## Appendix D Deflection Of Distributed Beam

Figure D. 1: Beam with distributed load

The distributed beam of Figure D. 1 can be described using the loading equation of (C.1), where q is the load intensity.

Integrating this 4 times give Equations (D.2) - (D.5), introducing the shear stress v, bending moment m, the slope equation dy/dx and the deflection curve y respectively v

El-d- = J M = -—(x - a) + -6\x - b) + C^-— + C2x + C3

Ely =--(x - a) + — (x - b) + Ci— + C2— + C3x + C4

Where

E = Young's modulus of material I = second moment of area of the beam Inserting the boundary condition that y=0 at x=0 and x=L into (D.5) gives (D.6) and (D.7) respectively.

Using the zero bending conidition at the built in ends gives theboundary conditions of dy/dx=0 at x=0 and x=L, substituted into (D.4) gives (D.8) and (D.9) respectively.

Re-arranging and equating (D.7) and (D.9) gives (D.10), resulting in (D.11) w ( 4 ^ 2CjL w /, 3 3] 3CjL

Substitution of (D.11) into (D.7) and re-arranging gives (D.12)

Maximum extension will be in the centre, x=L/2, substituted into (D.5) gives (D.13)

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