ENG1011 Chap.12 Bending Stress, Deflection and Beam Design
Bending Stress, Deflection and Beam Design
Bending Stress, Deflection and Beam Design is Week 9, where the bending moment from Week 8 is turned into a design. A sagging moment shortens the top fibres of a beam and stretches the bottom ones, and the layer between that keeps its length is the neutral axis, through the centroid of the section.
Bending stress varies linearly with distance from the neutral axis, as given by the bending stress formula, moment times distance divided by the second moment of area, so the largest stresses occur at the extreme fibres. Because the second moment of area of a rectangle grows with the cube of its depth, beams are much stronger and stiffer standing on edge.
Deflection is found from given formulas for standard cases, each with the elastic modulus times the second moment of area in its denominator. Design checks the ultimate limit state, whether the peak bending stress stays within the allowable stress, and the serviceability limit state, whether the deflection stays within its limit.
A beam can pass one and fail the other.
Week 9 brings the semester together: the reactions of Week 3, the moment diagram of Week 8 and the section properties of Week 7 all feed one design check. A typical question gives a span, a load and a section, then asks for the peak bending stress, the maximum deflection, and whether the beam passes.
The order matters, because the maximum moment from the diagram is the input to the stress check.
The same two checks run through the bridge project: a member can be strong enough and still deflect too much, and the report should say which limit controls the design.
What this chapter covers
- 01
The neutral axis and its position
- 02
The bending stress formula
- 03
Peak stress at the extreme fibres
- 04
Second moment of area and section depth
- 05
Given deflection formulas for standard beams
- 06
Ultimate limit state: strength
- 07
Serviceability limit state: deflection
- 08
Choosing a section that passes both
Worked example · free
Peak bending stress in a rectangular beam
- 1Second moment of area: 100 times 200 cubed over 12 = 66.7 times 10 to the sixth mm to the fourth.
- 1Extreme fibre distance: half the depth, 100 mm.
- 1Stress: 20 times 10 to the sixth N mm times 100, divided by 66.7 times 10 to the sixth, gives 30 MPa.
Key terms
- Neutral axis
- The line through a beam's section where bending causes no stress or strain.
- Bending stress
- The normal stress caused by a bending moment, proportional to distance from the neutral axis.
- Ultimate limit state
- The design condition concerned with strength, checked by comparing peak stress with allowable stress.
- Serviceability limit state
- The design condition concerned with performance in use, checked by comparing deflection with a limit.
- Flexural rigidity
- The product of elastic modulus and second moment of area, which controls how much a beam deflects.
- Extreme fibre
- The point of a cross-section farthest from the neutral axis, where bending stress is largest.
Bending Stress, Deflection and Beam Design FAQ
Why does a beam need checking for both strength and deflection?
The two checks depend on different properties. Strength depends on the bending moment and the section, while deflection also depends on the material's stiffness and grows steeply with span. A beam can be strong enough yet sag too much, or the reverse.
Which face of a beam is in tension?
Under a sagging moment the bottom face is in tension and the top in compression; under a hogging moment, such as at a cantilever's support, it is the reverse. Check the sign of the moment before deciding which face governs.
How should units be handled in deflection formulas?
Use newtons and millimetres throughout: load intensity in newtons per millimetre, span in millimetres, modulus in megapascals and second moment of area in millimetres to the fourth. The deflection then comes out in millimetres without further conversion.
How does turning a beam on its side change its strength?
For a rectangle, strength in bending depends on width times depth squared and stiffness on width times depth cubed. Swapping width and depth on a 50 by 150 mm section cuts its second moment of area to one ninth and triples its peak bending stress.
Exam move
Practise each beam problem in a fixed order: maximum moment, section properties, peak stress against the allowable value, then deflection against its limit. Note which check governs and what change would fix a failure. Keep a one line reminder of unit conversions beside you until they are automatic. For each practice beam, write a one-line verdict naming the governing limit state and the change that would fix any failure.
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