Deflections in beams in bending

1. Beam Deflection
BIOE 3200 - Fall 2015Watching stuff break

2. Learning Objectives
 Define beam deflection (δ) and identify
the factors that affect it
 Determine deflection and slope in
beams in bending using double
integration method
BIOE 3200 - Fall 2015

3. Deflection in beams
Deflection is
deformation
from original
position in y
direction
BIOE 3200 - Fall 2015

4. Recall relationships between shear
force, bending moment and normal
stresses
BIOE 3200 - Fall 2015
To balance forces within
beam cross section:
P = σx 𝑑𝐴
V =  𝑥𝑦 𝑑𝐴
𝑀 𝑧 = − σx 𝑦 𝑑𝐴
 𝑥𝑦

5. Bending moments balance normal
stresses across area of cross
section. This is how we relate
applied loads and deformation
(stress and strain).
BIOE 3200 - Fall 2015
Simplify: 𝑀𝑧 = 𝐸
𝑑2 𝑢 𝑜𝑦(𝑥)
𝑑𝑥2 𝑦2
𝑑𝐴
Recall that 𝐼x = 𝑦2
𝑑𝐴
So:
where u0y(x) is displacement in y direction

6. General formula for beam
deflection involves double
integration of bending moment
equation
 Governing equation for deflection:
𝑑2δ
𝑑𝑥2 =
𝑀(𝑥)
𝐸𝐼
, where deflection δ(x) = uoy(x)
 Solve double integral to get equation for δ(x)
(elastic curve), or the deflected shape
 Shape of beam determined by change in
load over length of beam
BIOE 3200 - Fall 2015

7. How to calculate beam deflection
using double integration method
BIOE 3200 - Fall 2015
δ(x)
EI = Flexural Rigidity
 Governing equation:
𝑑2δ
𝑑𝑥2 =
𝑀
𝐸𝐼
𝐸𝐼
𝑑δ
𝑑𝑥
= 0
𝑥
𝑀 𝑥 𝑑𝑥 + 𝐶1 Note:
𝑑δ
𝑑𝑥
= tan θ ≈ θ(x)
 𝐸𝐼θ(x) = 0
𝑥
𝑀 𝑥 𝑑𝑥 + 𝐶1
𝐸𝐼δ = 0
𝑥
[ 0
𝑥
𝑀 𝑥 𝑑𝑥 + 𝐶1 ] dx + 𝐶2
𝐸𝐼δ = 0
𝑥
𝑑𝑥 0
𝑥
𝑀 𝑥 𝑑𝑥 + 𝐶1x + 𝐶2 - general formula for
beam deflection

8. Sign conventions
for beam
deflection
 X and Y axes: positive to the right and
upward, respectively
 Deflection δ(x): positive upward
 Slope of deflection at any point
𝑑δ
𝑑𝑥
and
angle of rotation θ(x): positive when CCW
with respect to x-axis
 Curvature (K) and bending moment (M):
positive when concave up (beam is
smiling) BIOE 3200 - Fall 2015
δ(x)

9. What affects deflection?
 Bending moment
◦ Magnitude and type of loading
◦ Span (length) of beam
◦ Beam type (simply supported, cantilever)
 Material properties of beam (E)
 Shape of beam (Moment of Intertia I)
BIOE 3200 - Fall 2015

10. How to complete double
integration
𝐸𝐼δ = 0
𝑥
𝑑𝑥 0
𝑥
𝑀 𝑥 𝑑𝑥 + 𝐶1x + 𝐶2
 Find C1 and C2 from boundary conditions
(supports)
 Example: cantilever beam with load at free
end
BIOE 3200 - Fall 2015

11. Using boundary conditions to
calculate deflections in beams
 Other examples of boundary
conditions:
BIOE 3200 - Fall 2015

12. Pulling it all together
 Relation of the deflection δ with beam
loading quantities V, M and load
 Deflection = δ
 Slope = d δ / dx
 Moment M(x) = EI
𝑑2δ
𝑑𝑥2
 Shear V(x) = - dM/dx = - EI
𝑑3δ
𝑑𝑥3 (for
constant EI)
 Load w(x) = dV/dx = - EI
𝑑4δ
𝑑𝑥4 (for constant
EI)
BIOE 3200 - Fall 2015

13. Load, moment, deformation and
slope can all be sketched for a
beam
BIOE 3200 - Fall 2015

14. Procedure for calculating
deflection by integration method
 Select interval(s) of the beam to be used,
and set coordinate system with origin at
one end of the interval; set range of x
values for that interval
 List boundary conditions at boundaries of
interval (these will be integration
constants)
 Calculate bending moment M(x)
(function of x for each interval) and set it
equal to EI
𝑑2δ
𝑑𝑥2
 Solve differential equation (double
integration) and solve using known
integration constants BIOE 3200 - Fall 2015

15. Typical deflection equation
Simply supported beam under uniform
constant load:
δx =
𝑤 𝑥
24 𝐸 𝐼
(𝑙3
− 2 𝑙 𝑥2
+ 𝑥3
)(at any point x)
BIOE 3200 - Fall 2015
Loa
d
Material
Property
Shape
Propert
y
L
Δmax
Span

16. Examples of deflection
formulae
FBD for simply supported beam under
constant uniform load:
 δmax =
5 𝑤 𝑙4
384 𝐸 𝐼
(at midpoint)
 δx =
𝑤 𝑥
24 𝐸 𝐼
(𝑙3
− 2 𝑙 𝑥2
+ 𝑥3
)(at any point x)
BIOE 3200 - Fall 2015

17. Examples of deflection
formulae
 Simply supported beam, point load at
midspan
 δmax =
𝑃 𝐿 𝑜
3
48 𝐸 𝐼
(at point of load)
 δx =
𝑃 𝑥
48 𝐸 𝐼
(3 𝐿 𝑜
2
− 4 𝑥2
) (where x <
𝐿 𝑜
2
; symmetric about midspan)
BIOE 3200 - Fall 2015
x

18. Examples of deflection
formulae
 Cantilever beam loaded at free end
 δmax =
𝑃 𝐿3
3 𝐸 𝐼
(at free end)
 δx =
𝑃 𝑥
2
6 𝐸 𝐼
(3 𝐿 - 𝑥)(everywhere else)
BIOE 3200 - Fall 2015
P