Lecture 28 – Review for Exam 2

Instructor: Prof. Marcial Gonzalez

Fall, 2021ME 323 – Mechanics of Materials

News: Ready for the exam?

Exam 2- Wednesday November 3rd , 8:00-10:00 p.m.

A-H last name: FRNY G140 & I-Z last name: SMTH 108(please arrive 15 minutes before the exam and bring a picture ID)

- You will scan your exam and submit to Gradescope – come prepared!

- Formula sheet will be provided

- No lecture on Wednesday

- Start working on the lecture book!

Announcements

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https://www.purdue.edu/freeform/me323/additional-lecture-notes-2/prof-gonzalez-730/

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Exam 2 - Summary of topics

- Flexural and shear stresses in beams- Deflection in beams (indeterminate problems):

- second-order integration method, - superposition

- Energy method: Castigliano’s theorem: dummy load – but …… redundant loads not included!

- Homework 6-8- Lectures 15–26

Announcements

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Equation sheet for Exam 2 (on the blog)

Normal and 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!

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Flexural stress in beams (due to bending moment)

Equilibrium of beams

Moment-curvature equation – Flexure formula – In addition

Note: the y-coordinate is measured from the centroid!!!!

positivecurvature

negativecurvature

Shear stress in beams (due to shear forces)

- Jourawski Theory (or Collignon Theory)

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Equilibrium of beams

What about ?

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⇤

= =

Load-deflection equations

(constant cross-section and material properties)

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Deflection of beams

inclinationangle (~slope)

deflection

Shear-deflection eqn.

Load-deflection eqn.

Moment-curvature eqn.

(2nd order) (4nd order)

(follow sign conventions)

Boundary conditions

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Deflection of beams

(follow sign conventions)

= |

= |

Constrainedrotation end

>0

>0

Continuity conditions

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Deflection of beams

>0

>0

= |

= |

Outline for 2nd order method (determinate or indeterminate):– FBD– Equilibrium for external forces and couples– Find internal moment 𝑀(𝑥) for each section– Integrate moment-curvature equation 𝐸𝐼𝑣!! 𝑥 = 𝑀(𝑥)– Apply boundary and continuity conditions– Solve for unknowns– Check units!

Deflection of beams

Energy methods

Work and elastic strain energy

Work done by the force: Work done by the torque: Work done by the moment:

AA

A

AA

A

Stored elastic strain energy: Stored elastic strain energy: Stored elastic strain energy:

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v

PC

△C

Energy methods - Castigliano’s Second Theorem

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Consider an indeterminate linearly elastic deformable body acting upon by forces ,moments , and torques . Among all possible equilibrium configurations

of the body, the actual configuration is the one for which:

where generalized displacements ( ) correspond to and are in the direction of the load ( ), and the redundant or internal load (that do not do any external work).

Note: some of these loads could be dummy loads, with value zero, that will facilitate the calculation of a generalized displacement at their point of application.

Castigliano’s Second Theorem (final version)

(displacement – force)

(slope – bending moment)

(angle of rotation – torque)

(redundant or internal load)

All dummy loads = 0

Outline for Castigliano’s Second Theorem (dummy loads):– Is there a conjugate load for each generalized displacement to be

determined? No: a dummy load is needed!

– Apply Castigliano’s Theorem: e.g.,

– FBD (including the dummy load) – Equilibrium equations

– Find internal moment 𝑀(𝑥), axial force 𝐹(𝑥) , torque 𝑇(𝑥), and shear force 𝑉(𝑥) for each segment

– Compute the total strain energy 𝑈 of the assembly

– Tip #1: "#"$!

= %&' ∫(

)𝑀 "*"$!

𝑑𝑥 +⋯

– Tip #2: enforce dummy load equal to zero right before integrating

Energy methods - Castigliano’s Second Theorem

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Exam 2 - Summary of topics

- Flexural and shear stresses in beams- Deflection in beams (indeterminate problems):

- second-order integration method, - superposition

- Energy method: Castigliano’s theorem (dummy & redundant loads)

- Homework 6-8- Lectures 15–26

Review session – Exam 2

Energy methods

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Problem 54 (practice problem):Cantilever beam (L x L x L) comprised of three straight segments

I II III

A

B

C

D

Study hard!

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Review session – Exam 2