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CALCULATOR MODULE : Beam Bending   ±

Calculate beam bending shear force, bending moment, slope and deflection for general beams using the Euler Bernoulli beam equation.

The Euler Bernoulli beam equation is suitable for slender beams (it does not include the effect of shear), and for small angles (θ < 0.5 rad). The calculations are not valid past the beam end points. For combined loads, the shear force, bending moment, slope and deflection are assumed to be additive. The beam end conditions are of the form left end - right end (for example Pin-Fix is left end pinned and right end fixed). All distances are measured from the left end of the beam.

Beam end types include: free fixed (cantilever), guided fixed, pinned fixed, fixed fixed (built in or fixed), pinned pinned (simply supported), and guided pinned beam ends.

Combined loads include axial loads, point loads, distributed loads, weight loads, concentrated moments, angular displacements, lateral displacements, and uniform temperature gradient.

For beams with compressive axial loads the bending moment, slope and deflection tend to infinity as the axial load tends to the buckling load. For tension loads, the bending moment, slope and deflection decrease with increasing tension. The buckling load can be calculated using either the Euler equation (suitable for long beams), the Johnson equation (suitable for short beams), or the buckling load equation can be determined from the transition length.

The effective length factor should be used for beams on a soft foundation such as soil, where the beam ends are poorly defined. For defined beam ends, such as structures, the effective length factor should be set to one (fe = 1).

Use the Result Plot option to plot the bending moment, shear force, slope, deflection and stress versus position x. Refer to the figures and help pages for more details. Refer to the links below for other beam options.

Reference : Roark's Formulas For Stress And Strain, Warren C Young, McGraw Hill

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CALCULATOR MODULE : Pipe Beam Bending   ±

Calculate beam bending shear force, bending moment, slope and deflection for pipe beams using the Euler Bernoulli beam equation.

The Euler Bernoulli beam equation is suitable for slender beams (it does not include the effect of shear), and for small angles (θ < 0.5 rad). The calculations are not valid past the beam end points. For combined loads, the shear force, bending moment, slope and deflection are assumed to be additive. The beam end conditions are of the form left end - right end (for example Pin-Fix is left end pinned and right end fixed). All distances are measured from the left end of the beam.

Beam end types include: free fixed (cantilever), guided fixed, pinned fixed, fixed fixed (built in or fixed), pinned pinned (simply supported), and guided pinned beam ends.

Combined loads include axial loads, point loads, distributed loads, weight loads, concentrated moments, angular displacements, lateral displacements, and uniform temperature gradient.

For beams with compressive axial loads the bending moment, slope and deflection tend to infinity as the axial load tends to the buckling load. For tension loads, the bending moment, slope and deflection decrease with increasing tension. The buckling load can be calculated using either the Euler equation (suitable for long beams), the Johnson equation (suitable for short beams), or the buckling load equation can be determined from the transition length.

The effective length factor should be used for beams on a soft foundation such as soil, where the beam ends are poorly defined. For defined beam ends, such as structures, the effective length factor should be set to one (fe = 1).

For multi layer beams the concrete stiffness can be included in EI by multiplying EI by a factor (1 + CSF). The bending stress at the field joint should also be multiplied by the factor (1 + CSF) to account for stress localisation (select the pipe joint option for bending stiffness) . The concrete stiffness factor is calculated from the ratio of concrete EI over beam EI in accordance with DNVGL RP F105. The method is suitable for circular beams and pipes. For other profile shapes engineering judgement is required.

The stress check includes longitudinal stress, Tresca combined stress, and von Mises equivalent stress. The bending stress is calculated at the pipe mid wall. The hoop stress is calculated using the Barlow mid wall equation with the nominal wall thickness.

:

`Sh = (P - Pe) (OD - tn) / (2 tn) `

where :

Sh = hoop stress
P = internal pressure
Pe = external pressure
OD = pipe outside diameter
tn = pipe nominal thickness

Use the Result Plot option to plot the bending moment, shear force, slope, deflection and stress versus position x. Refer to the figures and help pages for more details.

Reference : Roark's Formulas For Stress And Strain, Warren C Young, McGraw Hill

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CALCULATOR MODULE : Maths Array And Vector   ±

Calculate vector (array sum, average, root mean square (RMS), and dot product) and matrix operations (matrix inverse, transpose, determinant, Crout factorisation (LU factorisation), and cross product). Enter each matrix row as a comma separated list, with a new line for each row. For large matrices the rows will sometimes wrap around to the next line. If the lines wrap, you must still enter a new line (enter key) for each new row. Matrices should have an equal number of elements in each row.

Solve a set of linear equations using matrix and vector operations. The matrix must be square (equal number of rows and columns), and must have an equal number of elements in each row. The matrix rows must be independent, and well conditioned.

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CALCULATOR MODULE : Matrix Transpose   ±

Calculate the transpose of a matrix (the columns and rows are swapped).

Enter each matrix row as a comma separated list, with a new line for each row. Matrices must have an equal number of elements in each row.

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CALCULATOR MODULE : Matrix Crout Factorisation   ±

Calculate the Crout or Cholesky LU factorisation of a square (n x n) matrix A.

Enter each matrix row as a comma separated list, with a new line for each row. Matrices must have an equal number of elements in each row.

The Crout LU factorisation is calculated so that

`L · U = A `

where :

L is a lower diagonal matrix
U is an upper diagonal matrix.

The L and U matrixes are combined into a single matrix. The U matrix has ones on the diagonal (not shown). The determinant is the product of the elements in the diagonal.

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CALCULATOR MODULE : Matrix Determinant   ±

Calculate the determinant of a square (n x n) matrix.

Enter each matrix row as a comma separated list, with a new line for each row. Matrices must have an equal number of elements in each row. The number of rows in the matrix must be equal to the number of columns. The determinant is calculated from the product of the diagonals of the Crout Cholensky LU factorisation.

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CALCULATOR MODULE : Matrix And Vector Dot Product   ±
CALCULATOR MODULE : Solve Simultaneous Equation   ±

Calculate the solution vector (A) for the linear simultaneous equations, from the linear equation matrix (M), and the result vector (C).

The simultaneous equations are solved using the Crout or Chelensky factorial

`L · U = M `
`M · A = C `
`L · D = C `
`U · A = D `

where :

L is a lower diagonal matrix
U is an upper diagonal matrix.
D is an intermediate result vector

Enter each matrix row as a comma separated list, with a new line for each row. Matrices must have an equal number of elements in each row. The number of rows in the matrix must be equal to the number of columns. The matrix equations must be independent (ie well conditioned - determinant not equal to zero). If the matrix determinant equals zero, the equations are indeterminant. Enter vectors as a comma separated list. The number of elements in the vector must equal the number of matrix columns.

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CALCULATOR MODULE : Matrix Inverse   ±

Calculate the inverse of a square (n x n) matrix.

The inverse of the matrix is calculated using the Crout or Chelenski factorials such that

`M^-1 · M = I `
`M · M^-1 = I `

where :

I = the identity matrix

Enter each matrix row as a comma separated list, with a new line for each row. Matrices must have an equal number of elements in each row. The number of rows in the matrix must be equal to the number of columns. The matrix equations must be independent (ie well conditioned - determinant not equal to zero). If the matrix determinant equals zero, the inverse is indeterminant.

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CALCULATOR MODULE : Vector Cross Product   ±

Calculate the cross product of two vectors.

The cross product of two vectors A and B is a vector C which is perpendicular to both of the input vectors. The cross product vector can be checked by calculating the dot products. The dot product of two perpendicular vestors equals zero.

`A · C = 0 `
`B · C = 0 `

Enter the arrays or vectors as comma separated lists. Both arrays must have three elements.

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CALCULATOR MODULE : Maths Linear Regression   ±

Calculate the best fit line for scatter data points using the least squares linear regression method. The curve does not have to pass through each data point.

For straight line or linear curves (Z = A x + B) the regression is performed directly on the X and Z data values. For power curves (Z = A x^B) the regression is performed on the ln(X) and ln(Z) values. For logarithmic curves (Z = ln(X)) the regression is performed on the ln(X) and Z values. For exponential curves (Z = A e^B) the regression is performed on the X and ln(Z) values. For the user defined transform (Z = A f(X) + B) the regression is performed on f(X) and Z where f(X) is the user defined transform.

The X and Z offsets can be used to change the origin for log values (ln(X - Xo) and ln(Z - Zo)) and user defined transform (f(X - Xo)). The offsets are not used for the X and Z values.

The Z unit value is applied for the log of negative Z values. The Z unit value is not applied for X and Z values, or for user defined transforms (user defined transforms should account for the sign of the data points).

The regression data and regression parameters are displayed in the output view at the bottom of the page. The correlation coefficient r is a measure of how well the curve fits the data points (close to one is better). Extrapolated values should be used carefully.

Enter vector data as X,Z pairs separated by a comma or tab, with each pair on a new line. Or copy and paste the data points from a spreadsheet. Enter array data X and Z values as separate comma or tab separated lists. Store file data to a text file as comma or tab separated pairs (X,Z), with each pair on a new line (or copy and past cells from a spreadsheet). Refer to the example text file in resources.

Use the data plot option on the plot bar to display the data points and the best fit line.

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CALCULATOR MODULE : Maths Curve Data Check   ±

Calculate smoothed data values from raw input data.

The calculator is intended for checking and smoothing digitised plot data. Use the Result Plot option to display the data value, slope and curvature. The value, slope and curvature curves should be smooth. Sudden changes in slope and or curvature indicate possible faulty values.

Options include data order (ascending or descending data), plot axies (Z versus X or X versus Z), smoothing type, and whether to smooth the maximum and minimum values. Smoothing uses a simple weighted mean value.

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CALCULATOR MODULE : Weibull Gumbel And Frechet Extreme Event Probability   ±

Calculate extreme event amplitude and return period from return period data using the Weibull, Gumbel and Frechet probability distributions.

A best fit line is calculated for the data points using the least squares linear regression method. The regression is calculated for X versus Z instead of Z versus X (the X and Z values are swapped). The three parameter distribution amplitude offset is a minimum amplitude. The regression data points and regression parameters are displayed in the output view at the bottom of the page. Use the Data Plot option on the plot bar to display the data points and the best fit line.

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