Lesson 2.4: Combined Bending and Torsion Loading

For a hollow circular tube with outer diameter D and inner diameter d:

1. Cross-sectional area:

2. Second moment of area (for bending stress):

3. Polar moment of inertia (for torsional stress):

4. Critical dimensions:

   • Outer radius: mm (where stresses are maximum)
   • Distance from neutral axis to outer fiber: mm

Design insight: The hollow tube maximizes bending and torsional stiffness while minimizing weight—critical for drone performance.

1. Calculate bending moment at fixed end (from cantilever beam analysis):

   For a cantilever with point load F at the free end:

2. Bending stress at outer fiber (maximum tension/compression):

   Using flexure formula :

   Location: Occurs at top and bottom of tube (tension on one side, compression on other)

3. Torsional shear stress at outer surface:

   Using torsion formula :

   Location: Maximum at outer surface, acts circumferentially

4. Identify critical stress location:

   The outer surface on the tension side of bending experiences:

   • Maximum normal stress: MPa (tension)
   • Maximum shear stress: MPa (torsion)

   These combine to create the highest equivalent stress.

Physical interpretation: The arm bends downward under rotor thrust (creating tension on top surface) while simultaneously twisting about its axis from motor torque.

1. Stress state at critical point (outer surface, tension side):

   • Normal stress: MPa
   • Shear stress: MPa
   • Other components: , (thin-walled assumption)

2. Von Mises equivalent stress formula (for combined bending + torsion):

   Substituting values:

3. Physical meaning:

   The combined effect of bending and torsion produces an equivalent uniaxial stress of 33.0 MPa. This is the “effective” stress that the material experiences.

Why von Mises? Carbon fiber composites in the fiber direction behave similarly to ductile materials, making von Mises theory appropriate for this analysis.

1. Calculate safety factor:

2. Design assessment:

   Criterion	Value	Status
   Von Mises stress	33.0 MPa	✅
   Material yield	400 MPa	✅
   Safety factor	12.1	✅ Excellent
   Typical drone SF target	3.0–5.0	✅ Exceeded

3. Design margin analysis:

   • Stress utilization: of material capacity
   • Available margin: Nearly 12× the actual stress
   • Conclusion: Design is highly conservative (very safe but possibly overdesigned)

4. Stress contribution analysis:

   From the von Mises stress formula , we can determine each component’s contribution:

   Bending contribution:

   Torsion contribution:

   For our drone arm:

   • Bending contributes: of von Mises stress
   • Torsion contributes: of von Mises stress
   • Key insight: Bending dominates — motor torque has minimal structural impact

5. Design recommendations:

   ✅ Current design is safe with excellent margins (SF = 12.1 >> 3.0 target)

   Potential optimizations (if weight reduction needed):

   • Could reduce wall thickness to t = 1.5 mm (stress increases ~33%, still SF > 8)
   • Could use smaller diameter (D = 18 mm) for weight savings
   • Current design provides excellent margin for dynamic loads, vibration, and fatigue

   When to keep current conservative design:

   • Crash resistance is critical (landing impacts, collisions)
   • Vibration and dynamic loads significant (rotor imbalance, wind gusts)
   • Fatigue life important (SF > 10 excellent for high-cycle loading from rotor operation)

For a simply supported beam with uniformly distributed load, the critical section is at mid-span where bending moment is maximum.

1. Calculate support reactions (symmetry):

   For uniform load w over length L:

2. Maximum bending moment location:

   For simply supported beam with uniform load, maximum moment occurs at center (x = L/2):

3. Calculate maximum bending moment:

   Converting to N·mm: N·mm

4. Torsional moment:

   Torque is constant along entire shaft:

Critical section identified: Mid-span (center of shaft) where bending moment is maximum and torsion is present.

For a hollow circular shaft with outer diameter D and inner diameter d:

1. Cross-sectional area:

2. Second moment of area (for bending stress):

3. Polar moment of inertia (for torsional stress):

4. Critical dimensions:

   • Outer radius: mm (where stresses are maximum)
   • Distance from neutral axis to outer fiber: mm

Design note: The 10 mm wall thickness provides good balance between strength and weight for power transmission applications.

At the critical section (mid-span, outer surface):

1. Bending stress at outer fiber:

   Using flexure formula :

   Location: Maximum at top and bottom of shaft (tension on bottom, compression on top under wind/gravity)

2. Torsional shear stress at outer surface:

   Using torsion formula :

   Location: Maximum at outer surface, uniform around circumference

3. Identify critical stress state:

   The outer surface on the tension side of bending experiences:

   • Normal stress: MPa (tension)
   • Shear stress: MPa (torsion)

4. Stress comparison:

   Notice that torsional stress dominates over bending stress:

   • MPa (much larger)
   • MPa (relatively small)

   This is typical for power transmission shafts where torque is the primary design consideration.

Physical interpretation: The shaft is primarily loaded in torsion (power transmission), with relatively minor bending from wind and rotor weight.

1. Stress state at critical point (outer surface, tension side at mid-span):

   • Normal stress: MPa
   • Shear stress: MPa
   • Other components: ,

2. Von Mises equivalent stress formula (for combined bending + torsion):

   Substituting values:

3. Physical meaning:

   The combined effect produces an equivalent uniaxial stress of 75.1 MPa. Notice this is dominated by the torsional component (43.2 MPa contributes most).

Key insight: When , the von Mises stress is approximately , confirming torsion dominates.

1. Calculate safety factor:

2. Design assessment:

   Criterion	Value	Status
   Von Mises stress	75.1 MPa	✅
   Material yield	310 MPa	✅
   Safety factor	4.13	✅ Excellent
   Typical turbine SF target	2.5–4.0	✅ Met

3. Design margin analysis:

   • Stress utilization: of material capacity
   • Available margin: 4.13× the actual stress
   • Conclusion: Design has appropriate safety margin for rotating machinery

4. Design evaluation:

   ✅ Design is adequate with good safety margins

   Why SF = 4.13 is appropriate:

   • Wind turbines experience fatigue loading (cyclic stress from rotation)
   • Dynamic loads from gusts and turbulence require margin
   • Longer service life (20+ years) needs conservative design
   • Consequence of failure is severe (turbine collapse, downtime)

   Stress contribution breakdown:

   • Torsion contributes:
   • Bending contributes:

   Design verification:

   • ✅ Shaft diameter appropriate for torque transmission
   • ✅ Hollow section provides strength with reduced weight
   • ✅ Safety factor adequate for fatigue and dynamic loads
   • ✅ Bending stresses negligible — design is driven by torsional capacity

For a rectangular cross-section (width b, height h), formulas differ from circular sections:

1. Cross-sectional area:

2. Second moment of area (bending about horizontal neutral axis):

   For bending about the axis parallel to width b:

3. Distance to extreme fiber:

4. Torsional constant for rectangular sections:

   For rectangles, torsional resistance is NOT given by polar moment J. Instead:

   For aspect ratio :

   Using torsional constant formula for rectangles:

   Where depends on aspect ratio. For , (from tables)

5. Maximum torsional shear stress location:

   For rectangular sections, maximum shear stress occurs at the midpoint of the longer side (h = 12 mm).

Key difference: Rectangular sections are less efficient at resisting torsion than circular sections of similar area.

1. Calculate bending moment at fixed end (cantilever with tip load):

   For a cantilever beam with point load F at free end:

2. Bending stress at extreme fiber:

   Using flexure formula :

   Location: Maximum at top and bottom surfaces at the fixed end (tension on one side, compression on other)

3. Critical location for bending:

   The outer fiber (top or bottom surface) at the fixed end experiences maximum bending stress.

Physical interpretation: The finger bends like a diving board when the grip force is applied at the tip.

Important: Torsion in rectangular sections uses different formulas than circular shafts!

1. Torsional shear stress formula for rectangular sections:

   Where is a coefficient depending on aspect ratio .

2. Determine coefficient α:

   For aspect ratio :

   Using standard tables for rectangular torsion,

3. Calculate maximum torsional shear stress:

   Location: Maximum at the midpoint of the longer side (middle of the 12 mm edge)

4. Alternative formula verification:

   Using torsional constant K:

   Note: Different formulas give different values because stress varies around the perimeter. We use the more conservative first formula: τ = 3.32 MPa.

Key insight: Rectangular sections have non-uniform shear stress distribution along the perimeter, unlike circular sections where it’s uniform.

1. Bending stress locations:

   • Maximum: Top/bottom surfaces at fixed end = 6.25 MPa
   • Zero: Neutral axis (center of section)

2. Torsional stress locations:

   • Maximum: Midpoint of long edges (h = 12 mm sides) = 3.32 MPa
   • Lower: Midpoint of short edges (b = 8 mm sides)
   • Zero: Corners of rectangle

3. Where do bending and torsion combine?

   The critical point is at the top or bottom surface, near the midpoint of the long edge, where:

   • Bending stress is near maximum: MPa
   • Torsional stress is maximum: MPa

   For conservative analysis, we assume both maximum values occur at the same point:

   • MPa
   • MPa

Note: In rectangular sections, exact critical location is more complex than circular sections, but this approach is conservative and standard for design.

1. Stress state at critical point:

   • Normal stress: MPa
   • Shear stress: MPa
   • Other components: ,

2. Von Mises equivalent stress:

   Substituting values:

3. Physical meaning:

   The combined bending and torsional loads create an equivalent uniaxial stress of 8.49 MPa in the gripper finger.

1. Calculate safety factor:

2. Design assessment:

   Criterion	Value	Status
   Von Mises stress	8.49 MPa	✅
   Material yield	275 MPa	✅
   Safety factor	32.4	✅ Extremely high
   Typical gripper SF target	3.0–5.0	✅ Far exceeded

3. Design margin analysis:

   • Stress utilization: of material capacity
   • Available margin: 32.4× the actual stress
   • Conclusion: Design is extremely conservative (vastly overdesigned)

4. Stress contribution analysis:

   • Bending contributes: of von Mises stress
   • Torsion contributes: of von Mises stress
   • Key insight: Nearly balanced loading — both bending and torsion are significant (unlike previous applications where one dominated)

5. Design recommendations:

   Current design: Vastly overdesigned with SF = 32.4

   Optimization opportunities:

   • Reduce cross-section: Could use b = 5 mm, h = 8 mm (stress increases ~2.5×, still SF > 12)
   • Use lighter material: Could switch to plastic or composite for weight savings
   • Increase grip force: Design can handle 20× current force safely

   When to keep current conservative design:

   • Repeated use: High-cycle fatigue from millions of grip operations
   • Impact loading: Accidental collisions or dropped objects
   • Safety critical: Handling fragile or expensive parts
   • Wear allowance: Material loss from abrasion over time

   Note on rectangular sections:

   • Rectangular torsion formulas differ from circular shafts
   • Non-uniform shear stress distribution around perimeter
   • Less torsion-efficient than circular sections of similar area
   • Simpler manufacturing and assembly in many gripper designs