Chapter 12 – Beam Deflection

Notice how deflection diagram (or elastic curve) is different for each beam. How does the support effect slope and deflection?

12.1 The Elastic Curve:

The moment diagram and sign can be a great aid in constructing the elastic curve!

Zero displacement and zero slope

Zero displacement

Displacement is max either where slope is zero or at end!!

Moment-Curvature Relationship (read 588 – 592):

Define x,

IE

M

1

Radius of curvature

Internal moment

Modulus of elasticity

Moment of inertia

IE

M

1

2

21

dx

d

Therefore:

EI

M

dx

d

2

2

EI = Flexural Rigidity assume constant throughout beam length

12.2 – Slope and Displacement by Integration!

)(4

4

xwdx

dEI

)(3

3

xVdx

dEI

Or

Or

)(2

2

xMdx

dEI

Any of these DE can be used to get (x). Just integrate and evaluate constants of integration using BC’s. This is called the integration method for getting beam displacement!

OR:

Careful! Need M(x) for each span (every time eqn for moment changes).

Sign convention – All Positive!!