1. Eng. 6002 Ship Structures 1 Hull Girder Response Analysis
Lecture 8: Review of Beam Theory (deformations)

2. Overview
In this lecture we will look at beam deformations caused by bending moments
We will also review the differential equations of beam theory and solve them using Maple

3. Deformations
In simple beam theory we make the assumption that only the bending moments cause distortions.
So we can assume that the beam is bent into a simple circular curve over any short length.

4. Deformations
The neutral axis (NA) does not stretch or contract. The upper and lower parts of the beam compress and/or stretch.
We can use the two known relationships

5. Deformations
For the top fibre of the figure we see that the strain can be represented by
Substituting, we have:

6. Deformations
Integrating, we have
For most sections EI is constant, so

7. Deflections
To find the deflections, v, we have:

8. Family of Differential Equations
Consider a beam between two supports, we describe the deflections with the variable v(x).
Recall:

9. Family of Differential Equations
Combining these equations gives:
We can describe all responses as derivatives of deflection

10. Family of Differential Equations
We can also express the relationships in integral form to give

11. Solving the Beam Equations
To solve beam equations, we need to know
the beam geometry and properties (L, E, I),
the loading conditions (length, shape and magnitude of load)
the boundary conditions (fixed, pinned, free, guided).
The boundary conditions are described in terms of what is known.
For a fixed end we know that the deflection and rotation are zero.
For a guided end we know that the shear (reaction) and rotation are zero.

12. Solving the Beam Equations
Four standard boundary conditions are shown
Each of the boundary conditions results in the deflection or one of its derivatives being equal to zero.