STABILITY OF STRUCTURES

Stability is more complex and in some manifestations more difficult to measure than  strength and stiffness but can be broadly defined as capacity to resist:

•  Displacement
•  Overturning 
•  Collapse
•  Buckling

Diagrams 1-3 give a theoretical definition;  all the other diagrams illustrate stability of conceptual structures.

1 Unstable
2 Neutral
3 Stable
4  Weak stability: high center of gravity, narrow base
5  Strong stability: low center of gravity, broad base
6  Unstable post and beam portal
7   Stable moment frame
8  Unstable T-frame with pin joint at base
9  Stable twin T-frames

 
Buckling stability 

Buckling stability is more complex to measure than strength and stiffness and largely based on empirical test data.. This introduction of buckling stability is intended to give only a qualitative intuitive understanding.

Column buckling is defined as function of slenderness and beam buckling as function of  compactness.  A formula for column buckling was first defined in the 18th century by Swiss mathematician Leonhard Euler.  Today column buckling is largely based on empirical tests which confirmed Euler’s theory for slender columns; though short and stubby columns may crush due to lack of compressive strength.

Beam buckling is based on empirical test defined by compactness, a quality similar to column slenderness.

1  Slender column buckles in direction of least dimension
2  Square column resist buckling equally in both directions
3  Blocking resists buckling about least dimension
4  Long and slender wood joist subject to buckling
5  blocking resists buckling of wood joist
6  Web buckling of steel beam
7  Stiffener plates resist web buckling

A  Blocking of wood stud
B  Blocking of wood joist
C  Stiffener plate welded to web
P Load

Buildings: Thermal examples

1 Curtain wall
Assume: 

Aluminum curtain wall, find required expansion joint

Thermal strain

2  High-rise building, differential expansion
Assume:

Steel columns exposed to outside temperature

Differential expansion

3  Masonry expansion joints 
(masonry expansion joints should be at maximum L = 100’)
Assume

STRUCTURES - STRAIN EXAMPLES

1 Elevator cables
Assume 

Elongation under load

2 Suspended building
3 Differential strain
Assume

4 Shorten hangers under DL to reduce differential strain, or prestress strands to reduce ∆L by half


Note: Differential strain is additive since both strains are downwards
To limit differential strain, suspended buildings have <= 10 stories / stack

STRUCTURES - PRINCIPLE STRESS

Shear stress in one direction, at 45 degrees acts as tensile and compressive stress, defined as principle stress.  Shear stress is zero in the direction of principle stress, where the normal stress is maximum.  At any direction between maximum principle stress and maximum shear stress, there is a combination of shear stress and normal stress.  The magnitude of shear and principle stress is sometimes required for design of details.  Professor Otto Mohr of Dresden University develop 1895 a graphic method to define the  relationships between shear stress and principle stress, named Mohr’s Circle.  Mohr’s circle is derived in books on mechanics (Popov, 1968).



Isostatic lines

Isostatic lines define the directions of principal stress to visualize the stress trajectories in beams and other elements.  Isostatic lines can be defined by experimentally by photo-elastic model simulation or graphically by Mohr’s circle.

1  Simple beam with a square marked for investigation
2  Free-body of square marked on beam with shear stress arrows
3  Free-body square with shear arrows divided into pairs of equal effect
4  Free-body square with principal stress arrows (resultant shear stress vectors)
5  Free-body square rotated 45 degrees in direction of principal stress
6  Beam with isostatic lines (thick compression lines and thin tension lines)

Note:

Under gravity load beam shear increases from zero at mid-span to maximum at supports. Beam compression and tension, caused by bending stress, increase from zero at both supports to maximum at mid-span.  The isostatic lines reflect this stress pattern; vertical orientation dominated by shear at both supports and horizontal orientation dominated by normal stress at mid-span.  Isostaic lines appear as approximate tension “cables” and compression “arches”.