Transcript Strength of Materials I EGCE201 กำลังวัสดุ 1
Strength of Materials I
EGCE201
ก ำลังวัสดุ
1 Instructor:
ดร
.
วรรณสิริ พันธ์อุไร
(
อ
.
ปู
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6391 Copyright © 2006 by Wonsiri Punurai
Deformations Under Uniform Axial Loading
• From Hooke’s Law
E
;
E
;
P A
• From the definition of strain
L
• Equating and solving for the deformation, ;
P AE
PL AE
Example
Steel bar (E=??) Cross-sectional area A=256 mm 2 Find the elongation of the bar.
Method: •Identify segments •Use equilibrium for internal axial Forces in each segment •Use constitutive relation to relate Forces to displacements •Use compatibility to get net displacement
Continuously varying bars
(load, area, modulus could be varying with position)
Composite bars
(Compound loading or discretely varying load/area/modulus )
Statically Indeterminate Problems
• “Statically Indeterminate” means the # of unknowns exceeds the number of available equations of equilibrium.
• Statics (equilibrium analysis) alone cannot solve the problem n R n E = # of reactions (or unknowns) = # of equilibrium equations • If n R = n E : statically determinate - forces in each member only depend on equilibrium.
• If n R > n E : statically indeterminate - too many unknowns, must invoke a constraint such as a deformation relation.
Statically Indeterminate Examples
Free body diagram Free body diagram
Statically Indeterminate: Load sharing Problem
A free body diagram
Not sufficient to determine P1 and P2!!!
If we consider the displacement of each member, additional information relating P1 and P2 becomes available 1
P
1
L
1
A
1
E
1 2
P
2
L
2
A
2
E
2
In order for the rigid end plate to remain vertical,
1 2
Statically Indeterminate: Load sharing Problem
Statically Indeterminate: Reaction
Rod AB fixed between 2 rigid walls ( ) compression (+) tension
Statically Indeterminate: Reaction
Statically Indeterminate: Superposition
• The method of superposition is an effective way to determine reactions when a structure is statically indeterminate (held by more supports than required to maintain equilibrium causing “
redundant reaction”
) • The solution to the statically indeterminate using this method is (1) Considering the deformations caused by the given load and redundant reaction separately.
(2) Superimposing the solutions for each upon one another to obtain the solution
Statically Indeterminate: Superposition
The reaction at C is redundant
Statically Indeterminate: Superposition
Temperature effect
Initial Stress/Strain and Temperature effect
Deformations under Thermal Effect
T
T L
• Coefficient of thermal expansion (CTE) m/m/ o F m/m/ o C
Deformations under Thermal Effect
Thin Walled Pressure Vessels
Spherical Vessels
Torsion of shafts ( แรงบิดภายในที่เกิดขึ้นด้ามเพลา
• Shafts are members with length greater than the largest cross sectional dimension used in transmitting torque from one plane to another
Learning objectives
• Understand the theory and its limitation and its applications for design and analysis of Torsion of circular shafts.
Internal Torque ( แรงบิดภายใน )
Consider circular shaft AC subjected to equal and opposite torques T and T’. A cutting plane is passed through the shaft at B.
M T
' 0 :
T
:
T T
'
dF dF
0 The FBD for section BC must include the applied torque and elementary shearing forces
dF
as . These forces are perpendicular To radius of the shaft and must balance to maintain equilibrium.
The axis of the shaft is denoted . Taking moments about the Axis of the shaft results in
dF is related to the shearing stress: dF =
t
dA
So the applied torque can be related to the shearing stress as
T
t
dA
T
t
dA
Shear stress can’t exist on one plane only.
The applied torque produces a shear stress to the axis of the shaft. The equilibrium require equal stresses on the faces This results in the shear stress distribution shown.
Theory of Circular Shafts
Theory Objectives • To obtain a formula for the relative rotation f 2 f 1 in terms of the internal torque T.
• To obtain a formula for the shear stress t x q terms of the internal torque T.
in f - angle of twist
Shearing Strain
Shearing Stress
Assume: Material is linearly elastic and isotropic
Recall
T
t
dA
Define
J
2
dA
Polar moment of inertia for the cross section
Torsion formula
Circular solid shaft Circular hollow shaft with outer radius R, inner radius r
J
2
Sign Convention
R
4
r
4
Relative rotation
f 2 f 1
torque T.
in terms of the internal Shear stress
t
x
q
in terms of the internal torque T.
Maximum occurs at shaft’s outer radius