Booklet #
MECE 3321
Mechanics of Solids
Exam Booklet
Fundamental Equations of Mechanics of Materials
Axial Load Bending
Normal Stress Normal Stress
𝑃 𝑀𝑦
𝜎= 𝜎=−
𝐴 𝐼
Displacement Unsymmetric bending
𝐿
𝑃(𝑥)𝑑𝑥 𝑀𝑧 𝑦 𝑀𝑦 𝑧 𝐼𝑧
𝛿=∫ 𝜎=− + , tan 𝛼 = tan 𝜃
𝐴(𝑥)𝐸 𝐼𝑧 𝐼𝑦 𝐼𝑦
0
𝑃𝐿 Shear
𝛿=∑
𝐴𝐸 Average direct shear stress
𝛿𝑇 = 𝛼∆𝑇𝐿
𝑉
𝜏𝑎𝑣𝑔 =
Torsion 𝐴
Shear stress in circular shaft Transverse shear stress
𝑇𝜌 𝑉𝑄
𝜏= 𝜏=
𝐽 𝐼𝑡
Shear Flow
where
𝜋 𝑉𝑄
𝐽 = 2 𝑐4 solid cross section 𝑞 = 𝜏𝑡 =
𝐼
𝜋
𝐽 = 2 (𝑐𝑜4 − 𝑐𝑖4 ) tubular cross section
Material Property Relations
Power
Poisson’s ratio
𝑃 = 𝑇𝜔 = 2𝜋𝑓𝑇 𝜀𝑙𝑎𝑡
𝜐=−
𝜀𝑙𝑜𝑛𝑔
Angle of Twist
𝐿
Generalized Hooke’s Law
𝑇(𝑥)𝑑𝑥 1
𝜙=∫ 𝜀𝑥 = [𝜎𝑥 − 𝜐(𝜎𝑦 + 𝜎𝑧 )]
0 𝐽(𝑥)𝐺 𝐸
𝑇𝐿 1
𝜙=∑ 𝜀𝑦 = [𝜎 − 𝜐(𝜎𝑥 + 𝜎𝑧 )]
𝐽𝐺 𝐸 𝑦
1
Average shear stress in a thin-walled tube 𝜀𝑧 = [𝜎 − 𝜐(𝜎𝑥 + 𝜎𝑦 )]
𝐸 𝑧
𝑇 1 1 1
𝜏𝑎𝑣𝑔 =
2𝑡𝐴𝑚 𝛾𝑥𝑦 = 𝜏 , 𝛾𝑦𝑧 = 𝜏 , 𝛾𝑥𝑧 = 𝜏
𝐺 𝑥𝑦 𝐺 𝑦𝑧 𝐺 𝑥𝑧
Shear Flow where
𝑇 𝐸
𝑞 = 𝜏𝑎𝑣𝑔 𝑡 = 𝐺=
2𝐴𝑚 2(1 + 𝜐)
2
Fundamental Equations of Mechanics of Materials
Stress in Thin-Walled Pressure Vessel Relations between w, V, M
𝑑𝑉 𝑑𝑀
Cylinder = 𝑤(𝑥), =𝑉
𝑑𝑥 𝑑𝑥
𝑝𝑟 𝑝𝑟
𝜎1 = 𝜎2 = Elastic Curve
𝑡 2𝑡
1 𝑀
Sphere =
𝜌 𝐸𝐼
𝑝𝑟
𝜎1 = 𝜎2 = 𝑑4𝑣
2𝑡 𝐸𝐼 = 𝑤(𝑥)
𝑑𝑥 4
Stress Transformation Equations
𝑑3 𝑣
𝜎𝑥 + 𝜎𝑦 𝜎𝑥 + 𝜎𝑦 𝐸𝐼 = 𝑉(𝑥)
𝜎𝑥 ′ = + cos 2𝜃 + 𝜏𝑥𝑦 sin 2𝜃 𝑑𝑥 3
2 2
𝑑2𝑣
𝜎𝑥 − 𝜎𝑦 𝐸𝐼 = 𝑀(𝑥)
𝜏𝑥 ′ 𝑦′ = − sin 2𝜃 + 𝜏𝑥𝑦 cos 2𝜃 𝑑𝑥 2
2
Principal Stress Buckling
Critical axial load
𝜏𝑥𝑦
tan 2𝜃𝑝 = 𝜋 2 𝐸𝐼
(𝜎𝑥 − 𝜎𝑦 )/2 𝑃𝑐𝑟 =
(𝐾𝐿)2
𝜎𝑥 + 𝜎𝑦 𝜎𝑥 − 𝜎𝑦 2 Critical stress
𝜎1,2 = ± √( 2
) + 𝜏𝑥𝑦 𝜋2𝐸
2 2 𝜎𝑐𝑟 = , 𝑟 = √𝐼/𝐴
(𝐾𝐿/𝑟)2
Maximum in-plane shear stress
Secant formula
(𝜎𝑥 − 𝜎𝑦 )/2 𝑃 𝑒𝑐 𝐿 𝑃
tan 2𝜃𝑠 = − 𝜎𝑚𝑎𝑥 = [1 + 2 sec ( √ )]
𝜏𝑥𝑦 𝐴 𝑟 2𝑟 𝐸𝐴
𝜎𝑥 − 𝜎𝑦 2
𝜏𝑚𝑎𝑥 = √( 2
) + 𝜏𝑥𝑦 Energy Methods
2 Conservation of energy
𝜎𝑥 + 𝜎𝑦 𝑈𝑒 = 𝑈𝑖
𝜎𝑎𝑣𝑔 =
2
Absolute maximum shear stress
𝜎𝑚𝑎𝑥 Strain energy
𝜏𝑎𝑏𝑠 𝑚𝑎𝑥 = 𝑓𝑜𝑟 𝜎𝑚𝑎𝑥 , 𝜎𝑚𝑖𝑛 𝑠𝑎𝑚𝑒 𝑠𝑖𝑔𝑛
2 𝑁2𝐿
𝜎𝑚𝑎𝑥 − 𝜎𝑚𝑖𝑛 𝑈𝑖 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑎𝑥𝑖𝑎𝑙 𝑙𝑜𝑎𝑑
2𝐴𝐸
𝜏𝑎𝑏𝑠 𝑚𝑎𝑥 = 𝑓𝑜𝑟 𝜎𝑚𝑎𝑥 , 𝜎𝑚𝑖𝑛 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑔𝑛
2 𝐿
𝑀2 𝑑𝑥
𝑈𝑖 = ∫ 𝑏𝑒𝑛𝑑𝑖𝑛𝑔 𝑚𝑜𝑚𝑒𝑛𝑡
0 2𝐸𝐼
𝐿
𝑓𝑠 𝑉 2 𝑑𝑥
𝑈𝑖 = ∫ 𝑡𝑟𝑎𝑛𝑠𝑣𝑒𝑟𝑠𝑒 𝑠ℎ𝑒𝑎𝑟
0 2𝐺𝐴
𝐿
𝑇 2 𝑑𝑥
𝑈𝑖 = ∫ 𝑡𝑜𝑟𝑠𝑖𝑜𝑛𝑎𝑙 𝑚𝑜𝑚𝑒𝑛𝑡
0 2𝐺𝐽
3
Geometric Properties of Area Elements
1 1
𝐼𝑥 = 𝑏ℎ3 𝐼𝑦 = ℎ𝑏 3 1 4 1
12 12 𝐼𝑥 = 𝜋𝑟 𝐼𝑦 = 𝜋𝑟 4
4 4
1
𝐼𝑥 = 𝑏ℎ3
36
1 4 1
𝐼𝑥 = 𝜋𝑟 𝐼𝑦 = 𝜋𝑟 4
8 8
4
Average Mechanical Properties of Typical Engineering Materials
U.S. Customary Units
Modulus Yield Strength Ultimate Strength % Coefficient of
Specific Modulus
of (ksi) σY (ksi) σ u Elongation Poisson's Thermal
Materials Weight of Rigidity,
Elasticity, in 2 in. Ratio, ν Expansion, α
(lb/in3) 3 G (103) ksi Tens Comp Shear Tens Comp Shear -6
E (10 ) ksi specimen (10 )/°F
Aluminum 2014-T6 0.101 10.6 3.9 60 60 25 68 68 42 10 0.35 12.8
Wrought Alloys 6061-T6 0.098 10.0 3.7 37 37 19 42 42 27 12 0.35 13.1
Gray ASTM 20 0.260 10.0 3.9 - - - 26 96 - 0.6 0.28 6.7
Cast Iron Alloys
Malleable ASTM A-197 0.263 25.0 9.8 - - - 40 83 - 5 0.28 6.6
Red Brass C83400 0.316 14.6 5.4 11.4 11.4 - 35 35 - 35 0.35 9.8
Copper Alloys
Bronze C86100 0.319 15.0 5.6 50 50 - 35 35 - 20 0.34 9.6
Metallic
Magnesium
AM 1004-T61 0.066 6.5 2.5 22 22 - 40 40 22 1 0.3 14.3
Alloy
Structural A-36 0.284 29.0 11.0 36 36 - 58 58 - 30 0.32 6.6
Structural A992 0.284 29.0 11.0 50 50 - 65 65 - 30 0.32 6.6
Steel Alloys
Stainless 304 0.284 28.0 11.0 30 30 - 75 75 - 40 0.27 9.6
Tool L2 0.295 29.0 11.0 102 102 - 116 116 - 22 0.32 6.5
Titanium Alloy Ti-6Al-4V 0.160 17.4 6.4 134 134 - 145 145 - 16 0.36 5.2
Low Strength 0.086 3.20 - - - 1.8 - - - - 0.15 6.0
Concrete
Nonmetallic
High Strength 0.086 4.20 - - - 5.5 - - - - 0.15 6.0
Plastic Kevlar 49 0.0524 19.0 - - - - 104 70 10.2 2.8 0.34 -
Reinforced 30% Glass 0.0524 10.5 - - - - 13 19 - - 0.34 -
Wood Select Douglas Fir 0.017 1.90 - - - - 0.3 3.78 0.90 - 0.29 -
Structural Grade White Spruce 0.130 1.40 - - - - 0.36 5.18 0.97 - 0.31 -
5
Average Mechanical Properties of Typical Engineering Materials
SI Units
Modulus Yield Strength Ultimate Strength % Coefficient of
Modulus
Density of (MPa) σY (MPa) σ u Elongation Poisson's Thermal
Materials of Rigidity,
3
(Mg/m ) Elasticity, in 50 mm. Ratio, ν Expansion, α
G GPa Tens Comp Shear Tens Comp Shear -6
E GPa specimen (10 )/°C
Aluminum 2014-T6 2.79 73.1 27 414 414 172 469 469 290 10 0.35 23
Wrought Alloys 6061-T6 2.71 68.9 26 255 255 131 290 290 186 12 0.35 24
Gray ASTM 20 7.19 67.0 27 - - - 179 669 - 0.6 0.28 12
Cast Iron Alloys
Malleable ASTM A-197 7.28 172 68 - - - 276 572 - 5 0.28 12
Red Brass C83400 8.74 101 37 70 70 - 241 241 - 35 0.35 18
Copper Alloys
Bronze C86100 8.83 103 38 345 345 - 655 655 - 20 0.34 17
Metallic
Magnesium
AM 1004-T61 1.83 44.7 18 152 152 - 276 276 152 1 0.3 26
Alloy
Structural A-36 7.85 200 75 250 250 - 400 400 - 30 0.32 12
Structural A992 7.85 200 75 345 345 - 450 450 - 30 0.32 12
Steel Alloys
Stainless 304 7.86 193 75 207 207 - 517 517 - 40 0.27 17
Tool L2 8.16 200 75 703 703 - 800 800 - 22 0.32 12
Titanium Alloy Ti-6Al-4V 4.43 120 44 924 924 - 1000 1000 - 16 0.36 9.4
Low Strength 2.38 22.1 - - - 1.8 - - - - 0.15 11
Concrete
Nonmetallic
High Strength 2.37 29.0 - - - 5.5 - - - - 0.15 11
Plastic Kevlar 49 1.45 131 - - - - 717 483 20.3 2.8 0.34 -
Reinforced 30% Glass 1.45 72.4 - - - - 90 131 - - 0.34 -
Wood Select Douglas Fir 0.47 13.1 - - - - 2.1 26 6.2 - 0.29 -
Structural Grade White Spruce 3.60 9.7 - - - - 2.5 36 6.7 - 0.31 -
6
Simply Supported Beam Slopes and Deflections
7
Simply Supported Beam Slopes and Deflections
8
Cantilevered Beam Slopes and Deflections
9
Cantilevered Beam Slopes and Deflections
10
Geometric Properties of Structural Shapes
11
Geometric Properties of Structural Shapes
12
Geometric Properties of Structural Shapes
13
Geometric Properties of Structural Shapes
14
Geometric Properties of Structural Shapes
15
Geometric Properties of Structural Shapes
16
Geometric Properties of Structural Shapes
17
Geometric Properties of Structural Shapes
18
Stress Concentration Factors
Axial Loading
19
Stress Concentration Factors
Torsional Loading
20
Stress Concentration Factors
Bending
21
