1.050 Engineering Mechanics I
Lecture 25: Beam elasticity – problem solving technique and examples
Handout
1
1.050 – Content overview
I. Dimensional analysis
1.
2.
On monsters, mice and mushrooms
Similarity relations: Important engineering tools
Lectures 1-3
Sept.
II. Stresses and strength
3.
4.
Stresses and equilibrium
Strength models (how to design structures,
foundations.. against mechanical failure)
Lectures 4-15
Sept./Oct.
III. Deformation and strain
5.
6.
How strain gages work?
How to measure deformation in a 3D structure/material?
Lectures 16-19
Oct.
IV. Elasticity
7.
8.
Elasticity model – link stresses and deformation
Variational methods in elasticity
Lectures 20-31
Oct./Nov.
V. How things fail – and how to avoid it
9.
10.
11.
Elastic instabilities
Plasticity (permanent deformation)
Fracture mechanics
Lectures 32-37
Dec.
2
1
1.050 – Content overview
I. Dimensional analysis
II. Stresses and strength
III. Deformation and strain
IV. Elasticity
Lecture 20:
Lecture 21:
Lecture 22:
Lecture 23:
Lecture 24:
Lecture 25:
Lecture 26:
…
Introduction to elasticity (thermodynamics)
Generalization to 3D continuum elasticity
Special case: isotropic elasticity
Applications and examples
Beam elasticity
Applications and examples (beam elasticity)
… cont’d and closure
V. How things fail – and how to avoid it
3
Beam bending elasticity
Governed by this differential equation:
d 4ξ z
fz
=
dx 4
EI
Integration provides solution for displacement
Solve integration constants by applying BCs
Note:
E = material parameter (Young’s modulus)
I = geometry parameter (property of cross-section)
fz = distributed shear force (force per unit length)
f z = pb0 where p0=pressure, b=thickness of beam in y-direction
4
2
4-step procedure to solve beam elasticity problems
• Step 1: Write down BCs (stress BCs and displacement BCs),
analyze the problem to be solved (read carefully!)
• Step 2: Write governing equations for ξ z , ξ x ...
• Step 3: Solve governing equations (e.g. by integration),
results in expression with unknown integration constants
• Step 4: Apply BCs (determine integration constants)
Note: Very similar procedure as for 3D isotropic elasticity problems
5
Difference in governing equations (simpler for beams)
Physical meaning of derivatives of ξ z
d 4ξ z
fz
=
dx 4
EI
3
d ξz
Q
=− z
3
dx
EI
M
d 2ξ z
=− y
2
dx
EI
dξ z
= −ω y
dx
d 4ξ z
EI = f z
dx 4
d 3ξ z
EI = Qz
−
dx 3
d 2ξ z
EI = M y
−
dx 2
dξ
− z =ω y
dx
ξz
ξz
Shear force density
Shear force
Bending moment
Rotation (angle)
Displacement
6
3
Step-by-step example
z
p = force/length
x
l length
Step 1: BCs
x=0
x=l
EI
ξ z (0) = 0
ω y (0) = 0
ξ z (l) = 0
M y (0) = 0
7
Step 2: Governing equation
d 4ξ z
f
= z
4
dx
EI
Step 3: Integration
ξ z'''' = −
p
EI
p applied in
negative zdirection
p
EI
p
ξ z'' = −
EI
p
ξ z' = −
EI
p
ξz = −
EI
ξ z''' = −
d 4ξ z
p
=
−
dx 4
EI
x + C1
x 2
+ C1 x + C2
2
x3
x 2
+ C1 + C2 x + C3
6
2
4
x
x3
x 2
8
+ C1 + C2
+ C3 x + C
4
24
6
2
4
Step 4: Apply BCs
p
EI
p
ξ z'' = −
EI
p
ξ z' = −
EI
p
ξz = −
EI
ξ z''' = −
x + C1 = −
Qz
EI
M
x2
+ C1 x + C2
= − y
2
EI
3
2
x
x
+ C1 + C2 x + C3 = −ω y
6
2
4
x
x3
x2
+ C1 + C2
+ C3 x + C4
24
6
2
Known quantities are marked
9
Step 4: Apply BCs (cont’d)
ξ z (0) = 0 → C4 = 0
ω y (0) = 0 → C3 = 0
p l4
l3
l2
ξ z (l) = 0 → −
+ C1 + C2 = 0
EI 24
6
2
p l2
M y (0) = 0 → −
+ C1l +C 2= 0
EI 2
⎛l
⎜
⎜⎜ 6
⎝l
3
⎛ l4 ⎞
l ⎞⎛ C ⎞ p ⎜ ⎟
⎟ 1
24
2 ⎟⎜⎜ C ⎟⎟ = EI ⎜ l 2 ⎟
⎜ ⎟
1 ⎟⎠⎝ 2 ⎠
⎜ ⎟
⎝ 2 ⎠
2
p 5
l
EI 8
p 1 2
C2 = −
l
EI 8
C1 =
10
5
Solution:
5 ⎞
⎛
Qz (x) = p⎜ x − l ⎟
8 ⎠
⎝
⎛ 1 2 x2 5 ⎞
M y (x) = p⎜⎜ l + − lx ⎟⎟
2 8 ⎠
⎝8
ω y (x) =
p
EI
⎞
⎛1 2
x3 5
⎜⎜ l x + − lx 2 ⎟⎟
6 16 ⎠
⎝8
p
ξ z (x) = −
EI
⎛ 1 2 2 x4 5 3 ⎞
⎜⎜ l x +
− lx ⎟⎟
16
24
48 ⎠
⎝
11
6