- 5.1 Effect of Bending
- 5.2 Pure Bending — 7 Assumptions, Neutral Axis, Equation of Pure Bending
- 5.3 Nature of Bending Stress
- 5.4 Sectional Modulus (Z)
- 5.5 Moment of Resistance (MOR)
- 5.6 Bending Stresses in Axially Loaded Beams + Middle Third Rule
- 5.7 Force on a Partial Area of a Section
- 5.8 Composite Beam — Equivalent Section Method
- 5.9 Flitched Beam (Top/Bottom and Side)
- 5.10 Beam of Uniform Strength
- 5.11 Biaxial Bending
5.1 Effect of Bending
In bending, the cross-sectional area abcd is rotated about transverse axes (z-z or y-y), whereas in twisting the cross-sectional area rotates about longitudinal axis (x-x). The transverse axis about which area cross-section rotates is called the Neutral Axis (NA).
- Above NA: compression (shortening) → lateral expansion occurs
- Below NA: tension (elongation) → lateral contraction occurs
- Rectangular section → trapezoidal section after bending
- Transverse deformations are small → area still treated as rectangular
The transverse axis about which the cross-section area rotates during bending is called the Neutral Axis (NA). The layer that does NOT change its length after bending = Neutral Layer.
5.2 Simply Bending (Pure Bending) — Theory and Derivation
Pure bending = bending moment is constant along the length. dM/dx = 0, i.e., shear force = zero throughout the span. In a SS beam with two equal point loads, the central portion (between loads) is under pure bending.
5.2.1 Seven Assumptions in Theory of Pure Bending (from source)
5.2.2 Neutral Axis
Layer EF that neither elongates nor shortens = Neutral Layer. Its intersection with cross-section plane = Neutral Axis.
Equal area axis may be different from centroidal axis or may coincide — depends on shape of cross-section.
5.2.3 Equation of Pure Bending — Derivation
Fig. 5.1 — Derivation of M/I=σ/y=E/R. SS beam (sagging): compression above NA, tension below. Cantilever (hogging): tension above, compression below. Stress distribution is always linear across the cross-section.
5.3 Nature of Bending Stress
| Beam Type & Loading | Above NA | Below NA |
|---|---|---|
| Simply Supported Beam with Sagging (+ve BM) | COMPRESSION (−ve) | TENSION (+ve) |
| Cantilever Beam with Hogging (−ve BM) | TENSION (+ve) | COMPRESSION (−ve) |
5.4 Sectional Modulus (Z)
Sectional modulus is the ratio of the moment of inertia of the beam cross-section about NA to the distance of the extreme fibre from NA. It represents the strength of the section.
Circle: πd³/32
Hollow circle: π(D⁴−d⁴)/32D
y_max is same on both sides → single Z value → single σ_max on both extreme fibres.
y_top ≠ y_bottom → two Z values → two different extreme stresses. Design uses lesser MOR.
5.5 Moment of Resistance (MOR)
MOR = resistance offered by the section against externally applied bending moment. It is the maximum bending moment that can be resisted by the section without failure.
M_R2 = σ_t × Z_t (tension)
Actual M_R = min(M_R1, M_R2)
5.6 Bending Stresses in Axially Loaded Beams + Middle Third Rule
Consider a cantilever beam with UDL and an axial tensile load P at free end. Total stress = sum of bending stress and direct stress.
Sign of My/I depends on location and type of BM
When Eccentric Axial Load is Applied (eccentricity e from axis):
Middle Third Rule (for Rectangular Cross-Section):
For a rectangular cross-section (b × d), if eccentric load lies within the middle third of the section, the resultant stress due to eccentric load will be of the same nature throughout the section (either all tension or all compression). This is called the Middle Third Rule.
No tension condition: P/A ≥ Pe·y/I
Core = middle ±b/6 in x, ±d/6 in y direction
5.7 Force on a Partial Area of a Section
Let σ_max be the maximum stress at y_max from NA. Stress at elementary strip dA at distance y:
Force on strip: dP = σ·dA = (σ_max / y_max) × y·dA
Total force on shaded area: P = σ_max × A·ȳ / y_max
[ ∵ ∫y·dA = A·ȳ, where ȳ = dist of centroid of shaded area from NA ]
5.8 Bending Stress in Composite Beam — Equivalent Section Method
In a composite beam of two materials firmly jointed, there is a common NA and they behave as a single beam. In composite beam, bending strain diagram is linear → strain in two materials at same vertical distance from NA is same = strain compatibility condition.
σ_1/E_1 = σ_2/E_2
σ_1/σ_2 = E_1/E_2 = m
σ_1 = m × σ_2
Steel/Wood: m = 10 to 25
Equiv. width in material 2:
b_equiv = m × b_original
5.9 Flitched Beam
A flitched beam is a composite beam of wood and steel in which wooden section is strengthened by metal plates either at top and bottom or at sides symmetrically. Used when a homogeneous wooden beam would require a very large cross-sectional area for same moment of resistance.
5.9.1 Top and Bottom Flitched Beam (Method I)
| Type | Stress Relation | Moment of Resistance |
|---|---|---|
| Top & Bottom Flitched (steel plates t thick) |
σ_s = m·σ_w (at same y from NA) |
M_R = σ_w·bd²/6 + σ_s·max·[b(d+2t)²/6 − σ_s·bd²/6] |
| Side Flitched (steel plates at sides) |
σ_s = m·σ_w (σ_s_max at extreme fibre) |
M_R = σ_w·bd²/6 + σ_s’·bd²/6 (steel plates same height d) |
5.10 Beam of Uniform Strength
For economical design, section of beam may be reduced towards supports as bending moment decreases. If at every section the extreme fibre stress reaches the permissible stress σ_p, the beam is called beam of uniform strength.
For uniform strength: M_Rx = M_x
σ·b·d_x²/6 = W·x/2
d_x = √(3Wx/σb²) ∝ √x
For uniform strength: M_Rx = M_x
σ·b_x·d²/6 = W·x/2
b_x = 3Wx/(σd²) ∝ x
5.11 Biaxial Bending
Consider a doubly symmetric cantilever beam with incline load P at angle θ with vertical y-axis. Components along y and z axes produce bending about z-axis and y-axis respectively.
M_y = P·sinθ·x (bending about y-axis)
σ_by = ±(M_y/I_y)·z
Effect of M_z: section ABZZ in tension, CDZZ in compression.
Effect of M_y: section ADYY in tension, BCYY in compression.
Net stress at any point = algebraic sum. Neutral axis for combined loading may be inclined.
Chapter 5: Bending Stress in Beams — Strength of Materials · MADE EASY Postal Study Package 2019