Stress on Inclines

Instructor Information

• Instructor: Hongsoo Choi, Ph. D.
• Email: hongsoochoi@umass.edu
• Course Code: BMED-ENG 241: Strength of Materials
• Semester: Spring 2026

Teaching Assistants (TAs) Information

• Seva Plotnichenko
  • Email: vvplotnichen@umass.edu
  • Office Hours: Fridays 2:30 PM - 3:30 PM
• Julian Lopez Ward
  • Email: jmlopez@umass.edu
  • Office Hours: Mondays 1:25 PM - 2:15 PM
• Location for Both TA Hours: Hasbrouck Lab Addition room 104A

Important Announcements

• Quiz 1:
  • Date: Monday, February 24, 2026
  • Time: 10:30 AM – 10:50 AM
  • Lecture will conclude by 10:10 AM, with a 10-minute preparation period before the quiz begins.
  • Accommodations:
  • If accommodations are needed, schedule at the Exam Proctoring Center through ClockWork. Contact your Access Coordinator to arrange testing.

Stresses on Inclined Sections

• This section builds on previous classes that examined normal, shear, and bearing stresses on planes that are parallel and perpendicular to axially loaded members.
• Focus now shifts to stresses acting on planes that are inclined with respect to the axis of an axially loaded bar.

Fundamental Concepts

• A prismatic bar subjected to an axial force $P$ applied at its centroid is under uniaxial loading.
• The normal stress on a cross section that is perpendicular to the axis is uniform and is given as follows:
  • Resultant Normal Stress:
  • Equal to the applied load $P$, acting along the bar's axis.
• Shear Stress:
  • No shear stress develops on this perpendicular cross section since the force acts normal to the surface.

Analysis of Inclined Sections

• Section a–a:
  • This is the only surface directly perpendicular to the applied force $P$; thus, inclined sections also necessitate consideration.
• Equilibrium Maintenance:
  • On inclined sections, stresses remain uniformly distributed to maintain equilibrium.
• Angle of Inclination:
  • Defined by angle $\theta$, measured between the bar axis (x-axis) and the normal to the plane (n-axis).

Force Component Resolution

1. Applied Force $P$:
   • Resolved into two components:
   • Normal Force (N):
     • $N = P \cos \theta$ (perpendicular to the plane)
   • Shear Force (V):
     • $V = -P \sin \theta$ (parallel to the plane)
2. Area of the Inclined Plane:
   • The area of the inclined plane is greater than the cross-sectional area and is expressed as:
     • $A_n = \frac{A}{\cos \theta}$
3. Calculation of Stresses:
   • Normal and shear stresses on the inclined plane are computed as follows:
     • Normal Stress ($\sigma_n$):
     • $\sigma<em>n = \frac{N}{A</em>n} = \frac{P \cos \theta}{A/\cos \theta} = \frac{P \cos^2 \theta}{A}$
     • Shear Stress ($\tau_{nt}$):
     • $\tau<em>{nt} = \frac{V}{A</em>n} = \frac{-P \sin \theta}{A/\cos \theta} = -\frac{P \sin \theta \cos \theta}{A}$

Stress Behavior with Angle

• Both normal stress ($\sigma<em>n$) and shear stress ($\tau</em>{nt}$) depend on angle $\theta$.
• Maximum Stress Conditions:
  • Maximum normal stress occurs at angles $\theta = 0^{\circ}$ and $180^{\circ}$.
  • Maximum shear stress occurs at angles $\theta = 45^{\circ}$ and $135^{\circ}$.
  • Moreover, the maximum shear stress is equal to one-half of the maximum normal stress.

Critical Planes and Stress States

• Normal stress is at maximum or minimum on planes where shear stress is zero.
• Conversely, shear stress is zero on planes where normal stress reaches its maximum or minimum.
• The concept of maximum normal stress and maximum shear stress extends to more general loading cases.
• The sign of shear stress changes when angle $\theta$ exceeds $90^{\circ}$.
• The magnitude of shear stress at angle $\theta$ equals that at angle $90^{\circ} + \theta$; only the direction changes.

Example Applications

• Discussion includes MecMovies 1.11.
• Context: An axially loaded prismatic bar is analyzed to understand stress behavior.
  • The load $P$ acts through the centroid, producing centric (uniaxial) loading.
  • Even under simple axial loading, multiple combinations of normal and shear stresses occur.
  • Stress magnitude and direction depend on the orientation of the plane in question.
  • A cut at section $a–a$ is performed for internal stress examination:
  • Since the surface is perpendicular to the applied force $P$, the stress is exclusively normal stress.
  • Normal stress is calculated as:
    • $\sigma = \frac{P}{A}$
  • Equilibrium dictates that the resultant internal force matches the applied force $P$, with no shear stress present due to the perpendicularity of the surface.

Stresses at Arbitrary Planes

• A cut at section $b–b$ assesses stresses on an arbitrary internal plane.
  • The free-body diagram reveals the internal forces on the inclined plane.
  • Stresses acting on the inclined surface maintain uniform distribution and must equate to the applied force $P$ for equilibrium, irrespective of the incline of the surface.
• Force Components on the Inclined Plane:
  • The orientation at this inclined surface is again defined by angle $\theta$ between the x-axis and the normal (n-axis).
  • Positive $\theta$ measures counterclockwise from the x-axis to the n-axis.
• The normal force component is given by:
  • $N = P \cos \theta$ (acting in the n-direction)
  • The shear force component is:
  • $V=-P \sin \theta$ (acting in the t-direction)

Changing Cross-sectional Area

• The cross-sectional area of the inclined plane alters as the angle $\theta$ changes.
  • The inclined area increases compared to the vertical cross-sectional area.
  • This area enlarges as the plane inclines further relative to the bar axis, affecting normal and shear stress magnitudes accordingly.

Engineering Considerations

• Numerous combinations of normal stress ($\sigma<em>n$) and shear stress ($\tau</em>{nt}$) may appear at any material point.
  • The magnitude and direction of these stresses hinge upon the orientation of the evaluated plane.
• When designing components, engineers must consider all potential stress combinations.
• Material Behavior:
  • Brittle materials generally fail due to high normal stress, fracturing on planes perpendicular to the load.
  • Ductile materials are more affected by shear stress, fracturing typically on planes at approximately 45°, where shear stress peaks.

Review Notes

• Reviewed the topic of stresses on inclined sections
• Emphasized the importance of selecting appropriate area, direction, and angle when assessing stresses
• Next discussion: Reading of section 1.6
• Topic for next class: Equality of Shear Stresses on Perpendicular Planes