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Dezi et. al (2010)
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Projectile motion
Verified by the CalcTree engineering team on June 27, 2024
Introducing our static analysis calculator for beams: a versatile tool tailored for engineers and students alike. With a capacity to analyze beams comprising up to 5 spans, it efficiently computes reactions, generating precise shear force and flexural moment diagrams.
This page was created using anastruct library.
Calculation
Inputs
Spans
L1
:2 m
L2
:3 m
L3
:3 m
L4
:2 m
L5
:1 mRestraints (nodes)
s1
:0
s2
:1
s3
:1
s4
:0
s5
:1
s6
:2Linear Loads
WL1
:30 kN / m
WL2
:10 kN / m
WL3
:5 kN / m
WL4
:15 kN / m
WL5
:10 kN / mVertical Point Loads
P1
:80.00 kN
P2
:0 kN
P3
:0 kN
P4
:40 kN
P5
:0 kN
P6
:0 kNOutputs
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Maximum shear demand
V_max
:140.00 kN
V_min
:-105.20 kN
Maximum moment demand
M_max
:220.00 kN m
M_min
:-43.09 kN m
Solution charts
Adjust the charts
zoom level
:1.8
font size
:0.2
height to width ratio
:0.6Explanation
Supports
Beams can have three types of idealised support conditions: pin, roller or fixed. Each support condition restrains the beam in different ways, thereby affecting the force development in the beam.
At a point of support, there are a maximum of 6 degrees of freedom, which describe the number of possible movements: 3 translations in x, y, z and 3 rotations about x, y, z.
If a certain degree of freedom is restrained at a support, then there is a corresponding reaction. For example, pinned supports allow rotation, so there will be no moment reaction. But since pinned supports restrict translation in both the x and y direction, there will be a reaction in both of those directions.
Internal Forces
When a beam is loaded, internal forces develop within it to maintain equilibrium. The internal forces in a beam have two components: shear forces in the vertical direction and normal forces in the axis of the beam.
We can represent the internal forces at any point in a beam with two resultant forces: a shear force, which is the resultant force of all internal vertical forces at the point of interest and a bending moment which is the resultant of the normal internal forces at the point of interest. Shear force and bending moment vary in magnitude along the length of the beam.
If you apply a load on the topside of a beam, the beam will be curving downwards, also known as sagging action. When a beam is sagging, the top of the beam will shorten and the forces at the top of the beam will be in compression, while the bottom side of the beam will stretch and the fibres at the bottom of the beam will be in tension. The inverse is known as hogging action, when the load is applied from the underside of the beam or when a beam runs continuously over supports.
Beam Analysis
To design a beam, you need to know what shear forces and bending moments will exist in the beam under your desired loading. Beam analysis is the process to determine these forces. A common method is to plot the Shear Force Diagram (SFD) and Bending Moment Diagram (BMD), which show you the shear force and bending moment at any point along the beam.
There are several ways to determine the points needed to plot a SFD and BMD. You can use engineering principles, for example that fact that a uniformly distributed load creates a parabolic bending moment profile. You can also analyse the internal forces at any point in a beam directly, by making an imaginary ‘cut’ at a point of interest in the beam and then solve the three equations of equilibrium: ΣFx = 0, ΣFy = 0 and ΣM=0 with the applied loads and reactions/internal forces.
Sometimes you cannot solve the three equations of equilibrium because there is more than three unknown variables, this means you have a statically indeterminate beam. You must therefore use Finite Element Analysis (FEA) to determine your internal forces by solving for your degrees of freedom in the stiffness equation F = k x. The maths quickly becomes too complex to do by hand, which is where computer FEA packages come into play!
Sign Convention
A common sign convention is as follows:
- Applied forces pointing downwards are negative (-)
- Reaction forces pointing upwards are positive (+)
- Reaction moments causing the beam to concave upwards and create a sagging action are positive (+). Remember, sagging = smiley face = positive.