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Frame element, values along the member

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Results.

ExampleReferenceRecordsLargest deviation
Simply supported beam, uniformly distributed load, single elementPublished source20%
Cantilever, point load at the tipPublished source20%
Beam fixed at both ends, uniformly distributed loadPublished source30%
Simply supported beam, distributed load in the lateral planePublished source20%
Propped cantilever with a member end release, distributed loadPublished source20%
Tapered beam, uniformly distributed loadIndependent calculation30.105%
Bar under distributed axial loadClosed form20%
Beam fixed at both ends, thermal gradientClosed form10%

Problem. The internal force and the deflection at every section along the element are tested. Simply supported, cantilever, fixed-fixed and end-released beams are covered, together with a thermal gradient and a tapered section.

Reference document. The closed-form solutions of Timoshenko, Strength of Materials, are used. For the tapered section the moment comes from statics and the deflection from the virtual work integral.

The moment field and the mid-span deflection of a simply supported beam are given by these relations.

M(x)=w x (L−x)2,δc=5wL4384 EIM(x) = \dfrac{w\,x\,(L-x)}{2},\qquad \delta_{c} = \dfrac{5wL^4}{384\,EI}

In the end-released beam the largest span moment occurs at x=5L/8x = 5L/8.

Mmax⁡=9wL2128,x=5L8M_{\max} = \dfrac{9wL^2}{128},\qquad x = \dfrac{5L}{8}

Numerical model. The section force comes from the equilibrium of the end forces and the load up to the section. The deflection is the double integral of the curvature M/EIM/EI, with the translations of the two ends as boundary conditions. In the end-released beam both supports are fully fixed and the bending rotation at end j of the element is released.

Simply supported beam, uniformly distributed load, single element

Parameters. L = 6 · w = 12 · sampling point = 21

Published source. Timoshenko, Strength of Materials

QuantityReferenceComputedDeviation
Mid-span moment wL2/8wL^2/854540%
Mid-span deflection 5wL4/384EI5wL^4/384EI-0.00125-0.001250%

Note. The single-element model gives the same diagram as the eight-element model.

Cantilever, point load at the tip

Parameters. L = 6 · P = 40

Published source. Timoshenko, Strength of Materials

QuantityReferenceComputedDeviation
Fixed-end moment PLPL-240-2400%
Tip deflection PL3/3EIPL^3/3EI-0.0177778-0.01777780%

Beam fixed at both ends, uniformly distributed load

Parameters. L = 6 · w = 12

Published source. Timoshenko, Strength of Materials

QuantityReferenceComputedDeviation
End moment wL2/12wL^2/12-36-360%
Mid-span moment wL2/24wL^2/2418180%
Mid-span deflection wL4/384EIwL^4/384EI-0.00025-0.000250%

Simply supported beam, distributed load in the lateral plane

Parameters. L = 6 · w = 12 · inertia = I₂₂

Published source. Timoshenko, Strength of Materials

QuantityReferenceComputedDeviation
Mid-span moment wL2/8wL^2/854540%
Mid-span deflection 5wL4/384EI5wL^4/384EI-0.005-0.0050%

Propped cantilever with a member end release, distributed load

Parameters. L = 6 · w = 12 · sampling point = 641

Published source. The classical propped cantilever solution

QuantityReferenceComputedDeviation
Maximum span moment 9wL2/1289wL^2/12830.37530.3750%
Location of the maximum, 5L/85L/83.753.750%

Tapered beam, uniformly distributed load

Parameters. L = 8 · hᵢ = 1.2 · hⱼ = 0.5 · b = 0.35 · w = -30

Independent calculation. Virtual work integral

δ(a)=∫0LM0(x) ma(x)EI(x) dx\delta(a) = \int_0^L \dfrac{M_0(x)\,m_a(x)}{EI(x)}\,dx
QuantityReferenceComputedDeviation
Mid-span deflection-0.00339638-0.00339638< 0.001%
Largest deflection inside the span-0.00348819-0.00348819< 0.001%
Where the largest deflection is, x/Lx/L0.5756060.5750.105%

Note. The peak is not at mid-span. The section weakens towards the shallow end, so the largest deflection shifts to that side.

Note. The value is read at 641 points, and the tolerance is the spacing between two points.

Bar under distributed axial load

Parameters. L = 6 · w = 25

Closed form. The closed-form solution for an axial bar

QuantityReferenceComputedDeviation
Axial elongation at mid-span w(Lx−x2/2)/EAw(Lx - x^2/2)/EA0.00006250.00006250%
Extension at the free end wL2/(2EA)wL^2/(2EA), the largest in the field0.00008333330.00008333330%

Beam fixed at both ends, thermal gradient

Parameters. ΔT = 12 · h = 0.6 · L = 6

Closed form. The closed form for fully restrained curvature

κ=α ΔT/h\kappa = \alpha\,\Delta T/h
QuantityReferenceComputedDeviation
Moment along the span M3=EI κM_3 = EI\,\kappa, constant32.432.40%