Beam Deflection 1

Beam Deflection (9.1-9.5)

MAE 314 – Solid Mechanics

Yun Jing

Beam Deflection 2

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 3

Differential Equation of Deflection

ydx

dy

ds

θ

tandx

dy

cosds

dx

sinds

dy

dsd

1

ds

d

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

Slope of the deflection curve

Beam Deflection 4

Assumptions Assumption 1: θ is small.

1. 2.

Assumption 2: Beam is linearly elastic. Thus, the differential equation for the deflection curve is:

dxdsdx

d

ds

d

1

tandx

dy

2

2

dx

yd

dx

d2

21

dx

yd

EI

M

1

EI

M

dx

yd

2

2

Beam Deflection 5

Diff. Equations for M, V, and w Recall from Ch. 5: So we can write: 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.

wdx

dV V

dx

dM

wdx

ydEI

4

4

Vdx

ydEI

3

3

Mdx

ydEI

2

2

Beam Deflection 6

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

Deflections and slopes at supports Known moment and shear conditions

0

0

B

B

M

V

0AM 0BM

Beam Deflection 7

Boundary Conditions Continuity conditions:

Displacement continuity Slope continuity

Symmetry conditions:

Section AC: yAC(x) Section CB: yCB(x)

)()( CyCy CBAC

)()()()( CCdx

dyCC

dx

dyCB

CBAC

AC

0dx

dy

Beam Deflection 8

Example ProblemFor the beam and loading shown, (a) express the magnitude and location of the maximum deflection in terms of w0, 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 w0 = 4.5 kips/ft, L = 18 ft, and E = 29 x 106 psi.

Beam Deflection 9

Statically Indeterminate Beams When there are more reactions than can be solved using statics, the beam is indeterminate. Take advantage of boundary conditions to solve indeterminate problems.Problem:Number of reactions: 3 (MA, Ay, By)Number of equations: 2 (Σ M = 0, Σ Fy = 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

x=0, y=0x=0, θ=0

x=L, y=0

Beam Deflection 10

Statically Indeterminate Beams

Problem:Number of reactions: 4 (MA, Ay, MB, By)Number of equations: 2 (Σ M = 0, Σ Fy = 0)+ 2 constants of integrationSolution: Boundary conditions

x=0, y=0x=0, θ=0

x=L, y=0x=0, θ=0

Beam Deflection 11

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

Beam Deflection: Method of Superposition 12

Beam Deflection: Method of Superposition (9.7-9.8)

MAE 314 – Solid Mechanics

Yun Jing

Beam Deflection: Method of Superposition 13

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 14

Superposition Consider sample problem 9.9 in text. Find reactions at A and C. Method 1: Choose MC and RC asredundant.

Method 2: Choose MC and MA as redundant.

Beam Deflection: Method of Superposition 15

Example ProblemFor the beam and loading shown, determine (a) the deflection at C, (b) the slope at A

Beam Deflection: Method of Superposition 16

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

Beam Deflection: Method of Superposition 17

Example ProblemFor the beam shown, determine the reaction at B.