simple span beam mathcad

7/28/2019 Simple Span Beam Mathcad

1/9

PTC_CE_BSD_1.1_us_mp.mcdx

Mathcad Enabled Content Copyright 2011 Knovel Corp.

Building Structural Design: Thomas P. Magner, P.E. 2011 Parametric Technology Corp.Chapter 1: Analysis of Beams

1.1 Simple Span Beams

Disclaimer

While Knovel and PTC have made every effort to ensure that the calculations, engineering solutions,diagrams and other information (collectively Solution) presented in this Mathcad worksheet aresound from the engineering standpoint and accurately represent the content of the book on which theSolution is based, Knovel and PTC do not give any warranties or representations, express or implied,including with respect to fitness, intended purpose, use or merchantability and/or correctness oraccuracy of this Solution.

Array or ig in:

RIGIN 1

Description

This application computes the reactions and the maximum bending moment for simple span beamsloaded with any practical number of uniformly distributed and concentrated loads. This application islimited to beams with all applied loads in the same direction. The user must divide the beam intosegments with each segment supporting a single uniformly distributed load over its length and/or aconcentrated load at the right end of each segment.

The user must enter the length of each segment, the uniformly distributed loads on each segment,and the concentrated loads at the right end of each segment.

Four sample problems are shown below to demonstrate use of the application. Each of the problems

demonstrates a different loading condition and alternate ways of entering input. The first problemshows the most flexible method of entering input which may be used for any combination of uniformand concentrated loads.

Sketches and text labels are included in this document to provide additional information about theapplication to the user. A user can save the application to another file and use that file as thecustomized working document. Note that any practical number of beams may be entered for analysis.

To do this the user would simply add to the Input and Summary sections.

The user should be familiar with subscript notation, entering numbers as vectors, and using thetranspose and vectorize operators on the pallete.

Input

Mathcad Enabled ContentCopyright 2011 Knovel Corp.

All rights reserved.

Page 1 of 9

Click to View Mathcad Document


2/9


Notation

n is the last segment

Input Variables

a segment lengths

w uniformly distributed loads on segments

P concentrated loads at right end of segments

Computed Variables

The following variables are calculated in this document:

L span length

RL beam reaction at left end

RR beam reaction at right end

XL distance from the left reaction to the point of maximum moment

Mmax maximum bending moment

Sample 1 This sample problem shows a series of uniform andconcentrated loads with differing load magnitudes andsegment lengths, with the values ofa, w and P entered astransposed vectors. This is the most flexible format sinceany beam with uniformly distributed and concentrated loadsmay be entered using this format.

Enter beam number, segment lengths, uniformly distributed loads,and concentrated loads starting from the left reaction:

Beam number: b 1

Segment lengths: a T

3.5 10 13.5 ftSpan length: L

b a b =L

b27 ft



Page 2 of 9


3/9


Uniformlydistributed loads:

w T

2.6 1.8 0.5 ipft

Concentrated

loads at the rightend of segments: P

T

7.8 10.3 kipSample 2 This sample problem shows a series of four equal

concentrated loads at uniform spacing with a singleuniformly distributed load over the length of the beam.

Beam number: b 2

Number of segments: n 5

Range variable ifor segments:

i 1 n

Segment lengths: a,i b

6 ft

Span length: Lb

a b =Lb

30 ftUniformlydistributed loads:

w,i b

60.0 plf

Range variable i1,for concentratedloads:

i1 1 n 1

Concentratedloads:

P,i1 b

7.5 kip



Page 3 of 9


4/9


Sample 3 This sample problem shows a single uniform load over thelength of the beam.

Beam number: b 3

Segment lengths: a,1 b

27 ft

Span length: Lb

a b =Lb

27 ft

Uniformlydistributed loads: w ,1 b 1.45

ipft

Sample 4 This sample problem shows a single uniform load over thelength of the beam with a single concentrated load not atmidspan. Since the load is not symmetrical, the "template"for Sample 1 is used with the single concentrated loadentered as a subscripted variable.

Beam number: b 4

Segment lengths: a b T

15 5 ft

Span length: Lb

a b =Lb

20 ft

Uniformlydistributed loads: w

b T

250 250 plfConcentratedloads at rightend of segments:

P,1 b

52.5 kip



Page 4 of 9


5/9


Solution

Beam reactions, location of the point of zero shear, and maximum bending moment are computedwithin this section.

Maximum number of segments entered: n rows a =n 5

Number of beams entered: b cols a =b 4

The following expressions adjusts the sizes of vectors P and w to the same size as vector a:

P,n b

if ,,


6/9


=TMR

15.9 274.9 930.7 930.7 930.71.1 49.3 144.7 287.3 477

528.5 528.5 528.5 528.5 528.528.1 312.5 312.5 312.5 312.5

ip ft

Left end reactions:

RLj

MR,n j

a j

=TRL 34.47 15.9 19.575 15.625 kipRight end reactions:

RRj

i

+w,i j

a,i j

P,i j

RLj

=TRR 17.48 15.9 19.575 41.875 kipShear at the left end of each segment:

VL,i j

RLj

VL,i j

=TVL

34.47 17.57 10.73 17.48 17.4815.9 8.04 0.18 7.68 15.5419.575 19.575 19.575 19.575 19.57515.625 40.625 41.875 41.875 41.875

kip

Shear at the right end of each segment:

VR ,i jRLj

VR,i j

=TVR

25.37 0.43 17.48 17.48 17.4815.54 7.68 0.18 8.04 15.9

19.575 19.575 19.575 19.575 19.57511.875 41.875 41.875 41.875 41.875

kip



Page 6 of 9


7/9


8/9


=TSL

0 3.5 13.5 27 270 6 12 18 240 27 27 27 270 15 20 20 20

ft

Distance from the left end reaction to the point of zero shear and maximum moment:

XLj+SL ,u

jj

a'j

=TXL 13.261 15 13.5 15 ftMaximum bending moment:

Mmaxj +f

,, >uj 1 M ,u

j1 j

0

VL ,uj

j a'j 1

2 w ,ujj a'j2

=TMmax 190.47 141.75 132.131 206.25 ip ftSummary

Distances fromthe left reactionto the point ofzero shear:

Beamnumbers:

Spanlengths:

LeftReactions:

RightReactions:

=j

1234

=Lj

27302720

ft =RLj34.4715.919.57515.625

kip =RRj17.4815.919.57541.875

kip =XLj13.2611513.515

ft

Maximummoment:

=Mmaxj

190.47141.75

132.131206.25

ip ft



Page 8 of 9


9/9


User Notices

Equations and numeric solutions presented in this Mathcad worksheet are applicable to thespecific example, boundary condition or case presented in the book. Although a reasonable effortwas made to generalize these equations, changing variables such as loads, geometries and

spans, materials and other input parameters beyond the intended range may make someequations no longer applicable. Modify the equations as appropriate if your parameters falloutside of the intended range.For this Mathcad worksheet, the global variable defining the beginning index identifier for vectorsand arrays, ORIGIN, is set as specified in the beginning of the worksheet, to either 1 or 0. IfORIGIN is set to 1 and you copy any of the formulae from this worksheet into your own, you needto ensure that your worksheet is using the same ORIGIN.



Page 9 of 9

simple span beam mathcad

Documents