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Initial strain

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ExampleReferenceRecordsLargest deviation
Bar restrained at one end only, uniform initial strainClosed form10%
Bar fixed at both ends, uniform initial strainClosed form10%
Beam fixed at both ends, strain gradient through the thicknessClosed form10%
Plate fully restrained in plane, uniform initial strainClosed form10%

Problem. Initial strain represents effects such as a temperature difference or shrinkage. The equivalent nodal load and the subtraction of the initial strain when internal forces are recovered are tested together.

Reference document. Closed-form solutions are used.

The initial strain is turned into an equivalent nodal load by this integral, and the stress is computed by subtracting the initial part from the total strain.

fe=∫BTD ε0 dV,σ=D (Bu−ε0)f_e = \int B^{\mathsf T} D\,\varepsilon_0\,dV,\qquad \sigma = D\,(B u - \varepsilon_0)

The elongation of a free bar, the axial force in a fully restrained bar, the moment produced by a constant curvature and the stress in a sheet restrained along two axes are given by these relations in turn.

u(L)=ε0L,N=−EA ε0,M=EI κ0,σ=Eε1−νu(L) = \varepsilon_0 L,\qquad N = -EA\,\varepsilon_0,\qquad M = EI\,\kappa_0,\qquad \sigma = \dfrac{E\varepsilon}{1-\nu}

Numerical model. The cases are a bar held at one end, a bar fixed at both ends, a beam fixed at both ends and a plate fully restrained in its plane.

Bar restrained at one end only, uniform initial strain

Parameters. ε₀ = 0.00025 · L = 6

Closed form. Free expansion

u(L)=ε0Lu(L) = \varepsilon_0 L
QuantityReferenceComputedDeviation
Free elongation0.00150.00150%

Note. In the same case the internal force is zero.

Bar fixed at both ends, uniform initial strain

Parameters. ε₀ = 0.00025 · EA = 2,400,000

Closed form. The closed form for a fully restrained bar

N=−EA ε0N = -EA\,\varepsilon_0
QuantityReferenceComputedDeviation
Axial force N6006000%

Beam fixed at both ends, strain gradient through the thickness

Parameters. κ₀ = 0.0004 · EI = 16,000

Closed form. The closed form at constant curvature, with the shear force staying zero

M=EI κ0M = EI\,\kappa_0
QuantityReferenceComputedDeviation
End moment M6.46.40%

Plate fully restrained in plane, uniform initial strain

Parameters. ε₀ = 0.0003 · t = 0.25 · ν = 0.2

Closed form. A sheet restrained along two axes

σ=E ε/(1−ν)\sigma = E\,\varepsilon/(1-\nu)
QuantityReferenceComputedDeviation
Membrane force N-2,812.5-2,812.50%