Semester 1 Flashcards
Variable Actions
Environmental actions
Hydrostatic actions
variable actions are Live loads associated with use of the building eg people and furniture
whereas environmental actions are accidental actions eg impacts or explosions
hydrostatic actions are pressure exerted by a fluid (or soil) at equilibrium
How does lateral pressure change in relation to depth?
LP increases in proportion to depth measured from surface due to the increasing weight of the retained material
What is a FBD?
A free body diagram, used to simplify and help solve a statics problem.
How to draw a FBD (5)
ground
add known forces
show orientation of principle coordinate system
add resistance forces
draw dimensions of structure
Types of load
Unit load
Uniformally distributed load (weight central)
Partially Distributed Load
Varying partially distributed load
how to calculate VPDL
Rectangle with triangle on top. With w1 being smaller side height and w2 being taller side height.
W1 = w1(X2 - X1) W2 = 0.5(W2-W1)(X2-X1)
External Supports and how many resistance forces
Rollers (1 VR) Pins (2R, V+H) Encastre (3R, V+H+M) Link Beam to column joint Connections in trusses
Link (external support)
Allows rotations and translation perpendicular to the direction of the link.
Internal Pins
Internal pin - only allows rotation
Internal roller - same as pin but situations between two beams
three types of stress in beams
Tensional - beam being pulled apart
Compressional - beam being pushed together
Shear stress - beam being pushed past each other, eg LHS pushed down RHS pushed upwards
Bending moment def.
The reaction induced in a structural element when an external force or moment is applied, causing it to bend.
Two ways of beam failure under loading.
Shearing the beam across its cross section
Bending by an excessive amount (causing tension in bottom of beam and compression on top)
Bending moments equation
MC = total ( external forces ) x Lever arm
MC = Va x L/2, Va being vertical reaction
How to calculate applied bending moment at any point of a beam
- Cut the beam at point of interest
- Bin, ignore everything to either the left or right of cut.
- One end in, Working from one end in sum moments about the cut
How to find shearing force
Cut at C
consider left or right
(total)V= 0 ( Vertical forces are in equilibrium)
e.g Va - (2 x P1) = 0
Va = 2P1
Sagging or hogging?
Sagging- Compression top, Tension bottom
Hogging - Tension top, Compression bottom
Shear force diagrams
Calculate shear force throughout beam and plot on graph, downwards forces causing sagging are positive.
start just after first force, using V = 0, then that force is constant until next applied load.
Bending moments diagram
Multiply shear force by distance to find bending moments
Bending moments known shapes
Concentrated Point load - straight lines w/ changes of direction at point of application of loads
Uniformly Distibuted Loads - Parabolic curves
Statically determinate structure
If a structure has less than or equal to 3 unknown reactions it can be analysed using equations of equilibrium and is externally statically determinate
if a structure is externally redundant/ externally statically indeterminate, we can’t yet analyse these structures
what does m + r = 2j tell us in pin jointed frames
- minimum number of constraints, yet doesn’t show how to configure them
- necessary condition for a statically determinate structure but it’s not sufficient
- there’s a need to check the reactions restrain the rigid body freedoms and that sections of the structure and not mechanisms
Types of trusses (6)
pratt howe warren parker king post (simplest) queen post truss
What do trusses do?
increase load carrying capacity while reducing material consumption. has a light open appearance and many shapes can be made from it.
assumptions w/ trusses (5)
Members are connected at nodes (ends) only
Connected by frictionless pins (don’t resist moments)
Pin jointed truss structure loaded only at nodes
The weight of the member may be neglected
All bars are two forces members, weight of members are neglected and members work in either tension or compression
what m + r >/= 2j tells us
m + r < 2j, statically unstable
m + r > 2j, statically indeterminate but stable
m + r = 2j, statically determinate and stable
what do the constants mean in m + r = 2j
m = members (lines) r = support reactions j = nodes (corners)
Method by section steps (4)
- check all equilibrium equations
- Cut the truss through members where forces are to be found.
- Draw the FBD of that part of the truss which has fewest unknowns
- write equilibrium equation to solve for unknown forces
Method of joints steps (5)
- Draw FBD of whole pin jointed frame
- Apply equations of equilibrium
- Determine the reactions
- Draw FBD of EACH JOINT in turn
- Determine the unknown forces
St Venant’s Principle
if self equilibriated loads are applied on one end of a long cylinder, the strain produced in the body are much larger near the loaded end than at points further away.
Assumptions made when developing the expression for direct stress
- Material has continuous and solid structure
- Material is homogeneous and isotropic
- There are no internal forces prior to loading
- The effect of a system of forces acting on the body is equal to the sum of the effects of those forces applied in succession and in any order.
- St Venants Principle applies
Tensile strain equation
change in L/L
Difference between brittle and ductile materials
Brittle materials like glass and concrete fail without yielding
Ductile materials like plastic or steel fail after yielding
Strain and stress sign conventions
tensile stresses and strains (positive) produce increase in length
compressive stresses and strains (negative) produce decrease in length
Poissons ratio
Elongation in the x axis of an object regulars in a reduction in y axis
Stress equation
applied force / area of cross section
centroid of section algebraic equation
y(bar) = [(int)ydA]/A
same for x bar replaying y with x
Centroid of standard rectangle
Middle
x(bar)= b/2
y(bar) = h/2
centroid of a Standard iscocoles triangle
x(bar) = b/2 y(bar)= h/3
centroid of standard scalene triangle w no lines of symmetry
x(bar) = (x coord of point) + b/3 y(bar) = h/3
centroid of standard circle
x(bar) = y(bar)= d/2
Second moment of area for rectangular section
I = (bh^3)/12
Second moment of area for a standard circle
I(x or y) = (pi.r^4)/4
standard second moment of area for a square
I (x or y) = (a^4)/12
Standard second moment of area for a hollow circle
I = (pi(d^4-d1^4))/64
radius of curvature 3 main equations
Length of AB before bending
= R x thetre
Length of AB after bending
= (R+y)thetre
Change in length
= y x thetre
R being distance between the neutral axis and the where the extended lines cross over
y being distance between bottom of beam and neutral axis
stress equations
weird 0 with flick is the symbol
= E x e (e bring strain) (E being modulus of elasticity for the material)
= F/A
Tension and Compression equation
since T = C
T = C = ((stress max)bd)/4
moment of resistance equation for rectangular section
M = (stress max x b x d^2)/6
Moment of resistance definition and it’s unit
the internal moment provides the beams resistance to the external bending moment resulting from the external load
measured in KN/m
engineers theory of bending equation? idk
M/I = stress/y
M being moment of resistance
I being second moment of area about Na
stress being in any fibre at distance y from NA
what 3 characteristics can be altered to increase resistance to shear and bending
shape
size
material
why do we find the centroid and what are the symbols
it’s the centre of mass/gravity, the neutral axis passes through it
how to find centroid of section for t and i section
split section into multiple sections
use a table to input the formula
y/x(bar) = (a1x1 + a2x2 + a3x3)/(a1+a2+a3) use y instead of x for y bar
neutral axis equations
Ixx = (integral of) y^2 da
Iyy = (integral of) x^2 da
which second moment of area/ inertia is the neutral axis for a section with two axis of symmetry
if the applied bending moments is in the vertical axis Ina = Ixx
if the applied bending moment is in the horizontal axis Ina = Iyy
which second moment of area is on the section with only one axis of symmetry
it’s perpendicular to the axis of symmetry and runs through the centroid
when do i use parallel axis theorem?
when it isn’t a standard shape
what is parallel axis theorem
when there’s a vertical line of symmetry
ixx = icc + Ah^2
icc being the second moment of area for the part, ixx being the second moment of area for the whole
what is the elastic section modulus
direct measure of strength in the beam
elastic section modulus equation for any non rectangular section
z = I/ymax
i being inertia (second moment of area)
ymax being distance from NA to top/bottom
elastic section modulus for rectangular section equation
Zyy = (bd^2)/6
Moment of resistance for non rectangular equation
M/I = stress/y
Moments of resistance equation with elastic section of modulus subbed in
M = stress max x Z
Moment of resistance equation for rectangular section
M = (stress max x b x d^2)/6
maximum value for bending moments for simply supported beam with distributed load
(wl^2)/8
w being force downwards
l being length of beam
