Beam Deflection

(9.1-9.5)

MAE 314 – Solid Mechanics Yun Jing Beam Deflection

1

Introduction

  Up to now, we have been primarily calculating normal and shear stresses.

In Chapter 9, we will learn how to formulate the deflection curve (also known as the elastic curve ) of a beam.

Beam Deflection 2

Differential Equation of Deflection



d

 

ds d



ds

 1  Recall from Ch. 4 that 1/ρ is the curvature of the beam.

y

θ ds dx dy

dy dx

 tan 

dx

 cos 

ds dy

 sin 

ds

Slope of the deflection curve Beam Deflection 3

Assumptions

 Assumption 1: θ is small.

 1.

ds



dx



d



ds

 1 

d

  2.

dy

 tan     

d



dx dx dx



d

2

y



dx

2 1  

d

2

y dx

2   Assumption 2: Beam is linearly elastic.

1  

M EI

Thus, the differential equation for the deflection curve is:

d

2

y



dx

2

M EI

Beam Deflection 4

Diff. Equations for M, V, and w

 Recall from Ch. 5:

dV dx

 

w dM



V dx

 So we can write:

EI d

2

y dx

2 

M EI d

3

y dx

3 

V EI d

4

y dx

4  

w

 Deflection curve can be found by integrating    Bending moment equation ( 2 constants of integration) Shear-force equation ( 3 constants of integration) Load equation ( 4 constants of integration)  Chosen method depends on which is more convenient.

Beam Deflection 5

Boundary Conditions

   Sometimes a single equation is sufficient for the entire length of the beam, sometimes it must be divided into sections.

Since we integrate twice there will be two each section .

constants of integration for These can be solved using boundary conditions .

 Deflections and slopes at supports  Known moment and shear conditions  

V B M B

  0 0   

M A

 0  Beam Deflection 

M B

 0  6

Boundary Conditions

  Continuity conditions: Section AC: y AC (x)   Displacement continuity

y AC

(

C

) 

y CB

(

C

) Slope continuity

dy AC

(

C

)  

AC

(

C

)

dx

Symmetry conditions: 

dy CB dx

(

C

)  

CB

(

C

) Section CB: y CB (x)

dy dx

 0 Beam Deflection 7

Example Problem

For the beam and loading shown, (a) express the magnitude and location of the maximum deflection in terms of w 0 , L, E, and I, (b) Calculate the value of the maximum deflection, assuming that beam AB is a W18 x 50 rolled shape and that w 0 = 4.5 kips/ft, L = 18 ft, and E = 29 x 10 6 psi.

Beam Deflection 8

Statically Indeterminate Beams

 

x=0, y=0 x=0, θ =0

When there are more reactions than can be solved using statics, the beam is indeterminate.

Take advantage of boundary conditions to solve indeterminate problems.

x=L, y=0

Problem: Number of reactions: 3 (M A , A y , B y ) Number of equations: 2 (Σ M = 0, Σ F y = 0) One too many reactions!

Additionally, if we solve for the deflection curve, we will have two constants of integration, which adds two more unknowns!

Solution: Boundary conditions Beam Deflection 9

Statically Indeterminate Beams

x=0, y=0 x=0, θ =0 x=L, y=0 x=0, θ=0

Problem: Number of reactions: 4 (M A , A y , M B , B y ) Number of equations: 2 (Σ M = 0, Σ F y = 0) + 2 constants of integration Solution: Boundary conditions Beam Deflection 10

Example Problem

For the beam and loading shown, determine the reaction at the roller support.

Beam Deflection 11

Beam Deflection: Method of Superposition

(9.7-9.8)

MAE 314 – Solid Mechanics Yun Jing Beam Deflection: Method of Superposition

12

Method of Superposition

 Deflection and slope of a beam produced by multiple loads acting simultaneously can be found by superposing the deflections produced by the same loads acting separately.

 Reference Appendix D (Beam Deflections and Slopes)  Method of superposition can be applied to statically determinate and statically indeterminate beams.

Beam Deflection: Method of Superposition 13

Superposition

 Consider sample problem 9.9 in text.

 Find reactions at A and C.

 Method 1: Choose M C redundant.

and R C as  Method 2: Choose M C and M A as redundant.

Beam Deflection: Method of Superposition 14

Example Problem

For the beam and loading shown, determine (a) the deflection at C, (b) the slope at A Beam Deflection: Method of Superposition 15

Example Problem

For the beam and loading shown, determine (a) the deflection at C, and (b) the slope at end A.

Beam Deflection: Method of Superposition 16

Example Problem

For the beam shown, determine the reaction at B.

Beam Deflection: Method of Superposition 17