√2 : 1

Let us assume that the two square beams have the same dimensions and are made of the same material.

Case 1: Two sides horizontal

In this case, the beam is supported by two edges and the load is applied at the center. The beam will bend along the vertical axis passing through the center. The maximum stress will be at the bottom of the beam and will be given by:

σ = (M*c)/I

where σ is the stress, M is the bending moment, c is the distance from the neutral axis to the bottom of the beam, and I is the moment of inertia of the beam.

The moment of inertia of a square beam about its horizontal axis passing through the center is given by:

I = (1/12)*b*h^3

where b is the width of the beam and h is the height of the beam.

In this case, b = h and c = h/2, so:

I = (1/12)*h*h^3 = (1/12)*h^4

The bending moment M can be calculated as:

M = F*(h/2)

where F is the load applied at the center of the beam.

Therefore, the maximum stress is:

σ = (F*h/2)*(h/2)/[(1/12)*h^4] = 6*F*h/[(h^3)]

Case 2: One diagonal vertical and other horizontal

In this case, the beam is supported by one edge and the load is applied at the center. The beam will bend along the diagonal axis passing through the center. The maximum stress will be at the bottom of the beam and will be given by:

σ = (M*c)/I

where σ is the stress, M is the bending moment, c is the distance from the neutral axis to the bottom of the beam, and I is the moment of inertia of the beam.

The moment of inertia of a square beam about its diagonal axis passing through the center is given by:

I = (1/6)*b*h^3

where b is the width of the beam and h is the height of the beam.

In this case, b = h and c = (h/2)*√2, so:

I = (1/6)*h*(h^3) = (1/6)*h^4

The bending moment M can be calculated as:

M = F*(h/√2)

Therefore, the maximum stress is:

σ = (F*h/√2)*[(h/2)*√2]/[(1/6)*h^4] = 3*√2*F/h^2

Ratio of flexural strengths:

Flexural strength is the maximum stress that a material can withstand before it breaks under bending. Therefore, the ratio of the flexural strengths of the two square beams is given by:

(Flexural strength of the beam with two sides horizontal)/(Flexural strength of the beam with one diagonal vertical and other horizontal) = (6*F*h/[(h^3)])/(3*√2*F/h^2) = 2√2 : 1

Therefore, the ratio of the flexural strengths of the two square beams is 2√2 : 1.

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