Download - Lecture 28 –Review for Exam 2
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Lecture 28 – Review for Exam 2
Instructor: Prof. Marcial Gonzalez
Fall, 2021ME 323 – Mechanics of Materials
News: Ready for the exam?
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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)
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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
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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⇤
= =
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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)
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Boundary conditions
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Deflection of beams
(follow sign conventions)
= |
= |
Constrainedrotation end
>0
>0
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Continuity conditions
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Deflection of beams
>0
>0
= |
= |
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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
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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:
12
v
PC
△C
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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)
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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
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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
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Study hard!
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Review session – Exam 2