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Symbolic maths

Video: Symbolic maths for a beam deflection

The calculation engine is numeric: it gives you numbers. The Symbolic dialog gives you formulas. It rearranges an equation, differentiates, integrates, solves for the constants of integration and substitutes values, all exactly, with fractions kept as fractions and symbols kept as symbols. Then you put the result into the sheet and let the engine check it with numbers.

Open it from Functions > Symbolic on the ribbon. The module loads the first time you run something, and it works offline.

Using the dialog

  1. Type an expression, or an equation with =. Write * for every product and ^ for powers. The functions are sin, cos, ln, exp and sqrt; the constants are pi and e. Enter runs it; Shift+Enter starts a new line.
  2. Choose the operation:
OperationWhat it needsWhat you get
Simplifythe expressionlike terms collected, powers combined
Expandthe expressionproducts and powers of sums multiplied out
Factorthe expressioncommon factors, and polynomials in one variable
Differentiatea variable, and an order (1 by default)the derivative
Integratea variable, and for a definite integral both limitsan antiderivative (with no constant), or the value between the limits
Solve forone variable for one equation (write =), or several variables separated by commas for a linear system whose equations are separated by ; or new linesthe solutions; a quadratic gives both roots
Substitutevalues such as x = 2, b = 3*athe exact result
Evaluatea number for every variablea decimal number to 12 digits

  1. The result is typeset (with KaTeX) and also shown as text, with a Copy button.

  2. Insert into worksheet writes the result at the cursor:

    • in a sheet: a display formula block ($$ ... $$, typeset in the preview) or a quantity line name: expression;
    • in a Calcpad worksheet: a calculation line name = expression, or a comment line.

    A quantity line is calculated by the engine, so every symbol in it needs a value in the sheet. Use the formula block to keep a result in terms of symbols.

A worked derivation: the deflection of a simply supported beam

The beam has span L, uniform load w and stiffness EI. The bending moment at x is M(x) = w L x / 2 - w x² / 2, and the deflection y, taken in the direction of the load, satisfies EI y’’ = -M(x). These are the dialog’s results for each step (the inputs are in the first column).

StepOperation and inputResult
1Integrate -(w*L*x/2 - w*x^2/2)/(E*I) for x-L*w*x^2/(4*E*I) + w*x^3/(6*E*I)
2Add a constant C1 yourself and Integrate again: -(w*L*x^2/4 - w*x^3/6)/(E*I) + C1 for x-L*w*x^3/(12*E*I) + w*x^4/(24*E*I) + C1*x
3Add C2. The supports give y(0) = 0 and y(L) = 0, so Solve for C1, C2 the equations C2 = 0; -L^4*w/(12*E*I) + L^4*w/(24*E*I) + C1*L + C2 = 0C1 = L^3*w/(24*E*I); C2 = 0
4Substitute x = L/2 into the curve -L*w*x^3/(12*E*I) + w*x^4/(24*E*I) + L^3*w*x/(24*E*I)5*L^4*w/(384*E*I)

The midspan deflection is 5 w L⁴ / (384 E I). That is the formula to put in the sheet. The dialog can also differentiate it (5*L^3*w/(96*E*I) with respect to L), or evaluate it for given numbers: with w = 12, L = 6, E = 200 and I = 250 the result is 0.00405 (in consistent units of your choice).

Putting the result into a sheet

In the sheet, write the derivation as prose and display formulas, then give the engine the numbers and check the formula against the curve:

## Step 4: midspan

Substituting $x = L/2$ into the curve and simplifying gives

$$y\left(\frac{L}{2}\right) = \frac{5 L^4 w}{384 E I}$$

## Numbers

w: 12 kN/m | design load
L: 6 m | span
E: 200 GPa | modulus of elasticity
I: 250000000 mm^4 | second moment of area
delta: 5 * w * L^4 / (384 * E * I) -> mm | midspan deflection from the formula

y(x): -L * w * x^3 / (12 * E * I) + w * x^4 / (24 * E * I) + L^3 * w * x / (24 * E * I) | the curve with its constants
y_mid: y(L / 2) -> mm | the curve at midspan
ratio: y_mid / delta | 1 when the formula and the curve agree

w = 12 kN/m    design load
L = 6 m    span
E = 200 GPa    modulus of elasticity
I = 250000000 mm⁴    second moment of area
δ = 5·w·L⁴/(384·E·I) = 5 · 12 kN/m · (6 m)⁴ / (384 · 200 GPa · 250000000 mm⁴) = 4.05 mm
y_mid = y(L/2) = y(6 m / 2) = 4.05 mm    the curve at midspan
ratio = y_mid / δ = 4.05 mm / 4.05 mm = 1

The two routes agree: the exact formula from the dialog and the curve with its constants give the same 4.05 mm. The sample Beam deflection derived symbolically (Samples, group Sheet syntax) is this sheet in full, with the deflection curve plotted:

L: 6 m
w: 12 kN/m
E: 200 GPa
I: 250000000 mm^4
y(x): -L * w * x^3 / (12 * E * I) + w * x^4 / (24 * E * I) + L^3 * w * x / (24 * E * I)
plot -y(x * 1 m) / (1 mm) for x in 0..6

(The plot divides by 1 mm to give the plot a plain number, and flips the sign so the beam sags downward on the page.)

Limits

The algebra is exact rational arithmetic, so a result that does not fit in 128 bits is an error rather than a rounded guess, and a refusal is shown in words with the library’s own message (“Cannot integrate: power of a non-linear base.”). Nothing is guessed. The limits:

  • no complex numbers;
  • only sin, cos, ln, exp and sqrt;
  • integration covers a small class of integrands;
  • solving covers a single equation of degree 1 or 2, and linear systems of up to four unknowns;
  • symbols are treated as positive real numbers, so sqrt(x^2) is x.

For anything else, use the numeric tools: $Root, $Area, $Slope, lsolve and friends.

Related pages: The beam deflection formula, derived. More pages are in Calculators and guides.