G.—ENGINEERING 115 
in a lamina of the part under consideration is usually considered, and 
in it the distribution of the stress follows the ‘ trapezium law,’ which is a 
particular case of Galileo’s solution of the beam problem. Thus if the 
line representing the centre of action of the load or thrust is on the centre 
of the section of the member, the stress intensity would be the same 
throughout the section. If the line of action is off the centre, then the 
intensity is increased on the side towards which the line has moved. 
The diagram representing the distribution of stress is a trapezium, the 
centre of gravity of which is on the line of action. 
In a pier or buttress which supports and at the same time resists the 
thrust of an arch, the line representing the resultant of the weight and 
thrust of the arch is deflected downwards by the weight of the buttress, 
and the buttress may be so shaped that the deflected line is everywhere 
near the centre giving a uniform intensity of stress in the masonry, and 
uniform pressure on the ground below the foundations. On the other 
hand, the balance may not be so good, and the line may be towards the 
outer side of the buttress, giving high concentration in the masonry and 
ground. 
The maximum intensity of stress in the masonry compared with the 
stress which will crush the particular material is a measure of the stability ; 
similarly with respect to the natural formation below the foundation, the 
comparison is between the maximum pressure it is considered capable 
of carrying, and the highest pressure with which the masonry bears 
on it. 
If the resulting line of action is anywhere outside the boundary of the 
part, then the part would be without stability unless the material of which 
it is composed were capable of withstanding tensile stresses. 
The tracing of the position of the line of action of the thrusts and loads 
in an arch, which must be kept well within its thickness, is the basis of the 
design and measure of the arch’s stability. 
Historically the problem of the masonry arch is extremely interesting. 
The arch form of construction has been known for thousands of years, 
and several magnificent arches built by the Romans are still in a very 
good state. Although the arch is a form of construction very generally 
used, their occasional failure in the past has kept alive a feeling of un- 
certainty, if not of mystery, as to their strength and stability. Real 
progress in the theory of the design and strength of the arch is com- 
paratively recent. Thus in 1870 the late Prof. W. C. Unwin, one of our 
greatest students of engineering construction wrote * :— 
«_. , itis but recently that a theory of the strength of arches has even seemed 
possible, and the theory has not yet been so far developed as to be applicable 
for practical purposes to the complex conditions of masonry bridges. 
Hence, in dealing with arched structures, the engineer is compelled to 
proceed in a manner not rigidly scientific. He adopts assumptions not in 
strict agreement with the nature of the materials he is using, if only such 
assumptions permit him to form a theory embracing the most essential 
circumstances of the problem, and if any error so introduced either favours 
1 Chatham Lectures. 
