Wall Reaction Forces
Snow Loading acting Through Top Purlins
Free Body Diagram of Roof Truss
with Snow Load Transfers to Truss
through Purlins at A, B and C.


Since the structure and loading is symmetrical, joints A and B will be the
same. This means, the reaction force from the wall at both joints will be equal,
or
F_{A} = F_{B} = (50 lb/ft^{2})
(2 ft) (6 ft) / 2 = 300 lb
Note that the truss spacing is 2 ft, so each truss must support a
2 ft wide section of roof loading. Also, only the vertical load is considered (50 lb/ft^{2}) over the span of 6 ft.
Next, the snow loading needs to be analyzed and replaced by equivalent
force loads at the joints. The snow load on the plywood
is supported by two purlins on each side of the roof. These purlins (one at the
top and the bottom of each side) hold the trusses in place and transfer the roof
load to the truss joints. The top purlins on each side of the roof will transfer
only half of the roof load. The two bottom purlins will transfer the other half
of the snow load.
The load at the top center is
F_{C} = (50 lb/ft^{2})(2
ft)(1.5 ft + 1.5 ft) = 300 lb
Half the load on each side of
the roof is taken by the center purlins, so the center 3 ft width section
of the roof is supported by the center
joint. The remaining snow load on the roof is transferred to joints
A and B.
The roof angle is
θ = tan^{1} (1/3) = 18.43^{o}

Free Body Diagram of Joint C


Before the bearing and shear stress at joint B can be determined, the
force in member CB must be determined. This can be done by analyzing the top
joint, C using static equilibrium equations,
ΣF_{x}
= 0
F_{CA} = F_{CB}
ΣF_{y} =
0
2 F_{CB} sin18.43  300 = 0
F_{CB} = 474.3 lb


The shear stress in the dowel pin at joint B is determined from the shear stress equation,
τ = V/A
= 474 / ((1/4)^{2}π)
= 474 lb/0.1963 in^{2}
= 2,414 PSI
The load in member F_{AB} ( = 474 cos18.4 = 450 lb) will also cause
a shear stress in the dowel, but since the load is less than the load in member
F_{CB},
the shear stress will also be lower.

Bearing Stress in Member CB at Joint B


The load in member AB is transferred into joint B through the dowel. This causes a bearing stress on the surface of the dowel against the member. The bearing stress is found by
σ_{b} = P/A_{b}
= 474 / ((3/4)(1/2)) = 474 lb/0.375 in^{2}
= 1,265 PSI 