Bending
Moment of Resistance And since We arrive at the well established Bending Equation;

Where: –

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Calculation of maximum Bending Stress in a Beam
Calculation of maximum Bending Stress in a Beam

For a given beam and a given loading situation we are interested in determining the maximum stress due to bending.
This will become part of the design or selection process for that beam, i.e. to ensure that the chosen beam will do the job.

1- 21

Calculation of maximum Bending Stress in a Beam
Calculation of maximum Bending Stress in a Beam To achieve this, the following steps should be followed: 1. Determine the maximum bending moment, Mmax for the beam. This will occur at some point in the beam and may require drawing the BM diagram and finding Mmax. In, symmetrical, simply loaded beams Mmax may be determined by inspection.

Second Moment of Area, I , for Rectangular and Circular sections

Calculation of maximum Bending Stress in a Beam

Rectangular sections Many beams are made up of rectangular and or circular sections so it is helpful to know the second moment of area, I , for these cross sections.

It can be shown (by integration) that

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Second Moment of Area, I , for Rectangular and Circular sections Circular sections

Calculation of maximum Bending Stress in a Beam

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Calculation of I for Beam sections
If symmetrical about the neutral axis then the cross sectional area of the beam can be split up into rectangles (or circles) and values of I can be added and subtracted (for cut outs)… … provided the neutral axis for each rectangle and circle also passes through the centroid of the beam cross section.

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Calculation of I for Beam sections – Tutorial Example
The following beam of symmetrical I section has the dimensions of; Flange 150 mm wide and 30 mm thick, total beam depth of 200 mm. Determine the second moment of area of the beam section about an axis through the centroid parallel to the flange face. If the beam is simply supported over a 2 m length and carries a uniformly distributed load of 6t/m run, calculate the maximum bending stress in the beam.

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Conclusions
Beams – Stresses due to Bending
• The fundamental Bending Equation • Pure bending

• •
• • •

The relationship between curvature and strain Position of the Neutral Axis
Moment of Resistance

Maximum Bending Stress in a Beam Second Moment of Area for: • Rectangular cross-sections • Circular cross-sections • I Beam sections

Further Reading
• Dodd, D.M. and Richardson R. (1993) – “Engineering Mechanics”, Trumps Copy Centre, Auckland University of Technology. Hannah, J. and Hillier, M.J. (1995) – “Mechanical Engineering Science”. (3rd Edition) Pearson Education Ltd, London (ISBN 0-58225632-1)

•

Ivanoff, V. (2010) – “Engineering Mechanics” McGraw Hill, (ISBN 007101003-3)

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