Mechanics of Materials: Stress-Strain Curve for Mild Steel and Related Concepts, Cheat Sheet of Mechanics of Materials

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Mechanics of Materials Definitions • Hooke’s Law 13-2a

• • Shear Modulus:Stress:
• • (^) PoissonStrain: ’s Ratio:
• • Normal stress or strain =Shear stress = || to the surface! " to the surface

Mechanics of Materials Stress and Strain Thin-Walled Tanks 13-3a

Hoop Stress: Axial Stress:

Mechanics of Materials Stress and Strain Transformation of Axes 13-3b

Mechanics of Materials Stress and Strain For examples 1 and 2, use the following illustration. 13-3d

Example 1 (FEIM) The principal stresses ( (A) (B) – 8462 000 kPa and 28 400 kPa and 14 σ000 kPa 2 400 kPa, σ 1 ) are most nearly (C) (D) 70112 000 kPa and 14 000 kPa and – 28 000 kPa 000 kPa

Mechanics of Materials Stress and Strain The center of Mohr’s circle is at 13-3d

!^ "max^ =^ (^30000 kPa)

(^2) + ( 24000 kPa) (^2) = 38 419 kPa !!^ "^ "^12 ==^ "" cc^ #+^ $#maxmax^ ==^ ((#$^2424000000 kPakPa^ #+^3838 419419 kPa)kPa)^ ==^ #^1462419 418 kPa^ kPa

Therefore, (D) is correct.

!^ Using the Pythagorean theorem, the radius of Mohr^ " c^ =^12 (" x^ +^ " y^ )^ =^12 (#^48000 kPa^ +^0 )^ =^ #^24000 kPa ’s circle (τmax) is:

Mechanics of Materials Stress and Strain General Strain 13-3e

!^ Note that^ " x^ is no longer proportional to^ # x^.

Mechanics of Materials Stress and Strain Static Loading Failure Theory 13-3f

Maximum Normal Stress: A material fails if • Or !!^ This is true of brittle materials.^ For ductile materials:^ •^ "^ "^ ##^ SSct Maximum Shear

!^ Distortion Energy (von^ T #max^ "=^ =^ max^21 %^ &^ '(^ #^ $^ %^ &^ &^1 "$^1 #^ #^22 ")^22 +,^ ("#^1 1 Mises#^2 $^ "^ #^33 ),^2 "Stress)^ +^2 (#^2 #^ "^2 3 $^ '^ (^ )^ )#^ >^3 ) S^22 yt (^ )^ *^ >^ Syt

Mechanics of Materials Stress and Strain Hollow, Thin-Walled Shafts 13-3h

Mechanics of Materials Beams 13-4a

Mechanics of Materials Beams Shear and Bending Moment Diagrams 13-4c

Example 1 (FEIM): Draw the shear and bending moment diagrams for the following beam.

Mechanics of Materials Beams 13-4c

! Shear is undefined at concentrated force points, but just short of x = 12 m

Therefore,^ R^ Rll^ +=^ R (^8 r^ )=^ (^ "^ #^ $ R^100 rR^ (4 l^ )^ mN= 533.3 N and^ =^ %^ &^ '^ (^0 16 m)^ =^1600 R^ r N = 1066.7 N

So the shear diagram is:

From 0 m to 12 m, V = Rl " # $ % (^100) mN^ & ' ( x = 533.3 N " # $ % (^100) mN^ & ' ( x ; 0 m < x < 12 m !

V ( 12 "^ ) = 533.3 N " # $ % (^100) mN^ & ' (( 12 m) = "666.7 N

! !^ V^ From 12 m to 16 m,^ =^1600 N^ "^ #^ $^ %^100 mN^ &^ '^ (^ Vx ;^^ =^12 V^^ (^12 <^ " x^ )^^ +)^ R^ 16 m r^ "^ (^100 N)( x^ "^12 )

• Mechanics of Materials Stress-Strain Curve for Mild Steel 13-
• Mechanics of Materials Beams The bending moment diagram is: 13-4c

Mechanics of Materials Beams Example 2 (FEIM): 13-4d

The vertical shear for the section at the midpoint of the beam shown is (A) 0 (B) (C) P (D) none of these Drawing the force diagram and the shear diagram, Therefore, (A) is correct.

21 P