ES196 Engineering Structures Flashcards
Reaction forces
Develop when an external force is applied to a body
If two forces have intersecting lines of action
They combine as vectors and act at the point of intersection
Principle of transmissibility
If a force is translated along its line of action - size of moment of force doesn’t change
Convenience of line of action
In practice, force acts at a point, but line of action allows force to be translated without affecting moment
Moment of a couple
Force x distance between points of application of force
(independent of reference point)
(r2-r1)xP
What causes an internal force
An external force haha!
Types of loads on a beam
Concentrated load (at a point)
Distributed load N/m
Reaction forces (forces holding beam up)
The moment of a force
describes the tendency of a rigid
body to rotate and is characterised by a force, a centre of
rotation and the perpendicular distance between centre of
rotation and point of application of the force
Static and Kinetic friction
Static friction is the friction present between two or more objects that are not moving with respect to each other. Kinetic friction is the friction present between two or more objects that are in motion with respect to each other.
Fs ≤ μsN
Fd = μkN
Usually μk is less than μs
https://www.youtube.com/watch?v=RIBeeW1DSZg last q
Fluid definition
A fluid is any continuous
substance which, when at
rest, exerts only normal
forces on its boundary
surfaces
Fluids may be either
liquids (hydrostatic
pressure) or gases (aerostatic pressure)
Types of loads
Dead - self weight
Live - people, cars, moveable furniture (assumed by designer)
Environmental - Loads that act due to weather/natural stuff
Types of support
Roller support - 1 unknown, orthogonal to surface its on (allows movement parallel to surface + rotation)
Pin joint - 2 unknowns, x and y components
(allow for rotation, no translation)
Fixed support - x,y and reaction moment
(no translations/rotations)
If a certain degree of freedom is restrained at a support
There will be a corresponding reaction force/moment
Why trusses are more efficient than beams (qualitative)
All material is being used close to failure stress, however beams act in bending and a lot of material is not under stress
Compression vs tension members
Compressive members break easier esp when long
Conditions for a truss system to not collapse
All connections between members must be “pins”, that is connections that allow members to
rotate relative to each other. If this is not the case, bending may occur in members
* All loads on a truss must be applied at the “nodes” where members meet. Again, if this is not
the case members will be in bending and inefficiently used.
* Trusses must consist of a number of triangles
In triangles, you cannot change the angle between pins (deflection) without changing the length of a side.
Deriving performance index
- Don’t want objective to depend on design properties (free variables), only materials
Relationship between sf and bm diagrams
dv/dx = -w where w is distributed load
So change in v = area under loading diagram
DM/dx = V
Change in M = area under sf diagram
Warren truss
Symmetrical
Bending moment at a pin joint
Zero bending moment - as rotation isn’t restricted, so there will be no moment reaction
Zero force members
3 members connected at a joint, with two aligned. Only one member has component in vertical direction, so vertical must carry no force.
Other when two members at a joint, and aren’t aligned, both must be zero force
True when no external load on members
Can remove zero force members from example
Used to prevent buckling of longer members, and unexpected loads won’t cause structures to fail
Condition for principle of superposition of forces for beams
Linear system (in linear elastic range)
WITHIN ELASTIC LIMIT
Lever rule
a1F1 = a2F2
Where a is respective distance to force on beam
Equilibrium conditions for body in 3d
6
Quick way to check statically indeterminacy
If a reaction can be removed without allowing any rigid body motion/mechanisms
Number of bars and joints in statically determinante planar truss
B=2j-3
Why there is no bm at pin and roller joints
Rotation is permitted
3 point bending
Simply supported beam w vertical force in middle
Characteristic strength
Maximum structural capacity
Types of stress
Direct stress σ (axial force and bending moment)
Perpendicular to cross section
Shortening/elongation
Shear stress τ (shear force and torsion)
Parallel to cross section
Distortion
Longitudinal line for beam
Neutral plane - remains at original length, no tension/compression
In positive bending, compression on top, tension under. Direct stress varies linearly above/below line
Neutral axis goes through com
Elastic theory
Objects deforms when a pair of forces are applied, if material is elastic it’ll return to its original shape when the forces are removed
Perfect elasticity is an approximation
Few materials remain purely elastic even after small deformations
Shear modulus/modulus of rigidity (G)
Object tendency to shear (deformation at constant volume) defined as shear stress/shear strain
Homogeneous and isotropic materials
Homogeneous - composition is uniform throughout
Isotropic - how material responds when certain force is applied (response changing depending on direction of load)
Homogeneous and isotropic - steel
Wood - heterogeneous and anisotropic
Concrete - heterogenous, isotropic
Nominal/engineering stress and strain
Stress is force over original area
Not true stress as area changes
Stress delta over og
Not true strain as “og” is changing
Nominal/engineering stress and strain
Stress is force over original area
Not true stress as area changes
Stress delta over og
Not true strain as “og” is changing
Stress and strain
In same direction
Modulus of rigidity
E/2(1+v) poissons ratio (measure of lateral deformation)
Yielding, strain hardening, necking
- sudden change in length at constant stress
- disproportionality
- cross section decreases then breaks
Ductility
Material that can be subjected to large strains before fracture
Percent elongation Lf-L0/L0 . 100
Perfect reduction in area A0-… U get
Brittle
Material that exhibits little/no yielding before failure
Isotropic Vs anisotropic materials
Showing physical behaviour (deformation) that doesn’t change depending on direction of load
Longitudinal Vs lateral strains
Longitudinal in the same direction as the load, lateral perpendicular
Poissons ratio
Negative ratio of lateral/longitudinal strain
Within elastic range of material this is constant
Unique for a homogeneous and isotropic material
Materials used in engineering practice 0≤v≤0.5
Assumptions in beams
Material is linear elastic homogenous and isotropic
Cross sections which are plane and normal to longitudinal axis remain so after deformation
Beam deflections are small
Beam is long and slender
length»width n depth
Structural rigidity
Resistance offered by a beam while undergoing deformation (EI)
Neutral plane
Goes through longitudinal axis
Moment of inertia/second moment of area
Measures an objects resistance to bending
(about a given axis) neutral axis in subscript
Rectangular cross section bh^3/12
b is in same direction of neutral axis
units length^4
Qualitative deduction of higher moi
Greater material further from bending axis - better at resisting bending motion - higher moi
Area moi reflects how area of a cross section is distributed relative to an axis
always positive due to square
can be added and subtracted e.g. rectangle w circular hole take I for rectangle minus I for hole
Kernel of a cross section
The area, within which a normal compressive
force will not produce tension in the cross-section, is called the kernel of the
cross-section
Relation between longitudinal axis and vertical plane along beam
Also any deformation of cross section in own plane due to bending moment is neglected
Always 90⁰
Why must a beam be adequately supported
To permit transfer of loads to supports
Neglections in bernouilli beam theory
Shear deformations
assumption that the shear
force Q(x) does not contribute directly to the deformation of the beam
deflections small and elastic
applies to slim beams (l/d>20)
Beam support conditons
(bernuly eq)
Free - M=0 V=0
simple supports M=0 Y=0
Fixed Y=0 θ=0
The beam must be supported by at least the number of constraints that
prevent free motion. However, many structures have more supports than
strictly needed to fix the structure in the plane or in space. Extra supports and
extra connections between structural elements typically increase the stiffness
of the structure, while rendering the structure statically indeterminate.
Relating rotation and vertical deflection
θ=-dy/dx
What allows for cross section to remain tangent to elastic line at a point
Shear stresses I think
What allows for cross section to remain tangent to elastic line at a point
Shear stresses I think
Values of shear in beam bending (wooden block model)
0 at top and bottom, non zero in middle
Failure in torsion ductile Vs brittle
Ductile tends to fail perpendicular to longitudinal axis, as they tend to fail in shear, so fracture along plane of maximum shear stress.
Brittle materials fail along 45 degrees. Brittle materials tend to fail along maximum tensile stress. Weaker in tension than in shear
