Mathematical Modelling (Statics) Flashcards
what are the two main fields in mechanics
fluid mechanics
solid mechanics
Fluid mechanics Example: (2)
Pressure around an airfoil.
Cardiovascular deseases
Two main types of Mechanics:
Static and dynamics
Static mechanics
Statics: loading and reactions are
independent of the time; nothing
moves
Dynamic mechanics
Dynamics: loading and reactions
depend on time.; the objects are
moving
Mass Point
Mass point: without any physical
extension but with mass.
Rigid Body
Rigid body: consists of several
masses but is undeformable; can
undergo rotation and translation
Single Force
Single force: a load, which acts on
a single point of an object.
Statics example:
Dynamics example:
Statics example:
Vertical deflection of the cathedral
in Strasbourg under gravity
Dynamics example:
Vibration of the steering wheel.
Sometimes so-called quasi-statics
are analysed, explain this condition:
The load velocity
is very small such that it can be
regarded as independent of time
For dynamic cases can failure occur, if so why?
For dynamic cases, failure may
occur even though no high loadings
are applied.
E.g. due to fatigue or excitation at
the natural frequency.
A force is determined by… (3)
- Magnitude;
- Direction;
- Point of action.
How can you distinguish between a scalar and vector quantity when representing them as letter
Remark: bold letters represent vectors or matrices while normal letters stand for scalar quantities. Vectors are also represented by letters with an arrow written over it.
Moving a force along its line of action does what?
Moving a force along its line of
action does not change the effect.
Scalar
A scalar is a positive or negative number; physical quantities described by scalars are for example: mass, volume, energy and temperature.
Vector
A vector is defined by direction, magnitude, direction and sense; it can be depicted by an arrow where the length of the arrow indicates the magnitude.
Forces, moments, displacements, velocity, acceleration are described via vectors.
Operations on Vectors
-Multiple or Divide vectors by scalars
-Add vectors together to get a resultant vector
-Subtract vectors
It’s a special case of addition (First multiply the vector
B by the scalar (-1) and then add to vector A)
Vectors representation
A vector can be decomposed into its
components in direction of the
coordinate axis:
a = ax + ay
* These components are mathematically expressed by the product of a scalar and the unit vector.
Components of a Vector (3D)
How would you find the magnitude?
In three dimensions, the vector F is
composed of three components:
F = Fx + Fy + Fz
Use Pythagoras to work out magnitude, by squaring each force, sum, then square root.
The order of the coordinates x-y-z is
given by…
The order of the coordinates x-y-z is
given by the “right-hand rule”
Thumb is x axis
Newton’s 3rd law states that:
Newton’s 3rd law states that:
“To every action (force exerted)
there is an equal and opposite
reaction”, which is:
A free body diagram means: (3)
A free body diagram means:
1. A closed cut around the object
- At each point where the object
was separated, reaction forces
(and eventually moments)
have to be inserted - The object should be totally
isolated.
To analyse a problem in mechanics you must… (2)
To analyze the problem, a plan of
location and a plan of forces has to
be drawn.
Moment
The Moment of a force is a measure of its tendency to cause a body to rotate about a specific point or
axis. In order for a moment to develop, the force must act upon the body in such a manner that the
body would begin to twist.
Varignon’s Theorem:
Moment of a force about any point is equal to the sum of the moments of the components of the force about the same point.
Moments
The Cross Product
M = r x F = (Fr x sin(alpha))u sub(M)
where r is the position vector, Fr sin(alpha) is the magnitude and u sub(M) is a unit vector in the direction of moment axis.
Direction of Moment
- In 3D, the orientation of the axis can be obtained by the “right-hand rule”
(similar to current in physics). The
moment is as well drawn as a “double arrow”.
A positive moment turns always
anti -clockwise.
Thus in 3D, there are three principal
directions for the moments each
rotating about one of the coordinate axis.
Mx;My;Mz
Moment
Couples
Two equal and opposite forces
separated by a distance d produce
moment M of magnitude
M = F(a + d) − Fa = Fd
Force-Couple System
Force-Couple systems: The translation
of a force perpendicular to its direction
creates a moment.
Full Free Body Diagram with Forces and Moments
- Definition: A free body diagram is a
sketch that shows a body “free” from
its surroundings with all the forces
and moments that act on the body. In
other words, “Isolate” the body ! - When isolated, the forces and
moments at the points where the
body was linked to other objects
should be considered. - Referring to Newton’s 3rd principle
“action = reaction”, the forces and
moments at the two sides where the
object was separated from the
environment have the same values
but opposite orientations.
Dot product of vectors
Dot product calculates the sum of the two vectors’ multiplied elements. Dot Product returns a
scalar number as a result
what is the dot product of vectors useful for?
The dot product is useful in calculating the projection of vectors.
Dot product in Python also determines orthogonality and vector decompositions.
Dot Product Equation:
You multiple the a1 with b1 and a2 with b2 and then get the sum
Cross Product of Vectors:
In mathematics, the cross product or vector product (occasionally directed area product, to
emphasize its geometric significance) is a binary operation on two vectors in three dimensional space. The magnitude of the product equals the area of a parallelogram with the
vectors for sides.
Degrees of Freedom: Definition
Any of the minimum number of coordinates required to specify completely the motion of a mechanical system.
Degrees of Freedom: Equation
Free Body Diagram – Procedure (4)
- Draw an outlined shape of the body
Imagines the body to be isolated of “free cut” from its
constrains and connections and sketch its outlined shape - Show all force and couple moments acting on the
body
Identify all the external forces and couple moments
acting on the body. Usually they are: a) applied loadings,
b) reactions occurring at the supports and points of
contact with other bodies, c) the weight of the body. - Label all the loadings and specify their direction
relative to the x and y axis. - Indicate the dimensions of the body necessary for
computing the moments of forces.
If a number of degrees of freedom are restricted (e.g. by an external support), you need to…
If a number of degrees of freedom are restricted (e.g. by an external support), you need
to insert the same number of reactions (forces or moments) to the free body diagram
Equilibrium means…
Equilibrium
means that
nothing is moving
(or is moving at
constant velocity
Internal forces balance out!
A system is in static equilibrium
if…
A system is in static equilibrium
if the effects of all forces and
moments create an overall balance;
This means: the resultant force and
the resultant moment are 0
what is meant by statically indeterminate?
In 2D-problems we have 3 independent equilibrium equations:
Sum Fix = 0; Sum Fiy = 0; Sum Miz = 0
- Hence we can determine three
unknown support reactions for a
structure consisting only of a single
part. If more supports are used, the
structure is called
statically indeterminate
what is meant by statically determinate?
If the number of support reactions
equals the number of equilibrium
equations, the structure is
statically determinate.
Improper Constraints
In some special cases, the
support reactions
a) all meet in one point
b) are all parallel
Then the structure can
move (rotate or translate)
AVOID THESE DESIGNS!
Degrees of Freedom
If a number of degrees of freedom are restricted (e.g. by an external support), you need
to insert the same number of reactions (forces or moments) to the free body diagram
The reactions (forces or moments) are calculated by applying the equilibrium conditions
Non-fixed support
- It can not move vertically
- It can move horizontally
- It can rotate
1 degree of
freedom restricted
= 1 reaction
Pin Connection
- It cannot move horizontally
- It cannot move vertically
- It can rotate
2 degrees of
freedom restricted
= 2 reactions
Clamped/Fixed support
It cannot move horizontally
- It cannot move vertically
- It cannot rotate
3 degrees of
freedom restricted
= 3 reactions
whats the difference between shear and normal force
Shear force is parallel to the cross sectional area, use V(x) to present it.
Normal is perpendicular, denoted by N(x)
Determination of Internal Forces and Moments: Procedure
- Before the body is “cut”, determine
the support reactions. - Pass an imaginary section through
the body, perpendicular to the axis, at
the point where the internal forces are
to be determined. - Draw a FBD including all distributed
loadings, couple moments and
external forces acting on the body, as
well as the internal forces at the cut. - Apply the equations of equilibrium to
calculate the internal forces Fn, Fs, M
The normal force is defined as
positive if it correspond to …;
… is hence related to a
negative normal force;
The normal force is defined as
positive if it correspond to traction;
compression is hence related to a
negative normal force;
What does the positive and negative bending moment result in for a beam
- A positive bending moment results
in sagging, a negative in hogging,
hence the bending moment is
related to the curvature of the
beam
Bending leads to tension and
compression at the different sides of
the beam (top and bottom part of each side),
which result in crumple,
snap and eventually in the
development of a crack
How can you tell if the direction you label a force is the right way or not
You follow the axis (if provided), if not use common sense
If you answer for the force turns out to be -ve then you know to alter the direction of the force.
Why do we need to know internal force and moment? (3)
-They can determine the properties of structures
-They reveal the internal loading conditions of
structures and can predict the failure of structure
-They can calculate the support reaction force and
moments
Positive bending moments will be those that…
Positive bending moments will be those that
put the lower section of the beam into tension
- Sagging moment is positive
- Hogging moment is negative
Torque
(what is it independent to?)
A torque (torsional moment) is a
moment rotating around the axis of
the beam;
The torque is independent from
normal, shear force or bending
moments
Define Stress
Stress is a quantity that describes
the distribution of internal forces
within a body
units Pa
Equation for Stress
Sigma = F / A
F- internal force
A- cross-sectional area
Name 2 types of stress
Tensile stress (+ve)
Compressive stress (-ve)
The engineering stress is defined as …
The engineering stress is defined as force
divided by the initial (perpendicular) area.
* It normally generates a change in volume of
the material which is measured as strain.
* The extension in length is normally related to
reduction in width
Tensile engineering strain
(does it have units)
- The engineering strain is the ratio
between change in dimension (delta l = l-l sub0)
to initial dimension. - Strain is always dimensionless.
Define Stiffness
Stiffness is the extent to which an object resists deformation in response to an applied force.
Define Young’s Modulus
Young’s modulus (Modulus of Elasticity) is a mechanical property that measures the tensile or compressive stiffness of a solid material when the force is applied lengthwise.
It quantifies the relationship between tensile/compressive 𝜎 (force per unit area) and axial strain 𝜀
(proportional deformation) in the linear elastic region of a material.
How to convert from m^2 to mm^2
x10^6
(x10^3)^2
Define Shear Stress
Stress acts parallel to the cross-section
- The shear stress is defined as shear
force divided by the initial (parallel) area. - It normally generates a change in shape
(square to parallelogram) which is
measured as shear strain.
Shear Stress (Equation):
tau = Fs [N/m²] / A0
Shear Stress = shear force / initial (parallel) area
Define Shear Strain (gamma)
- Shear strain corresponds to the
change of angle. - For small strains we have
tan(theta) = approximately theta
Shear Strain (gamma) Equation:
(Delta x) / y = tan(theta)
Shear modulus: (Equation)
G = tau / gamma
Second moment of area (Define):
The second moment of area is
THE characteristic quantity for
bending of beams.
Also named moment of inertia.
The Concept of Second Moment of Area
Consider a plate submerged in a liquid.
The pressure of a liquid at a distance z
below the surface is given by p = gamma x z,
where gamma is the specific weight of the
liquid.
The force on the area dA at that point is:
dF = p x dA.
The moment about the x-axis due to this force is z (dF). The total moment is:
Intergral sub(A) z dF = Integral Sub(A) gamma x z^2 dA = gamma x Integral sub(A) (z^2 x dA).
This sort of integral term also appears in solid mechanics when determining stresses and deflection.
This integral term is referred to as the moment of inertia of the area of the plate about an axis.
The second moment of area is:
- A geometric property of a section;
The second moment of area tells what?
- It tells us how the area of the
section is distributed about a
particular axis
The bending of a beam is mainly
influenced by: (2)
The bending of a beam is mainly
influenced by:
1. Young’s modulus,
2. Moment of inertia;
The 2nd moment of area is normally
calculated with respect to what?
The 2nd moment of area is normally
calculated with respect to the centre
of gravity or the middle axis of the
beam
- Sometimes a mixed 2nd moment of
area is used: I sub(xy) = Integral sub(A) xy dA
The first moments of area are
defined as: (2 equations for the x and y axis)
S sub(x) = Integral Sub(A) y dA
S sub(y) = Integral Sub(A) x dA
The first moments of area are
defined as:
S sub(x) = Integral Sub(A) y dA
S sub(y) = Integral Sub(A) x dA
The second moments of area are
hence obtained by:
I sub(x) = Integral Sub(A) x^2 dA
I sub(y) = Integral Sub(A) y^2 dA
The polar (rotational) second
moment of area is defined by: (equation)
I sub(p) = Integral sub(A) r^2 dA
= I sub(x) + I sub(y)
To calculate the second moment of
area, what process can be
applied
The differential area element can be
expressed as:
d A = l x d z
The equation for the second
moment of area becomes:
I sub(y) = Integral z^2 dA
= Integral z^2 l.dz
And the limits of integration are
from zmin to zmax.
* Attention: Here y, z are used as
coordinates ( the diagram on qmplus had axis z on the y, and y on the x)
Centroid
Centre
CoG
The Parallel-Axis Theorem states
that…
the second moment with
respect to an axis parallel to the
axis through the CoG but within a
distance can be computed as
follows:
The second moment of area with
respect to axis y through the CoG is
I sub(y) = Integral z^2 dA = Integral z^2 L.dz
The second moment with respect to
the translated axis (bar) y is:
I sub (bar y) = I sub(y) + (d sub(z) ^2) x A
where d sub(z) is the perpendicular distance between the two parallel axis
First moment of area
First Moment of Area is area times distance (to some reference line)
S sub(z) = A.d = A.(bar)y
If it is an arbitrary shape, we summarise all the differential elements:
Why do we use tubes or I-beams instead of solid structures?
Because they are related to the second moment of area I
sigma = M sub(e) x y / 𝐼 sub (z)
The farther the area of the section is from the neutral axis, the greater the bending section modulus of the section 𝐼 sub(z).
𝐼𝑧 reflects how its points are distributed with regard to a given axis.
Second Moment of Area (Area moment of inertia) in words
Second Moment of Area is area times distance squared (to some reference line).
