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### Uniform Temperature Differential Frame Deflections Equations and Calculator

**Beam Deflection and Stress Equation and Calculators**

Frame Deflections with Uniform Temperature Differential Between Outside and Inside Equations and Calculator.

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Frame Deflections with Uniform temperature Rise Calculator:

General reaction and deformation expressions with right and left ends pinned

Horizontal Deflection at A:

Reaction locations are pinned therefore, the displacements = 0 = δ_{HA}

Angular Rotation at A:

Where:

Loading Terms LP_{H} and LP_{M} are given below.

Reaction loads and moments V_{A} and V_{B}, and H_{B} can be evaluated from equilibrium equations after calculating H_{A} and M_{A}.

Note: t_{`}; t_{2}; and t_{3} are beam thicknesses from inside to outside

Reaction locations are pinned therefore, the Moments = 0 = M_{A} = M_{B}

Where:

Δ_{o} = Displacement (in, mm),

θ_{o} = Angular Displacement (radians),

W = Load or Force (lbsf, N),

w = Unit Load or force per unit length (lbs/in^{2}, N/mm^{2}),

M_{o} = Applied couple (moment) ( lbs-in, N-mm),

θ_{o} = Externally created angular displacement (radians),

Δ_{o}, = Externally created concentrated lateral displacement (in, mm),

T_{1} - T_{2} = Uniform temperature rise (°F),

T_{o} = Average Temperature (deg °F),

γ = Temperature coefficient of expansion [ µinch/(in. °F), µmm/(mm. °F) ],

T_{1}, T_{2} = Temperature on outside and inside respectively (degrees),

H_{A}, H_{B} = Horizontal end reaction moments at the left and right, respectively, and are positive clockwise (lbs, N),

I_{1}, I_{2}, and I_{3} = Respective area moments of inertia for bending in the plane of the frame for the three members (in^{4}, mm^{4}),

E_{1}, E_{2}, and E_{3} = Respective moduli of elasticity (lb/in^{2}, Pa) Related: Modulus of Elasticity, Yield Strength;

γ1, γ2, and γ3 = Respective temperature coefficients of expansions unit strain per. degree ( in/in/°F, mm/mm/°C),

*l*_{1}, *l*_{2}, *l*_{3} = Member lengths respectively (in, mm),

References:

Roark's Formulas for Stress and Strain