Chapter 5 Flex, sag and wobble: stiffnesslimited design

1
Chapter 5Flex, sag and wobble stiffness-limited
design
2

• Figure 5.1 (a) A tie with a cross-section A
  loaded in tension. Its stiffness is S F/d. (b)
  A beam of rectangular cross-section loaded in
  bending. The stress ? varies linearly from
  tension to compression, changing sign at the
  neutral axis, resulting in a
• bending moment M. (c) A shaft of circular
  cross-section loaded in torsion.

3
Although tensile elasticity is the simplest case
to think about, it is much more common to be
concerned with the response to bending or
torsion. In a bent beam, such as a floor joist
or an axle, there is a tensile side, a
compressive side and a neutral axis with no
stress. In 3-point bending the maximum tensile
stress, and usual breaking point, is in the
middle of the tensile side. This can be bad if
the material is variable as the measured strength
will also be variable. In 4-point bending, if
the loads and support points are right, there is
a larger area of high stress on the top surface
and variations average out. From the other
notes, you will see that the breaking load
depends on the shape of the cross-section rod,
cylinder, bar etc. not just on the cross-section
area. This is described by the moment of the
area, I.
4

• Figure 5.2 Cross-section area and second moments
  of sections for four section shapes.

5

• Figure 5.3 Elastic deflection of beams. The
  deflection d of a span L under a force F depends
  on the flexural stiffness EI of the cross-section
  and the way the force is distributed. The
  constant C1 is defined in equation (5.5) for
  stiffness of the beam defined as load/deflection.

6

• Figure 5.4 Elastic torsion of circular shafts.
  The stress in the shaft and the twist per unit
• length depend on the torque T and the torsional
  rigidity GK.
• Torsion is important for automobile drive shafts
  and any other motor-driven turning shaft or any
  screw or bolt. Most of the stress is at the
  outer surface.

7

• Figure 5.7 (a) A tie with cross-section area A,
  loaded in tension. Its stiffness is S F/d where
  F is the load and d is the extension. (b) A panel
  loaded in bending. Its stiffness is S F/d,
  where F is the total load and d is the bending
  deflection. (c) A beam of
• square section, loaded in bending. Its stiffness
  is S F/d, where F is the load and d is the
  bending deflection.

8
In design, we need to decide what is fixed
(constraints) and what we can vary (variables).
The objective is then what we seek to optimize
with the variables. A first approach to any
problem is to decide on the function(s),
constraints, variables and objective(s).
The optimum material will have the lowest weight
for a given stiffness. This corresponds to a
maximum value of stiffness/density, E/?
9
The optimum material will have the lowest weight
for a given stiffness, rewrite the bending
equation for a fixed stiffness, width and length
but variable thickness, t. Lht can be replaced by
Mass/? volume. The highest stiffness to weight
ratio occurs at a maximum of Mp
10
The optimum material will have the lowest weight
for a given stiffness, rewrite the bending
equation for a fixed stiffness and length. For a
square beam, bh and Lh2 can be replaced by
Mass/? volume. The highest stiffness to weight
ratio occurs at a maximum of Mb
11

• Figure 5.10 A schematic Er chart showing
  guidelines for three material indices for stiff,
• lightweight structures. The best will be
  furthest above the line of slope corresponding to
  the application (tension, beam or sheet).

12

• Figure 5.11 A schematic Er chart showing a grid
  of lines for the index E1/3/r. The units are
• (GPa)1/3/(Mg/m3).

13

• Figure 5.9 A schematic ERelative cost chart
  showing a lower limit for E and an upper one for
  Relative cost.

14

• Figure 5.12 Computer-aided selection using the
  CES software. The schematic shows the three types
  of selection window. They can be used in any
  order and any combination. The selection engine
  isolates the subset of materials that pass all
• the selection stages.

15

• Figure 5.15 The materials of a building are
  chosen to perform three different roles. Those
  for the structure are chosen to carry loads.
  Those for the cladding provide protection from
  the environment. Those for the interior control
  heat, light and sound. Here we explore structural
  materials.

16
The cost C of the beam is just its mass, m,
times the cost per kg, Cm, of the material of
which it is made C mCm AL?Cm The index for
a stiff beam of minimum cost
17

• Figure 5.16 The selection of materials for stiff
  floor beams. The objective is to make them as
  cheap as possible while meeting a stiffness
  constraint.

18

• Figure 5.13 The corkscrew lever from Chapter 3.
  It must be adequately stiff and, for traveling,
  as light as possible.

19
(No Transcript)
20

• Figure 5.14 Selection of materials for the lever.
  The objective is to make it as light as possible
  while meeting a stiffness constraint.