Page 38 - Mechanical Engineer's Data Handbook

STRENGTHS OF MATERIALS 27

3 - - 7 -
1
16 32 768
at wall at prop at load
(not maximum)
sw
16

1
1
- - 0.0054
8 48
at wall at prop 0.57851, from wall
3w
8

I A.3 Continuous beams n Spans:
Apply to each group of three supports to obtain (n - 2)
Most beam problems are concerned with a single span. simultaneous equations which can be solved to give
Where there are two or more spans the solution is the (n - 2) unknown bending moments.
more complicated and the following method is used.
This uses the so-called ‘equation of three moments’ (or Solution :
For cases (2) and (3). If M, and M3 are known (these
Clapeyron’s equation), which is applied to two spans are either zero or due to an overhanging load), then
at a time.
M, can be found. See example.
Clapeyron’s equation of three moments
Symbols used:
M = bending moment “Free EM’ diabram I
L =span
I = second moment of area
A =area of ‘free’ bending moment diagram treating
span as simply supported
%=distance from support to centroid C of A
y=deflections of supports due to loading
(1) General case:
MI LlIIl+2M,(Ll/11 +L,l~z)+M3L,II, = P+4-
6(Aixi/LiIi + A,x,/LzIz)+ 6Eb2IL1 + (YZ -Y~)/LzI
(2) Supports at same level, same I: Resultam BM diagram
Yl =YZ=Y3=’
MIL1 +2M,(L1 +L,)+ M,L, =6(A,x,/L1+ AZxJZ-2)
(usual case)
(3) Free ends, Ml=M3=O: I A4 Bending of thick curved bars
M2(4 + &)=3(A,xJb+ &41,2)
In these the calculation of maximum bending stress is
morecomplex, involving the quantity h2 which is given
for several geometrical shapes. The method is used for
loaded rings and the crane hook.

Yl Y2 Y3