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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
Youngβs modulus/Hookeβs Law: ............................................................................................................ 15
Poissonβs ratio: ...................................................................................................................................... 15
Axial loading: ......................................................................................................................................... 16
Stress concentration factor:.............................................................................................................. 16
Saint-Venantβs Principle: ................................................................................................................... 16
Axial load: DEFORMATION ................................................................................................................ 17
Stress and strain with deformation: ............................................................................................. 17
Hookeβs 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
CASTIGLIANOβs SECOND THEOREM: ................................................................................................. 25
Example: Castiglioneβs 2nd theorem: ............................................................................................. 26
Torsion: ................................................................................................................................................. 27
Shear stress: ...................................................................................................................................... 27
Shear strain ....................................................................................................................................... 27
Hookeβs 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
Macauleyβs 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
Stateβs 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
Mohrβs 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: πmax β€ ππ
Max displacement (Stiffness) π£max β€ π£π
Stability (eg. Buckling of columns) π < πππ
Statics: Statics concerns the equilibrium of bodies under external loadings
1. Beams:
Types of loading:
Force: - Concentrated
- Distributed:
πΉ = β« π€(π₯)ππ₯πΏ
0
(ππππππ‘π’ππ ππ πππππ)
π =β« π€(π₯)π₯ππ₯
πΏ
0
πΉ (πππππ ππ πππππβ²π πππ‘πππ)
Moment
π0 = πΉπ = β« π€(π₯)π₯ππ₯πΏ
0
Couple It consists of two forces equal in magnitude but opposite in direction whose line of action are
parallel but no collinear.
π
π€(π₯) πΉ
π
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:
ππ¦ = ππππππ πΉππππ (+= π‘πππ πππ; β = πππππππ ππ£π)
ππ₯; ππ§ = π βπππ πππππ
ππ₯ = πππππππ ππππππ‘
ππ§ = πππππππ ππππππ‘
ππ¦ = π‘πππ πππππ ππππππ‘
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)
βπΉπ₯ = 0
β΄ πΉπ₯ = 600 cos 60 sin 30
πΉπ¦ = 600 cos 60 cos 30
πΉπ§ = 600 sin 60
βπ = 0
β΄ ππ₯ = 600 sin 60 (. 4) ππ¦ = β600 cos 60 sin 30 (. 5)πππ‘
Example: internal loads of beam structure
Example Determine the resultant internal loadings at D, E and F
πΉπ
πΉπ₯ πΉπ¦
ππ§
ππ₯
ππ¦
Cutting at D:
β΄ πΉπ· = 0; ππ· = 0
Cutting at F:
β΄ β πΉπ₯ ; β πΉπ¦ ; β π = 0
β΄ πΉπΉπ¦ = 12 ππ; πΉπΉπ = 0; ππΉ = 4.8 πππ
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:
π = limΞπ΄β0
ΞπΉπ
Ξπ΄
Shear Stress:
π = limΞπ΄β0
ΞπΉπ‘
Ξπ΄
Average Normal Stress: (uniform, uniaxial stress)
πππ£ =πΉ
π΄
Average Shear stress:
πππ£ =π
π΄
Maximum normal and shear stresses As stress is in a specific plane, we can have many stresses through a specific point:
π = πΉ cos π ; π = πΉ sin π ; π΄ =π
cos π
β΄ π =πΉ
πcos2 π
π\max =πΉ
π (π = 0)
π =πΉ
2πsin 2π
πmax =πΉ
2π (π = 45Β°)
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.
β΄ π β€ ππππππ€ππ
Example: An 80 kg lamp is supported by a single electrical copper cable. if the maximum allowable stress for
copper is ΟCu,allow=50MPa, please determine the minimum size of the wire/cable from the material
strength point of view.
β΄ π =πΉ
π΄=
ππ
(π4 π2)
β€ ππΆπ’,πππππ€ππ
β΄ π β₯ β4ππ
πππΆπ’= 4.469 ππ
Factor of safety: πΉπ is a ration of the failure load πΉππππ divided by the allowable load πΉπππππ€
πΉπ =πΉππππ
πΉπππππ€=
πππππ
ππππππ€=
πππππ
ππππππ€
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 Οallow = 70 MPa, determine the required diameter of the pins at A and B.
1. Finding internal resultant loadings:
β πΉπ¦ : 72 = πΉπ΄π + πΉπ΅πΆ sin 60
β πΉπ : 0 = πΉπ΄π + πΉπ΅πΆ cos 60
β ππ΄ : 0 = πΉπ΅πΆ 2cos 60 (2.4 tan 60) + β72(1.2) = 0
πΉπ΅πΆ = 41.57 ππ
AT a-a:
β΄ π =πΉπ΅πΆ
π΄= 103.9 πππ
π =π
π΄= 0
At b-b:
π = πΉπ΅πΆπππ 60; π = πΉπ΅πΆ sin 60
β΄ π = 26 πππ; π = 46 πππ
General stress: As a beam is 3D; there is stressed in all 3 directions
β΄ πππ π ππ’ππ π‘βπππ πππ 9 π π‘πππ π ππ (6 normal; 3 shear
Strain:
Average normal strain:
ν =πΏππππππππ β πΏππππππππ
πΏππππππππ=
ΞπΏ
πΏ (π’πππ‘πππ π ; %)
Units of strain: Usually, for most engineering applications Ξ΅ is very small, so measurements of strain are in
micrometers per meter (ΞΌm/m) or (ΞΌ/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)
Youngβs modulus/Hookeβs Law: π = πΈν
Poissonβs ratio:
νππ₯πππ =πΏ
πΏ
νπππ‘ππππ =πΏπ
π=
Ξπ·
2πΏ
β΄ ππππ π ππβ²π πππ‘ππ:
π = β (νπππ‘ππππ
νππ₯πππ ) (π βππ’ππ ππ π β [0,1])
Axial loading:
Stress concentration factor:
πΎ =πmax
πππ£πππππ
Saint-Venantβs Principle:
ππβπ = πππ£ =πΉ
π΄
Axial load: DEFORMATION
Stress and strain with deformation:
π(π₯) =π(π₯)
π΄(π₯)
ν(π₯) =ππΏ
ππ₯
Hookeβs law with deformation: π(π₯)
π΄(π₯)= πΈ(π₯) [
ππΏ
ππ₯] βΉ ππΏ =
π(π₯)
π΄(π₯)πΈ(π₯)ππ₯
β΄ πΏ = β«π(π₯)
π΄(π₯)πΈ(π₯)ππ₯
πΏ
0
πΏ =ππΏ
πΈπ΄ (πππ π π ππππ π ππππππ‘ ππ ππππ π‘πππ‘ π, πΈ, π΄)
and P =EAΞ΄
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 ππ΄π΅ = 20 mm, ππ΅πΆ = 25 mm and
ππΆπ· = 12 mm. Take πΈππ’ = 126 GPa (p. 133)
πΉπ΅πΆ : β πΉπ(ππ π΅πΆ) = 36 + πΉπ΅πΆ β 45 = 0
πΉπ΅πΆ = 9 ππ
Segment AB:
πΏπ΄π΅ =πΉπ΄π΅πΏπ΄π΅
π΄π΄π΅πΈπ΄π΅=
Total displacement =πΏπ΄π΅ + πΏπ΅πΆ + πΏπΆπ·
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 π½ is the number of pin joints:
- 2π½ = number of equilibrium equaitons
o β πΉπ₯ = 0; β πΉπ¦ = 0
Unkown forces:
- π = ππ’ππππ ππ ππππππ ππππππ
- π =number of reaction forces
o ππ π + π = 2π½; frams is statically determinate
o ππ π + π > 2π½, frame is statically indeterminate
o ππ π + π < 2π½, frame is a mechanism
Mechanism:
Stress concentration factor:
πΎ =πmax
πππ£
πmax β€ ππππππ€ππ
Statically
determinate
mechanism