Calculating the safe load of a glued structure with given shear stress

[maitake91]

Homework Statement

A laminated wood beam made up of three 2cm*4cm plates glued together forming a rectangular cross section that is 4cm*6cm is given. The allowable shear stress in the glued joints is 5MPa. The beam is 10cm long and simply supported at both ends.
What is the safe load that can be carried at mid-span and what is the corresponding. maximum bending stress?

Relevant Equations

maximum shear stress in a rectangular beam = 1.5(F/A)

I have tried to calculate the safe load with the equation of maximum shear stress, A = 4*6*10^-4, and the given shear stress 5MPa, but I couldn't seem to get the right answer which is 18kN.

 

[256bits]

Quote
maximum shear stress in a rectangular beam = 1.5(F/A)

To where on the beam does the maximum shear stress apply?

 

[maitake91]

Thank you for your reply!

I think it is at x=0 and x=l, therefore the value F I obtain from the equation is P_max/2?
In that case I'm getting 5*10^6*(4*6*10^-4)*(2/3) = 8000, which is not 9000 as the given answer suggests and I don't know why.

 

[256bits]

Thank you for your reply!

I think it is at x=0 and x=l, therefore the value F I obtain from the equation is P_max/2?
In that case I'm getting 5*10^6*(4*6*10^-4)*(2/3) = 8000, which is not 9000 as the given answer suggests and I don't know why.

For your beam and the related moment and shear force diagrams, shear force acts along the whole beam from 0 to L, with the shear force changing direction at the point of application of the load point P in the centre of the beam.

A shear stress is set up on a face of cross section of the beam in the y-direction. Correspondingly, to make the beam statically in equilibrium, also in the x-direction. For a location y from the central axis, a cube of dimensions dx dy dz would have shear on both opposite faces being equal in magnitude.

For your beam, on this face of cross section, where does the maximum shear stress apply?

 

[maitake91]

For your beam and the related moment and shear force diagrams, shear force acts along the whole beam from 0 to L, with the shear force changing direction at the point of application of the load point P in the centre of the beam.

A shear stress is set up on a face of cross section of the beam in the y-direction. Correspondingly, to make the beam statically in equilibrium, also in the x-direction. For a location y from the central axis, a cube of dimensions dx dy dz would have shear on both opposite faces being equal in magnitude.

For your beam, on this face of cross section, where does the maximum shear stress apply?

Thank you very much, I was able to get the correct answer!

I finally realised that the maximum shear stress applies at y=0, but the shear stress provided is for the glued joints at y = 1cm, therefore I can't just use the equation for maximum shear stress.

 

[Chestermiller]

Let's see your complete analysis of the beam.