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Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
Lecture Note for Solid Mechanics- Essential Concept and Statics Review -
Prof. Sung Ho YoonDepartment of Mechanical EngineeringKumoh National Institute of Technology
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
Text book : Mechanics of Materials, 6th ed.,
W.F. Riley, L.D. Sturges, and D.H. Morris, 2007.
Prerequisite : Knowledge of Statics, Basic Physics, Mathematics, etc.
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
Objective of mechanics of materials
Relationships between loads applied to a nonrigid body and internal forces and deformations induced in the body
- Determination of load carrying capability of structural members
- Development of mathematical and experimental methods of analysis
Subject matter of this course
Given a certain function to perform, of what materials should be constructed, and what sizes and proportions of various elements? (Designer’s task)
Given the completed design, does it perform the function economically and without excessive deformation? (Checkers’ problem)
Given a completed structure, what is its actual load carrying capability? (Rating problem)
Essential Concepts and Statics Review
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
Classification of forces
Forces : action of one body on another(equal magnitude, opposite direction pairs)
Direct and indirect contact between two bodies Surface forces (direct contact)
- Concentrated force and distributed force (uniform or non-uniform)
Body forces (indirect contact)- Gravitational attraction force and electromagnetic force
unit : force/length (N/m)
Essential Concepts and Statics Review
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
Applied forces and reactions Applied forces
- Concentrated and/or distributed forces designed to carry Reactions
- Forces at supports to prevent movements
External forces and internal forces
Static forces and dynamic forces Static forces
- Forces which do not vary with time Dynamic forces
- Forces which vary with time such as impact load and repeated load
Essential Concepts and Statics Review
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
Equilibrium of a rigid body Rigid body
- Body that does not deform under the action of applied forces
Equilibrium condition- Resultant of the system of forces acting on the body is zero- ):( bodytheofforonoMandF o 00
00 oMandF
000
000
zyx
zyx
MMM
FFF
000 zyx MFF
00 yx FF
Free body diagram Free body : Body of interest separated from all other interacting bodies
Free body diagram : Free body with all external forces Use for determining the unknown forces acting on rigid bodies
(3-Dim)
(2-Dim)
(2-Dim)
Essential Concepts and Statics Review
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
(Example) 900kg mass is supported by a roller that can move along the beam. Determine (a) reactions at A and B neglecting the mass of beam and (b) reactions at A and B if the mass of the beam is 8.5kg/m.
(Sol)
000 zyx MFF(a) Using
03882980
088290
00
)()(BM
BAF
AF
A
yy
xx
NBNAA yx 331055200
000 zyx MFF(b) Using
04398383882980
03983888290
00
)().()()(
).(
BM
BAF
AF
A
yy
xx
NBNAA yx 364058500
(83.39 N/m x 8m)
Essential Concepts and Statics Review
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
(Example) Calculate the support reactions and the forces in the BD, DE, and CE. Assume that the truss supports one side of bridge and floor beams carry vehicle loads (W=3400 lb) to the truss joints.
(Sol)
00 MandF
Since the truss is a two-force member, it is subjected to either tension or compression.
lbElbAAEM
EAF
AF
yx
A
yy
xx
425127500)32(850)64(0
08508500
00
Half of the car’s weight (W=1700 lb) is carried by the truss and the other half is carried by the truss on the other side of the bridge.
Using
Essential Concepts and Statics Review
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
Free body diagram of pin E
lbTlbT CEDE 7.5048.659
0sin425sin0
0cos0
DEDEy
DECEx
TTEF
TTF
Since = tan -1 (16/19) = 40.10 O
Summing moments about point C
)(850850
0)32(425)16(0
ClblbT
TM
BD
BDC
26 ft
Essential Concepts and Statics Review
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
(Example) The weight of books on a shelf bracket is 75 lb. Determine all forces on all three members of this frame.
(Sol) 00 MandF
Using
lbClbAlbA
AF
CAF
CM
yx
yy
xx
A
0.1200.750.120
0750
00
0)75(850
lbBlbT
BM
TM
AC
A
ACB
0.1300.50
0)75(8135120
0)75(1212)75(40
Dismember the bracket and draw free body diagrams of each member
Essential Concepts and Statics Review
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
(Example) Determine reactions at support A neglecting the weights of the pipe and the wrench.
(Sol)
00 MandF(a) Using
00
01802000
0230035002000
02000
00
00
zz
yy
xx
zz
yy
xx
MM
MM
MM
AF
AF
AF
).(
)..(
NANANA zyx 20000
mNMmNMmNM zyx 00360116 ..
Essential Concepts and Statics Review
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
(Example) Determine the forces at A and B on rigid structure AB subjected to a 5000-lb force.
(Sol)
00 MandF
Using
0)4(5000)4)(45sin(0
0500045sin0
045cos0
TM
TAF
TAF
z
yy
xx
lbTlblbAlbA yx 7071004795.05000
Essential Concepts and Statics Review
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
Equilibrium of a deformable body
Interaction between loads acting on a body, deformations of the body, and the geometry of the free body diagram makes the solution of the deformable body problems more complex
Solution requires either a trial-and-error solution or a numerical solution or an iterative solution method
(Example) 5000 lb weight is to be supported by AB (rigid bar) and BC (nonrigid wire). Using TBC=k and =Lf-Li, determine the tension in the wire after the load is applied for k=5000 lb/in.
Essential Concepts and Statics Review
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
0cos4)2/45sin(40)cos4()sin4()2/45cos(
)cos4()2/45sin(0
0)2/45sin(0
0)2/45cos(0
WTWT
TM
WTAF
TAF
A
yy
xx
Using the trigonometric relation, BABABA sincoscossin)sin(
Force-deformation relation is introduced
inLL
inL
L
inftL
if
f
f
i
118867
886718867
18867
90484824848
8867657544222
22
sin.
.sin.
sin.
)cos(
..
Thus, cos)/sin(sin. Wk 245118867
If k is 5000 lb/in, =2.465deg, T=7221 lb, Ax= 4995 lb, Ay = -215 lb
(Example) Determine the tension in the wire if k=5000 lb/in.
(Sol)
Essential Concepts and Statics Review
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
Internal forces
- Forces that exist throughout the interior of a body
C : centroidR : resultant forceC : resultant couple
Internal forces : R & C
0:0
0:0
21
21
CMMM
RFFF
C
where M1 and M2 are the moments of F1 and F2 about the centroid.
We find the resultant of the internal force system using the equilibrium equations. However, we can not find the exact distribution of the internal forces without information of the deformation of the body.
Essential Concepts and Statics Review
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
000
000
zyx
zyx
MMM
FFF
Resolve the resultants R and C into components along and perpendicular to the section
P : Normal forceV : Shear force
T : Twisting momentM : Bending moment
Components of the internal force system can be found using the equilibrium equations
Essential Concepts and Statics Review
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
(Example) Determine (a) the support reactions and (b) the internal forces on a section 4m to the right of the support A.
(Sol)
00 MandF
(a) Using
03552100
05100
00
)().(AA
yy
xx
MM
AF
AF
mkNMkNAkNA Ayx 40150
(b)
0)1(5)2(8)4(15400
058150
00
zz
yy
x
MM
VF
PF
mkNMkNVkNP zy 120
Essential Concepts and Statics Review
Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수
(Homework)
(1-8), (1-9), (1-22), (1-34), (1-56), (1-61)
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