Structural Analysis Cheat Sheet PDF

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Summary

Cheat sheet for the course - tables for setting out work to ensure the correct method is followed....

Description

VIRTUAL WORK Real structure elongations Member 1 2 3

N [kN]

Virtual load – internal work product Member e = fN [mm] 1

Same as for the real structure

f [mm/kN]

e = fN [mm]

 [kN] 𝑵 Re-determine internal forces with only the virtual load – you will often get 0 bar forces

 e [kNmm] 𝑵

2 3 …

𝒆 𝑼 = ∑𝑵

Virtual load – external work product DOF 1

 [kN] 𝑷

u [mm]

Should all be 0 except for the DOF with the virtual load and the reaction force DOF

 𝒖 [kNmm] 𝑷

Should all be unknown Will nearly always except for the reaction sum to 1 x u where u displacements which is the unknown should be 0 displacement we want

2 3 …

𝑾 = ∑ 𝑷𝒖 𝑾=𝑼

Force/Flexibility Method (trusses) Bar

N0 [kN]

n1 [kN]

f [mm/kN)

n1fN0

n1fn1 = n12f

∑ 𝐧𝟏𝐟𝐍𝟎

∑ 𝐧𝟏𝐟𝐧𝟏

Real N = N0 + X1N1

By compatibility laws (using W = U and nothing that displacement of real supports = 0)

𝑿𝟏 =

−𝒏𝟏 𝒇𝑵𝟎 (𝒏𝟏 )𝟐 𝒇

We can then find internal elongations using 𝒆 = 𝒇𝑵 This is essentially 𝒆 = 𝒇(𝑵𝟎 + 𝑿𝟏 𝒏𝟏) THIS CAN BE EXTENDED TO TWO-FOLD INDETERMINACY BY ADDING COLUMNS FOR: 1. n2 2. n2fN0 3. n2fn2 = n22f Then

𝑿𝟐 =

−𝒏𝟐 𝒇𝑵𝟎 (𝒏𝟐 )𝟐 𝒇

The final solution is then that real N is given by:

N = N0 + X1N1 + X2N2

Force/Flexibility Method (Frames/Moments) The relevant formula for this is:

∫ Thus:

𝑀0 𝑚1 𝐸𝐼

. 𝑑𝑥 + 𝑋 ∫

𝑚1 𝑚1 . 𝑑𝑥 = 0 𝐸𝐼

𝑴 𝒎 −∫ 𝟎 𝟏 𝑿 = 𝒎 𝑬𝑰 𝟏 𝒎𝟏 ∫ 𝑬𝑰 Be careful that EI is not different for the beams before you cancel them out.

Element

∫

𝑀0 𝑚1 . 𝑑𝑥 𝐸𝐼

∫

𝑚1 𝑚1 . 𝑑𝑥 𝐸𝐼

1

2 … ∑ 𝑎𝑙𝑙 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠

∑ 𝑎𝑙𝑙 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠

Moment distribution method LAY OUT THE BEAM ON TOP OF THE TABLE RHS SUPPORT 1 Distribution factors

LHS 2

RHS 2

LHS 3

1 for fixed support

RHS 3

LHS 4 0 if roller/pin

Fixed end moments

Balance Carry over ( x 0.5)

Balance Carry over

… BALANCE

SUM

Always end on a balancing

Sum the total of each column. You finish when for all supports LHS = RHS

Approximation method for BMD

Frames and truss stability For truss stability: 𝑚 + 𝑟 ≥ 2𝑗 For frame stability: 3𝑚 + 𝑟 ≥ 3𝑗 + 𝑐 What constitutes a member and a joint in a frame?...