Formula Sheet Mechanics Materials

FORMULA SHEET FOR ENGINEERING 3016 PART 4 – MECHANICS OF MATERIALS Chapter 4 Stress, Strain, and Deformation: Axial Stress-Strain Relationships Low-carbon steel or ductile materials Loading F Normal stress is normal to the plane σ = , F is the A normal force, A is the cross-sectional area. V Shear stress is in the plane τ = , V is the shear A force, A is the cross-sectional area.

Average normal strain ε =

δn L

High-carbon steel or alloy steel

Average shear strain γ =

δs L

= tan φ

Hooke’s Law: for normal stress σ = Eε for shear stress τ = Gγ E is the Young’s modulus G is the shear modulus ε E , ν = − lat is Poisson’s ratio where ε lat G= 2(1 + ν ) ε long is strain in lateral direction and ε long is strain in longitudinal direction. . Deformation of Axially Loaded Members PL Member with uniform cross section δ = EA n PL Members with multiple loads/sizes δ = ∑ i i i =1 Ei Ai

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Chapter 8 Flexural Loading: Stress in Beams

Chapter 7 Torsional Loading: Shafts

Tc , J J is polar second moment of area

Shear stress at c, τ =

For solid cross section J =

π

2

For hollow cross section J =

c4

π 2

(c24 − c14 )

Torsional displacement or angle of twist

M r is the resultant of normal stress Vr is the resultant of shear stress The Elastic Flexural Formula My Normal stress at y : σ = − I Max. normal stress at upper surface y = c : σ = − For uniform shaft θ =

TL GJ

I is the second moment of area For a rectangular cross section n

For shaft with multi-step θ = ∑ i =1

Ti Li Gi J i

Work of a couple u = Cθ , C is couple, θ is angle of twist Power Transmission by Torsional Shafts Power = T ω , ω is angular velocity

For a circular cross section

2

M rc I

Chapter 9 Flexural Loading: Beam Deflections

Shear Forces and Bending Moments in Beams M I the max. bending stress σ max = r max where S = is S c the section modulus of the beam. If the beam is uniform cross section, S is constant. M r max is the max. bending moment in the beam as M r varies along the beam, to find M r max , need to draw the bending moment diagram. Shear force diagram shows the variation of the shear force Vr along beam Bending moment diagram shows the variation of the bending moment M r along beam Sign convention

Procedure 1. Find the reactions at supports. 2. Determine how to divide the beam into different segments. 3. Starting from the far left end, section the beam at an arbitrary location x within the chosen segment. 4. Draw FBD for the portion of the beam to the left. 5. Apply equilibrium equations. 6. Repeat the process for each different segment of the beam. 3

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