• To be updated Announcement

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Announcement

@Copyright Prof. Juhyuk Moon (문주혁), CEE, SNU

• To be updated

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Chap. 5 Stresses in Beams

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Chap. 5 Stresses in Beams

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Chap. 5 Stresses in Beams

5.1 Introduction

5.2 Pure Bending and Nonuniform Bending 5.3 Curvature of Beam

5.4 Longitudinal Strains in Beams 5.5 Normal Stress in Beams

5.6 Design of Beams for Bending Stresses 5.7 Nonprismatic Beams

5.8 Shear Stresses in Beams of Rectangular Cross Section 5.9 Shear Stresses in Beams of Circular Cross Section

5.10 Shear Stresses in the Webs of Beams with Flanges

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Introduction

Bending of a cantilever beam: beam with load and resulting deflection curve

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Pure Bending and Non-uniform Bending

<Simple beam in pure bending>

<Cantilever beam in pure bending>

<Simple beam in non-uniform bending>

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Curvature of a beam

𝜅 𝜌

Radius of curvature =

Reciprocal of the radius of curvature =

𝑑𝑠 = 𝑑𝑥

Assumption:

= 𝑚

₁

𝑂

^′

= 𝑚

₂

𝑂

^′

Sign convention for curvature

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Longitudinal Strains in Beams

elongation

Strain (=elongation / original length of dx)

Strain-curvature relation

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Longitudinal Strains in Beams

Strain

Strain-curvature relation

- Normal strains in a beam was derived solely from the geometry of the deformed beam.

- Properties of the material did NOT enter into the derivation.

- The strains in a beam in pure bending vary linearly with distance from the neutral surface regardless of the shape of the stress-strain curve of the material.

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Normal Stresses in Beams (Linearly Elastic Materials)

Strain in bending

Linearly Elastic Materials

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Normal Stresses in Beams (Linearly Elastic Materials)

Linearly Elastic Materials Strain in bending

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Normal Stresses in Beams (Linearly Elastic Materials)

But…Where is origin axis?

Neutral axis

- Neutral axis passes through the centroid of the cross-sectional area when the material follows Hooke’s law and there is no axial force acting on the cross section.

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Moment-Curvature Relationship

2^nd moment of inertia

Curvature-Moment

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Flexure Formula

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Flexure Formula

where Section moduli of the cross-sectional area

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Moment of Inertia (I) and Section Modulus (S)

Polar moment of inertia

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Moment of Inertia (I) derivation

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Moment of Inertia (I) derivation

𝐼 = 𝑏ℎ³

12 𝐼 = ℎ𝑏³

12

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Paper folding

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Engineering

APPENDIX E

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Customized section

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Customized section

Where is the neutral axis?

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Customized section

Moment of Inertia

(if neutral axis is different from the centroid of each components)

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