Stress resultants are simplified representations of the
stress
Stress may refer to:
Science and medicine
* Stress (biology), an organism's response to a stressor such as an environmental condition
* Stress (linguistics), relative emphasis or prominence given to a syllable in a word, or to a word in a phrase ...
state in
structural element Structural elements are used in structural analysis to split a complex structure into simple elements. Within a structure, an element cannot be broken down (decomposed) into parts of different kinds (e.g., beam or column).
Structural elements can ...
s such as
beams,
plates, or
shells.
The geometry of typical structural elements allows the internal stress state to be simplified because of the existence of a "thickness'" direction in which the size of the element is much smaller than in other directions. As a consequence the three
traction components that vary from point to point in a cross-section can be replaced with a set of
resultant force
In physics and engineering, a resultant force is the single force and associated torque obtained by combining a system of forces and torques acting on a rigid body via vector addition. The defining feature of a resultant force, or resultant for ...
s and resultant moments. These are the stress resultants (also called ''
membrane forces'', ''
shear forces'', and ''
bending moment
In solid mechanics, a bending moment is the reaction induced in a structural element when an external force or moment is applied to the element, causing the element to bend. The most common or simplest structural element subjected to bending mo ...
'') that may be used to determine the detailed stress state in the structural element. A three-dimensional problem can then be reduced to a one-dimensional problem (for beams) or a two-dimensional problem (for plates and shells).
Stress resultants are defined as integrals of stress over the thickness of a structural element. The integrals are weighted by integer powers the thickness coordinate ''z'' (or ''x''
3). Stress resultants are so defined to represent the effect of stress as a membrane force ''N'' (zero power in ''z''), bending moment ''M'' (power 1) on a beam or
shell (structure)
A shell is a type of structural element which is characterized by its geometry, being a three-dimensional solid whose thickness is very small when compared with other dimensions, and in structural terms, by the stress resultants calculated in ...
. Stress resultants are necessary to eliminate the ''z'' dependency of the stress from the equations of the theory of plates and shells.
Stress resultants in beams
Consider the element shown in the adjacent figure. Assume that the thickness direction is ''x''
3. If the element has been extracted from a beam, the width and thickness are comparable in size. Let ''x''
2 be the width direction. Then ''x''
1 is the length direction.
Membrane and shear forces
The resultant force vector due to the traction in the cross-section (''A'') perpendicular to the ''x''
1 axis is
:
where e
1, e
2, e
3 are the unit vectors along ''x''
1, ''x''
2, and ''x''
3, respectively. We define the stress resultants such that
:
where ''N''
11 is the ''membrane force'' and ''V''
2, ''V''
3 are the shear forces. More explicitly, for a beam of height ''t'' and width ''b'',
:
Similarly the shear force resultants are
:
Bending moments
The bending moment vector due to stresses in the cross-section ''A'' perpendicular to the ''x''
1-axis is given by
:
Expanding this expression we have,
:
We can write the bending moment resultant components as
:
Stress resultants in plates and shells
For plates and shells, the ''x''
1 and ''x''
2 dimensions are much larger than the size in the ''x''
3 direction. Integration over the area of cross-section would have to include one of the larger dimensions and would lead to a model that is too simple for practical calculations. For this reason the stresses are only integrated through the thickness and the stress resultants are typically expressed in units of force ''per unit length'' (or moment ''per unit length'') instead of the true force and moment as is the case for beams.
Membrane and shear forces
For plates and shells we have to consider two cross-sections. The first is perpendicular to the ''x''
1 axis and the second is perpendicular to the ''x''
2 axis. Following the same procedure as for beams, and keeping in mind that the resultants are now per unit length, we have
:
We can write the above as
:
where the membrane forces are defined as
:
and the shear forces are defined as
:
Bending moments
For the bending moment resultants, we have
:
where r = ''x''
3 e
3.
Expanding these expressions we have,
:
Define the bending moment resultants such that
:
Then, the bending moment resultants are given by
:
These are the resultants that are often found in the literature but care has to be taken to make sure that the signs are correctly interpreted.
See also
*
Shear force
*
Bending moment
In solid mechanics, a bending moment is the reaction induced in a structural element when an external force or moment is applied to the element, causing the element to bend. The most common or simplest structural element subjected to bending mo ...
*
Plate theory
In continuum mechanics, plate theories are mathematical descriptions of the mechanics of flat plates that draws on the theory of beams. Plates are defined as plane structural elements with a small thickness compared to the planar dimensions ...
*
Bending of plates
Bending of plates, or plate bending, refers to the deflection of a plate perpendicular to the plane of the plate under the action of external forces and moments. The amount of deflection can be determined by solving the differential equations of ...
*
Kirchhoff–Love plate theory
The Kirchhoff–Love theory of plates is a two-dimensional mathematical model that is used to determine the stresses and deformations in thin plates subjected to forces and moments. This theory is an extension of Euler-Bernoulli beam theory and ...
*
Mindlin–Reissner plate theory
The Uflyand-Mindlin theory of vibrating plates is an extension of Kirchhoff–Love plate theory that takes into account shear deformations through-the-thickness of a plate. The theory was proposed in 1948 by Yakov Solomonovich UflyandUflyand, Y ...
*
Vibration of plates
The vibration of plates is a special case of the more general problem of mechanical vibrations. The equations governing the motion of plates are simpler than those for general three-dimensional objects because one of the dimensions of a plate is ...
References
{{DEFAULTSORT:Stress resultants
Continuum mechanics
Mechanics
Solid mechanics
Composite materials