Solid Mechanics Spring 2007

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18 Ιουλ 2012 (πριν από 4 χρόνια και 9 μήνες)

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Solid Mechanics Spring 2007

1/13
Chapter 2 & 3 - Concept of Stress, Strain and Deformation













A
F


A
F





Objectives

- Distinguish between normal, shear and bearing stresses
- Analyze simple frames and structures, which consist of members that are pin-connected
- Calculate normal and shear stresses in a general plane
- Define strain, ε.
- Determine modulus of elasticity (E), yield strength and ultimate strength from a stress vs.
strain plot (σ vs. ε)
- Understand fatigue and how to consider it when designing structures that will be subjected to
cyclic loading
- Solve statically indeterminate problems in axial loading (mechanical load)
- Solve statically indeterminate problems in axial loading (thermal expansion)
- Use factor of safety to determine the maximum allowable loads under design conditions
P P
P
P
F
P
P
cut
F
Solid Mechanics Spring 2007

2/13
Normal Stress under Axial Load


Rod BC, length L, cross-sectional area A
Apply a load P

B




C









Single Shear


Rivet CD connects plates A and B, which are subjected to tension forces with magnitude, F.







_________________________ develops on the _______________________ of the rivet.

FBD - section of rivet




Double Shear


Splice plates C and D and bolts connect plates A and B.






F
F
C
D
F
F
C
D
A
B
Solid Mechanics Spring 2007

3/13

FBD of bolt FBD of section








Bearing stresses in connections


Bolts, pins and rivets exert forces on the surfaces upon which they contact. The forces exerted
on the bearing surfaces result in _______________________.






Pin exerts a force P onto the bearing surface of the plate. P is equal and opposite to the force
exerted by the rivet onto the plate.
Bearing stress σ
B
is defined as




Analysis of simple structures


Analyze 2-D structures by considering the normal, shear and bearing stresses in the various parts
of the structure (members, bolts, pins, etc.)

1. Normal stress in 2-force members
a) Determine reaction forces by drawing the FBD and writing equilibrium equations of the
entire structure
b) Consider equilibrium at various joints
c) Sometime it is useful to consider the FBD of a section of the structure
2. Determine the shear stresses in connectors (pin, bolts, rivets, etc.)
3. Determine bearing stresses at contact surfaces

Solid Mechanics Spring 2007

4/13
Example - Link BD is a steel bar 40 mm wide and 12 mm think. Knowing that each pin has a
diameter of 10 mm, determine a) the maximum value of the average normal stress in the link
when
0=
α
degrees, b) the shear stress in pin D and c) the bearing stress in the link at D.
Solid Mechanics Spring 2007

5/13
Stress on Inclined Planes

Axial forces cause both normal and shear stresses on planes that are not perpendicular to the
axis.

For an axially loaded member, consider stresses on the cross-section at an angle θ






Draw FBD of left section




Resolve P into components

N


V


Average normal and shear stresses on the incline plane are



The area of the inclined cross-section A
θ
is related to the normal cross-section A
o







The average normal stress is maximum at



The average shear stress is maximum at

Solid Mechanics Spring 2007

6/13
Strain in Axial Loading







BA
δ
δ
δ
+
=


ε
σ
E
=


EA
FL




So far, we've determined stresses that occur in axially loaded members and learned to design
them to avoid failure under specific loads.

Now, we consider deformation due to applied loads.
- deformations may be good or bad
- can use deformations to help solve statically indeterminate problems
F
A B
Solid Mechanics Spring 2007

7/13
Normal Strain under Axial Load


Rod BC, length L, cross-sectional area A
Apply a load P

B




C


Normalize deformation to determine normal strain

Stress-strain diagrams

Relation between σ and ε gives the mechanical properties of the material.
We can obtain a stress strain diagram by performing tensile tests on a material specimen.

Centrically load a specimen with a load P, and record δ
For each pair of readings (P,δ) compute

Stress

Strain

σ









ε

Hooke's Law
Solid Mechanics Spring 2007

8/13
Deformation Under Axial Loading


Recall that

Stress is

Strain is

So the relation between axial deformation and applied load is




If a rod has various cross sections, different materials, or loaded at places other than the end of
the rod, then you must sum over i sections


Solid Mechanics Spring 2007

9/13
Example

Known: Rod made of Aluminum with shown loads
Given: E
Al
= 70 x 10
9
Pa, P
1
= 100 N, P
2
= 75 N, P
3
= 50 N
Find: Deflection at points K and M
Assumptions: Neglect weight of bars, linearly elastic

Diagram:











1.75 m
1.25 m
1.5 m
P1
P2
P3
A = 0.8 m
2

A = 0.5 m
2

J
K
L
M
Solid Mechanics Spring 2007

10/13
Example


Known: Rigid beam rests on posts
Given: AC is steel, d
S
= 20mm, E
S
= 200 Gpa, BD is aluminum d
Al
= 40mm, E
Al
= 70 Gpa, force
at point F = 90 kN
Find: Displacement at point F


A
B
C
D
90 kN
400 mm
200 mm
300 mm
F
Solid Mechanics Spring 2007

11/13
Statically Indeterminate Problems


So far, solved problems by
1) Determining internal forces produced in members using
a) FBD's
b) Equilbrium equations
1) Determining stresses and deformations due to these forces

In the analysis of MANY engineering structures and machines, internal forces and often-external
reaction forces can not be determined with these principals alone. These problems are statically

indeterminate
.

Example - Bar JK of length L, fixed supports at J and K, centric load P at a distance L
1
from pt J,
cross-section area A.













Method of Superposition


Break statically intedeterminate problem into 2 statically determinate problems.




= +











J
K
P
Solid Mechanics Spring 2007

12/13
Problems Involving Temperature Change


Consider bar AB of uniform cross-section and length L, which rests freely on horizontal surface



If the temperature of the bar is increased ____, then the rod elongates by ____


Normalize elongation by ___________________ to define thermal strain ___




Statically indeterminate problem - thermal stress/strain


Consider bar JK with fixed supports at both ends




What happens if the temperature of the bar is raised?

As before, use superposition to determine the force P and the normal stress σ










Total elongation is


Thermal stress is


A B
J
K
L
Solid Mechanics Spring 2007

13/13
Factor of Safety

Even after careful design and analysis of a structure of machine, we typically design structures to
be stronger to "be on the safe side". Design with a factor of safety.

From material testing, determine a maximum load called the ultimate load, Pu, which is the
maximum load a material can handle.



If loading is centric, ultimate normal stress is



And ultimate shear stress is



Structures and machines will be designed so that the allowable load is considerably less than the
ultimate load. Thus only a fraction of the ultimate load carrying capacity is utilized when the
max allowable load is applied.

Factor of safety (F.S.) is defined as




How do we choose F.S.?




Trade offs

Low F.S.

High F.S.


Typically F.S. is chosen to be in the range of