Course Information

Course Code: AME 3353Instructor: Dr. Yijie JiangInstitution: School of Aerospace and Mechanical Engineering, University of Oklahoma

Bending Deformation of a Straight Beam

• Top portion experiences compression; bottom undergoes tension.

• Neutral surface: region with no change in length; cross-sections distort but remain planar.

Stress and Strain Variation in Beams

• Profile view shows normal strain and bending stress.

The Flexure Formula

For linear-elastic behavior in straight beams:

Equation: ( ( \sigma = -\frac{My}{I} ) )

• M: Internal bending moment about the neutral axis.

• y: Distance from neutral axis to the point of interest.

• I: Moment of inertia of the cross-section.

Neutral Axis Location

• Passes through the centroid of the cross-section; at the center for symmetric objects.

Moment of Inertia

• Defined about the centroidal axis.

• Parallel Axis Theorem: To calculate moment of inertia around a non-centroidal axis.

Shear in Straight Members

• Longitudinal shear stress: along the member’s length.

• Transverse shear stress: across the width/height.

• Shear strain varies, leading to warping.

The Shear Formula

Equation: ( ( \tau = \frac{VQ}{It} ) )

• τ: Shear stress at a specific location.

• V: Internal resultant shear force.

• t: Width at specific location.

Limitations of Shear Formula

• Applicable for straight members of homogeneous material, assumes uniform shear stress distribution.

• Not applicable to: “Short” or “flat” sections (b/h > 2), areas with stress concentrations, inclined boundaries.

Maximum Shear Stress Location

• Occurs where: Q is maximized (at neutral axis), t is minimized.

Thin-Walled Pressure Vessels

• Common structures: tanks, boilers, pipes.

• Thin wall: radius/thickness ratio > 10; uniform stress distribution.

• Gauge pressure: positive pressure above atmospheric pressure.

Cylindrical Vessels

Stress components:

• Circumferential (hoop) stress: ( ( \sigma_1 = \frac{pr}{2t} ) ).

• Longitudinal (axial) stress: ( ( \sigma_2 = \frac{pr}{t} ) ).

Spherical Vessels

• Only circumferential stress is applicable: ( ( \sigma = \frac{pr}{2t} ) ).

Practical Example

• Calculate max internal pressure for a cylindrical vessel with:

  • Inner diameter: 4 ft

  • Thickness: 0.5 inch

  • Max stress = 20 ksi.

Plane-Stress Transformation

• Defined stresses: normal stresses in x and y directions, shear stress in xy plane.

Orientation of Stress Element

• Normal and shear stresses vary based on element orientation; altering orientation changes new stress conditions.

General Equations of Plane Stress Transformation

Using statics and a defined sign convention for stress:

• Normal stresses outward: positive.

• Shear stresses in positive y direction: positive.

Principal Stresses

• Maximum and minimum normal stresses occur with zero shear stress at specific angles.

Maximum In-Plane Shear Stress

• Orientated 45 degrees away from principal stress.

Mohr’s Circle

• Graphical method to determine plane stress transformations; the circle’s center is at average stress.

Applications of Mohr’s Circle

• Determine state of stress for any orientation angle, principal stresses, and maximum in-plane shear stress.

Practical Problems

• Determine stress states and conditions based on provided stress diagrams.