Mechanics of solids

1. Compression Members, MOS-II
CED-UET-P Prepared By: Muhammad Bilal (16PWCIV4617)
Mechanics of solids
Compression members:
 Structural members subjected primary to compressive loads (columns)
 Bending and shearing stresses are also present but negligible
Assumptions
 Load is applied to centroid
 Material is homogenous
 Member is straight
 Material is isotropic
Modes of failure (for columns)
 Yielding/crushing (for short columns)
 Buckling/Bending out of plan (for long columns)
 Combination of these two (for intermediate lengths)

Buckling
Lateral deflection of members under axial loads OR lateral deflection of members when the
state of equilibrium changes from stable to unstable
Critical load
When vertical load is applied on a compression member after some point it will start buckling
that load is critical load, 𝑃𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙
 If load is applied and member comes back to its original state we say its stable
equilibrium P < 𝑃𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙
 If load is applied and member doesn’t comes back to its original state and regain a
new equilibrium position we call it unstable equilibrium P=𝑃𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙
 If P > 𝑃𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 then after removal of load it will continue to bend and a high deflection
is obtained
 Buckling is between stable and unstable Equilibrium

2. Compression Members, MOS-II
CED-UET-P Prepared By: Muhammad Bilal (16PWCIV4617)
Long columns by Euler formula / Euler Buckling
 His analysis is based on differential equation of the elastic curve, EI(
𝑑2 𝑦
𝑑𝑥2
) = 𝑀
 by solving the above equation with boundary conditions we get 𝑃𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 =
𝐸𝐼𝜋2
𝐿2
 Euler formula determine critical loads not working loads or safe loads it is therefore
necessary to divide it by a suitable factor of safety
 The critical stresses are 𝜎𝑐𝑟 =
𝜋2 𝐸
(
𝐿
𝛾
)2
 Where
𝐿
𝛾
is slender ratio
Effect of End Constraints
After buckling the length L changes the new length is called effective length and is equal to
L=KL the value of k is different for different supports
End Condition Effective Length
fixed ends 0.5L
one fixed end, the other hinged 0.7L
both ends hinged L
one end fixed, the other free 2L