AMME2301 Solid Mech NOTES Contents Introduction: ........................................................................................................................................... 4

Max stress: .......................................................................................................................................... 4

Max displacement (Stiffness) .............................................................................................................. 4

Stability (eg. Buckling of columns) ...................................................................................................... 4

Statics: ..................................................................................................................................................... 4

Types of loading: ................................................................................................................................. 4

Force: .............................................................................................................................................. 4

Moment .............................................................................................................................................. 4

Couple ................................................................................................................................................. 4

Support structures .............................................................................................................................. 5

2 force members: ................................................................................................................................ 5

Zero force meber: ............................................................................................................................... 5

Internal Resultant Loading: ................................................................................................................. 6

Stress ....................................................................................................................................................... 9

Normal Stress: ................................................................................................................................. 9

Shear Stress: .................................................................................................................................. 10

Average Normal Stress: (uniform, uniaxial stress) ............................................................................ 10

Average Shear stress: ........................................................................................................................ 10

Maximum normal and shear stresses ............................................................................................... 11

Allowable Stress: ............................................................................................................................... 11

Example: ........................................................................................................................................ 11

Factor of safety: ................................................................................................................................ 11

General stress: .................................................................................................................................. 13

Strain: .................................................................................................................................................... 14

Average normal strain:...................................................................................................................... 14

Units of strain: ............................................................................................................................... 14

Notes on strain: ............................................................................................................................. 14

Young’s modulus/Hooke’s Law: ............................................................................................................ 15

Poisson’s ratio: ...................................................................................................................................... 15

Axial loading: ......................................................................................................................................... 16

Stress concentration factor:.............................................................................................................. 16

Saint-Venant’s Principle: ................................................................................................................... 16

Axial load: DEFORMATION ................................................................................................................ 17

Stress and strain with deformation: ............................................................................................. 17

Hooke’s law with deformation: .................................................................................................... 17

Example: Axial Load internal force diagram ................................................................................. 18

Axial load- Deformation: Principle of Superposition: ....................................................................... 18

Statically determinaten/indeterminate structures:.......................................................................... 19

Analysis of pin joint frames ........................................................................................................... 19

Mechanism: ................................................................................................................................... 20

Stress concentration factor:.............................................................................................................. 20

Thermal strain and thermal stress .................................................................................................... 21

Combined thermal/force stress: ................................................................................................... 21

Compatibility condtions: ............................................................................................................... 21

ENERGY METHODS: AXIAL LOAD ...................................................................................................... 23

Axial Load: ..................................................................................................................................... 23

External Work: .............................................................................................................................. 24

Internal work (strain energy): ....................................................................................................... 24

Total strain energy in a deformable body: .................................................................................... 24

Principle of superposition: ............................................................................................................ 25

Energy methods: Virtual Force ......................................................................................................... 25

CASTIGLIANO’s SECOND THEOREM: ................................................................................................. 25

Example: Castiglione’s 2nd theorem: ............................................................................................. 26

Torsion: ................................................................................................................................................. 27

Shear stress: ...................................................................................................................................... 27

Shear strain ....................................................................................................................................... 27

Hooke’s law for shear: .................................................................................................................. 27

Internal torque and resultant shear stress ................................................................................... 28

Maximum shear stress: ..................................................................................................................... 29

Angle of twist: ............................................................................................................................... 29

Torsion: power transmission ............................................................................................................ 30

Summary of torsion and axial load: .................................................................................................. 31

Axial load: ...................................................................................................................................... 31

Torsion: ......................................................................................................................................... 32

Bending: ................................................................................................................................................ 32

Internal loading- bending moments and shear force ....................................................................... 32

Sign convention: ............................................................................................................................ 32

Formulation of bending moment: .................................................................................................... 33

Macauley’s notation: ........................................................................................................................ 34

Bending stress and strain: ..................................................................................................................... 35

Bending strain: .................................................................................................................................. 35

Flexure formula: ............................................................................................................................ 35

Neutal axis and moment of inertia: .................................................................................................. 36

Composite area: neutral axis ........................................................................................................ 37

Transformation factor ....................................................................................................................... 37

Transverse shear stress ......................................................................................................................... 38

State’s of stress from different loadings: .......................................................................................... 38

Combined Loadings: .............................................................................................................................. 39

Principle of superposition ................................................................................................................. 39

Total normal stress: ...................................................................................................................... 39

Total shear stress at N.A. .............................................................................................................. 40

Plane stress distribution: .................................................................................................................. 40

Principal stresses and maximum in-plane shear stress .................................................................... 41

Overall: principal stresses and maximum in plane shear stress: .................................................. 43

Mohr’s circle: .................................................................................................................................... 43

Thin walled pressure vessels ............................................................................................................. 46

Failure: .................................................................................................................................................. 46

Von misses: fail criterion: .................................................................................................................. 46

Displacement of beams: ....................................................................................................................... 46

Double integration method: ............................................................................................................. 46

Theorem of moment areas: .............................................................................................................. 47

Method of superposition: ................................................................................................................. 50

Statically indeterminatne beams: ................................................................................................. 51

Buckling of columns .............................................................................................................................. 51

Example: ........................................................................................................................................ 52

Mechanics of materials: 9th edition, 2013, RC Hibbeler, Prentice Hall international.

Assignments (5): 25%

Quiz (week 6) : 10%

Exam 65%

Introduction: Studies the internal effects of stress and strain in a solid body subjected to loads.

Max stress: 𝜎max ≤ 𝜎𝑝

Max displacement (Stiffness) 𝑣max ≤ 𝑣𝑝

Stability (eg. Buckling of columns) 𝑃 < 𝑃𝑐𝑟

Statics: Statics concerns the equilibrium of bodies under external loadings

1. Beams:

Types of loading:

Force: - Concentrated

- Distributed:

𝐹 = ∫ 𝑤(𝑥)𝑑𝑥𝐿

0

(𝑚𝑎𝑔𝑛𝑖𝑡𝑢𝑑𝑒 𝑜𝑓 𝑓𝑜𝑟𝑐𝑒)

𝑐 =∫ 𝑤(𝑥)𝑥𝑑𝑥

𝐿

0

𝐹 (𝑝𝑙𝑎𝑐𝑒 𝑜𝑓 𝑓𝑜𝑟𝑐𝑒′𝑠 𝑎𝑐𝑡𝑖𝑜𝑛)

Moment

𝑀0 = 𝐹𝑐 = ∫ 𝑤(𝑥)𝑥𝑑𝑥𝐿

0

Couple It consists of two forces equal in magnitude but opposite in direction whose line of action are

parallel but no collinear.

𝑂

𝑤(𝑥) 𝐹

𝑐

Support structures

2 force members:

If an element has pins or hinge supports at both ends and carries no load in-

between, it is called a two-force member. The reaction forces travels through the

beam

Zero force meber:

If two non-collinear members meet in an unloaded joint, both are zero-force members.

If three members meet in an unloaded joint of which two are collinear, then the third member

is a zero-force member.

Reasons for Zero-force members in a truss system

These members contribute to the stability of the structure, by providing buckling prevention

for long slender members under compressive forces

These members can carry loads in the event that variations are introduced in the normal

external loading configuration

Internal Resultant Loading:

𝑁𝑦 = 𝑁𝑜𝑟𝑚𝑎𝑙 𝐹𝑜𝑟𝑐𝑒 (+= 𝑡𝑒𝑛𝑠𝑖𝑙𝑒; − = 𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑖𝑣𝑒)

𝑉𝑥; 𝑉𝑧 = 𝑠ℎ𝑒𝑎𝑟 𝑓𝑜𝑟𝑐𝑒

𝑀𝑥 = 𝑏𝑒𝑛𝑑𝑖𝑛𝑔 𝑚𝑜𝑚𝑒𝑛𝑡

𝑀𝑧 = 𝑏𝑒𝑛𝑑𝑖𝑛𝑔 𝑚𝑜𝑚𝑒𝑛𝑡

𝑇𝑦 = 𝑡𝑜𝑟𝑠𝑖𝑜𝑛𝑎𝑙 𝑚𝑜𝑚𝑒𝑛𝑡

Example: internal resultant loadings in pipe:

1-27 The pipe assembly is subjected to a force of 600 N at B. Determine the resultant internal

loadings acting on the cross section at C. (p. 21)

∑𝐹𝑥 = 0

∴ 𝐹𝑥 = 600 cos 60 sin 30

𝐹𝑦 = 600 cos 60 cos 30

𝐹𝑧 = 600 sin 60

∑𝑀 = 0

∴ 𝑀𝑥 = 600 sin 60 (. 4) 𝑀𝑦 = −600 cos 60 sin 30 (. 5)𝑒𝑐𝑡

Example: internal loads of beam structure

Example Determine the resultant internal loadings at D, E and F

𝐹𝑍

𝐹𝑥 𝐹𝑦

𝑀𝑧

𝑀𝑥

𝑀𝑦

Cutting at D:

∴ 𝐹𝐷 = 0; 𝑀𝐷 = 0

Cutting at F:

∴ ∑ 𝐹𝑥 ; ∑ 𝐹𝑦 ; ∑ 𝑀 = 0

∴ 𝐹𝐹𝑦 = 12 𝑘𝑁; 𝐹𝐹𝑋 = 0; 𝑀𝐹 = 4.8 𝑘𝑁𝑚

Stress Stress: the intensity of the internal force on a specific plane passing through a point

This assumes that the material is continuous (no voids) and cohesive (no cracks, breaks and defects)

Can be either Tensile or compressive stress (positive/negative respectively)

Normal Stress:

𝜎 = limΔ𝐴→0

Δ𝐹𝑛

Δ𝐴

Shear Stress:

𝜏 = limΔ𝐴→0

Δ𝐹𝑡

Δ𝐴

Average Normal Stress: (uniform, uniaxial stress)

𝜎𝑎𝑣 =𝐹

𝐴

Average Shear stress:

𝜏𝑎𝑣 =𝑉

𝐴

Maximum normal and shear stresses As stress is in a specific plane, we can have many stresses through a specific point:

𝑁 = 𝐹 cos 𝜃 ; 𝑉 = 𝐹 sin 𝜃 ; 𝐴 =𝑆

cos 𝜃

∴ 𝜎 =𝐹

𝑆cos2 𝜃

𝜎\max =𝐹

𝑆 (𝜃 = 0)

𝜏 =𝐹

2𝑆sin 2𝜃

𝜏max =𝐹

2𝑆 (𝜃 = 45°)

Allowable Stress: When the stress (intensity of force) of an element exceeds some level, the structure will fail. For

convenience, we usually adopt allowable force or allowable stress to measure the threshold of

safety in engineering.

∴ 𝜎 ≤ 𝜎𝑎𝑙𝑙𝑜𝑤𝑒𝑑

Example: An 80 kg lamp is supported by a single electrical copper cable. if the maximum allowable stress for

copper is σCu,allow=50MPa, please determine the minimum size of the wire/cable from the material

strength point of view.

∴ 𝜎 =𝐹

𝐴=

𝑚𝑔

(𝜋4 𝑑2)

≤ 𝜎𝐶𝑢,𝑎𝑙𝑙𝑜𝑤𝑒𝑑

∴ 𝑑 ≥ √4𝑚𝑔

𝜋𝜎𝐶𝑢= 4.469 𝑚𝑚

Factor of safety: 𝐹𝑆 is a ration of the failure load 𝐹𝑓𝑎𝑖𝑙 divided by the allowable load 𝐹𝑎𝑙𝑙𝑜𝑤

𝐹𝑆 =𝐹𝑓𝑎𝑖𝑙

𝐹𝑎𝑙𝑙𝑜𝑤=

𝜎𝑓𝑎𝑖𝑙

𝜎𝑎𝑙𝑙𝑜𝑤=

𝜏𝑓𝑎𝑖𝑙

𝜏𝑎𝑙𝑙𝑜𝑤

Example: normal/shear stress and allowable stress

Example w = 30 kN/m. Member BC has a square cross section of 20 mm. a) Determine the average

normal stress and average shear stress acting at sections a-a and b-b; b) If the allowable shear stress

for the pins τallow = 70 MPa, determine the required diameter of the pins at A and B.

1. Finding internal resultant loadings:

∑ 𝐹𝑦 : 72 = 𝐹𝐴𝑌 + 𝐹𝐵𝐶 sin 60

∑ 𝐹𝑋 : 0 = 𝐹𝐴𝑋 + 𝐹𝐵𝐶 cos 60

∑ 𝑀𝐴 : 0 = 𝐹𝐵𝐶 2cos 60 (2.4 tan 60) + −72(1.2) = 0

𝐹𝐵𝐶 = 41.57 𝑘𝑁

AT a-a:

∴ 𝜎 =𝐹𝐵𝐶

𝐴= 103.9 𝑀𝑃𝑎

𝜏 =𝑉

𝐴= 0

At b-b:

𝑁 = 𝐹𝐵𝐶𝑐𝑜𝑠60; 𝑉 = 𝐹𝐵𝐶 sin 60

∴ 𝜎 = 26 𝑀𝑃𝑎; 𝜏 = 46 𝑀𝑃𝑎

General stress: As a beam is 3D; there is stressed in all 3 directions

∴ 𝑓𝑜𝑟 𝑎 𝑐𝑢𝑏𝑒 𝑡ℎ𝑒𝑟𝑒 𝑎𝑟𝑒 9 𝑠𝑡𝑟𝑒𝑠𝑠𝑒𝑠 (6 normal; 3 shear

Strain:

Average normal strain:

휀 =𝐿𝑑𝑒𝑓𝑜𝑟𝑚𝑒𝑑 − 𝐿𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙

𝐿𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙=

Δ𝐿

𝐿 (𝑢𝑛𝑖𝑡𝑙𝑒𝑠𝑠; %)

Units of strain: Usually, for most engineering applications ε is very small, so measurements of strain are in

micrometers per meter (μm/m) or (μ/m). Sometimes for experiment work, strain is expressed as a

percent, e.g. 0.001m/m = 0.1%.

Notes on strain:

Original geometries

Note: make sure that in questions use the original geometrical parameters, as the change is very

small comparatively

Rigid member:

Will not change under stress

Example: geometrical changes in beam:

2-9 If a force is applied to the end D of the rigid member CBD and causes a normal strain in the cable

of 0.0035 mm/mm, determine the displacement of point D. (p78)

Young’s modulus/Hooke’s Law: 𝜎 = 𝐸휀

Poisson’s ratio:

휀𝑎𝑥𝑖𝑎𝑙 =𝛿

𝐿

휀𝑙𝑎𝑡𝑒𝑟𝑎𝑙 =𝛿𝑟

𝑟=

Δ𝐷

2𝐿

∴ 𝑃𝑜𝑖𝑠𝑠𝑜𝑛′𝑠 𝑟𝑎𝑡𝑖𝑜:

𝜈 = − (휀𝑙𝑎𝑡𝑒𝑟𝑎𝑙

휀𝑎𝑥𝑖𝑎𝑙 ) (𝑠ℎ𝑜𝑢𝑙𝑑 𝑏𝑒 𝜈 ∈ [0,1])

Axial loading:

Stress concentration factor:

𝐾 =𝜎max

𝜎𝑎𝑣𝑒𝑟𝑎𝑔𝑒

Saint-Venant’s Principle:

𝜎𝑎−𝑎 = 𝜎𝑎𝑣 =𝐹

𝐴

Axial load: DEFORMATION

Stress and strain with deformation:

𝜎(𝑥) =𝑃(𝑥)

𝐴(𝑥)

휀(𝑥) =𝑑𝛿

𝑑𝑥

Hooke’s law with deformation: 𝑃(𝑥)

𝐴(𝑥)= 𝐸(𝑥) [

𝑑𝛿

𝑑𝑥] ⟹ 𝑑𝛿 =

𝑃(𝑥)

𝐴(𝑥)𝐸(𝑥)𝑑𝑥

∴ 𝛿 = ∫𝑃(𝑥)

𝐴(𝑥)𝐸(𝑥)𝑑𝑥

𝐿

0

𝛿 =𝑃𝐿

𝐸𝐴 (𝑓𝑜𝑟 𝑎 𝑠𝑚𝑎𝑙𝑙 𝑠𝑒𝑔𝑚𝑒𝑛𝑡 𝑜𝑓 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑃, 𝐸, 𝐴)

and P =EAδ

L

Example: Axial Load internal force diagram

4-2: the copper shaft is subjected the axial loads shown. Determine the displacement of end

A with respect to end D if the diameters of each segment are 𝑑𝐴𝐵 = 20 mm, 𝑑𝐵𝐶 = 25 mm and

𝑑𝐶𝐷 = 12 mm. Take 𝐸𝑐𝑢 = 126 GPa (p. 133)

𝐹𝐵𝐶 : ∑ 𝐹𝑋(𝑖𝑛 𝐵𝐶) = 36 + 𝐹𝐵𝐶 − 45 = 0

𝐹𝐵𝐶 = 9 𝑘𝑁

Segment AB:

𝛿𝐴𝐵 =𝐹𝐴𝐵𝐿𝐴𝐵

𝐴𝐴𝐵𝐸𝐴𝐵=

Total displacement =𝛿𝐴𝐵 + 𝛿𝐵𝐶 + 𝛿𝐶𝐷

Axial load- Deformation: Principle of Superposition: 1. The loading must be linearly related to the displacement or stress that is to be

determined

2. The loading must not significantly change the original geometry or configuration of

the member

Statically determinaten/indeterminate structures:

Analysis of pin joint frames - If 𝐽 is the number of pin joints:

- 2𝐽 = number of equilibrium equaitons

o ∑ 𝐹𝑥 = 0; ∑ 𝐹𝑦 = 0

Unkown forces:

- 𝑚 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑚𝑒𝑚𝑏𝑒𝑟 𝑓𝑜𝑟𝑐𝑒𝑠

- 𝑟 =number of reaction forces

o 𝑖𝑓 𝑚 + 𝑟 = 2𝐽; frams is statically determinate

o 𝑖𝑓 𝑚 + 𝑟 > 2𝐽, frame is statically indeterminate

o 𝑖𝑓 𝑚 + 𝑟 < 2𝐽, frame is a mechanism

Mechanism:

Stress concentration factor:

𝐾 =𝜎max

𝜎𝑎𝑣

𝜎max ≤ 𝜎𝑎𝑙𝑙𝑜𝑤𝑒𝑑

Statically

determinate

mechanism