Help needed on Structures Question, Simply supported beam?

Question

A simply supported steel beam of 5m span carries a uniformly distributed load made up of 5kN/m dead load plus 2.5kN/m imposed load. The beam is restrained at the supports only. Choose a suitable UB section using grade S275 steel for the beam, checking lateral torsional buckling, bending and shear considerations, deflection and that the section is class 1 compact. E=205,000 N/mm2.

Buckling resistance moment Mb=(pb x Sx)

Answer 1

You have two conflicting statements here. You said its a simply supported beam and the beam is restrained at the support.

Ok, I'll just help you on a simply supported beam. I'll show you how the math is done and you plug in the figures:

Let;

w = the unit load = unit live load or imposed load + unit dead load

V = SUM wdx ;(SUM means integral of)

V = wx + C ;C = constant of integration, V is shear

if x = 0, V= -R, R = wL/2, C = -R

Where R = reaction of the support,

L = unsupported length of the beam, Hence;

V = wx - R= wx - wL/2  Eq.(1)

M = SUM Vdx = SUM(wx-wL/2)dx  ;where M = bending moment

M = Wx^2/2 - wLx/2 + C

if x = 0, M = 0, C =0, hence;

M = wx^2/2 - wLx/2

For M = Mmaximum;

V = wx -wL/2 = 0

x = L/2, Therefore;

Mmax = wL^2/8 - wL^2/4 = -wL^2/8 Eq.(2)

EIB = SUM Mdx, where B = deflection angle, I = moment of inertia of beam cross section, E = modulus of elasticity of beam material, thu;

EIB = SUM(wx^2/2 - wLx/2)dx



EIB= wx^3/6 - wLx^2/4 + C

if x = L/2 B = 0 hence;

C = wL(L/2)^2/4 - w(L/2)^3/6 = wL^3/16 - wL^3/48

C = wL^3/24, hence;

EIB = wx^3/6 -wLx^2/4 + wL^3/24 Eq.(3)

EID = SUM Bdx, where D = deflection

EID = SUM(wx^3/6 - wLx^2/4 + wL^3/24)dx

EID = wx^4/24 - wLx^3/12 + wL^3x/24

if x = L/2, D = Dmax, hence

EIDmax = wL^4/384 - wL^4/96 + wL^4/48

EIDmax = wL^4(1- 4 +8)/384 = 5wL^4/384 Eq.(4)

Use Equations (1), (2) and (4) to find the shear force, bending moment and deflection.

For the shear stress of steel beams use

v = V/(th)

where:

v is the shear stress

V is the maximum shear force

t = the thickness of the web of the beam

h is the depth of the beam

For the bending stress use;

b = Mc/I = M/z

Where;

b is the bending stress

c is the distance from the neutral axis to the extreme fiber

I is the moment of inertia of the beam cross section

M is the bending moment

z is the section modulus = I/c

Just take care of the consistency of the units.