Lecture 17 – Shear stress in beams

Instructor: Prof. Marcial Gonzalez

Spring, 2020ME 323 – Mechanics of Materials

Reading assignment: 6.8

News: ____

Last modified: 2/7/20 8:57:21 AM

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Beam theory (@ ME 323)- Geometry of the solid body:

straight, slender member with constant cross sectionthat is designed to supporttransverse loads.

- Kinematic assumptions: Bernoulli-Euler Beam Theory

- Material behavior: isotropic linear elastic material; small deformations.

- Equilibrium:+ Pure bending ( )

Equilibrium of beams

Longitudinal Planeof Symmetry

(Plane of Bending)

Longitudinal Axis

Moment-curvature equation – Flexure formula

Note: the y-coordinate is measured from the neural axis!!

J. Bernoulli L. Euler

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Beam theory (@ ME 323)- Geometry of the solid body:

straight, slender member with constant cross sectionthat is designed to supporttransverse loads.

- Kinematic assumptions: Bernoulli-Euler Beam Theory

- Material behavior: isotropic linear elastic material; small deformations.

- Equilibrium:+ Pure bending ( )

- + What if ?Most real configurations will have a shear force.

Q: What is the distribution of shear stresses?

Equilibrium of beams

Longitudinal Planeof Symmetry

(Plane of Bending)

Longitudinal Axis

J. Bernoulli L. Euler

Shear stress in beams-

- Kinematic assumptions: Bernoulli-Euler Beam Theory- (from Lecture 15) cross sections remain plane and perpendicular to the

deflection curve of the deformed beam;(how is this possible if there are shear strains?)

- (now, in addition) the distribution of flexural stresses on a given cross sectionis not affected by the deformation due to shear. 5

Equilibrium of beams

What about ? Transverse and longitudinal

shear stress!

Shear stress in beams

- Jourawski Theory (or Collignon Theory)

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Equilibrium of beams

What about ? Transverse and longitudinal

shear stress!

where Q(y) is the first moment of area A’(y) with respect to the neutral axis.

Average transverse

shear stress

Q(y) =

Z

A0(y)⌘dA = A⇤y⇤

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Equilibrium of beams

Example 25:(a) Determine the maximum shear stressin the rectangular cross section.

(b) If the cross section has both bending momentand shear force as internal resultants, determine the state of stress at five points in the cross section.

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Equilibrium of beams

This equation can be used as an approximation for the maximumshear stress along the neutralsurface.

Example 26: Determine the maximum shear stressin a circular.

Watch: Example 10.15

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Equilibrium of beams

This equation can be used as an approximation for the maximumshear stress along the neutralsurface.

Example 27: Determine the maximum shear stressin a pipe ( ).

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Equilibrium of beams

This equation can be used as an approximation for the maximumshear stress along the neutralsurface.

Example 28: Determine the maximum shear stressin a web-flange beam.

Watch: Example 10.14

Any questions?

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Equilibrium of beams