Bài giảng Strength of materials - Chương 3: Axial forces - Nguyễn Sỹ Lâm
CHAPTER 3: AXIAL FORCES
3.1 Concept, Internal Forces
3.2 Normal Strain and Normal Stress
3.3 Stress – Strain diagram
3.4 Hooke’s law: Modulus of Elasticity
3.5 Deformation under axial loading
3.6 Safety Factor, Allowable Stress
3.7 Statically Indeterminate Systems
3.1 CONCEPT, INTERNAL FORCES
• Suitability of a structure or machine may depend on the deformations in
the structure as well as the stresses induced under loading. Statics
analyses alone are not sufficient.
• Considering structures as deformable allows determination of member
forces and reactions which are statically indeterminate.
• Determination of the stress distribution within a member also requires
consideration of deformations in the member.
• Chapter 3 is concerned with deformation of a structural member under
axial loading. Later chapters will deal with torsional and pure bending
loads.
3.1 CONCEPT, INTERNAL FORCES
A structural member under axial loading means:
Internal forces: only axial force N (along longitudinal axis of the bar)
Method of Section: the same as discussed in previous chapter
Imaginary cut is made through the body in the region where the internal
loading is to be determined.
The two parts are separated and a free body diagram of one of the parts is
drawn. Only then applying equilibrium would enable us to relate the
resultant internal force and moment to the external forces.
3.1 CONCEPT, INTERNAL FORCES
WHAT DOES IT LOOK LIKE?
3.1 CONCEPT, INTERNAL FORCES
WHAT DOES IT LOOK LIKE?
3.1 CONCEPT, INTERNAL FORCES
HOW CAN WE KNOW THAT IT IS A PURELY AXIAL LOADING
PROBLEM?
(1) Structure member is subjected to axial (longitudinal direction) forces only. For example:
OR
(2) Structure members are connected together by hinges (so-called nodes) and there is no
load on the member, just have loads at nodes. For example:
3.1 CONCEPT, INTERNAL FORCES
EXAMPLES: DERTEMINE INTERNAL FORCES OF STRUCTURES
BELOWS?
(1)
(2)
3.2 NORMAL STRAIN AND NORMAL STRESS
a force P produces a deformation . In engineering, we usually transform
this force into stress and the deformation into strain and we define these as
follows:
2P P
P
A
P
A
stress
2A A
L
2
normal strain
2L L
L
3.2 NORMAL STRAIN AND NORMAL STRESS
Strain has no unit’s since it is a ratio of length to length. Most engineering
materials do not stretch very mush before they become damages, so strain
values are very small figures. It is quite normal to change small numbers in to
the exponent for 10-6( micro strain).
Sign convention of normal stresses:
x > 0 : tensile stress
x < 0 : compression stress
3.2 NORMAL STRAIN AND NORMAL STRESS
Normal stress: at a section far from the bar ends, the normal stress
can be regarded uniform, or considered as average value
Stress has units of force per unite area (kN/m2; N/m2…)
Note: Most of engineering fields used kPa, MPa, GPa.
3.2 NORMAL STRAIN AND NORMAL STRESS
3.3 STRESS – STRAIN DIAGRAM
3.3 STRESS – STRAIN DIAGRAM
Ductile Materials
3.3 STRESS – STRAIN DIAGRAM
Brittle Materials
3.3 STRESS – STRAIN DIAGRAM
Stress Strain Diagram
3.3 STRESS – STRAIN DIAGRAM
Elastic domain: OA
linear stress–strain relationship
point A: proportional limit
Const slope: Modulus of Elasticity, E
Transition domain: AB
Region of Plasticity: BC
B: Elastic Limit(A B)
Deformation develops under constant yield stress,
C end of the region of plasticity
Strain-hardening & necking regions:
Nonlinear Stress-Strain Relationship
Point D: the ultimate stress,
Point E: Fracture of the specimen
3.4 HOOKE’S LAW: MODULUS OF ELASTICITY
• Below the yield stress
E
E Youngs Modulus or
Modulus of Elasticity
• Strength is affected by alloying,
heat treating, and manufacturing
process but stiffness (Modulus of
Elasticity) is not.
3.4 HOOKE’S LAW: MODULUS OF ELASTICITY
POISSON RATIO
• For a slender bar subjected to axial loading:
E
x
0
y z
x
• The elongation in the x-direction is
accompanied by a contraction in the other
directions. Assuming that the material is
isotropic (no directional dependence),
0
y
z
• Poisson’s ratio is defined as
lateral strain
axial strain
y
z
x
x
- Poisson’s ratio, determined by experiment; values from
0.2 – 0.5
3.5 DEFORMATION UNDER AXIAL LOAD
POISSON RATIO
• From Hooke’s Law:
P
E AE
E
• From the definition of strain:
L
• Equating and solving for the deformation,
PL
AE
• With variations in loading, cross-section or
material properties,
P L
i i
A E
i i i
3.5 DEFORMATION UNDER AXIAL LOAD
Example 3.01
SOLUTION:
• Divide the rod into components at
the load application points.
• Apply a free-body analysis on each
component to determine the
internal force
6
E 2910 psi
D 1.07 in. d 0.618in.
• Evaluate the total of the component
Determine the deformation of
the steel rod shown under the
given loads.
deflections.
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