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Positive direction for Reaction Forces is up, for Reaction Moments is by clockwise. beam@orlovsoft.com (for new scheme).
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Beam Bending Sample #4 Reactions: $$R_A=\frac{1}{2}\frac{Fb\left(2b^2+3a^2+6ab\right)}{b^3+3b^{2}a+3a^{2}b+a^3}$$ $$R_B=\frac{1}{2}\frac{\left(2a+3b\right)a^{2}F}{b^3+3b^{2}a+3a^{2}b+a^3}$$ $$M_A=-\frac{1}{2}\frac{\left(a+2b\right)baF}{a^2+2ab+b^2}$$ Shearing Forces: $$Q_1=\frac{1}{2}\frac{Fb\left(2b^2+3a^2+6ab\right)}{b^3+3b^{2}a+3a^{2}b+a^3}$$ $$Q_2=-\frac{1}{2}\frac{\left(2a+3b\right)a^{2}F}{b^3+3b^{2}a+3a^{2}b+a^3}$$ Bending Moments: $$M_1=-\frac{1}{2}\frac{\left(a^3-3a^{2}z+3a^{2}b-6azb+2b^{2}a-2z{b}^2\right)bF}{\left(a^2+2ab+b^2\right)\left(a+b\right)}$$ $$M_2=\frac{1}{2}\frac{\left(2a+3b\right)a^{2}F\left(-z+b\right)}{b^3+3b^{2}a+3a^{2}b+a^3}$$ Bending Angles: $${\varphi }_1=-\frac{1}{4}\frac{\left(2a^3-3a^{2}z+6a^{2}b-6azb+4b^{2}a-2z{b}^2\right)Fbz}{J_{x}E\left(b^3+3b^{2}a+3a^{2}b+a^3\right)}$$ $${\varphi }_2=\frac{1}{4}\frac{a^{2}F\left(-2z^{2}a-3z^{2}b+4azb+6z{b}^2-2b^3+a^{2}b\right)}{J_{x}E\left(b^3+3b^{2}a+3a^{2}b+a^3\right)}$$ Deflections: $$w_1=-\frac{1}{12}\frac{\left(3a^3-3a^{2}z+9a^{2}b-6azb+6b^{2}a-2z{b}^2\right)Fb{z}^2}{J_{x}E\left(b^3+3b^{2}a+3a^{2}b+a^3\right)}$$ $$w_2=-\frac{1}{12}\frac{a^{2}F\left(2z^{3}a+3z^{3}b-6z^{2}ba-9z^2{b}^2+6z{b}^3-3zb{a}^2+3a^2{b}^2+4b^{3}a\right)}{J_{x}E\left(b^3+3b^{2}a+3a^{2}b+a^3\right)}$$ Numeric Sample (Maple input data): a:=2: b:=1: F:=3000: E:=2*10^11: J[x]:=200*10^(-8): q:=1000:

Beam Bending Sample #5 Reactions: $$R_A=\frac{1}{8}\frac{qa\left(8b^3+24a{b}^2+20a^{2}b+5a^3\right)}{b^3+3a{b}^2+3a^{2}b+a^3}$$ $$R_B=\frac{1}{8}\frac{q{a}^3\left(4b+3a\right)}{b^3+3a{b}^2+3a^{2}b+a^3}$$ $$M_A=-\frac{1}{8}\frac{\left(a^2+4ab+4b^2\right)q{a}^2}{a^2+2ab+b^2}$$ Shearing Forces: $$Q_1=\frac{1}{8}\frac{q\left(-8z{b}^3-24za{b}^2-24z{a}^{2}b-8z{a}^3+8a{b}^3+24a^2{b}^2+20a^{3}b+5a^4\right)}{b^3+3a{b}^2+3a^{2}b+a^3}$$ $$Q_2=-\frac{1}{8}\frac{q{a}^3\left(4b+3a\right)}{b^3+3a{b}^2+3a^{2}b+a^3}$$ Bending Moments: $$M_1=-\frac{1}{8}\frac{\left(a^5-5a^{4}z+5a^{4}b-20a^{3}zb+4a^3{z}^2+8a^3{b}^2-24a^{2}z{b}^2+12a^2{z}^{2}b+4b^3{a}^2-8az{b}^3+12a{z}^2{b}^2+4z^2{b}^3\right)q}{\left(a^2+2ab+b^2\right)\left(a+b\right)}$$ $$M_2=\frac{1}{8}\frac{q{a}^3\left(4b+3a\right)\left(-z+b\right)}{b^3+3a{b}^2+3a^{2}b+a^3}$$ Bending Angles: $${\varphi }_1=-\frac{1}{48}\frac{\left(6a^5-15a^{4}z+30a^{4}b-60a^{3}zb+8a^3{z}^2+48a^3{b}^2-72a^{2}z{b}^2+24a^2{z}^{2}b+24b^3{a}^2-24az{b}^3+24a{z}^2{b}^2+8z^2{b}^3\right)qz}{\left(b^3+3a{b}^2+3a^{2}b+a^3\right)E{J}_x}$$ $${\varphi }_2=\frac{1}{48}\frac{q{a}^3\left(-12z^{2}b-9z^{2}a+24z{b}^2+18zba-8b^3+6a^{2}b+a^3\right)}{\left(b^3+3a{b}^2+3a^{2}b+a^3\right)E{J}_x}$$ Deflections: $$w_1=-\frac{1}{48}\frac{\left(3a^5-5a^{4}z+15a^{4}b-20a^{3}zb+2a^3{z}^2+24a^3{b}^2-24a^{2}z{b}^2+6a^2{z}^{2}b+12b^3{a}^2-8az{b}^3+6a{z}^2{b}^2+2z^2{b}^3\right)q{z}^2}{\left(b^3+3a{b}^2+3a^{2}b+a^3\right)E{J}_x}$$ $$w_2=-\frac{1}{48}\frac{q{a}^3\left(4z^{3}b+3z^{3}a-12z^2{b}^2-9z^{2}ba+8z{b}^3-6z{a}^{2}b-z{a}^3+a^{3}b+6a^2{b}^2+6a{b}^3\right)}{\left(b^3+3a{b}^2+3a^{2}b+a^3\right)E{J}_x}$$ Numeric Sample (Maple input data): a:=2: b:=1: F:=1000: E:=2*10^11: J[x]:=200*10^(-8): q:=1000:

Beam Bending Sample #6 Reactions: $$R_A=-\frac{1}{8}\frac{-8b^{3}qa-20b{a}^{3}q-24b^{2}q{a}^2-5q{a}^4+12b{a}^{2}F+8b^{3}F+24a{b}^{2}F}{b^3+3a^{2}b+3b^{2}a+a^3}$$ $$R_B=-\frac{1}{8}\frac{a^2\left(-4bqa+12bF-3q{a}^2+8aF\right)}{b^3+3a^{2}b+3b^{2}a+a^3}$$ $$M_A=\frac{1}{8}\frac{\left(-q{a}^3-4bq{a}^2-4b^{2}qa+4abF+8b^{2}F\right)a}{a^2+2ab+b^2}$$ Shearing Forces: $$Q_1=-\frac{1}{8}\frac{8qz{b}^3+24qz{a}^{2}b+24qz{b}^{2}a+8qz{a}^3-8b^{3}qa-20b{a}^{3}q-24b^{2}q{a}^2-5q{a}^4+12b{a}^{2}F+8b^{3}F+24a{b}^{2}F}{b^3+3a^{2}b+3b^{2}a+a^3}$$ $$Q_2=\frac{1}{8}\frac{a^2\left(-4bqa+12bF-3q{a}^2+8aF\right)}{b^3+3a^{2}b+3b^{2}a+a^3}$$ Bending Moments: $$M_1=-\frac{1}{2}q{z}^2-\frac{1}{8}\frac{\left(-8b^{3}qa-20b{a}^{3}q-24b^{2}q{a}^2-5q{a}^4+12b{a}^{2}F+8b^{3}F+24a{b}^{2}F\right)z}{b^3+3a^{2}b+3b^{2}a+a^3}+\frac{1}{8}\frac{\left(-q{a}^3-4bq{a}^2-4b^{2}qa+4abF+8b^{2}F\right)a}{a^2+2ab+b^2}$$ $$M_2=\frac{1}{8}\frac{a^2\left(-4bqa+12bF-3q{a}^2+8aF\right)z}{b^3+3a^{2}b+3b^{2}a+a^3}-\frac{1}{8}\frac{a^2\left(-4bqa+12bF-3q{a}^2+8aF\right)b}{b^3+3a^{2}b+3b^{2}a+a^3}$$ Bending Angles: $${\varphi }_1=\frac{1}{6}\frac{-q{z}^3-\frac{3}{8}\frac{\left(-8b^{3}qa-20b{a}^{3}q-24b^{2}q{a}^2-5q{a}^4+12b{a}^{2}F+8b^{3}F+24a{b}^{2}F\right)z^2}{b^3+3a^{2}b+3b^{2}a+a^3}+\frac{3}{4}\frac{\left(-q{a}^3-4bq{a}^2-4b^{2}qa+4abF+8b^{2}F\right)az}{a^2+2ab+b^2}}{E{J}_x}$$ $${\varphi }_2=-\frac{1}{48}\frac{a^2\left(12z^{2}bqa-36z^{2}bF+9z^{2}q{a}^2-24z^{2}aF-24qz{b}^{2}a+72z{b}^{2}F-18qz{a}^{2}b+48bzaF-6b{a}^{3}q+8b^{3}qa-24b^{3}F-q{a}^4+12b{a}^{2}F\right)}{\left(b^3+3a^{2}b+3b^{2}a+a^3\right)E{J}_x}$$ Deflections: $$w_1=\frac{1}{24}\frac{-q{z}^4-\frac{1}{2}\frac{\left(-8b^{3}qa-20b{a}^{3}q-24b^{2}q{a}^2-5q{a}^4+12b{a}^{2}F+8b^{3}F+24a{b}^{2}F\right)z^3}{b^3+3a^{2}b+3b^{2}a+a^3}+\frac{3}{2}\frac{\left(-q{a}^3-4bq{a}^2-4b^{2}qa+4abF+8b^{2}F\right)a{z}^2}{a^2+2ab+b^2}}{E{J}_x}$$ $$w_2=\frac{1}{6}\frac{\frac{1}{8}\frac{a^2\left(-4bqa+12bF-3q{a}^2+8aF\right)z^3}{b^3+3a^{2}b+3b^{2}a+a^3}-\frac{3}{8}\frac{a^2\left(-4bqa+12bF-3q{a}^2+8aF\right)b{z}^2}{b^3+3a^{2}b+3b^{2}a+a^3}-\frac{1}{8}\frac{a^2\left(-6b{a}^{3}q+8b^{3}qa-24b^{3}F-q{a}^4+12b{a}^{2}F\right)z}{b^3+3a^{2}b+3b^{2}a+a^3}+\frac{1}{8}\frac{\left(-q{a}^3-6bq{a}^2-6b^{2}qa+12abF+16b^{2}F\right)b{a}^3}{b^3+3a^{2}b+3b^{2}a+a^3}}{E{J}_x}$$ Numeric Sample (Maple input data): a:=2: b:=1: F:=3000: E:=2*10^11: J[x]:=200*10^(-8): q:=1000:

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