Contents:

• Introduction
  • About Module
  • Program Motivation
• Installation
• Documentation
  • Units and Conventions
  • Table of all classes and arguments
  • Beam
    • Methods in Beam class
  • Load
  • Point Load
  • Uniformly Distributed Load(UDL)
  • Uniformly Varying Load(UVL)
  • Reaction
  • Point Moment
  • Hinge
• Examples

Introduction

A beam is a structural element that primarily resists loads applied laterally to the beam's axis (an element designed to carry primarily axial load would be a strut or column). Its mode of deflection is primarily by bending. The loads applied to the beam result in reaction forces at the beam's support points. The total effect of all the forces acting on the beam is to produce shear forces and bending moments within the beams, that in turn induce internal stresses, strains and deflections of the beam. Source: Wikipedia

About the module

This module is aimed to solve Statically Determinate, Geometrically Stable two dimensional Beam just by using 3 equations of static equilibrium. $F_x = 0, F_y = 0 \text{ and } M_{point} = 0$.

This module supports:

• determinate 2D beam with Point Loads, Uniformly Distributed Loads, Uniformly Varying Loads, Point Moments
• Fixed or Hinged or Roller types of support conditions
• A Beam with Internal Hinge
• Bending Moment Diagram and Shear Force Diagram

Program Motivation

I studied about Beams first time in my second semester which had an included course of Applied Mechanics - I (Statics). About 10-15 marks question were sure from beams and frames in board examination. But, while practicing beam analysis questions, from past question collection, one thing that collection was lacking was solution to those questions. Not even reaction values were given. So, it would cost us 15 minutes for even simple reaction calculation error. So, I thought of making this library from that period. And, I turned this thought into action in my 3rd semester's vacation.

Installation and usage

pip install beamframe

Requirements

"numpy>=1.19",
"sympy>=1",
"matplotlib>=3"

Note All requirements will be installed automatically while installing this module.

• PyPi link

Importing the module

Any one way of importing can be adopted

from beamframe import beam
#or 
from beamframe.beam import *

Documentation

Units and Conventions

One can use and follow thier own sign conventions in this module. But, default values will have following units and sign conventions.

Sign Conventions:

• Positive x-axis for beam: increases in right hand side.
• Positive y-axis for beam: increases upward direction.
• Positive angle direction: Counter clockwise with respect to positive x-axis of beam.
• Positive moment: Counter clockwise

Standard Units:

• Length: meter
• Angle: degrees
• Load: kN
• Moment: kNm
• Load per meter: kN/m

List of all classes and arguments

Beam:

Beam is the main class to represent a beam object and perform various calculations.

Arguments

• length(float): length of a beam
• ndivs(int) : number of divisions of beam. This specifies number of points along beam in which shear force and bending moment values will be calculated.

Here are few optional keyword arguments

• E(float) = Modulus of Elasticity of beam material
• I(float) = 2nd moment of area of the cross section of beam

Methods

Note

Just first and second methods are sufficient to solve beam and generate graph. But, to keep track of ongoing process use other methods. Remember not to use fast_solve and other methods(excluding method no. 2). Doing this will re-add all those loads you've passed again.

Example

#importing the module
from beamframe.beam import *

# to create a beam of length 5m:
b = Beam(5)

Load

Arguments:

• pos(float): position of that netload with respect to beam coordinates's origin
• load(float): net load of that load type(for point load that is point load value, but it will be different for other loads like uvl and udl)
• inverted(bool)=False: Default direction of positive net load is in positive direction of y-axes
  • by default: inverted = False (Positive Load)
  • use inverted=True (indicates negative Load)

PointLoad

Description

Subclass of class Load

Arguments

• pos, load, inverted: inherit from super class Load
• inclination(float)=90: unit=degree represents angle made by direction of net load with positive direction of beam's x axis
  • inclination will be positive for counter clockwise direction
  • put negative inclination if you want to take angle in clockwise direction

Attributes

• load_x: component of net load value in positive direciton of beam's x-axis
• load_y: component of net load value in positive y-direciton(upward direction)

UDL

UDL(Uniformly Distributed Load) is type of load that is combinaiton of infinite points load over certain length acting transverse to beam

Arguments:

• start(float):Start position of UDL
• loadpm(float): Load Per meter of udl
• span(float): Total length of udl
• inverted(bool) = True: UDL facing downwards on beam
  • use inverted=False for upside udl

Attributes

• netload(float): total effective load of udl
• pos(float): position of effective load from - beam origin

UVL

It is that load whose magnitude varies along the loading length with a constant rate. Uniformly varying load is further divided into two types:

1. Triangular Load
2. Trapezoidal Load

Arguments

• start:float = Start position of uvl from beam's origin along x-axis of beam coordinate system
• startload:float = unit: kN/m = Starting load/m value of uvl
• span:float = Total length of uvl object
• endload:float = Ending load/m value of uvl object
• inverted:bool= True : Default=True Inverts the uvl object

Attributes

• end = End coordinate of uvl object
• tload = Net load value of upper triangular part of trapezoidal or triangular load
• rload = Net load value of lower rectangular part of trapezoidal load itself
• netload = Net load of whole uvl object itself. netload = tload + rload
• netpos = Net position(coordinates) where net load of uvl acts

Reaction

Reactions are given by supports. 3 types of supports are defined for now: hinge, roller and fixed support.

Arguments

• pos(float): position of reaction
• type(str): any one of ('roller','hinge','fixed') or ('r','h','f') Representing support condition at that point.
• pos_sym(str): Symbolic variable to represent support location name

Attributes

• rx_val, ry_val, mom_val: variables to store numerical values for reaction loads and moments
• rx_var, ry_var, mom_var: symbolic variables to store symbolic values for reactions

PointMoment

Pure moment that act at point

Arguments

• pos : location of that point moment from beam's origin
• mom : value of that point moment
• ccw(bool) = False : counterclockwise direciton is positive value of moment,
  • by defalut: ccw = False and given moment is positive

Hinge

Internal hinges are provided in a structure to reduce statical indeterminacy of the structure. Bending moment at internal hinge is always zero. Internal hinge makes structure more flexible. It allows structure to move which reduces the reactive stresses.

Arguments

• pos:float = Position of that internal hinge from origin of beam coordinate system
• side:str = 'l' : Accepted Values = ('r', 'right', 'l', 'left')
  • Default Value = 'l'
  • This side specifies which side of loads to take in order to take moment of that loads about hinge.

Examples

Example-1: Solving Simplest Beam

The simplest possible code to solve simply supported beam with pointload at middle of span.

#import module
from beamframe.beam import *

# create a beam of length 5m
b = Beam(5)

# create reaction and pointload objects
ra = Reaction(0, 'r', 'A')
rb = Reaction(b.length, 'h', 'B')
p = PointLoad(b.length/2, 10, inverted=True)

b.fast_solve((ra, rb, p))
b.generate_graph()

Graph:

Example-2: Cantilever beam with udl

from beamframe.beam import *

# create a beam of length 5m
b = Beam(5)

# create reaction and udl object
ra = Reaction(0, 'f', 'A')
udl = UDL(0, 5, 5)

b.fast_solve((ra, udl))
b.generate_graph()

Graph:

Example-3: Overhanging beam with internal hinge, and 3 supports

Note: In b.generate_graph: use details=True to show details like max and min bending moment, max and minimum shear force. save_fig=True to save figure of desired quality specified by quality in res

from beamframe.beam import *

b = Beam(10)
ra = Reaction(0, 'h', 'A')
p1 = PointLoad(1,100, inverted=True, inclination=40)
m1 = PointMoment(2.5,20, ccw=False)
rd = Reaction(3.5, 'r', 'D')
h = Hinge(5, side='r')
udl = UDL(5,10,2)
rf = Reaction(7, 'r', 'F')
p2 = PointLoad(b.length, 50, inverted=True)

lds = (ra, p1, m1, rd, h, udl, rf, p2)
b.fast_solve(lds)
b.generate_graph(which='both', details=True, save_fig=True, show_graph=False, res='h')

Graph:

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