The task is to provide basic theories and analytical methods of strength, rigidity and stability for rational design.
2. Fundamental Assumptions
1. Continuity The material is continuously distributed over its volume .
p=lim F/A
is shearing stress at K
A 0
Relationship: p2 = 2 + 2
SI unit of stress: 1 Pa =1 N/m2 , 1 MPa =106 Pa
5. Deformation and Strain
1. Normal strain
Method of F1
m
Fn
sections
F1 F2
F2 m y FR
m
F1 M
C m z
x
F2
y
m My Fy Mx
C
zMz Fmz
Fx
x
Fx = 0, Fy = 0, Fz = 0
Mx= 0, My= 0, Mz= 0
F1 F2
y
m My
Fy Mx
C
zMz Fmz
Fx
x
Fx : axial force
q
F
D ABC
aaa
Solution: BCD: MB= Βιβλιοθήκη , FD = qa/2 ABCD:
FBx
F
D
FBy B C
FD
MA= 0, MA = qa2
q MA
F
Fx = 0, FAx = 0 FAx A B C
D
Fy = 0, FAy = 3qa/2 FAy a
a
a FD
Problem 1.2
We can find : = E
E : modulus of elasticity
Unit of E :
1 GPa = 109 Pa = 103 MPa
Modulus of steel :
E = 200 ~ 220 GPa
Modulus of aluminum : E = 70 ~ 72 GPa
Find reactions of A and D, internal forces in left and right sections of C.(F=qa)
q
F
D ABC
aaa
Solution: ( FD = qa/2 ) C1D: Fy = 0, FS1 = qa/2
MC1= 0, M1 = - qa2/2
3. External Forces and Classification
According to the ways of their action 1. Surface force 2. Body force According to the cases of their distribution 1. Concentrated force 2. Distributed force: uniform or nonuniform According to their changes with time 1. Static load 2. Dynamic load: inertia, impact, repeated
C2D: Fy = 0, FS2 = - qa/2 MC2= 0, M2 = - qa2/2
F
M1 C1
FS1
D
FD
C2 M2
D
FS2 FD
Problem 1.3
World Financial Center building , 9. 11 of 2001
In New York of USA
Questions:
4. Internal Force and Stress
(1) Internal force
F1
m
Fn
F2
m
F1
m
F2
m
Internal force is a force set up within a body to balance the effect of the externally applied forces.
FN
Fy , Fz : shearing force FSy , FSz
Mx : torque moment T
My , Mz : bending moment My , Mz
(2) Stress
F1
m
F1
A F
K
F2
m
F2
m
p
K
m
Stress at K of the section: is normal stress at K
7 Analysis of Stress & Strain (8) 8 Strength of Combined Deformations (8) 9 Stability of Columns (4) ( Exam. B ) 10 Dynamic Load and Fatigue Strength (6) 11 Energy Methods (8) 12 Statically Indeterminate Members (6) 13 Experimental Stress Analysis (4)
Mechanics of Materials
Edited by Guo Ying-Zheng in 2013
CONTENTS
1 Introduction (2) 2 Tension & Compression (10) 3 Torsion (6) 4 Internal Forces in Bending (6) 5 Stresses in Bending (10) ( Exam. A) 6 Deformation in Bending (8)
Example 1.1
Find the internal forces at fixed end D of the bar
D
a
a
C
B
as shown in the right Fig.
Aa
Solution:
F
x
Translate F from A to C
Bar CD: Fx = 0, FN = F
Structure Laboratory
Load Test of Airplane
Chapter 1 Introduction
1. The Task of Mechanics of Materials
Strength: Capacity to resist break or yield. Rigidity: Capacity to resist over deformation. Stability: Capacity to keep in original equilibrium.
Large-scale Bridge
Column
Cable
Structure of Bridge Floor
Macao Bridge
Space Shuttle “Discovery”
Space Station “Peace”
High-speed Train
Nuclear Reactor
Mz= 0, M = Fa
F
m
n
a 90
a m
a
a
n
z ya F
M
m - m : Fy = 0, FS = F
x FS
My
Mz= 0, M = Fa x
F
z
Mx= 0, T = 2Fa
T FS
Problem 1.2
Find reactions of A and D, internal forces in left and right sections of C.(F=qa)
2. Homogeneity The material is homogeneously distributed over its volume .
3. Isotropy
The mechanical properties are the same in all directions at a point .
b
=
lim
x 0
u x
=
du dx
K x a u Normal strain at K along Ka
2. Shearing strain
b
is the change of a right angle
K x a
Unit of is radian ( rad )
6. Hookes law
It shows that :
FN
Mz
D
My
z
y
My= 0, My = Fa Fa
Fa
C
Mz= 0, Mz = - Fa F
6. Types of deformations
Fundamental deformations: 1. Tension or compression 2. Shearing 3. Torsion 4. Bending
Combined deformations 1. Tension (compression) and Bending 2. Bending and Torsion 3. Tension (compression) , Bending and Torsion