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Tapered element

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ExampleReferenceRecordsLargest deviation
The classical stiffness matrix at constant sectionClosed form10%
Tapered cantilever, tip loadIndependent calculation10%
Convergence of the Gauss rule, 3:1 taperIndependent calculation1< 0.001%
Tapered beam fixed at both ends, uniformly distributed loadIndependent calculation2< 0.001%
Simply supported tapered beam, a single elementIndependent calculation1< 0.001%
Simply supported tapered beam, distributed loadIndependent calculation2< 0.001%
Haunched member, haunches at both ends and a constant section betweenIndependent calculation2< 0.001%

Problem. A tapered element is a frame element whose section properties vary along the span. The stiffness matrix is built as the inverse of the flexibility matrix. The flexibility terms are integrals containing 1/EI(x)1/EI(x), taken numerically at Gauss points along the element.

Reference document. With a constant section the result is compared with the stiffness matrix written in closed form. With a varying section the reference is the virtual work integral and the force method.

Numerical model. The default number of Gauss points is eight. For a haunched element the start and end points of the haunches are given to the element, and the integral is split at those points.

The classical stiffness matrix at constant section

Parameters. Gauss points = 8

Closed form. Stiffness matrix of the 3D frame element, 12×12

QuantityReferenceComputedDeviation
Relative deviation of the stiffness matrix02.264×10−162.264 \times 10^{-16}0%

Tapered cantilever, tip load

Parameters. hᵢ = 1.2 · hⱼ = 0.5 · b = 0.35 · Gauss points = 10

Independent calculation. Virtual work integral

δ=P∫0L(L−x)2E I(x) dx\delta = P\int_0^L \dfrac{(L-x)^2}{E\,I(x)}\,dx
QuantityReferenceComputedDeviation
Tip deflection-0.0124872-0.01248720%

Convergence of the Gauss rule, 3:1 taper

Parameters. hᵢ/hⱼ = 3 · inertia ratio = 27

Independent calculation. Twenty-four point Gauss rule

QuantityReferenceComputedDeviation
Relative deviation of the stiffness, eight points under the default rule04.161×10−74.161 \times 10^{-7}< 0.001%

Tapered beam fixed at both ends, uniformly distributed load

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

Independent calculation. Force method

fij=∫0Lmi mjEI(x) dx,f X=−∫0Lmi M0EI(x) dxf_{ij} = \int_0^L \dfrac{m_i\,m_j}{EI(x)}\,dx,\qquad f\,X = -\int_0^L \dfrac{m_i\,M_0}{EI(x)}\,dx
QuantityReferenceComputedDeviation
Fixed-end moment at the deep end254.692254.692< 0.001%
Fixed-end moment at the shallow end-89.0096-89.0096< 0.001%

Simply supported tapered beam, a single element

Parameters. hᵢ = 1.2 · hⱼ = 0.5 · element count = 1

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%

Note. There is no node at mid-span, and the value is read from inside the element.

Simply supported tapered beam, distributed load

Parameters. hᵢ = 1.2 · hⱼ = 0.5 · element count = 4

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-0.00339638-0.003396380%
Largest deflection inside the span-0.00348819-0.00348819< 0.001%

Haunched member, haunches at both ends and a constant section between

Parameters. haunch length = 2 · h(end)/h(middle) = 2

Independent calculation. Force method

fij=∫0Lmi mjEI(x) dx,f X=−∫0Lmi M0EI(x) dxf_{ij} = \int_0^L \dfrac{m_i\,m_j}{EI(x)}\,dx,\qquad f\,X = -\int_0^L \dfrac{m_i\,M_0}{EI(x)}\,dx
QuantityReferenceComputedDeviation
End moment, one element with the haunch ends defined195.208195.207< 0.001%
End moment, the member divided into three elements195.208195.207< 0.001%